Chapter 3: Turbulence and its modelingopencourses.emu.edu.tr/file.php/27/Chapter_3.pdf · Chapter...

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1 Chapter 3: Turbulence and its modeling Ib hi S i Ibrahim Sezai Department of Mechanical Engineering Eastern Mediterranean University Fall 2012-2013 Is the Flow Turbulent? External Flows ρUL Re L where Internal Flows 5 10 5× x Re along a surface around an obstacle μ Re L wee Other factors such as free-stream turbulence, surface conditions, and disturbances may cause earlier transition to turbulent flow. L = x, D, D h , etc. ,300 2 h D Re 20,000 D Re I. Sezai – Eastern Mediterranean University ME555 : Computational Fluid Dynamics 2 Natural Convection where is the Rayleigh number

Transcript of Chapter 3: Turbulence and its modelingopencourses.emu.edu.tr/file.php/27/Chapter_3.pdf · Chapter...

1

Chapter 3:Turbulence and its modeling

Ib hi S iIbrahim SezaiDepartment of Mechanical Engineering

Eastern Mediterranean University

Fall 2012-2013

Is the Flow Turbulent?

External Flows ρULReL ≡where

Internal Flows

5105×≥xRe along a surface

around an obstacle

μReLw e e

Other factors such as free-stream turbulence, surface conditions, and disturbances may cause earlier transition to turbulent flow.

L = x, D, Dh, etc.

,3002 ≥hD Re

20,000≥DRe

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Natural Convectionwhere

is the Rayleigh number

2

IntroductionFor Re > Recritical => Flow becomes Turbulent.Chaotic and random state of motion develops.Velocity and pressure change continuously withVelocity and pressure change continuously with timeThe velocity fluctuations give rise to additional stresses on the fluid

=> these are called Reynolds stresses

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We will try to model these extra stress terms

3.1 What is turbulence?

( ) ( )u t U u t′= +

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( ) ( )

3

Two Examples of Turbulent FlowIn turbulent flows there are rotational flow structures called turbulent eddies, which have a wide range of length scales.

Larger Structures Smaller Structures

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In turbulent flow:A streak of dye which is introduced at a point will rapidly

break up and dispersed effective mixing

Give rise to high values of diffusion coefficient for;- Mass- Momentum

and heat

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- and heatAll fluctuating components contain energy across a wide

range of frequencies or wave numbers2 fwave number f frequencyUπ

= =

4

Fig.3.3 Energy Spectrum of turbulence behind a grid

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Scales of turbulence

• Largest eddies break up due to inertial forces • Smallest eddies dissipate due to viscous forces • Richardson Energy Cascade (1922)

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5

Large Eddies:- have large eddy Reynolds number,- are dominated by inertia effects - viscous effects are negligible

are effectively inviscid

vlυ

- are effectively inviscidSmall Eddies:- motion is dictated by viscosity- Re ≈ 1- length scales : 0.1 – 0.01 mm- frequencies : ≈ 10 kHz

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Energy associated with eddy motions is dissipated and converted into thermal internal energy

increases energy losses.- Largest eddies anisotropic- Smallest eddies isotropic

3.2 Transition from laminar to turbulent flowCan be explained by considering the stability of laminar flows to small disturbances

Hydrodynamic instabilityHydrodynamic stability of laminar flows;

a) Inviscid instability: flows with velocity profile having a point of inflection.

1. jet flows2. mixing layers and wakes3 b d l i h d di

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3. boundary layers with adverse pressure gradientsb) Viscous instability: flows with laminar profile having no

point of inflection - occurs near solid walls

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Fig. 3.4 Velocity profiles susceptible to

a) Inviscid instability and

b) Viscous instability

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Transition to TurbulenceJet flow: (example of a flow with point of inflection in velocity

profile)

Fig. 3.5. transition in a jet flow

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Boundary layers on a flat plate (Example with no inflection point in the velocity profile)

The unstable two-dimensional disturbances are called Tolmien-Schlichting (T-S) waves

Fig. 3.6 plan view sketch of transition processes in boundary layer flow over a flat plate

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Fig. 3.7 merging of turbulent spots and transition to turbulence in a natural flat platenatural flat plate boundary layer

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Common features in the transition process:(i) The amplification of initially small disturbances(ii) The development of areas with concentrated rotational

structures(iii) The formation of intense small scale motions(iii) The formation of intense small scale motions(iv) The growth and merging of these areas of small scale motions

into fully turbulent flowsTransition to turbulence is strongly affected by:- Pressure gradient- Disturbance levels- Wall roughness

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Wall roughness- Heat transferThe transition region often comprises only a very small fraction of

the flow domainCommercial CFD packages often ignore transition entirely (classify the flow as only laminar or turbulent)

3.3Effect of turbulence on time-averaged Navier-Stokes eqns

In turbulent flow there are eddying motions of a wide range of length scalesA domain of 0.1×0.1m contains smallest eddies of 10-100 μm sizeμm sizeWe need mesh pointsThe frequency of fastest events ≈ 10 kHz Δt ≈ 100μs neededDNS of turbulent pipe flow of Re = 105 requires a computer

9 1210 10−

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which is 10 million times faster than CRAY supercomputerEngineers need only time-averaged properties of the flowLet’s see how turbulent fluctuations effect the mean flow properties

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Descriptors of Turbulent FlowTime average or meanFirst we define the mean Φ of a flow property φ as follows:

1 ( )t

t dtt

ϕΔ

Φ =Δ ∫ (3.2)

The property φ can be thought of as the sum of a steady mean component Φ and fluctuation component φ'

The time average of the fluctuations φ' is , by definition, zero:

0tΔ ∫

( ) ( )t tϕ ϕ′= Φ +

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g φ , y ,

0

1 ( ) 0t

t dtt

ϕ ϕΔ

′ ′= ≡Δ ∫ (3.3)

Variance, r.m.s. and turbulence kinetic energyThe spread of the fluctuations φ' about the mean Φ are measured by

( ) ( )1/ 2

2 21 tΔ⎡ ⎤∫

(3.4a)( ) ( )2 2

0

1 t

dtt

ϕ ϕΔ

′ ′=Δ ∫variance

( ) ( )2 2

0

1rms dt

tϕ ϕ ϕ

⎡ ⎤′ ′= = ⎢ ⎥Δ⎣ ⎦

∫The rms values of velocity components can be measured (e.g. by hot-wire anemometer)The kinetic energy k (per unit mass) associated with the turbulence is defined as

2 2 21 ( )2

k u v w′ ′ ′= + + (3.5)

and root mean square (r.m.s.) (3.4b)

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( )2

The turbulence intensity Ti is linked to the kinetic energy and a reference mean flow velocity Uref as follows:

1/ 2(2 / 3 )i

ref

kTU

= (3.6)

10

Moments of different fluctuating variablesThe variance is also called the second moment of the fluctuations.If and with 0ϕ ϕ ψ ψ ϕ ψ′ ′ ′ ′= Φ + = Ψ + = =Their second moment is defined as

0

1 t

dtt

ϕ ψ ϕ ψΔ

′ ′ ′ ′=Δ ∫

If velocity fluctuations in different directions were independent random fluctuations, then would be equal to zero.

However, although are chaotic, they are not independent.

, and u v u w v w′ ′ ′ ′ ′ ′

, andu v w′ ′ ′

(3.7)

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, g , y pAs a result the second moments are non-zero.

, and u v w, and u v u w v w′ ′ ′ ′ ′ ′

Higher order moments

Third moment:3 3

0

1( ) ( )t

dtt

ϕ ϕΔ

′ ′=Δ ∫ (3.8)

Third moment is related to skewness (asymmetry) of the distribution of the fluctuations:

0

Fourth moment:1 tΔ

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4 4

0

1( ) ( )t

dtt

ϕ ϕΔ

′ ′=Δ ∫

Fourth moment is related to kurtosis (peakedness) of the distribution of the fluctuations:

(3.9)

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Correlation functions – time and spaceThe autocorrelation function based on two measurements shifted by time τ is defined as

1( ) ( ) ( ) ( ) ( )t

R t t t t dtτ ϕ ϕ τ ϕ ϕ τΔ

′ ′ ′ ′= + = +∫

( )Rϕ ϕ τ′ ′

(3 10)

( )Rϕ ϕ′ ′ ξ

0

( ) ( ) ( ) ( ) ( )R t t t t dttϕ ϕ τ ϕ ϕ τ ϕ ϕ τ′ ′ = + = +

Δ ∫

Similarly, the autocorrelation function based on two measurements shifted by a certain distance in function spaceis defined as

1( ) ( ) ( ) ( ) ( )t t

R t t t t dtξ+Δ

′ ′ ′ ′ ′ ′ ′+ +∫ξ ξ

(3.10)

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( ) ( , ) ( , ) ( , ) ( , )t

R t t t t dttϕ ϕ ξ ϕ ϕ ϕ ϕ′ ′ ′ ′ ′ ′ ′ ′ ′= + = +

Δ ∫x x ξ x x ξ

(3.11)

When τ = 0, or ξ = 0 and it is maximum because the two correlations are perfectly correlated. Since φ' is chaotic we expect that the fluctuations become increasingly decorrelated as τ or ξ→∞ so, Rφ'φ'(∞) → 0.

h ddi h f b l i d f l l

2(0)Rϕ ϕ ϕ′ ′ ′=

The eddies at the root of turbulence cause a certain degree of local structure in the flow, so there will be a correlation between the values of φ' at time t and a short time later or at a given location x and a small distance away.The decorrelation process will take place gradually over the lifetime (or size scale) of a typical eddy.The integral time and length scale represent concrete measures of the

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The integral time and length scale represent concrete measures of the average period or size of a turbulent eddy. They can be computed from integrals of the autocorrelation functions Rφ'φ'(τ) or Rφ'φ'(ξ).By analogy it is also possible to define cross-correlation functions Rφ'ψ'(τ) or Rφ'ψ'(ξ) between pairs of different fluctuations.

12

Probability density functionProbability density function P(φ*) is related to the fraction of time that a fluctuating signal spends between φ* and φ*+d φ*:

* * * * *( ) Prob( )P d dϕ ϕ ϕ ϕ ϕ ϕ= < < + (3.12)The average, variance and higher moments of the variables and its fluctuations are related to the probability density functions as follows:

( )P dϕ ϕ ϕ ϕ∞

−∞

= ∫ (3.13a)

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( ) ( ) ( )n n P dϕ ϕ ϕ ϕ∞

−∞

′ ′ ′ ′= ∫If n = 2 we get variance, if n = 3, 4,… we get higher order moments.

(3.13b)

3.4.1 Free turbulent flows

maxmin U UU UU y y yg f h−−⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

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b= cross sectional layer width, y=distance in cross-stream directionx=distance downstream the source

max max min max min

for jets for mixing layers for wakes

g f hU b U U b U U b

= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠ ⎝ ⎠

13

Eddies of a wide range of length scales are visibleFluid from surroundings is entrained into the jetentrained into the jet.

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If x is large enough functions f, g and h are independent of x. such flows are called self – preserving.

The turbulence structure also reaches a self-preserving state albeit after a greater distance from the flow source than thealbeit after a greater distance from the flow source than the mean velocity. Then

Uref= Umax – Umin for a mixing layer and wakes

2 2 2

1 2 3 42 2 2 2ref ref ref ref

u y v y w y u v yf f f fU b U b U b U b′ ′ ′ ′ ′⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

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Uref= Umax for jetsForm of functions f, g, h and fi varies from flow to flowSee Fig.3.10

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Fig. 3.10 mean velocity distributions and turbulence properties for

(a) two- dimensional mixing layer, (b) planar turbulent jet and (c) wake behind a solid strip

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In Fig.3.10:are maximum when is maximum

u' gives the largest of the normal stresses (v and w ) .

uy∂∂

2 2 2,u v and w and u v′ ′ ′ ′ ′−

2u′

Fluctuating velocities are not equal anisotropic structure of turbulenceAs mean velocity gradients tend to zero turbulence quantities tend to zero turbulence can’t be sustained in absence of shear

max0.15 0.40u

u= −

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The mean velocity gradient is also zero at the centerline of jets and wakes. No turbulence there.The value of is zero at the centerline of a jet and wake since shear stress must change sign here.

u v′ ′−

15

3.4.2 Flat plate boundary and pipe flow

y: distance away from the wallNear the wall (y small) is small viscous forces dominate

Re inertia forcesviscous forces

=

Re Uyv

=

ReNear the wall (y small) is small viscous forces dominateAway from the wall (y large) is large inertia forces dominate.Near the wall U only depends on y, ρ, μ and τ (wall shear stress), so

Dimensional analysis shows that

Re y

Re y

( , , , )wU f y ρ μ τ=

( )u yUu f f yτρ+ +⎛ ⎞= = =⎜ ⎟ (3 16)

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Formula (3.18) is called the law of the wall

( )u f f yuτ μ

= = =⎜ ⎟⎝ ⎠

(3.16)

( )1/2 1/2( / 2) friction velocityw fu V Cτ τ ρ ∞= =

Far away from the wall:

Dimensional analysis yields( , , , )wU g y independent ofδ ρ τ μ=

U y⎛ ⎞y y

The most useful form emerges if we view the wall shear stress as the cause of a velocity deficit Umax – U which decreases the closer we get to the edge of the boundary layer or the pipe centerline. Thus

U yu guτ δ

+ ⎛ ⎞= = ⎜ ⎟⎝ ⎠

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pipe centerline. Thus

This formula is called the velocity – defect law.

maxU U yguτ δ− ⎛ ⎞= ⎜ ⎟

⎝ ⎠(3.17)

16

Linear sub layer – the fluid layer in contact with smooth wall

Very near the wall there is no turbulent (Reynolds) shear stresses flow is dominated by viscous shear

For shear stress is approximately constant,( 5)y+ <

( ) wUyτ μ τ∂

= ≅

Integrating and using U = 0 at y = 0,

After some simple algebra and making use of the definitions of u+

and y+ this leads to

( ) wyy

μ∂

wyU τμ

=

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Because of the linear relationship between velocity and distance from the wall the fluid layer adjacent to the wall is often known as the linear sub-layer

u y+ += (3.18)

Log-law layer – the turbulent region close to smooth wall

Outside the viscous sublayer (30 < y+ < 500) a region exists where viscous and turbulent effects are both important.

The shear stress τ varies slowly with distance; it is assumed to be constant and equal to τwbe constant and equal to τw

Assuming a length scale of turbulence lm = κy where lm is mixing length, the following relationship can be derived:

κ = 0.4, B=5.5, E=9.8; (for smooth walls)

1 1ln ln( )u y B Eyκ κ

+ + += + = (3.19)

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, , ; ( )B decreases with roughnessκ and B are universal constants valid for all turbulent flows past smooth

walls at high Reynolds numbers.(30 < y+ < 500) log – law layer

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Outer layer – the inertia dominated region far from the wall

Experimental measurement show that the log-law is valid in the region 0.02 < y ⁄ δ < 0.2.

For larger values of y the velocity-defect law (3.19) provides the correct form.

In the overlap region the log-law and velocity-defect law have to become equal.

Tennekes and Lumley (1972) show that a matched overlap is obtained by assuming the following logarithmic form:

1U U y− ⎛ ⎞

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where A is constantSee Fig.3.11

max 1 lnU U y A Law of the Wakeuτ κ δ

⎛ ⎞= +⎜ ⎟⎝ ⎠

(3.20)

- The inner region: 10 to 20% of the total thickness of the wall layer; the shear stress is (almost) constant and eq al to the all shear stress Within this region there are three

Fig.3.11

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(almost) constant and equal to the wall shear stress τw. Within this region there are three zones

- the linear sub-layer: viscous stresses dominate the flow adjacent to the surface- the buffer layer: viscous and turbulent stresses are of similar magnitude- the log-law layer: turbulent (Reynolds) stresses dominate.

- The outer region or law-of-the-wake layer: inertia dominated core flow far from wall; free from direct viscous effects.

18

For y ⁄ δ > 0.8 fluctuating velocities become almost equalisotropic turbulence structure here. (far away the wall)

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For y ⁄ δ < 0.2 large mean velocity gradientshigh values of .(high turbulence production) .Turbulence is anisotropic near the wall.

2 2 2,u v and w and u v′ ′ ′ ′ ′−

Fluid entering from top into the CV will bring in higher momentum fluid and will accelerate the slower moving layer As a result the fluid

Rules which govern the time averages of fluctuation propertiesd′Φ + ψ ψ ′Ψ +

Control volume in a 2D turbulent shear flow

moving layer. As a result, the fluid layer will experience additional turbulent shear stresses which are known as the Reynolds stresses.

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and :

.

0; ; ;

; ; ; 0

ds dss sϕϕ ψ ϕ

ϕ ψ ϕψ ϕ ψ ϕ ϕ

∂ ∂Φ′ ′= = Φ = Φ = = Φ∂ ∂

′ ′ ′+ = Φ +Ψ = ΦΨ + Ψ = ΦΨ Ψ =

∫ ∫

ϕ ϕ′= Φ + ψ ψ= Ψ +

19

Since div and grad are both differentiations the above rules can be extended to a fluctuating vector quantity a = A + a' and its combinations with a fluctuating scalar :

( ) ( ) ( ) ( ); ;div div div div div divϕ ϕ ϕ′ ′= = = ΦΑ +a A a a a (3 22)

ϕ ϕ′= Φ +

To illustrate the influence of turbulent fluctuations on mean flow we consider

( ) ( ) ( ) ( ); ;

div grad div grad

ϕ ϕ ϕ

ϕ = Φ(3.22)

( )

01

1

divu pdiv u v div grad ut xρ

=∂ ∂

+ = − +∂ ∂∂ ∂

u

u (3.24a)

(3.23)

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Substitute

( )

( )

1

1

v pdiv v v div grad vt yw pdiv w v div grad wt z

ρ

ρ

∂ ∂+ = − +

∂ ∂∂ ∂

+ = − +∂ ∂

u

u

(3.24b)

(3.24c)

; ; ; ; u U u v V v w W w p P p′ ′ ′ ′ ′= + = + = + = + = +u U u

Consider continuity equation: note that this yields the continuity equation

A similar process is now carried out on the x-momentum equation (3.9a). 0div =U (3.25)

div div=u U

p q ( )The time averages of the individual terms in this equation can be written as follows:

S b i i f h l i h i

( ) ( ); ( )

1 1 ;

u U div u div U div ut t

p P vdiv grad u v div grad Ux xρ ρ

∂ ∂ ′ ′= = +∂ ∂

∂ ∂− = − =

∂ ∂

u U u

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Substitution of these results gives the time-average x-momentum equation

1( ) ( )

( ) ( ) ( ) ( ) ( )

U Pdiv U div u v div grad Ut xI II III IV V

ρ∂ ∂′ ′+ + = − +∂ ∂

U u (3.26a)

20

Repetition of this process on equations (3.9b) and (3.9c) yields the time-average y-momentum and z-momentum equations

1( ) ( )V Pdiv V div v v div grad V∂ ∂′ ′+ + = − +U u (3 26b)( ) ( )

(I) (II) (III) (IV) (V)1( ) ( )

(I) (II) (III) (IV) (V)

div V div v v div grad Vt y

W Pdiv W div w v div grad Wt z

ρ

ρ

+ + +∂ ∂

∂ ∂′ ′+ + = − +∂ ∂

U u

U u

(3.26b)

(3.26c)

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The process of time averaging has introduced new terms (III).Their role is additional turbulent stresses on the mean velocity components U, V and W.

Placing these terms on the rhs:

21( )U P u u v u wdiv U v div grad Ut x x y zρ

⎡ ⎤′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂+ = − + + − − −⎢ ⎥

∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦U

t x x y zρ∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦(3.27a)

21( )V P u v v v wdiv V v div grad Vt y x y zρ

⎡ ⎤′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂+ = − + + − − −⎢ ⎥

∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦U

(3.27b)

⎡ ⎤

Reynolds equations

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21( )W P u w v w wdiv W v div grad Wt z x y zρ

⎡ ⎤′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂+ = − + + − − −⎢ ⎥

∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦U

(3.27c)

21

The extra stress terms result from six additional stresses, three normal stresses and three shear stresses:

2 2 2xx yy zz

xy yx xz zx yz zy

u v w

u v u w v w

τ ρ τ ρ τ ρ

τ τ ρ τ τ ρ τ τ ρ

′ ′ ′= − = − = −

′ ′ ′ ′ ′ ′= = − = = − = = −

(3.28a)

(3.28b)-These extra turbulent stresses are termed the Reynolds stresses.-The normal stresses are always non-zero-The shear stresses are also non-zeroIf, for example, u' and v' were statistically independent fluctuations, the time

average of their product would be zeroSimilar extra turbulent transport terms arise when we derive a transport

2 2 2, andu v wρ ρ ρ′ ′ ′− − −, andu v u w v wρ ρ ρ′ ′ ′ ′ ′ ′− − −

u v′ ′

( )

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equation for an arbitrary scalar quantity. The time average transport equation for scalar φ is

*( ) ( ) u v wdiv div grad St x y z

ϕ ϕ ϕΦ Φ

⎡ ⎤′ ′ ′ ′ ′ ′∂Φ ∂ ∂ ∂+ Φ = Γ Φ + − − − +⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦

U (3.29)

In compressible flow density fluctuations are usually negligibleThe density-weighted averaged (or Favre-averaged) form of the compressible turbulent flow are:

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22

Closure problem – the need for turbulence modeling

In the continuity and Navier – Stokes equations (3.8) and (3.9 a-c) there are 6 additional unknowns

the Reynolds stressesySimilarly in scalar transport eqn (3.14) there are 3

extra unknowns:

Main task of turbulence modeling:,u v and wϕ ϕ ϕ′ ′ ′ ′ ′ ′

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is to develop computational procedures to predict the 9 extra terms. (Reynolds stresses and scalar transport terms).

3.4. Characteristics of simple turbulent flowsIn turbulent thin shear layers,

1

x yδ

∂ ∂<<

∂ ∂

<<

The following 2D incompressible turbulent flows with constant P will be considered:Free turbulent flows- Mixing layers- Jet- Wake

B d l lid ll

L

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Boundary layer near solid walls- Flat plate boundary layer- Pipe flow

Data for U, will be reviewed.These can be measured by 1) Hot wire anemometry 2) Laser doppler

anemometers

2 2 2, ,u v w and u vρ ρ ρ ρ′ ′ ′ ′ ′− − − −

23

3.5 Turbulence models

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We need expressions for :- Reynolds stresses:- Scalar transport terms :

2 2 2' , , , , ,u u v u w v v w w′ ′ ′ ′ ′ ′ ′ ′,u v and wϕ ϕ ϕ′ ′ ′ ′ ′ ′

Classical models: use the Reynolds eqn’s.(all commercial CFD codes)Large eddy simulation: the flow eqn’s are solved for the

- mean flow- and largest eddies

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and largest eddiesbut the effect of the smaller eddies are modeled- are at the research state- calculations are too costly for engineering use.

24

energy cascade (Richardson, 1922)

Turbulence Scales and Prediction Methods

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The mixing length and k-ε models are the most widely used and validated.They are based on the presumption that“there exists an analogy between the action of viscous stressesand Reynolds stresses on the mean flow”and Reynolds stresses on the mean flowViscous stresses are proportional to the rate of deformation. For incompressible flow:

Notation:

2 jiij ij

j i

uusx x

τ μ μ⎛ ⎞∂∂

= = +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠(2.31)

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Notation:i = 1 or j = 1 x-directioni = 2 or j = 2 y-directioni = 3 or j = 3 z-direction

25

For example

Turbulent stresses are found to increase as the mean rate of deformation increases.

1 212

2 1xy

u u u vx x y x

τ τ μ μ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂

= = + = +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

It was proposed by Boussinesq in 1877 thatReynolds stress could be linked to mean rate of deformation

where

23

jiij i j t ij

j i

UUu u kx x

τ ρ μ ρ δ⎛ ⎞∂∂′ ′= − = + −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

(3.33)

2 2 20.5( )k u v w′ ′ ′= + +

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This is similar to viscous stress equation

except that μ is replaced by μt, where μt = turbulent (eddy) viscosity

2 jiij ij

j i

UUsx x

τ μ μ⎛ ⎞∂∂

= = +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

Similarly; kinematic turbulent or eddy viscosity.Eqn.(3.23) shows that

/t tv μ ρ=

jiij i j t

j i

UUu ux x

τ ρ μ⎛ ⎞∂∂′ ′= − = +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

Turbulent momentum transport is assumed to be proportional to mean gradients of velocity

By analogy turbulent transport of a scalar is taken to be proportional to the gradient of the mean value of the transported quantity. In suffix notation we get

j i⎝ ⎠

i tuρ ϕ ∂Φ′ ′− = Γ (3 34)

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Where Γt is the turbulent diffusivity.We introduce a turbulence Prandtl / Schmidt number as

i ti

ux

ρ ϕ Γ∂

(3.34)

tt

t

μσ =Γ

(3.35)

26

Experiments in many flows have shown that

Most CFD procedures use Mixing length models:

1tσ ≈1tσ ≈

Mixing length models:Attempts to describe the turbulent stresses by means of simple algebraic

formulae for μt as a function of positionThe k-ε models:

Two transport eqn’s (PDE’s) are solved:1. For the turbulent kinetic energy, k

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2. For the rate of dissipation of turbulent kinetic energy, ε.

Both models assume that μt is isotropic (same for u, v, and w equations)(i.e. the ratio between Reynolds stresses and mean rate of deformation is the same in all

directions)

2 / ( constant)t C C k Cμ μμ ρϑ ρ ε= = =

This assumption fails in many categories of flow.It is necessary to derive and solve transport eqn’s for the 6 Reynolds stresses themselves. ( ).

These eqn’s contain:2 2 2' , , , , ,u u v u w v v w w′ ′ ′ ′ ′ ′ ′ ′

1. Diffusion 2. Pressure strain 3. Dissipationterms whose individual effects are unknown and cannot be measured

Reynolds stress equation models:Solves the 6 transport equations (PDE’s) for the Reynolds stress terms

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Solves the 6 transport equations (PDE s) for the Reynolds stress terms together with k and ε eqn’s., by making assumptions about

- diffusion- pressure strain- and dissipation terms.

27

Algebraic stress models:

Approximate the Reynolds stresses in terms of pp yalgebraic equations instead of PDE type transport equations.Is an economical form of Reynolds stress model.Is able to introduce anisotropic turbulence effects into CFD simulations

I. Sezai – Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 53

into CFD simulations

3.5.1 Mixing length model

tv Cϑ= (3.36)

2

: turbulent velocity scale

: length scale of largest eddies

tm m ms s

ν ϑ→ = ×

ϑ

where C is a dimensionless constant of proportionality.Turbulent viscosity is given byThis works well in simple 2-D turbulent flows where the only significant

Reynolds stress is and only significant velocity gradient is

xy yx u vτ τ ρ ′ ′= = −

t

t Cμ ρϑ=

Uy

∂∂

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For such flows

where c is dimensionless constant

y

Ucy

ϑ ∂=

∂ (3.37)

28

Combining (3.26) and (3.27) and absorbing C and c into a new length scale

This is Prandtl’s mixing length model.

(3.38)2t m

Uvy

∂=

m

The Reynolds stress is2

xy yx mU Uu vy y

τ τ ρ ρ ∂ ∂′ ′= = − =∂ ∂

(3.39)

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The transport of scalar quantity φ is modeled as

(3.40)tv

yρ ϕ ∂Φ′ ′− = Γ

where Γt = μt ⁄ σt and vt is found from (3.28).

σt = 0.9 for near wall flows

σt = 0.5 for jets and mixing layers Rodi(1980)

σ = 0 7 in axisymmetric jets

y∂

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σt 0.7 in axisymmetric jetsIn table 3.3:y : distance from the wallκ : 0.41 von Karman’s constant

29

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30

3.5.2. The k – ε model If convection and diffusion of turbulence properties

are not negligible (as in the case of recirculating flows), then the mixing length model is not applicable.

The k-ε model focuses on the mechanisms that affect the turbulent kinetic energy.Some preliminary definitions:

( )2 2 21 ki iK U V W

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( )

( )

2 2 2

2 2 2

mean kinetic energy2

1 turbulent kineticenergy2

( ) instantaneous kinetic energy

K U V W

k u v w

k t K k

= + +

′ ′ ′= + +

= +

To facilitate the subsequent calculations it is common to write the components of the rate of deformation sij and the stresses τij in tensor (matrix) form:

andxx xy xz xx xy xz

ij yx yy yz ij yx yy yz

s s ss s s s

τ τ ττ τ τ τ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟

= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

- Using gives zx zy zz zx zy zzs s s τ τ τ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠( )ij ij ijs t S s′= +

( ) ; ( ) ;

( ) ;

1 1( ) ( )

xx xx xx yy yy yy

zz zz zz

U u V vs t S s s t S sx x y y

W ws t S sz z

U V u vs t S s s t S s

′ ′∂ ∂ ∂ ∂′ ′= + = + = + = +∂ ∂ ∂ ∂

′∂ ∂′= + = +∂ ∂

′ ′⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂′ ′= + = = + = + + +⎢ ⎥ ⎢ ⎥

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( ) ( )2 2

1( ) ( )2

xy xy xy yx yx yx

xz xz xz zx zx zx

s t S s s t S sy x y x

U Ws t S s s t S sz x

= + = = + = + + +⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦∂ ∂⎡ ⎤′ ′= + = = + = + +⎢ ⎥∂ ∂⎣ ⎦

12

1 1( ) ( )2 2yz yz yz zy zy zy

u wz x

V W v ws t S s s t S sz y z y

′ ′∂ ∂⎡ ⎤+⎢ ⎥∂ ∂⎣ ⎦′ ′⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂′ ′= + = = + = + + +⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦

31

The product of a vector a and a tensor bij is a vector c

( )11 12 13

1 2 3 21 22 23

31 32 33

ij i ij

b b bb a b a a a b b b

b b b

⎛ ⎞⎜ ⎟≡ = ⎜ ⎟⎜ ⎟⎝ ⎠

a

The scalar product of two tensors aij and bij is evaluated as follows

1 11 2 21 3 31 1

1 12 2 22 3 32 2

1 13 2 23 3 33 3

T T

j

a b a b a b ca b a b a b c ca b a b a b c

+ +⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= + + = = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

c

11 11 12 12 13 13 21 21 22 22 23 23ij ija b a b a b a b a b a b a b⋅ = + + + + +

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Suffix notation:1 x – direction2 y – direction3 z – direction

31 31 32 32 33 33a b a b a b+ + +

Governing equation for mean flow kinetic energy K

Reynolds equations: (3.27a), (3.27b), (3.27c)

Multiply x – component Reynolds eqn. (3.27a) by UMultiply x component Reynolds eqn. (3.27a) by UMultiply y – component Reynolds eqn. (3.27b) by VMultiply z – component Reynolds eqn. (3.27c) by W

Then add the result together

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Then add the result together.After some algebra the time-average eqn. for mean

kinetic energy of the flow is

32

Or in words, for the mean kinetic energy K, we have

( )( ) ( ) 2 2

( ) ( ) ( ) ( ) ( ) ( ) ( )

ij i j ij ij i j ijK div K div P S u u S S u u StI II III IV V VI VII

ρ ρ μ ρ μ ρ∂ ′ ′ ′ ′+ = − + − − ⋅ + ⋅∂

U U U U

(3.41)

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Governing equation for turbulent kinetic energy k1- Multiply x, y and z momentum eqns (3.24a-c) by u´, v´ and w´,respectively2- Multiply x, y and z Reynolds eqns (3.27a-c) by u´, v´ and w´, respectively3- subtract the two of the resulting equationsRearrange:

i ld h i f b l ki i kyields the equation for turbulent kinetic energy k:

( ) 1( ) 2 22

( ) ( ) ( ) ( ) ( ) ( ) ( )

ij i i j ij i j ijij

k div k div p s u u u s s u u StI II III IV V VI VII

ρ ρ μ ρ μ ρ∂ ⎛ ⎞′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′+ = − + − ⋅ − ⋅ + ⋅⎜ ⎟∂ ⎝ ⎠U u u

(3.42)

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33

The viscous term (VI)

Gives a negative contribution to (3.32) due to the appearance of the sum of squared fluctuating deformation rate s'ij

( )2 2 2 2 2 211 22 33 12 13 232 2 2 2 2ij ijs s s s s s s sμ μ′ ′ ′ ′ ′ ′ ′ ′− ⋅ = − + + + + +

The dissipation of turbulent kinetic energy is caused by work done by the smallest eddies against viscous stresses.

The rate of dissipation per unit mass (m2/s3) is denoted by

The k – ε model equations

2 ij ijvs sε ′ ′= ⋅ (3.43)

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It is possible to develop similar transport eqns. for all other turbulence quantitiesThe exact ε – equation, however, contains many unknown and unmeasurable termsThe standard k – ε model(Launder and Spalding,1974) has two model eqns (a) one for k and (b) one for ε

We use k and ε to define velocity scale and length scale representative of the large scale turbulence as follows:

ϑ

3/ 21/ 2 kkϑ = =

Applying the same approach as in the mixing length model we specify the eddy viscosity as follows

kϑε

= =

2

tkC Cμμ ρϑ ρε

= = (3.44)

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where Cμ is a dimensionless constantε

34

The standard model uses the following transport equations used for k and ε( ) ( ) 2

k

tt ij ij

k P

k div k div grad k S St

ρ

μρ ρ μ ρεσ⎛ ⎞∂

+ = + ⋅ −⎜ ⎟∂ ⎝ ⎠U (3.45)

2( ) ( ) 2tdi di d C S S Cμρε ε ερε ε ρ⎛ ⎞∂

+ +⎜ ⎟U (3 46)In words the equations are

1 2( ) ( ) 2t

t ij ijdiv div grad C S S Ct k kε ε

ε

μρ ρε ε μ ρσ

+ = + ⋅ −⎜ ⎟∂ ⎝ ⎠U (3.46)

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The equations contain five adjustable constants . The standard k – ε model employs values for the constants that are arrived at by comprehensive data fitting for a wide range of turbulent flows:

1 20.09; 1.00; 1.30; 1.44; 1.92kC C Cμ ε ε εσ σ= = = = =(3.47)

1 2, , , and kC C Cμ ε ε εσ σ

To compute the Reynolds stresses with the k – ε model (3.44-3.46) Boussinesq relationship is used:

2 223 3

jii j t ij t ij ij

j i

UUu u k S kx x

ρ μ ρ δ μ ρ δ⎛ ⎞∂∂′ ′− = + − = −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

(3.48)

δij = 1 if i = j, δij = 0 if i ≠ jContraction gives

For incompressible flow Then which is the definition of k.

j i⎝ ⎠

2 22 23 3

i ii i t ii t

i i

U Uu u k kx x

ρ μ δ ρ μ ρ∂ ∂′ ′− = − = −∂ ∂

31 2

1 2 3

0 due to continuityi

i

U UU Ux x x x

∂ ∂∂ ∂= + + =

∂ ∂ ∂ ∂3

2i iu u k′ ′ =∑

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So an equal 1/3 is allocated to each normal stress component in eqn (3.48) to have –2ρk when summed. This implies an isotropic assumption for the normal Reynolds stresses.

1i=∑

( )1 1 2 2 3 31 where 2

ii i ik u u u u u u u u u u⎧ ⎫′ ′ ′ ′ ′ ′ ′ ′ ′ ′= + + =⎨ ⎬⎩ ⎭

35

B.c.’s for high Re model k and ε-equationsThe following boundary conditions are needed

For inlet, if no k and ε is available crude approximations can be obtained from the

- Turbulence intensity, Ti

- Characteristic length, L (equivalent pipe radius) :

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g , ( q p p )

The formulae are closely related to the mixing length formulae in section 3.5.1 and the universal distributions near a solid wall given below.

( )3/ 22 3/ 42 ; ; 0.07

3 ref ikk U T C Lμε= = =

Wall boundary conditions for high Re modelsNear the wall, at high Reynolds numbers, equations (3.44-3.46) of the

standard k – ε model are not integrated right to the wall. Instead, universal behaviour of near wall flows is used.

1 ln for 30 500u Ey y+ + += < < (3.19)

In this region measurements of the budgets indicates that:the rate of turbulence production = rate of dissipation

Using this assumption and , the following wall functions can be derived for a point P in the region 30 < y+ < 500:

ln for 30 500u Ey yκ

< < ( )

( )2 31 ln ; ; for 30 500P u uUu Ey k yτ τε+ + += = = = < < (3 49)

2 /t C kμμ ρ ε=

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whereκ = 0.41 (Von Karman’s constant)E = 9.8 (wall roughness parameter) for smooth walls

( )ln ; ; for 30 500P Pu Ey k yu yCτ μ

εκ κ

= = = = < < (3.49)

( )1/2 friction velocity, wu yu y τ

τρτ ρμ

+= =

36

Wall b.c.’s for high Re models: Momentum EqnsUsing k from Eqn. (3.49),

21/2 1/4 P P

uk u k CCτ

τ μμ

= → =1/2 1/4P PP

P

k C yu yy μτ ρρμ μ

+ = =

In the viscous sublayer near the wall (y+ < 5) the flow is laminar andIn the viscous sublayer, near the wall (y < 5) the flow is laminar and the velocity is given by

u y+ +=

The position of the interface between the laminar sublayer and the log law layer can be found by equating Eqns. (3.18) and (3.19):

(3.18)

int int int1 ln( ), 11.63y Ey y+ + += → =

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int int intln( ), 11.63y Ey yκ

if 11.631 ln( ) if 11.63

P P

PP P

y yu

Ey yκ

+ +

++ +

⎧ <⎪= ⎨

≥⎪⎩

Then, the velocity at a point P near the wall is found from

Wall b.c.’s for high Re models: k and ε-Eqnsk-equation, Eqn. (3.45), is integrated right to the wall. However, the source term of the k-equation given by

2k t ij ij ks S S Pμ ρε ρ ρε= ⋅ − = −is calculated at a near wall point P by assuming local equilibrium. Under this assumption,the rate of turbulence production = rate of dissipation.Thus, the rate of turbulence production is calculated from

2 1/2 1/41/4 1/2 , where , w

k w w w PUP u u k Cy C k y τ τ μ

ττ τ τ ρκρ

∂≈ = = =

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P Py C k y μμκρ∂

and ε is calculated from3/4 3/2

PP

P

C ky

μεκ

=

37

Wall b.c.’s for high Re models: k and ε-EqnsThe boundary condition for k imposed at the wall is

/ 0k n∂ ∂ =where n is normal to the wall.The ε-equation is not solved at the wall-adjacent cells but is computed q j pfrom

3/4 3/2P

PP

C ky

μεκ

=

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Wall b.c.’s for high Re models: Energy equationThe wall heat flux is given by

where T+ is calculated from the universal near wall temperature distribution valid at high Reynolds numbers (Launder and Spalding,

1/4 1/2 ( ) /w P P P wq C C k T T Tμρ += − −

g y ( p g,1974):

( ) ,,

,

,

,

with temperature at near wall point turbulent Prandtl number

wall temperature / Prandtl numberll h fl h l d i

T lw PT t

w T t

p P T t

w T l P T

T T C uT u P

q

T y

T C

τ σρσ

σ

σ

σ μ

+ +⎡ ⎤⎛ ⎞−

≡ − = +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦= =

= = Γ =

Γ i

(3.50)

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Finally P is the “pee-function”, a correction function dependent on the ratio of laminar to turbulent Prandtl numbers (Launder and Spalding, 1974)

wall heat flux thermal conductiw Tq = Γ = vityfluid specific heat and constant pressurepC =

38

Low Reynolds number k-ε modelsAt low Reynolds numbers (which is the case near the wall) the constants Cμ, C1ε and C2ε in Eqns. (3.52-3.53) are not valid so these equations cannot be integrated right to the wall. To integrate the governing equations right to the wall, modifications to the standard k-ε model is necessary.The equations of the low Reynolds number k model which replaceThe equations of the low Reynolds number k – ε model, which replace (3.44-3.46), are given below:

( ) ( ) 2tt ij ij

k div k div grad k S St

μρ ρ μ μ ρεσ

⎛ ⎞⎛ ⎞∂+ = + + ⋅ −⎜ ⎟⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠

U

2

tkC fμ μμ ρε

= (3.51)

(3.52)

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kt σ⎜ ⎟∂ ⎝ ⎠⎝ ⎠

2

1 1 2 2

( ) ( )

2

t

t ij ij

div div gradt

C f S S C fk k

ε

ε ε

μρε ρε μ εσ

ε εμ ρ

⎛ ⎞⎛ ⎞∂+ = +⎜ ⎟⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠

+ ⋅ −

U

(3.53)

Modifications:A viscous contribution is includedCμ, C1ε and C2ε are multiplied by wall damping functions fμ, f1 , f2

which are functions of turbulence Reynolds number or similarwhich are functions of turbulence Reynolds number or similar functions

As an example we quote the Lam and Bremhorst (1981) wall-damping functions which are particularly successful:

2 20.51 exp( 0.0165Re ) 1 ;Rey

t

fμ⎛ ⎞

⎡ ⎤= − − +⎜ ⎟⎣ ⎦⎝ ⎠ (3 54)

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In function the parameter is defined by . Lam and Bremhorst use as a boundary condition.0yε∂ ∂ =

3

21 2

0.051 ; 1 exp( Re )tf ffμ

⎛ ⎞= + = − −⎜ ⎟⎜ ⎟

⎝ ⎠

(3.54)

1/ 2 /k y vfμ Re y

39

Assessment of performance

Table 3.5 k-ε model assessment

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k-ε Model Implementation DetailsThe flow equations (3.27) can be written in tensor notation as

( ) ( ) ( )ji ii j i j

j j j i j i

UU U PU U u ut x x x x x x

UU P

ρρ μ ρ

⎡ ⎤⎛ ⎞∂∂ ∂∂ ∂ ∂ ∂′ ′+ = + − −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞∂∂∂ ∂

(3.54a)

jii j

j j i i

UU Pu ux x x x

μ ρ⎡ ⎤⎛ ⎞∂∂∂ ∂′ ′= + − −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

Inserting the Boussinesq relation (3.38)

23

jiij i j t ij

j i

UUu u kx x

τ ρ μ ρ δ⎛ ⎞∂∂′ ′= − = + −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

( ) *UU U Pρ ⎡ ⎤⎛ ⎞∂∂ ∂∂ ∂ ∂

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( ) ( ) ( ) ji ii j t

j j j i i

UU U PU Ut x x x x xρ

ρ μ μ⎡ ⎤⎛ ⎞∂∂ ∂∂ ∂ ∂

+ = + + −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦where P* = P + (2/3)ρk

(3.54b)

Note that Eqn. (3.44b) is the same as that of Navier-Stokes Eqns. given by (2.32a,b,c) with μ replaced by (μ + μt) and p by P*.

40

Rearranging( ) ( )

*

*

( )

( ) ( )

ji ii j t

j j j i i

ji

UU U PU Ut x x x x x

UU P

ρρ μ μ

μ μ μ μ

⎡ ⎤⎛ ⎞∂∂ ∂∂ ∂ ∂+ = + + −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤ ∂⎡ ⎤∂∂ ∂ ∂= + + + −⎢ ⎥ ⎢ ⎥ ( ) ( )

(

t tj j j i i

tj

x x x x x

x

μ μ μ μ

μ μ

= + + + −⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂⎢ ⎥ ⎣ ⎦⎣ ⎦

∂= +∂

*

*

source term

) ( ) ( )

( ) ( )

j jit

j j i j i i

jit t

j j j i i

U UU Px x x x x x

UU Px x x x x

μ μ

μ μ μ

⎡ ⎤ ∂ ∂⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂+ + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤ ∂⎡ ⎤∂∂ ∂ ∂= + + −⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂⎢ ⎥ ⎣ ⎦⎣ ⎦

Note that we have used:(3.54c)

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Note that we have used:

( )( ) 0 0

(due to continuity if = constant, and the fluid is incompressible)

j j j

j i j i i j i

U U Ux x x x x x x

μ μ μ μ

μ

⎛ ⎞∂ ∂ ∂⎡ ⎤ ⎛ ⎞∂ ∂ ∂ ∂= = = =⎜ ⎟⎜ ⎟⎢ ⎥ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎝ ⎠ ⎝ ⎠

The terms after the first parenthesis on the RHS of Eqn. (3.44c) are treated as a source .

k-ε Model Equations in Generic Form

*

( ) ( ) ( )

EquationContinuity 1 0 0

x-Momentum ff ff ff ff

div div grad st

s

U V W PU

φ

φ

ρφ ρ φ φ

φ

μ μ μ μ

∂+ = Γ +

Γ

∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

U

The low Reynolds number k-ε model equations can be written in generic form as

x Momentum

y-Momentum

eff eff eff eff

eff eff eff

Ux x y x z x x

U VVx y y

μ μ μ μ

μ μ μ

+ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞∂ ∂ ∂ ∂

+⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

*

*

2

z-Momentum

Energy 0

Turbulence ke

eff

eff eff eff eff

t

t

tk

k

W Py z y y

U V W PWx z y z z z z

T

k P

μ

μ μ μ μ

μμσ σ

μμ ρ ρεσ

⎛ ⎞ ⎛ ⎞∂ ∂ ∂+ −⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

+

+ −

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2

1 1 2 2Dissipation of ke tk

ij ij

C f P C fk k

US S

ε εε

μ ε εε μ ρ ρσ

+ −

∂=

( )( )

2 2 22 2 2

3* 21

2

1 1 12 2 2

(2 / 3) , , / , 1 0.05 / , 2 /

1 exp 0.0165Reeff t t k t ij ij

y

V W U V W U V Wx y z y x x z z y

P P k C f k f f P S S

fμ μ μ

μ

ρ μ μ μ μ ρ ε μ ρ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠= + = + = = + =

⎡ ⎤= − −⎣ ⎦ ( ) 2 22

1/21 2

1 20.5 / Re , 1 exp( Re ), Re /Re / , 0.09, 1.00, 1.30, 0.9, 1.44, 1.92, Pr

t t t

y k t

f kk y C C Cμ ε ε ε

ρ εμρ μ σ σ σ σ

+ = − − == = = = = = = =

41

Source term linearization in k-ε modelsWhen the source term is linearized as

C P Ps s s φ= +

1) SP must be negative for a convergent iterative solution, ) P g gwhich ensures diagonal dominance of the coefficient matrix. This is a requirement for the boundedness criteria discussed in chapter 5.

2) SC must be positive (and SP must be negative ) to obtain all positive φ values. (Patankar, 1980)

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Both k and ε are strictly positive quantities. So, to obtain always positive results for k and ε, the source terms should be formulated such that SC is positive and SP is negative for k and ε equations.

Source term linearization in k equationk-equation:Consider first term, ρPk. Inserting into Pk

k ks Pρ ρε= −2 /t C f kμ μμ ρ ε=

22

32 2k t ij ij ij ijkP S S C f S S C kμ μρ μ ρε

= = =ε

Linearization of ρPk in the form of sk = sPφP + sC would give . However, since C3 > 0 and k > 0, then sP > 0, which violates the boundedness criteria. We can conclude that positive terms (e.g. Pk) cannot be linearized using such an implicit manner. A l h ld b i

3 and 0P Cs C k s= =

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As a result, Pk should be put into sC.Consider the second term, –ρε. From

. Inserting Pk and ε into the source termgives:

2 / tC f kμ με ρ μ=

2 /t C f kμ μμ ρ ε=

k ks Pρ ρε= −

42

Source term linearization in k-equationk-equation:

2 2 2

2

/

where and /k k k t

k t

s P P C f k b ak

b P a C fμ μ

μ μ

ρ ρε ρ ρ μ

ρ ρ μ

= − = − = −

= =

N h i b 0 i b li i d i i i l i iNote that, since b > 0, it cannot be linearized impicitly as it gives a positive sP. For the second term, ak2, the linearization proposed by Patankar (1980) can be used:

( ) ( ) ( )*

* * * 2 * * * 2 *( ) 2 ( ) 2P P P P P P P P Pdss s k k b a k ak k k b a k ak kdk

⎛ ⎞ ⎡ ⎤= + − = − − − = + −⎜ ⎟ ⎣ ⎦⎝ ⎠

2

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2* 2 * 2

2* *

( ) ( )

2 2

C P k Pt

P P Pt

C fs b a k P k

C fs ak k

μ μ

μ μ

ρρ

μ

ρμ

= + = +

= − = −

Then,

where the superscript * refers to the previous iteration values.

Positivity of sC term in k-equationThe criteria that SC must be positive (and SP must be negative ) to obtain all positive φ values (Patankar, 1980) should be implemented for the k-equation. Rewriting the sC term:

2* 2 * 2( ) ( )

C fb k P kμ μρ

We observe that all terms are positive except , which may become negative if ε becomes negative during iterations. To ensure sC > 0, μt should be enforced to take positive values during iterations. The limiter used for enforcing positive values for μt is given later in Eqn (3 51b)

2 2( ) ( )C P k Pt

fs b a k P kμ μρ

ρμ

= + = +

2 /t C f kμ μμ ρ ε=

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later in Eqn. (3.51b).

43

Source term linearization in ε-equationε-equation:

22

1 1 2 2

where /

ks C f P C f b ak k

b C f P a C f k

ε ε εε ερ ρ ε

ερ ρ

= − = −

= =1 1 2 2where , /kb C f P a C f kkε ερ ρ= =

Note that, since b > 0, it cannot be linearized impicitly as it gives a positive sP. For the second term, aε2, the linearization proposed by Patankar (1980) can be used:

( ) ( ) ( )*

* * * 2 * * * 2 *( ) 2 ( ) 2P P P P P P P P Pdss s b a a b a ad

ε ε ε ε ε ε ε ε εε

⎛ ⎞ ⎡ ⎤= + − = − − − = + −⎜ ⎟ ⎣ ⎦⎝ ⎠

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* * 2* 2

1 1 2 2* *

**

2 2 *

( )( )

2 2

P PC P k

PP P

s b a C f P C fk k

s a C fk

ε ε

ε

ε εε ρ ρ

εε ρ

= + = +

= − = −

Then,

where the superscript * refers to the previous iteration values.

Source term linearization in ε-equationThe source terms sP and sC of the ε-equation can be written in terms of eddy viscosity μt using

2 2C f k C f k C fμ μ μ μ μ μρ ρ ρ εε εμ = → = = , tt tk k

με μ μ

= → = =

which yields

2 2 *1 1 2 2

*

k PC

t t

C C f f P C C f fs με μ ε μ μρ ρ ε

μ μ= +

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*2

2 22 PP

t

ks C C f fε μ μρ μ= −

44

Positivity of sC term in ε-equationsC term should be positive to obtain positive values of ε during iterations. Rewriting the sC term for the ε-equation:

2 2 *1 1 2 2k P

C

C C f f P C C f fs με μ ε μ μρ ρ ε

μ μ= +

We observe that, even if we enforce μt to take positive values, if ε*

happens to be negative during iterations, then sC will become negative.To ensure that the variable φ does not become negative, the source terms should be rearranged as

t tμ μ

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where * denotes the previous iteration values.

* (if 0)

0

CP P C

P

C

ss s s

→ + <

Limiting of eddy viscosity μtNote that both k and ε are strictly positive quantities.However, during iterations, undershoots may develop which will result in negative values for either k or ε which will give negative μ values in accordance with Eqn (3 51):give negative μt values in accordance with Eqn. (3.51):.

(3.51)Also if ε → 0, division by zero will occur in accordance with Eqn. (3.51).Also, sP and sC terms of both k and ε-equations contain 1/μtt th t if 0 di i i b ill T

2 /t C f kμ μμ ρ ε=

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term so that if μt → 0, division by zero will occur. To overcome these difficulties μt is limited to a small fraction of laminar viscosity μl by using

(3.51b)2

4max ,10t lkC fμ μμ ρ με

−⎛ ⎞= ⎜ ⎟

⎝ ⎠

45

3.5.3 Reynolds stress equation modelsThe most complex classical turbulence model is Reynolds stress equation model (RSM),

Also called :second order- second order

- or second – moment closure modelDrawbacks of k – ε model emerge when it is attempted to

predict - flows with complex strain field

fl ith i ifi t b d f

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- flows with significant body forcesThe exact Reynolds stress transport eqn can account for the

directional effects of the Reynolds stress field.Let (Reynolds stress)ij

ij i jR u uτρ

′ ′= − =

The exact solution for the transport of takes the following formij

ij ij ij ij ij

DRP D

Dtε= + − +Π +Ω (3.55)

ijR

Equation (3.45) describes six partial differential equations: one for

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q ( ) p qthe transport of each of the six independent Reynolds stresses( )Two new terms appear compared with ke eqn (3.32)

1. . : pressure strain correction term2. . : rotation term

2 2 21 2 3 1 2 1 3 2 3 2 1 1 2 3 1 1 3 3 2 2 3, , , , , since ,u u u u u u u and u u u u u u u u u u and u u u u′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′= = =

ijΠijΩ

46

The convective term is

The production term is

( )( )k i j

ij i jk

U u uC div u u

ρ′ ′∂

′ ′= =∂

U (3.56)

p

The rotation term is

(3.57)

(3.58)

j iij im jm

m m

U UP R Rx x

∂⎛ ⎞∂= − +⎜ ⎟∂ ∂⎝ ⎠

2 ( )ij k j m ikm i m jkmu u e u u eω ′ ′ ′ ′Ω = − +

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where rotation vector1 if , and are different and in cyclic order

1 if , and are different and in anti-cyclic order

0 if any two indices are the same

k

ijk

ijk

ijk

e i j k

e i j k

e

ω =

= +

= −

=

The diffusion term Dij can be modelled by the assumption that the rate of transport of Reynolds stresses by diffusion is proportional to gradient of Reynolds stress.

( )ijt tij ij

m k m k

Rv vD div grad Rx xσ σ

∂⎛ ⎞ ⎛ ⎞∂= =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

(3.59)

The dissipations rate εij is modeled by assuming isotropy of the small dissipative eddies. It is set so that it effects the normal Reynolds stresses (i = j) only and in equal measure. This can be achieved by

2

with ; 0.09 1.0

m k m k

t kkv C C andμ μ σε

⎝ ⎠ ⎝ ⎠

= = =

2ε εδ= (3 60)

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where ε is the dissipation rate of turbulent kinetic energy defined by (3.43). The Kronecker delta, δij is given by δij = 1 if i = j and δij = 0 if i ≠ j

3ij ijε εδ= (3.60)

47

For Πij term:The pressure – strain interactions are the most difficult to modelTheir effect on the Reynolds stresses is caused by two distinct physical processes:

1. Pressure fluctuations due to two eddies interacting with each other

2. Pressure fluctuations due to the interactions of an eddy with a region of flow of different mean velocityIts effect is to make Reynolds stresses more isotropic and to reduce the Reynolds shear stressesM t i di t th t

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Measurements indicate that;The wall effect increases the isotropy of normal Reynolds stresses by damping out fluctuations in the directions normal to the wall and decreases the magnitude of the Reynolds shear stresses.

A comprehensive model that accounts for all these effects is given in Launder et al (1975). They also give the following simpler form favoured by some commercial available CFD codes:

2 2C R k C P Pε δ δ⎛ ⎞ ⎛ ⎞Π ⎜ ⎟ ⎜ ⎟ (3 61)

Turbulent kinetic energy k is

1 2

1 2

3 3with 1.8; 0.6

ij ij ij ij ijC R k C P Pk

C C

δ δ⎛ ⎞ ⎛ ⎞Π = − − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= =

(3.61)

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( ) ( )2 2 211 22 33 1 2 3

1 12 2

k R R R u u u′ ′ ′= + + = + +

48

The six equations for Reynolds stress transport are solved along with a model equation for the scalar dissipation rate ε. Again a more exact form is found in Launder et al (1975), but the equation from the standard k – εmodel is used in commercial CFD for the sake of simplicity

2vDε ε ε⎛ ⎞ (3.62)1 1 2

1 2

2

where 1.44 and 1.92

tt ij ij

vD div grad C f v S S CDt k k

C C

ε εε

ε ε

ε ε εεσ⎛ ⎞

= + ⋅ −⎜ ⎟⎝ ⎠= =

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Boundary conditions for the RSM modelThe usual boundary conditions for elliptic flows are required for the solution of the Reynolds stress transport equations:inlet: specified distributions of Rij and εoutlet and symmetry: ∂Rij /∂n = 0 and ∂ ε/∂n = 0y y ij

free stream: Rij = 0 and ε = 0 are givenor, ∂Rij /∂n = 0 and ∂ ε/∂n = 0

solid wall: use wall functions relating Rij to either k or (uτ)2

e.g.In the absence of any information, inlet distributions are calculated from

2 2 21 2 3 1 21.1 , 0.25 , 0.66 , 0.26u k u k u k u u k′ ′ ′ ′ ′= = = − =

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where ℓ is the characteristic length of the equipment (e.g. pipe diameter)

( )3 / 22 3 / 4

2 2 21 2 3

2 ; ; 0 .0 7 ;3

1; ; 0 ( )2

re f i

i j

kk U T C L

u k u u k u u i j

με= = =

′ ′ ′ ′ ′= = = = ≠

49

Boundary conditions for RSM modelFor computations at high Reynolds numbers wall-function-type boundary

conditions can be used which are very similar to those of the k-ε model .Near wall Reynolds stress values are computed from formulae such as

ij i j ijR u u c k′ ′= =where cij are obtained from measurements.

Low Reynolds number modifications to the model can be incorporated to add the effects of molecular viscosity to the diffusion terms and to account for anisotropy in the dissipation rate in the Rij-equations.

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Wall damping functions to adjust the constants of the ε-equation and a modified dissipation rate variable

give more realistic modeling near solid walls.

1/ 2 2( 2 ( / ) )v k yε ε≡ − ∂ ∂

Similar models exist for the 3 scalar transport terms of eqn (3.32) in the form of PDE’s.

In commercial CFD codes a turbulent diffusion coefficient /t t ϕμ σΓ =

iuϕ′ ′

is added to the laminar diffusion coefficient, where

for all scalars.

t t ϕμ

0.7ϕσ =

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50

Assessment of RSM model

Table 3.6 Reynolds stress equation model assessment.

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Asssesment of performance of RSM model

Figure 3.16 Comparison of predictions of RSM and standard k-ε model with measurements on a high lift Aerospatiale aerofoil: (a) pressure coefficient; (b) skin friction coefficient

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high-lift Aerospatiale aerofoil: (a) pressure coefficient; (b) skin friction coefficient Source: Leschziner, in Peyret and Krause (2000)

51

Incorrect usage of the terms: High and low Reynolds numberHigh or low Reynolds number turbulence models do not necessarily refer to models which are used to simulate flows having high or low speeds. In this usage of the term, the Reynolds number is based on the distance from the wall. In the near-wall region the flow velocity and distance to the wall is low so that the Reynolds number is low Thendistance to the wall is low so that the Reynolds number is low. Then, low Reynolds number models refer to turbulence models which can simulate the flow in near-wall region where the viscous effects dominate, by integrating the equations right to the wall, without resorting to wall functions. High Reynolds number models on the other hand refer to models which can simulate the flow in the fully turbulent region only where

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which can simulate the flow in the fully turbulent region only, where the Reynolds number is high. In this region the turbulence effects dominate over the viscous effects. The model equations are not valid in the near wall region. As a result, high Reynolds number model equations are not integrated right to the wall. These models use wall functions to find the variables in the near wall region.

Shortcomings of two-equation models such as k-ε model

Low Reynolds number flows: Very rapid changes occur in the distribution of k and ε as we reach the buffer layer between the fully turbulent region and the viscous sublayer. To force the rapid changes to k and ε, the constants of the high Reynolds number model are

l i li d b i l d i f i hi h i lmultiplied by exponential damping functions which require large number of grid points to resolve the changes. As a result, the system of equations become stiff which makes the numerical solution difficult to converge.Rapidly changing flows: The Reynolds stress is proportional to Sij in two-equation models. This only holds in equilibrium flows

h h f d i d di i i f k hl i

i ju uρ ′ ′−

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where the rates of production and dissipation of k are roughly in balance. In rapidly changing flows this is not true.

52

Shortcomings of two-equation models such as k-ε model

Stress anisotropy: Two equation model predicts normal stresses which are all approximately equal to –⅔ρk if a thin shear layer is simulated. Experimental data presented in section 3.4 showed that this is not correct. In spite of this the k-e model performs well in such fl b h di f l b l

2iuρ ′−

2′flows because the gradients of normal turbulent stresses are small compared with the gradient of the dominant turbulent shear stress . In more complex flows the gradients of normal turbulent stresses are not negligible and can drive significant flows. These effects can not be predicted by the two equation models.Strong adverse pressure gradients and recirculation regions: This

bl i l ib bl h i f h di d l

2iuρ ′−

u vρ ′ ′−

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problem is also attributable to the isotropy of the predicted normal stresses of the k-ε model. The k-ε model overpredicts the shear stress and suppresses separation in flows over curved walls. This is a significant problem in flows over airfoils, e.g. in aerospace applications.

Shortcomings of two-equation models such as k-ε model

Extra strains: Streamline curvature, rotation and extra body forces all give rise to additional interactions between the mean strain rate and the Reynolds stresses. These physical effects are not captured by standard two-equation models.RSM model addresses most of these problems adequately, but at the cost of additional storage and computer time.Below we consider some of the more recent advances in turbulence modeling that seek to address some or all of the above problems.

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53

Advanced turbulence modelsAdvanced treatment of the near-wall region: two-layer k-ε

modelThe numerical instability problems associated with the wall damping functions used in low Reynolds number k- ε models are avoided byfunctions used in low Reynolds number k ε models are avoided by subdividing the boundary layer into two regions (Jongen,1997):1) Fully turbulent region:

In this region the standard k-ε model is used where the eddy viscosity is defined by Eqn. (3.44); 2) Viscous region, Red < Red

*:

* *Re / Re , Re 50 200d d dy k ν= ≥ = −

2, /t t C kμμ ρ ε=

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Only the momentum equations and the k-equation is solved (Eqn.3.45). Turbulent viscosity in this viscous region is calculated from

and ε is calculated fromwhere

3/2 /k εε =

1/2,t v C kμ μμ =

Two-layer k-ε modelThe length scales ℓμ and ℓε contain necessary damping effects in the near wall region and are taken from the one–equation model of Wolfshtein (1969):

1 exp( Re /l dC y Aμ μ⎡ ⎤= − −⎣ ⎦

where

y is the normal distance to the wall.In order to avoid instabilities associated with differences between μt,t

d h j i b h f ll b l d i i

[ ]1 exp( Re /l dC y Aε ε

⎣ ⎦= − −

3/4 1/2, 2 , Re /l l dC C A C k dμ εκ ν−= = =

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and μt,v at the join between the fully turbulent and viscous regions, a blending formula is used to evaluate the eddy viscosity in

2 2 / 3ij i j t ij iju u S kτ ρ μ ρ δ′ ′= − = −

54

The turbulent viscosity is calculated from

where

, ,(1 )t t t t vε εμ λ μ λ μ= + −

*Re Re1 1 tanh d dλ⎡ ⎤⎛ ⎞−

= +⎢ ⎥⎜ ⎟where

The blending function λε is zero at the wall and tends to 1 in the fully turbulent region when Red >> Red

*.The constant A can be adjusted in order to control the sharpness of the transition from one model to the other. For example A =1,…,10 leads

i i i i hi f ll ll l i i

1 tanh2 Aελ +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

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to transitions occurring within a few cells, smaller values giving sharper transitions.

Geometry of the diffuser.

Comparison of two-layer k-ε model with various turbulence models in predicting Cp = (P2–P1) / (ρU1

2) in a 2-D diffuser, (Jongen, 1997).

y(Jongen, 1997)

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Profiles of ε at the exit section of the diffuser (θ = 10), for different values of the switching parameter Red

* between the two layers, (Jongen, 1997).

55

Strain sensitivity: RNG k-ε modelRNG k - ε model (Renormalization Group). The RNG procedure:systematically removes the small scales of motion from the governing equations by expressing their effects in terms of large scale motions and a modified viscosity. (Yakhot et al 1992):

k d l i f hi h ld b flRNG k-ε model equations for high Reynolds number flows:

( )( ) ( ) k eff ij ijk div k div grad k S

tρ ρ α μ τ ρε∂

+ = + ⋅ −∂

U

( )2

*1 1 2 2

( ) ( )

with 2 2 / 3

eff ij ij

ij i j t ij ij

div div grad C f S C ft k k

u u S k

ε ε ερε ε ερε α μ ε τ ρ

τ ρ μ ρ δ

∂+ = + ⋅ −

∂′ ′= − = −

U

(3.65)

(3.66)

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2

1

and ;

and 0.0845; 1.39; 1.42;

eff t t

k

kC

C C

μ

μ ε ε

μ μ μ μ ρε

α α

= + =

= = = = 2 1.68C ε =

(3.67)

( ) ( )1/20*1 1 03

1 /and ; 2 ; 4.377; 0.012

1 ij ijkC C S Sε ε

η η ηη η β

βη ε−

= − = ⋅ = =+

Only the constant β is adjustable; the above value is calculated from near wall turbulent data. All other constants are explicitly computed as part of the RNG process.The ε equation has long been suspected as one of the mainThe ε-equation has long been suspected as one of the main sources of accuracy limitations for the standard version of the k-ε model and the RSM.It is, therefore, interesting to note that the model contains a strain-dependent correction term in the constant C1ε of the production term in the RNG model ε-equation

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production term in the RNG model ε-equation.Its performance is better than the k-ε model for an expanding duct, but actually worse for a contraction with the same ratio.

56

Spalart-Allmaras modelHas only one transport equation for kinematic eddy viscosity parameter . Turbulent eddy viscosity is computed from

(3.68)where fv1 is the wall damping function given by

ν

1t vfμ ρν=fv1 p g g y

which tends to unity for high Reynolds number flows ( ).and fv1 → 0 at the wall.The Reynolds stresses are computed from

3

1 3 31

; vv

fc

χ νχχ ν

= =−

tν ν→

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(3.69)12 jiij i j t ij v

j i

UUu u S fx x

τ ρ μ ρν⎛ ⎞∂∂′ ′= − = = +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

The transport equation for is as follows:ν2

2 1 1( ) ( ) ( ) ( )

(I) (II) (III) (IV) (V) (VI)

b b w wk k

div div grad C C C ft x x yρν ν ν νρν μ ρν ν ρ ρν ρ

κ⎡ ⎤ ⎛ ⎞∂ ∂ ∂

+ = + + + Ω−⎢ ⎥ ⎜ ⎟∂ ∂ ∂ ⎝ ⎠⎣ ⎦U

(3.70)Transport Transport of Rate of Rate ofRate of change of by by turbulent production dissipationof convection diffusion of of ν νν ν ν

+ = + −

22 ( )

where 2 mean vorticity

v

ij ij

fyνκ

Ω = Ω+

Ω = Ω Ω =

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where 2 mean vorticity

1 mean vorticity tensor2

ij ij

jiij

j i

UUx x

Ω Ω Ω

⎛ ⎞∂∂Ω = − =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ 1/66

32 6 6

1 3

11 , wall damping functions1

wv w

v w

cf f gf g c

χχ

⎡ ⎤+= − = ⎢ ⎥+ +⎣ ⎦

57

In the k-ε model the length scale is ℓ = k3/2/ε.In a one-equation model ℓ cannot be computed since there is no transport equation for k and ε. However, ℓ should be specified to determine the dissipation rate. Inspection of Eqn. (3.70) reveals that ℓ= κy has been used as a length scale. This is also the mixing length used in developing the log-law for wall boundary layers.The constants are:

2 21 2 1 1

12 / 3, 0.4187, 0.1355, 0.622, bv b b w b

v

CC C C Cσ κ κσ+

= = = = = +

⎡ ⎤

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The model gives good performance for flows with adverse pressure gradients. In complex geometries it is difficult to determine the length scale, so the model is unsuitable for more general internal flows.

62 2 32 2( ), min ,10 , 0.3, 2w w wg r C r r r C C

yνκ

⎡ ⎤= + − = = =⎢ ⎥Ω⎣ ⎦

Wilcox k-ω modelIn the k-ε model the kinematic eddy viscosity is expressed as

However, ε is not the only possible length scale determining variable. In fact, k-ω model (Wilcox,1994) uses ω = ε / k as the second

3/2, velocity scale, / length scalet k kμ ϑ ϑ ε→ = =

, ( , )variable. Then, the length scale is ℓ = (k)1/2 / ε and

(3.71)The Reynolds stresses are computed from Boussinesq expression:

(3.72)

/t kμ ρ ω=

2 223 3

jiij i j t ij ij t ij

j i

UUu u S k kx x

τ ρ μ ρ δ μ ρ δ⎛ ⎞∂∂′ ′= − = − = + −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

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The transport equation for k and ω for turbulent flow at high Reynolds is:

*( ) ( ) ( )tk

k

k div k div grad k P kt

μρ ρ μ β ρ ωσ

⎡ ⎤⎛ ⎞∂+ = + + −⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦

U

58

whereand

(3 74)

223

ik t ij ij ij

j

UP S S kx

μ ρ δ∂= ⋅ −

( ) ( ) ( )tdiv div gradt

μρω ρω μ ωσ

⎡ ⎤⎛ ⎞∂+ = +⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦

U(3.74)

21 1

2 23

iij ij ij

j

t

US Sx

ωσ

γ ρ ρω δ β ρω

∂ ⎝ ⎠⎣ ⎦⎛ ⎞∂

+ ⋅ − −⎜ ⎟⎜ ⎟∂⎝ ⎠Transport Transport of Rate of Rate ofRate of change of or by or by turbulent production dissipationof or convection diffusion of or of or

kkk k k

ω ωω ω ω+ = + −

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The model constants are*

1 12.0, 2.0, 0.553, 0.075, 0.09k ωσ σ γ β β= = = = =

The k-ω model initially attracted attention because integration to the wall does not require wall-damping functions in low Reynolds number applications.The value of k is set to zero at the wall.ω tends to infinity at the wall, → use very large value or use

at the near wall grid point P.At inlet boundaries k and ω must be specified.At outlet boundaries zero gradient conditions are used.In a free stream k→0 and ω→0, then μt→∞ from Eqn. (3.71). This

bl i f i S l f

216 / ( )P Pyω ν β=

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causes problems in free stream regions. So a non-zero value of ωshould be specified. Unfortunately, results of the model tend to be dependent on the assumed free stream value of ω.

59

Menter SST k-ω modelMenter (1992) noted that the results of the k-ε model are much less sensitive to the (arbitrary) assumed values in the free stream, but its near-wall performance is unsatisfactory for boundary layers with adverse pressure gradients. He suggested a hybrid model using(i) a transformation of the k-ε model into k-ω model in the near-wall region(ii) the standard k-ε model in the fully turbulent region far from the wall.The calculation of τij and the k-equation are the same as in Wilcox’s original k-ω model, but the ε-equation is transformed into an ω-

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g , qequation by substituting ε = kω. This yields

22 2

,1 ,2

( ) 2( ) ( ) 2 23

t iij ij ij

j k k

U kdiv div grad S St x x xω ω

μρω ρ ωρω μ ω γ ρ ρω δ β ρωσ σ ω

⎡ ⎤ ⎛ ⎞⎛ ⎞ ∂∂ ∂ ∂+ = + + ⋅ − − +⎢ ⎥ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

U

(3.75),1(II)(I)

(III)

( ) ( ) ( )

2

tdiv div gradt

U k

ω

μρω ρω μ ωσ

ρ ω

⎡ ⎤⎛ ⎞∂+ = +⎢ ⎥⎜ ⎟⎜ ⎟∂ ⎢ ⎥⎝ ⎠⎣ ⎦

⎛ ⎞∂ ∂ ∂

U

Comparison with Eqn. (3.74) shows that (3.75) has an extra source term (VI): the cross-diffusion term, which arises during the ε = kωtransformation of the diffusion term in the ε-equation.

22 2

,2(V)(VI)(IV)

2 2 23

iij ij ij

j k k

U kS Sx x xω

ρ ωγ ρ ρω δ β ρωσ ω

⎛ ⎞∂ ∂ ∂+ ⋅ − − +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠

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60

Revised Menter SST k-ω model (2003)Model constants:

Blending functions:The differences between the μ values computed from the standard k-ε

*,1 ,2 2 21.0, 2.0, 1.17, 0.44, 0.083, 0.09k ω ωσ σ σ γ β β= = = = = =

The differences between the μt values computed from the standard k-εmodel in the far field and the k-ω model near the wall may cause instabilities. To prevent this, blending functions are introduced in the equation to modify the cross diffusion term. Blending functions are also used for model constants:

(3.76)where C = constants in original k ω model (inner constants)

1 1 1 2(1 )C F C F C= + −

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where C1 = constants in original k-ω model (inner constants)C2 = costants in Menter’s transformed k-ε model (outer contants)F1 = a blending function

e.g. FC = FC(ℓt / y, Rey) is a function of the ratio of turbulence ℓt = (k)1/2/ω and distance y to the wall and a turbulence Reynolds number Rey = y2ω/ν.The functional form of FC is chosen such that(i) it is zero at the wall(ii) tends to 1 in the far field(iii) produces a smooth transition around a distance half way between

the wall and the edge of the boundary layer.In this way, the model combines the good near-wall behaviour of the k ω model with the robustness of the k ε model in the far field

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k-ω model with the robustness of the k-ε model in the far field.

61

LimitersEddy viscosity models tend to overpredict k in regions where Sij is high. This problem is sometimes referred to as “stagnation point anomaly” [Durbin, 1996]. To avoid such overprediction turbulent viscosity μ should be limited Theoverprediction, turbulent viscosity μt should be limited. The limiter proposed by Medic and Durbin [2002] is

min ,6

t kC k C Sμ μ

μ αρ ε

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

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where α = 0.61/2( )ij ijS S S=

Limiter in SST k-ω modelLimiters:The eddy viscosity is SST k-ω model is limited to give improved performance in flows with adverse pressure gradients and wake regions:

(3.77a)

The k-production is limited to prevent the build-up of turbulence in stagnation regions:

1

1 2

1 2

max( , )

where 2 , a constant, a blending function

t

ij ij

a ka SF

S S S a F

ρμω

=

= = =

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(3.77b)* 2min 10 , 23

ik t ij ij ij

j

UP k S S kx

β ρ ω μ ρ δ⎛ ⎞∂

= ⋅ −⎜ ⎟⎜ ⎟∂⎝ ⎠

62

Menter SST k-ω model: summary

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Menter SST k-ω model: summary

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63

Menter SST k-ω model: summaryEach of the constants is a blend of an inner (1) and outer (2) constants, blended via:

where C1 → inner (1) constants, and C2 → outer (2) constants.1 1 1 2(1 )C F C F C= + −

1 ( ) , 2 ( )Note that it is generally recommended to use a production limiter where P in the k-equation is replaced with

The boundary conditions recommended are:( )*min ,10P P kβ ρω=

2

610( )wallνω

β=

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21 1( )

0

wall

wall

yk

β Δ=

Assessment of turbulence models for aerospace applications

External Aerodynamics:The Spalart-Allmaras, k-ω and SST k-ω models are all suitableThe SST k-ω model is most general; it gives superior performance for zero pressure gradient and adverse pressure gradient boundary layers.p g p g y yGeneral purpose CFD:The Spalart-Allmaras model is unsuitable, but the k-ω and SST k-ωmodels can both be applied.They both have a similar range of strengths and weaknesses as the k-εmodel and fail to include accounts of more subtle interactions between turbulent stresses and mean flow compared with RSM

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turbulent stresses and mean flow compared with RSM.

64

The k-ε-v2-f turbulence modelIn the k-ε-v2-f model of Durbin, (1991, 1993, 1995) two additional equations, apart from the k and ε-equations, are solved:(i) the normal stress, (ii) a relaxation function f for the production of k, (f = Pk /k)

22v′

( ) f p , (f k )In usual eddy-viscosity models wall effects are accounted for through wall functions.In the k-ε-v2-f model the problem of accounting for the wall damping of is simply resolved by solving its transport equation similar to RSM model:

22v′

2 2( ) ⎡ ⎤⎛ ⎞′ ′∂

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2 22 22 2

2 2( ) ( ) t

k

v vdiv v div grad v kft k

μρ ρ μ ρ ρ εσ

⎡ ⎤⎛ ⎞′ ′∂ ′ ′+ = + + −⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦U

An equation for the production of k is formulated in terms of a function f = Pk /k:

222

1 2

2 / 3 /( ( )) (1 ) k

v k PL div grad f f C CT k

⎡ ⎤′−⎣ ⎦− = − −

where

Note that:

1 2T k1/2

max ,6 time scalekT νε ε

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦ 1/23 3

2 22, max , , length scaleL

kL C Cην

ε ε

⎡ ⎤⎛ ⎞⎢ ⎥= = =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

2 2 2 2/ / /divgrad x y z= ∇ ⋅∇ = ∇ = ∂ ∂ + ∂ ∂ + ∂ ∂

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The eddy viscosity is given by:Constants:Boundary conditions at the wall:

22/t C v Tμν μ ρ ′= =

1 20.19, 1.4, 0.3, 0.3, 70.0LC C C C Cμ η= = = = =

2 22 2 2

2 4

200, 0, 2 / , , normal distance to the wall vk v k y f yyνε νε

′′= = = = − =

65

A variant of the k-ε-v2-f turbulence model: ς-f modelIn the k-ε-v2-f model of Durbin (1995), f is proportional to y4 near the wall. As a result, the boundary condition for f makes the equation system numerically unstable. An eddy-viscosity model based on k-ε-v2-f model of Durbin is proposed by Hanjalic et.al. (2004), which solves a transport equation for the velocity scales ratio instead of Thus, the resulting model is more robust and less sensitive to grid non-uniformities.Eddy viscosity is defined as

22 /v kζ ′= 2

2v′

t C kTζμν ζ=

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where can be interpreted as the ratio of the wall-normal velocity time scale Tvand the general (scalar) turbulence time scale T:

μ

22 /v kζ ′=

22 / and /vT v T kε ε′= =

A variant of the k-ε-v2-f turbulence model: ς-f modelThe complete set of the model equations are:

( )

( ) ( ) 0

( ) ( ) eff Mx

divtU div U div gradU st

ρ ρ

ρ ρ μ

∂+ =

∂∂

+ = +∂

U

U

( )

( )

( ) ( )

( ) ( )

( ) ( )

( )

eff My

eff Mz

tk

tV div V div gradV stW div W div gradW st

div div grad f Pt k

k

ζ

ρ ρ μ

ρ ρ μ

μρζ ζρζ μ ζ ρ ρσ

∂∂

+ = +∂

∂+ = +

∂⎡ ⎤⎛ ⎞∂

+ = + + −⎢ ⎥⎜ ⎟⎜ ⎟∂ ⎢ ⎥⎝ ⎠⎣ ⎦

⎛∂

U

U

U

⎡ ⎤⎞

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( ) ( ) t

k

k div k divt

μρ ρ μσ

⎛∂+ = +⎜∂ ⎝

U

1 2

21 2

( ) ( )

1 2( ( )) 13

k

t k

k

grad k P

C P Cdiv div gradt T

PL div grad f f C CT

ε ε

ε

ρ ρε

μ ρ ρ ερε ρε μ εσ

ζε

⎡ ⎤⎞+ −⎢ ⎥⎟

⎠⎣ ⎦⎡ ⎤⎛ ⎞ −∂

+ = + +⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦⎛ ⎞⎛ ⎞′− = − + −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

U

66

where1/2

1/43/2 1/2 3

max min , ,6

max min

t

T

C kT

k aT CC S

k kL C C

ζμ

ζμ

μ ρ ζ

νε εζ

ν

=

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= ⎜ ⎟ ⎜ ⎟⎜ ⎟

1 2 2

max min , ,6

0.6, =0.22, =1.4(1+0.012/ ), =1.9, =0.65, =1, =1.3,

1

L

k

L C CC S

a C C C C

ηζμ

ζμ ε ε ε

ζ

ε εζ

ζ σ σ

σ

⎢ ⎥= ⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦′=

=*

*

.2, 6.0, 0.36, 85T L

Mx eff eff eff

My eff eff eff

C C C

U V W Psx x y x z x x

U V W Psx y y y z y y

η

μ μ μ

μ μ μ

= = =

∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂

= + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

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Mz eff eff

x y y y z y y

U Vsx z y z

μ μ

∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

*

*

2 2 22 2 2

, (2 / 3) , 2 / , 2

1 1 12 2 2

eff

eff t k t ij ij ij ij

ij ij

W Pz z z

P P k P S S S S S

U V W U V W U V WS Sx y z y x x z z y

μ

μ μ μ ρ μ ρ

∂ ∂⎛ ⎞ −⎜ ⎟∂ ∂⎝ ⎠

= + = + = =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + + + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Boundary conditions at the wall:

Note that both the nominator and the denominator of f ( f at the wall)

22

20, 0, 2 / , , normal distance to the wall k k y f yyνζζ ε ν= = = = − =

Note that both the nominator and the denominator of fw ( f at the wall) are proportional to y2 instead of y4 as in the Durbin’s v2-f model.This brings improved stability to computational scheme.

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67

3.5.4 Algebraic stress equation models (ASM)- ASM is an economical way of accounting for the

anisotropy of Reynolds stresses.- Avoids solving the Reynolds stress transport eqns.

I t d l b i t d l R ld- Instead, uses algebraic eqns to model Reynolds stresses.

The Simplest method is:To neglect the convection and diffusion terms altogether

A ll li bl h d i

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A more generally applicable method is:To assume that the sum of the convection and diffusion

terms of the Reynolds stresses is proportional to the sum of the convection and diffusion terms of turbulent kinetic energy

[ ]

( )

transport of ( . . )i j i jij

i ji j ij

Du u u u DkD k i e div termsDt k Dt

u uu u E

′ ′ ′ ′ ⎛ ⎞− ≈ ⋅ −⎜ ⎟⎝ ⎠

′ ′′ ′= ⋅ − ⋅ −

(3.78)

Introducing approximation (3.78) into the Reynolds stress transport equation (3.55) with production term Pij (3.57), modeled dissipation rate term (3.60) and pressure – strain correction term (3.61) on the right hand side yields after some arrangement the following algebraic stress model:

2 23 1 / 3

Dij i j ij ij ij

C kR u u k P PC P

δ δε ε

⎛ ⎞⎛ ⎞′ ′= = + −⎜ ⎟⎜ ⎟− + ⎝ ⎠⎝ ⎠(3.79)

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- appear on both sides (on rhs within Pij ). For swirling flows, CD=0.55, C1= 2.2

- eqn(3.79) is a set of 6 simultaneous algebraic eqns. for 6 unknowns,- solved iteratively if k and ε are known.- The standard k-ε model eqns has to be solved also (3.44 - 3.47)

i ju u′ ′

13 1 / 3C P ε ε+ ⎝ ⎠⎝ ⎠

i ju u′ ′

68

Algebraic stress model assessment

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Non-linear k-ε modelsThe standard k-ε model uses the Boussinesq approximation (3.33)

(3.33)and eddy viscosity expression (3.44).

(3 44)

23

jiij i j t ij

j i

UUu u kx x

τ ρ μ ρ δ⎛ ⎞∂∂′ ′= − = + −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

2 /t C C kμμ ρϑ ρ ε= = (3.44)Hence (8.80)This relationship implies that the turbulence adjusts itself instantaneously as it is convected through the flow domain. However, the viscoelastic analogy holds that the adjustment does not take place immediately. → τij should also be a function DSij / Dt. So,

t μμ ρ ρ

( , , , )i j ij ij iju u S kρ τ τ ε ρ′ ′− = =

ijDS

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In RSM, τij is actually regarded as a transported quantity.Bringing in a dependence on Dsij /Dt can be regarded as a partial account of Reynolds stress transport, which recognises that the state of turbulence lags behind the rapid changes.

( , , , , )iji j ij ij iju u S k

Dtρ τ τ ε ρ′ ′− = =

69

Elaborating this idea, a non-linear k-ε model is proposed. This approach involves the derivation of asymptotic expansions for the Reynolds stresses which maintain terms that are quadratic in velocity gradients.The nonlinear k-ε model (Speziale, 1987):

2

32

2

2 23

1 1 43 3

where ( ) and 1 68

ij i j ij ij

o oD im mj mn mn ij ij mm ij

ij jo i

ku u k C S

kC C S S S S S S

S UUS grad S S S C

μ

μ

τ ρ ρ δ ρε

δ δε

′ ′= − = − +

⎛ ⎞− ⋅ − ⋅ + −⎜ ⎟⎝ ⎠

∂ ∂⎛ ⎞∂+ +⎜ ⎟U

(3.82)

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The nonlinear k - ε model accounts for anisotropy.Accounts for the secondary flow in non-circular duct flows.

where ( ) and 1.68j jiij ij mj mi D

m m

S grad S S S Ct x x

= + ⋅ − ⋅ + ⋅ =⎜ ⎟∂ ∂ ∂⎝ ⎠U

The simplest non-linear eddy viscosity modelThe simplest non-linear eddy viscosity model relates the Reynolds stresses to quadratic tensor products of Sij and Ωij:

(3.83)2 231

ij i j t ij iju u S k

k

τ ρ μ ρ δ′ ′= − = −

⎫⎛ ⎞

where

( )

1

2

3

13

quadratic terms

13

t ik jk kl kl ij

t ik jk jk ik

t ik jk kl kl ij

kC S S S S

kC S S

kC

μ δε

με

μ δε

⎫⎛ ⎞− ⋅ − ⋅⎜ ⎟ ⎪⎝ ⎠ ⎪⎪

− ⋅Ω − ⋅Ω ⎬⎪⎪⎛ ⎞− Ω ⋅Ω − Ω ⋅Ω⎜ ⎟ ⎪

⎝ ⎠ ⎭

1/ 2( / / ), 1 / 2( / / )ij i j j i ij i j j iS U x U x U x U x= ∂ ∂ + ∂ ∂ Ω = ∂ ∂ −∂ ∂

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Quadratic terms allow Reynolds stresses to be different.The model has the potential to capture anisotropy effects.C1, C2 and C3 are constants in addition to the 5 constants of the original k-ε model.

70

Cubic k-ε modelCraft et al. (1996) demonstrated that it is necessary to introduce cubic tensor products to obtain the correct sensitising effect for interactions between Reynolds stress production and streamline curvature.They also included:Variable Cμ with functional dependence on Sij and Ωij

Ad hoc modification of the ε-equation to reduce the overprediction of the length scale, leading to poor shear stress predictions in separated flows.Wall damping functions to enable integration of the k- and ε-equations to the wall through the viscous sub-layer

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to the wall through the viscous sub layer.The performance of cubic k-ε model is very close to that of RSM.

Large Eddy Simulation (LES)Up to now, developing a general-purpose RANS turbulence model suitable for a wide range of practical applications have not been possible.This is due to differences in the behaviour of large and small eddies.Small eddies are nearly isotropic, but large eddies are anisotropic and their behaviour is dictated by the geometry, boundary conditions and body forces.Problem dependence of the largest eddies complicates the search for widely applicable models using RANS.An alternative approach to the problem is large eddy simulation

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An alternative approach to the problem is large eddy simulation (LES) which computes the larger eddies with a time dependent simulation, but tries to model only the smaller eddies.The universal behaviour of the smaller eddies is easier to capture with a compact model.

71

Instead of time averaging, LES uses a spatial filtering operation to separate the larger and smaller eddies.To resolve all those eddies with a length scale greater than a certain width, the method selects a filtering function and a certain cutoff width.Then, the filtering operation is performed on the time-dependent flow equations.During spatial filtering, information relating to the smaller, filtered-out turbulent eddies is destroyed.This and interaction effects between the larger resolved eddies and

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This, and interaction effects between the larger, resolved eddies and the smaller unresolved ones, gives rise to sub-grid-scale stresses or SGS stresses.Their effect on the resolved flow must be described by means of an SGS model.

In LES, - the time-dependent, space filtered flow equations- together with the SGS model of the unresolved stresses

are solved on a grid of control volumesare solved on a grid of control volumesThe solution gives:- the mean flow and - all turbulent eddies at scales larger than the cutoff width.

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72

Spatial filtering of unsteady Navier-Stokes equationsFilters are familiar separation devices in electronics that are designed to split an input into a desirable, retained part and an undesirable, rejected part.Filtering Functions:In LES spatial filtering is done by using a filter function G(x, x′,Δ):

(3.84)1 2 3( , ) ( , , ) ( , )

where ( , ) filtered function ( , ) original (unfiltered) function

t G t dx dx dx

tt

φ φ

φφ

∞ ∞ ∞

−∞ −∞ −∞

′ ′ ′ ′ ′= Δ

==

∫ ∫ ∫x x x x

xx

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In this section overbar indicates spatial filtering, not time averaging.Only in LES, the integration is not carried out in time but in three-dimensional space.

filter cutoff widthΔ =

Common filtering functions in LESTop hat filter:

(3.85a)

31/ if / 2( , , )

0 if / 2G

⎧ ′Δ − ≤ Δ⎪′ Δ = ⎨′− > Δ⎪⎩

x xx x

x x

Gaussian filter:

(3.85b)Spectral cutoff:

(3.85c)

23/2

2 2( , , ) exp , 6G γ γ γπ

⎛ ⎞′−⎛ ⎞′ ⎜ ⎟Δ = − =⎜ ⎟ ⎜ ⎟Δ Δ⎝ ⎠ ⎝ ⎠

x xx x

3 sin[( ) /( ) i ix xG′− Δ′ Δ ∏x x

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( )1

[( )( , , )( )

i i

i i i

Gx x=

′ Δ =′−∏x x

73

The top-hat filter is used in finite volume implementations of LES.Gaussian and spectral cutoff filters are preferred in research.Δ is intended as an indicative measure of the size of eddies that are retained in the computations and the eddies that are rejected.p jIn principle we can choose Δ to be any size, but in finite volume method it is pointless to choose Δ < grid size, since only a single nodal value of the variable is retained over the control volume, so all finer detail is lost anyway.The most common selection is

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(3.86)where Δx, Δy and Δz are the length of a cubic cell in x, y and z-directions.

1/33 ( )cellx y z VΔ = Δ Δ Δ =

Filtering in 1-D

I LES fil ( l ) h i I 1 D

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In LES we filter (volume average) the equations. In 1-D we get

0.5

0.5

1( , ) ( , )x x

x xx t t d

xφ φ ξ ξ

+ Δ

− Δ=Δ ∫

74

Filtered unsteady Navier-Stokes equationsThe unsteady Navier-Stokes equations for a fluid with constant viscosity μ are

(2.4)( ) 0div

tρ ρ∂+ =

∂u

(2.37a)

(2.37b)(2.37c)

( )

( )

( )

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

u

v

w

u pdiv u div grad u St xv pdiv v div grad v S

t yw pdiv w div grad w St z

ρ ρ μ

ρ ρ μ

ρ ρ μ

∂ ∂+ = − + +

∂ ∂∂ ∂

+ = − + +∂ ∂

∂ ∂+ = − + +

∂ ∂

u

u

u

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If the flow is also incompressible → div(u) = 0, → Su, Sv, Sw = 0

Filtering of the equations (2.4) and (2.37) yields:(3.87)

(3 88a)( )

( ) 0

( ) ( ( ))

divt

u pdiv u div grad u

ρ ρ

ρ ρ μ

∂+ =

∂∂ ∂

+ = +

u

u (3.88a)

(3.88b)

(3.88c)The overbar indicates a filtered flow variable.

( )

( )

( )

( ( ))

( ) ( ( ))

( ) ( ( ))

div u div grad ut xv pdiv v div grad vt yw pdiv w div grad wt z

ρ μ

ρ ρ μ

ρ ρ μ

+ = − +∂ ∂

∂ ∂+ = − +

∂ ∂∂ ∂

+ = − +∂ ∂

u

u

u

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The problem is, how to compute

The second term on the rhs is modeled.

( )div ρφu( ) ( ) [ ( ) ( )]div div div divρφ φ ρφ φ= + −u u u u

75

Substitution of the modeled into (3.88a-c) yields the LES momentum equations:

(3.89a)

( )div ρφu

( )( ) ( ( )) ( ( ) ( ))

( )

u pdiv u div grad u div u div ut xρ ρ μ ρ ρ∂ ∂

+ = − + − −∂ ∂

∂ ∂

u u u( )

(3.89b)(3.89c)

The filtered momentum eqns. are similar to RANS momentum eqns.

( )

( )

( ) ( ( )) ( ( ) ( ))

( ) ( ( )) ( ( ) ( ))

(I) (II) (III)

v pdiv v div grad v div v div vt yw pdiv w div grad w div w div wt z

ρ ρ μ ρ ρ

ρ ρ μ ρ ρ

∂ ∂+ = − + − −

∂ ∂∂ ∂

+ = − + − −∂ ∂

u u u

u u u

(IV) (V)

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(3.26a-c) or (3.27a-c).The last terms (V) are caused by the filtering operation similar to the Reynolds stresses in RANS equations that arose as a result of time averaging.They can be considered as a divergence of a set of stresses τij

The i-component of these terms can be written as follows:

(3.90a)

( ) ( ) ( )( ) i i i i i ii i

ij

u u u u u v u v u w u wdiv u ux y z

ρ ρ ρ ρ ρ ρρ ρ

τ

∂ − ∂ − ∂ −− = + +

∂ ∂ ∂∂

u u

( )

(3.90b)τij are termed as sub-grid-scale stresses.Remembering that a flow variable φ(x, t) can be decomposed as the sum of

where

j

j

ij i i i j i j

x

u u u u u uτ ρ ρ ρ ρ

=∂

= − = −u u

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(i) the filtered function with spatial variations that are larger than the cutoff width and are resolved by the LES computations and(ii) φ′(x, t), which contains unresolved spatial variations at length scales smaller than the filter cutoff width,we can write:

( , )tφ x

76

(3.91)Using this decomposition in eqn. (3.90b) we can write the first term on the far rhs as

( , ) ( , ) ( , )t t tφ φ φ′= +x x x

( )( )u u u u u u u u u u u u u uρ ρ ρ ρ ρ ρ′ ′ ′ ′ ′ ′= + + = + + +

Then, we can write SGS stresses as(3.92)

( )( )

( )i j i i j j i j i j i j i j

i j i j i j i j i j i j

u u u u u u u u u u u u u u

u u u u u u u u u u u u

ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ ρ

= + + = + + +

′ ′ ′ ′= + − + + +

LES ReynoldsLeonard stresses cross-stresses

( )ij ij

ij i j i j i j i j i j i j i j

L C

u u u u u u u u u u u u u uτ ρ ρ ρ ρ ρ ρ ρ′ ′ ′ ′= − = − + + +

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Leonard stresses, Lij, are due to effects at resolved scale. They are caused due to the fact that for space filtered variables, unlike in time averaging, where

S ey o dsLeonard stresses, cross stresses, stresses, ij ij

ijL C

R

φ φ≠( ) ( )t tϕ ϕ= Φ = Φ =

The cross stresses Cij are due to interactions between the SGS eddies and the resolved flow.LES Reynolds stresses Rij are caused by convective momentum transfer due to interactions of SGS eddies and are modeled with a so-called SGS turbulence model.The SGS stresses (3.92) must be modeled.

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77

Smagorinski-Lilly SGS modelSmagorinski (1963) suggested that, since the smallest turbulent eddies are almost isotropic, we expect that the Boussinesq hypothesis (3.33) might provide a good description of the effects of the unresolved eddies on the resolved flow.Thus in Smagorinsky’s model the local SGS stresses Rij are taken to be proportional to the local rate of strain of the resolved flow:

(3.93)

⅓R δ performs the same function as the term ⅔ρkδ in eqn (3 33)

1/ 2( / / ) :ij ij i j j iR S u x u x= ∂ ∂ + ∂ ∂∼

1 123 3

jiij SGS ij kk ij SGS kk ij

j i

uuR S R Rx x

μ δ μ δ⎛ ⎞∂∂

= − + = − + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

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⅓Rkkδij performs the same function as the term ⅔ρkδij in eqn (3.33).It ensures that the sum of the modeled normal SGS stresses is equal to the kinetic energy of the of the SGS eddies.Above model is used together with approximate forms of Leonard stresses Lij and cross stresses Cij.

In spite of the different nature of Lij and Cij, they are lumped together with Rij in the current versions of the finite volume method.Then, the whole stress τij is modeled as a single entity by means of a single SGS turbulence model:

⎛ ⎞ (3.94)

Smagorinsky-Lilly model is based on Prandtl’s mixing length model:

Length scale: ℓ = Δ is used since Δ fixes the size of SGS eddies.V l i l

1 123 3

jiij SGS ij kk ij SGS kk ij

j i

uuSx x

τ μ τ δ μ τ δ⎛ ⎞∂∂

= − + = − + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

SGS Cν ϑ=

h 2S S S Sϑ Δ

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Velocity scale:Substituting

(3.95)where

where 2 ij ijS S S Sϑ = Δ× =

and into and defining SGS SGS SGSϑ ν μ ρν=2 2( ) ( ) 2SGS SGS SGS ij ijC S C S Sμ ρ ρ= Δ = Δ

12

jiij

j i

uuSx x

⎛ ⎞∂∂= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

78

Near the wall, μSGS becomes quite large because velocity gradients are high there. However, since the SGS turbulent fluctuations near a wall go to zero, so must μSGS . To ensure this, μSGS is multiplied with a damping function fμ:

A more convenient way to dampen μSGS near the wall is simply to use the RANS length scale as an upper limit, i.e.

1 exp( / 26)f yμ+= − −

1/3⎡ ⎤

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n = distance to the nearest wallDisadvantage of the Smagorinsky model: the “constant” CSGS is not constant, but it is flow dependent. It is found to vary in the range from CSGS = 0.065 (Moin, 1982) to CSGS = 0.25 (Jones, 1995).

1/3min ( ) ,cellV nκ⎡ ⎤Δ = ⎣ ⎦

The difference in CSGS values is attributable to the effect of the mean flow strain or shear.This indicates that the behavior of the small eddies is not as universal as was surmised at first.→ Successful LES turbulence modeling might require

(i) a case by case adjustment of CSGS, or (ii) a more sophisticated approach.

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79

Higher order SGS modelsBoussinesq eddy viscosity hypothesis for Reynolds stresses given by Eqn. (3.93) assumes that changes in the resolved flow take place sufficiently slowly that the SGS eddies can adjust themselves instantaneously to the rate of strain of the resolved flow field.Instead of using a case by case adjustment of CSGS for different applications, another approach is to use the idea of RANS modeling to account for the transport effects. In this approach we defineLength scale: ℓ = ΔVelocity scale: where kSGS = SGS turbulent kinetic energyThen (3 96)

SGSkϑ =

C kμ ρ ′= Δ

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Then (3.96)where To account for the effects of convection, diffusion, production and destruction on the SGS velocity scale we solve a transport equation to determine the distribution of kSGS:

SGS SGS SGSC kμ ρ= ΔconstantSGSC′ =

(3.96)

Dimensional analysis shows that the rate of dissipation εSGS of SGS

( ) ( ) ( ) 2kSGS

SGS SGSSGS SGS SGS ij ij SGS

kP

k div k div grad k S St

ρ μρ μ ρεσ

⎛ ⎞∂+ = + ⋅ −⎜ ⎟∂ ⎝ ⎠

u

Dimensional analysis shows that the rate of dissipation εSGS of SGS turbulent kinetic energy is related to length and velocity scales as

(3.97)

This model is implemented in commercial CFD code STAR-CD.

3/2

constant

SGSSGS

kC

C

ε

ε

ε =Δ

=

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80

Advanced SGS modelsThe Smagorinsky model is purely dissipative: the direction of energy flow is from eddies at the resolved scales towards the sub-grid scales.Quarini (1979) have shown that there is also energy flow in reverse direction.Furthermore, modeled SGS stresses using the Smagorinsky-Lilly model do not correlate strongly with actual SGS stresses computed by accurate DNS. (Clark et al. (1979), McMillan and Ferziger, (1979)).These authors suggested that the SGS stresses should not be taken as proportional to the strain rate of the whole resolved flow field, but should be estimated from the strain rate of the smallest resolved

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eddies.Bardina et al. (1980) proposed a method to compute local values of CSGS based on the application of two filtering operations, taking the SGS stresses to be proportional to the stresses due to eddies at the smallest resolved scale. They proposed:

The Bardina model

where C′ is an adjustable constant.The term in the brackets can be evaluated from twice-filtered resolved

( )ij i j i jC u u u uτ ρ ′= −

The term in the brackets can be evaluated from twice-filtered resolved flow field information. The cutoff width of the test filter may be equal to that of first filter or it may be coarser.Bardina model has improved performance. But, the appearance of negative viscosities generated stability problems. They proposed adding a damping term with the form of the Smagorinsky model (3 94) (3 95) to stabilise calculations which

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Smagorinsky model (3.94)-(3.95) to stabilise calculations, which yields a mixed model:

(3.99)

C′ depends on the cutoff width used for the second filtering, but C′ ≈ 1

( ) 2 22ij i j i j SGS ijC u u u u C S Sτ ρ ρ′= − − Δ

81

Twice filteringLet’s filter velocity component once more at a node I using a cutoff width Δx. For simplicity we do it in 1D.

u

( )/ /1 1 1 Δ Δ⎛ ⎞

Box filter for a control volume. (Lars Davidson, Lecture notes Chalmers Univ.)

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( )/2 0 /2

/2 /2 0

1 1 1( ) ( ) ( )2 2

x x

I A Bx x

x xu u d u d u d u ux x x

ξ ξ ξ ξ ξ ξΔ Δ

−Δ −Δ

Δ Δ⎛ ⎞= = + = +⎜ ⎟Δ Δ Δ ⎝ ⎠∫ ∫ ∫Estimating at locations A and B by linear interpolation givesu

( )1 1 1 11 1 3 3 1 1 62 4 4 4 4 8I I I I I I I I Iu u u u u u u u u− + − +

⎡ ⎤⎛ ⎞ ⎛ ⎞= + + + = + + ≠⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Dynamic SGS modelGermano (1986) proposed a different decomposition of the turbulent stresses for 2 different filtering operations with cutoff widths Δ1and Δ2

(3.100)Second filter is coarser and is called the test filter with Δ2 = 2Δ1

(2) (1)ij ij ij i j i jL u u u uτ τ ρ ρ− = ≡ −

Eqn. (3.100) is evaluated from the resolved flow data.The SGS stresses are modeled using Smagorinsky’s model (3.94) –(3.95) assuming same CSGS for both filtering operations. This yields:

(3.101a)

h (3 101b)

213ij kk ij SGS ijL L C Mδ− =

2 22 2M S S S SΔ + Δ

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where (3.101b)Lilly suggested a least squares approach to evaluate local values of CSGS :

(3.102)Note that the product of two tensors is a scalar.

2 22 12 2ij ij ijM S S S S= − Δ + Δ

2 ij ijSGS

ij ij

L LC

M M=

82

The test filter

Control volume for grid and test

The test-filtered variables are computed by integration over the test filter. The test filter is twice the size of grid filter, i.e. Δ2 = 2Δ1.For example in 1D, is computed asu

( )1 1E P Eu udx udx udx= = +∫ ∫ ∫

Control volume for grid and test filter

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( )( )

( )

2 21 1

2 2 2 21 24

W W P

W P P Ew e

W P E

x xu u u uu x u x

x

u u u

Δ Δ+ +⎛ ⎞= Δ + Δ = +⎜ ⎟Δ ⎝ ⎠

= + +

∫ ∫ ∫

The test filter in 3D

A 2D test filter control volume

In 3D, filtering at the test level is carried out in the same way by integrating over the test cell assuming linear variation of the variables (Zang, et. al., 1993)

, , 1/2, 1/2, 1/2 1/2, 1/2, 1/21 (8I J K I J K I J Ku u u− − − + − −= +

A 2D test filter control volume. (Lars Davidson, Lecture notes at Chalmers Univ Tech., Sweden)

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1/2, 1/2, 1/2 1/2, 1/2, 1/2

1/2, 1/2, 1/2 1/2, 1/2, 1/2

1/2, 1/2, 1/2 1/2, 1/2,

8

I J K I J K

I J K I J K

I J K I J K

u u

u u

u u

− + − + + −

− − + + − +

− + + + + +

+ +

+ +

+ + 1/2 )

83

Initial and boundary conditions for LESInitial conditionsFor steady flows it is adequate to specify an initial field that conserves mass with superimposed Gaussian random fluctuations with the correct turbulence level or spectral content.Solid wallsNo-slip b.c. is used if equations are integrated to the wall. This requires fine grids with near-wall grid points y+ ≤ 1For high Re flows with thin boundary layers non-uniform grids clustered near the walls is necessary.Alternatively wall functions can be used

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Alternatively, wall functions can be used.

Implementation of Wall functions in LESThe wall function models used in LES are:1) The universal near wall modelIf the first near-wall node P is in the laminar sublayer use

1/2u y u yu ρ ρ

If node P is in the fully turbulent region use

If node P is within the buffer region the two above layers are blended

( )1/2 for 5 where friction velocity, wu y u yu y u y

uτ τ

ττ

ρ ρτ ρμ μ

+ += < = =

1 ln , 30 500, 9.793u yu E y Eu

τ

τ

ρκ μ

+⎛ ⎞= < < =⎜ ⎟

⎝ ⎠

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g yusing a function suggested by Kader (1981)

where

1/ , lam turbuu e u e u uuτ

+ Γ + Γ + += + =4( ) , 0.01, 5

1a y a b

by

+

+Γ = − = =+

84

Similarly, the general equation for the derivative du+/dy+ is

1/lam turbdu dudu e edy dy dy

+ ++Γ Γ

+ + += +

This approach allows the fully turbulent law to be easily modified and extended to take into account other effects such as pressure gradients or variable properties.This formula guarantees reasonable representation of velocity profiles in the cases where y+ falls inside the wall buffer region (3 < y+ < 10 ). F i i l i b l l d f 1/2 1/4k C

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Friction velocity can be calculated fromHowever, in most of the LES models k equation is not solved and kP is not available. Since uτ appears on both sides of the non-linear logarithmic law equation, then an iterative method should be used to solve for uτ.

1/2 1/4Pu k Cτ μ=

2) The Werner-Wengle model2) The Werner-Wengle model (Werner,1993)To avoid the iterative solution procedure required for determining uτ,Werner and Wengle proposed analytical integration of power-law near-wall velocity distribution resulting in the following expressions for the wall shear stress τw:

2/(1 )

21 11

2/(1 )1

2 for

2

1 1A + for 2 2

P BP

w B B BBBB

P P

uu A

z z

B B u u Az A z z

μ μρ

τμ μ μρρ ρ ρ

+ ++−−

⎧≤⎪ Δ Δ⎪⎪= ⎨

⎡ ⎤⎪ ⎛ ⎞ ⎛ ⎞− +≤⎢ ⎥⎜ ⎟ ⎜ ⎟⎪ Δ Δ Δ⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎣ ⎦⎩

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uP = velocity parallel to wall, A = 8.3, B = 1/7, Δz = near-wall control volume length scale (height if CV is rectangular).Then uτ is calculated from

⎪ ⎣ ⎦⎩

( )1/2wuτ τ ρ=

85

3) Spalding’s law of the wall methodThe universal velocity profile proposed by Spalding is a fit of the laminar, buffer and logarithmic regions of the equilibrium boundary layer into a single equation:

( ) ( )2 31 1 11uκ ++ + + + +⎡ ⎤

where κ = 0.42 and E = 9.1.If y+ = yPρuτ/μ and u+ = uP/uτ are inserted into the above equation, an equation of only one unknown uτ is obtained. This non-linear equation can be solved using the Newton-Raphson method (Villiers, 2006):

( ) ( )12 6

uy u e u u uE

κ κ κ κ+ + + + +⎡ ⎤= + − − − −⎢ ⎥⎣ ⎦

1 /nu u f f− ′

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where (n–1) → previous iteration value/u u f fτ τ= −

( ) ( )2 31 1 112 6

uf u y e u u uE

κ κ κ κ++ + + + +⎡ ⎤= − + − − − −⎢ ⎥⎣ ⎦

( ) ( )2 31 1 12

uf u y u uf e u uu u u E u u u u

κ

τ τ τ τ τ τ τ

κ κ κ κ+

+ + + ++ +⎡ ⎤∂′ = = − − + − + + +⎢ ⎥∂ ⎣ ⎦

The Newton-Raphson method converges rapidly to a tight tolerance when applied in the above procedure.By using initial values from the previous timestep, only 1 or 2 iterations are sufficient.The predicted wall shear is then calculated from

2w uττ ρ=

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86

Implementation of LESFor the most general case incompressible fluids where µ is not constant, LES momentum Eqns. (3.89) can be written as

(3.105)( ) ( ) ( ) ji i

i j ijj j j i j i

uu u pu ut x x x x x xρ

ρ μ τ⎡ ⎤⎛ ⎞∂∂ ∂∂ ∂ ∂ ∂

+ = + − −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦ ( )

The SGS turbulence model for τij is

(3.94)Then, the second term on the RHS of Eqn. (3.105) becomes

where

j j j j

ij i j i ju u u uτ ρ ρ⎝ ⎠⎣ ⎦

= −

1 123 3

jiij SGS ij ii ij SGS kk ij

j i

uuSx x

τ μ τ δ μ τ δ⎛ ⎞∂∂

= − + = − + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

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(3.106)( ) 13

jiij SGS kk

j j j i i

uux x x x x

τ μ τ⎡ ⎤⎛ ⎞∂∂∂ ∂ ∂ ⎛ ⎞− = + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

Substituting (3.106) into (3.105) gives

(3.106)( ) ( )

*

( ) ji ii j SGS

j i j j i

uu upu ut x x x x xρ

ρ μ μ⎡ ⎤⎛ ⎞∂∂ ∂∂ ∂ ∂

+ = − + + +⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

where

Note that Eqn. (3.106) is the same as that of the Navier-Stokes Eqns. given by (2.32a,b,c) with

μ replaced by (μ + μSGS) and

* 1 , ( 1,3)3 kkp p kτ= + =

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p replaced by *p

87

Smagorinsky-Lilly SGS Model Equations in Generic Form

( ) ( ) ( )

EquationContinuity 1 0 0

div div grad st

s

φ

φ

ρφ ρ φ φ

φ

∂+ = Γ +

Γ

U

The Smagorinsky-Lilly SGS model equations can be written in generic form as

*

Continuity 1 0 0

-Momentum

-Momentum

eff eff eff eff

eff eff eff

u v w px ux x y x z x x

u vy vx y y

μ μ μ μ

μ μ μ

∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞∂ ∂ ∂ ∂

+⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

*

*

-Momentum

eff

eff eff eff eff

w py z y y

u v w pz wx z y z z z z

μ

μ μ μ μ

⎛ ⎞ ⎛ ⎞∂ ∂ ∂+ −⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

I. Sezai – Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 173

2 2 1, ( ) ( ) 2 , , 2

jieff SGS SGS SGS SGS ij ij ij

j i

ij ij

uuC S C S S Sx x

uS Sx

μ μ μ μ ρ ρ⎛ ⎞∂∂

= + = Δ = Δ = +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠∂

=∂

2 2 22 2 2

* 1/3

1 1 1 ,2 2 2

1 ( 1,3), 0.065 0.25, ( )3 kk SGS cell

v w u v w u v wy z y x x z z y

p p k C Vτ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠= + = = − Δ =

References (in addition to that given in textbook)G. Medic and P. A. Durbin (2002) . Towards improved prediction of heat transfer on turbine blades, Journal of turbomachinery, 124:187-192.P. A. Durbin(1996). On the k-ε stagnation point anomaly. International journal of heat and fluid flow, 17(1):89–90.T. Jongen and Y. P. Marx (1997). Design of an unconditionally stable, positive scheme for the k-e and two-layer turbulence models, Computers and Fluids, vol. 26, No 5, pp. 469-487.Y. Zang, R.L., Street and J.R. Hoseff, 1993 . A dynamic subgrid-scale eddy viscosity model, Physics of fluids A, 5:3186-3196. B. Kader, Temperature and Concentration Profiles in Fully Turbulent Boundary Layers. Int. J. Heat Mass Transfer, 24(9):1541-1544, 1981H. Werner and H. Wengle. Large-Eddy Simulation of Turbulent Flow Over and Around a Cube in a Plate Channel In Eighth Symposium on Turbulent Shear

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Around a Cube in a Plate Channel. In Eighth Symposium on Turbulent Shear Flows, Munich, Germany, 1991.D.B. Spalding (1961). A single formula for the law of the wall. Journal of Applied Mechanics, Trans ASME, Series E, Vol. 28, pp. 455-458.E. Villiers (2006). The Potential of Large Eddy Simulation for the Modeling of Wall Bounded Flows. PhD thesis, Imperial College of Science, Technology and Medicine, Department of Mechanical Engineering.

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P. A. Durbin (1991). Near-wall turbulence closure modeling without damping functions, Theoretical and Computational Fluid Dynamics, vol. 3, No. 1, pp. 1-13.P. A. Durbin (1993). Application of a near-wall turbulence model to boundary layers and heat transfer, Int. J. Heat and Fluid Flow, vol. 14, no. 4, pp. 316-323.P A Durbin (1995) Separated flow computations with the k-ε-v2 model AIAAP. A. Durbin (1995). Separated flow computations with the k ε v model, AIAA Journal, vol. 33, no. 4, pp.659-664.K. Hanjalic , M. Popovac and M. Hadziabdic (2004). A robust near-wall elliptic-relaxation eddy-viscosity turbulence model for CFD, Int. J. Heat and Fluid Flow, vol. 25, pp. 1047-1051.P. Moin and J. Kim (1982). Numerical investigation of turbulent channel flow. Journal of Fluid Mechanics, vol. 118, pp.341–377.W P Jones and M Wille (1995) Large eddy simulation of a jet in a cross-flow In

I. Sezai – Eastern Mediterranean UniversityME555 : Computational Fluid Dynamics 175

W.P. Jones and M. Wille (1995). Large eddy simulation of a jet in a cross-flow. In10th Symp. on Turbulent Shear Flows, pages 4:1 – 4:6, The Pennsylvania StateUniversity.