TPFE MSc Advanced Turbulence Modelling I Most of the...

School of Mechanical Aerospace and Civil Engineering

TPFE MSc Advanced Turbulence Modelling

Lengthscale-Determining Equations

T. J. CraftGeorge Begg Building, C41

Reading:S. Pope, Turbulent FlowsD. Wilcox, Turbulence Modelling for CFDClosure Strategies for Turbulent and TransitionalFlows, (Eds. B.E. Launder, N.D. Sandham)Notes: Blackboard and CFD/TM web server:http://cfd.mace.manchester.ac.uk/tmcfd

- People - T. Craft - Online Teaching Material

Lengthscale-Determining Equations 2011/12 1 / 30

Introduction

Most of the modelling discussed so far has been based on the use of anε equation to provide a turbulence lengthscale.

However, as already seen, this is not the only possible route: one cansolve for one or more other quantities instead.

Even if one retains an ε equation, it is worth considering whether themodelled source/sink terms met so far could be improved upon.

In non-equilibrium flows it might be questionable whether the use ofsingle velocity and length scales is sufficient.

This lecture thus aims to:

Examine some weaknesses and alternative forms of the ε equation.

Explore the use of other variables in a second transport equation.

Briefly consider the use of multi-scale models.


Time & Length Scales in Turbulence Modelling

Most turbulence models require at least:

A velocity and a length scale (or time and length scales) Some tensorial information regarding the turbulence.

For example, a linear k -ε model represents the stress tensor via thestress-strain relation, and models the turbulent viscosity from acombination of velocity (k1/2) and length (k3/2/ε) scales.

uiuj =−νt

(∂Ui

∂xj+

∂Uj

∂xi

)+(2/3)kδij (1)

νt = cµk2/ε (2)

A stress transport model obtains the tensorial information from solvingtransport equations for uiuj , but uses time and length scales in modellingthe processes appearing in these equations.

Duiuj

Dt= Pij + φij − εij +dij (3)


The turbulent kinetic energy (obtained either directly from its transportequation or from summing the three normal stresses, u2

i /2) is generallyused to provide a velocity scale.

In one-equation models the lengthscaleis prescribed. In more advanced modelsit is obtained by solving a secondtransport equation.

l

y

ll=c y

The most widely-used choice is to solve for the turbulent kinetic energydissipation rate, ε, but other alternatives have also been proposed.

Before considering some of these alternatives, we first re-examine themodelling of the ε equation.


The ε Equation

Since ε = ν (∂ui/∂xj)(∂ui /∂xj), it is possible to derive an exact transportequation for ε by manipulating those for ui and uj .

However, this is not of great use in closing the equations, since mostterms in it are related to the fine-scale eddies, are not in closed form, andtherefore need to be modelled for practical applications.

In high-Reynolds-number turbulence it is often assumed that thelarge-scale flow structures (principally responsible for the turbulenceenergy) are generally unaffected by the details of the fine-scale eddies.

Consequently, if one assumes some kind of spectral equilibrium for theturbulence, one can argue that the level of dissipation can becharacterised by what happens in the larger scales.

As a result, most proposed ε equations are based on empirical modellingapproaches.


A widely-used (high Reynolds number) form of ε equation within theeddy-viscosity framework is:

DεDt

= cε1εPk

k−cε2

ε2

k+

∂∂xj

((νt/σε)

∂ε∂xj

)(4)

with coefficients cε1 = 1.44, cε2 = 1.92, σε = 1.22.

The coefficients have generally been tuned in order to ensure the modelpredicts certain basic flow features:

Decaying grid turbulence

The log-law in an equilibrium zero-pressure-gradient boundary layer

A simple shear flow (such as the plane jet, or mixing layer)

Most stress-transport schemes use a similar form, but often with a GGDH(Generalized Gradient Diffusion Hypothesis) model for the diffusion:

DεDt

= cε1εPk

k−cε2

ε2

k+

∂∂xj

(cεujuk (k/ε)

∂ε∂xk

)(5)

with cε around 0.18.


Some models have taken the coefficients cε1 and/or cε2 to not be simplyconstants.

For example, the RNG k -ε model makes cε1 sensitive to thenon-dimensional strain rate, taking

cε1 = 1.42− cµη3(1−η/ηo)

1+0.012ηcε2 = 1.68

with cµ = 0.0845, ηo = 4.38 and η = Sk/ε where S = (Sij Sij/2)1/2 andSij = ∂Ui/∂xj + ∂Uj/∂xi .

An advanced form of stress-transport model developed at UMIST madecε2 sensitive to local turbulence structure by taking it to be a function ofstress anisotropy invariants:

cε1 = 1.44 cε2 =1.92

1+0.7A1/22 A

where

A2 = aijaij A3 = aijajkaki A = 1− (9/8)(A2−A3)

aij = uiuj/k − (2/3)δij


Predicted Lengthscales with the ε Equation

Mixing-length and one-equation models rely on a prescribed lengthscale.

In fully-turbulent near-wall flows this isusually taken to increase linearly withdistance from the wall – to agree withexperimental data in equilibriumboundary layers.

l

y

ll=c y

Two-equation models have usually been tuned to return the samelengthscale in simple boundary layers, but are not constrained to do so inother flows.

Whilst this is a desirable feature, very large departures of the lengthscalefrom its equilibrium value can cause problems.

Two examples are predictions of flow and heat-transfer near reattachmentand impingement points.


Sudden Pipe Expansion

Computations with low-Re k -ε model.

The large lengthscale implies a highnear-wall turbulent viscosity, leading tohigh heat-transfer rates.


Impinging Jet

—- No Yap- - - Yap

Computed with low-Re k -ε scheme.

Predicted lengthscales near thestagnation point are much greater than inan equilibrium boundary layer.


Lengthscale Corrections

To overcome the above weaknesses, a “lengthscale correction” is oftenadded to ε equations.

This is generally an additional source term, designed to prevent themodel from returning excessively large lengthscales.

A simple form proposed by Yap (1987) is

Sε = cw(ε2/k)max[(l/le −1)(l/le)2,0

](6)

where l = k3/2/ε and le = cly with y the wall-normal distance and cw andcl are constants (typically 0.83 and 2.55).

One drawback with this is that it does require one to define the walldistance – which may be difficult in complex geometries.

Hanjalic (1996) proposed a form which compared the lengthscalegradient in the wall-normal direction to cl instead.


Iacovides & Raisee (1997) extended this to use the resultant lengthscalegradient, making the correction free of wall-geometry details.

They proposed

Ydc = cwε2

kmax

[F (F +1)2,0

]where

F =[((∂ l/∂xj )(∂ l/∂xj ))

1/2−dledy]/cl

dledy is the ‘equilibrium lengthscalegradient’, from differentiating theWolfshtein 1-equation formulation:

dledy = cl

[1−exp(−Bε Rt )

]+BεclRt exp(−Bε Rt )

with cl = 2.55, Bε = 0.1069, cw = 0.83and Rt = k2/(εν).

Heat Transfer in Ribbed Pipe:

- - - with Yap; —— with Ydc


Alternative Lengthscale-Determining Equations

Although it is the most widely-used second variable, ε is not the onlypossible choice.

Alternatives that have been proposed include

ω (≡ ε/k ) τ (≡ k/ε)

It is possible to formally convert a modelled ε equation into an exactequivalent for τ or ω .

For example, an equivalent ω equation can be obtained by writing

DωDt

=DDt

(ε/k) =1k

DεDt

− εk2

DkDt

(7)

and then substituting the source and diffusion terms from the k andmodelled ε equations into equation (7).

However, a model produced in this fashion would be expected to showexactly the same characteristics as the original modelled ε equation.


In practice, the equations adopted for other variables are slightly differentfrom such a formally converted ε one.

There may be different effective balances between generation anddestruction terms (equivalent cε1 and cε2 coefficients), and the modelleddiffusion terms are not generally entirely equivalent.

The expressions resulting from combining the diffusion terms from the kand ε equations into an ω equation are rather complex:

1k

∂∂xj

[(ν + νt/σε)

∂ε∂xj

]− ε

k2

∂∂xj

[(ν + νt/σk )

∂k∂xj

]

These are often replaced by a simple gradient-diffusion model:

∂∂xj

[(ν + νt/σω)

∂ω∂xj

]

The above replacement of “diffusion” terms can be quite significant –especially in the near-wall region.


Wilcox’s k -ω Models

Wilcox’s (1988) k -ω model took the form

DkDt

= Pk −ωk +∂

∂xj

[(ν + νt/σk)

∂k∂xj

](8)

DωDt

= cω1ωPk

k−cω2ω2 +

∂∂xj

[(ν + νt/σω)

∂ω∂xj

](9)

νt = cµk/ω (10)

with cω1 = 5/9, cω2 = 5/6, cµ = 0.09, σk = σω = 2.

Although this form did not include any specific “near-wall” terms, asubsequent proposal (Wilcox, 1991) did employ a number ofRt -dependent functions in order to improve its near-wall behaviour.

The model was reported to perform better than ε based schemes innear-wall adverse pressure gradient flows.


Sofialides’ Non-Linear k -ω Model

Sofialides (1993) testedWilcox’s k -ω model in animpinging jet flow.

Although stagnation heat transfer was overpredicted, the very sharp peakoften returned by ε-based schemes without a lengthscale correction (seeearlier slide) was not present.

As will be seen later, a linear EVM can be expected to show manyweaknesses in complex flows – whether it employs an ε, ω or any otherlengthscale-governing equation.

One of these weaknesses is an overprediction of stagnation heat transferin impinging flows.


Sofialides then developed a non-linear EVM (based on Suga’s 1995version – see later lecture) with an ω equation.

Plane Channel Flow

Impinging Jet Heat Transfer

Good impinging jet stagnation point heat transfer was obtained withoutany need for lengthscale-correction terms.

However, the model did not perform so well in other near-wall flows (forexample, by-pass transition).


Abe et al’s Non-Linear k -ω Model

Abe et al (2003) developed a non-linear EVM in which they employed anω equation.

This was based on an EASM (see laterlecture), although with additional empiricalterms to produce high levels of near-wallstress anisotropy.

They proposed two versions of the model: onewith an ω equation and one with an ε equation.

Flow over an array of 2-D hills:


ω-Equation Problems

One potential problem with ω is that it goes to infinity at a wall, which isnot a convenient boundary condition!

In practice, it is generally recommended that the value of ω should beprescribed at the near-wall node as

ω =6ν

cω2 y2 (11)

where y is the distance of the node from the wall.

This result arises from considering thenear-wall balance of terms in the modelledω equation.

Another problem – particularly in externalaerodynamic flows – is that the ω equationtends to show too much sensitivity tofree-stream turbulence levels.


Menter’s SST Model

One feature of Menter’s (1993) model is that it was designed to retain thenear-wall ω equation, but remove its sensitivity to free-stream turbulence.

It does this by using a blending function to switch from an ω equation towhat is equivalent to an ε equation far from a wall:

DωDt

= cω1ωPk

k−cω2ω2 +

∂∂xj

[(ν + νt/σω)

∂ω∂xj

]+2(1−F1)

σω2

ω∂k∂xj

∂ω∂xj

where F1 is a function designed to switch between the two models.

Adverse pressure gradientboundary layer over anairfoil.

However, this is still essentially a linear EVM, so does not performparticularly well in many complex flows (such as impinging ones).


Multi-Scale Models

As indicated earlier, the ε equation modelling is based on what ishappening in predominantly the large-scale eddies.

There is thus an implied assumption that the turbulence is in (or close to)spectral equilibrium. ie. the rate of transfer of energy from the smallestscales (dissipation) is equal to the rate at which it is transferred from thelarge to the intermediate scales.

This is unlikely to be true in complexflows. In rapidly evolving flows (in eitherspace or time), the turbulence spectrumwill not be in equilibrium.

The idea behind multiscale models is toattempt to account for this spectralnon-equilibrium.


The HLS Model

Hanjalic, Launder & Schiestel (1980) proposed a multiscale model:DkP

Dt=−uiuj

∂Ui

∂xj− εP +DkP

DkT

Dt= εP − εT +DkT

DεP

Dt=−CP1

εP

kPuiuj

∂Ui

∂xj+C′

P1kP∂Ul

∂xm

∂Ui

∂xjεlmk εijk −CP2

ε2P

kP+DεP

DεT

Dt= CT 1

εPεT

kT−CT 2

ε2T

kT+DεT

The motivation behind the model can be explained in terms of a series ofreservoirs:

Production scale energy, kP , istransferred to kT at a rate εP .

Transfer range energy, kT , isdissipated at a rate εT (≡ ε).

P

Pk

Pεk kT

ε = ε T


A further feature of the HLS model is that the generation rate of εP isbased not only on Pk , but on a more varied combination of mean strains.

The effect of this is to produce a stronger response to normal strainingcompared to shearing.

Application to the 4:1 sudden contraction of Uberoi (1957):

——: HLS k ; - - -: Single scale model;– –: Experiment


The Pulsed Jet

Bremhorst et al (2003) applied the HLS scheme to a pulsed round jet.

Both k -ε and RSM were also tested, but only the HLS schemereproduced some crucial features of the flow.

For example, the half-width variation through a cycle:x/D=50.0 x/D=56.0 x/D=60.0 x/D=66.0 x/D=70.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

7.5

10.0

12.5

15.0

17.5

20.0

t/tp

2 y1/2/d

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

7.5

10.0

12.5

15.0

17.5

20.0

t/tp

2 y1/2/d

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

7.5

10.0

12.5

15.0

17.5

20.0

t/tp

2 y1/2/d

k-ε TCL Model HLS Model

Further investigations showed that the modelling of spectralnon-equilibrium was not crucial in this case. Instead, it was the form ofgeneration terms in the εP equation that improved the predicted results.


Generation Rates of ε

The form of εP generation in the HLS model is similar to that tested byHanjalic & Launder (1980) within a 2-equation framework.

They proposed a generation term for ε of the form (in a 2-D flow)

−cε1εk

uv∂U∂y

−cε3εk

(u2 −v2 )∂U∂x

with cε1 = 1.44, cε3 = 4.44, andapproximated the difference (u2 −v2 ) by

u2 −v2 = 0.33k

Improved predictions were reported in theaxisymmetric jet, as well as adversepressure gradient boundary layers.

The modification as above is not in a coordinate-invariant form, (but couldbe formulated as one) and does not perform so well in some other flows.


Plane/Round Jet Anomaly

Measurements show a turbulent planejet in stagnant surroundings spreadsat a slower rate than the equivalentaxisymmetric jet.

Standard k -ε models generallyoverpredict the round jet spreadingrate dy1/2/dx .

Hanjalic & Launder reported improvedpredictions with the above model.

xy

U

1/2U m

m

1/2y

Plane Jet Round JetExpt. 0.11 0.093k -ε 0.107 0.112H-L 0.116 0.098

The streamwise velocity decay rate is greater in the round than in theplane jet.

The normal-strain related term is thus more important in the round jetcase, tending to increase ε, decrease k , and hence decrease thespreading rate.


Another plane/round jet solution was proposed by Pope (1978) as anadditional source term in the ε equation:

SεP = cεP(k2/ε)ΩijΩjkSki

where

Sij =∂Ui

∂xj+

∂Uj

∂xiΩij =

∂Ui

∂xj− ∂Uj

∂xi

In a plane jet flow SεP is zero, but it is non-zero in the axisymmetric case,increasing ε and hence decreasing the spreading rate.

However, these (and many similar) modifications do also have adverseeffects in some other flows.


References

Abe, K., Jang Y-J., Leschziner, M.A., (2003), An investigation of wall-anisotropyexpressions and length-scale equations for non-linear eddy-viscosity models, Int.J. Heat Fluid Flow, vol. 24, pp. 181-198.

Baughn, J.W., Shimizu, S., (1989), Heat transfer measurements from a surfacewith uniform heat flux and an impinging jet, ASME J. Heat Transfer, vol. 111, pp.1096-1098.

Baughn, J.W., Yan, X., Mesbah, M., (1992), The effect of Reynolds number on theheat transfer distribution from a flat plate to an impinging jet, Proc. ASME WinterAnnual Meeting.

Bremhorst, K., Craft, T.J., Launder, B.E., (2003), Two-time-scale turbulencemodelling of a fully-pulsed axisymmetric air jet, Proc. 3rd Int. Symposium onTurbulent Shear Flow Phenomena, Sendai, Japan.

Hanjalic, K., (1996), Some resolved and unresolved issues in modellingnon-equilibrium and unsteady turbulent flows, Proc. 3rd Int. Symp on EngineeringTurbulence Modelling and measurements, Crete.

Hanjalic, K., Launder, B.E., (1980), Sensitizing the dissipation equation toirrotational strains, it Trans. ASME Journal of Fluids Engineering, vol. 102, pp.34-40.


Hanjalic, K., Launder, B.E., Schiestel, R., (1980), Multiple-time-scale concepts inturbulent transport modelling, in Turbulent Shear Flows 2, (eds. L.J.S. Bradbury,F. Durst, B.E. Launder, F.W. Schmidt, J.H. Whitelaw), Springer-Verlag.

Iacovides, H., Raisee, M., (1997), Computation of flow and heat transfer in 2D ribroughened passages, Proceedings of the Second International Symposium onTurbulence, Heat and Mass Transfer, (eds. K. Hanjalic & T.W.J. Peeters), Delft.

Launder, B.E., Sharma, B.I., (1974), Application of the energy-dissipation modelof turbulence to the calculation of flow near a spinning disc, Lett. in Heat MassTransfer, vol. 1, pp. 131-138.

Menter, F.R., (1993), Zonal two equation k-ω turbulence models for aerodynamicflows, AIAA 24th Fluid Dynamics Conference, Orlando, Florida.

Pope, S.B., (1978), An explanation of the round-jet/plane-jet anomaly, AIAA J.,vol. 16, p. 279.

Sofialides, D., (1993), The strain-dependent non-linear k-ω model of turbulenceand its application, MSc Dissertation, Faculty of Technology, University ofManchester.

Suga, K., (1995), Development and application of a non-linear eddy viscositymodel sensitized to stress and strain invariants, PhD Thesis, Faculty ofTechnology, University of Manchester.

Uberoi, M.S., (1957), Equipartition of energy and local isotropy in turbulent flows,J. Appl. Phys., vol. 28, p. 1165.


Wilcox, D.C., (1988), Reassessment of the scale determining equation foradvanced turbulence models, AIAA J., vol. 26, pp. 1299-1310.

Wilcox, D.C., (1991), Progress in hypersonic turbulence modelling, Proc. AIAA22nd Fluid Dynamics, Plasmadynamics & Laser Conference, Honolulu.

Wolfshtein, M., (1969), The velocity and temperature distribution inone-dimensional flow with turbulence augmentation and pressure gradient, Int. J.Heat Mass Transfer, vol. 12, pp. 301-318.

Yap, C.R., (1987), Turbulent heat and momentum transfer in recirculating andimpinging flows, PhD Thesis, Faculty of Technology, University of Manchester.


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