Flaps with Wavy Leading Edges for Robust Performance agains Upstream Trailing Vortices

University of Southampton

Faculty of Engineering and the Environment

School of Engineering Sciences

Master of Science Dissertation

Flaps with Wavy Leading Edges

for Robust Performance against

Upstream Trailing Vortices

By

Rafael Pérez Torró

[email protected]

A dissertation submitted in partial fulfilment of the degree of Master in Science (MSc)

in Race Car Aerodynamics by taught course

First Supervisor: Dr. Jae Wook Kim

Second Supervisor: Prof. Bharathram Ganapathisubramani

September 2013

mailto:[email protected]

ii

Acknowledgements

Firstly I would like to thank the University of Southampton for providing the

computational resources needed to this project’s fulfilment and especially to my project

supervisor Dr. Kim who first gave me the freedom to complete this project in the way

that I so wished and ultimately helped me out to improve the quality of the written

work.

Secondly I would like to mention my colleagues with whom I had an amazing time here

in Southampton and who were always willing to help me out.

In third place I have a special mention to the Postinger family who made me feel as if I

was part of their family, cheering me up when I was sad and giving me support when I

was alone. A special mention needs to be dedicated to Hannah Postinger too who

always stood by my side and brought a little bit of love into my life. Thank you very

much Hannah.

Per acabar, m’agradaria dedicar també unes paraules en la meua llengua materna a la

meua família. Moltes gràcies per donar-me el vostre suport incondicional. Jo sé que

vosaltres ho donaríeu tot per a què jo poguera tenir el millor futur possible, però

realment a dins de mi jo també sé que si puc ser la meitat de feliç del que vosaltres sou

podeu estar tranquils, heu fet un gran treball. Moltes gràcies de tot cor, us estime molt.

iv

Abstract Computational Fluid Dynamic calculations have been carried out using URANS

turbulence models in order to test the performance of a two-element airfoil with Wavy

Leading Edge flap shape against upstream trailing vortices. The GA(W) – 1 airfoil

shape was used for the main element and the NACA 634-021 airfoil shape was used for

the flap element. Because of the lack of any previous studies for this configuration, a

parametric study of the elements relative position was first of all carried out. Once the

relative position of the elements was defined the sinusoidal shape was introduced into

the design with different amplitudes and wavelengths and simulations were performed

with usual clean upstream conditions. Results of this experiment showed that the flaps

with undulated leading edges performed worse than the plain leading edge flaps for the

configuration tested here in both terms of mean lift and drag since the relative position

of the elements was just optimised for the plain leading edge case. A second experiment

was carried out in which the effect of an upstream trailing vortex was modelled using a

vortex-train perpendicular to the streamwise direction. Results of this latter experiment

showed that with the conditions tested, the models with Wavy Leading Edges

performed in a more robust way than the straight leading edge flaps.

Contents Acknowledgements .......................................................................................................... ii

Abstract ............................................................................................................................ iv

Contents ........................................................................................................................... vi

List of Figures ................................................................................................................ viii

List of Tables .................................................................................................................... x

Nomenclature.................................................................................................................. xii

Chapter 1 : Introduction ........................................................................................... 1

1.1. Dissertation’s Organisation ............................................................................ 1

1.2. Introduction and motivation of the problem .................................................. 1

1.3. Approach to the Problem................................................................................ 2

1.4. Objectives ....................................................................................................... 3

1.5. Literature Review ........................................................................................... 3

Chapter 2 : Theoretical Background ........................................................................ 9

2.1. Basic Notions and Assumptions ..................................................................... 9

2.2. Turbulence .................................................................................................... 12

2.3. Reynolds Averaged Navier-Stokes Equations ............................................. 15

2.4. Eddy Viscosity Approximation .................................................................... 17

2.5. Linear Eddy Viscosity Models ..................................................................... 22

2.6. Unsteady RANS (URANS) .......................................................................... 25

2.7. Discretising the Equations: the Finite Volume Method (FVM) ................... 26

Chapter 3 : Problem Description ........................................................................... 29

3.1. Ambient Conditions ..................................................................................... 29

3.2. Tools Used.................................................................................................... 29

3.3. Model’s Geometry........................................................................................ 29

3.4. Mesh ............................................................................................................. 31

3.5. Domain Size and Boundary Conditions ....................................................... 33

3.6. Time Resolution ........................................................................................... 34

3.7. Turbulence Model ........................................................................................ 35

Chapter 4 : Results ................................................................................................. 37

4.1. Elements’ Relative Position Study ............................................................... 37

4.2. Experiment A: Flaps with Wavy Leading Edges ......................................... 41

vii

4.3. Experiment B: Flaps with Wavy Leading Edges against Upstream Vortex

Condition ................................................................................................................ 54

Chapter 5 : Final Remarks ..................................................................................... 65

Appendix A: Mesh Dependency Study ...................................................................... 69

Appendix B: Domain Size Dependency Study .......................................................... 71

Appendix C: Validation against S-A Turbulence Model ........................................... 73

Appendix D: Matlab, VBA and JAVA Codes ............................................................ 79

References .................................................................................................................. 91

viii

List of Figures Figure 1.1 Humpback Whale’s Flippers detail ................................................................. 2

Figure 2.1 Water Jet Turbulence: Instabilities quickly develop until fully turbulent flow

is achieved. ..................................................................................................................... 13

Figure 2.2 Typical Hot-Wire velocity measurement ...................................................... 15

Figure 2.3 2D Shear Flow Detail .................................................................................... 19

Figure 2.4 URANS Measurement of a Mean Quantity .................................................. 26

Figure 2.5 Volume Ω divided into three subvolumes Ω1, Ω2, and Ω3 ............................ 27

Figure 3.1 Sketch of the Elements’ relative position ..................................................... 30

Figure 3.2 Gap and Overlap Definition .......................................................................... 30

Figure 3.3 WLE Sketch .................................................................................................. 31

Figure 3.4 Volumetric Controls around the model ......................................................... 32

Figure 3.5 Boundary Layer Mesh Detail ........................................................................ 32

Figure 3.6 Wall y+ contour levels over the model .......................................................... 32

Figure 3.7 Wall y+ Distribution ...................................................................................... 33

Figure 3.8 Boundary Conditions and Domain Size Sketch (not to Scale) ..................... 33

Figure 3.9 Courant Number Detail ................................................................................. 35

Figure 4.1 Optimisation Cycle........................................................................................ 38

Figure 4.2 Response Surface with Red Dot indicating the Final Baseline Model ......... 40

Figure 4.3 Baseline geometry with Coordinate System Indications. From Top-Left to

Bottom-Right: Top View, Isometric View, Front View and Rear View. ....................... 40

Figure 4.4 CL and CD Coefficients for all the Models Tested ........................................ 42

Figure 4.5 CLflap Coefficient for all the Models Tested .................................................. 43

Figure 4.6 Wall Shear Stress [i] Contours on the Flap Surface plus Vorticity [i]

Contours on Several Spanwise Planes: a) model0, b) model1.1, c) model2.1 and d)

model3.1 ......................................................................................................................... 44

Figure 4.7 model0 Base Section, Velocity Slice non-dimensionalised by the Free Stream

Velocity .......................................................................................................................... 45

Figure 4.8 model1.1 Base Section, Velocity Slice non-dimensionalised by the Free

Stream Velocity .............................................................................................................. 45

Figure 4.9 model1.1 Peak Section, Velocity Slice non-dimensionalised by the Free

Stream Velocity .............................................................................................................. 46

Figure 4.10 model1.1 Valley Section, Velocity Slice non-dimensionalised by the Free

Stream Velocity .............................................................................................................. 46

Figure 4.11 model2.1 Base Section, Velocity Slice non-dimensionalised by the Free

Stream Velocity .............................................................................................................. 47

ix

Figure 4.12 model2.1 Peak Section, Velocity Slice non-dimensionalised by the Free

Stream Velocity .............................................................................................................. 47

Figure 4.13 model2.1 Valley Section, Velocity Slice non-dimensionalised by the Free

Stream Velocity .............................................................................................................. 47

Figure 4.14 CP Contours on the models surface: a) model0, b) model1.1, c) model2.1

and d) model3.1 .............................................................................................................. 48

Figure 4.15 Wall Shear Stress [i] Contours on the Flap Surface plus Vorticity [i]

Contours on Several Spanwise Planes: a) model1.4, b) model1.8, c) model2.4, d)

model2.8, e) model3.4 and f) model3.8 .......................................................................... 50

Figure 4.16 Velocity Slice non-dimensionalised by the Free Stream Velocity: a)

model1.4 Peak Section, b) model1.4 Valley Section, c) model2.4 Peak Section, d)

model2.4 Valley Section, e) model3.4 Peak Section and f) model3.4 Valley Section .. 51

Figure 4.17Velocity Slice non-dimensionalised by the Free Stream Velocity: a)

model1.8 Peak Section, b) model1.8 Valley Section, c) model2.8 Peak Section, d)

model2.8 Valley Section, e) model3.8 Peak Section and f) model3.8 Valley Section .. 52

Figure 4.18 Isosurface of Streamwise Vorticity Ω=±100 s-1

: a) model1.4, b) model1.8,

c) model2.4, d) model2.8, e) model3.4 and f) model3.8 ................................................ 53

Figure 4.19 CP Contours on the models surface: a) model1.4, b) model1.8, c) model2.4,

d) model2.8, e) model3.4 and f) model3.8 ..................................................................... 54

Figure 4.20 Detail of Pressure Calculation from Initial Conditions: a) Vorticity contours

at t=0s, b) CP Contours at t=0s, c) Vorticity Contours at t=0.1s, and d) CP Contours at

t=0.1s .............................................................................................................................. 55

Figure 4.21 Total CL History Comparison ..................................................................... 56

Figure 4.22 CLflap History Comparison ........................................................................... 56

Figure 4.23 CLmain History Comparison.......................................................................... 57

Figure 4.24 Total CL Statistic Values Comparison ........................................................ 57

Figure 4.25 CLflap Statistic Values Comparison .............................................................. 58

Figure 4.26 CLmain Statistic Values Comparison............................................................. 58

Figure 4.27 model0 CP rms contours .............................................................................. 59

Figure 4.28 model1.4 CP rms contours ........................................................................... 59




Figure 4.32 model0 Wall-Shear Stress [i] contours: a) t=0.2s, b) t=0.3s, c) t=0.4s, d)

t=0.5s, e) t=0.6s and f) t=0.7s. ........................................................................................ 62

Figure 4.33 model1.4 Wall-Shear Stress [i] contours: a) t=0.2s, b) t=0.3s, c) t=0.4s, d)

t=0.5s, e) t=0.6s, f) t=0.7s ............................................................................................... 62

Figure 4.34 model1.8 Wall-Shear Stress [i] contours: a)t=0.2s, b)t=0.3s, c) t=0.4s, d)

t=0.5s, e) t=0.6s, f) t=0.7s ............................................................................................... 63

x


t=0.5s, e) t=0.6s, f) t=0.7s ............................................................................................... 63


t=0.5s, e) t=0.6s, f) t=0.7s ............................................................................................... 64

Figure A.1 Mesh Dependency Study Convergence ....................................................... 69

Figure B.1 Domain Size Convergence ........................................................................... 71

Figure C.1 Velocity Contours Comparison .................................................................... 73

Figure C.2 Velocity Profiles Comparison Moving from 1 to 4 in the stream direction . 74

Figure C.3 Pressure Coefficient Distribution Comparison ............................................. 74

Figure C.4 Pressure Coefficient Contours Comparison ................................................. 75

Figure C.5 Skin Friction Coefficient Distribution Comparison ..................................... 76

Figure C.6 Wall Shear Stress X-Direction Contours Comparison ................................. 76

Figure C.7 Q-Criterion Iso-Surfaces Q = 100 Coloured by Dimensionless Velocity X-

Direction Comparison .................................................................................................... 77

List of Tables Table 3.1 Ambient Values .............................................................................................. 29

Table 4.1 Final Baseline Parameters .............................................................................. 39

Table 4.2 Models Parameters in terms of cflap ................................................................ 41

Table 4.3 Experiment A Results ..................................................................................... 42

Table A.1 Mesh Dependency Study Results .................................................................. 69

Table B.1 Domain Size Study ........................................................................................ 71

xii

Nomenclature

Symbols

Re = Reynolds Number

LC = Lift Coefficient

DC = Drag Coefficient

PC = Pressure Coefficient

LmainC = Lift Coefficient Main Element

LflapC = Lift Coefficient Flap Element

c = Chord

b = Elements’ Span

mainc = Main Element’s Chord

flapc = Flap Element’s Chord

t = Time

= Density

V = Volume

S = Surface

u

= Velocity Vector

f = Scalar Function

wandvu, = Velocity Components

zandyx ,, = Cartesian Coordinates

iu = Velocity Components Tensor Notation

ix = Cartesian Coordinates Tensor Notation

i

= Stress Vector

p = Pressure

ij = Kronecker Delta

xiii

ij = Viscous Stress Tensor

ijS = Rate-of-Strain

= Dynamic Molecular Viscosity

= Kinematic Molecular Viscosity

U = Free Stream Velocity

l = Characteristic Length

k = Turbulent Kinetic Energy

= Kolmogorov Length Scale

= Dissipation Rate of Turbulent Kinetic Energy

0l = Large Scales Length Scale

0u = Large Scale Velocity Scale

u = Turbulence Velocity Fluctuation

u = Time Average Velocity

jiuu = Reynolds Stress Tensor

ij = Reynolds Stress Tensor, General Stress Tensor

ijb = Reynolds Stress Isotropic Part

ija = Reynolds Stress Anisotropic Deviatoric Part

u = Molecular Velocity Fluctuations

xyp = Momentum Flux across XY Plane

T = Turbulent Eddy Viscosity

Tl = Turbulent Length Scale

Tu = Turbulent Velocity Scale

Tt = Turbulent Time Scale

m = Molecular Mass

mfpl = Mean Free Path Distance

thv = Thermal Velocity

xiv

n =

Number of Molecules per Unit Volume, Number of Waves along the

Flap’s Span

A = Wavy Leading Edge Amplitude

LEx = Flap’s Leading Edge x-coordinates

LEz = Flap’s Leading Edge z-coordinate

00 yandx = Flap’s Leading Edge Position

= Flap’s Deflection Angle

= Wavy Leading Edge Wavelength

= Specific Dissipation Rate of Turbulent Kinetic Energy

Kn = Knundsen Number

Ma = Mach Number

= Scalar Function

F

= Flux of a Scalar Function

= Rate-of-Rotation, Volume Control

tandn

= Normal and Tangential Unitary Vectors

x = Grid Resolution in the x-direction

t = Time Step

C = Courant Number

rms = Root-Mean-Square

w = Wall Shear Stress

00 yandx = Vortex Core Initial Position

r = Radial Vortex Coordinate

21 kandk = Vortex Equation Constants

L = Domain Length

)(r = Vortex Function

xv

Abbreviations

CFD = Computational Fluid Dynamics

RANS = Reynolds Averaged Navier-Stokes

CAA = Civil Aviation Authority

AoA = Angle of Attack

FAA = Federal Aviation Administration

AIM = Aeronautical Information Manual

WLE = Wavy Leading Edge

LE = Leading Edge Protuberances

TE = Trailing Edge

URANS = Unsteady RANS

S-A = Spalart Allmaras

SST = Shear Stress Transport

APG = Adverse Pressure Gradient

LEP = Leading Edge Protuberances

DES = Detached Eddy Simulation

LES = Large Eddy Simulation

DNS = Direct Numerical Simulation

LDV = Laser Doppler Velocimetry

AR = Aspect Ratio

RHS = Right Hand Side

LHS = Left Hand Side

NSE = Navier-Stokes Equations

HIT = Homogeneous Isotropic Turbulence

TKE = Turbulent Kinetic Energy

HOT = High Order Terms

LEVMs = Linear Eddy Viscosity Models

FVM = Finite Volume Method

FDM = Finite Difference Method

CAD = Computer Aided Design

PDAS = Public Domain Aeronautical Software

WLEF = WLE Flap

VC = Volumetric Control

VBA = Visual Basic for Applications

HA = High Amplitude

MA = Medium Amplitude

LA = Low Amplitude

HW = High Wavelength

MW = Medium Wavelength

LW = Low Wavelength

WSS = Wall Shear Stress

Chapter 1 : Introduction

1.1. Dissertation’s Organisation

Here we are going to introduce the following chapters and subsections in order to give

the reader an overall view of what can be found in this dissertation.

This chapter, the first one, describes the problem the project is focused on and tries to

give a general overview of the previous related work done. In the second chapter the

author reviews the fundamental equations over which all CFD codes are bases and more

specifically the RANS models. In the third chapter a deeper technical description of

how the problem is going to be tackled is given, including the tools and geometry used

as well as general simulation parameters such as mesh size and domain size. In the

fourth chapter the more relevant results obtained in this project are shown and explained

and finally in chapter five a general conclusion and recommendations can be found.

1.2. Introduction and motivation of the problem

According to the Civil Aviation Authority (CAA) there were nearly three million

airplane movements in 2012 in the United Kingdom1. This means nearly 8,000 flights

per day in the UK. Supposing that the same amount of planes that depart from the UK

destined to other countries equals the number of planes arriving UK from other

countries the number of airport operations (take-off and landing) can be roughly

approximated to 16,000 operations per day, only in UK airports.

Every time an airplane either takes off or lands leaves a turbulent wake behind. This

wake is mostly composed by the trailing vortices produced by the airplane wings. This

phenomenon is more pronounced because of the high-lift devices used during take-

off/landing. Rarely the wake produced by an aircraft can produce structural damages on

the aircraft encountering the wake. However it is more likely the vortices induce rolling

moments which can exceed the roll−control authority of the encountering aircraft.

Additionally the induced velocities could increase the effective angle of attack (AoA)

beyond the stall angle and thus causing a failure in the operation.

Due to the high number of take-off/landing operations the air in the runway’s

surroundings can be highly disrupted, and operations might be postponed until ambient

air has settled down. In fact the United States Federal Aviation Administration (FAA)

dedicates in its Aeronautical Information Manual (AIM) a whole chapter to Wake

Turbulence (AIM Chapter 7, section 3)2.

1

CAA, “Table 3.2 Aircraft Movements 2002 - 2012 (in Thousands),” 2012,

http://www.caa.co.uk/docs/80/airport_data/2012Annual/Table_03_2_Aircraft_Movements_2002_2012.pd

f. 2 United States. Federal Aviation Administration., “Aeronautical Information Manual : Official Guide to

Basic Flight Information and ATC Procedures,” Official Guide to Basic Flight Information and ATC

Procedures Aim (2012): v.

Chapter 1: Introduction 2

Since the ideal case would be a continuous landing/taking-off scenario there exist a

need in reducing the sensitivity of the airplane airfoils/wings to these trailing vortices

and hence increase the number of operations under safety conditions.

1.3. Approach to the Problem

Given that one of the biggest issues associated with trailing vortices is the premature

stall they provoke, the solution to the problem must be so that it delays the stall under

such turbulent conditions. It turns out that, in recent years, researchers such as Fish and

Launder3 have found that wavy leading edge (WLE) such as those the humpback

whale’s flipper possesses (see Figure 1.1) are an efficient passive method for delaying

stall.

Figure 1.1 Humpback Whale’s Flippers detail4

The objective of this project is to implement the WLE in a landing/take-off high-lift

configuration. Nonetheless all the studies published regarding WLE are based on single

element configurations, so in order to implement the WLE in a high-lift configuration

two airfoils in tandem will be studied. The rear airfoil of the tandem will be considered

as a flap, and it will be the one with the leading edge (LE) modifications.

All the tests that will be carried out in this project will be computer simulations of the

flow based on the Unsteady Reynolds Averaged Navier-Stokes (URANS) equations

with different turbulent closure models. These simulations will test if the WLE can

improve the performance of high-lift configurations against upstream trailing vortices

and if they can do it in a robust manner.

The process that will be carried out to test the WLE effect will be broken down into two

major experiments. 1) The first part involves the combination of the main element

followed by a second airfoil (flap) with the modified LE. In this part the effect of the

shape of the WLE is tested. 2) The second experiment tests the performance of different

3 F.E. Fish and G.V. Lauder, “Passive and Active Flow Control by Swimming Fishes and Mammals,”

Annual Review of Fluid Mechanics 38 (2006), doi:10.1146/annurev.fluid. 4

“Humpback Whale Speaks, Says ‘Thank You’,” Sea Monster, accessed September 19, 2013,

http://theseamonster.net/2011/07/humpback-whales-speaks-says-thank-you/.


WLE shapes against similar turbulent conditions such as those produced by trailing

vortices. The model is hit by vortex convected by the free stream flow, which emulates

the effect of the trailing vortices present during ground operations.

1.4. Objectives

The main objectives to be accomplished in this project are:

First to implement the promising Wavy Leading Edges shapes that have given

good results on airfoils near stall regime situations into a two-element high-lift

configuration and investigate how can they perform with clean upstream

conditions

Second investigate if the Wavy Leading Edges can be beneficial when used in a

two-element high-lift configuration against an upstream trailing vortex

condition.

1.5. Literature Review

In this project major attention is put into the ability of the WLE to delay stall, however

the leading edge modifications are applied to the second element’s leading edge of a

two-element high-lift configuration airfoil. Therefore it is worthwhile to gather some

information about these two major topics, i.e. High-Lift Aerodynamics and Wavy

Leading Edges.

High-Lift Aerodynamics

The paper published by A.M.O. Smith5 in 1975 might be the most referenced paper

regarding high-lift aerodynamics. The author notes that the use of multielement

configurations for high-lift has been present since early XX century. Although it has

been suggested that a single airfoil shape can obtain the same amount of lift as a multi-

element design, Smith states that properly designed multielement are more convenient

and that generally the greater the number of elements the greater the lift. Because of

Bernoulli equation it is well known that that if the surface is to lift the velocity over it

must be higher than that in the pressure surface. But when the flow reaches the trailing

edge the flow is decelerated to velocities even lower than the freestream. If this

deceleration is too stiff it may cause separation because of the high adverse pressure

gradient (APG). Two major aspects then arise when designing high-lift devices: 1) The

analysis of the boundary layer, prediction of separation, and determination of the kinds

of flows that are most favourable with respect to separation; and 2) analysis of the

inviscid flow about a given shape with the purpose of finding shapes that put the least

stress on the boundary layer.

Smith points out that one of the biggest misconceptions in multielement aerodynamics

is that slots supply a blowing type of boundary layer control. Instead the author

5

A. M. O. Smith, “High-Lift Aerodynamics,” Journal of Aircraft 12 (June 1, 1975): 501–530,

doi:10.2514/3.59830.


highlights the five major effects present in a common high-lift configuration using

multielement airfoils.

1) Slat effect: The circulation velocities of a former element, for example, a slat, run

counter to the velocities on a downstream element in its LE zone and so reduce pressure

peaks on the downstream element. The effect of this slat effect is to delay the stall

angle.

2) Circulation effect: The upstream element is also beneficiated by the presence of the

downstream element which causes the trailing edge of the adjacent upstream element to

be in a region of high velocity that is inclined to the mean line at the rear of the forward

element. Such flow inclination induces considerably greater circulation on the forward

element.

3) Dumping effect: Because the trailing edge of a forward element is in a region of

velocity appreciably higher than freestream, hence the flow in the boundary layer does

not suffer such a severe deceleration as it would happened in isolation. The higher

discharge velocity relieves the pressure rise impressed on the boundary layer, thus

alleviating separation problems or permitting increased lift.

4) Off-the-surface pressure recovery: The boundary layer from forward elements is

dumped at velocities appreciably higher than freestream. The final deceleration to

freestream velocity is done in an efficient manner. The deceleration of the wake occurs

out of contact with a wall. Such a method is more effective than the best possible

deceleration in contact with a wall.

5) Fresh-boundary-layer effect: Each new element starts out with a fresh boundary layer

at its LE. Breaking up a flow into several short boundary layer runs reduces the risk of

separation because thin boundary layers can withstand stronger adverse gradients than

thick ones and thus lift can be increased.

Rumsey and Ying6 wrote an exceptional review of different numerical investigations

regarding high-lift aerodynamics of multi-element airfoils. The authors point out that

most of the calculations of multi-element CFD calculations have been carried out using

structured-grid Reynolds Averaged Navier-Stokes (RANS) codes. The turbulence

models S-A and Menter’s SST arise as the best options for closing the RANS equations.

In this paper the work of Godin et al.7 is mentioned in which several calculations over

the GA (W)-1 2-element configuration using both S-A and Menter’s models were

conducted. For this case both the S-A and SST models gave very similar results for this

case, with the SST model giving pressures in the separated region on the flap in

somewhat better agreement with experiment. Velocity profiles, including the main

element wake, were predicted similarly by both models in reasonable agreement with

experiment. Turbulent shear stress profiles were also similar between the two models,

but did not agree quite as well with the experiment, particularly in the main element’s

wake region.

Rumsey and Ying also review some investigations done using the NLR-7301 geometry

(typical take-off configuration). Their major findings among others were that in general

6 Christopher L. Rumsey and Susan X. Ying, “Prediction of High Lift: Review of Present CFD

Capability,” Progress in Aerospace Sciences 32 (2002): 145–180. 7 P. Godin, D. W. Zingg, and T. E. Nelson, “High-Lift Aerodynamic Computations with One- and Two-

Equation Turbulence Models,” AIAA Journal 35 (February 1, 1997): 237–243, doi:10.2514/2.113.


terms the CL,max tended to be over predicted. In addition the k-ε model was found to

predict larger values of the skin friction and gave worse agreement with experimental

data. Compressible formulation showed better agreement with experiments in the post-

stall regimes. Finally drag was found to be sensitive to the far-field grid and boundary

conditions whereas lift coefficient tend to converge faster as regards mesh resolution

and domain length. A possible cause of this drag misprediction could be the poor

resolution of the grid resolving the wakes of the profiles as suggested by Anderson et

al.8.

Although most of the CFD studies in high-lift devices have been conducted in 2D

Rumsey and Ying state that 3-D RANS computations have recently gain a higher status.

They seem to generally be able to predict many complex multi-element flow fields at

angles of attack below stall. However, their performance near maximum lift conditions

has been less reliable, depending on particular configurations. The authors state that

more grid refinement is necessary in 3-D so results can be truly trusted. Despite this the

use of 3-D CFD calculations is highly encouraged by the authors since flows near

maximum lift possesses dominant three-dimensional effects.

Wavy Leading Edges

During recent years the astonishing agility and manoeuvrability of such a big maritime

mammal as the humpback whale has caught the attention of many researchers. The

humpback whale or “Megaptera novaeangliae” flipper is unique because of the

rounded protuberances located on its LE. This feature present in the whale flippers,

which are a result of the evolutionary Darwinian “natural selection”, act as a passive-

flow control that improves the flipper’s performance. The protuberances delay the stall

angle of the flipper therefore increasing its lift in the post stall regime without adding

additional drag9. Because of this beneficial effect that the LE tubercles have on the

whale’s flipper, it is thought that they might be used in aerodynamic airfoils to increase

its performance too.

The humpback whale’s flippers, which are the longest of any cetacean10

have been

studied both via experimental procedures and numerical computations.

Experimental investigations were conducted by Hansen et al.11

regarding the effect of

different amplitude and wavelengths of LE tubercles on two different airfoils, NACA

0021 and NACA 65-021 at Re = 1.2e5. Result revealed that protuberances were more

effective on the latter airfoil. It was postulated that this was because the position of the

maximum thickness in the NACA 65-021 airfoil was located further downstream which

extends the percentage of the laminar boundary layer. The tubercles, which according to

the authors act like vortex generators, increasing the rate of momentum exchange in the

laminar boundary layer. The authors found that increasing the amplitude leads to a

8 W. Kyle Anderson et al., “Navier-Stokes Computations and ExperimentalComparisons for Multielement

Airfoil Configurations” (presented at the Aerospace Sciences Meeting, Reno, Nevada, 1993). 9 Frank E. Fish et al., “The Tubercles on Humpback Whales’ Flippers: Application of Bio-Inspired

Technolog,” Integrative and Comparative Biology 51 (2011), doi:10.1093/icb/icr016. 10

F. E. Fish and J. M. Battle, “Hydrodynamic Design of the Humpback Whale Flipper.,” J. Morph 225

(1995): 51–60. 11

Kristy L. Hansen, Richard M. Kelso, and Bassam B. Dally, “Performance Variations of Leading-edge

Tubercles for Distinct Airfoil Profiles,” Journal of Aircraft 49 (2011): 185–194, doi:10.2514/1.J050631.


smoother stall; however this is accompanied by a smaller value of CL,max and stall

angles. In contrast, airfoils with smaller protuberance’s amplitude performed much

better than the unmodified profile in the post-stall regime without any significant

difference in the drag production. The paper also shows that although reducing the

wavelength proves to be beneficial, for a given value of the tubercles amplitude there is

a limiting wavelength value for which reducing its value does no longer give improved

characteristics. One key finding in this paper is the fact that for similar values of

tubercles amplitude/tubercle’s wavelength ratio the flow over the airfoil behaves

similarly. The authors suggest that tubercles could be beneficial for use on foils

operating near the stall point or in variable flow conditions.

Lohry et al.12

performed numerical computations over a NACA 0020 with WLE using

RANS models, in particular Menter’s SST model. The authors used both centered and

vertex based schemes developed by their own group in Princeton. Calculations were

performed over a range of Re between 62,500 and 500,000. The k-ω SST model proved

to be able to reproduce the experimental measurements and trends with reasonable

accuracy. This paper illustrates how the leading edge protuberances (LEP) act as vortex

generators which tend to reduce the CL,max but to mitigate the stall. Results indicated

that the variation in thickness along the span due to the WLE modification creates a sort

of channels that create spanwise fences that can be used to increase the CL,max if they are

proper optimised. The Reynolds effect study showed that as postulated by Hansen et

al.13

the WLE can be detrimental for very low Re.

Miklosovic et al.14

performed wind tunnel experiments over a wing model that

resembled to the humpback whale flipper. The model cross section was based on the

NACA 0020 airfoil. A comparison was made between a clean LE configuration and a

modified LE configuration of the model. Results showed that the flipper performance

relies in the presence of the LEP which allows the modified LE model to delay stall to

higher AoA values (40% delayed). The drag of the scalloped model was found to be

lesser than the clean one for most AoA tested leading to an expanded operating

envelope combined with drag reduction. Yet the scalloped wing gave lower values of

CL for some AoA in the pre-stall regime. It was also found that the CL was relatively

insensitive to Reynolds effects for Re > 4e5. Further experiments of the authors

15

included infinite span (2D case) wing models in order to quantify the importance of 3D

effects on the flippers. Results showed that the finite (3D case) span scalloped wings

have a largely 3D benefit that is a function of the planform shape and the Reynolds

number. For the 2D case it was found that the WLE are detrimental in the pre-stall

regime producing less lift and more drag. Nonetheless in post-stall regime the WLE

enhance the airfoil performance delaying and smoothing stall. The authors point out that

the WLE have a utility for lifting devices operating past their stall point.

12

Mark W. Lohry, David Clifton, and Luigi Martinelli, “Characterization and Design of Tubercle

Leading-Edge Wings” (presented at the Seventh International Conference on Computational Fluid

Dynamics (ICCFD7), Big Island, Hawaii, 2012). 13

Hansen, Kelso, and Dally, “Performance Variations of Leading-edge Tubercles for Distinct Airfoil

Profiles.” 14

D. S. Miklosovic et al., “Leading-edge Tubercles Delay Stall on Humpback Whale (Megaptera

Novaeangliae) flippers,” PHYSICS OF FLUIDS 16 (2004). 15

David S. Miklosovic, Mark M. Murray, and Laurens E. Howle, “Experimental Evaluation of Sinusoidal

Leading Edges,” Journal of Aircraft 44 (July 1, 2007): 1404–1408, doi:10.2514/1.30303.


Pedro and Kobayashi16

emulated the experimental tests using DES computations based

on the S-A turbulence model at Re of 5e5 for wings inspired in the humpback whale

flipper. The model tested had the same wing planform as the whale flipper with a

diminishing chord length from the root to the tip of the wing, sort of the same

configuration used in Miklosovic et al.17

. The major findings were that: 1) The Re

highly influences the type of separation, producing leading edge separation in the

outboard section and trailing edge separation otherwise. 2) The higher aerodynamic

performance for the scalloped flipper is due to the presence of stream-wise vortices

originated by the tubercles. The spanwise vortices bring momentum to the boundary

layer preventing the trailing edge separation and plus confine the leading edge

separation to the wing tip.

An improved performance at the post-stall regime was also found by Johari et al.18

who

conducted wind tunnel experiments comparing a plain NACA 634-021 airfoil with a

range of WLE modifications of it with different wavelengths and amplitude for the

protuberances. The experiments were carried out at Re = 1.83e5 using Laser Doppler

Velocimetry (LDV) and tufts.

Wind-tunnel tests of wings with low aspect ratios (AR) of 1 and 1.5, rectangular

planforms, and four distinct sinusoidal LE and one straight LE were conducted by

Guerreiro and Sousa19

at Reynolds numbers of 70,000 and 140,000. Various

combinations of protuberance’s amplitude and wavelength were analysed and the results

indicated that gains of the order of 45% in maximum lift can be achieved combining

large amplitude and wavelength, which contrasts with Hansen et al.20

findings, past the

stall angle of the airfoil. It is has to be remarked that the airfoil used in these

investigations was the LS(1) – 0417 which has significant differences with other more

common models used in investigations regarding WLE. As in all the papers reviewed

for Re 140,000 the WLE proved to be clearly beneficial for AoA greater than the stall

angle being even more important for the 1.5 AR models. For lower Reynolds numbers

(Re = 70,000) the performance of the WLE models was fairly maintained whereas the

clean LE model performance was clearly diminished. The authors concluded that

sinusoidal WLE are less prone to performance deterioration.

Malipeddi21

conducted numerical 3D calculations using the commercial CFD package

Fluent over a NACA 2412 airfoil. All wings tested had a chord and wing span of 0.1.

The calculations were performed at a Re of 5.7e5

using DES on SA and k-ω models.

Investigations were carried out as regards the influence of the undulations wavelength

and amplitude. The amplitude was found to be very important whereas the wavelength

played a minor role. Results showed that wings with WLE performed better than the

16

Hugo T. C. Pedro and Marcelo H. Kobayashi, “Numerical Study of Stall Delay on Humpback Whale

flippers” (presented at the 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 2008). 17

Miklosovic et al., “Leading-edge Tubercles Delay Stall on Humpback Whale (Megaptera

Novaeangliae) flippers.” 18

Hamid Johari et al., “Effects of Leading-Edge Protuberances on Airfoil Performance,” AIAA Journal 45

(November 1, 2007): 2634–2642, doi:10.2514/1.28497. 19

J. L. E. Guerreiro and J. M. M. Sousa, “Low-Reynolds-Number Effects in Passive Stall Control Using

Sinusoidal Leading Edges,” AIAA Journal 50 (February 1, 2012): 461–469, doi:10.2514/1.J051235. 20

Hansen, Kelso, and Dally, “Performance Variations of Leading-edge Tubercles for Distinct Airfoil

Profiles.” 21

Anil Kumar Malipeddi, “Numerical Analysis of Effects of Leading-Edge Protuberances on Aircraft

Wing Performance” (Master of Science Thesis, Wichita State University, 2011).


clean wing in post-stall regime giving higher values of CL with almost the same value of

CD. However the clean configuration proved to be better in pre-stall regime. It was

found that the protuberances along the LE produced streamwise vortices which carried

high momentum flown in the boundary layer keeping the boundary layer attached. It

was postulated that the streamwise vortices produced by the protuberances are the

reason why the modified airfoil produced 48% higher lift than the clean airfoil in the

post-stall regime.

Chapter 2 : Theoretical Background

In this chapter the student aims to set up the theory on which the entire project relies on.

As this is projects is based on Computational Fluid Mechanics calculations it is worth

reviewing the main theories this two sciences are based on.

2.1. Basic Notions and Assumptions

Whenever we look to any fluid in motion we see that it tends to develop series of very

complicated structures. One might think that these complicated structures must be

governed by an also complicated mathematical expression or a set of very complicated

expressions and in fact this turns out to be true. But if we look at them in a more general

way we only have to recall a more simple formulation that drives us back to the

secondary school level: Newton’s Second Law which states that:

Mass x Acceleration = Force

But in Fluid Mechanics it is rebuilt in the equivalent form of:

Rate of Momentum Change = Force

So now we find ourselves with an expression that relates the Rate of Momentum

Change with the forces present in the fluid motion. These forces are for a wide range of

Aerodynamic applications reduced to two: Pressure Forces and Viscous Forces. Here

we have obviated the Gravity Forces since the fluid we are dealing with is air for which

Gravity Forces are relatively much smaller than the Pressure and Viscous Forces due to

the air density.

In order to determine the Viscous Forces we have to assume that the fluid we are

dealing with is pure homogeneous, this is the case for air. For these pure homogeneous

fluids the constitutive law is given by the Newtonian Fluid Model which states that:

Viscous Stresses Rate-of-Strain

We also suppose that the air is a continuum matter which also fits with our everyday

experience where we think of fluids like air or water in a continuum perspective.

Although we know that matter is formed by smaller structures called molecules, in most

engineering applications where the scales of motion are much larger than the molecules

the continuum assumption seems to be a fair assumption. Plus in these engineering

applications the physical laws of conservation which refer to Mass, Momentum and

Energy conservation are also applied.

Mass Conservation

The Mass Conservation satisfies the well-known statement that matter cannot be created

or destroyed. For a fluid volume V enclosed by a surface S this means that the rate of

increase of mass inside the volume equals to the rate at which mass enters the volume

through the surface minus the rate at which mass leaves the volume through the surface.

Rate of Increase of Mass = Rate of Mass Entering – Rate of Mass Leaving

Chapter 2: Theoretical Background 10

Or

Rate of Increase of Mass – Net Mass Flow out of the Volume

The mathematical expression for this is:

SVdSnudV

t

(2.1)

Applying the Gauss divergence theorem combining and shrinking V to a point gives:

0

u

t

(2.2)

Given that the material derivative of a function f is given by:

fut

f

Dt

Df

(2.3)

The equation (2.3) can be rewritten as follows:

uDt

D

(2.4)

For incompressible flows where the density does not change following a fluid element

the equation (2.4) can be simplified to:

0 u

(2.5)

Or in tensor notation:

0

i

i

x

u

(2.6)

Equations (2.5) and (2.6) represent the Mass Conservation Equation in vector and tensor

notation respectively.

Momentum Conservation

As mentioned before the main law governing the mechanics of the fluid motion is the

Newton’s Second Law which applied to a volume V bounded by a surface S can be

resumed as:

Rate of Increase of Momentum –Net Flux of Momentum out of the Volume = Sum of

Forces Acting on the Surface

The mathematic expression for this can be written as follows:


V S Siii dSndSnuudVuu

t

(2.7)

Where is the stress vector acting on the surface S which is our representation of

unresolved molecular motions, i.e. momentum flux density of molecules crossing dS.

Applying the Gauss and shrinking V to the point equation is written as:

iiuut

u

(2.8)

Or in tensor notation:

i

ji

j

jii

xx

uu

t

u

(2.9)

Now the stress can be split into its normal and its tangential parts. The normal part is

due to the pressure forces whereas the tangential part is due to the viscous forces. The

stress tensor hence can be written as:

ijijij p (2.10)

In addition as the flow is assumed to be Newtonian the viscous stresses are only related

to the local strain rate, which for incompressible flows is:

i

j

j

iij

x

u

x

uS

2

1

(2.11)

And then:

i

j

j

iijij

x

u

x

uS 2

(2.12)

Substituting equation (2.12) in (2.11) and then replacing in (2.10) we have:

bleincompresifor

ji

j

jj

i

i

i

j

j

iij

ij

jii

xx

u

xx

u

x

p

x

u

x

up

xx

uu

t

u

0

22

(2.13)


But given that the divergence of the velocity is zero for incompressible flow equation

(2.13) reduces to:

jj

i

ij

jii

xx

u

x

p

x

uu

t

u

2

(2.14)

Now the LHS of the equation can also be simplified using the Continuity equation.

Dt

Du

x

uu

t

u

x

u

tu

x

uu

t

u

x

uu

t

u

i

j

ij

i

Continuityby

j

j

i

j

ij

i

j

jii

0

(2.15)

Finally the Momentum Equation can be written as:

TermViscous

jj

i

Termessure

i

TermLinearNon

j

ij

ii

xx

u

x

p

x

uu

t

u

Dt

Du

2

Pr

(2.16)

The Navier-Stokes Equations

Equations (2.6) and (2.16) all together constitute the Navier-Stokes Equations (NSE).

This set of non-linear equations governs all fluid flows. There are four equations, three

momentum equations plus the continuity equation, for four unknowns, three velocity

components plus the pressure. However, despite the system is closed, the non-linearity

present in them because of the LHS second term in equation (2.16) makes them almost

impossible to solve analytically for most industrial flows. Nonetheless, with the

computational power available nowadays, solutions can be found for simple geometries

and low Re numbers using Computational Fluid Dynamics or most commonly known as

CFD.

2.2. Turbulence

The Oxford Dictionaries defines Turbulence as “violent or unsteady movement of air or

water, or of some other fluid”22

. This violent or unsteady movement is due to the non-

linear terms in the NSE equations. We can see turbulence every day in our lives. When

the smoke comes out of a chimney we can clearly see how fluid structures rapidly

destabilise and break down into smaller structures. This behaviour can be seen in Figure

2.1 where the well-defined jet of water quickly breaks down into vortical structures.

This happens when the viscous term is no longer big enough to balance the non-linear

term in equation (2.16) and then instabilities arise. This imbalance happens when the

22

Oxford Dictionaries, “‘Turbulence’. Oxford Dictionaries.,” n.d., definition/english/.


viscous forces are much smaller than the inertial forces, or in other words, when the Re

is too high.

lUlU

ForcesViscous

ForcesInertial

Re

(2.17)

Where U∞ is the freestream velocity, l is a characteristic longitude and

is the

kinematic viscosity.

Figure 2.1 Water Jet Turbulence: Instabilities quickly develop until fully turbulent flow is

achieved.23

One can find defining the dimensionless variables:

2~;~;~;~

U

pp

l

Utt

l

xx

U

uu i

ii

i

(2.18)

The NSE can also be expressed as:

jj

i

ij

ij

i

xx

u

x

p

x

uu

t

u~~

~

Re

1~

~

~

~~

~

~ 2

(2.19)

Which shows that as the Re number increases the viscous term becomes less important

and the non-linear inertial term rules the equation.

23

Milton Van Dyke, “An Album of Fluid Motion,(1982),” Parabolic, Palo Alto (Calif.) (n.d.).


Basic Features of Turbulent Flows

Turbulent flows are by nature random and chaotic. But this does only mean that

variables are continuously fluctuating, i.e. they do not have a unique value. It is

unpredictable in detail, but with predictable statistical properties. They are also

inherently 3-Dimensional and unsteady, which does not mean that statistical values

cannot be steady though if they are averaged over large enough population sizes.

Turbulent flow contains vorticity, the curl of the velocity. These vortical and unsteady

properties make turbulence to contain a wide range of time, velocity and length scales,

which is commonly known as Energy Cascade. The famous Richardson’s quote

summarises it perfectly:

“Big whirls have little whirls,

That feed on their velocity;

And little whirls have lesser whirls,

And so on to viscosity.”

― Lewis Fry Richardson

What Richardson was trying to say was that turbulence, or energy is produced by large

scales, or low frequency scales, that then break down into smaller scales transferring

energy until it reaches a point where the viscosity forces are of the same order of

magnitude of the inertial forces and turbulence is then dissipated into heat. This scale,

where turbulence dissipates, is known as the Kolmogorov Scale24

in honour to Andréi

Kolmogórov. For Homogeneous Isotropic Turbulence (HIT) where Turbulent Kinetic

Energy Production ( ) equals the Turbulent Kinetic Energy Dissipation ( ) the

smallest scales happen where Re = 1, i.e. where inertial forces equal viscous forces.

Taking this into account and using some dimensional analysis the Kolmogorov Length

Scale η can be found to be:

4/13

(2.20)

And then the ratio between the largest scales l0, at which production of TKE happens,

and the smallest scales η, where energy is finally dissipated into heat, is:

4/3

4/3

00 Re

lulo

(2.21)

This means that the higher the Re number, the smaller the smallest scales are compared

to the largest scales and wider is the scale’s range. So to fully solve the NSE equations

the full range of scales from the biggest to the smallest has to be resolved. This is called

Direct Numerical Simulation (DNS). Another approach is to resolve the biggest scales

and model the effect of the smallest ones. This is called Large Eddy Simulation. And

24

Andrey Nikolaevich Kolmogorov, “The Local Structure of Turbulence in Incompressible Viscous Fluid

for Very Large Reynolds Numbers,” Proceedings of the Royal Society of London. Series A: Mathematical

and Physical Sciences 434, no. 1890 (1991): 9–13.


finally, the last option is to completely model the entire range of scales, which is called

Reynolds Averaged Navier-Stokes (RANS). This latter approach is the one chosen for

this project since it is the less computationally expensive. However this benefit of being

less computationally expensive comes with the drawback that it is also less accurate

than LES and DNS, in that order.

2.3. Reynolds Averaged Navier-Stokes Equations

As aforementioned in this project turbulence is going to be tackled using RANS. The

bases on what these equations are built are the NSE and the Reynolds Decomposition.

Reynolds Decomposition and Average

We have already said that turbulence is a random process for which statistical properties

can be extracted. Figure 2.2 shows a typical Hot-Wire velocity measurement. It can be

noted that velocity u(t) varies randomly in time but once the experiment is concluded

we can obtain an average or mean velocity quantity ū and a velocity fluctuation u’(t). Of

course this infinite limit only attempts to say that the variable has to be averaged over

large amounts of time to have a statistically converged value. Additionally we can

express now the velocity measurement as:

tututu (2.22)

Where the mean quantity ū is defined as:

T

iT

i dttuT

u0

)(1

lim

(2.23)

Figure 2.2 Typical Hot-Wire velocity measurement

This process described here is known as Reynolds Decomposition introduced by

Osborne Reynolds in 189525

. A fluctuating variable is decomposed into its mean and

fluctuating parts.

Although above we have presented a time averaged quantity, this is not the only way of

obtaining averaged quantities. Another option could be the ensemble averaging in

which a variable u is measured in N experiments at the same time t and position x. Then

the average quantity is given by:

25

Osborne Reynolds, “On the Dynamical Theory of Incompressible Viscous Fluids and the

Determination of the Criterion” 186, Philosophical Transactions of the Royal Society of London (1895):

123–164.


N

n

iN

i txuN

txu1

),(1

lim),( (2.24)

However this Ensemble averaging approach is more complex to be obtained in a

laboratory where measurements on a single point in space during time are more suitable,

i.e. Hot-Wire Anemometry.

From its definition the following rules can be obtained:

: The time average of an averaged time quantity is the average itself.

: The time average of a fluctuating quantity is zero.

(

)

(

)

: The time average of a time and/or space derivative

is the derivative of the time averaged quantity.

Applying the Reynolds Decomposition and Average to the Navier-Stokes

Equations

Now we have the necessary tools to define the Reynolds Averaged Navier-Stokes

Equations (RANS). Using the definition (2.22) in equation (2.6) and (2.16), applying

the averaging rules and averaging over time the equations we have:

0

i

i

x

u

(2.25)

And

jj

i

ij

ji

j

ij

i

xx

u

x

p

x

uu

x

uu

t

u

21

(2.26)

Or rearranging

ji

j

i

jij

ij

i uux

u

xx

p

x

uu

t

u

1 (2.27)

We note here that equations (2.25) and (2.27) are very similar to the unaveraged NSE

(2.6) and (2.16). The difference is that here in RANS equations we are solving for the

mean flow quantities and we have an extra term . The term , which comes

from the non-linear term of the Navier-Stokes equation, is a symmetric tensor, which

dimensionally speaking has units of stress, and is known as the Reynolds Stress Tensor,

ij.


2

2

2

wwvwu

wvvvu

wuvuu

ij

(2.28)

Additional stresses due to the fluctuating turbulent quantities are introduced by this

term. This term is the correlation between the velocity components and can be seen as

the average rate of momentum transfer and accounts for the extra dissipation that

characterises turbulent flows. Because it is symmetric, i.e. this term

introduces also 6 new unknowns which leave us now with 10 (6 components of the

Reynolds Stress Tensor plus 3 mean velocity components plus the mean pressure)

unknowns for still 4 equations. This cannot be a surprise since we have only performed

mathematical and algebraic operations over the N-S equations without introducing any

additional physical principles. This is known as the Closure Problem. The role of the

turbulence models is then to give an expression to model this term and then close the

problem.

The Reynolds Stresses diagonal components represent the normal

stresses whereas the off-diagonal components represent the shear stresses. The

Turbulent Kinetic energy (TKE or k) mentioned above is defined to be half the trace of

iiuuk 2

1

(2.29)

In the principal axis of the Reynolds Stress Tensor the shear stresses are zero and the

normal stresses are the eigen-values, which are non-negative. This means that that the

Reynolds Stress Tensor is a symmetric semi-positive defined tensor.

The Reynolds Stress Tensor can also be decomposed into its isotropic part:

ijij kb 3

2

(2.30)

And its anisotropic deviatoric part:

ijijij ka 3

2

(2.31)

2.4. Eddy Viscosity Approximation

The simplest way of modelling the Reynolds Stress Tensor is to prescribe a linear

relationship between the stress tensor and mean Rate-of-Strain as Boussinesq26

did in

1877. Boussinesq, similarly to the Stress-Rate-of-Strain relation for Newtonian Fluids

(2.12) defined a linear relationship between the anisotropic deviatoric part of the

26

Boussinesq, “Theorie de l’Ecoulement Tourbillant” 23, no. Mem. Presentes par Divers Sa- vants Acad.

Sci. Inst. Fr (1877): 46–50.


Reynolds Stress and the mean Rate-of-Strain. But in order to understand why he did so,

we might first consider a two-dimensional flow at the molecular level as illustrative

example, with x-velocity component u>0 and y-velocity component v=0.

For this 2D illustrative flow, at the molecular level, velocity can be Reynolds

decomposed into its mean velocity and its molecular random velocity perturbation so the velocity is expressed as . If the instantaneous flux of any fluid

property is calculated across a given horizontal plane we will have that the flux is

proportional to the velocity perpendicular to the plane, i.e. in this case. The x-

momentum flux dpxy across a differential surface area dS then will be:

dSvuudpxy

(2.32)

And then using an ensemble averaging over all molecules we have:

dSvupd xy

(2.33)

Given that the stress tensor acting on the plane we are looking at is σxy=dPxy /dS and

equation (2.10) we have that the viscous stress tensor τxy at this plane is:

vuxy

(2.34)

We can now clearly see how similar this expression is to the Reynolds stresses (2.28).

Here the difference is that the fluctuating velocity is due to the molecular movement and

in the Reynolds stresses is due to turbulence. In laminar flows, the flow is structured in

layers that slide over one another because of the existence of a velocity gradient

between them. The momentum exchange between layers of flow due to the fluctuating

velocities results into friction and ultimately energy dissipation. We have seen that

turbulence flows are much more dissipative than laminar flow and this is because of the

extra momentum exchange due to the turbulent fluctuations . At the molecular level

this is known as the viscous stresses, which are proportional to the velocity gradients, or

the strain rate, as equation (2.12) shows, whereas at the macroscopic level we can think

of this as the Reynolds stresses. Consequently because they both, i.e. the viscous stress

and the Reynolds stress, similarly extract energy from the flow we can relate the

Reynolds stresses with the velocity gradient as follows:

ijT

i

j

j

iTijji

S

x

u

x

ukuu

2

3

2

(2.35)

Where the positive scalar coefficient νT is the Eddy Viscosity, or the Turbulent

Viscosity.

Some clarifications need to be made at this point about the Eddy Viscosity.

The Molecular Viscosity ν is a constant property of the flow whereas the Eddy

Viscosity νT is a scalar function of space and time.

The Eddy Viscosity hypothesis implies that the anisotropy tensor is aligned with

the mean Rate-of-Strain.


If νT can somehow be prescript the Closure Problem is solved and the RANS

equations can be solved.

νT has dimensions of [L2T

-1], hence by dimensional analysis we can estimate the

eddy viscosity as or where c is a non-

dimensional constant, lT is a characteristic length scale (typically the size of the

large scale eddies) and uT is a characteristic velocity scale (typically the velocity

magnitude of the large scale eddies)

Velocity and Length Scales

Further investigation of the 2D molecular example we have just seen can give us an

expression of the molecular viscosity that we can then use to similarly define the eddy

viscosity.

So let’s consider again the 2D simple shear flow shown in Figure 2.3 where lmfp is the

mean-free-path, i.e. the average length travelled by a molecule between collisions. Each

particle coming from point P that crosses the x-axis brings a momentum deficit of

m[U(0)-U(-lmfp)] whereas molecules coming from point Q bring a momentum surplus of

m[U(lmfp)-U(0)] where m is the molecular mass.

Figure 2.3 2D Shear Flow Detail

If we suppose that the average vertical molecular speed is ⁄ , because mainly the

same amount of particles are coming from both sides of the y=0 plane, then the average

number of molecules crossing a unitary area in the positive vertical direction is

, where n is the number of molecules per unit volume and vth is the molecules

thermal velocity. Consequently the momentum flux due to molecules crossing y=0 from

below is given by:

mfpth lUUmvnP )0(4

1

(2.36)

And the momentum flux due to molecules crossing form above is:


0)(4

1UlUmvnP mfpth

(2.37)

If we then approximate the velocity gradient by:

...

0)(...

)0(TOH

l

UlUTOH

l

lUU

dy

dU

fmp

mfp

fmp

fmp

(2.38)

And we use the fact that ρ=m·n we can express the viscous stress as:

dy

dU

dy

dUlvPP mfpthxy

2

1

(2.39)

Where

is the molecular dynamic viscosity. In dimensional analysis

grounds we can think of vth and lmfp as the velocity and length scales at the molecular

level. By comparison we can also define the eddy viscosity vT as a product of a

turbulent velocity scale uT and a turbulent length scale lT. This turbulent scales have to

be defined using turbulence quantities in the same way the scales of the molecular

viscosity was defined using molecular quantities.

It seems reasonable to use the turbulent perturbation velocity to define the turbulent

velocity scale, and given that the turbulent kinetic energy is ⁄ , it seems

also reasonable to use the turbulent kinetic transport partial differential equation to

prescribe the turbulent velocity scale. In statistic terms k is nothing but a root mean

square, which gives the intrusion level between neighbour flow layers and thus the

interaction level between flow layers.

But the turbulent length scale remains still undefined. One of the main conclusions of

1980/1981 AFOSR-HTTM Stanford Conference on Complex Turbulent Flows was that

the great amount of uncertainty about turbulence models comes from the transport

equation used to model the length scale. Dimensional analysis has been the main tool to

define this length scale. On dimensional grounds, Kolmogorov (1942), related the

turbulence eddy viscosity, the turbulent kinetic energy, and turbulent length scale as

follows:

k ~k ~k ~ 21 TT l (2.40)

Where ε is the dissipation of turbulent kinetic energy and ω is the specific dissipation

rate of turbulent kinetic energy which has dimensions of (time)-1

. The inverse of the

specific dissipation rate ω-1

can be seen as the time scale on which dissipation of

turbulent kinetic energy occurs. Despite dissipation occurs at the smaller scales, the

dissipation rate is the rate at which energy is transferred from the large scales to the

small scales by the Energy Cascade process and therefore it is set by properties of the

large scales such as k and lT.


The Intrinsic and Specific assumptions behind Boussinesq Approximation

The Boussinesq approximation can be seen in two parts. In first place, the intrinsic

assumption that the Reynolds Stress anisotropy aij=σij – 2/3k δij is determined by the

local mean Rate-of-Strain , and secondly that the relation between the Reynolds

stresses anisotropy and the mean Rate-of-Strain are linear related by the scalar νT as

follows.

ijTij Sa 2 (2.41)

This two faces of the same problem are well explained in Pope27

. For the intrinsic

assumption, the example of an axisymmetric contraction is studied. At the contraction

zone the axial mean Rate-of-Strain is constant

⁄ and the lateral

compressive strain rates are ⁄ , whereas at the straight section

. Therefore, if the Boussinesq intrinsic assumption stands, there would only be non-

zero Reynolds stresses at the contraction part where the mean Rate-of-Strain is non-zero

too. However experiments show that at the straight section immediately after the

contraction section the Reynolds stresses created at the contraction are still present. In

the simple 2D shear flow example studied in the previous section the relation between

the molecular time scale vth/lmfp and the shear time scale ⁄ is of the

order of the Knundsen number Kn times the Mach number Ma.

Ma Kn~Sl

v

mfp

th

(2.42)

This is typically very small, which means that the molecular motions almost

instantaneously adapt to what is imposed by the strain and can therefore be directly

related to the local mean Rate-of-Strain. In contrast, for turbulent shear flows such as

the axisymmetric contraction, experiments have shown that the ratio between the

turbulence time scale tT=k/ε and the strain time scale S-1

can be of the order of the unity

or even bigger. This means that turbulence does not rapidly adjust to what the mean

strain says and hence cannot be in general directly related to the local strain rate.

A part from this assumption, the specific assumption is made by establishing a linear

relationship between the mean Rate-of-Strain and the Reynolds stress anisotropy tensor.

This assumption implies that both tensors, i.e. and σij, are aligned but also that the

Reynolds stresses are scaled by the same scalar, i.e. νT. In simple Turbulent Shear flows

the normal Rates-of-Strain are all equal to zero and yet measured

normal Reynolds stresses are not zero but also they are different from each other. This

shows that the specific assumption does not stand even for the simplest flows, where the

Reynolds Stress tensor might be misaligned with the mean Rate-of-Strain tensor.

Furthermore it has to be pointed out that this is not only a problem of different eigen

vectors, but also eigen values, because as just seen Reynolds stress components may

sometimes scale differently each other.

27

S. B. Pope, Turbulent Flows (Cambridge University Press, 2000).


2.5. Linear Eddy Viscosity Models

The Closure Models based on the Boussinesq approximation are called Linear Eddy

Viscosity Models (LEVMs) because of the linear relationship between the Reynolds

Stress Tensor and the Main-Rate-of-Strain established by the Boussinesq

Approximation. The goal of these LEVMs is to provide additional equations to either

directly prescript the Eddy Viscosity or prescribe a velocity scale uT and length scale lT

or time scale tT from which the Eddy Viscosity can then be obtained. The LEVMs can

be divided in:

Algebraic / Zero equation Models: the Boussinesq Approximation is used, with

where uT and lT are prescribed using algebraic expressions, e.g.

Baldwin-Lomax.

One-equation Models: The Boussinesq Approximation is used introducing an

extra Partial-Differential Transport Equation, e.g. Spalart-Allmaras model. In

Spalart-Allmaras a transport equation for νT itself is used.

Two-equation Models: The Boussinesq Approximation is used, with or

where two extra Partial-Differential Transport Equations are used to

prescribe uT and lT or tT respectively. Typically the TKE (k) transport equation is

used to prescribe the velocity scale, where √ , and models then differ in

which equation is used to prescribe the length scale or time scale. In the k-ε

model, the transport equation for the dissipation rate ε is used for the time

scale

definition whereas in the k-ω model the specific dissipation rate

ω transport equation is used instead.

In this project we are going to focus on the One-equation Spalart-Allmaras and the

Two-equation Menter’s k-ω SST models since these are the ones used in the

computations carried out.

Spalart-Allmaras Model

As aforementioned the Spalart-Allmaras is a One-equation model that instead of

prescribing νT using velocity and a length or time scales it has its own transport equation

to directly solve νT. Spalart and Allmaras28

describe their Standard model as:

ii

b

jj

tb

wwtb

j

jxx

cxxd

fc

fcSfcx

ut

~~~~1~

~~)1(

~~2

2

22

1121

(2.43)

Where the turbulent viscosity is computed from:

1~

vT f (2.44)

28

P. R. Spalart and S. R. Allmaras, “A One-Equation Turbulence Model for Aerodynamic Flows,”

Recherche Aerospatiale no. 1 (1994): 5–21.


Where

222

3

1

3

3

1

~~

~

v

v

v

fd

S

cf

(2.45)

And √ is the vorticity magnitude, d is the distance to the nearest wall and

i

j

j

iijttt

w

w

ww

v

v

x

u

x

uWccf

dSrrrcrg

cg

cgf

ff

2

1exp

10,~

~min

1

11

2

432

22

6

2

6

1

3

3

6

3

3

1

2

(2.46)

And the model constants are:

2

2

114

3132

21

15.0

2.11.723.0

41.0622.03

21335.0

bbwt

tvww

bb

cccc

cccc

cc

(2.47)

Menter’s k-ω SST Model

In his paper, Menter29

proposes two new Two-equation turbulence models. Both models

are based on former Two-equation models, i.e. the k-ε model30

and the k-ω model31

.

Menter takes the best of each of the two models and combines them into a new one with

some modifications. The k-ω model is used for the flow near the walls because, unlike

any other two-equation model, the k-ω model does not involve damping functions and

allows zero initial wall conditions to be specified and, because of its simplicity, the k-ω

model is more numerical stable with respect to other models. However, because of the

strong sensitivity of the k-ω model to freestream boundary conditions, the k-ε is

preferred out of the boundary layer regions. This integration of both models is achieved

29

F.R. Menter, “Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications,” AIAA

Journal 32, no. 8 (August 1994): 1598–1605. 30

B. E.Launder and B.I. Sharma, Letters in Heat and Mass Transfer, vol. 1 (New York ; Oxford:

Pergamon Press., 1974). 31

David C. Wilcox, “Reassessment of the Scale-Determining Equation for Advanced Turbulence

Models,” AIAA Journal 26, no. 11 (1988): 1299–1310.


by means of blending functions that activates the k-ω model near the walls and switches

to the k-ε in the outer region. To achieve this, the k-ε model is transformed into a k-ω

formulation. An additional cross-diffusion term is added to the original k-ω formulation

and the values of the constants are slightly modified.

The original k-ω is:

j

tk

j x

k

xkP

Dt

kD

1

*

(2.48)

j

t

jt xxP

Dt

D

1

2

11

(2.49)

Where the production term P and the stress tensor τij are:

j

iij

x

uP

(2.50)

ijij

k

kijtij k

x

uS

3

2

3

22

(2.51)

And then the transformed k-ε is:

j

tk

j x

k

xkP

Dt

kD

2

*

(2.52)

jjj

t

jt xx

k

xxP

Dt

D

12 22

2

22

(2.53)

Then equations (2.50) and (2.51) are multiplied by a function F1 and equations (2.52)

and (2.53) are multiplied by (1 - F1) and both sets of equations are added together to

give the new model:

j

tk

j x

k

xkP

Dt

kD

*

(2.54)

jjj

t

jt xx

kF

xxP

Dt

D

112 21

2 (2.55)

Where the constants of the new model ϕ are calculated from a combination of 2 sets of

constants ϕ1 (form the original model) and ϕ2 (from the transformed k-ε model).

2111 1 FF (2.56)


Where

4

11 argtanhF (2.57)

2

2

21

4;

500;

09.0maxminarg

yCD

k

yy

k

k

(2.58)

Where y is the distance to the next surface and CDkω is the positive portion of the cross-

diffusion term in equation (2.53).

The term arg1 goes to zero as the distance to the wall increases due to it is in inverse

proportion to the wall distance y. As arg1 goes to zero as it approaches the boundary

layer edge, so does F1 and then the k-ε model is recovered.

Finally, the eddy viscosity for the SST model is defined as:

21

1

;max Fa

kat

(2.59)

Where Ω is the absolute value of the vorticity and F2 is given by:

2

22 argtanhF (2.60)

22

500;

09.02maxarg

yy

k

(2.61)

The sets of constants to be used in (2.56) are:

*2

1

*

11

*

111k1

1

41.009.0

31.00750.05.085.0

:inner) (SST 1Set

a

(2.62)

*2

2

*

22

*

22k2

2

41.009.0

0828.0856.00.1

)-k(Standart 2Set

(2.63)

2.6. Unsteady RANS (URANS)

Because of the unsteadiness nature of both the two-element configuration and the Wavy

Leading Edges airfoils, and the fact that in the second part of the investigations the

effect of a vortex convected by the freestream impinging the models will be tested, the

use of Unsteady RANS equations was a must.

URANS differ from Steady RANS in that the transcient (unsteady) term of the NSE is

retained. So URANS still is based on the RANS Equations (2.25) and (2.27) which


means that the Reynolds decomposition (2.22) is used as well with a slightly modified

notation.

uUU (2.64)

Where a second prime has been added to the modelled turbulent fluctuation part.

However the dependent variables are no longer just a function of space but also of time.

That means that the velocity quantities, or the pressure quantity must now be defined as

ui=ui (x,y,z,t) or pi=pi(x,y,z,t) respectively.

URANS then take into account time variations of the mean quantities. Nevertheless we

still care about statistical values of such quantities. This can be easily seen in Figure 2.4

where represents the mean quantity ⟨ ⟩ represents its mean value and represents its

fluctuation around ⟨ ⟩ and the modelled turbulent fluctuation cannot be seen in the

figure.

Figure 2.4 URANS Measurement of a Mean Quantity

Here the mean quantity is time dependent and hence an average of it in time can be

extracted ⟨ ⟩. The difference between them is therefore and the real value U can be

expressed as:

uuUuUU

(2.65)

2.7. Discretising the Equations: the Finite Volume Method (FVM)

If Navier-Stokes Equations want to be numerically solved the equations must be

discretised. The most widely used approach, and the one used in this project is the Finite

Volume Method because of its conceptual simplicity an ease of implementation for

arbitrary structured and unstructured grids32

. The FVM is based on averaged volume

values contained in each cell whereas for example the Finite Difference Method (FDM)

is based on local function evaluations at each grid point. Hence in the FVM, integral

conservation laws are applied to each cell volume analogous to the control volume

concept used in fluid mechanics.

32

Charles Hirsch, Numerical Computation of Internal and External Flows: The Fundamentals of

Computational Fluid Dynamics (Butterworth-Heinemann, 2007).


For example, for a scalar φ with on a volume Ω discretised into smaller

volume controls/cells Ω1, Ω2, Ω3, as shown in Figure 2.5, where equation (2.66) applies.

SSdFd

t0

(2.66)

Figure 2.5 Volume Ω divided into three subvolumes Ω1, Ω2, and Ω3

We can have that (2.66) applied to the whole volume Ω equals applying it to the three

smaller volumes and adding them together.

AEDABDEBABCA

AEBCA

SdFdt

SdFdt

SdFdt

SdFdt

321321

(2.67)

Where the internal fluxes cancel because the flux in adjacent cells contributes in the

same amount but with opposite signs.

BAAB

dSdS

(2.68)

We can replace equation (2.66) by its discrete form (2.69) where the volume integrals

are substituted by the average volume values at each cell φi and the surface integral is

replaced by a sum over all the bounding faces of the considered cell volume Ωi.

0

Faces

ii SFt

(2.69)

It is worth mentioning that the coordinates of the cell are not present on equation (2.69)

explicitly and hence the value can be seen as the average value of the quantity inside the

cell, and grid coordinates are only needed to compute the cell volume and the bounding

surfaces. The equation below also expresses that the variation of the average quantity φ

over the time interval ∆t equals to the sum of the fluxes exchanged between

neighbouring cells. Finally, the FVM also allows to easily introducing boundary

conditions because it only needs to prescribe the desired flux values on the boundaries.

Chapter 3 : Problem Description

3.1. Ambient Conditions

The models will be simulated at Sea Level conditions using the following values.

Table 3.1 Ambient Values

Density ρ [Kg · m-3] 1.225

Pressure P [Pa] 101325

Dynamic Viscosity μ [Pa·s] 1.7885

Free Stream Velocity U∞ [m·s-1] 6.0

Re 493150

The Reynolds number Re is calculated using equation (3.1) with the chord of the model

c = cmain + cflap =1.2 m.

UcRe

(3.1)

3.2. Tools Used

In this project several software and computing languages have been used to carry out to

accomplish the project’s goals. For the Geometry creation the software used was

SOLIDWORKS because it was the CAD software provided by the University of

Southampton. As regards as the CFD computations STAR-CCM+ was the solver

chosen among the variety of tools provided by the University. This was mainly because

the author already had experience with the software, which is always an advantage

because it takes the learning curve out of the contest, but also because it provides with

an efficient meshing and CAD management environment which enables to have all the

CFD process of Pre-Processing, Solving and Post-Processing in a single tool.

Additionally Matlab was used in the Baseline geometry definition (see Chapter 4:

4.1Elements’ Relative Position Study) for implementing the optimisation scheme and

creating basic geometry that was then used in SOLIDWORKS to create the 3D models.

During this optimisation the need of automating the process lead to the use of 2 codes in

VBA and JAVA that took care of automating the 3D geometry creation and the CFD

simulations respectively.

3.3. Model’s Geometry

As aforementioned (see Chapter 1) the geometry will be defined by two tandem airfoils.

The upstream element will be regarded as the main element in a typical two-element

high-lift configuration whereas the second element will be considered as a flap. The

main airfoil used in the studies will be the GA(W) – 1 which is a typical airfoil used for

Chapter 3: Problem Description 30

landing flap configurations33

. The airfoil geometry used for the flap element is the

NACA 634-021 since it resembles the cross section of the humpback whale flipper34

.

The coordinates for the GA(W)-1 were obtained from the UIUC airfoil coordinates

database35

, whereas the coordinates for the NACA 634-021 were obtained from the

Public Domain Aeronautical Software (PDAS) web page36

. The chord for the main

element cmain is 1 m and the chord of the flap cflap is 0.2 m. The origin of coordinates is

located at the leading edge of the main element as shown in Figure 3.1. The AoA of the

main element remains constant at zero degrees for all simulations.

Figure 3.1 Sketch of the Elements’ relative position

The gap is defined as the minimum distance between the lower point of the main

element trailing edge and the flap profile whereas the gap is defined by the horizontal

distance between the closest point of the flap profile to the lowest point of the main

element trailing edge. A sketch of the gap and overlap definition can be seen in Figure

3.2.

Figure 3.2 Gap and Overlap Definition

The geometry of the Wavy Leading Edge Flap (WLEF) will be based on the baseline

model aforementioned. The leading edge of these models will follow (3.2) where is

the x-coordinate of the Flap’s LE, A is the amplitude of the protuberances, n is the

33

Rumsey and Ying, “Prediction of High Lift: Review of Present CFD Capability.” 34

Fish and Battle, “Hydrodynamic Design of the Humpback Whale Flipper.” 35

Department of Aerospace Engineering UIUC Applied Aerodynamics Group, “UIUT Airfoil

Coordinates Database,” 1995, http://www.ae.illinois.edu/m-selig/ads/coord_database.html#G. 36

Ralph Carmichael, “Public Domain Aeronautical Software,” 31, http://www.pdas.com/index.html.

-0.2

-0.1

0

0.1

0.2

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

Y

X

GA(W)-1

NACA 63(4)-021


number of waves present along the Flap’s Span, and zLE is the z-coordinate of the Flap’s

LE.

LELE znAx 2sin (3.2)

From (3.2) one can easily see that the WLE wavelength λ is just ⁄ where b is the

model’s span. It has to be pointed out that the use of a sine function allows the

protuberances to grow from the baseline LE, i.e. for the modified LE the average of

(3.2) still coincides with the baseline geometry with no protuberances at all (see Figure

3.3).

Figure 3.3 WLE Sketch

3.4. Mesh

The mesh strategy chosen for this project was an unstructured meshing algorithm based

on polyhedral elements because it is able to adapt to complicated geometries such as the

two-element plus the additional wavy shape on the Flap’s LE that was investigated in

this project. A generic base size was used for the entire domain and additional mesh size

control volumes were used to produce a finer mesh near the model and in the wake zone

in order to well-capture all the flow features. The volumetric controls (VC) can be seen

in Figure 3.4. Each VC has an element size which is a percentage of the general base

size (16%, 12%, 8% and 4% respectively) so every time the base size is reduced or

increased for purposes such as mesh dependency studies (see Appendix A: Mesh

Dependency Study), so do the VC, refining then the areas where the mesh needs more

resolution.


Figure 3.4 Volumetric Controls around the model

In terms of Near Wall Treatment, both turbulence models used, i.e. Menter’s SST and S-

A, need a well-defined mesh near the wall because no wall functions are used. The first

grid point must then lie on the viscous sub-layer, and hence the y+ value should be less

than 5. Figure 3.5 shows a detailed picture of the prism layer mesh in the boundary

layer zone for both elements.

Figure 3.5 Boundary Layer Mesh Detail

Figure 3.6 and Figure 3.7 show that the y+ value was almost always below the value of

5 unless at the stagnation points where the boundary layer is starting to develop and

hence its value increases considerably.

Figure 3.6 Wall y+ contour levels over the model


Figure 3.7 Wall y+ Distribution

The baseline mesh used for this computational study finally had around 5M elements.

The number of elements for the other meshes used for WLE geometries oscillates

among this value, but due to the geometry modifications, mainly due to curvature

refinement, lead to different amount of cells used.

3.5. Domain Size and Boundary Conditions

In any CFD calculation the real domain has to be truncated to a finite volume control

where ultimately the computations are going to be carried over. This is the

computational domain.

The size of the computational domain used in this project is summarized in Figure 3.8.

The dimensions used are a result of a domain size study (see Appendix B: Domain Size

Dependency Study).

Figure 3.8 Boundary Conditions and Domain Size Sketch (not to Scale)

Because of we are solving a set of partial differential equations; it is needed to provide

some information at the initial time and boundaries in order to have a well-posed

problem.

In Figure 3.8 we can also see the boundary conditions applied to the computational

domain. The inlet of the domain was modelled as a Velocity Inlet for which the


freestream velocity was prescribed, i.e. U∞ = 6 m/s, which gives a Reynolds number

based on the chord of the combined elements c = cmain + cflap = 1.2 m of Re = 493,151 ≈

500,000. Because of the extra equations needed to model the turbulence, also extra

boundary conditions have to be prescribed for the turbulence quantities. Because of two

different turbulence models were used, the turbulent viscosity ratio option was chosen.

The turbulence viscosity is a quantity present in both S-A and SST models and hence can

be used to prescribe the turbulent boundary conditions. In order to have a natural

transition on the model’s boundary layer a turbulent viscosity ratio of ⁄ was

used with a turbulence intensity of 0.5%.

The top and the bottom part of the domain were specified as Symmetry walls which is a

common strategy for defining freestream conditions. Here the flux across these two

boundaries is zero, which is equivalent to say that the velocity component on the normal

direction to these boundaries is set to zero. Furthermore, this implies that the following

gradients vanish.

0

0

0

nUtand

tUn

n

(3.3)

Where are the normal and the tangential unitary vectors to the boundary surface

respectively, φ is a scalar quantity and is the velocity vector.

As regards the outlet, a Pressure Outlet condition was used using the same turbulent

parameters as for the inlet because the flow is supposed to have similar conditions far

away from the model in both sides.

For a viscous flow that passes over a solid wall the relative velocity between the flow

and the solid wall is supposed to be zero at the wall surface. This is known as non-slip

boundary condition and is the main source of shear stress.

Finally, both sides of the domain were defined as periodic boundaries with a null

pressure jump. The solution obtained on one lateral side of the domain is mapped into

the opposite side every iteration. Using this configuration the effect of an infinite wing

is achieved but without taking out the cross-flow components as happens when using

the symmetry condition.

3.6. Time Resolution

As previously stated, in this project Unsteady RANS calculations have been used. When

the solution is allowed to vary on time it is necessary to define a time integration

method. In this project an implicit time scheme has been used. In theory, implicit time

schemes are more computationally expensive than explicit schemes because some

matrix inversion is needed during the solving process. However, while explicit schemes

are at the best conditionally stable as long as a sufficient small time step is used,

implicit time schemes are unconditionally stable, which allows for larger time steps

without risking the stability of the solution. Nevertheless it is very convenient to still

use time steps that are representatives of the flow time scale. For example for a flow

with velocity U∞=5 m/s using a grid resolution of ∆x=1 m and a time resolution of ∆t=1

s the convective courant number C would be:


5

x

tUC

(3.4)

This means that the a fluid particle would go through at least 5 cells in each time step,

which is not desirable to happen. So despite using an implicit time scheme, the time step

was set to ∆t=0.0005 s for all the simulations carried out in this project, and as seen in

Figure 3.9 the Courant number was kept very close to the unit almost everywhere.

However, at some regions where velocities were higher than the free stream velocity,

i.e. in the gap zone between the elements, values above 1 were registered.

Figure 3.9 Courant Number Detail

3.7. Turbulence Model

Simulations carried out in this project used the SST turbulence model described in

section 2.5 because it is recommended in many of the papers reviewed (see Chapter 1)

and because it is supposed to better perform in separate flow conditions such as those

we shall expect in this project. However given that no experimental or computational

data was available to validate the results shown here, it was decided to use the S-A

turbulence model to compare the results and see if there were major differences that

could show up modelling errors. The results of this validation study can be found in

Appendix C: Validation against S-A Turbulence Model.

Chapter 4 : Results

In this chapter we are going to investigate all the results extracted from the investigation

of the models in the different scenarios they have been tested. First we are going to

review the element’s relative position investigation in which the an optimisation scheme

was used to obtain a configuration of the two-element airfoil with high lift performance

such as what it is usually used on landing and taking-off configurations. Second, a study

of different WLEs shapes over the baseline’s flap’s LE was carried out. And finally,

third, a selection of modified WLE models were tested against an upstream vortex

condition and compared with the baseline case to test the robust performance of the

WLEs models in comparison with the baseline.

4.1. Elements’ Relative Position Study

The two airfoils used in this study have never been tested in combination in a two-

element configuration as far as the student knowledge. The initial baseline model tested

for this project can be seen in Figure 3.1 and poorly produced less lift than the main

element on its own. Therefore it was necessary to perform a study of the relative

position of both elements in order to achieve a high-lift configuration which is

ultimately the main goal of a two-element airfoil. Investigating the relative position

involves mainly three variables: 1) The flap’s X-Coordinate position xo, 2) The flap’s

Y-Coordinate position yo and 3) The flap’s deflection angle δ.

In order to efficiently quantify the benefits of one particular configuration a surrogate

modelling approach was taken. The surrogate modelling method used was Kriging37

,

which seems to be the most effective and versatile method for use in engineering design

and more particularly for intensive CFD based calculations38

. The benefits of using such

method is that not only provides a response surface which can accurately predict the

effect of the variables tested, but also can predict the expected improvement that can be

obtained by testing new design points.

It has to be mentioned that the simulations performed to obtain the results needed to fill

the response surface were obtained in 2D Steady RANS CFD simulations. The reason

for that is that exploring a 3D variable space can take long and then the computer

efficiency of 2D Steady RANS calculations, despite its inaccuracy where the final

approach taken. It is also worth mentioning that the surrogate models were built just

based on lift results, i.e. CL coefficient.

Initial Sample

In order to investigate the effect of these three variables, the Dr Andreas Sobester’s

Latin Hypercube39

code was used for designing the initial population for 30 models, 10

37

D. G. Krige, “A Statistical Approach to Some Mine Valuation and Allied Problems on the

Witwatersrand” (MSc thesis., University of the Witwatersrand, 1951), (G00907503). 38

Alex Forrester, “SESG6019 Notes: Design Search and Optimization - Case Study 1:Fast Global

Optimization” (2013). 39

Andreas Sobester, Bestlh.m (Southampton: University of Southampton, n.d.).

Chapter 4: Results 38

per each variable. The Latin Hypercube Sampling is a statistical method for generating a

sample of plausible collections of parameter values from a multidimensional

distribution which is often used to construct computer experiments.

In the context of statistical sampling, a square grid containing sample positions is a

Latin square if (and only if) there is only one sample in each row and each column. This

means that using this sampling method variables are sampled only once in each variable

space given then an initial population that can represent the model with accuracy

without repeating any variable value in the experiments. Applied to this project this

means that the effect of a given flap’s deflection angle is tested just one reducing the

general computational costs since the representation of the response surface obtained

has been examined better than it would have been with a simple random sampling plan

for example.

Automation of the process

In order to have a completely automated optimisation process some code has to be

created to: 1) Create the geometric models, 2) Run the models in a CFD environment, 3)

Extract the results that are going to be analysed, and 4) Analyse the Results, i.e. create

the surrogate model, and suggest new design points.

Figure 4.1 Optimisation Cycle

All the process is summarized in Figure 4.1. First, the base geometry, i.e. the wireframe

of the 3D model was created in Matlab. This enables to modify the relative position of

the flap’s by translating and rotating the flap’s points coordinates, additionally it also

allows to modify the shape of the trailing edge by changing the parameters A and n in

equation (3.2). The wireframe geometry then is stored in coordinate files with extension

.txt so we have a txt file per every line present in the model. Second SOLIDWORKS is

started and a VBA macro code is played so it creates the 3D models without any user


intervention and saves them in a format compatible with STAR-CCM+. These two

programs communicate themselves by means of a main txt file called “Data.txt” which

stores the main characteristics of the model, i.e. the model id, the flap’s X and Y

Coordinate position, the flap’s deflection angle and the value of the parameters A and n

(in this case both A and n are equal to zero because we are working with the baseline

model which does not have WLEs). Third, once the 3D model is ready, it is imported

into STAR-CCM+ and it is prepared and simulated again without user intervention

using a JAVA macro code. This macro names the geometry, creates the computational

domain, meshes, converts the mesh into 2D to save computational power, simulates and

finally extracts the CL coefficient to a text file named “Results.txt”. In this file every

time a design is added, a new line with the model characteristics and results is added to

the end of the file. Finally, the results file is read by Matlab again which updates the

surrogate model and based on it suggests a new design point and the cycle starts again.

Final Baseline Model

After 52 calculations (30 initial samples plus 22 surrogate model updates) the best

model was model 35 with a CL of 1.60 which is 220% better than the single element

configuration which had a CL of 0.50. The geometric values for model 35 where: xo =

0.9955 m, yo = -0.02540 m and δ = 33º. Despite model 35 was the best design model

tested it was not finally chosen to be the baseline model because the flap’s LE was too

close to the main element’s TE so the space left for further flap’s LE shape

modifications would not have been enough. However a very similar configuration was

taken for the final baseline model with the following parameters:

Table 4.1 Final Baseline Parameters

xo 0.995 m

yo -0.028 m

δ 33º


Figure 4.2 Response Surface with Red Dot indicating the Final Baseline Model

Figure 4.2 shows how despite the baseline is not in the predicted maximum zone of CL

it is still a good design near the maximum area. In fact the CL for this Baseline is 1.40

which is still 180% better than the single element configuration and much better than

the initial two-element configuration which produced less lift than even the single-

element one.

The final 3D geometry is shown in Figure 4.3. It can be seen in the front view that the

space left for the WLEs shape is still enough.

Figure 4.3 Baseline geometry with Coordinate System Indications. From Top-Left to Bottom-

Right: Top View, Isometric View, Front View and Rear View.


4.2. Experiment A: Flaps with Wavy Leading Edges

Once the baseline geometry was defined it was time to begin the investigation of

different wavy shapes on the flap’s LE. In total 9 extra cases were prepared to be

simulated using Unsteady RANS with the Menter’s SST closure model. Given that in

the papers reviewed concerning the WLE effect reveal the importance of the amplitude

and the wave length of the protuberances on the performance of the modified airfoil

geometry it was decided to try different combinations of both parameters. Three

different amplitudes, i.e. high, medium and low, and three different wavelengths, i.e.

high, medium and low, were combined to produce 9 models. The High Amplitude (HA)

models had A = 3 cm, the medium amplitude (MA) A = 2 cm and the low amplitude

(LA) had A = 1 cm. The High Wavelength (HW) models had only one wave over the

flap’s span n = 1 which means that λ = 50 cm. The Medium Wavelength (MW) models

had four waves n = 4 with λ = 12.5 m, and the Low Wavelength (LW) had eight waves

n = 8 with λ = 6.25 cm. Table 4.1 summarises all the models parameters in terms of the

chord of the flap cflap including the baseline case (model0) as it is usually done.

However from now onwards we will refer to them by their model names. For each

model calculations were done until results were statistically converged and then

statistical values were taken such as for example the mean velocity or the pressure

coefficient variance.

Table 4.2 Models Parameters in terms of cflap

Model

Name

WLE

Amplitude

A

WLE

Wavelength

λ

model0 0 cflap 0 cflap

model1.1 0.05 cflap 2.5 cflap









Results

In Table 4.3 it can be seen how none of the models tested had a better performance than

the baseline case for clean upstream conditions. All the modified models have less lift

and more drag than the baseline case in this particular configuration. The reason for this

happening is that the baseline model was the result of the elements’ relative position

optimisation. The introduction of the WLE to the flap’s LE modifies the basic

parameters of a multielement wing, i.e. the gap and the overlap, consequently some


sections of the flap are outside that optimal configuration and hence do not perform as

good as the baseline case with a straight LE for the specific configuration tested here.

But not only that, the WLE add additional flow features on the flap zone as we will see

in former sections which promotes flow separation or attachment in different sections of

the flap’s span.

Table 4.3 Experiment A Results

Model CL CD CLflap

model0 1.4059 0.0320 0.1178

model1.1 1.1767 0.0410 0.0862

model1.4 1.2166 0.0427 0.0940

model1.8 1.2471 0.0442 0.1004

model2.1 1.1662 0.0442 0.0884

model2.4 1.1803 0.0466 0.0929

model2.8 1.1751 0.0474 0.0938

model3.1 1.1330 0.0462 0.0866

model3.4 1.1463 0.0498 0.0914

model3.8 1.1081 0.0468 0.0851

Figure 4.4 show that model1.8 is the best model with LE modifications in terms of

overall lift at the same time that is not the worse model in terms of drag. Model1.4,

model 2.4 and model2.8 are the best ones in this order. We can clearly see in Figure 4.5

that the models just mentioned are also the ones with the highest flap’s lift coefficient

although still lesser than model0, the baseline.

Figure 4.4 CL and CD Coefficients for all the Models Tested


Figure 4.5 CLflap Coefficient for all the Models Tested

Wall-Shear Stress Streamwise Direction Contours for n=1 and n=0

In order to investigate the effect of the WLE on the flap element the first magnitude

observed is the Wall Shear Stress (WSS) in the streamwise direction. The WSS can be

expressed as:

0

y

wy

u

(4.1)

Figure 4.6 a) shows the WSS contours on the baseline model. In this figure we can

observe that only a small stripe of flow is attached to the flap’s upper surface at the first

¼ of its chord. Further downstream two reattachment zones can be founded at ¼ and ¾

of the flap’s span. If we observe the vorticity slices just after this reattachment zones we

can isolate a structure with a “V” shape composed by two counter-rotating vorticity

contours. We will see in the following paragraphs that this “V” shape appears also on

the WLE models again and it can be associated with zones of attached flow.


Figure 4.6 Wall Shear Stress [i] Contours on the Flap Surface plus Vorticity [i] Contours on

Several Spanwise Planes: a) model0, b) model1.1, c) model2.1 and d) model3.1

Figure 4.6 b) shows the WSS contours for model1.1 and we can clearly see that the

modification of the flap’s LE has completely modified the flow behaviour on the flap

element. The flow is now detached in the entire chord length at the valley zones, i.e.

where the chord length is lesser, and it is completely attached at the peak zones, i.e.

where the chord length is bigger, which agrees with the papers reviewed40, 41

. We can

observe again the “V” shape on the vorticity contours where the flow is attached. The

flow tends to travel from the peak zones with more airfoil thickness to the valley zones.

This movement creates streamwise vortices that enclosures the flow in-between valleys

and separates zones of attached flow from zones of detached flow. Figure 4.6 c) and

Figure 4.6 d) show that the flow behaviour is very similar between models with the

same WLE’s wavelength but also showing that the area of attached flow is increased

with higher amplitudes. However if we recall Table 4.1we can see that the flap’s lift

increases from the model with A=1 cm (model1.1) to the model with A=2 cm

(model2.1) but the model with the higher amplitude (model3.1) with A=3 cm has

however less lift than model2.1 and hence pointing out that the increment in amplitude

is no longer beneficial for this particular wavelength past a certain value of A>2 cm.

Additionally we can see at the upper-right corner of the aforementioned figures that the

WLE modifications on the flap also affects the main element flow near its trailing edge

where a negative zone of the WSS can be founded. This zone becomes bigger with

higher amplitudes which explains why the overall lift coefficient diminishes as the

amplitude increases for n=1 (λ=2.5 cflap). It can be as well observed that due to the

amplitude rise the two counter-rotating vortices that form this “V” tend to separate

leaving then more space in between them were the flow can remain attached and as a

result the area of attached flow becomes wider.

40

Johari et al., “Effects of Leading-Edge Protuberances on Airfoil Performance.” 41

Malipeddi, “Numerical Analysis of Effects of Leading-Edge Protuberances on Aircraft Wing

Performance.”


Velocity Contours at Peak, Valley and Base Sections for n=1 and n=0

We can see in the following figures (Figure 4.7 to Figure 4.13) how for the baseline

model (Figure 4.7) that the gap between both elements forces the flow to accelerate

giving more momentum and making the boundary layer to attach near the flap’s LE,

something that would not occur if the flap element, recall with more than 30º deflection

angle, would have been tested in isolation. We can also observe that the stagnation point

in the flap element creates a zone of low velocity and hence high pressure under the

main element trailing edge which contributes to a higher overall CL.

Figure 4.7 model0 Base Section, Velocity Slice non-dimensionalised by the Free Stream

Velocity

Figure 4.8 model1.1 Base Section, Velocity Slice non-dimensionalised by the Free Stream

Velocity

In contrast, we can see how in Figure 4.8, Figure 4.10, Figure 4.11 and Figure 4.13 for

the WLE models the sections with shorter chord length, i.e. the base and valley sections,

the flow separates from the very beginning whereas in Figure 4.9 and Figure 4.12 the

flow is attached at the peak sections of the flap. Nonetheless, despite the flow is

attached at the peaks in model1.1 and model2.1 (model3.1 is not shown here because of

its similar behaviour with the other cases shown) the velocity of the flow is lower than


that of the baseline model at its attached zone. This explains how, despite having a

wider area of attached flow, the lift produced by the modified flap models is still lower

than the baseline flap. Finally, the pressure increase created by the presence of the flap

element is less significative for the cases with undulated LEs which explains why the

total lift is lesser.

Figure 4.9 model1.1 Peak Section, Velocity Slice non-dimensionalised by the Free Stream

Velocity

Figure 4.10 model1.1 Valley Section, Velocity Slice non-dimensionalised by the Free Stream

Velocity


Figure 4.11 model2.1 Base Section, Velocity Slice non-dimensionalised by the Free Stream

Velocity

Figure 4.12 model2.1 Peak Section, Velocity Slice non-dimensionalised by the Free Stream

Velocity

Figure 4.13 model2.1 Valley Section, Velocity Slice non-dimensionalised by the Free Stream

Velocity


Pressure Coefficient Contours for n=1 and n=0

Figure 4.14 clearly shows that the WLE were not particularly beneficial for this

configuration. Here we will investigate the pressure coefficient Cp, which can be

expressed as:

221

U

ppC p

(4.2)

The area covered by low pressure is smaller and unstructured for the flap element and

additionally include detrimental effects on the suction surface of the main element near

its trailing edge which finally leads to, as stated before, a lower performance in both lift

and drag.

If we compare the baseline against the modified models we can see that for the baseline

there exists a zone of very low pressure at the entire flap’s LE span whereas for the

other models, not only the pressure is higher but also it is for a smaller area. Near the

valley zones, the main element experiments a more intense negative pressure gradient.

This pressure increase on the suction surface of the main element becomes even more

intense with higher amplitudes as seen in Figure 4.14 c) and Figure 4.14 d). This is due

to the fact that when the valley is deeper, its stagnation point moves further downstream

getting away from the main element and consequently, the increase in pressure does no

longer only affect the main’s element pressure surface but also its the suction surface.

Figure 4.14 CP Contours on the models surface: a) model0, b) model1.1, c) model2.1 and d)

model3.1

Wall Shear Stress Streamwise Direction Contours for n=4 and n=8

Increasing the number of waves n along the flap’s span also increases the amount of

streamwise vortices. The combined effect of increasing n and A results in more


interaction between the vortices created by the protuberances. We can see in Figure 4.15

a) at the middle section of the flap’s span that inside the characteristic “V” pair of

vortices now there is another pair of counter-rotating vortices. We can see in Figure

4.15 c) how the interaction between these four vortices becomes stronger as the

amplitude is increased and so does the strength of the vortices. The effect of what just

mentioned keeps increasing till it reaches a point where they become so important that

vortex fences are created in the streamwise direction completely separating the positive

zones from the negative zones of WSS as shown in Figure 4.15 e). This behaviour leads

to a more patterned WSS contours that tend to group in pairs of two peaks with a valley

separating them.

As the number of waves n increases, the wavelength λ decreases, joining the vortices

together and forcing them to interact in a more intensive way. Particularly, in Figure

4.15 e) and Figure 4.15 f) we can see how the positive zones of WSS tend to gather for

n=8 into groups of three and two. This pattern is repeated along the spanwise direction.

The flow coming from three peaks at the LE joins into a single strip of positive WSS

that finally reaches the TE. The flow that joins two peaks at the LE however can only

reach the TE together for the higher amplitude case. Nevertheless if we compare the lift

coefficients of these models against the baseline case only model1.8 can barely resist

the comparison. It is hence postulated that the WLEs tend to tidy up the flow in the

streamwise direction due to the fences effect created by the streamwise vortices, but at

the same time it debilitates the flow reducing its streamwise velocity.


Figure 4.15 Wall Shear Stress [i] Contours on the Flap Surface plus Vorticity [i] Contours on

Several Spanwise Planes: a) model1.4, b) model1.8, c) model2.4, d) model2.8, e) model3.4 and

f) model3.8

Velocity Contours at Peak and Valley Sections for n=4 and n=8

Following what was previously done with the n=1 models we now explore the velocity

contours at the most important sections. In Figure 4.16 we can observe that for n=4

models, although the flow might be attached in bigger regions for the WLEs models

than for the baseline, as stated before the velocity in these regions is considerably lower.

For example at Figure 4.16 c) and Figure 4.16 e) it can be seen that a large percentage

of the flap’s chord is having attached flow as the WSS positive sign indicates. We can

also confirm that negative areas of WSS can be associated with separated flow regions

as seen in Figure 4.16 f) where the flow is detached along the entire chord length. This,

results in a more intense and unstable wake and ultimately in a higher drag than the

baseline.

At the same time, due to the introduction of the streamwise vortices produced by the LE

protuberances, the flow is highly three-dimensional. The flow tends to move from the

peak zones to the valley zones promoting reattachment in the valley sections as seen in

Figure 4.16 b) and Figure 4.16 d).


A similar behaviour can be observed for the n=8 models shown in Figure 4.17. Again

we see that velocities at the suction surface of the flap are lower compared to the

baseline model where velocities of nearly 1.5U∞ were reached at the very beginning of

the flap. It can be appreciated that light blue colours are predominant in Figure 4.17,

which means that velocities on the suction surface of the flap are even lower than the

freestream velocity. Because of this reduction on the modified models boundary layer’s

edge velocity, the undulated models show a CLflap drop of almost 30% in the worst case

(model3.8). However there are regions of completely attached flow (from the flap’s LE

to the TE) when the protuberances are introduced (recall that the flap’s deflection angle

is 33º) in contrast with the just almost 1/8 attached zone for the baseline case.

Consequently it can be postulated that the WLEs models would be much more resistant

to upstream perturbations because although they have slower boundary layers they have

zones of attached boundary layers along the entire flap chord.

Figure 4.16 Velocity Slice non-dimensionalised by the Free Stream Velocity: a) model1.4 Peak

Section, b) model1.4 Valley Section, c) model2.4 Peak Section, d) model2.4 Valley Section, e)

model3.4 Peak Section and f) model3.4 Valley Section


Figure 4.17Velocity Slice non-dimensionalised by the Free Stream Velocity: a) model1.8 Peak

Section, b) model1.8 Valley Section, c) model2.8 Peak Section, d) model2.8 Valley Section, e)

model3.8 Peak Section and f) model3.8 Valley Section

Isosurfaces of Vorticity ±100 s -1

for n=4 and n=8

In previous sections it has been stated that the WLEs introduce streamwise vortices

increasing the cross-flow velocity components. Previously we saw vorticity contours in

different sections of the flap, here in Figure 4.18 it can be seen iso-surfaces of constant

vorticity in the streamwise direction. It is shown in the figure below that streamwise

vorticity is only created by the LE protuberances. A pair of counter-rotating vortices is

created in each peak because of the flow moving from the peak to its two adjacent

valleys. If we recall Figure 4.16, where we had vorticity contours on several planes

along the flap’s chord, we can now see how these vortices evolve. We can relate zones

of flow attachment to zones where the vortices tend to join and separated zones where

the vortices tend to separate. Vortices are most likely to be attracted by the attachment

zones where the pressure is lower rather than the separated regions from which they

tend to move away. At the same time we see that the WLEs also introduce streamwise


vorticity on the main airfoil near its trailing edge. This fact increases the cross flow

component of it, which might not be the most desirable effect when high lift coefficients

are pursued.

Figure 4.18 Isosurface of Streamwise Vorticity Ω=±100 s-1

: a) model1.4, b) model1.8, c)

model2.4, d) model2.8, e) model3.4 and f) model3.8

Pressure Coefficient Contours for n=4 and n=8

In Figure 4.19 how the lower pressure zones coincide with the zones where vortices

shown in Figure 4.18 joined. The low pressure zones are always located on the zones

where positives values of the WSS appear. However similarly with what happened with

the high wavelength models the pressure coefficient tends to be higher on the suction

surface of the main element than it was on the baseline model, and given that the main

element is the one that produces the most part of the lift the medium and low

wavelength models are also less efficient than the baseline in terms of overall lift.


Figure 4.19 CP Contours on the models surface: a) model1.4, b) model1.8, c) model2.4, d)

model2.8, e) model3.4 and f) model3.8

4.3. Experiment B: Flaps with Wavy Leading Edges against

Upstream Vortex Condition

In this second experiment 4 modified models plus the baseline case were tested against

an upstream vortex condition. The vortex was added to the Experiment A flow

conditions and then the new solution was used as Initial Condition for Experiment B.

The vortex was based on equations (4.3) extracted from42

with some minor

modifications to adapt to the case studied here. The strength of the vortex is controlled

by the constants k1 and k2 and the initial position is controlled by the constants x0 and y0.

Constant L represents the domain length and U∞ as usual represents the freestream

velocity in the x-direction.

The procedure to incorporate the vortex to previous solutions was tedious but simple.

The solutions were extracted from STAR-CCM+ into coma separated values files and

then modified using a MATLAB code with equations (4.3).

42

J.W. Kim, “Quasi-disjoint Pentadiagonal Matrix Systems for the Parallelization of Compact Finite-

difference Schemes and Filters,” Journal of Computational Physics 241 (2013): 168–194.


sm

kk

myx

L

rkkr

yyxx

rLxxkUvtyxv

rLyykUutyxu

6U

m36L

35.0,400,

25.0,2,

2

1exp

r

with)()0,,(

)(0,,

21

00

2

22

12

2

0

2

0

01

01

(4.3)

It is worth saying that (4.3) does not include modifications to the pressure field, the

kinetic energy field or the specific dissipation field, and hence identical values to the

non-vortex solutions were used. As the solver main variable are velocity’s components,

and all the rest of the variables highly depend on them, the solver computed at the first

iteration values for all the rest quantities needed to initialise the flow field based on the

velocity field provided. This can be seen in Figure 4.20 where we can observe that no

pressure due to the vortex appears at the initial time because the pressure field was not

modified in by the vortex equations. However we see how after some iterations the

solver has computed the new pressure field according to the new velocity field in Figure

4.20 d). The author nevertheless acknowledges that proper calculations of these non-

computed variables would have result in a smoother evolution of the solution.

Figure 4.20 Detail of Pressure Calculation from Initial Conditions: a) Vorticity contours at t=0s,

b) CP Contours at t=0s, c) Vorticity Contours at t=0.1s, and d) CP Contours at t=0.1s

Results

In this section the results obtained for the Experiment B are investigated. We focus on

the total lift coefficient of the model, the main element’s lift coefficient and the flap’s

lift coefficient. Simulations time begins when the vortex has been placed in front of the

model and then is convected downstream until it hits the models. In Figure 4.21 we can

observe that t ≈ 0.325s the vortex hits the LE of the main element, however since the

very beginning the inclusion of the vortex has a direct effect on the lift produced by the

models. On one hand, it is noticed that the performance of the baseline model (model0)

is more affected by the presence of the vortex and quickly starts decreasing its value.

Past t=0.4s the baseline model never recovers the initial CL value of 1.4 and ends up


producing even less lift than model1.8. On the other hand, although the models with

WLE are also affected by the introduction of the vortex condition they seem to have a

more continuous trend during all the simulation time. WLEs models end up having very

similar CL coefficients to the ones they started or even with slightly higher values.

Figure 4.21 Total CL History Comparison

Figure 4.22 CLflap History Comparison

The same trend can be observed in Figure 4.22 for the flap’s lift coefficient. However in

this case all the models with modified LE end up producing even more lift at t=0.7s

than at t=0s and what is more, more lift than the baseline flap in contrast with what

happened with clean freestream conditions on Experiment A. An interesting feature to

look at Figure 4.22 is that the models higher number of waves along the span, or lower

wavelengths, tend to have more stable behaviour after t=0.2s, which is even before the

vortex has encountered the main element.

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Tota

l CL

Simulation time [s]

model0

model1.4

model1.8

model2.4

model2.8

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

CLf

lap

Simulation Time [s]

model0model1.4model1.8model2.4model2.8


As regards the main element, the curves showing the CLmain history in Figure 4.23 are

almost identical to the ones showed in Figure 4.21. Again the WLEs models experiment

less variations than the baseline model during the simulated time frame.

Figure 4.23 CLmain History Comparison

At this point the reader might be interested in how big the variations were experimented

by the models. Figure 4.24, Figure 4.25 and Figure 4.26 show that the difference

between the time averaged CL values and both maximum and minimum values reached

during the tested time frame is much bigger for the baseline. The root-mean-square

(rms) gives us an idea of how big where the variations in the lift coefficient suffered by

the models, and we can clearly see that for the baseline model were considerably higher,

being twice as big as the rest of the models if we look at Total CL and CLmain values.

Figure 4.24 Total CL Statistic Values Comparison

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

CLm

ain

Simulation Time [s]

model0

model1.4

model1.8

model2.4

model2.8

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

model0 model1.4 model1.8 model2.4 model2.8

Tota

l CL

rms

Tota

l CL

Average CL Max CLMin CL CL rms


Figure 4.25 CLflap Statistic Values Comparison

Figure 4.26 CLmain Statistic Values Comparison

Cp rms contours

We can confirm all what was afore stated if we investigate how the pressure has varied

on the surface of the models by investigating the rms of the pressure coefficient.

Comparing the following figures (Figure 4.27 to Figure 4.31) we observe that the

highest values of the CP rms are present on the baseline model at the flap’s LE. If we

recall the last section (4.2 Experiment A: Flaps with Wavy Leading Edges) the flow on

the baseline was just attached at this part of the flap, and because of that we may

conclude that flow finally separated because of the presence of the vortex (this

affirmation will be soon confirmed).

There are although some variations as well on the WLEs models as depicted by Figure

4.28 to Figure 4.31. This is however inevitable because the vortex tends to destabilise

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14


CLf

lap r

ms

CLf

lap


0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40


CLm

ain r

ms

CLm

ain



the whole flow field, and yet the ability to maintain the flow attached on the modified

models highly depend on the patterned vortical structure created because of the

sinusoidal protuberances. Nonetheless the CP rms contours on the modified models’

main element show that because of the more regular behaviour of the modified flaps the

flow on the main element is much more stable.

Figure 4.27 model0 CP rms contours

Figure 4.28 model1.4 CP rms contours


Wall-Shear Stress Streamwise Direction Contours

In this section we are going to analyse instantaneous values of the WSS in the

streamwise component in order to clarify what happened when the vortex collided with

the models. For the reader’s sake of ease in the following figures, a part form the WSS

contours a detail of the vorticity contours has been added to show the location of the

vortex at each time location.

In Figure 4.32 we can confirm what aforementioned: the small strip of attached flow

present in the simulations with clean upstream conditions is no longer present when the

vortex condition is introduced. The flow structure on the flap rapidly brakes down and

only the two recirculation zones remain, for a short period of time. Ultimately we

observe that at t=0.6 the WSS on the flap’s suction surface is all negative, which is an

indication of separated flow.

More difficult is to comment on Figure 4.33 because the presence of the vortex in the

upstream flow brakes the well-patterned flow we used to have with clean upstream

conditions. The “V” shape pair of counter-rotating vortices is almost unappreciable and

as a result, regions of attached flow are more irregularly distributed. Nevertheless we

can say that this patterned flow organisation is finally recovered at t=0.6s. At t=0.7s the

counter-rotating pair of vortices is again present in the middle section of the flap’s span,

and consequently positive values of the WSS can be seen. We can conclude that this is

the reason why in Figure 4.21 and Figure 4.23 the models with sinusoidal LEs end up

having almost the same lift value that the one they started with.

If the number of waves is increased we see that the flow features depicted by the WSS

contours are much similar earlier on. For example in Figure 4.34 and Figure 4.36 the

WSS distribution on the flap for model1.8 and model2.8 starts to be qualitatively stable

for t=0.5s and t=0.4s respectively whereas for both model1.4 and model2.4 this

happens at t=0.6s.

Finally, if we have a look at the zone where the WLE are strictly located, i.e. form the

LE with +A to –A (being A the amplitude of the sinusoidal wave), we see that the

contours of the WSS are almost constant during the simulated time. But also if we

compare them to former figures in section 4.3 (Figure 4.15) we see that they are very

alike as well. In contrast, if we have a look at the same percentage of the flap’s chord on

the baseline model, we can depict big differences. Given that this part of the flap

element is the one that higher interacts and influences the flow and performance of the

main element, we can conclude that the more stable performance on the modified flaps’

LE region is the responsible for the overall robust performance of the main element in

terms of lift production.


Figure 4.32 model0 Wall-Shear Stress [i] contours: a) t=0.2s, b) t=0.3s, c) t=0.4s, d) t=0.5s, e)

t=0.6s and f) t=0.7s.

Figure 4.33 model1.4 Wall-Shear Stress [i] contours: a) t=0.2s, b) t=0.3s, c) t=0.4s, d) t=0.5s, e)

t=0.6s, f) t=0.7s


Figure 4.34 model1.8 Wall-Shear Stress [i] contours: a)t=0.2s, b)t=0.3s, c) t=0.4s, d) t=0.5s, e)

t=0.6s, f) t=0.7s


t=0.6s, f) t=0.7s



t=0.6s, f) t=0.7s

Chapter 5 : Final Remarks

Unsteady 3D RANS CFD calculations have been performed over a two-element airfoil

configuration with a flap’s sinusoidal leading edge shape. Models with flap’s LE

modifications were tested in two different environments. In the first part of the

experiments, the modifications were introduced to an already optimised two-element

case and were tested against upstream clean conditions and then compared to the

straight LE case. In the second part of the experiments, some modified models and the

baseline model with straight LE were tested against an upstream upcoming vortex in

order to test if the WLE shapes can contribute to have a more robust performance when

using multi-element airfoils in turbulent conditions such as those usually present in the

airports.

For the first experiment (see 4.2 Experiment A: Flaps with Wavy Leading Edges) the

main conclusion that can be obtained is that for clean upstream conditions the WLEs are

not beneficial in terms of both lift and drag. Consequently the aerodynamic efficiency

drops when the sinusoidal LEs are used.

Positive effects that one may obtain with the modified LEs cannot be compared with the

negative effect that the modifications on the flap’s LE has on the overall performance of

the two-element airfoil. Because of the sinusoidal shape of the flap’s LE, two out of

three main parameters that have direct relation with the lift produced by the airfoil are

also modified, i.e. the gap and overlap between the two elements varies along the flap’s

spanwise direction. These two parameters are the main responsible of the second

element beneficial effect present on multi-element configurations and if they are not set

properly the lift produced by the main element drops.

Additionally, if we just look at the lift produced only by the flap element, the WLE

neither have higher lift coefficients than that of the straight model. This is not a surprise

for the author, because although it has been extensively proven that the WLE perform

much better than straight LE, all the experiments to the date have been carried out with

the WLE in isolation. It is also known that the presence of the upstream wake of the

main element completely modifies the flow field that those WLE are facing.

Nevertheless many of the flow characteristics depicted by former studies of the WLEs,

i.e. the spanwise vortex fences separating the attached regions from the detached

regions, are still present.

At the same time, it might be possible that, what it is an optimum flap position for the

straight LE, might not be an optimum position for the sinusoidal edges. This statement

does not mean that there might not be one, however a proper optimisation of the

elements’ relative position taking into account the amplitude and the wavelength of the

protuberances will require an optimisation of five variables: flap’s x-position, flap’s y-

position, flap’s deflection angle, WLE amplitude, and WLE wavelength. This

optimisation is then nothing but at least computationally expensive because of its three-

dimensional and unsteady nature.

Finally it is worth saying that both the flap’s deflection angle and the overall angle of

attack have remained constant during all the calculations. The author acknowledges that

these two parameters can have a noticeable impact on the flow behaviour and hence

they would have been also tested. However, there existed limited time and

computational resources available for this project, and the student consequently decided

Chapter 5: Final Remarks 66

to carry on with the second part of the experiments because it could at least show if the

WLE could be beneficial when an upstream vortex impinges on the airfoils.

For the second experiment (4.3 Experiment B: Flaps with Wavy Leading Edges against

Upstream Vortex Condition) the major finding was that the wavy shapes on the flap’s

LE make the models more resistant to dirty upstream conditions. The variations the lift

coefficient suffers are smaller for the modified models than for the baseline. In some

cases the lift is even higher once the vortex has completely passed over the entire

model. After careful investigation of the results, the student concludes that reducing the

wavelength of the WLE results into more stable behaviour.

The vortex added in front of the models, highly affects the flow field downstream

immediately because of the flow has been treated as incompressible. The baseline

model, which only has a small percentage of the chord with attached flow, is in a very

unstable condition, and the perturbations added upstream of it leads to finally the flow

to separate all over the flap’s upper surface. When the vortex has passed, the baseline

model is not able to recover its initial condition with partially attached flow near its

leading edge. In contrast, although also affected initially by the upstream perturbations,

the WLEs models show much more robust performance during the simulated time.

Their patterned flow field produced by the sinusoidal LE seems to be more stable, and

consequently the models are able to recover the same flow field they used to have

before the introduction of the vortex.

At the same time, the main element, which has not been modified at any point during

the entire project, benefits form of the WLEs robust performance. The baseline flap

completely flow field completely changes when the upstream vortex condition is faced,

and this change is bigger at its LE. The flap’s LE zone is the one that interacts the most

with the main element and consequently big changes in former will led to big changes

in the latter. In contrast, for the modified models, we have seen that the flow remains

almost exactly the same in the first 10% of the flap’s chord. This almost constant

behaviour results in fewer variations on the main element and ultimately an overall

stable performance.

Major Findings and Conclusions

The WLEs did not perform better than the baseline case for clean upstream

conditions for the configuration studied in this project. However a proper

optimised design that includes the WLE parameters in the optimisation may

show different results.

When an upstream vortex impinges the multi-element airfoil, the models that

use WLE have more robust performance than the baseline with straight LE.

The protuberances at the flap’s LE introduce streamwise vortices that separate

the attached regions from the detached ones. The attached zones can be found

where those vortices tend to join, and the separated zones where the vortices

tend to separate.

The amplitude A of the protuberances is related with the intensity of those

streamwise vortices and the wavelength λ is related with the interaction that

those vortices have with themselves.

The behaviour we see when the WLE are used in a single element configuration

with clean upstream conditions, i.e. flow attached at the peak sections and


detached at valley section, cannot be assured when they are used in a two-

element configuration because of the effect of the upstream element.

The two-main objectives of the project have been fulfilled. The WLE have been

successfully implemented into a two-element configuration and the student has

gained some insight in how they perform in that situation. The performance

against upstream trailing vortices has been tested via Experiment B and results

showed that WLE offer more robust performance than straight LE.

Further Work and Recommendations

Both the angle of attack α and the flap deflection angle δ, have remained

constant during this project. It is recommended that the effect of these two

variables has to be tested to fully understand the flow produced by the WLE

when used in the flap element.

An optimisation of the models is needed to conclude that there is no way that

WLE can offer more lift than straight models. It is recommended that optimal

values of flap’s relative position and WLE shape are founded and then compared

with the straight LE optimum design.

The flow field created by the WLE shapes has multiple features that might be

too complicated to capture by a RANS model. Especially we have seen that

models based on the linear eddy viscosity assumption can be a poor

approximation of the real flow when sudden changes in the Strain rate appear.

This is the case in the flap region where we might expect massive separation. It

seems then more convenient to use more accurate models such as DES or LES

to confirm the findings obtained in this project.

Given that there are no academic papers that investigate the WLE in a two-

element configuration, a wind tunnel experiment will give at least some data to

validate against.

Finally, we have seen in this project, and in the literature review, that the WLE

can offer higher lift coefficients at high angles of attack than straight LE. The

maximum lift coefficient is nevertheless reduced, but there are some

applications where the author believes that having lift at very high AoA can be

very beneficial. For example in wind turbines, where high AoA means high

momentum produced by the turbine blades. It might be that the maximum lift is

lower for these WLE wind turbine blades; however the component of the force

they produce that contributes to make the wind turbine spin is much bigger

because of the very high AoA.

Appendix A: Mesh Dependency Study 69

Appendix A: Mesh Dependency Study

In order to quantify the amount of error in the simulations coming from numerical

errors/uncertainties, e.g. round-off error, iterative error or discretization error, a mesh

dependency study was carried on the baseline geometry. Modifying the base size of the

mesh the number of elements used for the calculations was changed for 6 cases from the

coarsest mesh with 3 Million elements to the finest mesh with 12.4 Million elements.

Results show that even for the finest mesh tested there still were minor variations on the

aerodynamic values investigated. Given that the computational power available is not

infinite the mesh with 4.7 Million cells was finally chosen because it was the best trade-

off between accuracy and computational efficiency. The number of simulations that

have been carried out in combination with the fact that because of the nature of

unsteady calculations, the Courant number must always be close to the unit, increased

the importance of having a computationally efficient mesh.

Table A.1 Mesh Dependency Study Results

# Elements (Millions)

Cl Error

% Cd

Error %

Cl main

Error% Cl Flap Error%

3.0 1.3969 -0.9% 0.0328 5.8% 1.2800 -0.9% 0.1169 -0.2%

3.6 1.4008 -0.6% 0.0327 5.4% 1.2836 -0.6% 0.1172 0.0%

4.7 1.4037 -0.4% 0.0322 3.8% 1.2861 -0.4% 0.1176 0.3%

5.8 1.4057 -0.2% 0.0316 1.8% 1.2888 -0.2% 0.1169 -0.3%

7.6 1.4061 -0.2% 0.0312 0.7% 1.2893 -0.2% 0.1168 -0.3%

12.4 1.4089 - 0.0310 - 1.2917 - 0.1172 -

Figure A.1 Mesh Dependency Study Convergence

With a 4.7 Million mesh simulations already took around 4 days to converge in the

worst cases. This means that if the author would have used the finest mesh as a base for

further calculations the cost in terms of time would have been unaffordable. It can be

seen in Table A.1 that the mesh with 4.7 Million cells has an error below 0.5% with

respect to the finest mesh in terms of CL coefficient. The author acknowledges that time

and computational power would not have been a request, they always are, and the best

option would have been taken the most accurate grid. However it has also to be pointed

Appendix A: Mesh Dependency Study 70

out that coarser meshes although would have been more computationally efficient, all

engineering problem need at least a minimum of accuracy so the results, could be

trusted. As a consequence the author decided that the grid with 4.7 Million elements had

to be the baseline mesh for further calculations.

Appendix B: Domain Size Dependency Study 71

Appendix B: Domain Size

Dependency Study

To be sure that the size of the domain was not influencing too much the computational

simulations a domain size study was carried out. Table B.1 shows the three different

domain sizes tested in this study. The first one, the smallest one, with 6 m before the

model and 12 m after the model with a height of 12 m as well. It can be seen that results

do not differ that much in any cases showing that the smallest domain was already a

good choice. However, given that it was tested that the domain with 12 m + 24 m x 24

m was computationally affordable and that it can be translated in chord units to 10c+20c

x 20c which is a common strategy in CFD calculations was the final domain size

selected to be the baseline for further calculations. Into account was also taken that a

vortex was planned to be introduced in the domain, which needs some space to be

placed away from the model and from the inlet because of computational stability.

Table B.1 Domain Size Study

Domain Dimensions

# Elements (Millions)

Cl Error % Cd Error % Cl main Error% Cl Flap Error%

6 + 12 x 12 3.1 1.4058 0% 0.0337 3% 1.2874 0.4% 0.1184 0.8%

12 + 24 x24 4.7 1.4037 0% 0.0322 -2% 1.2861 0.3% 0.1176 0.1%

15 +30 x30 8.0 1.3995 - 0.0328 - 1.2821 - 0.1175 -

Figure B.1 Domain Size Convergence

Appendix B: Domain Size Dependency Study 72

Appendix C: Validation against S-A Turbulence Model 73

Appendix C: Validation against S-A

Turbulence Model

In order to validate the results obtained using the SST model the baseline case was

simulated using the S-A model as well and the results were compared. In Figure C.1 we

can see that results obtained are similar between both turbulence models. However the

S-A model a larger zone of attached flow over the flap and a weaker wake combined

with a higher velocity above the main airfoil with lower velocity below it, which leads

to a higher performance in terms of lift coefficient. But if we have a look at Figure C.2

we can see how inside the boundary layer the velocity profiles are quite similar. Even

though at the very beginning, at Profile 1, there seems that S-A, as mentioned before

predicts a higher flow velocity, further downstream at Profiles 2, 3 and 4 the velocity

profiles are almost identical showing that the modelling of the attached boundary layer

is similar for both turbulence models

Figure C.1 Velocity Contours Comparison


Figure C.2 Velocity Profiles Comparison Moving from 1 to 4 in the stream direction

If we have a look at the pressure distribution over the model’s surfaces in Figure C.4 we

can confirm that the same trends stand for both models although as stated before the S-A

model predicts slightly bigger differences between the pressure and the suction surfaces

of the model. In both models it can be appreciated how intense is the pressure gradient

on the flap’s suction surface forcing the boundary layer to detach near ¼ of the flap’s

chord. We can further investigate the pressure on the model by looking at Figure C.3

where it is shown that the S-A pressure coefficient difference between pressure and

suction surfaces is slightly bigger on the main element. The pressure distribution on the

flap element also shows what has been just stated. However the differences are only

quantitative different showing in both cases the same pressure distribution shape with

some differences in the flap element where more turbulence modelling is present due to

the wake of the main element.

Figure C.3 Pressure Coefficient Distribution Comparison


Figure C.4 Pressure Coefficient Contours Comparison

Figure C.6 shows the Wall Shear Stress in the streamwise direction. We can see that

again the main differences appear in the flap element where the SST model predicts a

smaller strip of positive Wall Shear Stress and hence a smaller part of the flap’s flow is

attached. These differences can be further observed in Figure C.5 where we can see that

the Skin Friction coefficient predicted by the one equation model remains above zero

for more space at the first ¼ of the flap’s chord. Additionally we can see that the

presence of the flap creates a zone at the last ¼ of the main element’s pressure surface

where the pressure created by the stagnation point of the flap induces an adverse

pressure gradient on the main element flow that deteriorates its boundary layer making

the skin friction coefficient to get very close to zero on the pressure side of the main

element.


Figure C.5 Skin Friction Coefficient Distribution Comparison

Figure C.6 Wall Shear Stress X-Direction Contours Comparison


Figure C.7 Q-Criterion Iso-Surfaces Q = 100 Coloured by Dimensionless Velocity X-Direction

Comparison

Finally in Figure C.7 we can see the iso-surfaces for the second invariant of the velocity

gradient at a value of 100. Positive values of this Q-Criterion represent the parts of the

flow where rotation dominates strain and hence is a very common tool to identify vortex

regions as can be seen in (C.1) where Ω is the Rate-of-Rotation tensor and S is the Rate-

of-Strain tensor.

22

2

1SQ

(C.1)

We can see that the same structures appear in both cases but with a smaller size in the

case of the SST model. We can clearly see the recirculation created by the presence of

the flap on the pressure side of the main element near its TE and how once the flow is

detached in the flap bigger structures start to develop.

All in all, although minor quantitative differences we see that both structures and trends

predicted by both models are qualitatively similar making the point of this validation

study.

Appendix D: Matlab, VBA and JAVA Codes 79

Appendix D: Matlab, VBA and JAVA

Codes

All over this project, but mainly during the baseline elements’ relative position

investigation (see section 4.1), some codes have been used. The most important ones

can be founded in this appendix.

Matlab Code for Creating Base Wireframe Geometry

The code here creates the wireframe geometry needed to build the 3D models. Each

curve is saved in a text file in space separated values. It depends on the function

rotateFlap.m which is not included here.

function

createNew(ID,FlapOriginX,FlapOriginY,FlapAoA,Amplitude,wNumber)

% close all;

main1 = importdata('HomeDirectory\Data\Main1.txt',' ');

flap1 = importdata('HomeDirectory\Data\Flap.txt',' ');

flap2 = zeros(length(flap1),3);

main2 = zeros(length(main1),3);

wleZ = 0:0.001:0.5;

wle = zeros (length(wleZ),3);

modelID = ID;

A = Amplitude;

n = wNumber;

alfa = - FlapAoA;

flap1 = rotateFlap(flap1,alfa);

for i=1:length(wleZ)

wle(i,:) = [A*sin(2*pi*wleZ(i)*n/0.5) 0 wleZ(i)];

end

wle = rotateFlap(wle,alfa);

flapOriginX = FlapOriginX;

flapOriginY = FlapOriginY;

flapOrigin = [flapOriginX flapOriginY 0];

data = [modelID flapOriginX flapOriginY -alfa A n];

for i=1:length(flap1)

flap1(i,:) = flap1(i,:) + flapOrigin;

flap2(i,:) = flap1(i,:) + [0 0 0.5];


end

for i=1:length(main1)

main2(i,:) = main1(i,:) + [0 0 0.5];

end

for i=1:length(wleZ)

wle(i,:) = wle(i,:) + flapOrigin;

end

flapTE1 = [flap1(1,1) flap1(1,2), flap1(1,3);

flap1(1,1) flap1(1,2),0.5];

flapTE2 = [flap1(length(flap1),1) flap1(length(flap1),2),

flap1(length(flap1),3);

flap1(length(flap1),1) flap1(length(flap1),2),0.5];

flapTE3 = [flap1(1,1) flap1(1,2), flap1(1,3);

flap1(length(flap1),1) flap1(length(flap1),2),

flap1(length(flap1),3)];

flapTE4 = [flap1(1,1) flap1(1,2), 0.5;

flap1(length(flap1),1) flap1(length(flap1),2), 0.5];

mainTE1 = [main1(1,1) main1(1,2), main1(1,3);

main1(1,1) main1(1,2),0.5];

mainTE2 = [main1(length(main1),1) main1(length(main1),2),

main1(length(main1),3);

main1(length(main1),1) main1(length(main1),2),0.5];

mainTE3 = [main1(1,1) main1(1,2), main1(1,3);

main1(length(main1),1) main1(length(main1),2),

main1(length(main1),3)];

mainTE4 = [main1(1,1) main1(1,2), 0.5;

main1(length(main1),1) main1(length(main1),2), 0.5];

save HomeDirectory\Data\Main1.txt -ascii main1

save HomeDirectory\Data\Main2.txt -ascii main2

save HomeDirectory\Data\Flap1.txt -ascii flap1

save HomeDirectory\Data\Flap2.txt -ascii flap2

save HomeDirectory\Data\WLE.txt -ascii wle

save HomeDirectory\Data\FlapTE1.txt -ascii flapTE1




save HomeDirectory\Data\MainTE1.txt -ascii mainTE1




dlmwrite('HomeDirectory\Data\Data.txt',data,...

'delimiter',' ','precision', 8)


disp(horzcat('Model ',num2str(modelID),' Data.txt Saved'))

Matlab Code for the Optimisation Process

This code here uses Kriging43

and other functions developed by Dr Andreas Sobester44

for creating a response surface and suggest new design points. The code here depends

on many matlab functions which were all obtained from SESG6019: Design Search and

Optimisation module at University of Southampton. A text file with the results is used

to input the data needed to perform the optimisation.

addpath('HomeDirectoryOptimisation');

for index=1:1

clear all; close all;

maxval = inf;

acuracy = 51;

best=0;

scrsz = get(0,'ScreenSize');

global ModelInfo

f=zeros(acuracy,acuracy,acuracy);

ei=zeros(acuracy,acuracy,acuracy);

pos = zeros(3,1);

normpos = zeros(3,1);

results = importdata('HomeDirectoryData\Results.txt',' ');

var = results(:,2:4);

[n,k]=size(var);

Xplot = zeros(acuracy,k);

newPoints=zeros(1,k);

minlim = [0.975 -0.05 0];

maxlim = [1.05 -0.025 40];

for i=1:k

ModelInfo.X(:,i)=(var(:,i)-minlim(i))./(maxlim(i)-minlim(i));

end

val = results(:,5);

for l=1:n

label(l) =cellstr((num2str(results(l,1))));

end

ModelInfo.y = -val(:);

varname = 'Cl';

while (maxval==inf)

% Set upper and lower bounds for search of log theta

UpperTheta=ones(1,k).*2;

LowerTheta=ones(1,k).*-3;

% Run GA search of likelihood

[ModelInfo.Theta,MinNegLnLikelihood]=ga_DSO(@likelihood,k,[],[],[],[],

LowerTheta,UpperTheta);

43

Krige, “A Statistical Approach to Some Mine Valuation and Allied Problems on the Witwatersrand.” 44

Sobester.


% Put Cholesky factorisation of Psi into ModelInfo

[NegLnLike,ModelInfo.Psi,ModelInfo.U]=likelihood(ModelInfo.Theta);

% Plot surrogate models

X=0:1/(acuracy-1):1;

for i=1:acuracy

for j=1:acuracy

for l=1:acuracy

ModelInfo.Option='Pred';

f(i,j,l)=predictor([X(j) X(l) X(i)]);

ModelInfo.Option='NegLogExpImp';

ei(i,j,l)=predictor([X(j) X(l) X(i)]);

end

end

end

%A=10.^-ei;

A=-f;

for i=1:n

if(val(i)>best)

best=val(i);

bestID=results(i,1);

end

end

[maxval maxloc] = max(A(:));

[pos(2) pos(1) pos(3)] = ind2sub(size(A), maxloc);

maxvalCl=-f(pos(2),pos(1),pos(3));

end

for i=1:k

normpos(i) = (pos(i)-1)/(acuracy-1);

newPoints(i) =normpos(i) * ((maxlim(i))-minlim(i))+ minlim(i);

for g=1:acuracy

Xplot(g,i) = X(g) * ((maxlim(i))-minlim(i))+ minlim(i);

end

end

newPoints

modelID = results(end,1)+1;

disp(horzcat('Cl for Model',num2str(modelID),' expected:

',num2str(maxvalCl)))

disp(horzcat('Best model so far is: model',num2str(bestID),' with Cl =

',...

num2str(best)))

end


VBA Code for Geometry Creation in SOLIDWORKS

With this code, the wireframe geometry created before in Matlab and stored in text files

was imported into SOLIDWORKS and then using geometric operations the final 3D

models were created.

Sub main()

Set swApp = _

Application.SldWorks

Set Part = swApp.NewDocument("C:\ProgramData\SolidWorks\SolidWorks

2011\templates\Pieza.prtdot", 0, 0, 0)

swApp.ActivateDoc2 "Part2", False, longstatus

Set Part = swApp.ActiveDoc

Dim myModelView As Object

Set myModelView = Part.ActiveView

myModelView.FrameState = swWindowState_e.swWindowMaximized


Part.InsertCurveFileBegin

Open "HomeDirectory\Main1.txt" For Input As #1

Do While Not EOF(1)

Input #1, X, Y, Z

boolstatus = Part.InsertCurveFilePoint(X, Y, Z)

Loop

Close #1

boolstatus = Part.InsertCurveFileEnd()


Open "HomeDirectory\Main2.txt" For Input As #6

Do While Not EOF(6)

Input #6, X, Y, Z


Loop

Close #6


[…] ‘some code has been omitted here referring to adding more

wireframe curves


Open "HomeDirectory\MainTE4.txt" For Input As #6

Do While Not EOF(6)

Input #6, X, Y, Z


Loop

Close #6



boolstatus = Part.Extension.SelectByID2("Curve1", "REFERENCECURVES",

0, 0, 0, True, 0, Nothing, 0)


0, 0, 0, True, 0, Nothing, 0)

Part.ClearSelection2 True


0, 0, 0, False, 1, Nothing, 0)



0, 0, 0, True, 1, Nothing, 0)

Part.InsertCompositeCurve


0, 0, 0, True, 0, Nothing, 0)


0, 0, 0, True, 0, Nothing, 0)





0, 0, 0, True, 1, Nothing, 0)



0, 0, 0, True, 0, Nothing, 0)


0, 0, 0, True, 0, Nothing, 0)





0, 0, 0, True, 1, Nothing, 0)



0, 0, 0, True, 0, Nothing, 0)


0, 0, 0, True, 0, Nothing, 0)





0, 0, 0, True, 1, Nothing, 0)


boolstatus = Part.Extension.SelectByID2("CompCurve3",

"REFERENCECURVES", 1.18843647886268, 6.04510587053255E-02,

0.09993755764566, True, 0, Nothing, 0)


0, 0, 0, True, 0, Nothing, 0)


0, 0, 0, True, 0, Nothing, 0)


0, 0, 0, True, 0, Nothing, 0)



"REFERENCECURVES", 1.214, -0.0232152, 0.5, False, 1, Nothing, 0)


"REFERENCECURVES", 1.214, -0.0232152, 0, True, 1, Nothing, 0)


0, 0, 0, True, 4098, Nothing, 0)


0, 0, 0, True, 8194, Nothing, 0)


0, 0, 0, True, 12290, Nothing, 0)

Part.FeatureManager.InsertProtrusionBlend False, True, False, 1, 0, 0,

1, 1, True, True, False, 0, 0, 0, True, True, True



"REFERENCECURVES", 0, 0, 0, True, 0, Nothing, 0)


"REFERENCECURVES", 0, 0, 0, True, 0, Nothing, 0)



0, 0, 0, True, 0, Nothing, 0)


0, 0, 0, True, 0, Nothing, 0)



"REFERENCECURVES", 1, -0.00074, 0, False, 1, Nothing, 0)


"REFERENCECURVES", 1, -0.00074, 0.5, True, 1, Nothing, 0)


0, 0, 0, True, 4098, Nothing, 0)


0, 0, 0, True, 8194, Nothing, 0)

Part.FeatureManager.InsertProtrusionBlend False, True, False, 1, 0, 0,

1, 1, True, True, False, 0, 0, 0, False, True, True

Part.ViewZoomtofit2


boolstatus = Part.Extension.SelectByID2("Loft1", "SOLIDBODY", 0, 0, 0,

False, 0, Nothing, 0)

boolstatus = Part.Extension.SelectByID2("Loft2", "SOLIDBODY", 0, 0, 0,

True, 0, Nothing, 0)

longstatus = Part.SaveAs3("HomeDirectory\model.x_t", 0, 0)

swApp.ExitApp

End Sub

JAVA Code for Simulation Automation in STAR-CCM+

This JAVA code imports the geometry, names it, sets up the case with both mesh and

simulation parameters and finally saves results to a text file. However the code is too

long to be included here and just some parts of it are included here. Hence the empty

methods such as for example createBlock() have been taken out of the code because of

printing expenses

// STAR-CCM+ macro: macro.java

package macro;

import java.util.*;

import java.lang.*;

import java.io.*;

import star.common.*;

import star.base.neo.*;

import star.meshing.*;

import star.resurfacer.*;

import star.dualmesher.*;

import star.prismmesher.*;

import star.vis.*;

import star.flow.*;

import star.segregatedflow.*;

import star.metrics.*;

import star.saturb.*;


import star.turbulence.*;

import star.material.*;

import star.base.report.*;

public class macro extends StarMacro {

public void execute() {

Simulation mySim = getActiveSimulation();

String[] modelData = readData();

String ID = modelData[0];

mySim.println("Model " + ID + " succesfully loaded" );

String simulationName = "model" + ID;

importGeo(simulationName);

prepareGeo();

createBlock();

performSubstract();

createRegion();

createVolCont();

createMeshCont();

applyVolCont();

createPhyContSA();

applyLocalSet();

createDervParts();

createReports();

createMonitors();

createPlots();

editFieldFun();

parallelMeshing(true);

mySim.saveState(resolvePath("HomeDirectory" + simulationName +

".sim"));

meshModel();

convert2D();

runSimulation(0);

saveResults(modelData);

}

private void importGeo(String modelName) {

Simulation simulation_0 =

getActiveSimulation();

PartImportManager partImportManager_0 =

simulation_0.get(PartImportManager.class);

partImportManager_0.importCadPart(resolvePath("HomeDirectoryCAD\\"

+ modelName + ".stp"), "AllEdges", 30.0, 4, true, true);

simulation_0.println("Geometry imported succesfully");

}


private void prepareGeo() {



MeshPartFactory meshPartFactory_0 =

simulation_0.get(MeshPartFactory.class);

CompositePart compositePart_0 =

((CompositePart)

simulation_0.get(SimulationPartManager.class).getPart("Part1"));

CadPart cadPart_0 =

((CadPart)

compositePart_0.getChildParts().getPart("PartBody.1"));

CadPart cadPart_1 =

((CadPart)

compositePart_0.getChildParts().getPart("PartBody.2"));

meshPartFactory_0.combineMeshParts(cadPart_0, new

NeoObjectVector(new Object[] {cadPart_1}));

compositePart_0.explode(new NeoObjectVector(new Object[]

{compositePart_0}));

PartSurface partSurface_0 =

cadPart_0.getPartSurfaceManager().getPartSurface("PartBody");

cadPart_0.splitPartSurfaceByPatch(partSurface_0, new IntVector(new

int[] {34}), "flapElement");


int[] {35}), "flapElementTE");

PartSurface partSurface_1 =

cadPart_0.getPartSurfaceManager().getPartSurface("PartBody 2");


int[] {4}), "mainElement");


int[] {5}), "mainElementTE");

simulation_0.println("Geometry prepared succesfully");

}

private void createBlock() {}

private void performSubstract() {}

private void createRegion() {}

private void createVolCont() {}

private void createMeshCont() {}

private void applyVolCont() {}

private void createPhyContSA() {}


private void applyLocalSet() {}

private void createDervParts() {}

private void createReports() {}

private void createMonitors() {}

private void createPlots() {}

private void editFieldFun() {}

private void meshModel() {



MeshContinuum meshContinuum_0 =

((MeshContinuum)

simulation_0.getContinuumManager().getContinuum("Mesh 1"));

MeshPipelineController meshPipelineController_0 =

simulation_0.get(MeshPipelineController.class);

meshPipelineController_0.generateVolumeMesh();

String simulationName = simulation_0.getPresentationName();

simulation_0.println("Model meshed succesfully, Saving 3D

mesh...");

simulation_0.saveState(resolvePath("HomeDirectory" +

simulationName + "_Mesh.sim"));

}

private void runSimulation( int numberIterations) {



StepStoppingCriterion stepStoppingCriterion_0 =

((StepStoppingCriterion)

simulation_0.getSolverStoppingCriterionManager().getSolverStoppingCrit

erion("Maximum Steps"));

int currentIteration =

stepStoppingCriterion_0.getMaximumNumberSteps();

stepStoppingCriterion_0.setMaximumNumberSteps(numberIterations +

currentIteration);

simulation_0.getSimulationIterator().run();

}


private void convert2D() {}

private void parallelMeshing(boolean state) {}

private String[] readData(){

Simulation sim = getActiveSimulation();

File f = new File(("HomeDirectoryData\\Data.txt"));

Scanner s;

String[] dataArray = new String[6];

try {

s = new Scanner(f);

while (s.hasNextLine()) {

String linea = s.nextLine();

Scanner sl = new Scanner(linea);

sl.useDelimiter("\\s");

dataArray[0] = (sl.next());






}

s.close();

} catch (Exception e) {

sim.println("file not foud");

}

return dataArray;

}

private void saveResults(String[] modelData){

Simulation mySim = getActiveSimulation();

ForceCoefficientReport cdReport =

((ForceCoefficientReport)

mySim.getReportManager().getReport("Cd"));

ForceCoefficientReport clReport =

((ForceCoefficientReport)

mySim.getReportManager().getReport("Cl"));

double cdValue = cdReport.getReportMonitorValue();

double clValue = clReport.getReportMonitorValue();

double efValue = clValue/cdValue;

String cd = String.valueOf(cdValue);

String cl = String.valueOf(clValue);

String ef = String.valueOf(efValue);

try{

FileWriter fstream = new

FileWriter("HomeDirectoryData\\Results.txt",true);

BufferedWriter fbw = new BufferedWriter(fstream);


fbw.write(modelData[0] + " " + modelData[1] + " " +

modelData[2] + " " + modelData[3] + " " + cl + " " + cd + " " + ef);

fbw.newLine();

fbw.close();

}catch (Exception e){

mySim.println("Error happened during results saving

operation");

}

mySim.println("Model " + modelData[0] + ": results saved");

}

}

References 91

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Flaps with Wavy Leading Edges for Robust Performance agains Upstream Trailing Vortices

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Transcript of Flaps with Wavy Leading Edges for Robust Performance agains Upstream Trailing Vortices