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Advanced CFD Modelling of Road-Vehicle Aerodynamics

A thesis submitted to the University of Manchester

Institute of Science and Technology for the degree of

Doctor of Philosophy

May 2001

Christopher M.E. Robinson

Department of Mechanical,

Aerospace and Manufacturing Engineering

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Declaration

No portion of the work referred to in this thesis has been submitted in support of an applica-

tion for another degree or qualification of this or any other university, or other institution of

learning.

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Acknowledgements

I would like to express my sincere gratitude to Professor B.E. Launder, my supervisor,

for his invaluable advice, guidance and constructive criticism throughout the course of this

research. I am honoured to have been given this opportunity to work in his research group.

I would also like to thank Dr T.J. Craft and Dr H. Iacovides for their continuous help

and advice. The many discussions I have had with them have benefitted considerably my

understanding of turbulent flows and turbulence modelling.

Funding for this project was provided by the European Union BRITE/EURAM Project

BE-97-4043: Models for Vehicle Aerodynamics (MOVA). Without this support, the present

study would not have been possible. I would like to thank all the members of the MOVA

project group for their advice on turbulence modelling and experiments, and the insight to

industrial applications of turbulence modelling which they have provided. In particular I

would like to thank: Professor D. Laurence (UMIST & Electricité de France), Professor K.

Hanjalic (TU Delft), Dr H. Lienhart (LSTM, Erlangen), Dr L. Elena (PSA, Peugeot-Citroën)

and Dr B. Basara (AVL, Graz).

My thanks to friends and colleagues at UMIST for providing advice, discussion, enter-

tainment and coffee - thanks to all members of the TM & CFD Group, especially: Simon

Gant, Aleksey Gerasimov and Dr Rob Prosser. Also, very special thanks go to my friends:

Dougie & Sharon, Martin, Ajay & Kerry and Simon for their support and encouragement,and providing me with many much-needed distractions.

Last, but by no means least, I would like to thank Mum, Dad, Tim & Jo, without whose

love, none of this would be possible.

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Abstract

This thesis is concerned with the application of a cubic non-linear k ε model to flows per-

tinent to road-vehicle aerodynamics. Three simple test cases are initially considered and a

simplified car model - the Ahmed body - is used as a final test case. The performance of the

non-linear k ε model is compared to a linear k ε model and, for one test case, a realizable

differential stress model. As the principal thrust of the work is to mimic current industrial

practices, wall functions are used to bridge the near-wall flow. Both established log-law wall

functions and a new analytical wall function are used.

The three simple test cases are: flow around a cylinder of square cross-section placed near

a wall, flow in a U-bend of square cross-section and flow in a plane diffuser. In calculating

the flow around the square cross-sectioned cylinder, the non-linear k ε model performs

generally better than the linear k ε model. In the case with a steady wake (g

d 0

25; Re

13 600) both the non-linear model and the linear model calculate too long a recirculating flow

region behind the cylinder. The linear model calculates velocity profiles in the wake more

accurately but the non-linear model calculates the Reynolds stresses better. Flow with vortexshedding in the wake has also been calculated (g

d 0

50 and 0.75).

In the square cross-sectioned U-bend, the non-linear k ε model is compared with a

differential stress model. This case allows comparison of the models in a flow with strong

streamline curvature and streamwise vorticity. The radius of curvature of the bend is Rc

D

3 35 and the Reynolds number of the flow is Re

58

000; there is no separation of the flow.

The non-linear k ε model calculates the mean velocities almost as well as the differential

stress model. It calculates the low momentum region in the central part of the duct well

and also the complex secondary flow pattern in the U-bend. The Reynolds stresses are not

calculated as well by the non-linear k ε model as they are by the differential stress model,

because the non-linear k ε model calculates these from local velocity gradients rather than

transport equations.

The plane diffuser is used to test both the cubic non-linear k ε model and the new ana-

lytical wall function. The flow is calculated at Re 20 000, the diffuser has one 10o inclined

wall and separation occurs part way along this wall. When the analytical wall function is used

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v

in conjunction with a linear k ε model, a small amount of separation is calculated. A linear

k ε model with a log-law wall function calculates no separation. The flow calculation is

improved further by the non-linear k ε model; the most accurate combination of turbulence

model and wall function tested for this case is the non-linear k ε model and analytical wall

function.The final test case is flow around a simplified car geometry - the Ahmed body - which

is a commonly used test case in industry. The Ahmed body is a bluff body mounted near a

ground-plane; the most significant feature of the body is the angle of the rear slant. Below a

critical angle (30o , the flow over the rear slant is predominantly attached and is characterised

by high drag and the formation of strong vortices at the side edges of the slant. Above the

critical angle, the flow is completely separated, there is relatively low drag and only weak

side-edge vortices form. Calculations of the flow are made for the Ahmed body with two

rear-slant angles which bracket the critical angle (25o and 35o).

For the body with the 25o rear-slant, the linear k ε model calculates the attached flow

with the strong side-edge vortices reasonably well; the non-linear k ε model does not cal-

culate this flow well. Only weak side-edge vortices are calculated by the non-linear model

which do not draw enough fluid out of the boundary layer over the slant to cause the flow

to attach. The influence of the choice of wall function, time-dependent effects, realizability,

near-wall length-scale correction, convection scheme, development of streamwise vorticity

and sensitivity to features of the non-linear k ε model are discussed. Over the 35o rear slant

both the linear and non-linear k ε models produce separated flow. Drag is calculated most

accurately by the non-linear model in conjunction with the analytical wall function.From the three initial test cases, it is concluded that the cubic non-linear k ε model is, in

general, able to calculate simple flows better than a linear k ε model and, in one case, almost

as well as a differential stress model. The failure of the non-linear model to calculate attached

flow over the 25o rear slant of the Ahmed body is attributed to the coefficients in the non-

linear stress-strain relationship which are not tuned for this class of flow and the functional

form of c µ. With the 35o rear-slant, both the non-linear and linear k ε models calculate

the separated flow which is observed in experiments. When used in conjunction with the

analytical wall function, the non-linear k ε model is able to calculate drag to sufficient

accuracy for industrial purposes.

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Contents

Nomenclature xi

1 Introduction and Literature Survey 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Calculating Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Direct Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.4 Turbulence Modelling:

Reynolds Averaged Navier-Stokes Methods . . . . . . . . . . . . . . 6

1.2.5 Near-Wall Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.1 Cylinder of Square Cross-Section Close to a Wall . . . . . . . . . . 15

1.3.2 U-bend of Square Cross-Section . . . . . . . . . . . . . . . . . . . . 18

1.3.3 Plane Diffuser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.4 Road-Vehicle Aerodynamics . . . . . . . . . . . . . . . . . . . . . . 26

1.4 Study Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.5 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Mathematical Models 37

2.1 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Two-Equation Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.1 Linear k ε Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.2 A General Non-linear Eddy-Viscosity Model . . . . . . . . . . . . . 41

2.2.3 Cubic Non-Linear k ε Model . . . . . . . . . . . . . . . . . . . . . 42

2.2.4 Realizability Conditions . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3 Differential Stress Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.1 Basic DSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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CONTENTS vii

2.3.2 Cubic DSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4 Near-Wall Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4.2 Basic Wall Function . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4.3 Chieng & Launder Wall Function . . . . . . . . . . . . . . . . . . . 532.4.4 Analytical Wall Function . . . . . . . . . . . . . . . . . . . . . . . 54

2.4.5 Note on Near-Wall Distance . . . . . . . . . . . . . . . . . . . . . . 56

3 Numerical Implementation 58

3.1 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1.1 Discretization of a General PDE . . . . . . . . . . . . . . . . . . . . 58

3.1.2 Convection Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.1.3 Source Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.1.4 Three-Dimensional, Discretized, General PDE . . . . . . . . . . . . 65

3.1.5 Calculation of Pressure . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.1.6 The SIMPLE Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 67

3.1.7 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.1.8 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2 Codes Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.1 TEAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2.2 TOROID-SE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2.3 STREAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Cylinder of Square Cross-Section Placed Near a Wall 77

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 Models Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Domain, Grids, Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 79

4.3.1 Domain and Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Calculated Flow Results for g

d 0 25 . . . . . . . . . . . . . . . . . . . . 80

4.4.1 High Dissipation Inlet Condition ( νt

ν

10) . . . . . . . . . . . . . 804.4.2 Low Dissipation Inlet Condition ( νt

ν 100) . . . . . . . . . . . . . 82

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Flow in a U-bend of Square Cross-Section 86

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Models Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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CONTENTS viii


5.3.1 Domain and Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3.2 Upstream Boundary Condition . . . . . . . . . . . . . . . . . . . . . 88

5.4 Calculated Flow Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.1 Comparison to Calculated Data . . . . . . . . . . . . . . . . . . . . 895.4.2 Inlet Flow Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.3 Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.4 Reynolds Stress Profiles . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4.5 Streamwise Velocity Contours and Secondary

Velocity Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Calculation of Flow in a 10o Plane Diffuser 94

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2 Models Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


6.3.1 Domain and Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


6.4 Calculated Flow Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4.1 Calculations with the Linear k ε Model . . . . . . . . . . . . . . . 96

6.4.2 Calculations with the Non-linear k ε Model . . . . . . . . . . . . . 98

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Ahmed Body Flow Calculation 100

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2 Models Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102


7.3.1 Domain and Coordinate System . . . . . . . . . . . . . . . . . . . . 103

7.3.2 Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104


7.4 25

o

Slant - Flow Field Results . . . . . . . . . . . . . . . . . . . . . . . . . 1067.4.1 Flow Upstream and Impinging on Body Nose . . . . . . . . . . . . . 106

7.4.2 Boundary Layer Flow on Body Mid-Section . . . . . . . . . . . . . . 108

7.4.3 Flow Over Rear Slant . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.4.4 Vortex Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.4.5 Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.4.6 Pressure on Base and Slant . . . . . . . . . . . . . . . . . . . . . . . 118

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7.5 35o Slant - Flow Field Results . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.5.1 Flow Over Rear Slant and In Wake . . . . . . . . . . . . . . . . . . . 119

7.5.2 Pressure on Base and Slant . . . . . . . . . . . . . . . . . . . . . . . 121

7.6 Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.6.1 Measured Drag Variation with Rear-Slant Angle . . . . . . . . . . . 1227.6.2 Calculated Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 23

7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8 Conclusions and Recommendations for Future Work 131

8.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.3 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . 134

Appendices 137

A Realizability Condition for an EVM 137

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 37

A.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

A.3 Implementation in STREAM . . . . . . . . . . . . . . . . . . . . . . . . . . 140

A.4 Comparison with c µ Function . . . . . . . . . . . . . . . . . . . . . . . . . . 140

B Derivation of the Analytical Wall Function 142

B.1 Specification of Analytical Wall Function . . . . . . . . . . . . . . . . . . . 142

B.1.1 Dimensionless Simplified Momentum Equation . . . . . . . . . . . . 142

B.1.2 Analytical Solution of Equations . . . . . . . . . . . . . . . . . . . . 143

B.1.3 Specification of Wall Shear Stress, τ w . . . . . . . . . . . . . . . . . 146

B.1.4 Average Production of Turbulent Kinetic Energy, Pk . . . . . . . . . 146

B.1.5 Average Turbulence Dissipation Rate, ε . . . . . . . . . . . . . . . . 148

B.1.6 Summary of Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

B.2 Concerning Implementation of Analytical Wall Function . . . . . . . . . . . 150

B.2.1 Wall Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

B.2.2 Turbulent Kinetic Energy & Dissipation . . . . . . . . . . . . . . . . 151

B.3 Further Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

C Cylinder of Square Cross-Section with Vortex Shedding 152

C.1 Calculated Flow Results for g

d 0

50 . . . . . . . . . . . . . . . . . . . . 152

C.2 Calculated Flow Results for g

d 0

75 . . . . . . . . . . . . . . . . . . . . 154

C.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

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CONTENTS x

D Cylinder of Square Cross-Section with Modified c µ 157

D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 57

D.2 Calculated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

D.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

E Calculation of Drag for the Ahmed Body 160

References 162

Figures 174

CONTENTS

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Nomenclature

Symbols

A Flatness parameter, 1

9

8

A2 A3

A1 Constant term in analytical wall function

A2 Second invariant of anisotropy, ai jai j

A3 Third invariant of anisotroy,

aik ak ja ji

Ae w

n

s

t

b Areas of cell faces

Ai j Tensor expressing relationship between S i j and Ωi j (Equation 2.23)

Am Damping function in one-equation k l model (Equation 5.2)

a E W

N

S

T

B

P Coefficients in discretised equations

ai j Anisotropic stress,

uiu j

k

2

3δi j

C 1 C 2 Non-equilibrium constants in analytical wall function

C D Coefficent of drag

C DP C DF Coefficients of drag due to pressure and friction

C F Coefficient of friction

C L Coefficient of lift

C P Coefficient of pressure

C B

C K

C S Components of pressure drag on Ahmed body, respectively due to:

base, nose cone and rear slant

C R Friction drag on Ahmed body

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NOMENCLATURE xiii

E Integration constant used in wall functions (=9.793 for smooth walls)

ei

ei Cartesian unit vectors

F e w

n

s

t

b Convective mass flux

F i Body force acting on fluid

f Frequency

f

Non-dimensional frequency,

f d

U o

f 1

f 2

f µ Damping functions used in low-Reynolds-number model (Equation

2.17)

f RS Damping function used in c µ2

G

n Coefficients in the general non-linear stress-strain relationship (Equa-

tion 2.25)

g Distance of square cross-sectioned cylinder from wall

gc Critical distance of square cross-sectioned cylinder from wall at which

vortex shedding starts to occur

gi

gi Base vectors tangential and normal to ξi

H Inlet height of plane diffuser

H ε Combined turbulent production and destruction term in ε transport

equation (Equation 2.12)

J Jacobian of transform matrix for curvilinear coordinate system

k Turbulent kinetic energy,

1

2

uu

vv

ww

k p Value of turbulent kinetic energy stored at near-wall node

k v Value of turbulent kinetic energy at the edge of the viscous sub-layer

k w Value of turbulent kinetic energy at the wall extrapolated from k p and

next adjacent node

L Integral length-scale

NOMENCLATURE

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NOMENCLATURE xiv

L Reference length used in Ahmed body flow ( L is height of Ahmed

body)

l Length-scale

lm Mixing length-scale

lo Largest length-scale in a flow

lt Turbulent length-scale

l µ lε Length-scales in k l model (Equations 2.57 & 2.58)

Ma Mach number

N Number of computational nodes

Nu Nusselt number

ni Unit vector normal to a wall

P Mean pressure

P Instantaneous pressure

Pi j Production term in uiu j transport equation (Equation 2.17)

Pk Production of turbulent kinetic energy

Pk Average production of turbulent kinetic energy in near-wall cell

Po Pressure at a given reference point

Pε3 Gradient production term in LRN ε transport equation (Equation 2.17)

Pe Peclet number,

ρU ∆ x

Γ

p Fluctuating pressure

q11 q22

q33 Coefficients in curvilinear coordinate system transport equations

Rc Radius of curvature in U-bend

Rm

Rφ Mass and general variable residual

Rv Sub-layer Reynolds number,

k 1 2v yv

ν

NOMENCLATURE

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NOMENCLATURE xv

Rt

˜ Rt Turbulent Reynolds number,

k 2

νε

k 2

νε

˜ Rto Limiting value of turbulent Reynolds number used to control the in-

fluence of Pε3 (Equation 2.34)

Re Reynolds number

U o d

ν

r Ratio used in slope limiter function of a TVD scheme

S

S Strain invariants,

k

ε

S i jS i j

2

k

ε

S i j S i j

2

S i j Strain tensor,

∂U i

∂ x j

∂U j

∂ xi

S C Contributions to linearized source term which are constant

S 1CD

S 2CD Cross-diffusion source terms in curvilinear coordinate system

S P Contributions to linearized source term which are a function of the

dependent variable

S Q Additional source term in discretized equations for contributions from

nodes not directly adjacent to cell-face under consideration

S ε Source term in ε transport equation due to “Yap correction” (Equation

7.4)

S φ Linearized source term for φ

St Strouhal number, f

d

U o

T Mean temperature

T

1

i j

T

10

i j Products of S i j and Ωi j in non-linear stress-strain relationship (Equa-

tion 2.26)

t Time

t

Non-dimensional time,

t U o

d

U V

W

U i Mean velocities

U

V

W

U i Instantaneous velocities

NOMENCLATURE

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NOMENCLATURE xvi

U

V

W

U i Velocities in curvilinear coordinate system

U n Resultant velocity in direction of flow along the wall, at near-wall

cell-face, opposite the wall in analytical wall function (Equation B.22)

U o Bulk or characteristic velocity used in definition of Reynolds number,etc

U re f Reference velocity used to non-dimensionalise variables in STREAM

U τ Friction velocity, τw

ρ

U Non-dimensional velocity,

U

U τ

u

v

w

ui Turbulent velocities

u v w ui Periodic velocities

ui Sub-grid velocity in LES

uu

vv

ww

uv uw

vw

uiu j

Reynolds (turbulent) stresses

uu vv

uv

uiu j

Periodic stresses

ut ut

vt vt

wt wt

ut iut

j

Total stresses, ut ut

uu uu etc.

V t Turbulent velocity-scale

W b Bulk velocity in U-bend

x

y

z

xi Coordinate directions

y Non-dimensional distance to the wall,

yU τ

ν

y

Non-dimensional distance to the wall,

y

k

ν

yn Distance from near-wall cell face to the wall

y p Distance from near-wall node to the wall

yv Distance from the edge of the viscous sub-layer to the wall

NOMENCLATURE

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NOMENCLATURE xvii

Greek Symbols

α Constant in analytical wall function

clc µ

α Angular distance in U-bend

α1 α2 α3 Values used to implement realizability condition (Equations A.25,

A.26, A.27)

β Rear-slant angle of Ahmed body

βc Critical rear-slant angle on Ahmed body at which drag crisis occurs

Γ φ Diffusion coefficent for φ

γ Constant in analytical wall function (Equation (B.38)

∆ Denotes change in given variable

∆t Time-step

∆t Non-dimensional time-step, ∆t

U o

d

∆ x ∆ y

∆ z Cell dimension (ie. distance between cell-faces)

δ x Cell dimension associated with a cell-face (ie. distance between ad-

jacent nodes)

δi j δi j δ

ji Kroneker’s delta

ε Rate of dissipation of turbulent kinetic energy, ν∂u j

∂ xi

∂u j

∂ xi

ε Average rate of dissipation of turbulent kinetic energy in near-wall

cell

ε Homogeneous part of turbulence energy dissipation, ε 2 ν

∂k 1 2

∂ x j

2

ε1 ε2

ε3 Values of ε used in realizability condition (Equations A.21, A.22,

A.23)

εi j Dissipation term in uiu j transport equation (Equation 2.37)

η Kolmogorov length-scale

NOMENCLATURE

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NOMENCLATURE xviii

η max

S Ω

θ Weighting factor in time-discretization

κ von Karman’s constant (=0.42)

λ Function used in Johnson & Launder wall function (1982)

λ Taylor micro-scale

µ Molecular viscocity

µt Turbulent (eddy) viscosity

ν Kinematic viscosity

νt Kinematic turbulent (eddy) viscosity

ξ η ζ ζi Curvilinear coordinate directions

ρ Density

ρre f Reference density used to non-dimensionalise variables in STREAM

σk σε Empirical constants in k and ε transport equations (Equations 2.7 &

2.12)

τi j Viscous stresses, ν∂U i∂ x j

τw Wall shear stress

φ General variable

φi j Pressure-strain correlation

φi j1 “Slow” part of the modelled pressure-strain correlation

φi j2 “Rapid” part of the modelled pressure-strain correlation

φW i j Wall-reflection term in the modelled pressure-strain correlation

ϕ Slope-limiter function in TVD scheme

Ω

Ω Vorticity invariants,

k

ε

Ωi jΩi j

2

k

ε

Ωi jΩi j

2

NOMENCLATURE

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NOMENCLATURE xix

Ω x Ω y Ω z Components of vorticity in Cartesian directions, Ω x

∂V

∂ z

∂W

∂ y

etc.

Ωi j Vorticity tensor,

∂U i

∂ x j

∂U j

∂ xi

ω Specific rate of dissipation of turbulent kinetic energy,

k

ε

Subscripts

1 Region 1: the viscous sub-layer in derivation of AWF

2 Region 2: outside the viscous sub-layer in derivation of AWF

E W

N

S

T

B

EE WW NN SS

T T BB

p

e

w

n

s

t

b

Node and face values of variables

b Bulk value

body Pertaining to the Ahmed body without the stilts

in Inlet value

NL Non-linear

nb Neighbouring nodes

o Free-stream value

re f Reference value used to non-dimensionalise variables in STREAM

rms Root-mean-square

stilt Pertaining to the stilts of the Ahmed body

tot Total

w Wall value

NOMENCLATURE

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NOMENCLATURE xx

Superscripts

0 Value at original time-step

1 Value at new time-step

Non-dimensional near-wall value scaled by U τ

Non-dimensional near-wall value scaled by

k

Guessed values in SIMPLE algorithm

Non-dimensional variables used in STREAM scaled by U re f d re f ρre f

Correction values in SIMPLE algorithm

Acronyms

ASM Algebraic Stress Model

AWF Analytical Wall Function

CFD Computational Fluid Dynamics

CPU Computer Processing Unit

DNS Direct Numerical Simulation

EARSM Explicit Algebraic Reynolds Stress Model

ERCOFTAC European Research Community on Flow, Turbulence and Combus-

tion

EVM Eddy-Viscosity Model

GMTEC General Motors research code

HRN High Reynolds Number

HWA Hot Wire Anemometry

LDV Laser Doppler Velocimetry

LES Large Eddy Simulation

LRR Launder, Reece & Rodi (1975)

NOMENCLATURE

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NOMENCLATURE xxi

LRN Low Reynolds Number

LSTM Lehrstuhl für Strömungsmechanik

MLH Mixing-Length Hypothesis

MOVA Models for Vehicle Aerodynamics

NLEVM Non-Linear Eddy-Viscosity Model

NWR Near-Wall Resolution

PDE Partial Differential Equation

PLDS Power Law Differencing Scheme

PSL Parabolic Sub-Layer

QUICK Quadratic Upwind Interpolation for Convection Kinematics

RANS Reynolds-Averaged Navier-Stokes

RMS Root-Mean-Square

RNG Re-Normalisation Group

SCL Simplified Chieng & Launder

SGS Sub-Grid Scale

SIMPLE Semi-Implicit Method for Pressure-Linked Equations

SSG Speziale, Sakar & Gatski (1991)

STREAM Simulation of Turbulent Reynolds-averaged Equations for All Mach

numbers

TEAM Turbulent Elliptic Algorithm - Manchester

TVD Total Variation Diminishing

UMIST University of Manchester Institute of Science and Technology and

Upstream Monotonic Interpolation for Scalar Transport

NOMENCLATURE

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Chapter 1

Introduction and Literature Survey

1.1 Background

The flow of air around road vehicles (cars, buses, trucks) under normal operating conditions is

principally turbulent. It is typically characterised by large-scale separation and recirculation

regions, a complex wake flow, long trailing vortices and the interaction of boundary layer

flows on the vehicle and ground. In developing a new road vehicle it is essential for the

designer to understand thoroughly the structure of the flow around the vehicle. This will

have influence on such principal features as: the shape of the vehicle, aerodynamic drag,

fuel consumption, noise production and road handling. Traditionally, vehicle designers have

gained their understanding of the air flow around a vehicle through extensive wind tunnel

testing.

More recently (within the last 10 years), Computational Fluid Dynamics (CFD) has ma-

tured sufficiently as a technology to enable it to calculate such quantities as drag and lift for

a road vehicle without resort to wind tunnel testing. However, the computational models are

very large and even with state-of-the-art processors it may take several days of CPU time to

gain a solution. In order to reduce this to a time scale which is acceptable to a vehicle designer

(within a day), it is necessary to use a simplified computational technique and adopt a model

to describe the mean effect of turbulence. Unfortunately, simple turbulence models often fail

to calculate the flow properly - eg. the position of flow separation on a rear slant is crucial indetermining the aerodynamic drag but it is an extremely difficult feature to calculate using a

simple turbulence model. Hence, to road vehicle manufacturers, CFD is currently a subject

of research rather than a design tool, and the key to understanding vehicle aerodynamics is

still the wind tunnel.

Over the past thirty years, a heirarchy of computational models for turbulence with vary-

ing levels of complexity has been developed. These can be broadly categorised into four

1

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CHAPTER 1. Introduction and Literature Survey 2

groups: Direct Numerical Simulation (DNS), Large Eddy Simulation (LES), Second-Moment

Closure and Eddy-Viscosity Models (EVM). Of these, only Direct Numerical Simulation cal-

culates the flow without resorting to a numerical model. It resolves the flow to sufficiently

fine detail to capture the motion of the smallest eddies and the briefest time-scales. This is

extremely computationally expensive and only currently feasible for very simple geometries(pipes, channels) at moderate Reynolds numbers or complex geometries with low Reynolds

number flows ( Re

102). In road vehicle flow the Reynolds number1 is typically Re

106

and DNS cannot be used to calculate the flow at road-going speeds. Large Eddy Simulation

uses the suppostition that most of the flow energy is contained in the largest eddies, and only

flow features on the scale of the largest eddies are calculated. A model is used to account for

the stresses generated in the flow by eddies which are smaller than this scale.

Rather than calculating instantaneous flow parameters (as in DNS and LES) Second-

Moment Closure and Eddy-Viscosity-Model methods both calculate mean flow parameters.

In the averaging process used to generate mean flow equations, information is lost (ie. the

information regarding turbulent fluctuations). This loss of information is manifested by the

second moments of fluctuating velocity or “Reynolds stresses” which appear explicitly in the

mean flow equations. The task of the turbulence modeller is to find an adequate numerical

representation of these Reynolds stresses.

A Second-Moment Closure model (also known as Reynolds Stress Model or Differen-

tial Stress Model, DSM) solves a separate transport equation for each fluctuating velocity

correlation (or Reynolds stress). In complex flows with features such as separation, recircu-

lation, curvature, swirl and impingement, the stress field is anisotropic and can vary rapidy.By solving a separate transport equation for each of the Reynolds stresses, it is, in principle,

possible to capture accurately the physical processes in the flow (albeit in mean, rather than

instantaneous quantities). Many schemes have been proposed and Second-Moment Closure

is considered capable of calculating complex flows accurately enough for many industrial

applications. Solving mean flow parameters is not a significant disadvantage, as in many

applications, engineers tend to prefer to work with mean quantities than to have a set of

time-dependent instantaneous results. The disadvantage of Second-Moment Closure is that it

introduces six transport equations (one for each Reynolds stress) which increase the compu-

tation time to a level which is unacceptable for many industrial applications.

Eddy Viscosity Models use the hypothesis that turbulent eddies act on the flow in the

same manner as molecular viscosity. A functional relationship is assumed between the stress

and strain fields which allows the Reynolds stress to be calculated. The most popular forms

1 Re U od ν; for typical values of vehicle speed 15ms

1 (55kph , length-scale 1m and kinematic viscosity

of air 14.65x10

6m2s

1, Reynolds number is Re 1x106

1.1. Background

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of this relationship require two transport equations to be solved - one each for length-scale

and velocity-scale. These “two-equation” models are relatively cheap and stable, and have

become the most commonly used turbulence models throughout industry in the past thirty

years. However, the stress-strain relationship used in most of these “two-equation” models is

linear. This is not an adequate assumption for complex flows and, although a two-equationmodel may provide a rapid and stable solution, the solution will be inaccurate. Recent ad-

vances have proposed new forms of the stress-strain relationship which include higher-order

terms to enable the stress anisotropies to be calculated more accurately. These “non-linear”

eddy viscosity models (NLEVM) have shown a lot of promise for relatively complex flows,

as they provide relatively quick, stable and accurate solutions.

Treatment of the wall boundary condition can cause additional problems. Near a wall

the profiles of velocity and Reynolds stresses vary rapidly thus requiring a very fine compu-

tational grid to resolve the variations which adds to the computational expense of the flow

calculation. Often a “wall function” is used to bridge the near-wall region where the rapid

changes occur and to provide average values over this region instead. The advantage of this

technique is that the fine computational cells are no longer required and the rapid near-wall

variations are not calculated explicitly; this results in a faster, more stable calculation. How-

ever, the assumptions used to define wall functions are typically only valid for simple shear

flows. Where there is skewing of the flow, adverse pressure gradient, separation, reattachment

or body forces acting on the flow, traditional wall functions will be inaccurate.

The work presented in this thesis develops a recently proposed non-linear eddy viscosity

model for flows pertinent to road-vehicle external aerodynamics. In addition to the turbulencemodel, a new “wall function” treatment is adopted for calculating mean quantities in the

near-wall flow regime. The remainder of this chapter describes the background to turbulence

modelling in greater detail and describes the test cases used to assess the models which are

studied.

1.2 Calculating Turbulent Flow

1.2.1 Governing Equations

The Navier-Stokes equations express the continuity equation and momentum equations for a

fluid with the stress tensor as a product of velocity gradients and viscosity. When considering

an incompressible, isothermal, Newtonian fluid flow, there are four equations with four un-

knowns (three components of velocity and pressure) and as such they form a mathematically

closed set. However, analytical solutions are in general only possible for very simple geome-

1.2. Calculating Turbulent Flow

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tries such as fully developed flow between planes or in straight pipes. Other simplifications

are possible depending on the physical characteristics of the flow. For example, if the fluid

can be considered as inviscid, the Navier-Stokes equations can be reduced to the Euler equa-

tions; a technique sometimes used for compressible flow at high Mach numbers. Similarly,

if the flow is inviscid and irrotational a Laplace equation can be solved for velocity potentialor if the Reynolds number is very low, convection can be assumed to be negligible and the

equations are solved for Stokes flow. For the majority of industrial flows, such simplifications

cannot be made. Above a critical Reynolds number the flow is turbulent and characterised by

unsteady, chaotic motion and in this regime the flow is calculated by a numerical approach.

1.2.2 Direct Numerical Simulation

Direct Numerical Simulation (DNS) is the solution of the governing equations without resort

to a mathematical model. This is conceptually the most straightforward method of solving

the equations, but the method is restricted as it is necessary to ensure that a wide range

of length and time-scales are resolved. A typical flow of engineering interest might have a

Reynolds number in the order Re

105 and a valid simulation must capture the largest eddies

which occur at the integral length-scale (say L 0 1m) and the smallest eddies at which

dissipation of kinetic energy occurs at the Kolmogorov length-scale (say η 10 µm). The grid

resolution must be at least L

η; DNS calculations are necessarily three-dimensional and for

the examples given this would result in a computational grid with 10 12 nodes. Furthermore,

the highest frequencies encountered in this flow may well be of the order of 10 kHz, requiring

a timestep of 100 µs. Pope (2000) estimates that if such a flow were calculated on a one

gigaflop machine, the solution would require several thousand years. In spite of this, some

progress has been made in calculating flows at Reynolds numbers upto Re

104 albeit for

very simple geometries or at low resolution.

The amount of data produced by a DNS calculation is excessive for most industrial engi-

neering purposes. Engineers do not need to know the velocity and pressure fluctuations for

all timesteps and would usually prefer to work with mean quantities. DNS is valuable though,

for providing flow detail which is difficult or impossible to measure experimentally, such as

pressure fluctuations and details of near-wall flow. It is also a useful tool to aid understanding

of the effects of compressibility and combustion on turbulence.

1.2.3 Large Eddy Simulation

In a highly turbulent flow it is possible to categorise the eddies into two classes with distinct

characteristics. Firstly, there are the large eddies which contain most of the energy, interact


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with the mean flow, are diffusive, anisotropic, long-lived, inhomogeneous, ordered, depen-

dent on boundaries and difficult to model analytically. Secondly, there are the small eddies

which are dissipative, isotropic, short-lived, homogeneous, random and lend themselves to

theoretical modelling. In a DNS calculation a very large proportion of the effort is devoted to

resolving the flow of the smallest eddies (Pope, 2000). A reasonable approach would there-fore be to calculate only the large eddies in the flow and to model the effects of the small

eddies: this is known as Large Eddy Simulation (LES).

In a Large Eddy Simulation a “filter” is specified with an associated length-scale, to de-

compose the velocity field into a resolved component U i and a residual (or sub-grid-scale,

SGS) component, ui. The resolved velocity field represents the motion of the large eddies

and is calculated numerically. The equations which describe this motion are derived from

the Navier-Stokes equations and contain a “SGS stress” tensor in the momentum equation

to account for the sub-grid-scale component. The SGS stress tensor is modelled to provide

closure. (Note: the “grid” referred to in “sub-grid scale” is not necessarily the computational

grid which is used to calculated the resolved velocity).

Smagorinski (1963) proposed a SGS stress model which is an eddy viscosity model 2

which includes a “constant” C S . However, C S may vary between flows of different Reynolds

number and may even vary from point to point within a given flow. Particularly, C S must

be reduced by an order of magnitude in shear flows, and even more so near walls and near-

wall damping functions are sometimes used. Alternative proposals by McMillan & Ferziger

(1980) and Yakhot & Orszag (1986) reduce C S and hence the eddy viscosity as the local SGS

Reynolds number decreases. C S must also be modified according to Froude or Richardsonnumber in stably stratified flows and flows with strong curvature.

To provide a model which would be more generally applicable, Germano et al (1991)

proposed a “dynamic Smagorinsky” model in which C S is calculated at every spatial location

and at every time step. The model can lead to rapid variations in eddy viscosity and even

cause the eddy viscosity to become negative. This is not a problem physically, as a negative

eddy viscosity represents “backscatter” - energy transfer from small to larger eddies. How-

ever, negative eddy viscosity can lead to numerical instabilities. Smagorinsky and dynamic

Smagorinsky models calculate the SGS stress tensor locally for a particular timestep. To

incorporate history and non-local effects a transport equation can be adopted. Examples of

these are Deardorff (1974) which solves a transport equation for the SGS stress tensor and

the models of Deardorff (1980) and Davidson (1993) which solve a transport equation for

sub-grid-scale kinetic energy.

2Eddy viscosity model concepts are discussed in more detail in Section 1.2.4.


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1.2.4 Turbulence Modelling:

Reynolds Averaged Navier-Stokes Methods

Large Eddy Simulation, although computationally faster than Direct Numerical Simulation

is still in general too time consuming for the majority of industrial applications. Instead, the

approach which is normally taken, is to decompose the instantaneous velocities and pressure

( U i

P) into mean (U i, P) and fluctuating components (ui

p). Reworking the Navier-Stokes

equations with this decompostition of the variables results in an equation set expressed in

terms of the mean velocities and pressures. These equations, known as the Reynolds Aver-

aged Navier-Stokes (RANS) equations after Reynolds (1895), are particularly attractive for

engineering purposes, as it is often the case that only mean values of velocity, pressure, force,

degree of mixing, etc, are required. The averaging process which is used to define the mean

variables results in a loss of information from the equations and the momentum equations

now contain a new tensor, uiu j, which cannot be expressed uniquely in terms of the mean

velocities. This tensor is usually termed a “turbulent stress” or “Reynolds stress” - it is not

actually a stress but acts on the equations in the same manner as the viscous stresses. With

the appearance of the Reynolds stress, the RANS equations are no longer a closed set; they

cannot be solved directly and require a model. The aim of the turbulence model is to express

the Reynolds stress in known or calculable quantities. As RANS models are the most widely

used and as this thesis is in the most part concerned with RANS based turbulence models, the

development of these turbulence models will be discussed in more depth.

The two principal types of turbulence model which provide closure for the RANS equa-

tions are: Eddy Viscosity Models (EVM) and Second Moment Closure (or “Reynolds Stress

Model” or “Differential Stress Model”, DSM). Eddy Viscosity Models use the turbulent vis-

cosity hypothesis of Boussinesq (1877) to define a relationship between the shear stress and

strain rate for a simple shear flow:

uv νt

∂U

∂ y (1.1)

where νt is the “turbulent kinematic viscosity”. This statement for the shear stress can be

extended to describe the complete Reynolds stress tensor:

uiu j νt

∂U i

∂ x j

∂U j

∂ xi

2

3k δi j (1.2)

where k is the turbulent kinetic energy and δi j is Kroneker’s delta. The implicit assumptions


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in this statement are that the Reynolds stress anisotropy:

ai j

uiu j

k

2

3δi j (1.3)

can be determined by the local velocity gradients and that the stress-strain relationship islinear. Neither of these assumptions are valid for cases other than simple shear. However, the

simplicity of the model and its ability to predict a wide range of flows with numerical stability

and reasonable accuracy have lead to its widespread use, particularly in two-equation models

(see below).

The turbulent viscosity, νt , acts in the same fashion as the molecular viscosity, ν, although

unlike the molecular viscosity, it is a property of the fluid’s motion and not a bulk property

of the fluid itself. EVMs need a suitable method of specifying the turbulent viscosity which

can be done algebraically, for example by Prandtl’s (1925) analogy with the kinetic theory of

gases which supposes:

νt ∝ lt V t (1.4)

where lt is the turbulent length-scale and V t as the turbulent velocity-scale. This is known as

the Mixing Length Hypothesis (MLH). Although Prandtl did not conceive of it in these terms,

the MLH may be arrived at by assuming that in a simple shear flow turbulence is dissipated

where it is generated. This ignores transport effects, and is only generally applicable near

walls. Near a wall there is only one significant Reynolds stress component and velocity

gradient, thus:

νt lmV t (1.5)

where lm is the mixing length-scale. V t is determined dimensionally from the local mean

velocity gradient, V t

lm ∂U

∂ y to give:

νt l2m

∂U

∂ y

(1.6)

Prandtl assumed that for boundary layer flow the mixing length would be proportional to

the distance from the wall and observed experimentally that lm

κ y where κ is Karman’s

constant. However, this definition of mixing length does not apply all the way to the wall

and refinements are often used to provide near-wall damping. For example Van Driest (1956)

proposed the following form for the mixing length:

lm κ y 1 exp

y

26

(1.7)


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where y

U τ y

ν is the non-dimensional distance from the wall U τ τw

ρ is the friction

velocity and τw is the wall shear-stress. The specification of mixing length is the downfall of

these models: it is generally geometry specific and the model is not able to handle important

features such as flow separation and recirculation. Where such features do not occur (such as

in attached boundary layers), mixing length models can be very successful and their use inthe aeronautical industry is well established.

The mixing length model described in Equation 1.6 bases the turbulent velocity scale on

the local velocity gradient. In several classes of flow this is not valid, for example along the

centreline of a pipe or round jet the velocity gradient is zero and yet the turbulent velocity

scale is non-zero. Prandtl (1945) and Kolmogorov (1942) independently suggested that the

turbulent velocity-scale could be taken from the turbulent kinetic energy, V t ∝

k which is a

reasonable assumption if turbulent transport and molecular transport are analogous and gives

an expression for turbulent viscosity:

νt c µ kl (1.8)

c µ is a constant. Prandtl and Kolmogorov both proposed that a differential transport equation

should be calculated to provide k , and models of this type are referred to as k l or “one-

equation” models. However, as the length-scale, l must still be specified empirically, one-

equation models are only slightly more general than mixing length models.

This problem can be overcome by calculating a differential transport equation for length-

scale as well as velocity-scale. Such models are known as “two-equation models”. Many

two-equation models have been proposed, which differ principally in the variable from which

the length-scale is derived. Kolmogorov (1942) proposed a transport equation for the mean

frequency of the most energetic motion, f k 1 2 L

Rotta (1951) proposed transport equations

for integral length-scale, L and the turbulent kinetic energy and integral length-scale com-

bined, kL Wilcox (1988) proposed a model with the turbulent length-scale derived from a

transport equation for ω k

l2. Chou (1945), Davidov (1961), Harlow & Nakayama (1968)

and Launder & Sharma (1974) have all proposed models with the turbulent length-scale de-

rived from a transport equation for dissipation of turbulent kinetic energy, ε ∝ k 3 2

l. The

choice of ε for the length-scale determining transport equation is a logical one as it is a phys-ical quantity and it appears in the k -transport equation. The high Reynolds number k ε

model has become the most widely used turbulence model throughout industry over the past

twenty-five years. It is able to provide stable and reasonably accurate solutions for a wide

range of industrially relevant flows without the need to modifiy the empirical constants in the

turbulence transport equations. In the k ε model the turbulent viscosity is calculated by:


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νt c µ

k 2

ε (1.9)

where c µ is a constant of proportionality which is normally defined empirically by considering

flow under local equilibrium.

A particular shortcoming of the k ε model and indeed EVMs in general, is that the

Reynolds stresses are calculated from local values of velocity gradients. Furthermore, the

stress-strain relationship commonly employed is linear (Equation 1.2). If a higher level of

closure is provided by calculating differential transport equations for the Reynolds stresses,

then it is possible to include non-local and history effects in the calculation. Such models are

known as Second-Moment Closure models, Reynolds Stress Models or Differential Stress

Models (DSM). Launder, Reece and Rodi (1975) developed such a model which has become

widely used and is sometimes referred to as the LRR model. In a three-dimensional calcu-

lation, six differential-transport equations are required for the Reynolds stress, u iu j. In thesetransport equations, production is calculated in its exact form, diffusion is modelled by the

gradient diffusion model of Daly & Harlow (1970) and dissipation is modelled by assum-

ing isotropy of the time-scale dissipative eddies (this requires the solution of an additional

transport equation for turbulent energy dissipation, ε). There is also a term known as the

“pressure-strain” or “redistribution” term, as its overall effect is to redistribute energy among

the normal stresses and reduce the shear stress.

There are two distinct processes which affect the pressure-strain term, φ i j. Firstly, there

is the pressure fluctuation due to the interaction of two turbulent eddies, φi j1, which is some-

times referred to as the “slow” term. Secondly, there is the pressure fluctuation due to the

interaction of turbulent eddies with the mean strain of the flow, φ i j2, sometimes referred to

as the “rapid” term. Hanjalic & Launder (1972) suggested a model for pressure-strain and

the subsequent models of Launder et al (1975), Jones and Musonge (1983) and Speziale et

al (1991) are all developments of that model. In addition to the rapid and slow contributions,

pressure-strain is affected by wall proximity. Models for “wall reflection”, φwi j

which de-

scribe this effect have been proposed by Shir (1973) and Gibson & Launder (1978). In spite

of the importance of wall proximity on pressure-strain, these models have the undesirable

feature of introducing the distance to the wall and the wall normal direction, which make themodel difficult to apply in complex geometries.

The large number of equations which are required for DSM and the uncertainties over

modelling the pressure-strain process have led to a slow take-up of the technique by indus-

try. The former obstacle is gradually being redressed as available computing speed increases;

the latter has led to the development of improved techniques. A failure of LRR type stress

models is that the rapid part of the pressure-strain is modelled by products of the mean strain


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and linear combinations of the Reynolds stresses. This treatment is not valid in flows with

large degrees of curvature or strong anisotropy, as demonstrated by Li (1992) in his study

of flow in non-circular cross-sectioned ducts and Fu (1988) in a study of swirling flows.

Speziale, Sarkar and Gatski (1991) proposed a model (known as the SSG model) which in-

cludes quadratric terms in the slow pressure-strain. This model reproduces the near-wallanisotropy without the need for wall topology dependent correction terms. Whilst this model

is not entirely successful, it is a distinct improvement on linear pressure-strain models. Laun-

der & Li (1994) and Craft et al (1996a) took this approach further by including up to cubic

terms in the rapid pressure-strain model, and defined new models for the rapid pressure strain

and εi j. Rather than developing non-linear pressure-strain models, an alternative approach

was taken by Durbin (1993). In his “elliptic relaxation” model a higher level of closure is

used to define the pressure-strain term from the solution of an elliptic equation, hence incor-

porating non-local effects into the pressure-strain.

Eddy Viscosity Models tend to fail because of the linear stress-strain relationship which

they use to calculate Reynolds stress. Differential Stress Models, on the other hand, are

unattractive because of the large number of differential transport equations required for the

Reynolds stresses. This can lead to high CPU demands which cannot be satisfied for indus-

trial calculations. A compromise solution is to calculate the anisotropic stress tensor (Equa-

tion 1.3) without resort to additional transport equations. One such approach is the Algebraic

Stress Model (ASM) of Rodi (1972) which expresses the Reynolds stresses in a set of six

implicit algebraic equations. Although it is a successful model in terms of improving EVM

calculations, it is not widely used as the implicit nature of the algebraic equations leads toproblems of numerical “stiffness” and high CPU demands.

An alternative approach is to express the anisotropy tensor explicitly from non-linear

polynomials of the mean velocity gradients. Early work was done on this by Rivlin (1957)

and Lumley (1970). Pope (1975) defined a general expression for the anisotropy tensor which

included up to quartic relationships of the normalized mean strain and vorticity tensors, S i j

and Ωi j, which are themselves defined from the local velocity gradients:

S i j

∂U i

∂ x j

∂U j

∂ xi

; Ωi j

∂U i

∂ x j

∂U j

∂ xi

(1.10)

This provides a framework from which a number of non-linear EVMs can be defined and in

the trivial case it can be reduced to a linear k ε type model. However, Pope did not evaluate

the coefficients of the non-linear polynomial required for a three-dimensional model. This

was achieved later by Taulbee (1992) and Gatski & Speziale (1993).

In practice, the Non-Linear Eddy Viscosity Model (NLEVM) was not taken up as a line


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of development until the late 1980’s. Speziale (1987) defined a model which he calibrated for

square duct flows and pipe U-bends; this model was later successfully applied to a backward

facing step with good results by Thangam & Speziale (1992). Nisizima & Yoshizawa (1987)

defined a NLEVM which they applied to channel and Couette flow, Rubinstein & Barton

(1990) developed a NLEVM from re-normalization group theory. Myong & Kasagi (1990)defined a model and applied it to boundary layer and square duct flows and Shih et al (1993)

applied their NLEVM to rotating flow and a backward facing step. The one unifying feature

of all these models is that they all defined the anisotropy tensor as a polynomial expression

with up to quadratic relationships of S i j and Ωi j (the strain and vorticity tensors). However,

as the models were calibrated for different classes of flow, the coefficients of the terms in the

polynomial vary widely between the models. They are not significantly more applicable to

general industrial flows than are linear EVMs. Moreover, as the stress-strain relationships

employed cease at the quadratic level, these models are not able to predict the viscous effects

of Reynolds stresses due to streamline curvature or swirl.

In response to the deficiencies in quadratic NLEVMs, a number of cubic NLEVMs

have been proposed recently. Suga (1995) developed two forms of a low Reynolds num-

ber NLEVM; a two equation k ε model and a three equation k ε A2 model, in which a

transport equation for A2 (the second invariant of anisotropy) was used to improve the near-

wall flow prediction. These models are also presented in Craft et al (1996b) and Craft et al

(1997) respectively. Suga’s models are based on the low-Reynolds-number linear k ε model

of Launder & Sharma (1974) and use Pope’s (1975) algebraic relationship for the anisotropy

tensor and stress-strain relationship. The coefficients in the stress-strain relationship wereselected by consideration of a number of test cases which isolated different aspects of the

model: homogeneous shear, fully developed swirling shear flow and flow with streamline

curvature. These test cases showed that the NLEVM performs consistently better than the

Launder & Sharma linear EVM with only a 10% increase in computing time. More recently,

Suga et al (2000) have applied the three equation k ε A2 version of the NLEVM to several

flows which are pertinent to the road vehicle industry and obtained good results in complex

three-dimensional flows.

Apsley & Leschziner (1998) developed a low Reynolds number k ε NLEVM deriving

the stress-strain relationship from successive iterations to the DSM of Launder, Reece and

Rodi (1975), truncating the process at the third iteration to provide the cubic (in strain and

vorticity tensors) relationship. The usual low-Reynolds-number procedure of applying the

same damping function, f µ to all the stresses does not take into account the different be-

haviour of the Reynolds stresses near the wall. These difference were incorporated into the

Apsley & Leschziner model. The coefficients of the non-linear relationship can in principle


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be defined from the parent DSM from which the NLEVM was defined. However, Apsley &

Leschziner noted that the DSM which they used to develop the model used a linear pressure-

strain correlation (φi j) and did not include wall reflection effects. They therefore used the

parent DSM to provide the relationship between the coefficients and calibrated the coeffi-

cients against DNS data for a plane channel flow. Apsley & Leschziner tested their model ina number of complex two-dimensional flows. They found that it performed well in compar-

ison to a number of linear, quadratic and cubic EVMs in flows in which separation was not

determined by the geometry (high lift aerofoil and plane diffuser). However, the new model

did not perform so well in a backward facing step: the recirculation length was over-predicted

due to insufficient turbulence energy generation in the curved shear layer.

Wallin & Johansson (2000) proposed an Explicit Algebraic Reynolds Stress Model3 (EARSM)

which represents a solution of the implicit ASM of Rodi (1972) in which the production to

dissipation ratio is obtained as a solution of Pope’s (1975) non-linear algebraic expression.

This was a low-Reynolds-number model and incorporated a modified wall treatment, which

was based on the van Driest damping function (Equation 1.7) and ensured realizability of

the Reynolds stresses near the wall. As the model calculates the production to dissipation

ratio of turbulent kinetic energy with the correct asymptotic profile for high strain rates, it re-

quired less wall damping than is generally used in low-Reynolds-number models. In testing

their model, Wallin & Johansson found that it improved results gained from linear EVMs and

gave a reasonable repetition of experimental measurements for axially rotating pipe flow and

compressible flows with Mach number upto Ma 5.

1.2.5 Near-Wall Effects

The treatment of wall boundary conditions requires particular attention in turbulence mod-

elling. Viscous stresses in the flow remote from a wall boundary are in general negligible

in comparison to the turbulent stresses. However, as the wall is approached the turbulent

shear stress is damped and the viscous stresses become more important. This results in sharp

gradients in the velocity, Reynolds stresses and other modelled quantities such as k and ε

In DNS this specific problem does not arise as the calculation domain will be sufficiently

resolved to capture these gradients. This is also true for a sub-class of LES, known as LES-

NWR (Near Wall Resolution), in which more than 80% of the flow kinetic energy is contained

in the resolved velocity field and there is sufficient resolution to calculate the near-wall gradi-

ents (Pope, 2000). If there is insufficient near-wall resolution of the grid or less than 80% of

3The terms Explicit Algebraic Reynolds Stress Model (EARSM) and Non-Linear Eddy Viscosity Model

(NLEVM) are synonymous. The adoption of one rather than the other merely emphasises the prarticular line of

development of the model.


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the flow kinetic energy is contained in the resolved velocity field, modelling techniques such

as those applied with EVMs must be used (see following discussion of wall functions).

DSMs and EVMs in which the RANS and turbulence equations are calculated right up

to the wall on a computational grid with sufficiently fine resolution to capture the steep,

near-wall gradients are known as “low-Reynolds-number” models. Examples of these are:Launder & Sharma’s (1974) k ε model which includes additional terms in the momentum

and turbulence equations to account for viscous effects, Wilcox’s (1988) k ω model, and

more recently Craft’s et al (1997) k ε A2 cubic NLEVM which uses the additional A2

(second invariant of anisotropy) transport equation to improve the near-wall calculation.

The near-wall boundary layer flow is often considered as consisting of distinct regions.

These are the viscous (laminar) wall layer, the “log-law” layer (between the core flow and

the laminar wall layer where the velocity profile is described by a logarithmic profile) and the

“buffer zone” which blends these regions. A potential problem arises in that the height of the

boundary layer is inversely proportional to the Reynolds number of the flow. Thus at high

Reynolds numbers the boundary layer is thin and fine computational cells are required near

the wall to calculate the flow. This can lead to numerical stiffness and a large computational

expense, particularly in three-dimensional computations. Instead of retaining a DSM or two-

equation EVM right upto the wall, a technique which is sometimes used is to revert to a

one-equation model in the near-wall region and specifying a length-scale. This is known as

zonal modelling.

An alternative approach is to adopt a “high-Reynolds number” model which uses a “wall

function” to bridge the solution between the wall and the fully turbulent core flow. Launder& Spalding (1972) outlined a technique which has become popular for specifying wall func-

tions. The first computational node is placed outside the log-law region, in the fully turbulent

flow at a non-dimensional distance from the wall 30 y

300, where the non-dimensional

distance is defined by y

U τ

ν y and the friction velocity is U τ τw

ρ. The local equi-

librium condition (ie. the production of turbulent kinetic energy equals the dissipation) and

the “universal” log-law of velocity:

U

1

κ

ln

Ey

(1.11)

(where E and κ are constants and U the non-dimensional velocity) are then used to define

the local wall shear-stress, τw. This wall shear-stress is used as a source term in the mo-

mentum equations to account for the frictional force of the wall on the flow. As the wall

function is commonly employed in a two-equation EVM it is also necessary to define values

for average production of turbulent kinetic energy (Pk ) and average dissipation rate (ε) in the


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near-wall cell. Pk is defined by assuming constant shear stress across the near-wall cell and

the value of ε calculated at the near-wall node is assumed to be the average value across the

near-wall cell.

The assumptions used in Launder & Spalding’s wall function are rather crude and Chieng

& Launder (1980) attempted to improve them. They allowed the shear stress to vary in thenear-wall cell, assuming that it would be zero in the portion of the cell which spanned the

viscous sub-layer and would vary linearly in the remainder of the cell (which spans fully

turbulent flow). Also, Chieng & Launder based the calculation of the wall shear stress on the

turbulent kinetic energy at the edge of the viscous sub-layer (k v) rather than at the near-wall

node and defined the height of the near-wall sub-layer by the sub-layer Reynolds number,

Rv

k 1 2v yv

ν 20. Johnson & Launder (1982) noted that there are several classes of flow for

which a constant viscous sub-layer height was not valid. For example, in rapidly accelerating

boundary layers, the magnitude of the turbulent shear stress falls rapidly with distance from

the wall resulting in an increased sub-layer thickness. Similarly, in reattaching flow where

there are low levels of wall shear-stress, but high levels of turbulent kinetic energy near the

wall, the height of the sublayer is decreased. To account for these variations Johnson &

Launder defined a functional form of the sub-layer Reynolds number:

Rv

20

1 3 1λ ; λ

k v k w

k v(1.12)

The problem with all the above forms of the wall function is that they assume the log-law

profile for velocity and either a constant or linear variation of total shear stress. No account ismade of pressure gradient, convective transport or body forces. Hence, developing flows, flow

in adverse or positive pressure gradient or flows subjected to heating, magneto-hydrodynamic

forces, etc. will not be properly represented.

Recently, a programme of work has been undertaken at UMIST to develop improved

wall functions. Gant (2000) has developed a “sub-grid wall function” in which the near-wall

cell is sub-divided to allow simplified one-dimensional transport equations to be calculated

and provide profiles of velocity and turbulence values. These profiles are then integrated

to provide the usual wall function parameters. Gerasimov (1999) calculates an analytical

solution of a simplified momentum equation in the near-wall cell assuming a viscosity profile.

Preliminary results have shown improvements in the calculation of impinging jet flows (Gant,

2000) and flows with strong buoyancy forces (Gerasimov, 1999).


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1.3 Test Cases

1.3.1 Cylinder of Square Cross-Section Close to a Wall

This test case is useful in developing models for the calculation of road-vehicle aerodynamics

as it contains certain features in common with the road vehicle’s aerodynamics. For example,

the impinging flow and strong curvature of the streamlines at the front face and around the

front edges of the bluff body are experienced by all but the most highly streamlined road

vehicles; there is large-scale separation at the rear of the vehicle and bluff body; complex

wake structures are observed in both cases which are affected by the proximity of the wall

(or ground).

Experimental Studies Early experiments on flows past two dimensional square cylinders

tended to concentrate on free stream flows without any wall influence. Vickery (1966) mea-sured fluctuating lift and drag coefficients and found that a square cylinder gives more lift

than a circular cross sectioned cylinder and that fluctuating lift increases with the turbulence

intensity. Lee (1975) measured mean and fluctuating pressure fields around a square cylinder

and found that weaker vortices were shed behind the cylinder at higher turbulence intensities

due to the thickening of the shear layers and intermittent reattachment of the shear layers

on the cylinder sides. Namarinian & Gartshore (1988) repeated Vickery’s and Lee’s exper-

iments but measured higher levels of fluctuating lift coefficient over a range of turbulence

intensities and Reynolds numbers. More recently Cheng et al (1992) also measured the flow

around a square cylinder in free stream conditions. They found that although the mean drag

decreased and RMS fluctuating drag increased as free-stream turbulent intensity increased,

the RMS fluctuating lift coefficient remained constant as the free-stream turbulence intensity

increased. However, the RMS fluctuating lift did decrease with increasing length-scale of

free-stream turbulence.

Devarakonda & Humphrey (1996) studied turbulent flow past two-dimensional square

cylinders in tandem (effectively in free stream conditions) and single cylinders placed at

various distances from a wall. The measurements were taken for a range of Reynolds num-

bers from Re

10

000 to 27,500 and showed that on single cylinders near a wall there wassome variation in mean drag with Reynolds number but no variation in mean lift. The non-

dimensional vortex shedding frequency or Strouhal number is given by

St

f

d

U o(1.13)

where f is shedding frequency, d the characteristic dimension of the bluff body and U o is the

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free stream velocity. Devarakonda & Humphrey found that the Strouhal number increased

as the distance to the wall decreased and St 0

154 at the nearest measured cylinder-wall

separation, g

d 0

95 (see Figure 1.1). The effect of the tandem prisms (with a smaller

prism placed in front of the main prism) was to decrease the drag and increase the frequency

of vortex formation on the main prism.Vortex shedding does not occur when the cylinder is mounted on the wall and periodic

oscillations in the wake are suppressed (Dimaczek et al, 1989, Schulte & Rouve, 1986, Good

& Joubert, 1968). As the square cylinder is moved progressively away from the wall, a crit-

ical cylinder-to-wall gap height is reached (gc) at which the steady wake ceases and regular

vortex shedding occurs. For the cases of circular, triangular and square cylinders respectively,

Bearman & Zdravkovich (1978), Kamento et al (1984) and Tanigushi et al (1990) obtained

critical gap heights of 0.3, 0.35 and 0.55. Tanigushi et al also measured a constant vortex

shedding frequency, St 0 14 for g

gc. Tanigushi & Miyakoshi (1990) concluded that vor-

tex shedding was suppressed when the cylinder was close enough to the wall for the boundary

layers of the wall and cylinder to interact. The exchange of vorticity of opposite sign in these

boundary layers reduces the vorticity in both, and highly turbulent eddies from the outer

boundary layer penetrate the separated shear layer and further weaken vorticity there.

Durao et al (1991) presented measurements for the flow past a square cylinder at Re

13 600 and a number of cylinder-wall separations. By analysing the power spectra of fluc-

tuations in the flow, they concluded that there was no vortex shedding for g

d 0

35

They

presented detailed mean velocity and stress results for two cases: one with steady flow

(g

d

0

25) the other with vortex shedding periodic flow (g

d

0

50) and the Strouhalnumber was measured as St

0

133 for this latter case. They concluded that proximity to the

wall increases the recirculation length behind the cylinder and causes asymmetric velocity

distributions. Also, the large regions of turbulence anisotropy which were measured would

require turbulence models which considered the individual Reynolds stresses separately, if

the problem were to be calculated numerically.

As part of a larger study of flow past square cylinders in general, Bosch et al (1996)

studied the effect of the cylinder-wall separation on the vortex shedding behaviour. Previ-

ous work in this programme had concentrated on free-stream measurements of the turbulent

near wake (Lyn et al 1995) and the flapping shear layer formed behind the cylinder leading

edges (Lyn & Rodi, 1994). Bosch et al confirmed that there was no vortex shedding below

g

d 0 35, while regular vortex shedding occurred only above g

d 0 50. Between these

limits, there was a mixture of steady flow interspersed with vortex shedding flow. Detailed

measurements of the flow were made at g

d 0

75 with the vortex-shedding frequency being

established as St 0

139 and the measured stresses being split into the turbulent and periodic

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component parts. Bosch et al noted some similarities with the free-stream flow studied by

Lyn et al (1995) but also measured asymmetries in the flow which were due to the presence

of the wall. They found that the downstream recirculation length increased as the cylinder

was moved closer to the wall, which was in agreement with Durao et al (1991).

Computational Studies Most numerical studies on flows past square cylinders which have

been reported in the literature have concentrated on the vortex shedding flow past the cylinder

in free-stream conditions. There are fewer studies which report the influence of proximity of

a wall in the calculations. Lee (1997) compared the performance of three k ε EVMs (linear

model with wall functions, RNG with wall functions and linear LRN) on the flow past a

square cylinder in a free stream at Re 22

000. He concluded that the RNG and LRN models

are superior but that the solution is equally dependent on accurate time and spatial resolution

and the choice of convection scheme. Franke & Rodi (1991) compared measurements with

calculations obtained from models using wall functions and a zonal 1-equation model in

the near-wall region. In the flow away from the walls they used both a linear k ε model

and a DSM. No vortex shedding was produced by the EVM with wall functions and only

poor shedding was calculated when this model was employed with the zonal approach. The

DSM models gave a good reproduction of experimental behaviour although in all cases there

appeared to be too much viscous damping which was attributed to an incorrect specification

of inlet turbulence length-scale.

Kato & Launder (1993) recognised that the cause of poor predictions by the linear k ε

model was due to this model’s excessive production of turbulent kinetic energy in an imping-ing flow. This could be avoided by redefining the turbulence production as a function of the

product of strain and vorticity invariants, so that the turbulence production was reduced to

zero at stagnation. This simple modification caused a great improvement in the calculation

of lift and drag forces, and on the mean turbulent flow patterns in the wake.

LES calculations were carried out by Sohankar (1998) with three different sub-grid scale

(SGS) models: a Smagorinsky model, a dynamic Smagorinsky model, and a 1-equation

model (Davidson, 1993). This last model was found to give the best agreement with the

experimental results of Lyn et al (1995). A more comprehensive study of RANS and LES

results for the same case was conducted by Rodi (1997) and included results from an LES

workshop. The models which were compared were: (i) a “standard” linear k ε, (ii) Kato-

Launder (1993) linear k ε (both (i) and (ii) were applied with, separately, wall functions

and a zonal approach), (iii) DSM and (iv) various LES models including Smagorinsky and

dynamic Smagorinsky SGS models. Of the RANS models, the two-layer approach and Kato-

Launder model were both found to be improvements on the standard linear k ε model;

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using the Kato-Launder model and zonal wall treatment combined was beneficial. However,

the DSM model gave the best of the RANS results, in terms of the mean drag coefficient,

wake recirculation length and turbulent kinetic energy levels in the wake. The LES models

were found to behave quite differently from the RANS models due to their ability to detect

low frequency oscillations in the flow and include these in the turbulence calculation. How-ever, there was a wide difference between the LES results indicating that care must be taken

with the grid resolution and SGS model specification.

Bosch & Rodi (1996) calculated the flow past a square cylinder at two distances from a

wall, g

d 0

25 and 0.50. Using a linear k ε model and a Kato-Launder k ε model, the

calculations attempted to reproduce experimental results of Bosch et al (1996). The Kato-

Launder k ε model was found to be an improvement over the standard k ε model: it

was able to calculate a vortex shedding flow at g

d 0

50 (as is measured experimentally)

whereas the standard k ε model produced steady flow which was incorrect.

1.3.2 U-bend of Square Cross-Section

Flows in square-sectioned ducts with a large degree of curvature undergo significant straining

and are subject to radial pressure gradients which induce secondary, cross-stream motions.

Secondary motions are also induced by the Reynolds stresses which are affected by the cur-

vature in the duct and by the square cross-section shape of the duct itself. Far from being

of academic interest only, these flows are important in many industrial applications such as

heat exchangers and cooling passages in turbine blades. The test cases are also useful in the

development and study of models for calculating flow past road vehicles. The streamline

curvature and streamwise vorticity generated in the U-bend is present in the motor vehicle

external aerodynamic flow, particularly around the front edges of the vehicle and in its wake.

The test cases allow these flow features to be studied in isolation. As the flow in the square

cross-section U-bend is very sensitive to the Reynolds stresses, the experimental and numer-

ical details of past studies are important and will be discussed in some depth.

Chang et al (1983a) proposed a searching test case for turbulence models when they mea-

sured the flow in a 180o U-bend with a radius of curvature of Rc

D 3

35 (see Figure 1.2;

D is the duct’s hydraulic diameter) and Reynolds number of Re 56 700. A long upstreamsection was used to develop an essentially fully developed flow at the inlet in which the

boundary layers completely filled the duct. Previously measurements had been made in a 90 o

bend with square cross-section by Humphrey et al (1981) using a fully developed flow at the

inlet plane and by Taylor et al (1982) using a bend inlet condition with thin boundary lay-

ers. Humphrey et al and Taylor et al showed that flow with a fully developed inlet condition

had high levels of turbulence and strong secondary motions induced in the bend. Humphrey

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et al (1981) carried out computations of the case with the fully developed inlet flow using a

two-equation linear k ε model; Buggeln et al (1980) had previously calculated the case with

the thin inlet boundary layers using a one-equation k l model. Both calculations showed

reasonable agreement with the measured data up to 45o, thereafter Buggeln et al (1980) have

better agreement. This is not an indication that the one-equation model is superior, as bothmodels use the isotropic eddy viscosity assumption. Moreover, the case with the fully devel-

oped inlet boundary (Humphrey et al) has higher levels of Reynolds stresses which cannot

be calculated accurately by either model and hence Humphrey et al have the less accurate

prediction.

In Chang et al’s (1983a) experiment the U-bend was extended to 180o, a longer period

of straining was introduced and a more complex secondary flow pattern developed. Most

notable amongst Chang et al’s measurements was the “hole” which develops in the stream-

wise velocity

W

W b profile near the inside (convex) wall between 90o and 130o (shown in

Figure 1.3). This is an indication that there is a significant secondary motion which is differ-

ent to the “double vortex” pattern often shown in textbooks. From the bend entry to the 90 o

plane the destabilizing effects of the curvature at the concave wall were shown by the increase

in the measured Reynolds stresses in this region. Between 90o and 180o striking variations

in the Reynolds stresses were measured in the radial direction. Localized increases in ww

(streamwise normal stress) and corresponding decreases in uu (cross-stream normal stress)

in the core flow were attributed to the large shearing motions induced by inviscid forces in

the core flow. Chang et al used a two-equation high-Reynolds-number linear k ε model to

calculate the flow, using wall functions to specify the near-wall velocity and turbulence con-ditions. To reduce the computational requirements, a semi-elliptic procedure was adopted

which required only the pressure to be stored over the whole domain. Other variables (U

V k

ε) were calculated on a plane by plane basis, with the code “sweeping” through the

planes in the streamwise direction. QUICK was employed as the cross-stream differencing

scheme whereas UPWIND differencing was used in the streamwise direction4, hence only

two adjacent streamwise planes needed to be stored at any time during the calculation. As

the numerical model employed was isotropic, the redistributive effects of the cross-stream

normal stresses were not present and the minima in the streamwise velocity (W

W b) between

90 130o were not predicted (Fig.1.3). Chang et al also failed to calculate the vortex at the

convex (inner) wall corner which is associated with local separation.

Further measurements and calculations on the square sectioned U-bend were carried out

by Chang et al (1983b). The experimental measurements were repeated using air as the work-

ing fluid and taking measurements with hot-wire anemometry (as opposed to water and LDA

4Convection schemes are discussed in Chapter 3.

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in the original experiments). These measurements confirmed the development of the “hole”

in the W

W b profile at 90o. The calculations were repeated, again using the semi-elliptic ap-

proach with only the pressure being stored in three dimensions and in these calculations both

a simple log-law type wall function and a more elaborate wall function (Johnson & Launder,

1982) were employed. Calculations were also attempted with an ASM turbulence model aswell as the linear k ε model. Again Chang et al failed to predict the “hole” in the stream-

wise velocity profile, with either turbulence model or wall function. The coarse streamwise

grid and use of wall functions rather than integrating the flow through the viscous wall-region

were presumed to be the cause of the models’ failure to predict the streamwise velocity pro-

file accurately. One would have expected the ASM to give more accurate predictions than

the EVM as the individual Reynolds stresses were calculated. Surprisingly, the results from

the ASM were in poorer agreement with the measured values than the results from the k ε

model. This was thought to be due to the first order accurate UPWIND convection scheme

which had to be used with the ASM for computational efficiency. (The linear k ε model

was used with a higher-order accurate QUICK scheme).

Heat transfer measurements were made in a square sectioned U-bend by Johnson &

Launder (1985) in an attempt to shed further light on the secondary motions in the U-bend

( Rc

D 3 35; Re

56 000). From the temperature contours measured, they were able to in-

fer that at the 90o position, the secondary flow path near the mid-plane of the duct is brought

to a halt and the flow is displaced towards the lower wall (Figure 1.4) It was noted that this

mechanism is also present in circular cross-section pipes and is the cause of the “hole” in the

streamwise velocity profile between 90

130o

.Azzola et al (1986) conducted measurements and calculations of flow in a 180 o bend with

circular cross section

Rc

D 3

375; Re

57

400 . The flow produced in this configuation

was quite different from the flow in a square U-bend. In a circular cross-section U-bend the

anisotropies between the normal stresses (generated at the corners of a square cross-section

duct) which modify the secondary motion are not present. However, the strong cross-stream

pressure gradients remain. Azzola et al incorporated some important improvements to their

numerical model, over that used previously by Chang et al (1983a,b). The semi-elliptic linear

k ε model was retained, but the wall functions were dropped and a zonal model was applied,

extending a fine grid right up to the wall. In the wall-region the mixing length hypothesis

was employed with a Van Driest damping term (Equation 1.7). To reduce the computer

storage required, the so-called “PSL” approximation was adopted. This assumes that pressure

variations near the wall are small enough such that the pressure in the near-wall fine grid cells

can be assumed to equal the pressure just outside that region. Thus pressure did not need to

be stored in the fine near-wall cells. The experiments showed that between 45 135o the

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cross-stream secondary flow undergoes a reversal and is redirected back towards the inner

wall (in the inner half of the flow). Hence, there were four secondary flow vortices; two each

side of the symmetry plane. On the whole, the semi-elliptic linear k ε model calculated the

flow, including the secondary motions, reasonably well. As expected the levels of secondary

flow induced in the U-bend were less than those present in a square sectioned U-bend.The improvements in the numerical model used by Azzola et al (1986) were incorporated

into a numerical model of the square sectioned U-bend by Choi et al (1989) and further adap-

tions were made which were found to be beneficial to the flow calculation. In calculating the

“hole” in the streamwise velocity profile, only a slight improvement was found over previous

calculations which employed wall functions. Choi et al argued that although the pressure

variations near the wall are small, they become significant at the corners. By dropping the

PSL approximation and calculating the pressure right across the near-wall region, a signifi-

cant improvement was made in the calculated streamwise velocity profile. Calculations using

an ASM in place of the linear k ε model improved the predicted flow further. At the 130o

station in the bend the model with the ASM now predicted a complex flow pattern with four

vortices either side of the symmetry plane. (The linear k ε /MLH model predicted three;

the linear k ε /wall function model predicted two). Whilst the experimental measurements

were too coarse to compare the measured and calculated secondary flow profiles in detail,

it seemed highly likely that the four-vortex pattern was indeed accurate due to the improve-

ments in predicting the streamwise velocity profiles.

The measurements and calculations by Chang et al (1983a,b), Johnson & Launder (1985),

Azzola et al (1986) and Choi et al (1989) all used a fully developed inlet profile with theboundary layers entirely filling the duct at the entrance to the U-bend. Measurements by

Taylor et al (1982) and Humphrey et al (1981) had shown that in a square duct with a 90 o

bend, when thin boundary layers are present at the entry to the bend, the Reynolds stresses

and secondary motions induced are much smaller than when fully developed flow is specified

at the bend inlet. In order to establish whether this trend would continue when the flow was

subjected to a longer period of straining, Iacovides et al (1990) carried out measurements

and calculations on the square sectioned U-bend with thin boundary layers (0 15 D) at the

bend inlet. The same numerical model was employed as used by Choi et al (1989) retaining

the semi-elliptic ASM treatment with MLH and Van Driest damping at the walls. A similar

“hole” in the streamwise velocity profile was found between 90 135o as occurs in the flow

with the fully developed inlet flow and had been initially measured by Chang et al (1983a).

The flow profile was calculated at least as well as by Choi et al (1989), when they had calcu-

lated the flow with fully developed inlet conditions. However, the secondary flow plot at 135 o

now showed five vortices either side of the symmetry plane, and so despite the lower amount

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of Reynolds stress anisotropy and secondary flow at the inlet to the bend, the flow seemingly

breaks down into an even more complex secondary flow pattern. Choi et al noted that this

breakdown of the classic single secondary vortex occurs later in the calculations than in the

measurements.

The semi-elliptic treatment was dropped by Iacovides et al (1996) in their calculation of flow in the same square sectioned U-bend ( Rc

D 3

37; Re

56

700 . They used a fully el-

liptic solver which calculated and stored all the flow variables (U V

W

P

k

ε T ) across the

whole computational domain. One of the advantages of employing a fully three-dimensional

solver was that a third-order accurate convection scheme (QUICK) could be applied in the

streamwise direction. This improves the accuracy of the streamwise velocity and stress cal-

culations and reduces the number of streamwise planes required. Three numerical schemes

were employed: the same ASM as used by Iacovides et al (1990), a basic DSM employing

“wall reflection” terms in the pressure-strain correlation (φW i j ), and a new DSM with a cubic

model for the pressure-strain correlation

φi j which enabled “wall reflection” to be dropped.

The cubic DSM had already been shown to be effective in near-wall flows by Launder &

Tselepidakis (1993) and Launder and Li (1994).

Slight improvements were seen in the calculated streamwise velocity (W

W b) , shear

stress (uw

W 2b ) and heat transfer ( Nu results by the basic DSM compared to the ASM.

Although the DSM required a longer time to solve each iteration of the transport equations

(due to the additional equations required for the Reynolds stresses), the ASM was numerically

less stable and required a greater number of iterations to achieve an adequately converged

solution. Iacovides et al concluded that an ASM no longer gave any advantage over a DSM,as it produced less accurate results in a comparable time to the basic DSM. They found

little difference in the calculated streamwise velocity profiles between the basic and cubic

DSM. There were improvements in the Reynolds stress calculations, especially where the

turbulence was enhanced by streamline curvature at the outer (concave) wall of the bend.

Improvements in the Nu calculation were more obvious. (As there is no equivalent to the

pressure gradient in the temperature equation, temperature field calulations are particularly

sensitive to the accuracy of the calculated Reynolds stresses and hence the turbulence model

employed.)

In a study of applications of the cubic non-linear k ε A2 model of Craft et al (1997),

Suga et al (2000) chose the square cross-section U-bend with a radius of curvature of Rc

D

3 35 as one of their test cases. The cubic k ε A2 model is a low-Reynolds-number model,

in which all three equations are integrated through the viscous near-wall region, right up to the

wall. Suga et al used a Launder & Sharma (1974) linear low-Reynolds-number k ε model

for comparison. They showed that the non-linear k ε A2 model calculated the streamwise

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velocity and velocity fluctuation profiles considerably better than the linear k ε model in

comparison to Chang’s et al (1983a) measurements. The “hole” in the velocity profile was

predicted by the non-linear k ε A2 model but not the linear k ε model and Suga et al

concluded that the quality of results from the non-linear model are as good as can be achieved

with a DSM (although no evidence was given for this).Choi et al (1990) have repeated the experiment of Chang et al (1983a) to confirm the

velocity profiles and record the Reynolds stresses in greater detail ( Rc

D 3 357; Re

56 700). In Choi’s et al experiment the working fluid was air and measurements taken by

hot wire anemometry, as opposed to water and LDA in Chang’s et al original experiment.

Some differences in the results are apparent. The results from these two experiments are

compared in Figures 1.5-1.6 for the only two stations around the bend where the authors’ re-

sults coincide (45o and 90o) at approximately three-quarters of the distance around the bend

(Chang: 130o; Choi: 135o - W -velocity only) and at the nearest measurement station to the

downstream exit from the bend (Chang: 177o; Choi: 180o - uu and ww stresses only). The

mean streamwise velocity (W

W b), Figure 1.5, shows the same principal features for both

cases; notably the “hole” in the streamwise velocity at α 90o

However, the gradients of

the streamwise velocity across the duct appear lower at this position in Choi’s results and

there is some discrepancy in the depth of the velocity “hole”, particularly on the centreline at

α 135o.

Secondary motions in the bend are principally driven by the strong curvature induced

pressure gradient. The imbalance between the cross-stream normal stresses (uu vv) also

contributes to the secondary motions, producing features such as the velocity “hole”. Figure1.6 shows a markedly different profile for uu (cross-stream in plane of U-bend) between

the two sets of measurements at α 90o. Although the vv (cross-stream, normal to plane

of U-bend) normal stress is not available in Chang et al, the ww streamwise normal stress

(Figure 1.7) shows a similar lack of consistency between the two sets of measurements. It

is easy to understand from this that the secondary motions induced by the normal stresses

will be different in the two cases. Figure 1.8 shows comparisons of the measured shear stress

(uw). Chang et al ’s results show a greatly increased generation of turbulence at the outside

(concave) wall at α 45o; on the centreline at α

90o Chang et al and Choi et al show shear

stresses of opposite sign over half the duct width.

There is clearly some discrepancy between the two sets of published measured results.

This may be due in part to differences in the measurement techniques, or perhaps some tran-

sient “sloshing” in the U-bend, or minor differences in the geometries and inlet conditions.

Whatever the cause of the discrepancy, it is an indication of the sensitivity of this test case

and an issue which must be borne in mind when analysing computed results.

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1.3.3 Plane Diffuser

The plane diffuser has become a popular test case for turbulence models in recent years.

Much of this interest stems from the work of Obi et al (1993) in which measurements were

made of a turbulent flow in a plane diffuser which has one plane wall, one inclined wall and

a diffuser angle of 10o. It is an interesting case because the flow separates from the inclined

wall between one-third and half-way along the diffuser due to the adverse pressure gradient.

It then reattaches some distance downstream of the diffuser. The plane diffuser is a relevant

test case for road vehicles as it has some similarities with the flow over the rear slant of a

vehicle. As in a diffuser flow, there is curvature of the streamlines and separation due to the

adverse pressure gradient may occur, depending on the angle of the slant (Ahmed et al, 1984).

Buice & Eaton (1997) noted that most turbulence models have difficulty predicting the

separation point and fail to predict accurately the reattachment point. Most experiments

address either separation or attachment, but not both. For example, in the classic backward-facing step experiment, separation is well defined at the step corner and reattachment can be

studied without having to measure or calculate the separation point. Obi et al’s case has two

other important features. Firstly, it has a well defined inlet condition - fully developed channel

flow. This allows the computation to be made with the certainty that the inlet conditions are

accurate. Also, it provides sufficient detail for inlet conditions to be specified for higher-order

computations such as LES. Secondly, the flow in the plane diffuser is two-dimensional, which

reduces the computational expense required for testing turbulence models.

Obi et al (1993) took LDV measurements in a diffuser with a length of 21 H and height

at the diffuser exit of 4 7 H where H was the height of the inlet. The upstream section was

a plane channel which was long enough to provide fully developed channel flow at the inlet

of the diffuser. The channel downstream of the diffuser extended to 40 H to allow sufficient

distance for pressure recovery. The Reynolds number of the flow was Re 20 000 based

on the inlet height ( H and the centreline velocity. The two-dimensionality of the flow was

examined and the variation in mean velocity profile across the span of the channel was found

to be less than 15% over 90% of the inlet span and less than 5% over 60% of the outlet

span. Separation was found to occur part-way along the length of the diffuser, at 11 H from

the diffuser inlet and only on the inclined surface. Reattachment occurred in the uniformarea section downstream of the diffuser at 26 H from the diffuser inlet (5 H from the end of

the diffuser). Obi et al also calculated the flow using a linear k ε model of Launder &

Sharma (1974) and a basic DSM of Gibson & Launder (1978). These models were used

with a simple log-law wall function (Launder & Spalding, 1974). The linear k ε model was

found to perform poorly in terms of the coefficient of pressure at the walls and the velocity

and stress profiles which were calculated. No separation was calculated with this model. The

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DSM performed better: a small amount of separation was calculated with this model and the

coefficient of pressure, and velocity and stress profiles were calculated more accurately.

Buice & Eaton (1997) reviewed Obi’s et al work and found some deficiencies in the

experimental method. Significantly, they noted that there was an increase in mass flow of 15%

between the inlet and outlet of Obi et al’s diffuser. Buice & Eaton concluded that this was dueto large wall-end boundary layers forming in the channel and three-dimensional effects in the

diffuser. They repeated the 10o plane diffuser experiment paying particular attention to the

two-dimensionality of the flow. Using splitter plates in the channel upstream of the diffuser

they removed the end-wall boundary layers. Consequently, the pressure gradient immediately

upstream of the diffuser was 40% less than expected. Following Simpson (1996), Buice &

Eaton defined the location of the separation point in two ways. “Transitory detachment”

was defined as the point at which instantaneous black-flow occurred 50% of the time and

“detachment” was defined as the point at which time averaged wall shear stress τw 0.

They measured a longer separated flow region than Obi et al; transitory detachment and

detachment coincided at 7 H and reattachment occured at 29 H . Buice & Eaton measured the

flow using hot-wire techniques which allowed them to measure the flow close to to wall. In

the recovering channel flow downstream of reattachment they found that the velocity profiles

in the near-wall flow fell well below the standard log-law of the wall.

In conjunction with Buice & Eaton’s measurements, Durbin (1995) calculated the flow us-

ing his k ε v2 model (Durbin, 1991) and Kaltenbach et al (1999) calculated the flow using

LES and a dynamic sub-grid scale model. Durbin calculated the coefficient of pressure well,

but the calculated separation region was somewhat longer than measured, extending from4 H to 35 H . Kaltenbach et al could not improve Durbin’s coefficient of pressure calculations

but their calculation of the coefficent of friction was significantly better. Also, Kaltenbach

et al calculated separation and reattachment at the positions measured in Buice & Eaton’s

experiment (7 H and 29 H respectively). Velocity and stress profiles calculated by Kaltenbach

et al were in general better than those calculated by Durbin. However, the velocity peak in

the separated flow region was slightly underpredicted as were the turbulence intensities after

reattachment.

The 10o plane diffuser was used as one of the test cases for the 8th ERCOFTAC Workshop

on Refined Turbulence Modelling (Hellsten & Rautaheimo, 1999). Among the contributions

were two-equation linear EVMs, two-equation NLEVMs, ASM, EARSMs, DSMs and LES.

The NLEVMs used were the Craft et al (1996b) non-linear k ε model and the Apsley &

Leschziner (1998) model, both applied as low Reynolds number models. Overall, the two

NLEVMs performed equally well. The peaks in the stress profiles calculated by both models

tended to be too high in the diffuser section and too low in the recovering flow section.

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The high pressure gradient just inside the diffuser was calculated accurately by both models

although the maximum coefficient of pressure calculated was about 10% too high in both

cases. Differences between the NLEVMs were more apparent in the calculated locations

of the separation (detachment) and reattachment (τw 0). The Craft et al (1996b) model

performed better in this respect as it calculated separation and reattachment at 8 H and 25 H compared to 12 H and 26 H calculated by the Apsley & Leschziner (1998) model.

1.3.4 Road-Vehicle Aerodynamics

Hucho & Sovran (1993) conducted a review in which they discussed road-vehicle aerody-

namic design, vehicle attributes affected by aerodynamics, typical aerodynamic characteris-

tics and methods of calculating road-vehicle flows. They described a road vehicle as being

essentially a bluff body in very close ground proximity. The geometry of the vehicle is

complex, the flow around it is fully three-dimensional, the boundary layers are turbulent,

flow separation is common and there are large turbulent wakes in which longitudinal trailing

vortices are common. As is typical for bluff bodies, the principal contribution to drag experi-

enced by a road vehicle is pressure drag and a major objective of vehicle aerodynamic design

is the avoidance, reduction or control of flow separation. However, Hucho & Sovran pointed

out that whereas the design of an aircraft wing and fuselage or a turbine blade is driven by

a required aerodynamic performance, the prime considerations in road-vehicle design are:

function, economics and aesthetics. Hence, the characteristics of a particular road vehicle

are often not intentional but a consequence of the vehicle’s shape. Vehicle attributes which

are affected by the aerodynamic characteristics include: performance and fuel economy, han-

dling, crosswind sensitivity and “functionals”. In 1993, a typical mid-sized US car used 18%

of its required tractive energy to overcome drag in an urban cycle and upto 51% in a highway

cycle (ie at higher speeds). Reduced drag reduces fuel consumption, allows for increased

acceleration and increased top speed. In most passenger cars, lift tends to be positive and the

coefficient of lift C L

0 3. This results in a reduction in weight of about 3% at 60 mph and

10% at 120mph, which is not particularly significant. Of greater significance are the pitching

moments induced by the lift, as these modify the front-to-rear weight distribution: increased

weight over the front axle will promote oversteering. Crosswinds cause the flow around aroad vehicle to become asymmetric and create side forces, yawing and rolling moments on

the vehicle. “Functionals” affected by aerodynamic attributes include: body panel flutter,

wind noise generated from aerials and wing mirrors, body-surface water flow and soiling and

interior flow systems.

Bearman (1980) conducted a review of bluff body flows applicable to vehicle aerody-

namics in which he reviewed the then current knowledge of three-dimensional flows and

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discussed problems encountered when the body is brought close to the ground. In order to

reduce the drag behind a bluff body Bearman identified two methods. Firstly, “boat-tailing”

or a tapering of the rear end of the body will reduce the drag . It is not necessary to continue

the taper to a point but it can be truncated without significant detrimental effect. (Hucho &

Sovran, 1993, noted that long, aerodynamically-optimised road vehicles still have a flow sep-aration at the rear end. The vehicle’s body can be truncated upstream of the separation point

without incurring a drag penalty.) Secondly, air can be bled into the bluff body’s wake to re-

duce drag. However, any benefit resulting from reduced drag must be offset against the power

requirements of the bleed-air system. Viscous-inviscid flow interaction can be difficult to pre-

dict for any bluff body separated flow and this is particularly so when there are ground effects

present. Bearman used a cube near the ground as an example. As the cube is brought closer

to the ground, flow around the cube creates a downforce (negative lift) which is controlled by

the underbody flow. In contrast, the drag is almost independent of the ground-distance as it

is controlled by the outer flow.

One of the most significant advances in road vehicle aerodynamics research was the iden-

tification of streamwise vortices which are often generated and their effect on drag. The front

section of a road vehicle typically makes only a small contribution to drag and the flow is

easy to control. In contrast, the rear of the vehicle can make a large contribution to drag and

the slant angle at the rear of the vehicle is critical in determining the mode of the wake flow

and the drag experienced by the vehicle. The critical influence of the rear slant angle was

first identified by Janssen & Hucho (1975) who found that a maximum drag was obtained

for a vehicle with a rear slant when the rear slant angle β 30

o

. Under this condition, theflow over the top of the vehicle remained partially attached as it passed over the rear slant

and longitudinal trailing vortices were formed at the edges of the slant. For β 30o the flow

became fully separated over the rear slant and there was a drop in drag. Morel (1978) saw

that there was a need to investigate this effect further, using a simplified geometry. He used

a slender, axi-symmetric cylinder with its principal axis aligned with the flow direction, a

well-rounded nose and various base (rear-end) slants with angles between 20 o

β 90o.

The drag versus rear-slant angle which he measured is shown in Figure 1.9. Morel defined

two flow regimes separated by a critical rear slant angle, βc 42o. At rear slant angles below

βc (which Morel called Regime II) edge vortices form and roll-up over the slant surface. Low

pressure at the vortex centres caused high drag; as the vortices increased in strength with

increasing slant angle, the drag increased. Air was supplied by the edge vortices to the rear

slant, relieving the downstream pressure rise and causing the flow over the slant to remain

attached up to the relatively large critical angle, βc 42o. Eventually, the supply of air from

the sides of the body was insufficient to maintain the attached flow over the slant, and for

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β βc, the flow over the slant became fully separated and with only very weak edge vortices

being formed (Regime I). In further tests, Morel studied the influence of free-stream turbu-

lence, Reynolds number dependence and ground proximity. By increasing the free-stream

turbulence from an initial value of 0.1% to 6% he found that there was a 25% increase in

drag through the range but no change in the critical slant angle. For Reynolds numbers inthe range 22

000

Re 112

000 he found no change in the drag behaviour or βc

To study

the effect of ground proximity, Morel used a simplified vehicle-like model, Figure 1.10. As

with Bearman’s example of a cube, Morel found that the drag experienced by the vehicle-like

model was only affected a small amount by variations in the distance to the wall but the lift

was affected much more, becoming strongly negative near the wall. When the distance from

the wall was h

Deq 0

12 (h is the distance to the wall, Deq

2x

Area) which Morel stated

was a typical value for a road vehicle, the critical angle was βc 30o.

Ahmed (1981) followed the work of Janssen & Hucho (1975) and Morel (1978) in a

study of the time-averaged wake of three vehicle models: estate, fast-back and notch-back.

He found longitudinal trailing vortices in the wakes of all three vehicle types which he con-

sidered to be instrumental in drag formation. To study the influence of the base slant on wake

structure and drag, Ahmed (1983) used a single vehicle model with nine interchangeable rear

sections. The rear sections had slant angles varying between 0o β 40o. A variety of

visualisation techniques were used to help understand the flow around the vehicle model and

velocity measurements were made along the centreline and in transverse planes in the wake.

The overall variation of drag with slant angle was similar to Morel’s measurement for an

axi-symmetric cylinder (Figure 1.9). However, Ahmed (1983) measured a decrease in dragin the range β

0o to 15o which was attributed to the rear-slant edge vortices providing air

to the slant surface and aiding the pressure recovery. The minimum and maximum (critical)

drag occurred at β 15o and 30o respectively.

To continue this work, Ahmed et al (1984) defined a simplified vehicle-like body, Fig-

ure 1.11, similar to that used by Morel to conduct a detailed study of the surface pressure

distribution, wake structure and how the wake structure is modified by varying rear-slant an-

gle. As Morel had found with his simplified vehicle-like model, Ahmed et al measured the

critical rear-slant angle βc 30o. Ahmed et al measured the contributions to drag due to the

nose, rear slant, base and friction for rear-slant angles of 5 o

12 5o

30o (high drag) and 30o

(low drag) using a vertical splitter plate in the wake to encourage the low drag (separated

flow) condition at 30o to form. Ahmed’s et al (1984) measurement of this drag breakdown is

shown in Figure 1.12. Here, C S is the drag coefficient due to the rear slant, C

B is due to the

base5, C K is due to the nose cone, C

R is the friction drag and C W is the total drag coefficient.

5The “base” is the rear-surface of the body

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Throughout the range of rear slant angles studied, C R and C

K remained approximately con-

stant. This was because the body had a sufficiently long mid-section to prevent downstream

effects at the slant from affecting the flow at the nose and the relative insensitivity of the drag

coefficient to changes in the surface shear-stress compared to changes in surface pressure

(Grün, 1996). Up to β

20o

the major contribution to drag was C B due to the low pressure

experienced on the base. Between 20o

β 30o the strength of the vortices generated at

the rear-slant edges increased causing lower pressure over the slant and C S became the major

contributor to C W . Measurements in the wake allowed Ahmed et al to define schematics for

the low and high drag flows for rear slant angles less than βc. Figure 1.13 shows the wake

structure defined by Ahmed et al for the low drag flow with rear-slant angle β

20o. There

are horseshoe vortices ( A & B) behind the base and a relatively weak longitudinal trailing

vortex emanating from the slant edge. The flow remained attached over the slant, promoting

pressure recovery and low drag. Figure 1.14 shows the wake structure defined for βc 30o

under the high drag condition, immediately before the flow totally separates above the rear

slant. The edge vortex was much stronger and the low pressure in the vortex decreases the

pressure above the slant and increases drag. Also the flow was beginning to separate at the

centre of the slant ( E ) which further lowers the pressure over the slant. This contributed to

the rapid rise in drag shown in Figure 1.12 between 25o β 30o. Once the rear-slant angle

increased beyond βc 30o the separation ( E on the rear slant merged with the upper of the

two horseshoe vortices behind the base and large-scale separation occurred. Low drag was

thus re-established which was characterised by the lack of strong longitudinal vortices.

A computational study of the Ahmed body was conducted by Han (1989) using a finite-volume code solving the RANS equations and employing the linear k ε of Launder & Spald-

ing (1974). With rear-slant angles between 0o

β 20o Han was able to calculate the cor-

rect wake features (recirculation regions and longitudinal vortices) described by Ahmed et

al (1984). However, the drag coefficients calculated by Han were approximately 30% too

high throughout this range, which Han attributed to too low base pressure. For β 30o no

separation was calculated and the steep rise in drag between 25o

β 30o shown in Figure

1.12 was not calculated. Wilcox (1993) used Han’s work as a demonstration of the need for

appropriate turbulence models. In a discussion of the relative performance of linear k ε and

k ω models in separated flows, Wilcox noted that a linear k ε model would underpredict

the length of flow separation behind a backward-facing step by 16% whereas a k ω model

would calculate this length to within 3%. Using Han’s (1989) Ahmed body calculations as

a more complex flow example, Wilcox provides the following failure mechanism. As in the

backward-facing step, the linear k ε model used failed to respond in a physically realistic

manner to the adverse pressure gradient. This led to too high skin friction on the base and too

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high vorticity which diffused into the wake flow. The vortices in the separated wake behind

the base were thus too strong, resulting in reduced pressure on the base and attached flow

over the slant. However, Wilcox’s explanation neglects several issues. Ahmed et al (1984)

recorded the free-stream turbulence intensity in their wind tunnel (<0.5%) but no turbulence

length-scale data. Han did not describe the free-stream turbulence values which he used inhis calculation. It is possible that he specified a level of turbulent kinetic energy which was

too high or a turbulence energy dissipation rate which was too low. Moreover, it is known

that a linear eddy viscosity model (irrespective of the choice of second variable ε or ω will

calculate too much production of turbulent kinetic energy (Pk ) on the flow-impingement sur-

face of a bluff body such as the front of the Ahmed body (Kato & Launder, 1993). Either

effect could lead to too much turbulent kinetic energy in the flow over the slant. This would

increase the turbulent viscosity (Equation 1.9) encouraging the flow to remain attached. Fur-

thermore, Han used a basic wall function in his calculation (Launder & Spalding, 1974). This

in itself could be the cause of too high shear stress on the base and following Wilcox’s analyis

too low base-pressure; ie the wall function employed may not be sufficiently accurate. Grün

(1996) demonstrated the importance of correct calculation of pressure in separated flow re-

gions. He showed that the influence of static presssure in separated flow regions is two orders

of magnitude greater than the influence of velocity and shear stress.

Angelis et al (1996) conducted an experimental and numerical study of flow over a two-

dimensional car body. (The experimental model used a cross-section profile from a typical

passenger car. The width of the model was approximately seven times its length and the cross-

sectional profile was kept constant over the width.) Pressure measurements were recorded onthe car’s upper and lower surface and LDA measurements of two velocity components around

the car were made to validate the calculated flow. Calculations were made with a RANS code

using a linear k ε model and wall function (Launder & Spalding, 1972). Angelis et al stated

that in the numerical methods used by most CFD codes, the computational effort required is

proportional to N 2 where N is the number of grid nodes. However, in the multi-grid method

which they used to solve the discretized RANS equations the computational effort is reduced

to approximately N . The main theme of the work is the application of the multi-grid method.

Angelis et al recognised that errors which were observed between the measured and computed

results are likely to be due to the linear k ε model used. Their calculated values of velocity

were reasonable over the front section of the car but only in qualitative agreement with the

measured values over the rear of the car where flow separation occurs. Angelis et al’s work is

unusual in road vehicle aerodynamic research as it ignores the inherent three-dimensionality

of the vehicle’s wake. However, it is an interesting application of CFD to a complex bluff

body close to a wall.

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Flow-field calculations of three simplified vehicle shapes were carried out by Han et al

(1996). The calculations were aimed at assessing the General Motors Research CFD code,

GMTEC and the calculated results were compared to detailed wind-tunnel measurements.

GMTEC solved the RANS equations which were calculated with either a linear k ε model

(Launder & Spalding, 1972) or an RNG k

ε model (Yakhot & Orszag, 1986). The con-vection scheme employed was a blend of central and upwind differencing. The three vehicle

shapes which were studied varied only in their rear-end detail and approximated a square-

back (estate), fast-back and optimised fast-back car. (The optimised fast-back used “boat-

tailing” or “planform taper” to achieve the lowest possible drag coefficient.) Most of the

calculations were made on a computational grids with approximately 150,000 nodes. Addi-

tional calculations were carried out with grids having 75,000 and 300,000 nodes. Han et al

showed that the drag coefficient decreased with increasing grid refinement and consequently

their results were not grid independent. Calculations of the flow around the optimised fast-

back model with the 150,000 node grid and standard k ε and RNG k ε models gave drag

coefficients which were 18% and 8% too high respectively. Repeating these calculations with

the refined 300,000 node grid reduced the inaccuracies to give drag coefficients 14% and 3%

too high. The calculation which used the RNG k ε model benefitted the most from the grid

refinement. The 3% error in drag coefficient represents an inaccuracy in drag coefficient of

∆C D 0 005 which would be acceptable to most vehicle manufacturers (Grün, 1996). Han et

al (1996) did not include any detail of the wake structures or estimation of the critical angle of

the rear slant. The measured drag coefficients were 0.30, 0.25 and 0.15 for the square-back,

fast-back and optimised fast-back models respectively.Kobayashi & Kitoh (1992) conducted a review of CFD methods and their application

to road-vehicle aerodynamics and Hashiguchi (1996) conducted a review of methods used

for turbulence simulation in the Japanese automotive industry. Kobayashi & Kitoh cate-

gorised four main areas of interest: panel methods, RANS calculations, LES and “quasi-

DNS”. Panel methods solve the Laplace equation for velocity potential and are only suitable

for inviscid, attached flow and hence do not have wide applicability in the automotive in-

dustry. Kobayashi & Kitoh considered RANS codes were the most promising of the four

categories for industrial flows and discussed RANS applications with linear k ε models in

some depth. They did however, note the deficiencies of this method for calculating separated,

reattaching and swirling flows. The “quasi-DNS” work which Kobayashi & Kitoh reviewed

solves the Navier-Stokes equations by direct simulation using third-order upwind schemes,

but without resolving the grid sufficiently to calculate the smallest turbulent eddies. As no

academic test cases had been carried out on simple flows using this method, Kobayashi &

Kitoh questioned its validity. In reality, “quasi-DNS” using a coarse grid is LES, ignoring

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the sub-grid scale motions and their effect on the resolved flow. In his review, Hashiguchi

concentrated on “quasi-DNS” with third-order upwinding as he considered this superior to

RANS calculations (there being no uncertainty due to the turbulence model). However, the

RANS caculations which he reviewed used linear k ε models, which are not generally con-

sidered to be appropriate for the three-dimensional, impinging, separating, reattaching andswirling flow typical of road-vehicle aerodynamics. Moreover, the “quasi-DNS” calculations

which he reviewed typically used grids with 106 nodes and a smallest cell size of the order of

10mm. These flows were calculated at Reynolds numbers of the order Re

106, in which the

smallest eddies would have been at the Kolmogorov lengthscale, approximately η

10 µm(6).

It is clear that the grids used in the calculations reviewed by Hashiguchi are too coarse for the

computational method used.

In spite of the apparent deficiencies, “quasi-DNS” appears to be a popular technique in

Japanese industry and some good results have been obtained. Kataota et al (1991) calculated

the drag of a sports car to within 5%. Ono et al (1992) calculated the external and internal flow

through a simplified vehicle model with internal compartments and calculated the internal

flow-rate to within 3% and drag to within 7% of the measured values. (They were not able to

repeat these levels of accuracy for a realistic car shape.) Horinouchi et al (1995) investigated

several drag reduction techniques on a realistic car shape using “quasi-DNS” and an over-laid

grid system in which the whole computational domain was covered with multiple grid blocks

which over-lapped one another. Each grid block was generated around the local boundary

shape to satisfy the best possible gridding practice.

Two novel approaches for calculating road-vehicle external aerodynamics are providedby Grün (1996) and Anagnost et al (1997). Grün used a zonal model in which the inviscid

part of the flow was calculated using a first-order panel method and the viscous part used an

integral boundary-layer code. This approach aimed to calculate the influence of separation

on the surface pressure distribution rather than calculate the wake structure in detail. Good

results were obtained for the pressure distribution in the attached flow region over a realistic

car geometry but the pressure distribution in the separated flow region was not so good. The

main advantage of this method over a RANS-type calculation was its speed of operation: it

was possible to achieve a turn around time of two to three days for model generation and

analysis of five to ten variants. Anagnost et al (1997) used a discrete particle method (which

was based on an extension to lattice gas theory) to calculate the flow over the Morel (1978)

body. The discrete particle method can be considered as a discrete version of the kinetic

theory of a dense gas in which the mean behaviour of the microscopic model of the flow can

6From Pope (2000): η lo

Re

3 4 where lo is the largest length-scale in the flow which is of the order

lo

1m for a road vehicle.

1.3. Test Cases

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be shown to agree with the macroscopic governing equations (the Navier-Stokes equations).

Anagnost et al calculated the flow over the body for several rear-slant angles. The calculated

drag coefficient was good but always slightly high (10% too high at β 5o and 5% too high at

β 10o). The shape of the drag versus rear-slant angle plot was reproduced well, showing the

drag minimum at β

12

5o

, steep rise near β

25o

and critical rear-slant angle of βc

30o

.The bi-stable wake behaviour at βc

30o which Ahmed et al (1984) discovered with the flow

tending to either high or low drag was reproduced by Anagnost et al. The wake would form

in one or other state depending on the initial flow conditions.

1.4 Study Objectives

The aim of the work presented in this thesis is to apply the two-equation cubic NLEVM

of Craft et al (1996b) to flows pertinent to road-vehicle aerodynamics. The NLEVM has notpreviously been tested for this class of flows. In developing the model, Craft et al (1996b) and

Suga (1995) originally tested the model in homogeneous flows (homogeneous shear, plane

strain, axisymmetric contraction and expansion), fully developed pipe and channel flows,

fully developed rotating pipe flow, fully developed curved channel flows, by-pass transitional

flows and impinging jets. The model has not been tested for separated and reattaching flow,

flow with periodic vortex shedding or flow with strong streamwise curvature. The perfor-

mance of the NLEVM is compared to a “standard” linear k ε model (Launder & Spalding,

1974) and a cubic DSM (Craft et al, 1996a).

This work is part of a larger study sponsored by the European Union7 in which it is aimed

to improve turbulence models used by the automobile industry. As the NLEVM is principally

tested to assess its suitablility for industrial flows, it has been used in conjunction with wall

functions (Launder & Spalding, 1972, Chieng & Launder, 1980). Wall functions are in gen-

eral necessary for industrial-scale calculations as low-Reynolds-number techniques are com-

putationally too expensive. In addition to the existing log-law wall functions, the “analytical

wall function” which is being developed at UMIST (Gerasimov, 1999) has been included in

a three-dimensional, non-orthogonal code. Previously, this analytical wall function has been

implemented in a two-dimensional orthogonal code and tested for flow with mixed convec-

tion in vertical pipes. In this work the analytical wall function is used to calculate separating

and reattaching flow for the first time.

The cumulation of the work is the calculation of flow around the Ahmed body using the

NLEVM and analytical wall function. The flow is calculated for the Ahmed body with two

rear-slant angles: 25o and 35o These have been chosen as they lie either side of the critical

7EU-BRITE-EURAM III, Project No. BE97-4043, Contract No. BRPR-CT98-0624

1.4. Study Objectives

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angle (βc 30o) at which the flow switches from the principally attached (high-drag) mode,

to the fully separated (low-drag) mode. This flow is also calculated using a linear k ε

model (Launder & Spalding 1974) and log-law wall function (Chieng & Launder, 1980) and

compared to recent measurements taken by LSTM, Erlangen.

1.5 Outline of Thesis

In Chapter 2 the mathematical models which have been used are described. These include the

various turbulence models and near-wall modelling techniques. The numerical implementa-

tion of these models is described in Chapter 3. Results from three test cases are presented in

Chapters 4, 5 and 6. Respectively, the test cases are:

Two-dimensional, time-dependent flow around a square cross-section cylinder placed

near a wall. Three cases with different distances to the wall are considered which

include steady and periodic vortex-shedding wake behaviour.

Three-dimensional, steady-state flow in a square cross-section U-bend with strong

streamwise curvature. The turbulence models’ ability to calculate accurately the sec-

ondary motion induced in the U-bend is assessed.

Two-dimensional, steady-state flow in a plane diffuser with diffuser angle 10 o. This

case is used to test the analytical wall function in a non-orthogonal code and its ability

to calculate flow with separation and reattachment.

In Chapter 7, the results of the calculation of flow around the Ahmed body are presented and

the models’ abilities to calculate the different flow modes are discussed. Finally, in Chapter

8 the main findings of the thesis are summarized.

1.5. Outline of Thesis

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Chapter 2

Mathematical Models

2.1 Navier-Stokes Equations

The instantaneous Navier-Stokes equations are derived from the principles of conservation of

mass and momentum. They can be written for an incompressible, isothermal flow as follows:

∂ U i

∂ xi 0 (2.1)

∂ U i

∂t

U j∂ U i

∂ x j

1

ρ

∂ P

∂ xi

ν ∂2 U i

∂ x j∂ x j

(2.2)

where U i is the instantaneous velocity component, P is the instantaneous pressure, ρ is the

density of the fluid and ν its kinematic viscosity. It is possible to solve these equations

analytically for simple cases such as flow in a pipe or between planes. For more complex

flow, the equations can be solved directly (DNS) but this has limitations for turbulent flow

as discussed in Chapter 1. In industry, the approach generally adopted to calculate turbulent

flows is to use time-averaged versions of Equations 2.1 and 2.2. The dependent variables U i

and P can be considered as consisting of a time mean (U i

P and a fluctuating component

(ui

p): U i

U i

ui and P

P

p. Decomposing the equations in this way results in the

Reynolds-Averaged Navier Stokes (RANS) equations:

∂U i

∂ xi 0 (2.3)

∂U i

∂t U j

∂U i

∂ x j

1

ρ

∂P

∂ xi

∂

∂ x j ν

∂U i

∂ x j

∂

∂ x juiu j

F i (2.4)

37

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CHAPTER 2. Mathematical Models 38

where F i is an external force applied to the fluid. Information is lost in the averaging process

and due to the appearance of the Reynolds (or turbulent) stresses, uiu j, Equations 2.3 and 2.4

no longer form a closed set. A turbulence model is used to provide values of the Reynolds

stresses.

2.2 Two-Equation Turbulence Models

2.2.1 Linear k ε Model

High-Reynolds-Number Form A simple way of approximating the Reynolds stresses

is the Eddy Viscosity Model (EVM), first proposed by Boussinesq (1877). The Reynolds

stresses are not actually stresses but are so-called as they act in the same manner as viscous

stresses (τi j ν∂U i

∂ x j). By analogy to the viscous stresses, Boussinesq’s EVM defines the

Reynolds stresses (stated in Equation 1.2 and repeated here for clarity):

uiu j

2

3δi jk

νt

∂U i

∂ x j

∂U j

∂ xi

(2.5)

where δi j is Kronecker’s delta, k is the turbulent kinetic energy of the fluid and νt is the

turbulent kinematic viscosity. Unlike the kinematic viscosity, νt is a property of the fluid

motion rather than an intrinsic property of the fluid itself and must be calculated from known

variables. Equation 2.5 implies that the relationship between the Reynolds stresses and strain

rate is linear. The linear k ε model of Launder and Spalding (1974) defines the turbulent

kinematic viscosity for high-Reynolds-number flows as (Equation 1.9 repeated for clarity):

νt

c µk 2

ε (2.6)

where c µ is a dimensionless constant which must be specified. For local equilibrium it can

be defined as

uv

k

2

and its value has been measured in simple shear: c µ

0 09

The

model is completed by solving transport equations for the turbulent kinetic energy, k and the

turbulent kinetic energy dissipation rate, ε. The transport equation for the turbulent kinetic

energy is derived from the Navier-Stokes equations (Equation 2.1 & 2.2). An exact transport

equation for turbulence energy dissipation rate can also be derived (Davidov, 1961) but it is

more usual to use a transport equation for energy dissipation rate defined by analogy to the

turbulent kinetic energy equation.

2.2. Two-Equation Turbulence Models

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k - equation:

The turbulent kinetic energy equation may be written:

dk

dt

U j

dk

dx j

d k

Pk ε (2.7)

In high Reynolds number flows, the turbulent diffusion is modelled by analogy to the molec-

ular diffusion (Prandtl, 1945) and the pressure diffusion is assumed to be negligible (Kol-

mogorov, 1942), giving a modelled diffusion term:

d k

∂

∂ x j

νt

σk

∂k

∂ x j

(2.8)

where σk is an empirical constant. The production term is represented exactly as:

Pk

uiu j∂U

i∂ x j

(2.9)

and, by application of the Boussinesq assumption (Equation 2.5) is modelled as:

Pk νt

∂U i

∂ x j

∂U j

∂ xi

∂U i

∂ x j(2.10)

The rate of dissipation of turbulent kinetic energy by viscous action is defined as:

ε

ν∂u j

∂ xi

∂u j

∂ xi(2.11)

In the k ε model this value is provided by the transport equation for ε.

ε - equation:

The corresponding equation for dissipation of turbulent kinetic energy is:

d ε

dt

U jd ε

dx j

d ε H ε (2.12)

Hanjalic & Launder (1972) proposed that the modelled terms for the ε-equation for high

Reynolds numbers should take the form:

d ε

∂

∂ x j

νt

σε

∂ε

∂ x j

(2.13)

H ε cε1

ε

k Pk

cε2ε2

k (2.14)


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c µ cε1 cε2 σk σε

0.09 1.44 1.92 1.0 1.3

Table 2.1: Empirical constants used in the high Reynolds number k ε model

Modelling for the diffusion term, d ε is analogous to the diffusion term in the k -equation.

The H ε term contains the combined effects of turbulent production and destruction of ε. The

empirical constants used in the model are shown in Table 2.1.

When the Reynolds number of the flow is sufficiently high (ie of the order Re

105)

the turbulent viscosity is several orders of magnitude greater than the molecular viscosity.

No dependence on the effect of molecular viscosity is included in the high-Reynolds-number

version of the k ε model. However, near to a wall boundary the turbulent viscosity is re-

duced and molecular viscosity has a significant effect. This causes steep gradients in velocity

and turbulence profiles. A wall function is used to bridge this near-wall region and provide

average values of velocity and turbulence to the calculation. Wall functions are discussed in

Section 2.4.

Low-Reynolds-Number Form The low-Reynolds-number form of the Launder & Sharma

(1974) k ε model includes dependence on the molecular viscosity. The model equations are

integrated right up to the wall boundary without resorting to a wall function to provide values

for the viscous-affected region. This usually results in fine computational grids being used

near the wall to resolve accurately the steep gradients found in this region. The equations fork and ε are similar to Equations 2.7 and 2.12:

dk

dt U j

dk

dx j

∂

∂ x j

ν

νt

σk

∂k

∂ x j

d k

Pk

ε (2.15)

The turbulence energy dissipation rate, ε is non-zero at the wall. In order that a simple

boundary condition may be set, a new turbulence energy dissipation rate, ε is defined. This

may be considered as the homogeneous part of the turbulence energy dissipation rate and is

defined as:

ε ε 2 ν

∂k 1 2

∂ x j

2

(2.16)


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The transport equation for ε in the low-Reynolds-number form of the model is then:

d ε

dt

U jd ε

dx j

∂

∂ x j

ν

νt

σε

∂ε

∂ x j

d ε

cε1 f 1ε

k Pk

cε2 f 2ε2

k

Pε3

2 ννt

∂2U i

∂ x j∂ xk

H ε

(2.17)

Turbulent viscosity is zero at the wall and its value must be damped in the model to ensure

that it does not rise too quickly in the near-wall region:

νt c µ f µ

k 2

ε (2.18)

The damping functions f µ

f 1 and f 2 in Equations 2.17 and 2.18 were expressed by Launder

& Sharma (1974) as functions of the turbulent Reynolds number, ˜ Rt :

˜ Rt

k 2

νε (2.19)

f µ exp

3 4

1

˜ Rt

50

2

(2.20)

f 1 1

0 (2.21)

f 2 1

0 0

3exp

˜ R2t

(2.22)

2.2.2 A General Non-linear Eddy-Viscosity Model

Equation 2.5 shows the assumed linear relationship for the anisotropy tensor, ai j

uiu j

k

2

3δi j

which provides a method for calculating the Reynolds stresses in the linear k ε model. A

more general expression for the anisotropy is:

ai j

Ai j

S i j Ωi j

(2.23)

where Ai j is a second order tensor expressing the relationship between the strain and vorticity

tensors which are themselves defined by:

S i j

∂U i

∂ x j

∂U j

∂ xi; Ωi j

∂U i

∂ x j

∂U j

∂ xi(2.24)

Pope (1975) expressed the most general possible form of Equation 2.23 which includes


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up to quintic products of the strain and vorticity tensors and can be written:

ai j

10

∑n

1

G

n T

n

i j (2.25)

T

1

i j

S i j

T

2

i j

S ik Ωk j Ωik S k j

T

3

i j

S ik S k j

1

3S lk S kl δi j

T

4

i j Ωik Ωk j

1

3Ωlk Ωkl δi j

T

5

i j Ωil S lmS m j S il S lmΩm j

T

6

i j Ωil ΩlmS m j S ilΩlmΩm j

2

3S lmΩmnΩnlδi j

T

7

i j Ωik S kl ΩlmΩm j Ωik Ωkl S lmΩm j

T

8

i j S ik Ωkl S lmS m j

S ik S kl Ωlm S m j

T 9

i j Ωik ΩklS lm S m j S ik S kl ΩlmΩm j 2

3S kl S lm ΩmnΩnk δi j

T

10

i j Ωik S kl S lmΩmnΩn j Ωik Ωkl S lmS mnΩn j

(2.26)

Setting the coefficients G

1

k

εc µ, G

n

0 for n 1, returns the linear stress-strain re-

lationship used in the Launder-Sharma k ε model (Equation 2.5). If the coefficients are

defined for G

n

1

a non-linear turbulence model is produced. Pope (2000) states that for

every algebraic stress model (and consequently, every differential stress model 1) there is a

corresponding non-linear eddy-viscosity model. The coefficients G

n can be derived from

the “parent” DSM and the cubic non-linear models of Apsley & Leschziner (1998) and Wallin

& Johansson (2000) use this approach. Taulbee (1992) and Gatski & Speziale (1993) have

proposed quintic models in which all ten G

n coefficients are non-zero.

2.2.3 Cubic Non-Linear k ε Model

High-Reynolds-Number Form The cubic non-linear expression for anisotropy developed

by Craft et al (1996b) is:

ai j

uiu j

k

2

3δi j (2.27)

The terms with coefficients c1

c2

c3

c4 and c5 correspond to the T

3

i j , T

2

i j , T

4

i j , T

5

i j and

T

6

i j terms in Equation 2.26 respectively. The c6 and c7 terms are both developed from the

T

1

i j term. Craft et al defined the coefficients for Equation ?? by “tuning” the model to a

number of reference flows:

1See Section 2.3 for a discussion of DSMs


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c1 c2 c3 c4 c5 c6 c7

0.1 0.1 0.26 -10c2 µ 0 5c2 µ 5c2 µ

Table 2.2: Constants used in the non-linear EVM

Homogeneous shear flow: As ∂U

∂ y is the only velocity gradient the only non-zero com-

ponents of ai j are a11 a22

a33 and a12. This gives the condition that c5 c6

c7 0 and

c2 and

c1

c3 were optimized with the values 0.1 and 0.26 respectively by comparison to

experimental and DNS data.

Fully developed swirling shear flow: This is a fully developed pipe flow in which the

pipe rotates about its own axis. The U V and W velocities are defined in the axial ( x), radial

(r ) and circumferential (ψ ) directions. Although experiments show a non-linear increase of

circumferential mean velocity with radius (Cheah et al, 1993), any linear EVM (where vw

νt S 23 and S 23

∂W

∂r

W

r ) gives a linear variation of circumferential (swirl) velocity

with radius. No quadratic term appears in the nonlinear expression of vw, and consequently

a quadratic non-linear model will not improve the calculation of swirl velocity.

Flow with streamline curvature: In a fully developed curved channel flow, the curvature

leads to increased mixing near the concave wall and reduced mixing at the convex wall (Ellis

& Joubert, 1974). This leads to an asymmetric velocity profile across the channel. The

normal stresses do not affect the mean streamwise velocity or turbulence profiles but the

shear stress (uv) does. Furthermore, the expression for uv from Equation ?? contains linear

and cubic correlations of S i j

and Ωi j

, but no quadratic terms. As with the fully developed

swirling shear flow, this demonstrates that the non-linear model must be extended to at least

cubic level to improve the calculation of flow with streamline curvature.

The coefficients which Craft et al defined from the reference cases are shown in Table

2.2. The c5 coefficient is set to zero as no simple test case could be found to distinguish its

effect. (It acts in a similar manner to the c4 term in swirling flow and the c7 term in curved

channel flow.)

Craft et al also defined a modelled form of c µ, based on the strain and vorticity invariants:

c µ

0 3

1 0

35

max

S Ω

1 5

1

exp

0 36exp 0 75 max

S Ω

(2.28)

where the dimensionless strain and vorticity invariants are defined:

S

k

ε

S i jS i j

2; Ω

k

ε

Ωi jΩi j

2(2.29)


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The maximum of S and Ω is used to increase the model’s sensitivity to streamline curvature.

Near a convex wall the vorticity invariant is larger than the strain invariant and near a concave

wall it is smaller. EVMs tend to overpredict the level of turbulence near a convex wall and

by adopting the larger invariant, greater damping of turbulence will be achieved.

Low-Reynolds-Number Form Further adjustments to the model are required if it is to be

applied to near-wall flows as a low-Reynolds-number model. Craft et al (1997) proposed that

the damping function used in the calculation of νt should be modified from Equation 2.20 to:

f µ 1 exp

˜ Rt

90

1 2

˜ Rt

400

2

(2.30)

In the current work, it was found that this provided insufficient damping of νt close to the

wall causing stability problems in the solution. Increasing the strength of the damping very

close to the wall (ie at low values of turbulent Reynolds number) was found to be beneficial:

f µ 1 exp

˜ Rt

300

1 2

˜ Rt

400

2

(2.31)

Craft et al retained the form of c µ described in Equation 2.28 for the low-Reynolds-

number model. However, in the current work, rapid oscillations occured in the S and Ω

fields near the walls and in other regions where the computational grid had a high aspect

ratio. These instabilities were fed back into the Reynolds stresses and momentum equations,

resulting in unsatisfactory convergence of the calculations. Better numerical convergence wasachieved when the sensitivity of c µ to the reciprocal of strain (or vorticity) was decreased. A

new variant of c µ has been proposed by Craft et al (1999):

c µ2

1 2

1 3 5η f RS

(2.32)

where

f RS 0 235

max

0 η 3 333

2exp

˜ Rt

400

(2.33)

and η max

S

Ω

. A comparison of the different c µ functions, DNS data (Rogers & Moin,

1987, Lee et al, 1990) and experimental data (Champagne et al, 1970, Tavoularis & Corrsin,

1981) is shown in Figure 2.1.

Craft et al also made a minor modification to the ε equation. They removed the Reynolds

number dependency from gradient production term, Pε3 (Equation 2.17) and replaced it by


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1

S , to give:

Pε3 (2.34)

in which the effect is limited to the near wall region by ˜ Rto 250 and the empirical constant

is set as: cε3 0

0022.

2.2.4 Realizability Conditions

Schumann (1977) defined a set of realizability conditions for turbulence models. He high-

lighted that the Reynolds stresses must not exceed the limits set by the Schwarz inequality,

normal stress should not become negative nor should negative turbulent kinetic energy be

allowed during the calculation. Lumley (1978) considered that these were perhaps mathe-

matical niceties but recognized the importance of imposing the realizability conditions, sincea frequent cause of aborted calculations is the occurrence of negative energies. (These may

occur as a consequence of poor initial conditions or where one component of the Reynolds

stress is suppressed such as in stably stratified flow.)

Both Schumann (1977) and Lumley (1978) discuss the realizability conditions in the

context of a DSM. The realizability conditions may also be usefully applied in a linear EVM

in which the Reynolds stresses are determined by Equation 2.5. For example, in regions of

smooth flow, the values of k and ε will approach zero. Depending on how the lower limits

of k and ε are controlled in the calculation, this may result in large values of νt c µ

k 2

ε

.

If there is an inviscid deflection of the flow in this region, the velocity gradients will become

non-zero and when amplified by νt , may result in negative normal stresses. May (1998)

described a method of implementing the realizability conditions in an EVM which provides a

limiting maximum value for νt . This is described in Appendix A. It is not necessary to apply

the condition to the NLEVM of Craft et al (1996b) as when k 2

ε becomes very large, the

strain and vorticity invariants (S , Ω will also be large. As the NLEVM uses the functional

form of c µ (Equation 2.28) which contains the reciprocal of max

S Ω , c µ and consequently

νt will be reduced in regions of very high k 2

ε

2.3 Differential Stress Models

2.3.1 Basic DSM

In Section 2.2 it is shown how the Reynolds Averaged Navier-Stokes equations (Equation 2.3

& 2.4) can be closed by using an eddy-viscosity model to calculate the Reynolds stresses. A

2.3. Differential Stress Models

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higher level of closure can be achieved if separate transport equations are calculated for the

individual Reynolds stresses: this is a differential stress model. The transport equations for

Reynolds stress include all the physical processes which act on the Reynolds stresses; they

are capable of accounting for history and non-local effects. This is not possible with an EVM

or NLEVM which rely on local velocity gradients to calculate the Reynolds stresses.Launder et al (1975) proposed a Reynolds stress transport equation:

Duiu j

Dt

uiuk

∂U j

∂ xk

u juk

∂U i

∂ xk

Pi j

φi j εi j

d i j

∂

∂ xk

ν∂uiu j

∂ xk

d νi j

(2.35)

The mean-flow convection Duiu j

Dt , production term Pi j and viscous diffusion term d νi j are

all in closed form and require no further modelling. The pressure-strain φ i j, dissipation εi j

and diffusion due to pressure-transport and turbulent convection d i j are:

φi j

p

ρ

∂ui

∂ x j

∂u j

∂ xi

(2.36)

εi j 2 ν∂ui

∂ xk

∂u j

∂ xk

(2.37)

d i j

∂

∂ xk

uiu juk

pui

ρ δ jk

pu j

ρ δik

(2.38)

All these terms require modelling. Local isotropy is assumed for the dissipative term:

εi j

2

3δi jε (2.39)

and a transport equation is solved for ε:

Dε

Dt

∂

∂ x j

cε

k

εuiu j

δi j ν

∂ε

∂ x j

1

2cε1

ε

k Pkk

cε2ε2

k (2.40)

The constants are: cε 0 18 cε1 1 44 and cε2 1 92. The generalized-gradient-diffusion

hypothesis of Daly and Harlow (1970) is adopted for the diffusion term (pressure-transport

and turbulent convection):

d i j

∂

∂ xk

csuk ul

k

ε

∂uiu j

∂ xl

(2.41)

with the coefficient cs 0

22.

The trace of the pressure-strain term (φii is zero and hence pressure-strain does not pro-

duce or destroy turbulence. Instead, it redistributes turbulent energy between the Reynolds


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stresses, causing the normal stresses to become isotropic and the shear stresses to vanish. It

is sometimes referred to as the return-to-isotropy term. In the absence of body forces the

pressure strain term is usually modelled in three parts:

φi j

φi j1

φi j2

φ

w

i j (2.42)

φi j1 describes the interaction of turbulent eddies with other turbulent eddies, whereas φi j2

describes the interaction of the mean strain with turbulent eddies. As φi j2 reacts to rapid

distortions in the flow, it is often referred to as the “rapid” term. Correspondingly, φi j1 is

referred to as the “slow” term as energy changes due to rapid distortions in the flow must cas-

cade down to the turbulent eddies before their effect contributes to φi j1. Pressure fluctuations

in the flow are deflected by walls and consequently they damp velocity fluctuations normal to

the wall. This effect is modelled by the wall-reflection term, φwi j which impedes the transfer

of energy to the Reynolds stress normal to the wall. The scheme proposed by Launder et al

(1975) uses Rotta’s (1951) model for φi j1 and Naot’s et al (1970) model for φi j2

φi j

c1ε

k

uiu j

2

3δi jk

φi j1

c2

Pi j

2

3Pk

φi j2

φw

i j (2.43)

where the constants c1 1

8 and c2

0 6. Gibson & Launder (1978) proposed the following

model for φwi j:

φwi j

c

1

ε

k

uk umnk nmδi j

3

2uk uink n j

3

2uk u jnk ni

l

yn

c

2 φkm2nk nmδi j

3

2φki2nk n j

3

2φk j2nk ni

l

yn

(2.44)

where the turbulence length-scale l

k 3 2

cl ε and the constants cl

c

1and c

2 take the values 2.55,

0.2 and 0.12 respectively. Problems arise when using this model in the definition of the wall

normal-distance ( yn) and the unit vector normal to the wall ( ni). The Gibson & Launder

derivation assumes that the wall is plane and infinite which makes the model awkward toapply (and invalid) to systems with curved walls, corners or more complex wall topography.

2.3.2 Cubic DSM

To overcome the wall-topography dependency of φwi j, Launder and Li (1994) proposed a

model which rigorously enforces the two-component limit (v2

0 at the wall) and eliminates


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vides et al (1996) used a zonal method replacing the second-moment closure with a mixing-

length hypothesis (MLH) in in the near-wall region. In the MLH the Reynolds stress (uiu j),

turbulent viscosity ( νt ) and mixing length (lm) are defined as:

uiu j

2

3k δi j

νt

∂U i

∂ x j

∂U j

∂ xi

(2.51)

νt

l2m

∂U i

∂ x j

∂U j

∂ xi

∂U i

∂ x j

1 2

(2.52)

lm κ y 1 exp

y

26

(2.53)

where k is determined from local equilibrium conditions:

uvk

2

c µ (2.54)

y is the normal distance to the wall (or smallest normal distance if near more than one wall),

the non-dimensional distance to the wall is y

y τw

ρ

ν and κ 0

42. The Reynolds

stresses determined the boundary condition between the cubic DSM and the MLH model. For

the normal stresses, the boundary condition on the cubic DSM was that gradients of the nomal

stresses should be the same as the gradient of turbulent kinetic energy, ie ∂

u2i

k

∂n 0

For the shear stresses, the values at the cubic DSM-MLH boundary had to be equal.

Iacovides (1998) used more sophisticated approach with a one-equation k

l model(Wolfshtein, 1969) in the near-wall region. This solved the transport equation for turbulent

kinetic energy (Equation 2.15) and the calculated the Reynolds stresses by the eddy-viscosity

model (Equation 2.51). The kinematic turbulent viscosity, νt was:

νt c µl µ k (2.55)

The dissipation rate ε:

ε

k 3 2

lε(2.56)

and the length-scales were:

l µ

cl y

1 exp

0 016 y

(2.57)

lε cl y

1 exp

0 263 y

(2.58)

where y is the wall-distance normalised by y

y

k

ν and the constant cl

2 55


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2.4 Near-Wall Models

2.4.1 Introduction

In high-Reynolds-number flow remote from a wall, viscous stresses are small in compari-

son to the Reynolds (turbulent) stresses. As the wall is approached, the Reynolds stresses

diminish and viscous stresses become more influential and their effects must be included

in the momentum equations. Two approaches for this method have already been discussed:

low-Reynolds-number modelling in the context of a k ε model (Section 2.2.1) and zonal

modelling using either a mixing-length or k l model (Section 2.3.2). As these techniques

integrate the momentum equations right up to the wall, they require fine computational grids

in order to capture the steep velocity and turbulence profiles near the wall. This results in

additional computational expense and is often not practicable for large three-dimensional

calculations. An alternative method is to “bridge” the near-wall region with a wall function.The first computational cell spans the near-wall region and extends into the fully turbulent

flow. The wall shear-stress is used to account for the frictional force of the wall on the flow

and average values are calculated for production of turbulence and dissipation rate in the

near-wall cell.

The three wall functions which are used in the present study are described in this section.

These are the basic wall function used in the TEAM code (Huang et al, 1983) described in

Section 2.4.2 and the Chieng & Launder (1980) wall function used in the STREAM code

(Lien & Leschziner, 1994a)2 described in Section 2.4.3. The new Analytic Wall Function

(Gerasimov, 1999) which has been developed and implemented during this work is described

in Section 2.4.4.

2.4.2 Basic Wall Function

Log-Law of the Wall

At a wall the no slip condition is applied (U i 0 . Immediately adjacent to the wall ( y

5) is the viscous sublayer in which viscous stresses are significant and the velocity varies

linearly with the non-dimensional distance from the wall ( y

. Further away from the wall(30

y

300) there exists the inner turbulent region, and further still ( y

300) is the

outer turbulent region. The viscous sublayer and inner turbulent region blend in the buffer

layer (5

y

30). A schematic diagram of the near-wall velocity profile is shown in Figure

2.2.

2See Chapter 3 for detailed descriptions of the TEAM and STREAM codes.

2.4. Near-Wall Models

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In the derivation of conventional wall functions such as Launder & Spalding (1972) and

Chieng & Launder (1980) a steady, two-dimensional boundary layer with zero pressure gra-

dient is assumed. The momentum equation for the boundary layer reduces to:

∂

∂ y

µ

∂U

∂ y

ρuv

0 (2.59)

In the inner turbulent region the viscous stresses are negligible and Equation 2.59 reduces to:

ρuv τw (2.60)

where the constant τw is the shear stress at the wall. Assuming local equilibrium (turbulence

is dissipated where it is produced) provides:

ρuv

ρl

2

m

∂U

∂ y

2

(2.61)

where the mixing length is proportional to the distance from the wall, lm κ y and the constant

is κ 0

42. Substitution of Equation 2.61 into Equation 2.60 yields the logarithmic velocity

profile for the inner turbulent region:

U

1

κ ln

Ey

(2.62)

where the non-dimensional wall-distance y

yU τ

ν

, the friction velocity U τ τw

ρ and

the integration constant E 9 793 for smooth walls. Due to the logarithmic velocity profile,

the inner turbulent region is sometimes referred to as the “log-law region”.

Wall Shear Stress, τw

When turbulence is in local equilibrium the ratio

uv

k is constant which provides the

definition of c µ :

c µ

uv

k

2

0 09 (2.63)

where the value 0 09 is suggested by measurements in the equilibrium regions of boundary

layer flows. The turbulent kinetic energy is assumed to be constant in the log-law region and

the value at the near-wall node (k p is taken in the basic wall function. The friction velocity

can be expressed in terms of k p:

k p

uv

c1

2

µ

U 2τ

c1

2

µ

(2.64)


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and hence the friction velocity, U τ c

1 4 µ k

1 2 p can be used in Equation 2.62 to provide an

explicit expression for the wall shear stress:

τw

ρκ c1 4

µ k 1 2 p U p

ln Ec1

4

µ k 1

2

p y p

ν

(2.65)

where U p is the wall tangential velocity at the near-wall node which is at distance y p from the

wall.

Average Production of Turbulent Kinetic Energy, Pk

The transport equation for k is solved in the near-wall cell. However, as the production of

turbulent kinetic energy Pk varies rapidly near the wall it is not appropriate to use the value

of Pk at the near-wall node (cell centre). Instead a cell-average value Pk is calculated and

used in the k transport equation for the near-wall cell. Turbulent stress across the near-wall

cell is assumed to be constant and is taken as being equal to the wall shear stress τ w. The

velocity gradient is assumed to be linear and is calculated from the near-wall node values.

The average production of turbulent kinetic energy is then:

Pk

1

yn

yn

0

uiu j∂U i

∂ x j

dy (2.66)

yn is the location of the near-wall cell face opposite the wall. Pk is modelled as:

Pk τw

U p

y p(2.67)

Figure 2.3 shows the near-wall cell with locations of p n, etc.

Average Dissipation Rate of Turbulence Kinetic Energy, ε

As ε varies rapidly near the wall, the transport equation for ε is not solved in the near-wall

cell and a cell-average value is specified. From the local equilibrium assumption:

ρuv τw

µt ∂U

∂ y(2.68)

and as above, assuming a linear velocity-gradient profile:

τw

ρ

c µ

k 2

ε

U p

y p(2.69)


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The non-dimensional near-wall velocity at node p is U p

c1 4

µ k 1 2 p U p

τw

ρ

which when substi-

tuted into Equation 2.69 provides a value for the dissipation rate at the near-wall node, ε p:

ε p

c3 4

µ k 3 4 p U

p

y p (2.70)

The cell-averaged value ε is assumed to be equal to the cell-centre value ε p

2.4.3 Chieng & Launder Wall Function

Simplified Version

In the Chieng & Launder (1980) wall function it is recognised that velocity changes inside the

viscous sublayer (where the turbulent stresses are zero) do not lead to production of turbulent

kinetic energy. The wall shear stress is calculated in the same way as in the basic wall function

(Section 2.4.2) but the average production of kinetic energy is calculated only from that part

of the near-wall cell which is outside the viscous sublayer. The turbulent stress is set via

the local equilibrium assumption (Equation 2.61) and the velocity gradient is supplied by the

differential of the log-law (Equation 2.62):

Pk

1

yn

yn

yv

τw

τw

κ c1 4

µ ρk 1 2 p y

dy

τ2w

κ c1 4

µ ρk 1 2 p yn

ln

yn

yv

(2.71)

A constant sublayer Reynolds number is used to specify the sublayer thickness yv:

Rv

yvk 1 2 p

ν

20 (2.72)

To calculate the average dissipation in the near-wall cell, ε is assumed to be constant and

equal to the wall value in the viscous sublayer. In the log-law region ε is assumed to vary

according to the equilibrium length-scale:

ε 1 yn

yv 2 νk p y2v

yn

yv

c3 4

µ

k 3 2 p

κ y dy 1

yn

2k

3 2

p

k 1

2

p yv

ν c

3 4

µ k

3 2

pκ

ln yn yv

(2.73)

Full Version

In order to reduce the influence of the near-wall cell size (location of y p), the full Chieng &

Launder (1980) wall function uses the value of turbulent kinetic energy at the edge of the


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viscous sublayer k v to define the wall shear stress:

τw

ρκ c1 4

µ k 1 2v U p

ln

Ec1 4

µ k 1 2v y p

ν

(2.74)

The value of k v is extrapolated from k p and k N . As there is a maximum in the turbulent

kinetic energy profile near to the wall (Figure 2.4) care must be taken in positioning p and N

to ensure the correct value of k v is extrapolated.

To calculate Pk , as in the simplified version, the velocity gradient is taken from the log-

law and the turbulent stress is assumed to be zero in the viscous sublayer. However, in the

fully turbulent region the turbulent shear stress is taken to vary linearly with wall-distance:

Pk

1

yn

yn

yv

τw

τn τw

yn y

τw

κ c

1 4

µ ρk

1 2

v y

1

ydy

τ2w

κ c1

4

µ ρk 1

2

v yn

ln

yn

yv

τw

τn τw

κ c1

4

µ ρk 1

2

v y2n

yn yv

(2.75)

Average dissipation rate is calculated using the same assumptions as in the simplified

model but with a linear interpolation for k in the fully turbulent region:

ε

1

yn

yv

2 νk p

y2v

yn

yv

c3

4

µ

κ

k n

k n k p

yn y p

yn y

3 2

dy

(2.76)

2.4.4 Analytical Wall Function

A major weakness of the log-law type wall functions described so far is the assumed loga-

rithmic velocity profile in the inner turbulent region. As was demonstrated in Section 2.4.2

this condition is based on the assumption that the boundary layer is two-dimensional, there

is zero pressure-gradient and the flow is in local equilibrium. For many applications these

assumptions are not valid. For example if the boundary layer is being accelerated or deceler-

ated near a flow reattachment or separation, or if there is an external force applied to the fluid

such as a buoyancy force from a heated wall.

In the analytical wall function neither the log-law velocity profile nor the constant or

linear variation in shear stress is assumed. A simplified momentum equation is specified in

the near-wall cell:

ρ∂UU

∂ x

ρ∂UV

∂ y

∂ p

∂ x

∂

∂ y

µ µt

∂U

∂ y

(2.77)


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A prescribed viscosity profile is also adopted:

µt 0 for y yv

ρc µcl k 1

2

p

y yv for y

yv (2.78)

which is the Prandtl-Kolmogorov condition (Equation 1.8) and in which cl 2

55. The

simplified momentum equation is integrated to find analytical values of velocity gradient,

wall shear stress and average production of turbulent kinetic energy for the near-wall cell.

The derivation of these terms for an iso-thermal wall function is described in Appendix B

and they are summarised below. (The prime suffix denotes non-dimensional variables; the

subscript “re f ” denotes far-field reference values used to normalise the variables. The wall

function was derived in dimensionless terms for the version of STREAM in which it was

used.)

Wall Shear Stress:

τ w

k 1 2

p

ρre f

A1

Average Production of Turbulent Kinetic Energy:

Pk

ραk 1

2

p

yn

yn

yv

y

y v

A1

C 1 y v

C 2

y

yv

1 α

y

yv

2

dy

Constant Terms:

A1

αU n

C 2

yn

yv

C 1 y v

C 2α

ln

1 α

yn

yv

αC 1 y

2v

2

α yv

ln

1 α

yn

yv

C 1

ν

k p

∂ p

∂ x

γ U

∂U

∂ x

γ V

∂U

∂ y

C 2

ν

k p

∂ p

∂ x

U

∂U

∂ x

V

∂U

∂ y

Average Dissipation Rate:

ε

k 3 2

p

yn

2 ν

yε

1

clln

yn

yε


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γ is an empirical constant which is used to control the influence of convection inside the

laminar sub-layer, α

c µcl . The turbulence and dissipation viscous sublayer thicknesses yv

and yε are 10.8 and 5.1 respectively

2.4.5 Note on Near-Wall Distance

Two different methods are commonly used for the normalisation of the near-wall distance, y.

In the basic wall function (Section 2.4.2) the near-wall distance is normalised by the friction

velocity at the wall, U τ

τw

ρ, thus:

y

U τ y

ν (2.79)

In the derivation of the analytical wall function an alternative method is used, in which y is

normalised by using the turbulent kinetic energy at the near-wall node for the velocity scale:

y

k 1 2 p y

ν (2.80)

This method is used in the full Chieng & Launder wall function but with the turbulent kinetic

energy evaluated at the edge of the viscous sublayer (k v) rather than at the near-wall node

(k p). Equation 2.64 provides an alternative definition of U τ c

1 4

µ k 1

2

p , which allows y to be

re-cast in the same fashion as y :

y

c1

4

µ k 1

2

p y

ν (2.81)

Hence, y and y differ by a factor of c

1 4 µ

0 55

For a wall function to be applicable, the near-wall node must be placed outside the buffer

layer which blends the viscous sublayer and turbulent inner region, and within the upper limit

of the turbulent inner region (ie. the “log-law” region) shown in Figure 2.2. It is generally

held that the distance of the near-wall node from the wall should fall in the range 30

y

300. This equates to 55 y

550.


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Chapter 3

Numerical Implementation

3.1 Finite Volume Method

3.1.1 Discretization of a General PDE

The RANS equations and turbulence equations described in Chapter 2 must be discretized

and organized within a numerical solution scheme for the spatial variation of the dependent

variables to be calculated. Several methods are commonly used; that adopted in the present

study is the Finite Volume Method. The essence of the FVM is to divide the domain over

which the flow is to be calculated into discrete (or “finite”) volumes. Pressure, velocities and

turbulence values are calculated at the central node of each of these volumes by considering

the fluxes of momentum and turbulence quantities between adjacent volumes. The array of

volumes is usually referred to as the computational “grid” or “mesh” and the discrete volumes

are sometimes referred to as “grid cells”. Although it is convenient, it is not necessary to use

the same grid for each variable. (The benefits of using separate grids are discussed later.)

A general, steady-state, partial differential equation can be written:

ρU j∂φ

∂ x j

convection

∂

∂ x j Γ φ

∂φ

∂ x j

dif fusion

S φ (3.1)

where φ is a general variable, Γ φ is a diffusion coefficient and S φ contains the source terms

for the equation. The continuity, momentum and turbulence equations can be recovered by

setting φ 1 U

V

W

k

ε respectively and giving appropriate values to Γ φ and S φ. Equation

3.1 can be integrated over a control volume (eg a grid cell - Figure 3.1) as follows:

volρU i

∂φ

∂ xi

dV

vol

∂

∂ x j

Γ φ∂φ

∂ x j

dV

volS φdV (3.2)

58

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CHAPTER 3. Numerical Implementation 59

Then, by applying Gauss’ divergence theorem and grouping convection diffusion terms through

the same faces, one obtains:

area ρU φ Γ φ

∂φ

∂ x

dydz

e

w

area ρU φ Γ φ

∂φ∂ y

dxdz

n

s

area ρU φ Γ φ

∂φ

∂ z

dxdy

t

b

vol S φdV

(3.3)

Assuming that the variables are constant over each cell face, and defining the face areas as:

Ae w

dydz

e w; An

s

dxdz

n s; At

b

dxdy

t b then integration over the faces results in:

ρU φ A

e

ρU φ A

w

Γ φ A

∂φ

∂ x e

Γ φ A∂φ

∂ x w

ρV φ A

n

ρV φ A

s

Γ φ A ∂φ

∂ y n

Γ φ A ∂φ∂ y s

ρW φ A

t

ρW φ A

b

Γ φ A

∂φ

∂ z t

Γ φ A∂φ

∂ z b

vol S φdxdydz

(3.4)

The convection

ρU φ A

i diffusion Γ φ A∂φ

∂ x i

and source term need to be approxi-

mated for each grid cell in terms of the cell’s nodal value and the values at the surrounding

nodes. These terms are described initially for the one-dimensional computational cell (Figure

3.2) and then extended to three-dimensions.

Diffusion The diffusion term is calculated by the central difference scheme which assumes

that there is a linear variation in φ between adjacent nodes and constant gradient:

Γ φ∂φ

∂ x e

Γ φ

δ x e

φ E φP

(3.5)

(Note that in the one-dimensional system Ae w

1 and is not shown in Equation 3.5).

Convection It is well known that the central difference scheme is not generally suitable for

calculating the convection terms. It is not bounded1 and when the Peclet number2 Pe 2

oscillations in the value of φ between nodes can occur causing the solution to diverge. A

better treatment of convection is required.

1Scarborough (1958) proposed the following “boundedness” criteria for convection schemes (i) In the ab-

sence of sources, internal nodal values of φ should be bounded by the boundary values. (ii) All coefficients of

the discretized equations should have the same sign.2The Peclet number is the ratio expressing the relative magnitude of the convection across the grid cell to

the diffusion. Pe ρU ∆ x Γ where ∆ x is the grid-cell dimension.

3.1. Finite Volume Method

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3.1.2 Convection Schemes

UPWIND

The central difference scheme assigns equal weighting to the influence of the adjacent E and

W nodes without regard to the flow direction. If there is substantial convection, then node Pwill be more stongly influenced by conditions at the upstream node. The UPWIND scheme,

first proposed by Courant et al (1952), senses the flow direction and the value of φ at the

cell-face is given the value of the upstream node:

φe φP for U e

0

φe φ E for U e 0

A similar expression can be written for the w cell-face and both can be expressed compactly

by the expression:

F eφe φP

F e 0

φ E

F e 0 (3.6)

where the operator

a b denotes the maximum of a and b. A one-dimensional discretized

equation which incorporates the diffusion and source terms (Equations 3.5 & 3.24) can now

be written:

aP S P

φP a E φ E

aW φW S C (3.7)

where

a E

De

F e 0

aW

Dw

F w 0

aP a E

aW

F e F w

and for the cell-faces i

e

w : F i

ρUA

i and Di Γ φ

δ x i

The UPWIND scheme is stable and bounded but is only first-order accurate. A significant

downfall of the scheme is that it gives rise to “false diffusion” (also known as “artificial

viscosity”), more so in a 2 or 3-dimensional flow which is not aligned with the grid. This

often acts initially to stabilize a flow calculation by reducing gradients in φ, but will producean incorrect solution unless a very fine grid is used.

Power Law Differencing Scheme (PLDS)

The PLDS was proposed by Patankar (1980) as a close fit to the exact solution of the convection-

diffusion problem (Equation 3.1) with no sources. It is possible to form a convection scheme


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from the exact solution (known as the “exponential convection scheme”) but it is expensive to

compute and is not exact for 2 and 3-dimensional problems or non-zero sources. The PLDS

is sensitive to the direction and relative strength of the flow expressed via the Peclet number:

Pee

10 ;

a E

De

Pee

10

Pee 0 ;

a E

De

1 0

1Pee

5 Pee

0 Pee 10 ;

a E

De

1 0 1Pee

5

Pee 10 ;

a E

De

0

which can be written:

a E

De

0

1

0 1 F e

5

De

0

F e

(3.8)

A similar expression can be defined for aW and these can be used to define a new version of

Equation 3.7. Like UPWIND, PLDS is first-order accurate but it is unconditionally bounded.

Quadratic Upstream Interpolation for Convection Kinetics (QUICK)

The QUICK scheme developed by Leonard (1979) uses a three point upstream-weighted

quadratic interpolation for the cell-face values, which fits through the two nodes either side

of the face and the next node upstream (Figure 3.3). The cell-face values are determined

from:

φe

3

8φ E

6

8φP

1

8φW for U e

0

φe

3

8φP

6

8φ E

1

8φ EE for U e 0

Thus the coefficients can be written:

a E De

3

8

F e 0

3

8F e (3.9)

aW

Dw

3

8

F w 0

3

8F w (3.10)

When using the QUICK scheme, the discretized PDE contains an additional source term

S Q which accounts for contributions from the nodes which are not directly adjacent to the

cell-face under consideration:


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aP S P

φP

a E φ E

aW φW

S C

S Q (3.11)

where

S Q 18

F e 0

φW

F e 0

φ EE

F w 0

φWW

F w 0

φ E

1

8

F n 0 φS

F n 0 φ NN

F s 0 φSS

F s 0 φ N

1

8

F t 0 φ B

F t 0 φT T

F b 0 φ BB

F b 0 φT (3.12)

and

aP aW

a E

F e F w (3.13)

QUICK is third-order accurate with respect to its Taylor series truncation error and is

considerably more accurate than either UPWIND or PLDS. However, stability problems can

occur when using QUICK as it is not guaranteed to be bounded. This can lead to slight

under and over-shoots in the solution which can be a problem particularly when calculating

turbulent kinetic energy, dissipation or Reynolds normal stress (none of which can physically

be negative).

Upstream Monotonic Interpolation for Scalar Transport (UMIST)

From Equation 3.1 it is apparent that convection is represented by a first-order derivative

and its representation by a first-order scheme is sound on both physical and mathematical

grounds. However, unless calculated on very fine grids, first-order convection schemes do not

provide sufficient accuracy. As the interpolation polynomial on which all first-order schemes

are based is truncated at the second-order term, the second-order derivative provides the

largest error. Even-order derivatives are associated with the diffusion process, thus first-order

schemes tend to introduce “false diffusion”.

Improved accuracy can be obtained by using schemes having an order greater than two,

such as the QUICK scheme already described. However, such schemes introduce spurious

oscillations in the region of steep gradients of the transported variable when the Peclet number

is high. This is a problem for turbulence modelling in particular, as it can result in negative

values of the Reynolds stresses and turbulent dissipation. A method of overcoming these

oscillations is to add a component to the scheme which introduces, or strengthens, the bias

towards the upstream node. To ensure that monotonicity is preserved, the upstream biasing

must be controlled by the oscillatory features of the solution and this results in non-linear

schemes.


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Harten (1983) proposed the Total Variation Diminishing (TVD) concept which was ex-

pressed for general conservation laws by Sweby (1984): Given a basic scheme which pre-

serves second-order accuracy in space and time, but which is unbounded, an appropriate

limiter is introduced which diminishes the oscillation-provoking, anti-diffusive truncation er-

ror on the basis of the TVD constraint. TVD schemes based on QUICK have been proposedby Leonard (1988) and Gaskell & Lau (1988). Both of these involve several condition state-

ments which lead to high computational expense. Lien & Leschziner (1994b) proposed a

continuous and highly compact QUICK-based scheme, the UMIST scheme, which is now

described.

For the iterative solution of statistically steady flows, the cell-face flux can be written:

φe φP

1

4

1 κ

φ E φP

1 κ

φP φW

(3.14)

where the values at the one-dimensional cell face (φe) and the surrounding nodes (φ E φP φW )

are defined in Figure 3.3 for flow in the direction west to east. Thus φe is given the upwind

value, φP, and an anti-diffusive component consisting of φ E φP and φW . The order of the

scheme can be set by κ . Setting κ 1

0and0

5 returns the central difference, UPWIND and

QUICK schemes respectively.

The variables are non-dimensionalised by φ

φ φW

φ E φP

and to make Equation 3.14 mono-

tonic, a slope limiter is introduced ϕ

r , in which:

r

φP

1 φP(3.15)

Equation 3.14 can now be expressed:

φe

φP

1

4

1 κ ϕ

r

1 κ r ϕ

1

r

1 φP

(3.16)

In order to ensure that symmetry is preserved - ie. that forward and backward gradients are

treated in the same manner - the following condition is imposed:

ϕ

r

r ϕ

1

r

(3.17)

and Equation 3.16 now becomes:

φe

φP

ϕ

r

2

1 φP

(3.18)

which is independent of κ . Sweby (1984) has shown that TVD arises from the constraint

ϕ

r

min

2r 2 for r

0 and ϕ

r

0 for r 0. Substituting the value for r from Equation


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3.15 gives the following constraints for TVD:

φe 1

φe 2φP

φe

φP for 0

φP 1 (3.19)

φe

φP for φP 0 or φP 1

The limiter function proposed by Lien & Leschziner (1994b) for a symmetric, monotonic

form of QUICK is:

ϕ

r

max

0 min

2r 0

25

0

75r

0

75

0

25r

QUICK

2 (3.20)

which gives the following cell-face values:

φe

φP (3.21)

0

5max

0 min

2φP

1 φP

0

25

0 75φP

1 φP

0

75

0 25φP

1 φP

QUICK

2

1 φP

The matrix of equations for each variable is set up in the same fashion as the unbounded

QUICK scheme (Equation 3.11), only the additional source term is now given by:

S Q 0

5

F e 0 ϕ

r e

F e 0 ϕ

r e

φ E φP

F w 0

ϕ

r w

F w 0

ϕ

r w

φP

φW

F n 0 ϕ

r n

F n 0 ϕ

r n

φ N φP

F s 0 ϕ

r s

F s 0 ϕ

r s

φP φS

F t 0 ϕ

r t

F t 0 ϕ

r t

φT φP

F b 0 ϕ

r

b

F b 0 ϕ

r b

φP φ B

(3.22)

and

r e

φP φW

φ E φP

(3.23)

3.1.3 Source Term

The source term from Equation ?? is linearized as:

S φ

S C S Q

φPS P (3.24)


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where S C usually contains constant contributions to the source term and S P the corresponding

sources which are a function of the dependent variable. To ensure stability of the solution it

is essential to maintain diagonal dominance in the matrices of discretized equations. As S P is

subtracted from the leading diagonal of the matrices, negative sources should be included in

S P, resulting in the addition of a positive value to the leading diagonal.

3.1.4 Three-Dimensional, Discretized, General PDE

Once all the coefficients have been specified according to the convection scheme, the full,

three-dimensional, general PDE can be written (cf. Equation 3.7):

aPφP ∑nb

anbφnb b (3.25)

a p ∑nb

anb S P∆ x∆ y∆ z

b

S C S Q

∆ x∆ y∆ z

where nb indicates the neighbouring nodes

E W

N

S

T

B and the coefficients anb are

specific to the convection scheme used. This defines the value of φ at a single node P; for

the whole grid, a matrix is assembled which is solved by a line-iterative method using a

Tri-Diagonal Matrix Algorithm (TDMA) with alternating sweep directions.

3.1.5 Calculation of Pressure

In the general PDE discussed so far (Equation 3.25) the pressure has not been stated explic-

itly but included in the source term. From the RANS equations there is no direct method of

specifing an equation for pressure. However, it is determined indirectly by way of the conti-

nuity equation: if the correct pressure field is used to determine the solution of the momentum

equations, then the continuity equation will be satisfied.

For a control volume in a staggered grid (Figure 3.4) the x-direction U -momentum equa-

tion for the node on face e can be written:

aeU e ∑nb

anbU nb

PP P E

Ae (3.26)

which is derived from the general PDE (Equation 3.25). The pressure force term which acts

on the control volume is

PP P E

Ae. A staggered grid is used with separate control volumes

for scalar quantities, U V and W (although only two dimensions representing U and V are


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shown in Figure 3.4). The node at the centre of the U V and W control volumes is placed at

the centre of the relevant face of the scalar control volume. Similarly, the node at the centre

of the scalar control volume is placed on the faces of the U

V and W control volumes. In

this fashion the values of pressure which are used to calculate the pressure gradient across

a velocity control volume are stored at its faces. If the velocity and pressure were stored onthe same grid, the pressure gradient across a control volume would be calculated from the

pressures stored at the adjacent nodes. Consequently the calculation of the pressure gradient

across a particular control volume would be dissociated from the pressure actually stored

in that control volume. This can lead to “chequerboarding” in the solution and unrealistic

pressure fields which satisfy the continuity equation.

To solve Equation 3.26 and momentum equations for the other coordinate directions, an

initial pressure field must be provided and is usually guessed. This guessed pressure P

when used to solve the momentum equations will result in an imperfect solution for velocity:

U

V

W . The resulting momentum equations are:

aeU e ∑ anbU nb

P P P E

Ae (3.27)

anV n ∑ anbV nb

P P P N

An (3.28)

at W t ∑ anbW nb

P P P T

At (3.29)

In order to find the correct pressure P, the guessed pressure P

must be amended with a

pressure-correction P .

P

P

P

(3.30)

Similarly, the velocity fields must be corrected by velocity increments: U U

U

, V

V

V , W W

W . The momentum equations can be expressed in terms of the guessed

velocity and pressure-correction:

U e

U e

d e

PP

P E

(3.31)

V n V

n d n

PP

P N

(3.32)

W t

W t

d t

PP

PT

(3.33)

where d e

n

t Ae

n

t

ae

n

t . The contributions due to the velocity-correction in the neighbour-

ing nodes ∑ anbU nb etc, have been dropped as they play no part in the converged solution.

Equations 3.31 to 3.33 are the velocity-correction equations which indicate how the guessed

velocity field, U

V

W is modified by the pressure-correction P

. The continuity equation


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(Equation 2.3) is discretized and expressed in terms of the pressure-corrections:

aPP P a E P E

aW P W a N P N

aS P S aT P T

a BP B b (3.34)

where

a E ρd e Ae ; aW

ρd w Aw

a N ρd n An ; aS ρd s As

aT ρd t At ; a B ρd b Ab

and

aP

a E

aW

a N

aS

a E

aW

b

ρU A

w

ρU A

e

ρV A

s

ρV A

n

ρW A

b

ρW A

t

3.1.6 The SIMPLE Algorithm

The SIMPLE algorithm (Semi-Implicit Method for Pressure Linked Equations) provides the

procedure for calculating the velocities, pressure and other scalar variables in an iterative

scheme (Patankar & Spalding, 1972). The procedure is as follows:

1. Guess a pressure field P .

2. Solve momentum equations (Equations 3.27 to 3.29) to obtain the guessed velocity

field U V

W

.

3. Solve the continuity equation (Equation 3.34) to obtain the pressure-correction P .

4. Calculate the corrected pressure field P

P

P .

5. Calculate the corrected velocity field U

V

W from the velocity-correction equations(Equations 3.31 to 3.33).

6. Solve additional transport equations for turbulence (k ε and any other scalars.

7. Treat the corrected pressure P as the new guess P and repeat the procedure from 2

until a converged solution is obtained.


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3.1.7 Convergence

The iterative procedure described in Section 3.1.6 is carried out until the normalised residuals

of the various equations have reached suitably small values. At this point the equations are

said to be converged. The residual can be considered as the amount of change in a particular

variable’s solution between iterations and the absolute normalised residual is specified in

TEAM and TOROID-SE3 by:

Rφ

∑all nodes ∑ anbφnb S C

aPφP

F inφin(3.35)

F in is usually set as the mass flow rate at the inlet to the domain. The convergence in the

pressure field is assessed through the normalized mass imbalance:

Rm

∑all nodes b

F in (3.36)

In STREAM, the residuals are assessed somewhat differently. The variables are already

non-dimensional

φ

and the momentum and scalar residual is expressed as:

Rφ

1

n∑

n

∑ anbφnb S C

aPφP

∑ anb

2

(3.37)

and the mass residual:

Rm

1n ∑n

b∑ anb

2

(3.38)

where n is the total number of nodes and b is taken from Equation 3.34. These RMS cal-

culations tend to give a smaller value of the absolute residual than the expressions used in

Equations 3.35 and 3.36. Moreover, the absolute residual is now independent of grid size

(number of nodes) which allows the relative degree of convergence between different calcu-

lations to be assessed.

3.1.8 Time DiscretizationThe discretization of the general PDE in Sections 3.1.1 and 3.1.2 considers a steady-state

convection-diffusion problem. This can be extended to an unsteady problem:

ρ∂φ

∂t

ρU j∂φ

∂ x j

∂

∂ x j

Γ φ∂φ

∂ x j

S φ (3.39)

3The codes TEAM, TOROID-SE and STREAM are discussed in Section 3.2


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For the purpose of derivation of the unsteady discretization, a simple one-dimensional diffu-

sion problem is studied. This will then be extended to the full convection-diffusion problem

and Equation 3.25 will be modified to a general PDE with fully implicit time-discretization.

The one-dimensional unsteady diffusion problem can be written:

ρ∂φ

∂t

∂

∂ x

Γ φ∂φ

∂ x

(3.40)

where Γ φ is the diffusion coefficient. Time can be considered as a grid coordinate which only

extends in one direction (Figure 3.5). The flow-field solution is calculated at time t and then

“marched” forward in time by a given timestep ∆t , and the flow-field is recalculated at time

t ∆t . Variables at the original time t are denoted by φ0

i and are known when calculating

variables at the new time t ∆t which are denoted by φ1

i . Integrating Equation 3.40 over the

control volume shown in Figure 3.5:

ρ

e

w

t

∆t

t

∂φ

∂t dtdx

t

∆t

t

e

w

∂

∂ x Γ φ

∂φ

∂ x

dxdt (3.41)

Assuming that the value φP is constant throughout a control volume at a given time and by

adopting the same discretization for diffusion as used in Section 3.1.1, then the unsteady

equation becomes:

ρ∆ x φ1

P φ0

P

t

∆t

t

Γ φ

δ x e

φ E φP

Γ φ

δ x w

φP φW

dt (3.42)

The profile of φ between φ0 and φ1 is assumed to take the general form:

t

∆t

t φPdt

θφ1

1 θ φ0 ∆t (3.43)

where θ is a weighting factor. Equation 3.42 can now be written:

ρ∆ x

∆t

φ1P

φ0P

θ

Γ φδ x

e

φ1 E

φ1P

Γ φδ x

w

φ1P

φ1W

(3.44)

1

θ

Γ φ

δ x e

φ0 E

φ0P

Γ φ

δ x w

φ0P

φ0W

On the understanding that variables at the old time

φ0

are known and that the variables at

the new time φ1

are unknown, the superscript “1” is dropped. The equation is rearranged


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to give:

aPφP a E

θφ E

1 θ φ0 E

aW

θφW

1 θ φ0W

a0P

1 θ a E

1 θ aW φ0

P (3.45)

where

a E

Γ φ

δ x e

The value of θ which is chosen determines the nature of the time-discretization; three par-

ticular values, θ 0 0 5 and 1 give rise to the explicit, Crank-Nicholson and fully implicit

schemes respectively. In the explicit scheme, φP is not related to the values of φ E or φW at the

new time t ∆t and is expressed solely in terms of known variables at time t . A necessary

condition for accuracy in the explicit scheme is ∆t ρ

∆ x

2

2Γ φ. Hence when ∆ x is decreased to

improve spatial accuracy, ∆t must be decreased much more. The Crank-Nicholson scheme is

second-order accurate and unconditionally stable but requires ∆t

ρ

∆ x

2

Γ φto ensure bound-

edness. The fully implicit scheme is unconditionally stable for any size of time-step but re-

quires relatively small time-steps for accuracy. It has been recommended for time-dependent

calculations by Patankar (1980) and Versteeg & Malalasekera (1995) and all time-dependent

calculations carried out as part of the current work have used the fully implicit scheme.

The fully implicit scheme for a three-dimensional unsteady convection-diffusion problem

is (cf. Equation 3.25):

aPφP ∑nb

anbφnb b (3.46)

where

a p ∑anb a0 p

S P∆ x∆ y∆ z

a0 p ρ0

P

∆ x∆ y∆ z

∆t

b S C ∆ x∆ y∆ z

a0

Pφ0P

3.2 Codes Used

The standard UMIST codes have been adapted for the present research. Their capabilities

and structures are briefly summarized below.

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3.2.1 TEAM

TEAM (Turbulent Elliptic Algorithm - Manchester, Huang & Leschziner, 1983) is a two-

dimensional FVM code which uses a staggered grid system and SIMPLE pressure-correction

in two-dimensions. Time-dependent calculations are made with the fully implicit scheme

(Section 3.1.8). Several convection schemes are available: UPWIND, PLDS and QUICK.

In general, initial solutions for a steady-state problem are calculated with UPWIND. Time-

dependent calculations or final steady-state calculations are made with either QUICK on all

variables or with a combination of QUICK for the momentum equations and PLDS for the

turbulence equations. For calculations with the NLEVM, the shear stresses

uv are stored at

the scalar nodes (ie. they are not staggered as would be the normal practice in a DSM code).

This provides consistency between c µ and S Ω when calculating the functional form of c µ

(Equation 2.28) and is especially important in high aspect-ratio cells.

3.2.2 TOROID-SE

TOROID-SE calculates elliptic, turbulent flow and is principally intended for flow in curved

ducts. It is written in toroidal coordinates, which can be reduced to spherical, cylindrical

and Cartesian coordinates. The TEACH code on which it is based is very similar to TEAM

using a staggered grid with the SIMPLE algorithm for pressure-correction and including the

QUICK convection scheme. The code has both EVM and NLEVM turbulence models and

in a separate version a DSM. In the DSM version, shear stresses are stored at the corners

of the scalar cells, normal stresses at the scalar cell-centres. In the NLEVM version, all theReynolds stresses are stored at the scalar cell-centres as they are in the NLEVM version of

TEAM.

3.2.3 STREAM

Introduction

Simulation of Turbulent Reynolds-averaged Equations for All Mach numbers (Lien & Leschziner,

1994a), STREAM is a three-dimensional, fully elliptic, turbulent flow solver. It calcu-

lates flow using non-dimensional variables, on general non-orthogonal grids, using a non-

staggered (collocated) FVM. It uses the SIMPLE pressure-correction algorithm and has op-

tions for unsteady and compressible flow, and the UPWIND, QUICK and UMIST convection

schemes. STREAM is written in curvilinear coordinates in order to calculate the flow in non-

orthogonal grids. It uses the Rhie & Chow (1983) interpolation for pressure smoothing and

multi-block grids for efficient representation of complex geometries.

3.2. Codes Used

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Curvilinear Coordinates

Before the conservation equations are stated in curvilinear coordinates, a two-dimensional

coordinate system is described to convey the basic ideas. Figure 3.6 shows three different

velocity decompositions, where

gi are the base vectors tangential to ξi (

gi is called the

natural basis) and

gi are the base vectors nomal to ξi (called the dual basis). The Cartesian

unit vectors are

ei

ei

. For the 2-dimensional system in this description, let i 1and2

such that the notation can be defined:

e1

i

e2

j

x1

x

x2

y

ξ1

ξ

ξ2

η

(3.47)

The rate of change of the Cartesian framework with respect to the curvilinear framework

is given by the Jacobian:

J

∂

x y

∂

ξ η

xξ xη

yξ yη

(3.48)

The natural and dual vectors can be expressed as:

gi

∂ xm

∂ξi

em ; (3.49)

which can be expanded to:

g1

g2

xξ

i yξ

j

xη

i yη

j

;

g1

g2

ξ x

i ξ y

j

η x

i η y

j

(3.50)

Reciprocity follows from Equation 3.49:

gi

g j δ

ji , where δ

ji δi j

δi j

is Kroneker’s

delta. The component relationship between the natural and dual base vectors is given by the

inverse of the Jacobian matrix:

J

1

∂

ξ η

∂

x

y

(3.51)

where J xξ yη

xη yξ or:

ξ x

yη J

; (3.52)

Extending this framework to three dimensions gives:

J

xξ yη zζ

xζ yξ zη

xη yζ zξ

xξ yζ zη

xζ yη zξ

xη yξ zζ

(3.53)

3.2. Codes Used

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or simply:

ξ x

zζ

J (3.54)

Lien & Leschziner (1994a) point out that for simplicity, if one assumes ∆ξ ∆η

∆ζ

1,

then J is in fact the volume of the cell under consideration. This provides a convenient means

of evaluating J .

The mass continuity equation can now be expressed for the curvilinear coordinate system:

ρ∂ J

∂t ρ

∂

U

∂ξ ρ

∂

V

∂η ρ

∂

W

∂ζ 0 (3.55)

also a general scalar-transport equation can be written :

ρ∂φ J

∂t

transient

ρ

∂

∂ξ

U φ

ρ

∂

∂η

V φ

ρ

∂

∂ζ

W φ

convection

∂

∂ξ Γ φ Jq11

∂φ

∂ξ

∂

∂η Γ φ Jq22

∂φ

∂η

∂

∂ζ Γ φ Jq33

∂φ

∂ζ

dif fusion

JS CD

1 JS CD

2 JS φ

source

(3.56)

where the contravariant velocities U

V

W are given by:

U

J

U ξ x

V ξ y

W ξ z

V

J

U η x

V η y

W η z (3.57)

W

J

U ζ x

V ζ y

W ζ z

and the coefficients q11

q22

q33 are given by:

q11 ξ xξ x ξ yξ y ξ zξ z

q22 η xη x η yη y η zη z (3.58)

q33 ζ xζ x ζ yζ y ζ zζ z

The terms JS CD1

and JS CD2

are “cross-diffusion” sources. When Γ φ is a constant, JS CD2

can be

neglected. However, in an eddy-viscosity model Γ φ in the momentum equations is replaced

by the turbulent viscosity νt which varies across the flow-field and JS CD2

must be retained.

(Full definition of these source terms is provided by Lien, 1992)

3.2. Codes Used

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Rhie-Chow Interpolation

It is often more convenient to collocate the flow variables at the same nodes in a non-staggered

grid. This is particularly true when using a non-orthogonal grid and it is the practice adopted

in STREAM. A problem with this practice is that the pressure gradient for a given cell must

be calculated from the adjacent cells. Thus the pressure gradient used in the momentum

equations for a given cell is dissociated from the actual value of pressure in that cell. This

can lead to “chequerboard” oscillations in which physically unrealistic pressure and velocity

fields will satisfy continuity. To prevent the pressure and velocity from becoming dissoci-

ated, Rhie & Chow (1983) proposed a scheme in which the pressure gradient used in the

momentum equations is corrected to capture pressure variations between adjacent nodes.

The non-staggered grid arrangement is shown in Figure 3.7. The equation can be written

for the one-dimensional situation is:

U P

∑nb anbU nb

aP

S C

H P

Pw Pe

J ∂ξ

∂ x

aP

DU P

(3.59)

which may also be written:

U P H P

DU P

Pw Pe

(3.60)

The Rhie & Chow interpolation for cell-face velocity is:

U e

1

2

U P

DU

P

Pw

Pe

P

U E

DU

E

Pw

Pe

E

1

2

DU P

DU E

PP P E

(3.61)

which may be written:

U e

1

2

U P

U E

linearinterpolation

(3.62)

1

2

DU

P

DU E

PP P E

DU P

Pw Pe

P DU

E

Pw Pe

E

pressuresmoothing

Equations 3.61 and 3.62 provide different ways of interpreting the Rhie & Chow interpola-

tion. In Equation 3.61 the velocity at the east face (U e) is directly linked to the two adjacent

pressure nodes

PP P E

. A different point of view is conveyed by Equation 3.62 which

states that the interpolation practice consists of a centred approximation for U e with an addi-

tional pressure smoothing term.

3.2. Codes Used

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Multi-Block Strategy

The multi-block strategy used in STREAM allows efficient grids to be generated for complex

geometries. The blocks are topological hexahedra and each block constitutes a self-contained

grid of cells. Adjacent blocks must match precisely along their common block-faces. Hence

all cells that lie on a block-face which adjoins another block, must align with a cell in the

adjacent block. “Hanging” nodes4are not permitted (Figure 3.8). Similarly, complete block-

faces must match complete block-faces - ie. it is not permitted to join a portion of a block-face

to an adjoining block-face, even if all the individual interfacing cells match precisely (Figure

3.9). “C” and “O” grids are permitted but have not been used in the present work.

Before calculation of the flow-field, STREAM extends each block by an additional two

nodes for each cell on a block-face which adjoins another block 5. The computational cells for

these additional nodes are referred to as “halo cells” and each halo cell is associated with the

relevant interior cell in the adjoining block. This association between halo cells and interiorcells of adjoining blocks is used to transfer flow-field and other calculation variables between

the blocks.

During the calculation of the flow-field, the variables stored at the halo cells are updated

by assigning the revelant value from the associated interior cell of the adjacent block. For

each iteration, the momentum equations are calculated for all the cells in the grid (this is Step

2 in the SIMPLE algorithm, Section 3.1.6). The halo cells are then updated with the values

of U V andW from the relevant interior nodes. After the pressure has been calculated and

the velocities corrected (Step 5 in the SIMPLE algorithm) the values of U

V

W and P in the

halo cells are updated.

4A “hanging” node is the term given to a boundary node on the face of block which does not share the

same spatial location as a node in an adjoining block. In this arrangement the grid lines do not match between

adjacent blocks.5Two cells are required for calculation of higher-order convection schemes such as QUICK.

3.2. Codes Used

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Chapter 4

Cylinder of Square Cross-Section Placed

Near a Wall

4.1 Introduction

Flow past a square cross-section cylinder is a popular test case and several groups have taken

measurements and made calculations of flow around the cylinder in a free-stream (as has been

discussed in Chapter 1). More recently, a number of researchers have taken measurements

and made calculations of the flow around square cylinders placed near to a wall. Notable

amongst these are the measurements of Durao et al (1991) and Bosch et al (1996) and the

calculations of Franke & Rodi (1993) and Bosch & Rodi (1996).

The popularity of this test case is due to the complex flow which is created by a rela-

tively simple geometry. The cylinder is a bluff body, there is stagnation of the flow at the

impingement point on the leading face and separation as the flow passes the leading edges.

Bending of the flow around the leading edges creates streamwise curvature and there is large-

scale separation and recirculation in the wake. Depending on how close the cylinder is to

the wall, the wake flow will either be steady or there will be periodic shedding of vortices

from the trailing edges. Durao et al and Bosch et al have shown that below a wall distance

of g

d 0 35 (Figure 1.1) the wake is steady and above g

d 0 50 there is regular vortex

shedding. Between these distances the wake fluctuates between steady and vortex sheddingbehaviour.

Although the flow past a square cylinder provides an essentially two-dimensional test

case, it mimics several features of flow around road vehicles: impingement, streamwise cur-

vature, separation and wake recirculation. The square cross-sectioned cylinder is thus an

appropriate test case for turbulence models which are intended for use in calculating the flow

around road vehicles.

77

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CHAPTER 4. Cylinder of Square Cross-Section Placed Near a Wall 78

The flow around the cylinder at three distances from the wall has been calculated: g

d

0 25

0

50 and 0

75. These three distances were chosen to cover the range of flow types: at

g

d 0

25 the wake is steady, at g

d 0

50 the cylinder is just far enough away from the

wall to generate periodic vortex shedding and at g

d 0 75 the periodic vortex shedding is

well established. These three distances coincide with the distances used in the measurementsby Durao et al (1991) and Bosch et al (1996). Durao et al measured the flow with g

d 0

25

and 0.50 at a Reynolds number Re 13 600; Bosch et al measured the flow with g

d 0 75

at a Reynolds number of Re 22 000.

In general there is no vortex shedding in the wake of a road vehicle, thus the calculation

with the steady wake, g

d 0

25, is the most relevant. Only this case is discussed in the

remainder of this Chapter; the discussion of the calculations of the remaining two cases is

included in Appendix C. When calculating the flow around the Ahmed body, a modified

form of the c µ function used in the non-linear k ε model was tested (Chapter 7). The flow

around the square cylinder has been recalculated for the case with g

d 0 25 using the non-

linear model with the modified c µ function to establish the effect of that modification on this

flow. The results of this calculation are discussed in Appendix D.

4.2 Models Used

Two turbulence models have been used to calculate the flow around the square cross-section

cylinder: the “standard” linear k ε model of Launder & Spalding (1974) and the non-linear

k

ε model of Craft et al (1996b) which are both described in Section 2.2. All calculations

have been made using the QUICK scheme to calculate convection of the mean velocities and

PLDS scheme for convection of turbulent kinetic energy and dissipation (Section 3.1.2). The

basic wall function described in Section 2.4.2 is used to “bridge” the viscosity-affected flow

in the near-wall regions around the cylinder and lower wall. All cases were calculated using

the implicit time-discretization scheme1 (Section 3.1.8). As this scheme is only first-order

accurate, a relatively small time-step was used to ensure accuracy, ∆t =0.009 to 0.014. This

corresponded to there being between 50 and 100 time-steps per vortex shedding cycle for

the cases g

d 0

50 and 0

75. All calculations were made using the TEAM code (Section

3.2.1).

1Durao et al and Bosch et al have both shown that with a wall-distance of g d

0

25 the wake is steady

and there is no vortex shedding. However, this case was calculated as a time-dependent flow to ensure that the

steady wake was calculated and not imposed by the solver.

4.2. Models Used

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4.3 Domain, Grids, Boundary Conditions

4.3.1 Domain and Grid

The general domain used for calculation of all three cases is shown in Figure 1.1. It extended

from 10d upstream of the leading face of the cylinder to 20d downstream (d is the diameter

of the cylinder). The lower wall extended 6d in front of the cylinder in accordance with the

measurements of Bosch et al (1996) and the height of the domain was 10 d . A sample of the

grid used for the case g

d 0 25 is shown in Figure 4.1 (showing every second grid line only

for clarity). The grids used with each value of g

d had 102x72 cells which was comparable

to that used by Bosch & Rodi (1996) which had 106x75 cells. To establish that a grid-

independent solution could be obtained with this number of cells, preliminary calculations

were carried out using the linear k ε model for the case with g

d 0 25 and using the

102x72 cell grid and a refined grid with 132x96 cells. No significant differences were foundbetween the mean velocities, turbulent kinetic energy or dissipation calculated on the two

grids. All further calculations were made with the 102x72 cell grid.

4.3.2 Boundary Conditions

Inlet boundary conditions were set for each case according to the relevant set of measure-

ments. The cases with g

d 0

25 and 0.50 were to be compared with the measurements of

Durao et al (1991) and used an inlet Reynolds number Re 13

600 (based on d ) and inlet

turbulence intensity of 6%. The case with g

d

0

75 was to be compared with the mea-surements of Bosch et al (1996) and used an inlet Reynolds number of Re

22 000 and 4%

turbulence intensity. Neither Durao et al nor Bosch et al provide sufficient information to be

able to specify an inlet value for turbulence energy dissipation rate. Following the method

used by Bosch & Rodi (1996) for their calculations, the inlet dissipation rate was specified

for all three cases by assuming νt

ν 10 at the inlet. To test the sensitivity of the models to

this value, a second set of calculations for g

d 0

25 were carried out assuming νt

ν 100

at the inlet (following the method used by Franke & Rodi, 1991).

All values at the inlet were specified with uniform profiles. A zero-gradient condition was

imposed on all variables at the downstream boundary and symmetry planes were specified on

the upper domain boundary and the initial 4d of the lower domain boundary (ie. the portion

of the lower boundary which was upstream of the wall).

4.3. Domain, Grids, Boundary Conditions

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4.4 Calculated Flow Results for g d 0 25

All calculated and measured results are normalised by the inlet streamwise velocity U o. The

measured data of Durao et al (1991) were not available in digital form and were therefore

re-digitized from the journal paper with a consequent loss in accuracy.

4.4.1 High Dissipation Inlet Condition ( νt

ν 10)

Drag and Lift

Coefficients of drag (C D) and lift (C L) are shown in Figure 4.2. The large oscillations in both

coefficients for time t 35 are due to the formation of starting vortices (all calculations were

started from a zero velocity inital flow-field). These are convected downstream and do not

influence the flow around the cylinder after t

45.

C D and C L calculated by the linear k ε model settle to constant values, indicating that the

calculated flow is steady: C D 1

77 and C L

0 28. However, there are small oscillations in

both C D and C L calculated by the non-linear k ε model throughout the period of calculation.

The mean value of C D calculated by the non-linear model is the same as that calculated by

the linear model (C D 1

77) but the mean value of C L is somewhat lower (C L

0 46).

Although Durao et al did not present detailed results for lift and drag, they did present power

spectra of vv for different distances of the cylinder from the wall. Durao et al use these to

show that there is no dominant frequency in the flow when g

d 0

5 and hence no vortex

shedding. However, in the power spectrum for g

d

0

25, there is a small peak whichoccurs at a non-dimensional frequency, f

f

d

U o

0 51. The frequency of the oscillations

in C D and C L calculated by the non-linear k ε model is f

0 40. The reasons why the

linear model does not calculate this oscillation are presented with the discussion of calculated

Reynolds stresses below2.

Velocity Profiles

Profiles of instantaneous U and V -velocity are shown in Figures 4.3 and 4.4 respectively.

Both models calculate the separation of the flow above the cylinder but the non-linear k

εmodel calculates a taller region of separated flow, with weaker reverse flow than the linear

model. Also, the non-linear model calculates a steeper velocity gradient than the linear model

in the upper shear layer above the cylinder and in the wake. In general, the non-linear model

2It is noted that the similarity in f between the measurements and non-linear calculation in itself does not

prove that the oscillations are due to the same effect. However, it provides some confidence that the non-linear

k ε model has calculated the flow accurately.

4.4. Calculated Flow Results for g

d 0 25

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does not calculate the downstream wake as accurately as the linear model. The length of the

recirculating-flow region behind the cylinder, calculated by either model, is too long.

The steep velocity gradients and long recirculating flow regions calculated by the non-

linear k ε model have been shown for similar cases by Suga et al (2000) and Craft et al

(1999). Suga et al calculated the flow over a surface mounted cube using the low-Reynolds-number k ε A2 form of the non-linear model (Suga, 1995) in which A2 is the second

invariant of the anisotropic stress tensor3. They found that the calculated reattachment length

behind the cube was nearly twice as long as that measured. Also the downstream velocity

profile calculated by the k ε A2 model was rather worse than that calculated by a linear

k ε model.

Craft et al (1999) used the low-Reynolds-number form of the non-linear k ε model

(Section 2.2.3) to calculate the flow in an abrupt pipe expansion. The pipe expansion had

a step height 60% of that of the radius of the larger (downstream) pipe. Craft et al found

that the non-linear and linear k ε models calculated the same reattachment length but that

this was somewhat longer than the measured length. Comparison of the velocity profiles in

Figure 4.3 with the work of Suga et al and Craft et al provides confidence that the non-linear

model has calculated the flow correctly, to within the limitation of the model.

Reynolds Stress Contours

Durao et al (1991) present contour plots of three Reynolds stresses ( uu vv

uv) around the

cylinder. These are reproduced in Figures 4.5 to 4.7 with plots showing the Reynolds stresses

calculated by the linear and non-linear k ε models (all three figures use the same contour

values). The uu-stress calculated by both models (Figure 4.5) is in general too low. The

measured uu-stress has a normalised, peak value of 0.125 in the upper shear layer at x

d

4 0. The non-linear model calculates the magnitude of the peak uu-stress more accurately

than the linear model but this is too far downstream at x

d

7 0. The same trends for the

calculated vv and uv stresses are shown in Figures 4.6 and 4.7.

In the Figure 4.2 it was noted that whereas the linear k ε model calculated steady values

of C D and C L, a small oscillation was calculated by the non-linear model. In Figure 4.5, it

can be seen that the linear model calculates much higher levels of Reynolds stress aroundthe cylinder, particularly in the shear layer immediately above the cylinder. The linear model

calculates the Reynolds stresses from the product of turbulent viscosity and velocity gradients

(Equation 2.5). Above the cylinder, the linear model calculates less steep velocity gradients

than the non-linear model (Figure 4.3) and yet it calculates higher values of Reynolds stress.

3A transport equation is solved for A2 with the principal effect of improving the near-wall flow which in the

current work is bridged by the wall function.


d 0 25

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This implies that the turbulent viscosity calculated by the linear k ε model is higher than

that calculated by the non-linear model. It is this increased level of νt which suppresses the

small oscillations in C D and C L calculated by the linear k ε model.

The increased level of νt is itself due to the high level of turbulent kinetic energy which

the linear k

ε model calculates around the cylinder. The linear model calculates too muchPk at the flow impingement; the subsequent high level of turbulent kinetic energy is convected

around the cylinder. In contrast, the non-linear k ε model reduces c µ in regions of high S

and Ω (Equation 2.28) and has a more sophisticated stress-strain relationship (Equation 2.27).

In calculating the impinging flow at the front of the cylinder, it does not produce the too high

levels of Pk and k which are calculated by the linear model. Due to the relatively large amount

of dissipation which has been assumed at the inlet for this calculation, relatively low levels

of turbulent kinetic energy are calculated in the flow which impinges on the cylinder. Hence,

the relatively high levels of turbulent kinetic energy calculated by the linear k ε model at

impingement do not show particularly well on Figures 4.5 to 4.7. The effect is much more

evident in the calculations with νt

ν 100 (Section 4.4.2).

Durao et al (1991) used an LDV technique to measure the flow. This was not able to

distinguish between “stresses” due to turbulent fluctuations (ie. Reynolds stresses u iu j) and

“stresses” due to any small oscillations in the shear layers or other periodic motions in the

flow. The periodic velocity can be defined by:

ui

U i ui

(4.1)

where U i is the instananeous velocity and ui

is the ensemble average. Thus time average

“stress” due to periodic motion measured by Durao et al is ui ui. If there was indeed a small

oscillation in the shear layers measured by Durao et al, then the values shown by them as

“Reynolds stresses” are in fact the time average total stresses:

ut iut

j uiu j

ui u j (4.2)

This could account for the higher levels of “Reynolds stress” which are measured, in com-

parison to the calculated levels.

4.4.2 Low Dissipation Inlet Condition ( νt

ν 100)

Drag and Lift

C D and C L for the case with a low rate of turbulence energy dissipation specificied at the

inlet are shown in Figure 4.8. After the starting effects have diminished both the linear and


d 0 25

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non-liner k ε models calculate steady values for C D and C L. The low dissipation rate causes

higher levels of turbulent kinetic energy to remain in the flow as it reaches the cylinder.

This leads to higher levels of turbulent viscosity which are sufficient to suppress the small

oscillations in the shear layers and wake which were previously calculated by the non-linear

k

ε model.The non-linear k ε model calculates C D

1 81 and C L

0 48, which are close to

the mean values calculated with the high level of inlet dissipation. Similarly, the linear k ε

model calculates C D 1 86 but there is a marked difference in lift: C L 0 05 (cf. C L

0 28

for high dissipation case).

Velocity Profiles

Figure 4.9 shows the U -velocity profiles calculated by the two models with the low inlet dis-

sipation rate. The U -velocity calculated by the non-linear k

ε model is virtually unchangedin comparison to the high inlet dissipation rate calculation (Figure 4.3). However, the pro-

files calculated by the linear model are in still closer agreement with the measured values,

particularly in the downstream wake.


Reynolds stress contours (uu) are shown for the two models in Figure 4.10; differences are

apparent in the values calculated by both models compared to the high inlet dissipation case

(Figure 4.5). The non-linear k

ε model calculates a higher level of uu-stress in the shearlayer immediately above the cylinder. The location of the peak value in the upper shear layer

is calculated closer to the cylinder ( x

d 5

5) and closer to the location of the measured peak

value ( x

d 4

0). It is the increased level of Reynolds stress (and hence turbulent viscosity)

in the shear layer above the cylinder which is responsible for the suppression of periodic

oscillations in this region.

The high levels of uu-stress calculated by the linear k ε model on the leading face

of the cylinder demonstrate the process which was described in Section 4.4.1. The linear

model calculates too-high levels of Pk at impingement, leading to high levels of turbulent

kinetic energy which are convected around the cylinder, producing the high levels of uu

stress in Figure 4.10. The non-linear k ε model calculates production of kinetic energy at

impingement more accurately due to the functional form of c µ (Equation 2.28) and the non-

linear stress-strain relationship (Equation 2.27). The Reynolds stresses calculated in the shear

layers by the non-linear model are primarily due to the velocity gradients present rather than

high levels of turbulent kinetic energy convected from upstream. The linear k ε model was


d 0 25

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developed by reference to simple shear flows and where the flow more closely matches this

situation (eg. in the wake) the excessive levels of Reynolds stress soon dissipate.

4.5 Conclusions

The comparison of the calculated results is compromised somewhat by not knowing the inlet

dissipation rate which was used in either Durao’s et al or Bosch’s et al experiments. The linear

k ε model calculates the mean velocity profiles more accurately than the non-linear model.

Also, the accuracy of the mean velocity profiles calculated by the linear model improves

when the lower dissipation rate inlet condition is used, whereas there is very little difference

in the profiles calculated by the non-linear model when considering either inlet condition. In

contrast, the Reynolds stresses calculated by the linear k ε model are considerably worse

than those calculated by the non-linear model and worsen when the lower dissipation rateinlet condition is used.

There is no experimental data available for the drag and lift generated on a square cylinder

placed near a wall and it is not possible to determine whether the linear or non-linear k ε

model has performed better in this respect. However, the coefficients of drag and lift are of

use in demonstrating differences between the models. When the high dissipation rate inlet

condition is used, the non-linear k ε model calculates a minor oscillation in drag and lift.

There is some justification for this oscillation presented by Durao et al (1991). When the low

dissipation rate inlet condition is used, this oscillation is attenuated by the higher levels of

turbulent viscosity, but the mean values of C D and C L are approximately the same for both

cases calculated by the non-linear model. There is a large change in C L calculated by the

linear model with the different inlet conditions and its sign changes from negative to positive

when the inlet dissipation rate is reduced.

The non-linear k ε model calculates Reynolds stresses more accurately than the linear

model. As the inlet boundary is sufficiently far upstream in this test case, the calculation of

the flow around the cylinder by the non-linear model is not greatly affected by alterations

in the turbulence boundary condtition. With its less accurate stress-strain model, the linear

k ε model is very sensitive to the upstream turbulence boundary condition. In industrial

calculations, full details of the upstream inlet condition are not usually known and must

instead be estimated. In such instances, it is far more desirable that the turbulence model

used is insensitive to minor changes in the upstream boundary condition if the flow is likewise

insensitive.

Calculations of the flow with g

d 0 50 and 0 75 are presented in Appendix C. With the

cylinder at these distances from the wall, vortices are shed periodically in the wake. Both the

4.5. Conclusions

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linear and non-linear k ε models are capable of calculating this vortex-shedding behaviour.

When considering the calculation of time-averaged mean velocities, both models calculate the

flow with greater accuracy than the flow with g

d 0

25 and a steady wake. The frequency

of vortex shedding is calculated more accurately by the non-linear model.

The non-linear k

ε model calculates time-averaged total stresses more accurately thanthe linear model. However, the calculated distribution of the total stress between the “pe-

riodic” and Reynolds stresses is the opposite to that measured. The calculated total stress

consists of a large “periodic stress” and small Reynolds stress; the measured total stress con-

sists of a small “periodic stress” and large Reynolds stress.

4.5. Conclusions

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Chapter 5

Flow in a U-bend of Square Cross-Section

5.1 Introduction

The flow in a U-bend of square cross-section provides a searching test case for turbulence

models. Through the U-bend the flow is subjected to significant straining, and the radial

pressure gradient which is present induces secondary motion across the streamwise direction.

This secondary motion is further modified by anisotropies in the Reynolds stresses which are

generated at the corners of the duct. Indeed, these Reynolds stress anisotropies are known to

induce secondary motion even in a straight duct. The streamwise curvature of the U-bend and

streamwise vorticity generated are both present in road vehicle external flows, particularly

around the front edges of the vehicle and in its wake.

In this Chapter, calculated results are presented for the flow in a square cross-section U-

bend with strong curvature. The Reynolds number of the flow is Re 58

000 based on the

streamwise bulk velocity, W b and hydraulic diameter, D; the radius of curvature of the bend

is Rc

D 3

35. With this radius of curvature, the flow does not separate from the inner wall

of the U-bend. This is the test case for which Chang et al (1983a) and Choi et al (1990) have

carried out measurements and for which calculations have been made by several groups (see

discussion in Chapter 1).

5.2 Models Used

Two turbulence models have been used to calculate the flow in the U-bend: the non-linear

k ε model of Craft at al (1996b) and the cubic DSM of Craft et al (1996a). The linear k ε

model has not been used for this case as the flow is very sensitive to the Reynolds stress

anisotropies. Calculations by Chang et al (1983a,b), Johnson & Launder (1985), Azzola et al

86

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CHAPTER 5. Flow in a U-bend of Square Cross-Section 87

(1986) and Choi et al (1989) have all shown the ineffectiveness of the linear k ε model in

calculating this flow (Section 1.3.2). The current calculations with the non-linear k ε model

are compared instead to calculations with the cubic DSM1.

Although the final method for calculating the flow around the Ahmed body (Chapter 7)

will use wall functions, they have not been used for this test case. Chang et al (1983a,b)showed that a linear k ε model with wall functions is particularly poor at calculating this

flow. Azzola et al (1986) showed that a low-Reynolds-number model using the MLH in the

fine near-wall cells improved the calculation of flow in a circular cross-section duct. Also,

Choi et al (1989) showed that near-wall pressure variations are significant at the corners

of the duct; these cannot be calculated by the coarse near-wall cell employed by the wall

functions described in Section 2.4. The calculations presented in this Chapter should hence

be considered as a test of the non-linear k ε model in isolation and not the combined non-

linear model and wall function method.

The flow has been calculated with low-Reynolds-number models using a zonal approach.

A one-equation k l model was used to calculate the near-wall flow and either the non-linear

k ε model (Craft et al, 1996b) or cubic DSM (Craft et al, 1996a) was used to calculate

the core flow. (This is slightly different from the approach used by Iacovides et al (1996)

which used the cubic DSM with a MLH model in the near-wall zone.) In the near-wall zone,

the one-equation model uses the eddy viscosity hypothesis (Equation 2.5) with the turbulent

viscosity, νt , provided by:

νt

c µ

k

lm (5.1)

and the mixing length, lm:

lm 2 4Y 1 exp

y

Am

(5.2)

where Y is the smallest normal distance to the wall, y

Y

k

ν, and Am is a damping

function set according to the particular solution strategy adopted. Preliminary calculations

showed that if the original form of the near-wall turbulence damping in the one-equation

model was retained (Equation 5.2; Am 62 5 as used by Iacovides et al., 1996), then a con-

stant c µ 0 09 had to be used in the k l model across the near-wall zone to give the correct

wall profile of the streamwise velocity. This model is hereinafter refered to as NLEVM-1.

When the functional form of c µ (Equation 2.28) was used in the near-wall zone, it was found

that the damping function in Equation 5.2 had to be reduced to Am 31

25 to retain the cor-

1The cubic DSM is the most advanced turbulence model used with the steady RANS equations, currently

employed at UMIST.

5.2. Models Used

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rect near-wall profile of streamwise velocity (Figure 5.1). This model is hereinafter refered

to as NLEVM-2. Calculated results are presented for both NLEVM-1 and NLEVM-2.

In the cubic DSM calculation, the Reynolds stress transport equations were solved across

the core region only. The k -transport equation was solved in the near-wall region to provide

values of turbulent kinetic energy for the one-equation model.All calculations were made using the TOROID-SE code (Section 3.2.2) and the QUICK

convection scheme (Section 3.1.2).



A sketch of the U-bend used for the flow measurements by Chang et al (1983a) and Choi et

al (1990) is shown in Figure 1.2. The z-coordinate is in the streamwise direction. Negative

values of z are used to indicate the distance in the entry section upstream of the start of the

curved section; positive values of z indicate distance from the end of the curved section in the

downstream tangent. Angular distance around the curved section is denoted by α.

A second sketch of the calculated domain in Figure 5.2 shows the lengths of the upstream

and downstream tangents (6 125 D and 12

9 D respectively, where D is the hydraulic diam-

eter of the duct). Only half the duct was calculated, with a symmetry plane placed at the

mid-height of the duct ( y 0). The grid used for all calculations had 67 x 35 nodes in the

cross-stream planes (Figure 5.3). 87 planes were used in the streamwise direction of which:25 were used in the upstream tangent, 42 in the U-bend itself and the remaining 20 in the

downstream tanget (Figure 5.4). This grid has been tested previously and calculations on it

using the cubic DSM found to be grid independent (Iacovides, 1999). The only modification

to the grid previously tested was to extend the grid in the tangent downstream of the bend.

The original grid used a constant inter-plane distance in the streamwise direction. The modi-

fication adopted in this study was to apply a geometric expansion of 1.1 to successive planes

downstream of the bend. In this way the downstream tangent was extended, whilst main-

taining the same number of streamwise planes. The one-equation k l model was applied

over the first twelve nodes adjacent to the wall in the cross-stream plane. The zone interface

between the one-equation model and the NLEVM or DSM in the core flow was at y

100.

5.3.2 Upstream Boundary Condition

To define the fully-developed, upstream inlet boundary condition, a pre-calculation in a

straight duct was carried out for each model. The straight duct used cyclic boundary condi-


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tions for mean velocities and turbulence variables at its inlet and outlet. The fully developed

profiles resulting from this pre-calculation were prescribed at the inlet to the upstream tangent

of the U-bend.

5.4 Calculated Flow Results

All calculated and measured results are normalised by the bulk streamwise velocity W b.

5.4.1 Comparison to Calculated Data

Two comprehensive data sets for the square U-bend experiment are readily available in the

literature: Chang et al (1983a) and Choi et al (1990). Different experimental methods were

used, with LDA being adopted by Chang et al and hot-wire anemometry being adopted by

Choi et al. Discrepancies between these sets of data are descibed in Section 1.3.2. In the

following discussion, calculated results are compared to Choi’s et al data. This is not because

Choi’s et al results are thought to be more accurate, rather that these results are available in

electronic form on the ERCOFTAC database. (The results of Chang et al would have had

to be re-digitised from a journal with subsequent loss of accuracy.) The exception is where

values are compared upstream of the U-bend ( z

D

1). Here Choi et al did not take any

measurements and calculated values are compared to Chang’s et al data.

5.4.2 Inlet Flow Profiles

Inlet profiles of the streamwise velocity (W ), secondary velocities (U

V ) and cross-stream

normal stresses (uu, vv) are shown in Figure 5.5 for the NLEVM-2 and cubic DSM calcu-

lations. These are the fully developed flow profiles on the duct centreline ( y 0) from the

pre-calculations in the straight duct with cyclic boundary conditions. The NLEVM-2 model

calculates a small difference between uu and vv and the weak secondary flow which is in-

duced can be seen in the U -velocity directing flow from the walls to the centre of the duct.

The cubic DSM calculates a much larger difference in the normal stresses which in turn

induces a larger secondary velocity in the inlet profile.

5.4.3 Velocity Profiles

The streamwise (W

W b) and cross-stream (U

W b) velocity profiles are shown in Figures

5.6 and 5.7. There is very close agreement between both sets of NLEVM calculation and

5.4. Calculated Flow Results

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the cubic DSM. Where there are differences between the NLEVM-1 and NLEVM-2, the

calculations using NLEVM-2 are in general slightly closer to the DSM results.

The “hole” in the W -velocity profile which appears between α 90o

135o is calculated

reasonably well, except on the centreline where the peak and trough near the inside wall is

not calculated. In Section 1.3.2 the differences between the measured data of Chang et al andChoi et al are discussed and the W -velocity profiles are compared in Figure 1.5. Chang et

al measured a much lower level of W -velocity on the centreline at α 130o than Choi et al

did at α 135o; the calculated W -velocity from all three models is in closer agreement with

Chang’s et al data than with Choi’s et al at this location.

Also shown in Figure 5.6 is the W -velocity calculated by Choi et al (1989) using a linear

k ε model with a MLH model in the near-wall region. (This is shown for two positions

around the bend: α 90o and 135o. It should be noted that the results shown at α

135o were

actually calculated at α 130o by Choi et al.) The W -velocity calculated by the NLEVMs is

in much closer agreement with that calculated by the cubic DSM than is the linear k ε /MLH

calculation of Choi et al. Also the NLEVMs and cubic DSM calculate the W -velocity more

accurately than the linear k ε /MLH in comparison to the measured data. This is particularly

notable at the centreline (2 y

D 0) but differences between the models lessen towards the

top wall (2 y

D 1 0).

The cross-stream U -velocity is calculated in reasonable agreement with the results of

Choi et al results up to α 90o

At the half-way point in the bend, the measured results

suggest that there is a positive U -velocity (fluid flowing from the inside to the outside of the

bend) from the centreline (2 y

D

0) to near the top wall of the duct (2 y

D

0

75). However,the calculated results show that the negative return flow is already present at 2 y

D 0

75

This would imply that in the measured flow, the return flow path must be confined to a small

region very close to the top wall of the duct and by continuity this must be relatively fast

moving fluid. In contrast, the calculations predict a much deeper region of return flow, with

consequently lower velocity. By the end of the bend (α 180o) the measurements show that

there is still a strong secondary motion across the duct, whereas the calculations predict a

much weaker overall secondary motion.

5.4.4 Reynolds Stress Profiles

Profiles of the Reynolds stresses are shown for the three normal stresses (uu, vv, ww) and

one shear stress (uw) in Figures 5.8 to 5.11 respectively. Differences between the Reynolds

stresses calculated by the NLEVM models and cubic DSM are more significant than the

differences in the mean velocities discussed in Section 5.4.3. As the cubic DSM solves a

separate transport equation for each of the Reynolds stresses (Equation 2.35), it incorporates


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non-local effects through convection and diffusion of the stresses. In contrast, the NLEVMs

calculate the Reynolds stresses from local velocity gradients (Equation 2.27).

Considering the normal stresses in Figures 5.8 to 5.10, in places the NLEVMs calculate

better agreement with Choi’s et al measurements than the cubic DSM does (eg. at the start

of the bend Figure 5.8a, uu stress, α

45o

) but this trend does not persist. Between α

90 135o the NLEVMs calculate normal stress profiles which fluctuate erratically across the

duct with several peaks and troughs across each profile. These fluctuations are not apparent

in the measured results and tend not to be calculated to such a great extent by the cubic DSM.

They can be understood by considering the mean velocity development. In the latter half of

the U-bend, the secondary motion develops four distinct vortices either side of the centreline

(Section 5.4.5). With this complex flow pattern high velocity gradients are generated. As

the NLEVMs rely solely on the local velocity gradients to calculate Reynolds stresses, the

complex flow profile causes the NLEVMs to calculate the erratic fluctuations in the Reynolds

stresses. The one Reynolds shear stress included (uw) shows the same trends as have been

noted for the Reynolds normal stresses.

5.4.5 Streamwise Velocity Contours and Secondary

Velocity Vectors

The development of the flow in the U-bend calculated by the NLEVM-2 and cubic DSM

models is shown by the streamwise velocity contours (W

W b) and the secondary velocity

vectors in Figure 5.12 (NLEVM-2) and Figure 5.13 (cubic DSM). Calculated results fromthe NLEVM-1 model are not included here as it has already been shown in Section 5.4.4

that there is very little difference in the results calculated by the NLEVM-1 and NLEVM-2

models.

At the upstream station, z

D

1, the NLEVM-2 model calculates a small amount of

flow across the duct. This is due to the influence of the bend affecting the flow upstream

of the bend itself. The cubic DSM calculates stronger secondary motion in the straight duct

(Figure 5.5) and this can also be seen by the four vortices shown by the secondary velocity

vectors calculated by the cubic DSM at z

D

1 (Figure 5.13). At the start of the bend,

between α 0 45o, a single vortex is calculated either side of the symmetry plane by both

the NLEVM-2 and the cubic DSM. At this distance around the bend, the location of the high

streamwise velocity is shifted towards the inside of the bend.

By α 90o a small vortex is calculated by both models on the inner (convex) wall, due to

the separation of the flow from the wall as it returns to the outer wall along the duct centreline.

The streamwise velocity contours are now altered considerably. The secondary flow along the


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top (and bottom) walls of the duct, along the inner wall and to the centreline, has convected

low W -momentum fluid from the near-wall region into the centre of the flow. This accounts

for the “hole” in the W -profiles between α 90 135o (Figure 5.6b) and low W -velocity

contour (W

W b 0 8) on the centreline at α 90o (Figures 5.12 and 5.13)

Four secondary flow vortices are calculated either side of the symmetry plane by bothmodels at α

135o. One of these vortices is located close to the symmetry plane; the flow

from the inner to the outer wall is deflected away from the symmetry plane by this vortex

causing the “mushroom” shape in the streamwise velocity contours. By α 180o the com-

plexity of the secondary flow profile has diminished with only two vortices remaining on

either side of the symmetry plane. The region of highest streamwise velocity has however

shifted from the inner to the outer wall, with the cubic DSM calculating a larger streamwise

velocity at the outer wall than the NLEVM-2 model.

5.5 Conclusions

Flow in a square cross-section U-bend has been used to demonstrate the ability of the non-

linear k ε model of Craft et al (1996b) to calculate flow with strong streamwise curvature

and streamwise vorticity. The flow has also been calculated using the cubic DSM of Craft et

al (1996a). Calculations by other authors have shown that the details of this flow cannot be

calculated accurately using a linear k ε model and wall functions. Choi et al (1989) showed

that a linear k ε model with a MLH model near-wall treatment could calculate the flow

reasonably well. The current work has shown that the non-linear k

ε model with 1-equation

(k l) near-wall treatment calculates the flow still better.

The work presented in this Chapter also shows that the non-linear k ε model is virtually

as good as the cubic DSM in calculating mean velocities and the complex, four-vortices

flow pattern around the U-bend. However, the non-linear k ε model is not as good as the

cubic DSM in calculating the Reynolds stresses as it relies on the local velocity gradients

to calculate the stresses and does not include non-local effects. In this flow, the presence

of the strong radial pressure gradient has a much greater effect on the mean velocity than

anisotropies in the Reynolds stresses. Hence, the failure of the non-linear k ε model to

calculate accurately the Reynolds stresses does not adversely affect the calculation of the

mean velocities.

The non-linear k ε model and cubic DSM were both applied as low-Reynolds-number

models with a zonal approach using a k l model to calculate the near-wall flow. In the

non-linear k ε calculations, turbulent viscosity in the near-wall model was calculated by

two approaches. Firstly, a constant value of c µ 0 09 was used in the near-wall region

5.5. Conclusions

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with the damping term in the expression for mixing length specified as Am 62 5; secondly,

the functional forn of c µ was used (Equation 2.28) with a reduced near-wall damping term,

Am 31

25. There was almost no difference between either the mean velocities or Reynolds

stresses calculated by the different methods.

Iacovides et al (1996) have studied this flow using an ASM in comparison with differentDSMs. It was found that although the ASM calculated mean velocity and Reynolds stress

profiles which were in fair agreeement with the DSM results, the instability and slow conver-

gence of the ASM meant that obtaining these results required as much computing resource as

the DSM version. This is not the case with the non-linear k ε model of Craft et al (1996b)

which is only slightly more computationally demanding than a standard k ε model and

reaches a converged solution 3 - 4 times faster than the cubic DSM.

5.5. Conclusions

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Chapter 6

Calculation of Flow in a 10o Plane

Diffuser

6.1 Introduction

Measurements of flow in an asymmetric plane diffuser with an angle of 10 o were presented

by Obi et al (1993). The experiment has proved useful and popular for assessing turbulence

models due to the smooth-wall flow separation which occurs. The diffuser angle is not so

large as to cause separation of the flow on entry to the diffuser and yet it is large enough to

generate an adverse pressure gradient which will cause a turbulent flow to separate. Separa-

tion from the inclined wall only occurs at about one-third of the distance along the diffuser.

Buice & Eaton (1997) found some deficiencies in Obi’s et al work and repeated the measure-

ments.

The flow has been calculated by Obi et al (1993), Durbin (1995) and Kaltenbach et al

(1999); it was also one of the test cases which was used at the 8th ERCOFTAC Workshop on

Refined Turbulence Modelling (Hellsten & Rautaheimo, 1999). These calculations and de-

tails of the two sets of measurements are descibed in Chapter 1. The flow has been calculated

previously with both the linear k ε model of Launder & Sharma (1974) and the non-linear

k ε model of Craft et al (1996b) which are used in the present study. The test case is in-

cluded here: firstly, as the flow has in common some of the features of flow over the rear slantof a road vehicle and secondly, as a test case for the new analytical wall function (AWF). (Pre-

vious calculations with the turbulence models used herein have been low-Reynolds-number

calculations or have used basic log-law wall functions).

94

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CHAPTER 6. Calculation of Flow in a 10o Plane Diffuser 95

6.2 Models Used

Three turbulence models have been used to calculate the flow in the diffuser: the high-

Reynolds-number linear k ε model of Launder & Spalding (1974), the low-Reynolds-

number linear k ε model of Launder & Sharma (1974) and the non-linear k ε model of

Craft et al (1996b). The realizability constraint described in Appendix A has not been used.

The linear and non-linear k ε models have been used as high-Reynolds-number models with

the simplified Chieng & Launder (SCL) wall function described in Section 2.4.3 and the new

analytic wall function (AWF) described in Section 2.4.4 and Appendix B. Calculations have

also been carried out using the low-Reynolds-number forms of the linear and non-linear k ε

models.

All calculations were run as steady-state with the first-order UPWIND convection scheme

used to provide an initial solution and the higher-order UMIST convection scheme to obtain

a final solution. The flow calculations were in two dimensions, all calculations were made

using the STREAM code (Section 3.2.3) and the flow was taken to be converged once the

residuals for mass, velocities and turbulence values had been reduced to less than 10

6.



A sketch of the 10o plane diffuser is shown in Figure 6.1. The dimensions of the diffuser

are normalised by the inlet height, H The diffuser is 21 0 H long, the outlet is 4 7 H high

and the corners at the inlet and oulet of the diffuser are rounded with radius 9 7 H . The

origin of the coordinate system used for the calculations is located on the plane wall, at the

inlet of the diffuser. The x-direction extends along the diffuser and the y-direction across the

diffuser. The model extends from x

H 11

0 upstream of the diffuser inlet to x

H 81

0

downstream.

An assessment of grid independence was carried out using grids with 125x25 cells ( x

x y-direction) and 145x50 cells which were both high-Reynolds-number calculations and

145x80 cells for a low-Reynolds-number calculation. There were only minor differences inthe calculated U -velocity and Reynolds stresses between the grids. These were attributed

to differences in the high and low-Reynolds-number models rather than the grids used. The

145x50 cell grid was adopted for the high-Reynolds-number calculations and the 145x80 cell

grid for the low-Reynolds-number calculations.

A sample of the 145x50 cell grid in the region of the diffuser is shown in Figure 6.2. (The

x-axis has been compressed for clarity.) A relatively large near-wall cell was needed due

6.2. Models Used

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to the low Reynolds number used ( Re 20 000) and the remaining cells across the diffuser

were spaced uniformly. The grid was expanded in the x-direction, upstream and downstream

of the diffuser.


The Reynolds number used in the calculations was Re 20 000, the same as used by Obi

et al (1993) and Buice & Eaton (1997). In the experiments, the flow at the inlet of the dif-

fuser was fully developed. This was reproduced for the calculations by a pre-calculation

of flow between parallel plates using periodic inlet and outlet boundaries. This provided

fully-developed profiles of U -velocity and uu-stress for the inlet of the main calculation. (A

separate pre-calculation was made for each combination of turbulence model and wall treat-

ment). At the upstream boundary ( x

H

11 0) pressure was set by extrapolation down-

stream; at the downstream boundary ( x

H 81 0) zero-gradient was set for all variables. As

the STREAM code was used, all variables in the calculation were normalised by the bulk

velocity (U o) and diffuser inlet-height ( H ). Viscosity was defined by the reciprocal of Re.

6.4 Calculated Flow Results

All calculations presented in Section 6.4 are compared to the data provided by Buice & Eaton

(1997). As the principal aim of this test case is to compare the different wall treatments (rather

than the turbulence models) and also for clarity, the linear k

ε model and non-linear k

εmodel calculations are presented separately.

6.4.1 Calculations with the Linear k ε Model

Velocities and Stresses

Profiles of U -velocity, the normal (uu, vv) and shear (uv) stresses are shown in Figures 6.3 to

6.6 respectively. Results from three calculations are shown all using the linear k ε model but

with different wall treatments: simplified Chieng & Launder (SCL, Section 2.4.3), analytical

wall function (AWF, Section 2.4.4) and low-Reynolds-number model (LRN, Section 2.2.1).

There is a small amount of variation apparent in the U -velocity and stress profiles cal-

culated by the different models. This is particularly visible in the latter half of the diffuser.

However, the variation between the profiles calculated by the different wall treatments is

much less significant than the error in comparison to the measured data.


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Coefficients of Pressure and Skin Friction

The coefficients of pressure (C P and skin friction (C F ) are calculated at each cell on the plane

and inclined wall as follows:

C P P

Po0 5

ρU 2o

(6.1)

C F

τw

0 5

ρU 2o

(6.2)

where P and τw are the pressure and shear stress calculated at a given cell on the wall, ρ is

the density (constant in the incompressible calculation) and U o and Po are reference velocity

and pressure. C F is a useful indicator of flow separation. As discussed in Chapter 1, Simpson

(1996) defines flow separation in a number of ways; one of these is the point at which the

time-averaged value of τw is zero (and hence C F 0 . This is a particularly useful indicator

in high-Reynolds-number calculations where the near-wall cell is relatively large. If there is

a small amount of separation, flow reversal will occur between the near-wall node and the

wall, and C F will be less than zero. However, the velocity calculated at the near-wall node

will still be greater than zero.

Figures 6.7 and 6.8 show calculated C P and C F for the inclined and plane walls respec-

tively. The upstream pressure gradient calculated by the linear k ε model with all three wall

treatments is greater than the measured pressure gradient. This is because Buice & Eaton

used splitter plates in the channel upstream of the diffuser to remove the end-wall boundary

layers for their measurements. The adverse pressure gradient is calculated well in the firstpart of the diffuser but calculated too high in the latter part and in the downstream plane

channel. The coefficient of skin friction (C F ) plot for the inclined wall shows a small but

definite improvement due to the AWF wall treatment. Neither the SCL wall function nor the

LRN model calculates separation, whereas the AWF wall function calculates a small negative

amount for C F in the latter part of the diffuser indicating that the flow just separates. One

might expect that the LRN model would perform better as it integrates the flow right up to

the wall. However, the LRN model must calculate the near-wall length-scale which it is not

able to do accurately. The wall functions specify near-wall length-scale through ε (SCL uses

Equation 2.76, AWF uses Equation B.37) and the AWF wall function is the more accurate of

the wall functions in this region.


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6.4.2 Calculations with the Non-linear k ε Model

Velocities and Stresses

Profiles of U -velocity and Reynolds stress calculated by the non-linear k ε model and the

three wall treatments are shown in Figures 6.9 to 6.12. The profiles of U -velocity in Figure6.9 show an immediate improvement in comparison to the U -velocity profiles calculated by

the linear k ε model (Figure 6.3). Indeed, all three wall treatments calculate a small amount

of flow separation at the inclined wall in the latter part of the diffuser, although none of them

calculate the full height or magnitude of the flow reversal. Similar improvements are apparent

in the calculation of the Reynolds stresses (Figures 6.10 to 6.12). As with the calculations

using the linear k ε model, differences in the U -velocity and Reynolds stress profiles due to

the three wall treatments are not large.

Coefficients of Pressure and Skin Friction

Coefficients of pressure and skin friction calculated by the non-linear k ε model on the

inclined and plane walls are shown in Figures 6.13 and 6.14. There is a clear improvement

in the calculation of C P and C F on the inclined wall due to the use of the non-linear model

(Figure 6.13) in comparison to the linear model (Figure 6.7). C P calculated by the non-linear

k ε model in the latter part of the diffuser and downstream section reproduces the measured

data well and the values of C F calculated by all three wall treatments show flow separation

between x

H

9 and x

H

25. With each wall treatment the calculated separation occurs

slightly later than the measured separation ( x

L

7) and the calculated reattachment occurs

slightly earlier than the measured reattachment ( x

H

29).

The AWF wall function performs slightly better than the SCL wall function in calcu-

lating C F on the inclined wall through the diffuser and in the plane channel immediately

downstream of the diffuser. Neither wall function performs as well as the LRN model which

calculates the magnitude of C F in the diffuser and the rise in C F at reattachment more accu-

rately. (The LRN model does however, still calculate a too-short region of separated flow.)

Along the plane wall there is a similar improvement in the calculation of C P and C F due to

the non-linear model (Figure 6.14) compared to the linear model (Figure 6.8).It is particularly noteworthy that the LRN model calculation is the most accurate of the

non-linear k ε model calculations, whereas with the linear k ε model the calculation with

the AWF wall function was more accurate than the LRN model. This demonstrates that

integration of the flow up to the wall is a more accurate technique than using a wall function

for this case of smooth wall separation. However, to achieve this result it is essential to

use a sufficiently accurate stress-strain relationship such as employed by the non-linear k ε


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model.

6.5 Conclusions

Flow in a plane diffuser has been used to demonstrate improvements in the flow calculation

due to the new analytical wall function (AWF). Also, this test case has been used to demon-

strate the relative capabilities of the linear and non-linear k ε models to calculate smooth

wall separation and reattachment in an adverse pressure gradient.

Calculations with the linear k ε model do not reproduce measured profiles of velocity or

Reynolds stress particularly well. In only one of the cases considered (calculation with AWF

wall treatment) was the linear model able to calculate separation of the flow on the inclined

wall of the diffuser. The non-linear model calculates improved profiles of U -velocity and

Reynolds stress through the diffuser; it also leads to flow separation on the inclined wall forall wall treatments considered herein.

Calculations with the new AWF wall function show a definite improvement over calcula-

tions with the SCL wall function. The AWF wall function calculates some flow separation in

the diffuser where the SCL wall function is not able to calculate any (when calculated with

the linear k ε model). When the wall functions are used in conjunction with the non-linear

k ε model flow separation is calculated by both the AWF and SCL wall functions but the

AWF calculates C F better on the inclined surface of the diffuser.

6.5. Conclusions

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Chapter 7

Ahmed Body Flow Calculation

7.1 Introduction

The Ahmed Body is a simplified car geometry which is widely used as a test case for compu-

tational models in the automotive industry. The geometry was originally defined by Ahmed

et al (1984); the development of bluff body test cases for the automotive industry is discussed

in Chapter 1. The principal features of the geometry are: a bluff nose cone with well-rounded

front edges, a long mid-section and interchangeable rear sections with varying slant angles.

The well-rounded front edges prevent separation of the flow at the front of the body. The

long mid-section has two functions: firstly, to allow the development of turbulent boundary

layers and secondly, to reduce the influence of the upstream impinging flow on the flow over

the slant. Interchangeable rear sections with different slant angles were used by Ahmed et

al (1984) to study the influence of rear-slant angle on drag and to find the critical angle at

which the drag crisis occurs. The body is supported on stilts and is in close proximity to

a ground-plane. The Ahmed Body is shown in Figure 1.11; Figure 1.12 shows Ahmed’s et

al breakdown of the measured drag coefficient and Figures 1.13 and 1.14 show the complex

wake formed at β

20o and 30o respectively.

Ahmed et al showed that a drag crisis occurs for this geometry when the rear slant angle

is βc 30o. At rear slant angles less than this, the flow over the slant is predominantly

attached (Figure 1.13) and characterised by strong side-edge vortices. At rear-slant angleslarger than the critical value, the flow over the rear slant is fully separated. In order to establish

whether these two flow modes could be calculated accurately, two rear slant angles, β 250

and 35o were here chosen for calculation. These were thought to be close enough to the

critical angle to show the flow processes which determine separated or attached flow, and

yet sufficiently far from the critical angle that converged solutions could be obtained without

the risk of the flow switching between modes in mid-calculation. In conjunction with this

100

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CHAPTER 7. Ahmed Body Flow Calculation 101

work and under the auspices of the same EU project1, LSTM Erlangen have conducted new

flow measurements using the original Ahmed Body (Lienhart et al, 2001). LSTM have used

laser-doppler anemometry (LDA) to measure the mean velocities and velocity fluctuations

for the Ahmed body with rear-slant angles of 25o and 35o at a Reynolds number of 7.86x105

(based on a mean upstream velocity, 40 ms

1

and the body height, 0

288 m). A detailed setof measurements has been obtained for the body with 25 o rear-slant angle and a reduced set

for the body with the 35o rear-slant angle. In addition to the LDA measurements, hot-wire

anemometry (HWA) was used to measure velocities upstream of the body and in the body’s

boundary layers. LSTM also measured pressure on the rear slant and base of the body.

In this chapter, flow-field variables (velocities, pressures, turbulence values) calculated

for geometries with 25o and 35o rear-slant angles are compared to the recent measurements

by LSTM Erlangen. Comparison of the calculated flow with the detailed measurements taken

by LSTM for the body with the 25o slant are made, including:

inlet profiles

upstream flow on the centreline

flow in the boundary layer over the mid-section of the body

flow on the centreline over the slant

off-centreline flow over the slant

flow in the wake

pressure on the slant and base of the body

Less detailed measurements were taken of the flow around the body with 35 o slant and the

following comparisons between calculated and the LSTM flow measurements are made:

flow on the centreline over the rear of the body, slant and wake

pressure on the slant and base of the body

In addition, bulk flow quantities, in particular the breakdown of coefficient of drag, are com-

pared to the original measurements of Ahmed et al (1984).

1EU-BRITE-EURAM III, Project No. BE97-4043, Contract No. BRPR-CT98-0624

7.1. Introduction

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7.2 Models Used

Two turbulence models have been used to calculate the Ahmed body flow with each rear

slant: the linear k ε model of Launder & Spalding (1974) and the non-linear k ε model of

Craft et al (1996b). The linear k ε model used the realizability constraint on µt (Appendix

A). This prevented µt from becoming excessively large in regions of very low turbulence and

inviscid deflection of the flow. The viscosity limiter is not required with the non-linear model

as this uses a functional form of c µ (Equation 2.28) which reduces µt in regions where the

strain and vorticity invariants (S and Ω) are high.

Both the linear and non-linear calculations have been made using the simplified Chieng

& Launder (SCL) wall function described in Section 2.4.3 and the new analytic wall function

(AWF) described in Section 2.4.4 and Appendix B2. Consequently there are in total, eight

calculations of the Ahmed body flow using all combinations of: two geometries, two tur-

bulence models and two wall functions. All calculations were initially run as steady-state,

using the first-order UPWIND convection scheme (Section 3.1.2). Once the residuals for all

variables had been reduced by two to three orders of magnitude, the higher order UMIST

convection scheme was used to obtain a final solution.

In some cases it was not possible to gain a stable solution using the UMIST scheme to

calculate both the velocity and turbulence variables. There are two possible reasons why

a stable solution could not be obtained. Firstly, as discussed in Section B.2.1, it was not

possible to implement the analytical wall function in the solver as robustly as log-law type

wall functions (eg the simplified Chieng & Launder wall function). The contribution to the

momentum equations of the AWF wall function is placed in the constant (S C ) rather than

the coefficient (S P) part of the linearized source. Secondly, the grid contained some skewed

cells, with corner angles of approximately 45o. Skewed cells increase the contributions to

the “cross-diffusion” sources S CD1

and S CD2

(Section 3.2.3). Skewed cells in the grids were

necessary due to the limited number of cells which could be used (see Section 7.3.2) and the

complexity of the geometry. The skewed cells were mainly in front of the body in the region

between the nose cone and the floor. There were also some near the side edge of the rear

slant, which is a critical region for the formation of the longitudinal vortices (more over the

β 35o rear slant than the β 25o rear slant). These source terms (S C S CD1 S CD2 ) reduce the

diagonal dominance of the matrices of discretized equations, thus reducing the stability of

the calculation.

2Low-Reynolds-number calculations have not been made for the Ahmed body as they were for the flow in

a U-bend of square cross-section (Chapter 5) and flow in a plane diffuser (Chapter 6). This was to reflect the

current industrial practice for calculating road vehicle flows using wall functions and very large grids (

106

cells).

7.2. Models Used

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Rear Slant Model Wall Function Mean Velocities Turb. Variables

25

Linear SCL UMIST UMIST

AWF UMIST UMIST

Non-linear SCL UMIST UMIST

AWF UMIST UPWIND

35 Linear SCL UMIST UMISTAWF UMIST UPWIND

Non-linear SCL UMIST UPWIND

AWF UMIST UPWIND

Table 7.1: Convection schemes used in final calculations of Ahmed body flow.

For the cases in which these numerical instabilities occurred, converged solutions were

obtained using the UMIST scheme to calculate mean velocities and the UPWIND scheme to

calculate the turbulence variables. The schemes used for each case are summarized in Table

7.1.

In a couple of cases (which will be identified later) the solution gained from using the

UMIST convection scheme was run on further as a time-dependent calculation using a non-

dimensional time-step, t 0 1. This attempted to reduce mean flow and turbulence residuals

still further and to establish whether there were any periodic fluctuations in the flow which

were affecting the calculations’ convergence.

All calculations were made using the STREAM code (Section 3.2.3).


7.3.1 Domain and Coordinate System

The experiments by Ahmed et al (1984) and recent measurements by LSTM show that there

is no large-scale periodicity in the flow, and no vortex shedding. Hence it is possible to

calculate the flow using a half-domain with a symmetry plane along the centreline of the

body. The half-domain used for all calculations extends one body length in front of the nose

of the body, and four lengths downstream of the base of the body. It is one body-length wideand one body-length high. The domain is shown in Figure 7.1. The origin of the coordinate

system is on the floor and centreline, at the base of the body. (This is the same reference

origin as used by Ahmed et al and LSTM). In Figure 7.1 and throughout this chapter, all

dimensions are normalised by the body height (0.288m).


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No. Blocks No. Cells No. Nodes

25 standard 22 331,000 545,000

35

standard 22 355,000 577,000

25

coarse 22 158,000 294,000

Table 7.2: Ahmed Body Grids

7.3.2 Grids

Separate grids were generated for the two geometries; these varied principally in the grid

structure over the rear slant, but there were also minor differences in the grid upstream and

around the front of the body. The accurate representation of the nose cone of the Ahmed

body is important; it ensures that the approaching flow is distributed accurately around the

top, side and underneath of the body, without separation from the front edges of the body.

To this end, the original surface data of the nose cone was obtained3 and used to define the

surface of the body for generation of the grids.

The block structure used in each grid was identical with a total of 22 blocks used. The

total number of cells and nodes used in each grid are shown to the nearest thousand in Table

7.2. Note that there are significantly more computational nodes than cells in these calcula-

tions. STREAM assigns additional “halo” cells around each block to provide the inter-block

connectivity (Section 3.2.3). The “legs” (or “stilts”) on which the model is supported in the

wind-tunnel experiments have not been included in the grid geometries. These would have

caused complications for the block and grid generation strategy and further increased thenumber of nodes required. The effects on the flow of ignoring the legs are addressed later in

this chapter.

Due to the large number of nodes required in the Ahmed body calculations, it was not

possible to refine the grids and establish grid independence. However, an additional, coarse

grid was generated for the 25o rear-slant geometry to investigate the effect of grid coarsening

and provide some information regarding grid independence.

After initial calculations all the grids were modified to ensure that the y values of as

many as possible near-wall cells surrounding the body lay in the region 55 to 550. (It is not

possible to maintain this criterion in regions of stagnation, separation or reattachment). The

y values of the near-wall cells adjacent to the ground plane were not controlled. Forcing

y 550 in these cells, particularly far downstream of the body, results in very high aspect

ratio cells which affects the stability of the calculation. Samples of the final grid used for the

25o rear-slant calculations are shown in Figures 7.2-7.4.

3The data file was provided by LSTM Erlangen (Lienhart et al, 2001).


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The floor of the domain and the Ahmed body itself are treated as wall boundary conditions,

using either the simplified Chieng & Launder wall function (SCL) or analytical wall function

(AWF) described in Section 2.4. The domain boundary along the centreline of the body

( y

L 0 0), the opposite boundary at the outside limit of the domain ( y

L 3 625) and

the upper domain boundary ( z

L 3

625) were all treated as symmetry planes. Ideally the

upper domain boundary and domain boundary opposite the centre-plane would be treated

as “entrainment” boundaries. Symmetry planes were used instead to provide a more stable

calculation and are justifiable as there is very little deflection of the flow at these boundaries 4.

The downstream outlet is set with zero-gradient for all variables. At the upstream inlet,

pressure is set by linear extrapolation from the flow domain.

As STREAM is a non-dimensional code and the flow is incompressible, both the in-

let U -velocity and density are given the value 1

0. Viscosity is set by the inverse of theReynolds number and inlet values are specified for turbulent kinetic energy (k in

and tur-

bulence dissipation rate (εin). Profiles of velocity and normal stress 400mm in front of the

nose ( x

L

5 014 measured according to the coordinate system described in Section 7.3.1)

were provided by LSTM. As the calculated results were to be compared primarily to LSTM’s

measurements, these profiles were used to define the inlet Reynolds number and turbulence

values. The velocity profiles showed that at x

L

5 014 there was a deflection of the flow

due to the blockage effect of the body. This resulted in a lower mean U -velocity (38.51ms

1)

approaching the body across the height of the calculated domain than the experimental bulk

U -velocity (40.0ms

1 . The bulk Reynolds number for the calculation was adjusted accord-

ingly to Re 7 57x105 although the Reynolds number is sufficiently high that this minor

adjustment should not have significantly influenced the results.5

Inlet turbulent kinetic energy was calculated from LSTM’s normal stress profiles by:

k in

1

2

uu vv

ww (7.1)

The mean values of uu

vv

ww over the measured inlet profiles were used to calculate a

non-dimensional value for inlet turbulent kinetic energy: k in 4 39x10

6. Inlet turbulence

4Whereas a symmetry boundary condition applies zero gradient to all variables except the normal velocity

which is set to zero, the entrainment boundary condition sets zero gradient on all variables.5Ahmed et al (1984) used a bulk Reynolds number of Re

1 18x106 based on the body’s height for their

measurements.


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dissipation rate was calculated by:

εin

c3 4

µ k 3 2

in

l(7.2)

The standard value of c µ 0 09 and, in the absence of any other data, an arbitrary value forl

0

07 L as suggested by Versteeg & Malalasekera (1996) were used. L is the reference

length used to calculate the Reynolds number of the flow - ie the height of the body. (In this

non-dimensional case, L 1 0). This provided a value: εin 2 16x10

8. It is somewhat

unrealistic to link the upstream dissipation rate to the dimensions of the body. However,

a cross-check was made by examining the turbulent to molecular viscosity ratio: νt

ν

60

which was thought to be a reasonable value. As LSTM had provided profiles at x

L

5 014

and the inlet was actually located at x

L

7 097, the inlet turbulence kinetic energy was

modified to k in 4

44x10

6 assuming a constant value of εin between the inlet and profile

locations. k in and εin were both set as flat profiles at the inlet.

At a later date, LSTM provided an estimate of the Taylor micro-scale of the flow upstream

of the body: λ 2

6mm, where the turbulence dissipation rate is defined:

ε

2k

λ2 ν (7.3)

The inlet dissipation rate was recalculated: εin 1 427x10

7 and the viscosity ratio: νt

ν

10.

This information was received too late in the project to implement the new boundary condi-

tion for all the calculations. It was used on one test calculation and shown to have no impacton the calculated flow around the body (demonstrated in Section 7.4.2).

7.4 25o Slant - Flow Field Results

7.4.1 Flow Upstream and Impinging on Body Nose

Upstream of the body, profiles of velocities and turbulent kinetic energy for three calcu-

lations (linear k ε /SCL; non-linear k ε /AWF; linear k ε /SCL/coarse grid) are shown

in Figures 7.5 and 7.6, compared to LSTM’s measured inlet data. The inlet profiles are at

x

L

5 014 - ie. 400mm in front of the body - and at three locations across the flow:

y

L 0 00 0 694 1 389. The location of the outside edge of the body is y

L 0 675.

The calculated profiles are all in good agreement with the measured data, providing con-

fidence that the inlet conditions and free-stream calculations are accurate. The U -velocity

profiles show the inviscid deflection of the flow, particularly on the centreline ( y

L 0

00)

7.4. 25 o Slant - Flow Field Results

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which is caused by the blockage effect of the body. The measured V -velocity on the cen-

treline shows a slight asymmetry in the flow, as there is a small measured amount of flow

across the y

L 0

00 plane. Naturally, this cannot be calculated as a symmetry condition is

imposed at the centreline.

Figures 7.7 to 7.9 show profiles for the same three calculations (linear k

ε /SCL; non-linear k ε /AWF; linear k ε /SCL/coarse grid) showing profiles of U and W -velocity and

turbulent kinetic energy on the centre-plane at various locations upstream and on the front

portion of the body. The velocity profiles all match the measured data well throughout this

region, with very little (if any) deviation between the calculations. The close agreement

between the standard and coarse grid calculations demonstrates grid independence in the

upstream region.

Differences between the linear and non-linear calculations become readily apparent in

the turbulent kinetic energy profiles (Figure 7.9). Both calculations which use the linear k ε

model show an increase in turbulent kinetic energy in the profile immediately upstream of

impingement ( x

L

3 7). This increased level of turbulent kinetic energy is convected

downstream and accounts for the high levels of turbulent kinetic energy near the surface of

the body in the last two profiles in Figure 7.9. The increase in turbulent kinetic energy is due

to the linear k ε model’s inability to calculate Pk accurately at impingement. This has been

discussed in detail for the case of flow around a square cylinder in Chapter 4.

Two sets of LSTM’s measured data are shown in Figure 7.9. The triangles show LDA

measurements which were made on the centre-plane whereas the crosses show the upstream

data which was measured using HWA in a separate experiment. These two datasets shareone common location: x

L

5 014; y

L 0

00. The data shown as triangles (centre-plane

data) is about an order of magnitude greater than that shown as crosses (upstream profiles) at

this location. Lienhart et al (2001) attribute the differences in the measured turbulent kinetic

energy to the different measurement techniques used. LDA is better suited to measurements

in highly turbulent regions, including reverse flow, whereas HWA is better suited to regions

of low turbulence. Upstream of the body, there is little turbulence in the flow and the HWA

measurements will be more accurate than the LDA measurements. The calculations are in

better agreement with the HWA data.

The flow upstream of the body, at impingement and on the front section of the body

is essentially inviscid. This is shown by the close agreement between the velocity profiles

for the different calculations despite the different turbulence models used and differences in

calculated turbulent kinetic energy.


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7.4.2 Boundary Layer Flow on Body Mid-Section

Profiles of U -velocity and uu normal stress in the boundary layer on the upper surface of

the body, part-way between the nose and slant ( x

L

1 736) and at three locations across

the body width ( y

L 0 00 0 278 0 556) are shown in Figure 7.10. LSTM’s HWA data

is compared to four calculations (linear k ε /SCL; non-linear k ε /SCL; non-linear k ε

/AWF; non-linear k ε /AWF /modified εin described in Section 7.3.3). In each plot the line

at z

L 1

174 denotes the upper surface of the body.

There is almost no variation between the different calculations of U -velocity although

the calculated velocities outside the boundary layer are somewhat greater than the measured

velocity. This is due to differences between the wind tunnel and the calculation domain.

The wind tunnel used for the measurements is a 3/4 open-section wind tunnel - ie. the only

wall bounding the flow is the floor. In this configuration mass is not necessarily conserved

between the wind tunnel outlet (upstream of the body) and wind tunnel intake (downstreamof the body). On the contrary, the domain used for calculations is bounded by the floor and

three symmetry planes and mass is conserved between the inlet and outlet of the computa-

tional domain. Around the body, the calculated flow is accelerated due to the reduced cross-

sectional area, which accounts for the calculated U velocity being higher than the measured

U -velocity around the body.

Differences between the calculations are apparent in the uu profiles. The increased tur-

bulent kinetic energy which is calculated by the linear k ε model and convected around

the body (Figure 7.9) has dissipated. Outside the boundary layer

z

L 1

25 , all the cal-

culations of uu are in good agreement with the HWA data providing confidence that tur-

bulence has been calculated accurately in this region and that the upstream high levels of

LDA-measured turbulent kinetic energy are indeed inaccurate (the triangles in Figure 7.9).

The measured profiles of uu normal stress show that there is a thickening of the boundary

layer at the centreline of the body ( y

L 0 00) compared to near the side ( y

L 0 556).

Differences in uu calculated by the different turbulence models are also apparent. At the cen-

treline, the non-linear model calculates the peak value of uu in the boundary layer reasonably

well. With the SCL wall function, the peak value is calculated slightly too high and with

the AWF wall function, it is calculated slightly too low. However, all the models calculatea too thick boundary layer. Towards the side of the body, differences in the thickness of the

boundary layer calculated by each model are less apparent.

The calculated results for uu in the boundary layer suggest that the near-wall node was

too far from the wall. Over the top surface of the body, the distance of the near-wall node

from the wall was 100

y

200 for all cells and all calculations. Whilst this is towards

the higher end of the range which is normally considered acceptable, it is certainly within the


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range which is generally used for three-dimensional calculations.

Figure 7.11 shows profiles for the same four calculations (linear k ε /SCL; non-linear

k ε /SCL; non-linear k ε /AWF; non-linear k ε /AWF with the modified εin) underneath

the body. z

L 0 00 is the location of the ground plane and the line at z

L 0 174 shows

the location of the lower surface of the body. Similar trends in the U -velocity and uu-stressprofiles as noted above the body are apparent.

A significant difference between the calculated and measured results can be seen in the

profiles at y

L 0 556. This position is towards the outside edge of the body - the outside

edge is located at y

L 0

675. Here the calculated U -velocity is too high and the calculated

uu-stress is too low. In the experiments the body’s stilts are located at x

L

2 924 (front

stilts) and 1 292 (rear stilts) and y

L 0

568. Hence this profile is between the front and

rear stilts and at the approximate lateral location of the stilts. As the stilts are not included

in the calculation their effect in reducing U velocity and increasing turbulence levels cannot

be calculated.

There is virtually no difference between values of U -velocity or uu-stress calculated by

the non-linear model with the original and modified values of εin (Figure 7.10 and 7.11).

Thus it is assumed that the difference in this inlet condition does not have a significant effect

on the flow around the body. Calculations with the modified value of εin described in Section

7.3.3 will not be considered any further.

7.4.3 Flow Over Rear Slant

Comparisons with Detailed Measurements on the Centreline

Note: Detailed LDA measurements over the rear slant were taken on the centre-plane ( y

L

0 00). Only two components of velocity (U W ) and fluctuating velocity were measured,

providing uu ww

uw Reynolds stresses only. (It was not possible to measure the lateral

velocity fluctuation, v, close to the surface, Lienhart et al, 2001)

Calculations with SCL Wall Function Figures 7.12 and 7.13 show velocity profiles over

the rear slant for three sets of calculations (linear k ε /SCL; non-linear k ε /SCL; linear

k ε /SCL/coarse grid). The measured results show a small amount of separation near the

start of the slant but by x

L

0 35 the flow is fully attached.

Regarding the U -velocity profiles in Figure 7.12, the linear k ε model performs well

over the whole slant, both within the boundary layer and in the free-stream region. However,

this calculation shows a very slight tendency to separated towards the end of the slant. In

comparison to the standard grid calculation, the linear k ε model using the coarse grid


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performs slightly less well in the boundary layer on the first half of the slant. On the latter

half of the slant, the flow calculated with the the coarse grid shows less tendency to separate

than the standard grid calculation. Refining the grid improves the calculated flow on the

upper half of the slant, but appears to worsen it on the lower half. Apparently, in this region

the “artificial viscosity” induced by the coarse grid helps maintain attached flow.The non-linear k ε model performs much less well than the linear k ε model; there is

a large amount of separation along virtually the whole slant. Separation is initially calculated

by the non-linear model near the start of the slant which is also apparent in the measurements.

However, instead of reattachment occurring by x

L

0 35 (as the measurements show),

the separation region grows. The excessive amount of separation calculated by the non-linear

model is similar to that observed for the square cylinder described in Chapter 4.

The W -velocity profiles in Figure 7.13 show the same trends as are seen for the U -velocity

profiles. The linear k ε calculation performs well; the non-linear k ε calculation performs

less well. Due to the large separated flow region calculated by the non-linear model, the

velocity vectors are nearly horizontal. Consequently, the magnitude of the W -velocity is

much smaller than that calculated by the linear model.

Three Reynolds stresses (uu

ww

uw) are shown in Figures 7.14 to 7.16 respectively. For

all three sets of calculations, the Reynolds stresses are not well predicted, which goes some

way to explaining the tendency to calculate flow separation towards the end of the slant. The

measurements show a large amount of Reynolds stress generated between x

L

0 64 and

0 50 due to the shear layer originating at the start of the slant. The high Reynolds stress is

convected downstream and gradually dissipates towards the end of the slant.Figure 7.10 shows that both the linear and non-linear k ε models calculate the stream-

wise normal stress (uu) reasonably accurately in the boundary layer over the top of the body

(certainly to the correct order of magnitude). It is peculiar that there should develop so rapidly

an order of magnitude difference between the measured and calculated Reynolds stresses at

the start of the slant. It is also hard to imagine that the linear k ε model can calculate such a

good agreement with the measured velocity profiles if the computed shear stress is so much

in error.

In an attempt to understand this better, comparison is made to a simpler flow in an adverse

pressure gradient. In Table 7.3 comparison is made of the approximate values of maximum

normalised Reynolds stresses6 which have been measured in the 10o plane diffuser (Chapter

6: Figures 6.10 to 6.12) with the values measured over the rear slant of the Ahmed body

(Figures 7.14 to 7.16). For all the stresses, the values measured over the rear slant of the

6The vertical direction in the plane diffuser is the y-direction and z-direction in the Ahmed body hence

vv Di f f user equates to ww Ahmed etc.


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be detected by the LDA measurements. The sampling rate was too low in comparison to the

expected frequency of oscillation and the LDA technique would not be able to distinguish

between turbulent fluctuations and periodic motion.

Calculations with AWF Wall Function U -velocity and uu-stress profiles are shown inFigures 7.17 and 7.18 for two sets of calculations (linear k ε /AWF; non-linear k ε /AWF).

As would be expected, there is no readily apparent difference between the calculated velocity

profiles shown in Figures 7.17 and 7.18, and those for the corresponding models using SCL

(Figures 7.12 and 7.14). In the plane diffuser (Chapter 6) it was shown that the AWF was

able to improve the calculation of separated flow over the SCL wall function treatment. The

plane diffuser is a relatively simple, two-dimensional flow whereas the flow over the rear

slant has more complex three-dimensional effects acting. As well as being controlled by the

shear layer originating at the start of the slant, the flow in this region is strongly influenced

by the side-edge vortices (discussed in Sections 7.4.4 and 7.4.5). Differences due to the wall

functions are more apparent when considering surface effects such as wall shear-stresses and

pressure coefficients used in the calculation of drag (Section 7.6). Both the SCL and AWF

wall functions assume that the mean velocity vector at the near-wall node is in the same

direction as the surface shear stress. Hence no account for near-wall skewing of the flow is

made and a sensitivity study of the flow compared with the thickness of the near-wall cell

would be required to assess the influence of near-wall skewing.

Time-Dependent Calculations Experience from the square cross-sectioned cylinder cal-culations (Chapter 4) showed that minor fluctuations in the wake, which are attenuated by the

linear k ε model, may remain and cause a lack of convergence in a steady-state calculation

with the non-linear model. To establish if this were also the case with the Ahmed body, time-

dependent calculations were made using the linear and non-linear k ε models (both with

SCL). These calculations used the fully implicit time-discretisation scheme (Section 3.1.8)

with a non-dimensional time-step of ∆t 0 1. Improvements to the levels of convergence

which were obtained are shown by typical residual values (Table 7.5) which were recorded

at the end of the steady-state and time dependent calculations.

In general the time-dependent calculations were able to reduce the residuals by an order

of magnitude on most variables (two orders of magnitude on k and ε). U -velocity and uu-

stress profiles are shown in Figures 7.19 and 7.20 for the following calculations: linear k ε

time-dependent, non-linear k ε steady-state and non-linear k ε time-dependent (all with

SCL). There is no improvement in the linear calculation by switching to time-dependent cal-

culations, but a small change in the velocity and Reynolds stress profiles from the non-linear


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Calculation Velocity Mass Turbulence

linear k ε; steady 7x10

5 4x10

6 2x10

5

linear k ε; time-dependent 1x10

5 7x10

7 2x10

7

non-linear k ε;steady 1x10

4 5x10

6 1x10

5

non-linear k ε; time-dependent 8x10

5 1x10

7 3x10

7

Table 7.5: Comparison of residual values for linear and non-linear, steady-state and time-

dependent calculations.

k ε time-dependent calculation. This implies that there is a small amount of fluctuation in

the wake of the Ahmed body calculated by the non-linear k ε model which inhibits con-

vergence of the steady-state calculation, and demonstrates that this cannot be considered as

a true steady-state problem. (Note: Figures 7.19 and 7.20 show instantaneous rather than

time-averaged quantities.)

Tests on Realizabililty and Yap Correction As the non-linear k ε model performed

poorly, it was necessary to study the influence of non-standard features which had been in-

cluded in the model. The two non-standard features used in the non-linear k ε model were

the realizability condition (described in Appendix A) and a “Yap correction”. The realizabil-

ity condition which prevents excessively large values of turbulent viscosity from occurring

is not required for the non-linear model. The non-linear k ε model uses the functional

form of c µ (Equation 2.28) which effectively provides realizability by reducing the value of

c µ (and hence turbulent viscosity) in regions of high strain rate or vorticity. The “Yap cor-

rection” (Yap, 1987) is used in low-Reynolds-number k ε models to reduce the near-wall

length-scale by introducing an additional source term to the ε-equation:

S ε max

0 83

k 3 2

2 5ε y

1

k 3 2

2 5ε y

2

ε2

k 0

(7.4)

Due to the use of the wall-normal distance, y, the “Yap correction” is only effective very

close to the wall. When a wall function is used in a high-Reynolds-number model, the “Yap

correction” should not be particularly influential.

Two calculations were made with the non-linear k ε model (SCL, time-dependent) to

establish whether these non-standard features had affected the calculated flow. The first re-

moved the realizability constraint; the second removed both the realizability constraint and

the “Yap correction”. There were no significant differences between these calculations and

the original non-linear k ε time-dependent calculation. (No figure is shown). The inclusion


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of the realizability constraint and “Yap correction” did not adversely effect the non-linear

k ε calculation.

Tests on Convection Scheme Initial calculations were made with the first-order UPWIND

convection scheme to establish the basic flow pattern before switching to the higher-orderUMIST convection scheme. Little difference in the flow around the body was noted with the

linear k ε model when switching between the UPWIND and UMIST convection schemes.

There was however, a marked difference when the non-linear k ε model was used.

Figure 7.21 shows U -velocity profiles for the steady-state linear k ε /UMIST, non-linear

k ε /UMIST and non-linear k ε /UPWIND calculations. Being only first-order accurate,

the UPWIND scheme is more diffusive than the UMIST scheme, adding artificial viscosity

to the flow. Hence the flow calculated by the non-linear model and UPWIND scheme re-

mains attached at the start of the slant, whereas the non-linear model and UMIST scheme

calculates a small separation. More significantly, the flow calculated by the non-linear model

and UPWIND scheme remains attached along the whole length of the slant, whereas with the

UMIST scheme it is separated.

The UMIST convection scheme is formally more accurate than the UPWIND scheme.

This comparison is made to demonstrate the action of viscosity (“real” or “artificial”) on

the flow over the slant, rather than to advocate the use of first-order convection schemes. In

Chapter 1 the critical influence of the side-edge vortices on determining whether the flow over

the slant separates is discussed. To help explain these differences, the formation of this vortex

due to mean strain bending is discussed in Section 7.4.4 with reference to the calculation of vorticity and mean strain bending by the different turbulence models and convection schemes.

Sensitivity to c µ and Non-Linear Components of Reynolds Stresses The two features of

the non-linear k ε model of Craft et al (1996b) which make it different from the standard

linear k ε model of Launder & Spalding (1974) are the functional form of c µ (Equation 2.28)

and the inclusion of the non-linear components in the Reynolds stresses (Equation 2.27). To

establish whether one or other of these features were responsible for the separated flow over

the rear slant, two tests were made. Firstly, the c µ function was modified to the form which

is effectively used in the linear k ε model when the realizability condition is employed

(Equation A.30). This form of the function does not reduce c µ so greatly in regions of high

S or Ω as shown in Figure 2.1. Secondly, the original c µ function proposed by Craft et al

(1996b) was retained, and the non-linear components were set to zero.

The U -velocity profiles over the rear slant calculated by the two modified models are

shown in Figures 7.22 (modified c µ) and Figure 7.23 (uiu j NL 0). For each modification


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there is a small improvement in the calculated velocity profiles, but neither modification

calculates attached flow along the slant. It would appear that either the original form of c µ

or the inclusion of the non-linear components in the Reynolds stresses alone, is sufficient to

cause the flow to separate over the slant.

Comparisons with Measurements in the Wake and off the Centreline

LSTM have taken LDA measurements of mean and fluctuating velocity components for a

number of profiles on the wake of the Ahmed body and off-centreline. These lie in planes

between the centreline and the outside edge of the body: y

L 0

00

0

347

0

625 and 0

675.

(This last plane being the plane of the outside edge of the body.) In general these confirm the

results discussed in Section 7.4.3.1, and only a brief selection is now shown.

Figures 7.24 and 7.25 show U and W -velocity profiles on the centreline compared to

the LDA masurements. Calculated values from the linear and non-linear k

ε models areshown both using the SCL wall function and higher order UMIST convection scheme. (Figure

7.24 is essentially the same as Figure 7.12 but extended into the wake.) The U -velocity

profiles in Figure 7.24 show that the linear k ε model calculates the mean velocity more

accurately than the non-linear model in the wake, as well as over the slant. The non-linear

model calculates U -velocity profiles in the wake similar to those seen in the wake of the

square cylinder (Chapter 4): the recirculation region is too high in the z-direction and the

magnitude of the reverse velocity is too low.

Reynolds normal and shear stresses (uu and uw) are shown at the same locations as the

velocities in Figures 7.26 and 7.27. Over the rear section of the slant, the linear k ε model

calculates the correct shape of the distribution of uu and uw, although the magnitudes of the

Reynolds stresses are too low. The non-linear model calculates a separated flow in this region

and the peak in the Reynolds stresses above the slant is due to the shear layer in the region of

the separation streamline.

Figures 7.28 to 7.30 show profiles of U , V and W -velocity for the same two models at

y

L 0 625 (near the outer edge). The notable feature is the near-wall “bulge” in the U -

velocity profiles towards the end (Figure 7.28) and the more vigorous motion shown by the

V -velocity profiles (Figure 7.29). These indicate the presence of the side-edge vortex. Again,the linear k ε model calculates this feature more accurately than the non-linear model,

although the magnitude of the calculated V -velocity is too low, probably due to the use of

wall functions. Formation of the side-edge vortex is discussed in Section 7.4.4. Reynolds

stresses (uu and uw) are shown in Figures 7.31 and 7.32. In general the Reynolds stresses

are much lower towards the side of the body than at the centreline. The non-linear model

calculates levels of Reynolds stress in the wake at y

L 0

625 slightly better than the linear


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model.

7.4.4 Vortex Formation

It is known that the occurrence of attached or separated flow on the rear slant of the Ahmed

body is strongly influenced by the formation of the side-edge vortex (Ahmed et al, 1984).

From the results discussed so far in Chapter 7, it is apparent that the calculation of separated

or attached flow, and consequently the calculation of the side-edge vortex, is dependent upon

the turbulence model and convection scheme used. To help understand the formation of the

vortex, it is useful to consider the equation for streamwise vorticity, Ω x:

DΩ x

Dt

(7.5)

In the above, vorticity components are Ω x ∂V

∂ z ∂W

∂ y

, Ω y ∂W

∂ x ∂U

∂ z

and Ω z ∂U

∂ y ∂V

∂ x

The principal source term for the streamwise vorticity is the mean strain bending. There will

also be contributions from turbulent normal stress and turbulent shear stress generation in

the non-linear k ε model (which are zero in a linear model). Figure 7.33 shows the mean

strain bending at the start of the slant ( x

L

0 701) calculated by the linear k ε model,

and the non-linear k ε model using the UMIST and UPWIND convection schemes. Both

the linear model and the non-linear model/UPWIND calculate higher values of mean strain

bending than the non-linear model/UMIST. Figure 7.34 shows the calculated mean strain

bending half-way along the slant ( x

L

0

347). Here, the difference is more marked withthe non-linear model/UMIST calculating significantly less mean strain bending than the other

two models.

As a relatively high value of the source term for streamwise vorticity is calculated by

the linear model and non-linear model/UPWIND, these two models calculate more stream-

wise vorticity than the non-linear model/UMIST computation. Figures 7.35, 7.36 and 7.37

show the calculated streamwise vorticity for the three models at the start of the slant ( x

L

0 701), half-way along the slant ( x

L

0 347) and at the end of the slant/base

x

L

0 00). It is apparent that, despite calculating separated flow over a large portion of the rear

slant, the non-linear k ε model/UMIST does calculate a well defined side-edge vortex. At

the end of the slant (Figure 7.37) the vortex calculated by the non-linear model with the higher

order UMIST scheme is almost as strong as that calculated using the UPWIND scheme, al-

though it is less diffuse. The stronger vortex calculated by the linear k ε model and the

diffuse vortex calculated by the non-linear model/UPWIND draw fluid out of the boundary

layer on the slant near the centreline of the the body. This is sufficient to maintain attached


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flow over the slant. In contrast, the weak and well-defined vortex calculated by the non-linear

model/UMIST does not draw a sufficient amount of fluid out of the boundary layer and the

flow separates.

To help visualise the region of attached flow and influence of the side-edge vortex, Figure

7.38 shows velocity vectors calculated at the near-wall nodes by the linear k

ε model andnon-linear k ε model/UMIST. (The top surface and slant are shown viewed from above with

the side of the body “unfolded”.) The separated (reversed) flow calculated by the non-linear

model/UMIST extends across most of the rear slant with only a small region of attached flow

near the side edge, due to the influence of the vortex.

7.4.5 Wake

Secondary velocity vectors in the wake are shown at three planes downstream of the body

( x

L 0 277 0 694 1 736) for the linear k

ε model (SCL/steady) in Figure 7.39 and for

the non-linear k ε model (SCL/steady) in Figure 7.40 (both using the UMIST scheme).

These figures show the outline of the Ahmed body and location of the rear edge of the slant.

Calculated vectors are shown in the right half-plane with LSTM measured vectors reflected

onto the left half-plane.

The secondary vectors calculated by the linear model in Figure 7.39 show good develop-

ment of the wake-vortex, both in terms of its position and magnitude. Flow across the lower

half of the base of the body ( x

L 0 277) which is associated with the horseshoe vortices

(Figures 1.13 and 1.14) is not calculated. The wake vortex develops from the side-edge vortex

which is discussed in Section 7.4.4. When the flow is calculated with the non-linear model,

a relatively weak side-edge vortex is calculated which is confined to a narrow region at the

side edge of the slant. Figure 7.40 shows how the lack of a strong side-edge vortex in the

non-linear calculation affects the wake. The non-linear model calculates a small amount of

downwash in the wake at x

L 0 277 and 0.694 but the strong wake-vortex which is mea-

sured by LSTM and calculated by the linear model does not appear. At x

L 1 736 the large

measured vortex is calculated by the non-linear model but not so well as it is calculated by

the linear model.

In Section 7.4.3.1 and Figure 7.21 it is shown that by reverting to the first-order, diffusive,UPWIND convection scheme with the non-linear k ε model, an attached flow over the slant

could be calculated. This then produces a strong side-edge vortex (Figures 7.35 to 7.37)

which develops the wake vortex shown in Figure 7.41. The strong wake-vortex which is

calculated by the non-linear/UPWIND model is, in general, in as good agreement with the

measurements as the linear model with the UMIST convection scheme. This demonstrates

the effectiveness of the non-linear terms in the stress-strain relationship (Equation 2.27) in


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improving flow with streamwise vorticity, albeit with a first-order convection scheme.

Development of turbulent kinetic energy in the wake is shown for the same three calcu-

lations: linear/UMIST, non-linear/UMIST, non-linear/UPWIND (all using SCL) in Figures

7.42 to 7.44 respectively. None of the calculated turbulent kinetic energy plots show the

peaks in the turbulent kinetic energy at the centre of the vortex at the rear edge of the slant( x

L 0

277). The linear/UMIST calculation reproduces the shape and magnitude of the tur-

bulent kinetic energy contours reasonably well (Figure 7.42); this is slightly improved upon

by the non-linear/UPWIND calculation (Figure 7.44). As would be expected from the poor

calculation of the wake secondary-velocity, the non-linear/UMIST calculation of turbulent

kinetic energy is also poor (Figure 7.43).

7.4.6 Pressure on Base and Slant

Coefficient of pressure (C P) contours on the surface of the body’s base and rear slant are

shown in Figures 7.45 and 7.46 for the linear k ε model (SCL, AWF) and non-linear k ε

model (SCL, AWF) calculations. Measured coefficient of pressure is shown on the left half

of the body and calculated pressure on the right half for each case. Rather than showing the

slant in projection, it has been “unfolded” hence the base and slant have different z

L scales.

Coefficient of pressure is calculated by:

C P

P P0

0 5ρU 2o

(7.6)

where Po is the pressure at a reference point in the flow, ρ is the density and U o is the upstream

(undisturbed) velocity used in the calculation of Reynolds number.

For the linear k ε calculations, Figure 7.45 shows the effect of the side-edge vortex

on C P contours at the right-hand edge of the slant. The pressure gradient shown by these

contours is not so great as the pressure gradient due to the measured side-edge vortex (left-

hand edge), indicating that the calculated vortex is not as strong as the measured vortex.

Conversely, the calculated pressure gradient at the top edge (start) of the slant is much greater

than the measured pressure gradient. Although the shape of the calculated C P plot is similar

to the measured data, the differences in the magnitude of C P will effect the accuracy of thecalculated drag. There is a very minor improvement in the calculation of pressure when using

the AWF wall function. Across the base of the body there is little variation in the measured

or calculated coefficients of pressure, indicating that the flow is quiescent in this region.

C P

0 190 across the base for the measured and both calculated sets.

The separated flow calculated by the non-linear/UMIST model is evident from relatively

small change in C P along the slant (Figure 7.46). The weak vortex which is calculated has


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a small effect on the contours at the right-hand edge. C P calculated using the non-linear

model and AWF does not show the weak vortex. When using the AWF wall function with the

non-linear model it was not possible to use the UMIST convection scheme to calculate both

the mean velocities and turbulence values as was used with the SCL wall function (Section

7.2). Instead, the UMIST scheme was used for the mean velocities and the UPWIND schemefor the turbulence variables (Table 7.1). Subsequently the calculated side-edge vortex is not

strong enough to show any effect on the calculated C P over the slant. The calculation of base

pressure is improved slightly by using the AWF rather than the SCL wall function. (On the

base the flow is fully separated and both the linear and non-linear models calculate separated

flow here).

7.5 35o Slant - Flow Field Results

7.5.1 Flow Over Rear Slant and In Wake

For the Ahmed body with the 35o rear-slant angle, flow is fully separated over the slant

and strong side-edge vortices do not form (Ahmed et al, 1984, Lienhart et al, 2001). The

measured data used in this section to assess the calculated results is from a preliminary study

by Lienhart et al (2001). There is less data available than for the body with the 25o rear slant

and it was released before consistency checks had been carried out.

Calculations with the SCL Wall Function Figure 7.47 to 7.49 show the U and W -velocityprofiles and turbulent kinetic energy profiles on the centreline for LSTM’s measured data

and two calculations: linear k ε and non-linear k ε both using the SCL wall function.

Although the data is a lot more sparse for the 35 o case, it is clear that both the linear and

non-linear models calculate separated flow over the body’s rear slant in accordance with the

measurements (Figure 7.47).

The calculated wake flow is similar to that behind the square cross-sectioned cylinder

(Chapter 4). The linear k ε model calulates the shape of the U and W -velocity profiles well

and the length of the recirculating flow region is calculated reasonably accurately by this

model. As was seen with the square cylinder, the non-linear model calculates a recirculation

region which is too high ( z-direction) and too long ( x-direction). The U -velocity gradients

calculated by the non-linear model in the upper and lower shear layers are too steep.

Both the linear and non-linear k ε models calculate too much turbulent kinetic energy in

the wake (Figure 7.49). In the lower shear layer of the wake and close to the body, the turbu-

lent kinetic energy ought to be similar for calculations with the 25 o and 35o slants. (Here, the


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Data 25

Rear Slant 35

Rear Slant

Measured

0 10

0 015

Calculated by linear k εmodel

0 07

0 08

Calculated by non-linear k εmodel

0.05

0 04

Table 7.6: Measured and claculated values of k in the lower shear layer close to the base of

the Ahmed body with 25o and 35o rear slants.

flow is relatively unaffected by the behaviour of the flow over the slant.) Measured data is not

available at the same locations in the wake for the 25o and 35o bodies. Instead, a comparison

is made of turbulent kinetic energy at x

L

0 3 for the 25o rear-slant body (Figure 7.50)

and x

L

0 5 for the 35o rear-slant body (Figure 7.49); measured and calculated values of

turbulent kinetic energy are shown in Table 7.6:

Although there is reasonable agreement in the calculated turbulent kinetic energy betweenthe 25o and 35o bodies when using either turbulence model, there is a considerable difference

in the levels of measured turbulent kinetic energy. It is worth re-iterating that the measured

data provided by LSTM for the 35o body (Lienhart et al, 2001) was preliminary data, which

was issued before the measurements were completed in full and before consistency checks

were made. Without the complete 35o rear-slant measured dataset and a full study of the

calculated versus measured data, it is not possible to judge the accuracy of the calculated

turbulent kinetic energy.

Calculations with AWF Wall Function Calculated profiles of U -velocity and turbulent

kinetic energy using the AWF wall function are shown in Figures 7.51 and 7.52. As would

be expected the change in the wall treatment has little effect on the wake flow, which is dom-

inated by free-stream effects. The minor differences that do occur (for example, in the linear

calculation of the U -velocity profile at x

L

1 77) are due to time-dependent fluctuations

in the wake, which in this steady-state calculation are manifested as minor differences in the

level of convergence.

Tests on Realizabililty and Yap Correction Han (1989) was not able to calculate a sepa-

rated flow for the Ahmed body with rear slant 35o using a linear k ε model. Indeed, Hucho

& Sovran (1993) state that at the time of writing, no RANS code had reproduced the large

changes in flow pattern that take place at the critical slant angle. To attempt to understand

what features of the current linear k ε model cause the improvement, tests were carried out

on the non-standard features which were used. These were the “Yap correction” used in the

ε-equation (see discussion in Section 7.4.3.1) and the realizability condition (Appendix A).


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Figures 7.53 and 7.54 show the U -velocity and turbulent kinetic energy profiles for the

linear k ε calculation (SCL) with the “Yap correction” switched on and the calculation with

the realizability condition switched off. (In the non-linear calculations the “Yap correction”

was implemented as a standard feature of the model; in the linear calculations it had to be

implemented specifically.) As the “Yap correction” only has an effect very near the wall andis principally intended to improve low-Reynolds-number calculations, no change in either the

U -velocity of turbulent kinetic energy profile is gained when using it in this high-Reynolds-

number (SCL) calculation. (Apart from some minor discrepancies in the downstream wake

due to differences in convergence.)

A significant change is noted when the realizability condition is removed from the model.

Although separation still occurs over the rear slant, the velocity gradients in the shear layers

downstream of the body are much less steep. This results in a much shorter recirculation

region behind the body. When the realizability condition is used: the turbulent viscosity is

prevented from rising to too high levels in regions of high strain rate, the velocity gradients

in the wake are not reduced too greatly and the recirulation region is calculated accurately.

This demonstrates that the realizability condition is essential for accurate calculation of this

flow with a linear k ε model.

7.5.2 Pressure on Base and Slant

Coefficient of pressure on the base and rear slant is shown in Figures 7.55 for the linear k ε

model calculations (SCL, AWF) and in Figure 7.56 for the non-linear k ε model calcula-

tions (SCL, AWF). As with the coefficient of pressure plots shown for the 25o case (Figures

7.45 and 7.46) the rear slant has been “unfolded”, rather than being shown in projection.

The measured contours are shown on the left-hand side of Figures 7.55 and 7.56, with the

calculated contours on the right-hand side.

A significant difference is seen in the measured coefficient of pressure between the cases

with rear-slant angles 25o (Figure 7.45) and 35o (Figure 7.55). With the 25o rear-slant angle,

the adverse pressure gradient in the attached flow over the slant is visible in the increasing

C P contours along the slant (top-edge to base-edge). The side-edge vortex can be seen in

the contours at the left-hand side of the slant and the relatively quiescent flow at the baseis demonstrated by the lack of variation in coefficient of pressure C P on the base. With the

35o rear-slant angle, the flow is fully separated over the slant. The measured coefficient of

pressure contours show that there is no side-edge vortex influencing the pressure on the slant.

There is however, a slight pressure gradient along the slant from the slant-base edge to the

slant-top edge, due to the flow reversal. (Note that the change in C P along the slant for the

separated, reverse flow with the 35o rear-slant is ∆C P

0 03, whereas for the attached flow


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with the 25o rear-slant it is much more pronounced: ∆C P

1 18). The recirculating flow

region above the 35o slant is accompanied by a second recirculating flow region, with the

opposite sign of rotation, located behind the base. This creates upward flow at the base and a

pressure gradient across the base, ∆C P

0 08.

The calculated coefficient of pressure C P contours from the linear k

ε model (SCL &AWF) shown in Figure 7.55 appear to reproduce the shape of the measured contours reason-

ably accurately. However, the direction of the measured pressure gradient is reversed. There

is a pronounced difference between the measured and calculated C P contours on the base.

The measured data shows a pressure gradient from the bottom edge to the slant edge of the

base, demonstrating that the lower recirculation region behind the body causes upward flow

across the base. In contrast, the calculated results show a pressure gradient from the centre of

the base towards the slant, side and bottom edges indicating that the calculated flow moves

radially outward from the centre of the base.

Calculated C P contours from the non-linear k ε model (SCL & AWF) using the UMIST

convection scheme (Figure 7.56) again show the reversed direction pressure gradient but the

pressure gradient is not as strong. Similarly the radial pressure gradient across the base is not

as pronounced.

7.6 Drag

7.6.1 Measured Drag Variation with Rear-Slant Angle

The calculated drag is compared principally to the original measurements taken by Ahmed

et al (1984). Ahmed’s et al drag breakdown is shown in Figure 1.12, with the contributions

due to the rear slant (C S ), base (C

B), nose cone (C K ) and friction (C

R) all shown separately.

The total drag is denoted by C W . From the measurements of coefficient of pressure provided

by LSTM (Figures 7.45 and 7.55), new values of C S and C

B have been calculated and are

included in this section for comparison.

A description of drag generated around the Ahmed body is given in Chapter 1, though it

is worth re-iterating the salient points here. Firstly, although Ahmed et al measured the total

drag with 11 rear-slant angles varying between 0o β

40o, the drag breakdown was only

calculated from the measured surface pressures for 4 rear-slant configurations (5o

12 5o

30o

high drag and 30o low drag). Hence, between 12 5o and 30o where there is a rapid rise in drag,

Ahmed et al have assumed that the contributions due to friction and the nose cone remain

constant (a reasonable assumption). More importantly, the relative contributions of drag at the

base and the rear slant are assumed. Hence, there is no basis in fact for the relative magnitudes

7.6. Drag

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of C S and C

B between 12 5o and 30o shown in Figure 1.12 other than the interpolation from

the values at β 12

5o and 30o. By comparing the components of calculated drag to the

measurements at β 25o, uncertainties over the accuracy of the comparison will be inherent.

(This is also true at β 35o though to a lesser extent as there is little variation in drag for

β

30o

)The flow over the rear slant at angles β

30o is predominantly attached. As was shown

by the plane diffuser in Chapter 6, one would normally expect separation of a turbulent flow

in an adverse pressure gradient at slant (diffuser) angles significantly less than 30 o. It is the

action of the side-edge vortices which draw fluid out of the boundary layer on the rear slant

which cause the flow to remain attached up to β 30o. Attached flow over the slant ought to

promote pressure recovery and reduce drag. However, there is low pressure generated in the

side-edge vortices which acts to increase drag on the rear slant and this is partly the reason for

the steep rise in C S between 12 5o

β 30o (Figure 1.12). A contribution to the steep rise in

drag is also due to the small amount of separation which occurs at the start and centreline of

the slant as the rear-slant angle approaches 30o (Figure 1.14). This inhibits pressure recovery.

7.6.2 Calculated Drag

The methods by which the drag is calculated and modifications to account for the stilts are

described in Appendix E. Note that throughout this Section, measured values of drag taken

from Ahmed et al (1984) are stated to the nearest 0 005 and calculated values of drag are

stated to the nearest 0 001

The values of C

S

and C

B

calculated from the LSTM data are

stated to the nearest 0 001.

Total Drag

The total drag calculated by the linear and non-linear k ε models with SCL and AWF wall

functions is shown in Table 7.7 for the body with rear-slant angle β 25o and in Table 7.8

for the body with rear-slant angle β 35o.

With the 25o rear slant angle, the linear k ε model calculates the flow-field with a rea-

sonable degree of accuracy (Section 7.4). This is shown in the drag calculations for this case.

There is no significant improvement in the drag calculation when using the AWF instead of

the SCL wall function; it was noted in Section 7.4.6 that the wall function had only a very

small influence on the pressure field over the slant. Grün (1996) states that for industrial

calculations the required accuracy for drag is 2%; drag calculated by the linear model for the

Ahmed body with 25o does not meet this criterion.

The non-linear k ε model calculates an incorrect separated flow (Figure 7.12) and poor

7.6. Drag

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Case C K C

S C B C

R

Ahmed et al (1984) 0.020 0.140 0.070 0.055

LSTM - 0.158 0.116 -

linear k ε;SCL 0.048 0.139 0.103 0.004

linear k ε;AWF 0.049 0.139 0 .100 0 .005

non-linear k ε;SCL 0.047 0.105 0.111 0.004

non-linear k ε;AWF 0.051 0.083 0.114 0.004

Table 7.9: Drag breakdown for Ahmed Body with β 25o rear slant

Drag Breakdown

Assessment of the calculated drag coefficients is somewhat compromised by the available

data. The flow-field data for the 25o and 35o rear-slant bodies has been compared to the

recent measurements by Lienhart et al (2001). These authors only measured pressure drag

on the rear slant and base of the Ahmed body. Total drag coefficient and the contribution due

to pressure on the nose of the body were measured by Ahmed et al (1984). However, Ahmed

et al did not measure the separate contributions to drag for the rear-slant angles which have

been calculated in the current work and the measured values have been interpolated to provide

data for comparison. Pressure drag on the rear slant and base measured by Lienhart et al is

somewhat higher than the values interpolated from Ahmed’s et al measurements (Tables 7.9

and 7.10). It is not possible to quantify whether this is due to interpolation error or differences

in experimental error.

The component parts of the drag are pressure drag on the nose cone (C K ), slant (C

S ) and

base (C B) and the skin friction drag (C

R). These are listed for the Ahmed body with 25o rear

slant in Table 7.9 and for the Ahmed body with 35o rear slant in Table 7.10. All four com-

ponents are shown from the Ahmed et al measurements with the two available components

from the LSTM measurements. For both rear-slant angles, LSTM measured higher values of

C S and C

B than Ahmed et al but the same trends are shown: C S

C B for the 25o rear slant

and C S

C B for the 35o rear slant.

For the Ahmed body with either rear-slant angle, the mid-section of the body is suffi-

ciently long that the flow over the rear slant and base does not affect the flow at the nose

cone. The coefficient of drag at the nose cone, C K , is the same (and calculated somewhat too

high) for both geometries.

With the 25o rear slant, the linear k ε model calculates C S and C B reasonably accurately

in comparison to both the Ahmed et al and LSTM measurements. The non-linear k ε model,

which does not calculate the flow over the slant correctly, calculates C S too low. However, its

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Case C K C

S C B C

R

Ahmed et al (1984) 0.020 0.095 0.090 0.055

LSTM - 0.121 0.129 -

linear k ε;SCL 0.046 0.134 0.102 0.004

linear k ε;AWF 0.047 0.112 0 .103 0 .005

non-linear k ε;SCL 0.047 0.088 0.093 0.004

non-linear k ε;AWF 0.052 0.088 0.099 0.004

Table 7.10: Drag breakdown for Ahmed Body with β 35o rear slant

calculation of the base pressure-drag coefficient, C B, is accurate in comparison to the LSTM

measured value. (The flow is fully separated behind the base when calculated by either the

linear or non-linear model.)

When using the SCL wall function with the non-linear k ε model, the UMIST convec-

tion scheme was used to calculate both the mean velocity and turbulence variables; the model

with AWF wall function used the UPWIND scheme for turbulence variables (Table 7.1). The

difference in a accuracy of these schemes is partly the cause of the improved calculation of

C S when using the SCL wall function.

Although they show quite different magnitudes, both sets of measurements show C S

C B

for the 35o rear-slant geometry. All the calculations (except linear k ε with SCL) show this,

but as the calculations with the linear model agree better with the LSTM measurements and

the calculations with the non-linear model agree better with the Ahmed et al measurements,

it is hard to draw further conclusions.

Skin friction drag, C R, is principally generated on the mid-section of the body and is

approximately the same for both rear-slant angles. However, it is calculated significantly

too low - by an order of magnitude - for both geometries and an assessment of the accuracy

of Ahmed et al’s calculation of C R is required. Ahmed et al used a strain-gauge balance to

measure the total coefficient of drag (C W ) and calculated C K

C S and C

B by integrating the

measured pressure over the relevant surfaces. The skin friction drag coefficient was then

calculated by:

C

R

C W

C

K

C

S

C

B

(7.7)

Clearly, any inaccuracy in the total drag measurement or calculation of the pressure drag

components will be absorbed into C R. Taking the β

25o geometry as an example, the

values of C S and C

B calculated from LSTM’s measurements (Table 7.9) are somewhat larger

than those calculated by Ahmed et al. If these values are substituted into Equation 7.7 then

assuming that Ahmed et al’s measurement of C W is accurate, C K

C R 0 011 This is half

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the value of the measured C K alone and emphasises the incompatibility between the two sets

of measurements. The calculated values of C R shown in Tables 7.9 and 7.10 may indeed be

the correct order of magnitude, but more detailed measurements would be required to prove

this.

7.7 Conclusions

The results presented in this chapter have shown that the flow around the Ahmed body can be

calculated to a reasonable degree of accuracy using the RANS equations and two-equation

turbulence models. Both the attached flow with strong vortices (25o rear slant) and fully

separated flow (35o rear slant) can be calculated. Moreover, the calculation of the body’s

drag with the 35o rear-slant angle using the non-linear k ε model and AWF wall function

is sufficiently accurate for industrial purposes (in comparison to Ahmed’s et al, 1984, data).However, the calculation of the flow over the body with the 25 o rear slant is not sufficiently

accurate. Furthermore, the current formulation of the non-linear k ε model is not appropri-

ate for calculating the flow over the body with 25o rear-slant angle.

There are a number of specific conclusions which can be drawn from the calculation of

the Ahmed body flow and which are addressed separately.

Boundary Layers on Body Mid-Section The boundary layers calculated on the mid-

section of the body are thicker than the measured boundary layers. This makes the flow

more susceptible to separation in an adverse pressure gradient such as over the rear slant.

The thickening of the boundary layers is most probably due to the use of large near-wall cells

in which 100

y

200. Also in this region, the calculated outer flow is accelerated due to

the blockage effect of the body and the use of symmetry planes at the domain boundary. If

further calculations are to be made on the Ahmed body, an investigation of the effect of reduc-

ing y should be made and the domain boundary condition altered to prevent the acceleration

of the outer flow.

Side-Edge Vortex Formation Flow in a 25o expansion would normally be expected to

separate from the walls due to the adverse pressure gradient encountered. However, mea-

surements by Ahmed et al (1984) and Lienhart et al (2001) show that flow over the Ahmed

body with a 25o rear slant remains attached. Strong longitudinal vortices are formed at the

side edges of the slant, which draw fluid out of the boundary layer on the slant and cause the

flow to remain attached. The linear k ε model calculates a vortex which is strong enough

to retain the attached flow. The non-linear model calculates a weaker vortex. When using

7.7. Conclusions

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a first-order convection scheme (UPWIND), this vortex is sufficiently diffuse to affect the

whole rear slant and produce attached flow. When using a higher-order convection scheme

(UMIST), the vortex is confined to a smaller region. It does not draw fluid out of the boundary

layer over a sufficiently large region of the slant to prevent separation.

Components of Non-linear Model The weak side-edge vortex calculated on the 25o rear

slant by the non-linear model was caused by the mean strain bending being calculated too low

in the region of the side edge. It was not possible to identify the specific feature of the non-

linear model which caused this, as both the functional form of c µ and the inclusion of non-

linear components in the Reynolds stresses was sufficient to cause separated flow. Craft et al

(1996b) tuned the coefficients of the non-linear model by testing the model in homogeneous

shear flow, fully developed swirling flow and flow with streamline curvature. In this case,

however, the flow is characterised by streamwise vorticity and an adverse pressure gradient.

The calculation of streamwise vorticity is influenced by the Reynolds normal and shear stress

generation (the 3rd and 4th terms of the right-hand side of Equation 7.5). These terms are zero

in the linear calculation but may well be significant in the non-linear k ε model calculation7.

It is therefore possible, that the model coefficients need re-tuning if it is to be used to calculate

flow in an adverse pressure gradient (or separated flow) with streamwise vorticity.

Similarly, it is possible that the functional form of c µ causes too great a reduction in

turbulent viscosity over the rear slant. (The c µ function is defined from measurements and

DNS calculations in homogeneous shear flow). With the thickened boundary layer over the

top surface of the body, the flow is susceptible to separation in adverse pressure gradients.The combined effect of low turbulent viscosity and a thick boundary layer then cause flow

separation, which inhibits the side-edge vortex formation.

Effect of Realizability in the Linear Calculation The realizability condition, which re-

duces turbulent viscosity in regions of high strain rate, was used in the calculations with the

linear k ε model. In comparison to the linear model without the condition, it significantly

improved the calculation of mean velocity and length of the recirculating flow region in the

wake of the body with 35o rear slant.

Calculation of Drag and Influence of Wall Function The calculation of drag induced

by flow over the body with the 25o rear slant is reasonable, but not sufficiently accurate

for industrial purposes. Although calculations for this case with the linear model produce

attached flow over the slant, the base pressure and overall drag are too high in comparison to

7There is insufficient measured data in the relevant areas around the slant to prove this conclusively.

7.7. Conclusions

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Ahmed’s et al (1984) measurement. The non-linear model calculates separated flow over the

rear slant and the distribution of drag between the rear slant and base is poor.

Both the linear and non-linear model calculate separated flow over the 35o rear slant. Drag

calculated by the linear model is too high for this case but drag calculated by the non-linear

model is much more accurate. Indeed, when the non-linear model is used with the AWF wallfunction, the calculated drag is sufficiently accurate for industrial use.

In all the cases where the correct flow profile is calculated, use of the AWF wall function

improves the calculation of drag in comparison to the SCL wall function. In all the calcu-

lations, the component of drag due to friction is an order of magnitude lower than the value

given by Ahmed et al. (It should be noted that Ahmed et al did not measure friction drag

directly, but calculated it from the difference between the total drag and the pressure drag.

Consequently, any errors in the measured pressure and total drag are passed on to the friction

drag.)

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Chapter 8

Conclusions and Recommendations for

Future Work

8.1 Preliminary Remarks

The aim of the work presented in this thesis has been to test the cubic non-linear k ε model

of Craft et al (1996b) in a number of flows pertinent to road vehicle aerodynamics. These are

flows which contain features such as separation, streamline curvature, streamwise vorticity

and vortex shedding. A secondary aim has been to test the suitability of a newly developed

analytical wall function. The non-linear turbulence model was tested in three flows with

simple geometries which had at least some of the flow complexity found in the flow around

road vehicles. In these computations, the non-linear model was shown on the whole to be

more accurate than a linear k ε model. The final test case, the flow around a simplified

model of a car was calculated with both linear and non-linear k ε models.

Conclusions have already been presented on a case-by-case basis at the end of each of

the flow calculation Chapters. It is not intended to repeat those conclusions here, but to

re-emphasise the salient points and to make recommendations for future work.

8.2 Conclusions

Flow Around a Cylinder of Square Cross Section Close to a Wall

In the flow with the steady wake

g

d 0 25 , velocity profiles calculated in the wake

of the cylinder by the non-linear k ε model are not so accurate as those calculated

by the linear model. In contrast, Reynolds stresses are calculated more accurately by

the non-linear k ε model than the linear model. The linear model is very sensitive

131

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CHAPTER 8. Conclusions and Recommendations for Future Work 132

to changes in the upstream value of turbulence dissipation rate specified at the inlet,

whereas the non-linear model is not. In a flow which is similarly insensitive to changes

in inlet turbulence intensity, this insensitivity is a desirable feature of the non-linear

model. It is often the case in industrial calculations that the inlet values of turbulence

are not known.

When the cylinder is placed sufficiently far from the wall that periodic vortex-shedding

occurs

g

d 0

50and0

75 , the non-linear k ε model calculates the frequency of

the vortex-shedding more accurately than the linear model. The non-linear model

calculates the time-averaged total stresses in the wake

ut iut

j

too high. Moreover,

the relative magnitudes of the contributions to total stress are the wrong way round:

the time-averaged periodic motion stresses ui u j

are too high and the time-averaged

Reynolds stresses

uiu j

are too low.

Flow in a U-bend of Square Cross Section

The non-linear k ε model is virtually as accurate as a cubic DSM in calculating the

mean velocities and complex secondary flow pattern in this flow with strong streamline

curvature and streamwise vorticity.

The non-linear k ε model calculates the Reynolds stresses reasonably well but is

not so accurate as the cubic DSM. In this flow, transport of the Reynolds stresses is

significant; the non-linear k ε model does not accurately account for Reynolds stress

transport.

Flow in a 10o Plane Diffuser

The non-linear k ε model calculates the mean velocities and Reynolds stresses signif-

icantly better than the linear model. Indeed when a log-law wall function is used, the

non-linear model correctly calculates flow separation on the inclined wall of the dif-

fuser whereas the linear model does not. However, the magnitude of the flow reversal

calculated by the non-linear model is not as great as that measured.

The ability of the non-linear k ε model to calculate the velocity more accurately than

the linear model in a separated flow is contrary to the findings of the square cross-

sectioned cylinder close to a wall (steady flow) test case. The difference between the

two cases is that in the plane diffuser, the adverse pressure gradient is relatively weak

and the non-linear model is the more sensitive to this. In the wake of the square cross-

sectioned cylinder the adverse pressure gradient is much stronger and there are other

8.2. Conclusions

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effects influencing the flow calculation, such as impingement at the leading face and

stronger curvature of the streamlines.

Flow Around the Ahmed Body

For the Ahmed body with the 25o rear slant the non-linear k ε model does not calcu-

late the flow over the slant correctly. It calculates separated, rather than attached flow

over the majority of the slant. In contrast, the linear model does calculate attached flow

over the whole slant but is not able to calculate the initial separation at the leading edge

of the slant.

The failure of the non-linear k ε model to calculate attached flow on the slant is

principally due to its failure to calculate a sufficiently strong side-edge vortex. This

is somewhat surprising as the non-linear model is well able to calculate flows with

streamwise vorticity, as is shown by the swirling-flow tests of Suga’s (1995) original

study of this turbulence model and by the square cross-sectioned U-bend test case in

the present study. It should be noted that unlike flow around the Ahmed body, the U-

bend is an enclosed flow and influenced by the strong radial pressure gradient. In light

of the findings of the current work, this may well have a bearing on the performance of

the non-linear model.

For the Ahmed body with the 35o slant both the non-linear and linear k ε models

calculate the correct, separated flow over the rear slant. In common with the square

cross-sectioned cylinder case, the velocity profiles in the wake are calculated more

accurately by the linear model. However, the non-linear model calculates drag on the

Ahmed body more accurately than the linear model. When used in conjunction with

the analytical wall function, the non-linear model calculates the drag on the body to the

degree of accuracy required by industry.

Analytical Wall Function

Calculations of the 10o plane diffuser flow with the AWF show that this wall function

calculates near-wall effects more accurately than a log-law type wall function (SCL).When used with the the linear k ε model, the AWF improves the flow calculation

to the extent that a small amount of separation is calculated (whereas separation is

not calculated with an SCL wall function). When used with the non-linear model, the

AWF improves the calculation of the separated flow. Of the models tested in this work,

the combination of the non-linear k ε model and AWF provides the most accurate

calculation for this case.

8.2. Conclusions

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Calculations of the flow around the Ahmed body with the AWF show that when the

correct mode of flow is calculated1 the AWF improves the calculation of drag. No

significant changes in the mean velocity profiles are apparent when using the AWF

rather than the SCL wall function. As with the plane diffuser, the non-linear k ε

model and AWF provide the most accurate combination of the models tested. It shouldbe noted that the distance from the wall of the near-wall nodes used in the Ahmed body

calculations was relatively large. This reduces the potential for the AWF to improve

the calculation and it would be reasonable to expect the AWF to improve further the

near-wall flow if the near-wall cell size were refined.

Realizability Condition

The realizability condition was initially introduced to the Ahmed body flow calcula-

tions as a mathematically nicety; it prevents the spurious growth of turbulent viscosityin regions of low turbulence intensity and improves the stability of the calculations.

However, it has also been shown that the use of the realizability condition improves the

calculation of the wake-velocity profiles.

8.3 Recommendations for Future Work

Ahmed Body Calculations

Future calculations of the Ahmed body flow should include a grid refinement study.

In particular, a sensitivity study of the distance to the body of the near-wall nodes

should be made. This will show whether the boundary layers around the body can be

calculated more accurately than in the present study and help to determine whether

thickening of the boundary layers is a cause of the excessive flow separation calculated

by the non-linear k ε model. It will also help to determine the influence of the near-

wall skewing of velocity over the rear slant.

The current work uses symmetry planes to represent the open boundaries of the wind

tunnel which leads to an acceleration of the flow around the body that is not measured.

Future calculations should modify this boundary condition to an “entrainment” type

boundary or expand the grid in the region of the cylinder to prevent the calculation of

the favourable pressure gradient which accelerates the flow.

1ie. when the linear k ε model is used to calculate the flow around the body with the 25o rear slant or when

either the linear or non-linear model is used to calculate the flow around the body with the 35 o rear slant

8.3. Recommendations for Future Work

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Non-linear k ε Model

As the non-linear k ε model appears not to be able to calculate sufficient streamwise

vorticity in a free-stream flow, a simple test case should be sought to study this feature.

The coefficients in the stress-strain relationship of the non-linear k

ε model were not

originally tuned for separated flows. As the current work shows conflicting results for

the accuracy of the non-linear model in relatively weak and strong adverse pressure

gradients, future development of the non-linear model should attempt to define which

elements of the model need to be re-cast to provide consistent performance in sepa-

rated flows. In particular, the relevant influence of the c µ function and the non-linear

contribution to the Reynolds stresses should be studied in separated flow. Either of

these features in isolation in the non-linear model are sufficient to cause separation of

the flow over the 25o rear-slant Ahmed body.

Further study should be made of the non-linear k ε model’s ability to calculate pe-

riodic flows. Such a study would aim to establish why the time-averaged Reynolds

stresses are calculated too low and the time-averaged periodic motion stresses are too

high in the wake of the square cylinder .

Analytical Wall Function

The current form of the analytical wall function is based on the solution of a simplified

momentum equation in the near-wall cell which does not include non-linear contribu-

tions to the Reynolds stresses. For consistency, when the analytical wall function is

used with the non-linear k ε model, it should be be re-cast to include contributions of

the non-linear stresses to the simplified momentum equation.

Stability enhancing measures should be sought for the analytical wall function to miti-

gate the effect of placing the momentum source in the constant (S C ) rather than coeffi-

cient

S P) part of the linearised source term.

8.3. Recommendations for Future Work

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Appendix A

Realizability Condition for an EVM

A.1 Introduction

The realizability conditions by which the Reynolds stresses are constrained are:

uiu j 0 ; i

j (A.1)

uiu j2

uiui

u ju j (A.2)

det

uiu j

0 (A.3)

Schumann (1977) discusses these conditions and some of their consequences for turbulence

modelling; Lumley (1978) discusses their application to Differential Stress Models in greater

detail. The realizability conditions are more readily applied in a DSM than an EVM as

transport equations are solved for the various turbulent stresses and limiting values can be

applied. In an EVM the turbulent stresses are not calculated explicitly, but turbulent viscosity

is calculated from the turbulent kinetic energy, k and dissipation rate, ε If these are not

controlled in an EVM flow calculation, it is possible that negative values of k and ε may

occur leading to numerical instability. By setting a lower limit for k (eg k 10

30), negative

values can be avoided without any detrimental numerical problems. A more sophisticated

approach is necessary for ε. In a k ε EVM the turbulent viscosity depends on the reciprocal

of ε : µt ρc µ

k 2

ε (A.4)

Thus very small values of ε may create erroneously large values of µ t . A simple method of

controlling ε would be to set its lower limit to an arbitrarily low value such as: εmin 10

20εo

(where εo is the inlet or far-field value); this can still lead to unacceptably high levels in µ t .

Moore & Moore (1999) conducted a review of methods used to control excessive levels of

137

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APPENDIX A. Realizability Condition for an EVM 138

µt . These use a functional form of c µ which decreases its value in regions of flow with high

strain rates, controlling the growth of µt . These functional forms of c µ are generally formu-

lated without reference to the concept of realizability. May (1998) reworked the realizability

conditions for an EVM to obtain a method by which µt could be limited. This Appendix de-

scribes how the realizability condition can be defined for an EVM. Throughout the derivation,the summation convention for terms containing repeated indices is not applied.

A.2 Mathematical Formulation

If one takes the Schwarz inequalities (Equation A.1 & A.2) and uses the eddy-viscosity hy-

pothesis for a linear EVM:

ρu

iu

j

2

3 ρδi jk

µ

t

∂U i

∂ x j

∂U j

∂ xi

(A.5)

then for the case i 1; j

2 the Reynolds stress components may be written:

ρuv

2

µ2t A2 (A.6)

ρuu

µt B C

0 (A.7)

ρvv

µt D C

0 (A.8)

and the terms A

B

C

D are:

A

∂U

∂ y

∂V

∂ x

(A.9)

B

2∂U

∂ x (A.10)

C

2

3ρk (A.11)

D

2∂V

∂ y(A.12)

Substituting Equation A.6-A.8 into the Schwarz inequality (Equation A.2):

A2 µ2t

Bµt C

Dµt C (A.13)

This can be re-written as:

C µt α

C µt β

0 (A.14)

where:

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α

1

2

B D

B D 2

4 A2

1 2

β

1

2

B D

B D 2 4 A2

1 2

(A.15)

To satisfy the inequality in Equation A.14, there are two cases which need to be considered.

Case 1:

C µt α ; C

µt β (A.16)

and Case 2:

C µt α ; C

µt β (A.17)

Substitution of β from Equation A.15 into the second part of the Case 2 condition (Equation

A.17) gives:

2C

µt

B

D

µt

B D

2

4 A2

1 2(A.18)

and summing the inequalities given with the normal stresses in Equations A.7 and A.8, gives:

2C µt

B D

0 (A.19)

Now Equation A.19 determines that the left hand side of Equation A.18 must be positive.

If both sides of Equation A.18 are multiplied by

1 then the inequality reverses and this

equation states that the square root of the term in square brackets must be less than or equal

to zero. This is not possible for non-zero, real numbers and hence Case 2 (Equation A.17)

cannot be true for any value but zero. For Case 1, the first inequality of Equation A.16

includes the second inequality and hence the only condition which must be satisfied is:

C

µt α (A.20)

Replacing C

µt α with their respective values from Eqns.A.11, A.4 and A.19, gives:

ε 34

c µk

B D

B D

2 4 A2

1 2

This and similar analyses for the cases: i 1; j

3 and i

2; j

3 provide three criteria for

the minimum value of ε:

ε1

3

2c µk

∂U

∂ x

∂V

∂ y

∂V

∂ y

∂U

∂ x

2

∂U

∂ y

∂V

∂ x

2

1 2

(A.21)

A.2. Mathematical Formulation

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ε2

3

2c µk

∂V

∂ y

∂W

∂ z

∂W

∂ z

∂V

∂ y

2

∂V

∂ z

∂W

∂ y

2

1 2

(A.22)

ε3

3

2c µk

∂W

∂ z

∂U

∂ x

∂U

∂ x

∂W

∂ z

2

∂W

∂ x

∂U

∂ z

2

1 2

(A.23)

A.3 Implementation in STREAM

The realizability condition has been implemented in STREAM immediately after µt has been

updated from the newly calculated values of k and ε by applying the criterion:

µt min

µt

ρk

max

TINY α1

α2 α3

(A.24)

where T INY is an arbitrarily small value and α1 α2 α3 are derived from Equation A.15 and

take the values from the outer set of parentheses in Equations A.21 to A.23 as follows:

α1

3

2

∂U

∂ x

∂V

∂ y

∂V

∂ y

∂U

∂ x

2

∂U

∂ y

∂V

∂ x

2

1 2

(A.25)

α2

3

2

∂V

∂ y

∂W

∂ z

∂W

∂ z

∂V

∂ y

2

∂V

∂ z

∂W

∂ y

2

1 2

(A.26)

α3

3

2

∂W

∂ z

∂U

∂ x

∂U

∂ x

∂W

∂ z

2

∂W

∂ x

∂U

∂ z

2

1 2

(A.27)

A.4 Comparison with c µ Function

The realizability condition described in this Appendix acts in a similar way to the functional

forms of c µ described in Equations 2.28 and 2.32: both act to reduce the turbulent viscosity

in regions of high velocity gradient. Indeed the realizability condition can be re-cast as a

functional form of c µ. If one considers a homogeneous simple shear where the only velocity

gradient is ∂U

∂ y and taking the latter part of the minimum function in Equation A.24:

µt

2

3

ρk

∂U

∂ y(A.28)

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From Equation 2.29, in a simple shear ∂U

∂ y

S ε

k , and using Equation A.4 for the turbulent

viscosity:

c µρk 2

ε

2

3

ρk

S ε

k (A.29)

which provides the following function for c µ:

c µ

2

3S (A.30)

The performance of this new c µ compared with the original versions (Equations 2.28 and

2.32) is shown in Figure 2.1. It is apparent that the effective c µ defined by the realizability

condition is not reduced in the presence of high strain rates as much as the other forms.

A.4. Comparison with c µ Function

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Appendix B

Derivation of the Analytical Wall

Function

B.1 Specification of Analytical Wall Function

B.1.1 Dimensionless Simplified Momentum Equation

A simplified momentum equation can be written for the near-wall region as follows:

ρ∂UU

∂ x ρ

∂UV

∂ y

∂P

∂ x

∂

∂ y

µ µt

∂U

∂ y

(B.1)

where ρ

µ

µt

P are the density, laminar viscosity, turbulent viscosity and pressure respec-tively, U and V are the mean velocity of the flow tangential and normal to the wall respec-

tively, x and y are the coordinates tangential and normal to the wall respectively (not nec-

essarily the Cartesian directions). This momentum equation retains the simple shear flow

assumption that diffusion in the x-direction (tangential to the wall) is negligible compared

to diffusion in the y-direction (normal to the wall). In a fully developed flow, convection

of momentum tangential to the wall is zero as there is no gradient in the tangential velocity

(∂U

∂ x) - this is not the case for developing flows. Inclusion of convection normal to the

wall is important when considering separation and reattachment of the flow where there are

significant velocities towards and away from the wall.

The simplified momentum equation is converted to a dimensionless form by application

of the reference velocity

U re f , distance

d re f and density

ρre f . These are the values which

are used to detemine the Reynolds number of the flow and their application provides dimen-

sionless velocities

U

V

, distances

x , y

, density

ρ

, viscosity

µ

and pressure

P

.

The version of the STREAM code into which this model is incorporated is non-dimensional,

142

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APPENDIX B. Derivation of the Analytical Wall Function 143

and hence it is convenient to express this model in non-dimensional variables which are de-

noted by a prime.

U

U

U re f ; V

V

U re f ; (B.2)

Substituting these into Equation B.1 and reworking, gives the following form of the momen-

tum equation:

U

∂U

∂ x

V

∂U

∂ y

∂P

∂ x

∂

∂ y

µ

µ t

∂U

∂ y

(B.3)

A code-specific simplification has been used here. The incompressible, non-dimensional

version of STREAM sets the density as: ρ ρ

ρre f

1 0. This has been used to simplify

Equation B.3.

B.1.2 Analytical Solution of Equations

When considering the near-wall effects it is more appropriate to use a wall-distance scaled

by the near-wall properties ( y

), rather than the bulk flow conditions ( y ). Hence, the wall

distance is substituted and the terms of Equation B.3 are re-arranged:

y

yk 1 2

ν

y k

1 2 p

ν

∂

∂ y

1

µt

µ

∂U

∂ y

ν

k p

∂P

∂ x

U

∂U

∂ x

V

∂U

∂ y

C (B.4)

C is a constant, ν

ν

d re f U re f is the non-dimensional kinematic viscosity and k

p is the

non-dimensional turbulent kinetic energy at the near-wall node p (Figure 2.3). It is assumed

that k varies quadratically in the viscous sublayer and remains constant in the fully turbulent

region. (This is the same assumption that is used in the simplified version of the Chieng &

Launder, 1980, wall function.)

A more sophisticated approach would be to use k

v - ie the value of turbulent kinetic energyat the edge of the viscous sublayer. This is the value used in the full Chieng & Launder (1980)

wall function which makes the value of τ w less dependent on the size of the near-wall cell and

the location of the near-wall node. k v is calculated by assuming a linear profile for k

in the

fully turbulent region, a constant sublayer Reynolds number, k

1 2

v y v

ν

20 and extrapolating

a value from the two near-wall nodes, p and N . Johnson & Launder (1982) used a similar

B.1. Specification of Analytical Wall Function

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technique but allow variation in the sublayer thickness:

R v

k 1

2

v yv

ν

20

1 3 1λ λ

k v

k w

k v

(B.5)

where k

w is the wall value of turbulent kinetic energy and k

v and k

w are extrapolated fromnodes p and N . The extrapolation of k

v and k w can cause problems. There is a maximum in

the turbulent kinetic energy profile near to the wall and depending on resolution of the grid,

the relative positions of nodes p and N may result in greatly differing values of k w and k

v

(Figure 2.4). The value of k p is also sensitive to the position of node p. However, by using

constant k k

p in the near-wall cell the additional uncertainty of extrapolating values for k v

and k w is removed. This is considered a more robust practice to adopt during the development

of the new wall function.

Solution Within the Viscous Sub-layer

In the viscous sublayer (referred to herein as Region 1) the flow is laminar and the following

conditions are specified:

y

y v ; µ t 0 (B.6)

where yv is the non-dimensional thickness of the viscous sub-layer in wall units. Equation

B.4 is then reduced to:

∂

∂ y

∂U

1

∂ y

C 1 (B.7)

∂U

1

∂ y

C 1 y

A1 (B.8)

U

1

C 1 y

2

2

A1 y

B1 (B.9)

where the subscript on U 1, A1

B1 and C 1 denotes Region 1: the viscous sub-layer. At the wall

the following boundary conditions apply: y

0, U

1 0 hence, B1 0, and:

U

1

C 1 y

2

2

A1 y (B.10)

Solution Outside the Viscous Sub-layer

Outside the viscous sub-layer in the fully turbulent region (Region 2) an analytical solution

is again sought for Equation B.4. The turbulent viscosity, µt must be retained in this region


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and it is specified from the Prandtl-Kolmogorov condition, µt ρc µk 1 2l. As this is to be

applied in the near-wall region the turbulent length scale, l , is specified as; l

cl

y yv and

the turbulent kinetic energy is given the near-wall cell-centre value, k p Writing µt in terms

of the wall distance, y gives:

µt ρk

1 2

p α

y

y v

ν

k 1

2

p

(B.11)

which reduces to the following non-dimensional form:

µ t µ α

y

y v (B.12)

where α c µcl and the constants c µ and cl take the values 0.09 and 2.55 respectively. In the

fully turbulent region, Equation B.4 now becomes:

∂

∂ y

1 α

y

y v

∂U

2

∂ y

C 2 (B.13)

which can be integrated to find U

2 :

∂U

2

∂ y

C 2 y

1 α

y

yv

A2

1 α

y

yv

(B.14)

U

2 C 2

y

1 α

y

yv

dy

A2

dy

1 α

y

yv

(B.15)

U

2

C 2 y

α

A2

α

C 2

α y ν

1

α2

ln

1 α

y

y v

D2 (B.16)

(Note: the subscript “2” denotes the region outside the viscous sub-layer.) Boundary condi-

tions can be set at the edge of the viscous sub-layer:

y

y v ; U

1

U

2 ;∂U

1

∂ y

∂U

2

∂ y

(B.17)

Applying the boundary conditions to Equation B.14 gives:

A2 A1

y v

C 1 C 2 (B.18)

and to Equation B.16 gives:

D2

C 1 y

2v

2

A1

C 2

α

y v (B.19)


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and the velocity in the fully turbulent region may now be written as:

U

2

C 2

α

y

y v

A1

α

C 1 y ν

α

C 2

α2

ln

1 α

y

y v

C 1 y

2v

2

A1 y v (B.20)

It is necessary to set one further boundary condition for the fully turbulent flow region at the

near-wall cell face opposite the wall:

y

y n ; U

2 U n (B.21)

U n is the resultant velocity in the direction of flow along the wall, located at the cell face

between the near-wall cell and the adjacent cell on the flow side of the domain. In a cell-

centred code, this must be found by interpolation of the resultant velocities at the centres of

the near-wall cell and the adjacent cell. By applying these conditions to Equation B.20 andrearranging, an expression for the integration constant A1 can be recovered:

A1

αU n

C 2

yn

yv

C 1 y v

C 2α

ln

1 α

yn

yv

αC 1 y

2v

2

α yv ln

1 α

yn

yv

(B.22)

B.1.3 Specification of Wall Shear Stress, τ w

A non-dimensional shear stress can be defined as:

τ w

µ

∂U

∂ y

(B.23)

and can be written in terms of the wall distance, y :

τ w

k 1

2

p

ρre f

∂U

∂ y

(B.24)

At the wall (ie within the viscous sub-layer), the wall distance y

0 and the velocity gradient

is specified by Equation B.8. The wall shear stress can now be defined:

τ w

k 1

2 p

ρre f

A1 (B.25)

B.1.4 Average Production of Turbulent Kinetic Energy, Pk

The average value of production of turbulent kinetic energy in the near-wall cell can be spec-

ified in non-dimensional units as:


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Pk

1

yn

y

n

y

v

ρ uv

∂U

2

∂ y

dy

1

yn

yn

yv

ρuv

∂U

2

∂ y

dy (B.26)

The velocity gradient is given by Equations B.14 & B.18, and the integral can thus be written:

Pk

ρ y

n

y n

yv

uv

A1 C 1 y v C 2

y

y v

1 α

y

yv

dy (B.27)

The shear stress is defined from the eddy viscosity hypothesis:

uv νt

dU

dy (B.28)

Recasting in dimensionless terms:

uv

ν t

dU

dy

ν t

dU

dy

k 1

2

p

ν

(B.29)

and substituting Equation B.12 for νt

ν :

uv

α

y

y v

dU

dy

k 1

2

p (B.30)

In the fully turbulent region the velocity gradient is given by Equation B.14, resulting in the

final expression for shear stress:

uv

k 1

2 p α

y

y v

A1

C 1 y

v

C 2

y

y

v

1 α

y

yv

(B.31)

The production of turbulent kinetic energy can now be defined as:

Pk

ραk 1

2

p

yn

yn

yv

y

y v

A1

C 1 y v

C 2

y

yv

1 α

y

yv

2

dy (B.32)

This can either be calculated numerically, within the CFD scheme, or by the analytical solu-

tion:

Pk

ρk 1 2

p

α yn

y n y v

C 22

2α2

2 α

y n y v

C 2

2 A1 2C 1 y v

3C 2

α

A1 C 1 y v

C 2

α

A1 C 1 y v

3C 2

4

ln

1 α

y n y v

(B.33)


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α

yn

yv

A1 C 1 y v

C 2α

2

1 α

yn

yv

B.1.5 Average Turbulence Dissipation Rate, ε

The average dissipation rate specified in the near-wall cell used with the analytical wall func-

tion is a simplified version of the term proposed by Chieng and Launder (1980). This is

derived from the assumption that close to the wall, in the viscous wall layer, the dissipation

is constant and equal to its wall limiting value given by: ( y yε)

ε

1 2 ν

∂k

1 2

∂ y

2

2 ν k

p

y

2ε

(B.34)

In the Chieng & Launder scheme the height of the dissipation sub-layer is taken to be: yε

yv.Further away from the wall, in the fully turbulent flow region, dissipation rate is given by:

( y

yε)

ε

2

k 3 2

p

cl y

(B.35)

Integrating these two values over their respective sections of the near-wall cell and summing

gives the average dissipation rate:

ε

1 y

n

y

ε

0

2 ν

k

p

y

2ε

dy

y

n

y

ε

k

3 2

pcl y

dy

1 y

n

2 ν

k

p y

ε k

3 2

pcl

ln y

n y

ε

(B.36)

In wall units this becomes:

ε

k 3

2

p

yn

2 ν

yε

1

cl

ln

yn

yε

(B.37)

In the Chieng & Launder (1980) wall function, the height of the viscous sub-layer is defined

as yv 20 ε is constant in the viscous sublayer and and varies according to the equilibrium

length-scale outside the viscous sublayer (Figure B.1). If the transition between these regionsis specified as y

v 20 then there is a discontinuity in ε

For the analytical wall function,

Gerasimov (2000) has defined a continuous function for ε from Equations B.34 and B.35,

which results in a new height being defined for the dissipation viscous sub-layer: yε 5 1.

To reproduce the log-law of the wall (in simple shear) the height of the turbulence viscous

sub-layer must be reduced to yv

10 8.


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B.1.6 Summary of Model

Wall Shear Stress: (Equation B.25)

τ w

k 1

2

p

ρre f

A1

Average Production of Turbulent Kinetic Energy: (Equation B.32)

Pk

ραk 1 2 p

yn

yn

yv

y

y v

A1

C 1 y v

C 2

y

yv

1 α

y

yv

2

dy

Constant Term: (Equation B.22)

A1

αU n

C 2

yn

yv

C 1 y v

C 2α

ln

1 α

yn

yv

αC 1 y

2v

2

α y v ln

1 α

y n

y v

Non-Equilibrium Constants: (Equation B.4)

C 1

ν

2

k p

∂P

∂ x

γ U

∂U

∂ x

γ V

∂U

∂ y

(B.38)

C 2

ν

2

k p

∂P

∂ x

U

∂U

∂ x

V

∂U

∂ y

(B.39)

Average Dissipation Rate: (Equation B.37)

ε

k 3

2

p

yn

2 ν

yε

1

cl

ln

yn

yε

γ is an empirical constant which is included to provide a degree of control over the in-

fluence of convection inside the laminar sublayer. One would normally expect the mean

convection in the viscous sub-layer to be appreciably less than in the region of the near-wall

cell outside the viscous sublayer. However, by testing the wall function in a plane shear flow,

γ has been optimised with the value γ 1 0.

Model with Pressure Gradient Only

To simplify the model for testing purposes and also to calculate fully developed flows in

which there is no convection tangential to the wall in the boundary layer (such as pipe or


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channel flows), the convection term U ∂U

∂ x

V ∂U

∂ y can be set to zero, hence:

C 1 C 2

ν

2

k p

∂P

∂ x

(B.40)

B.2 Concerning Implementation of Analytical Wall Func-

tion

B.2.1 Wall Shear Stress

As in a log-law type wall function, the shear stress in the analytical wall function is used

to calculate the wall-shear force acting on the fluid, F s

τw

Area. This wall-force is then

supplied as a source term to the momentum equation but care must be taken in how this

source term is specified. The usual method is to linearise the source term as follows:

S

S C φS P (B.41)

S

S C and S P are the total source term and contributions from the constant part and coefficient

respectively. φ is the transported variable. For numerical stability, it is necessary to ensure

that S P remains negative (Patankar, 1980) and negative source terms, such as the wall-shear

force, are usually included in S P. In the Chieng & Launder (1980) scheme (Equation 2.74),

and other log-law wall-functions, this is convenient as τw contains a power of φ (ie U V

W ).

It is only necessary to reduce this power by 1 before the shear stress (wall shear force) isincluded in S P.

In the analytical wall function, the velocity is not directly present in the specification of

τw (Equation B.25) and if τw is to be included in S P then it must be first multiplied by φ

1 (ie

reciprocal of velocity). This does not pose a problem in parabolic and non-separated flows

where the velocity in the near-wall cell is never zero. However, at separation and reattachment

points the velocity in the near-wall cell is zero (or virtually zero). This will cause very large

values of S P and consequently, instabilities in the solution. Hence, it is necessary to move

away from the normal practice and place F s

τw Area in the constant (S C ) part of S

When the analytical wall function is applied to a non-orthogonal grid in a cell-centred

scheme, attention must be given to the degree of skew of the near-wall cell. Figure B.2

shows a low aspect ratio, skewed near-wall cell. The wall shear stress (τw) is calculated

from the velocity U n at n, which is itself determined from the velocities at P and N (U P

and U N respectively). This is a potential source of error as velocities used to calculate τw are

somewhat displaced from the location of τw. Strictly, the velocity U n should be re-interpolated

B.2. Concerning Implementation of Analytical Wall Function

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to be aligned with the location of τw. However, the degree of non-orthogonality shown in

Figure B.2 is unlikely to occur if care is taken in generating the computational grid, and the

potential error will be small.

B.2.2 Turbulent Kinetic Energy & Dissipation

The turbulent kinetic energy in the near-wall cell is calculated in the same manner as log-law

type wall functions; that is, the transport equation for k is calculated in the near-wall cell

using the cell-averaged values of Pk and ε. The ε-equation is not solved in the near-wall cell;

the cell-average value ε is supplied directly.

B.3 Further Perspectives

In the derivation of the analytical wall function, the eddy viscosity hypothesis is used to

determine the Reynolds stress (shear stress). This introduces a possible consistency error

in that the eddy viscosity hypothesis assumes a linear stress-strain relationship. Hence the

analytical wall function could be described as a linear wall function. However, the wall

function is used with both linear and non-linear eddy viscosity models. To be thoroughly

consistent, the wall function should be derived with higher-order terms in the stress-strain

relationship (Equation 2.27), to produce a non-linear wall function. This has not been done

within the current project and could be considered for future development.

B.3. Further Perspectives

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Appendix C

Cylinder of Square Cross-Section with

Vortex Shedding

C.1 Calculated Flow Results for g d

0 50

Bosch et al (1996) and Durao et al (1991) have measured periodic vortex shedding in the wake

of a square cylinder placed at g

d 0

50 from a wall. At this distance, the cylinder is only

just far enough away from the wall to prevent the flow reverting to a non-vortex shedding

mode. Durao et al (1991) present measurements for this flow and the current calculations

are compared to their results. The flow has been calculated with a Reynolds number, Re

13 600 and 6% turbulence intensity at inlet, as used in the measurements. Durao et al did not

record any information about inlet turbulence dissipation rate which has been specified for

the calculation by the assumed viscosity ratio, νt

ν 10.

All calculated and measured results are normalised by the inlet streamwise velocity U o.

The measured data of Durao et al (1991) was not available in digital form and was re-digitized

from the journal paper with a consequent loss in accuracy. All velocities shown are time-

averaged velocities, with the average being calculated over one complete vortex-shedding

cycle. As discussed in Chapter 4, Durao’s et al LDV measurements were not able to distin-

guish between turbulent velocity fluctuations (ui) and the periodic “flapping” velocity

ui .

Hence all stresses shown in this Section are “total stresses”, being the sum of the Reynoldsstresses and periodic motion “stress” (Equation 4.2).

Drag and Lift

Coefficients of drag and lift are shown in Figure C.1 and demonstrate the fluctuations in C D

and C L due to vortex shedding. The mean and root-mean-square (RMS) values of C D and C L

152

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APPENDIX C. Cylinder of Square Cross-Section with Vortex Shedding 153

C D C D

RMS

C L C L

RMS

linear k εmodel 2.19 2.21 -0.27 1.23

non-linear k εmodel 2.09 2.10 -0.21 0.95

Table C.1: Lift and drag coefficients for square cylinder with g

d

0

50

St

Experiments 0.133-0.140

linear k εmodel 0.149

non-linear k εmodel 0.147

Table C.2: Strouhal numbers for square cylinder with g

d 0

50

are summarised in Table D.1.

Several groups have measured the non-dimensional frequency or Strouhal Number, St , of

the vortex shedding. (St f

d

U o - Equation 1.13). The values of St measured by Taniguchi

et al (1983), Durao et al (1991) and Bosch et al (1996) and the calculated values are sum-

marised in Table D.2. There is a clear improvement in the calculation of the vortex shedding

frequency when using the non-linear k ε model although the calculated value is still some-

what high.

Velocity Profiles

Time-averaged U and V -velocity profiles are shown with Durao’s et al (1991) measured data

in Figures C.2 and C.3 respectively. Both the linear and non-linear k ε models calculate

the time averaged velocities reasonably well, with the non-linear model being slightly more

accurate. The time-averaged recirculation length behind the cylinder is much shorter for the

present case than the steady wake-flow case (g

d 0

25 - Chapter 4). It extends to x

d

2 5

and this length is calculated well by both models.

Total Stress Contours

Contours of total stresses, ut ut vt vt and ut vt are shown in Figures C.4 to C.6 respectively.

The shape of the contours for each stress is generally calculated well by both models. The

non-linear k ε model tends to calculate higher values of total stress than the linear k ε

model and maximum values of ut ut and ut vt are calculated reasonably accurately by this

model. The maximum value of vt vt calculated by both models is considerably too high.

C.1. Calculated Flow Results for g

d 0 50

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C D C D

RMS

C L C L

RMS

linear k εmodel 2.09 2.10 -0.24 1.04

non-linear k εmodel 2.74 2.77 -0.39 1.96

Table C.3: Lift and drag values for square cylinder with g

d

0

75

C.2 Calculated Flow Results for g d

0

75

When the square cylinder is placed g

d 0 75 from the wall, vortex shedding behaviour

in the wake is well established (Bosch et al, 1996). Bosch et al measured this flow and the

current calculations are compared with their measurements. The flow has been calculated

with a Reynolds number, Re 22

000 and 4% turbulence intensity at inlet, as used in the

measurements. Bosch et al did not record any information about inlet turbulence dissipation

rate which has been specified for the calculation by the assumed viscosity ratio, νt

ν

10.All calculated and measured results are normalised by the inlet streamwise velocity U o.

As with the g

d 0 50 case, the calculated and measured velocities are shown as time-

averaged values, averaged over one complete vortex-cycle. Bosch et al separated their mea-

surements of the total stress into the separate Reynolds stress and “periodic stress” compo-

nents. Calculated values of stress are shown principally as total stress; some discussion of

the Reynolds stress and “periodic stress” contributions is included.

Drag and Lift

Coefficients of drag and lift are shown in Figure C.7. Whereas the linear k ε model cal-

culates simple variations in both C L and C D, the non-linear model only calculates a more

complex variation in C D. The mean and root-mean-square (RMS) values of C D and C L are

summarised in Table C.3.

The mean and RMS values of C D and C L calculated by the linear k ε model do not

change greatly as the cylinder wall-distance is increased from g

d 0

50 to 0.75. In contrast,

the coefficients calculated by the non-linear k ε model do change, in particular the RMS

value of C L increases from 0.95 to 1.96.

The values of Strouhal number measured by Taniguchi et al (1983), Durao et al (1991)

and Bosch et al (1996) and the calculated values are summarised in Table C.4. As with the

calculation at g

d 0 50, there is a clear improvement in the vortex shedding frequency when

using the non-linear k ε model. However, it should be noted that the calculated Strouhal

number is now too low, rather than too high, as it was for g

d 0

50.

C.2. Calculated Flow Results for g

d 0 75

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St

Experiments 0.140

linear k epsilon model 0.164

non-linear k epsilon model 0.134

Table C.4: Strouhal numbers for square cylinder with g

d 0

75

Velocity Profiles

Time-averaged U and V -velocity profiles are shown in Figures C.8 and C.9. The linear k

ε model calculates the recirculation length behind the cylinder and the U -velocity profiles

in the lower shear layer of the wake better than the non-linear model. The linear model

also calculates V -velocity more accurately than the non-linear model, although neither model

calculates the strong negative V -velocity immediately above the cylinder.

Reynolds Stress Profiles

Profiles of total stresses, ut ut

vt vt and ut vt are shown in Figures C.10 to C.12 respectively.

The non-linear k ε model calculates all three stresses more accurately than the linear model,

although it tends to calculate the stresses somewhat high (whereas the linear model calculates

them somewhat low).

In Figure C.13 the total normal (ut ut ) stress and “periodic stress” contribution (uu), cal-

culated by the non-linear k ε model, and measured values are shown for two locations in the

wake. From the measured values it is clear that the “periodic stress” is the lesser contributor to

the total stress and consequently the Reynolds stress must be relatively large. The calculated

stresses show the opposite: the “periodic stress” is large and the Reynolds stress relatively

small. Indeed, it is the low level of Reynolds stress, and associated low value of turbulent

viscosity, calculated by the non-linear model which allow a vigorous periodic motion in the

wake. This vigorous periodic motion generates large values of uu, resulting in a reasonably

accurate calculation of total stress, despite the error in the relative sizes of “periodic stress”

and Reynolds stress.

C.3 Conclusions

Although they do not form part of the key areas of investigation in this thesis, the flows

around the square cylinder at g

d 0

50and0

75 from the wall are quite informative. Both

the linear and non-linear k ε models are capable of calculating the periodic, vortex-shedding

C.3. Conclusions

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behaviour; the non-linear model calculates the shedding freqency more accurately and to

within 5%.

The ability to calculate vortex-shedding of the linear model in particular, is no doubt in-

fluenced by the upstream inlet condition. For the calculations of the flow around the cylinder

at g

d

0

25 from the wall (Chapter 4) it was demonstrated that changing the inlet condi-tion from νt

ν 10 to 100 had a dramatic effect in increasing the Reynolds stresses around

the cylinder. If this were repeated for the cylinder at g

d 0 50, the increase in calculated

Reynolds stress (and consequent increase in νt at the cylinder) would greatly inhibit vortex

shedding.

The measured time-average recirculation region behind the cylinder is much shorter for

the vortex-shedding flows (g

d 0

50

0

75) than steady flow (g

d 0

25). Both the linear

and non-linear models calculate a too long recirculation region behind the cylinder for steady

flow (Chapter 4). However, for the periodic vortex-shedding flow, both models calculate the

time-averaged recirculation length with a reasonable degree of accuracy, but contrary to the

steady case, these are slightly too short.

The non-linear model calculates too much total stress in the wake of the cylinder for both

the g

d 0

50 and 0.75 cases. From the measured data at g

d 0

75 it is apparent that

the non-linear model calculates too little Reynolds stress. Due to there then being too little

turbulent viscosity, the “flapping” motion is too strong and the periodic contribution to the

total stress is too high.

C.3. Conclusions

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Appendix D

Cylinder of Square Cross-Section with

Modified c µ

D.1 Introduction

The realizability condition which is developed in Appendix A can be considered as equivalent

to using a standard k ε model with a functional form of c µ (Equation A.30). Where there

are high strain rates or vorticity, this form does not reduce the level of c µ as much as other

forms which have been used in this work (Figure 2.1). Regarding the flow over the Ahmed

body with the 25o rear slant, the non-linear k ε model calculates separated flow over a

large portion of the slant (Section 7.3.2.1). Also, it appears that either the original functional

form of c µ (Equation 2.28) or the non-linear contributions to the Reynolds stresses alone are

sufficient to cause this separation. It may therefore be necessary to use a “weakened” function

for c µ in future calculations of the Ahmed body flows.

To assess the effect of the modified c µ on the non-linear model in a simpler flow, the flow

around the cylinder of square cross-section placed near a wall has been recalculated at the end

of the research programme. The wall-distance used is g

d 0 25, which produces a non-

vortex shedding wake. The inlet and boundary conditions used are as described in Chapter

4: Re 13 600, inlet turbulence intensity is 6% and inlet disipation rate is calculated from

νt

ν

10.

157

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APPENDIX D. Cylinder of Square Cross-Section with Modified c µ 158

D.2 Calculated Flow

Drag and Lift

C D and C L are shown in Figure D.1 with the plots for the linear k ε model and standard

non-linear k

ε model (Craft et al, 1996b). The small oscillation in C D and C L which was

calculated by the standard non-linear model is not calculated by the non-linear model with

modified c µ. The value of the coefficient of drag calculated by the modified non-linear model,

C D 1

81, is similar to the values calculated by the linear and standard non-linear models

(both calculate C D 1

77). However, the coefficient of lift shows a larger change, calculated

as C L

0 22 by the modified non-linear model and C L

0 46 by the standard non-linear

model.

Velocity Profiles

U -velocity profiles calculated by the standard and modified non-linear k ε models are shown

in Figure D.2. The profiles calculated by the modified non-linear model are a closer match to

the measured data. The velocity gradients in the shear layers are not so steep when calculated

by the modified non-linear model and the recirculation length is not so long.


Contours of Reynolds stress (uu only) are shown in Figure D.3 (measured contours and con-

tours calculated by the standard non-linear model are shown also for reference). Two notablefeatures are apparent. Firstly, the locations of the peak values of uu-stress in the upper shear

layer above the cylinder and in the wake more closely match the measured locations, al-

though the values are still somewhat low. Secondly, too much uu-stress is generated at the

impingement point on the leading surface.

D.3 Conclusion

In the modified function, c µ

is not decreased as much as in the original version in the presence

of high strain rates. This allows the modified non-linear model to calculate higher values of

turbulent viscosity and Reynolds stress in the shear layers around the cylinder and in the

wake. This has the effect of reducing the steepness of the calculated velocity gradients and

reduces the length of the downstream recirculation region. However, these improvements are

at a cost. The reduction in c µ at impingement is not sufficient to prevent the production of

turbulent kinetic energy from being calculated too high. Whilst this is not a serious defect in

D.2. Calculated Flow

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APPENDIX D. Cylinder of Square Cross-Section with Modified c µ 159

this calculation, if the quantity to be calculated were more sensitive to the Reynolds stresses

(eg. if heat transfer were involved) the errors induced could be significant.

D.3. Conclusion

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Appendix E

Calculation of Drag for the Ahmed Body

The drag force experienced by the Ahmed body consists of two components: the pressure

drag and the skin friction drag. The coefficient of drag due to pressure is calculated for the

nose cone, rear slant and base by:

C DP

∑ncelli

1

C Pi A xi

A x

body

(E.1)

where for a given surface: C DP is the coefficient of drag due to pressure, A xi is the wall-area

of cell i projected in the x-direction, C Pi is the coefficient of pressure and ncell is the number

of cells which comprise the surface.

A x

body is the area of the whole Ahmed body projected

in the x-direction. Similarly the coefficient of drag due to friction is calculated for the nose

cone, top, side, bottom and slant surfaces by:

C DF

∑ncelli

1

C F

xi Ai

A yz

body

(E.2)

where C DF is the coefficient of drag due to friction,

C F

xi is the coefficient of friction calcu-

lated from the x-direction component of shear stress, Ai is the wall-area of a given cell i, and

A yz

body is the total surface area of the body on which friction drag acts. The coefficient of

friction is calculated for each cell by:

C F

xi

τw

xi

0 5ρU 2o(E.3)

τw

xi is the x-direction component of wall shear-stress and U o is the upstream (undisturbed)

velocity used in the calculation of Reynolds number.

As the stilts on which the Ahmed body is supported are not included in any of the calcu-

160

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APPENDIX E. Calculation of Drag for the Ahmed Body 161

lations, it is necessary to adjust the calculated drag to account for their effect. Massey (1989)

states that the coefficient of drag for an infinitely long cylinder 1 with a turbulent boundary

layer is C D

0 7. The wall functions used throughout this work impose turbulent boundary

layers, thus it is consistent to let the drag coefficient of the stilts C Dstilt 0 7.

To adjust the calculated total drag for the body to account for the stilts the followingformula was used:

C W C Dbody

Abody

Atot

C Dstilt

Astilt

Atot (E.4)

where Abody and Astilt are the projected frontal areas of the Ahmed body (without stilts) and

the stilts alone and Atot Abody

Astilt .

1It is acknowledged that the stilts supporting the Ahmed body are not infinitely long cylinders and that the

presence of the floor and underside of the body will affect the flow around the stilts. Horseshoe vortices will

form at the ends of the stilts which will increase the drag. The drag increase due to these vortices has not been

included in the modification as their overall effect would be small compared to the drag of the body itself.

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