Publication 97/2

An Introduction to TurbulenceModels

Lars Davidson, http://www.tfd.chalmers.se/lada

Department of Thermo and Fluid DynamicsCHALMERS UNIVERSITY OF TECHNOLOGYGoteborg, Sweden, November 2003

Nomenclature 3

1 Turbulence 51.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Turbulent Scales . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Vorticity/Velocity Gradient Interaction . . . . . . . . . . . . . 1.4 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . .

2 Turbulence Models 122.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Boussinesq Assumption . . . . . . . . . . . . . . . . . . . . . 2.3 Algebraic Models . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Equations for Kinetic Energy . . . . . . . . . . . . . . . . . .

2.4.1 The Exact Equation . . . . . . . . . . . . . . . . . . . 2.4.2 The Equation for !#"%$'&($)&+* . . . . . . . . . . . . . . . ,2.4.3 The Equation for !#".-$/&.-$)&+* . . . . . . . . . . . . . . .

2.5 The Modelled Equation . . . . . . . . . . . . . . . . . . . . !02.6 One Equation Models . . . . . . . . . . . . . . . . . . . . . . . 1

3 Two-Equation Turbulence Models 223.1 The Modelled 2 Equation . . . . . . . . . . . . . . . . . . . . . 33.2 Wall Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3 The 452 Model . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4 The 476 Model . . . . . . . . . . . . . . . . . . . . . . . . . 33.5 The 458 Model . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Low-Re Number Turbulence Models 284.1 Low-Re :452 Models . . . . . . . . . . . . . . . . . . . . . . . 34.2 The Launder-Sharma Low-Re 452 Models . . . . . . . . . . ;34.3 Boundary Condition for 2 and 4.5 The low-Re 476 Model . . . . . . . . . . . . . . . . . . . . . ;3

4.5.1 The low-Re 476 Model of Peng et al. . . . . . . . . . ;34.5.2 The low-Re 476 Model of Bredberg et al. . . . . . . . ;3

5 Reynolds Stress Models 385.1 Reynolds Stress Models . . . . . . . . . . . . . . . . . . . . . ;35.2 Reynolds Stress Models vs. Eddy Viscosity Models . . . . . . >905.3 Curvature Effects . . . . . . . . . . . . . . . . . . . . . . . . . >.5.4 Acceleration and Retardation . . . . . . . . . . . . . . . . . . >3>

!#"%$'&)(+*,-/.1032)4

568795;:= constants in the Reynolds stress model5@?

6

795@?

: constants in the Reynolds stress model5@A 6 795BA : constants in the modelled 2 equation5BCD68795BCE: constants in the modelled 6 equation5@F constant in turbulence modelG

energy (see Eq. 1.8); constant in wall functions (see Eq. 3.4)H

damping function in pressure strain tensor turbulent kinetic energy ( I

6

: J

&

J

&

)$ instantaneous (or laminar) velocity in K -direction$ & instantaneous (or laminar) velocity in K & -direction-

$ time-averaged velocity in K -direction-$)& time-averaged velocity in K& -directionJL

7

JM shear stressesJ fluctuating velocity in K -directionJ

:

normal stress in the K -directionJ

& fluctuating velocity in K & -directionJ

&

JON Reynolds stress tensorP

instantaneous (or laminar) velocity in Q -direction-

P

time-averaged velocity in Q -directionL fluctuating (or laminar) velocity in Q -directionL

:

normal stress in the Q -directionLEM shear stressR

instantaneous (or laminar) velocity in S -direction-

R

time-averaged velocity in S -directionM fluctuating velocity in S -directionM

:

normal stress in the S -direction

TVUXWDWZY

*,-/.102[4

\

boundary layer thickness; half channel height2 dissipation] wave number; von Karman constant ( ^ 0O_ >. )` dynamic viscosity`ba dynamic turbulent viscosityc kinematic viscosityca kinematic turbulent viscosityd

& instantaneous (or laminar) vorticity component in K & -direction

;

-

d

& time-averaged vorticity component in K & -direction6 specific dissipation ( 2?! )6/& fluctuating vorticity component in K & -direction turbulent Prandtl number for variable 8 laminar shear stress8

a a total shear stress8

a turbulent shear stress

*. 4 U &$

centerlineM wall

>

($ U 0 $X& 0(

Almost all fluid flow which we encounter in daily life is turbulent. Typicalexamples are flow around (as well as in) cars, aeroplanes and buildings.The boundary layers and the wakes around and after bluff bodies such ascars, aeroplanes and buildings are turbulent. Also the flow and combustionin engines, both in piston engines and gas turbines and combustors, arehighly turbulent. Air movements in rooms are also turbulent, at least alongthe walls where wall-jets are formed. Hence, when we compute fluid flowit will most likely be turbulent.

In turbulent flow we usually divide the variables in one time-averagedpart -$ , which is independent of time (when the mean flow is steady), andone fluctuating part J so that $ ^ -$ J .

There is no definition on turbulent flow, but it has a number of charac-teristic features (see Tennekes & Lumley [41]) such as:

%

D

. Turbulent flow is irregular, random and chaotic. Theflow consists of a spectrum of different scales (eddy sizes) where largesteddies are of the order of the flow geometry (i.e. boundary layer thickness,jet width, etc). At the other end of the spectra we have the smallest ed-dies which are by viscous forces (stresses) dissipated into internal energy.Even though turbulence is chaotic it is deterministic and is described bythe Navier-Stokes equations.

. In turbulent flow the diffusivity increases. This meansthat the spreading rate of boundary layers, jets, etc. increases as the flowbecomes turbulent. The turbulence increases the exchange of momentumin e.g. boundary layers and reduces or delays thereby separation at bluffbodies such as cylinders, airfoils and cars. The increased diffusivity alsoincreases the resistance (wall friction) in internal flows such as in channelsand pipes.

!

D#" !3%$& ('

. Turbulent flow occurs at high Reynoldsnumber. For example, the transition to turbulent flow in pipes occurs that)#*+-,

3;!030 , and in boundary layers at)#*./,

030303030 .

102

43!5

O6%3Z

. Turbulent flow is always three-dimensional.However, when the equations are time averaged we can treat the flow astwo-dimensional.

024

66%7 %

. Turbulent flow is dissipative, which means that ki-netic energy in the small (dissipative) eddies are transformed into internalenergy. The small eddies receive the kinetic energy from slightly larger ed-dies. The slightly larger eddies receive their energy from even larger eddiesand so on. The largest eddies extract their energy from the mean flow. Thisprocess of transferred energy from the largest turbulent scales (eddies) tothe smallest is called cascade process.

3'

O% Z

0 %b

. Even though we have small turbulent scales in theflow they are much larger than the molecular scale and we can treat theflow as a continuum.

U . 2 W ($V*O"2 W 4

As mentioned above there are a wide range of scales in turbulent flow. Thelarger scales are of the order of the flow geometry, for example the bound-ary layer thickness, with length scale and velocity scale . These scalesextract kinetic energy from the mean flow which has a time scale compara-ble to the large scales, i.e.

-$

Q

^ "

6

*^ " *

The kinetic energy of the large scales is lost to slightly smaller scales withwhich the large scales interact. Through the cascade process the kinetic en-ergy is in this way transferred from the largest scale to smaller scales. Atthe smallest scales the frictional forces (viscous stresses) become too largeand the kinetic energy is transformed (dissipated) into internal energy. Thedissipation is denoted by 2 which is energy per unit time and unit mass( 2 ^

:

). The dissipation is proportional to the kinematic viscosity c

times the fluctuating velocity gradient up to the power of two (see Sec-tion 2.4.1). The friction forces exist of course at all scales, but they arelarger the smaller the eddies. Thus it is not quite true that eddies whichreceive their kinetic energy from slightly larger scales give away all of thatthe slightly smaller scales but a small fraction is dissipated. However it isassumed that most of the energy (say 90 %) that goes into the large scalesis finally dissipated at the smallest (dissipative) scales.

The smallest scales where dissipation occurs are called the Kolmogo-rov scales: the velocity scale , the length scale and the time scale 8 . Weassume that these scales are determined by viscosity c and dissipation 2 .Since the kinetic energy is destroyed by viscous forces it is natural to as-sume that viscosity plays a part in determining these scales; the larger vis-cosity, the larger scales. The amount of energy that is to be dissipated is2 . The more energy that is to be transformed from kinetic energy to inter-nal energy, the larger the velocity gradients must be. Having assumed thatthe dissipative scales are determined by viscosity and dissipation, we canexpress , and 8 in c and 2 using dimensional analysis. We write

^

c

2

^

:

:

"( *

where below each variable its dimensions are given. The dimensions of theleft-hand and the right-hand side must be the same. We get two equations,

0 Z! 0 O b $&O% % DX

one for meters

^ #

7

"( 9*

and one for seconds

4 ^ 4 45;

7

"( ;9*

which gives ^ ^ > . In the same way we obtain the expressions for and 8 so that

^ "

c

2?*

6

7

^

c

2

6

7

8 ^

c

2

6 :

"( > *

0 U $X& & $8,

W 2[0 & $8, TVU " & W (3$

($ WDU " $X& 03(

The interaction between vorticity and velocity gradients is an essential in-gredient to create and maintain turbulence. Disturbances are amplified the actual process depending on type of flow and these disturbances,which still are laminar and organized and well defined in terms of phys-ical orientation and frequency are turned into chaotic, three-dimensional,random fluctuations, i.e. turbulent flow by interaction between the vor-ticity vector and the velocity gradients. Two idealized phenomena in thisinteraction process can be identified: vortex stretching and vortex tilting.

In order to gain some insight in vortex shedding we will study an ide-alized, inviscid (viscosity equals to zero) case. The equation for instanta-neous vorticity (

d

&b^

-

d

& 6/& ) reads [41, 31, 44]

$

N

d

&

N

^

d

N

$)&

N

c

d

&

N N

d

&

^

&

N

$

N

"( 9*

where & N is the Levi-Civita tensor (it is : if ! , " are in cyclic order, 4 if! , " are in anti-cyclic order, and 0 if any two of ! , " are equal), and where" _ *

N denotes derivation with respect to K N . We see that this equation is notan ordinary convection-diffusion equation but is has an additional term onthe right-hand side which represents amplification and rotation/tilting ofthe vorticity lines. If we write it term-by-term it reads

d

6

$

6

6

d

:

$

6

:

d

$

6

"( 9*

d

6

$

:

6

d

:

$

:

:

d

$

:

d

6

$

6

d

:

$

:

d

$

The diagonal terms in this matrix represent vortex stretching. Imagine a0

Z $#

> 3&

slender, cylindrical fluid element which vorticity % . We introduce a cylin-drical coordinate system with the K 6 -axis as the cylinder axis and K : as the

0 Z! 0 O b $&O% % DX

d

6

d

6

radial coordinate (see Fig. 1.1) so that % ^ "d

6>7

0

7

0 * . A positive $ 6 6 willstretch the cylinder. From the continuity equation

$

6

6

"

$

:

*

:

^ 0

we find that the radial derivative of the radial velocity $ : must be negative,i.e. the radius of the cylinder will decrease. We have neglected the viscositysince viscous diffusion at high Reynolds number is much smaller than theturbulent one and since viscous dissipation occurs at small scales (see p. 6).Thus there are no viscous stresses acting on the cylindrical fluid elementsurface which means that the rotation momentum

:

d

"( 3*

remains constant as the radius of the fluid element decreases (note that alsothe circulation

d

:

is constant). Equation 1.7 shows that the vortic-ity increases as the radius decreases. We see that a stretching/compressingwill decrease/increase the radius of a slender fluid element and increase/decreaseits vorticity component aligned with the element. This process will affectthe vorticity components in the other two coordinate directions.

The off-diagonal terms in Eq. 1.6 represent vortex tilting. Again, take a0

Z $#

slender fluid element with its axis aligned with the K : axis, Fig. 1.2. The ve-locity gradient $ 6 : will tilt the fluid element so that it rotates in clock-wisedirection. The second term

d

:

$

6

: in line one in Eq. 1.6 gives a contributiontod

6 .Vortex stretching and vortex tilting thus qualitatively explains how in-

teraction between vorticity and velocity gradient create vorticity in all threecoordinate directions from a disturbance which initially was well definedin one coordinate direction. Once this process has started it continues, be-cause vorticity generated by vortex stretching and vortex tilting interactswith the velocity field and creates further vorticity and so on. The vortic-ity and velocity field becomes chaotic and random: turbulence has beencreated. The turbulence is also maintained by these processes.

From the discussion above we can now understand why turbulence al-ways must be three-dimensional (Item IV on p. 5). If the instantaneousflow is two-dimensional we find that all interaction terms between vortic-ity and velocity gradients in Eq. 1.6 vanish. For example if $

I 0 and allderivatives with respect to K

are zero. If $

I 0 the third line in Eq. 1.6vanishes, and if $ &

I0 the third column in Eq. 1.6 disappears. Finally, the

,

3 7'

3 7'

3 7'

($ U 0 $X& 0(

When the flow is turbulent it is preferable to decompose the instantaneousvariables (for example velocity components and pressure) into a mean valueand a fluctuating value, i.e.

$/&3^ -$)&

J

&

^ -

3_

"+ *

One reason why we decompose the variables is that when we measure flowquantities we are usually interested in the mean values rather that the timehistories. Another reason is that when we want to solve the Navier-Stokesequation numerically it would require a very fine grid to resolve all tur-bulent scales and it would also require a fine resolution in time (turbulentflow is always unsteady).

The continuity equation and the Navier-Stokes equation read

"

$

&

*

&

^ 0 "+ 9*

$/&

"

$/&%$

N

*

N

^ 4

&

`

$/&

N

$

N

&4

;

\

&

N

$

N

"+ ;9*

where " _ * N denotes derivation with respect to K N . Since we are dealing withincompressible flow (i.e low Mach number) the dilatation term on the right-hand side of Eq. 2.3 is neglected so that

$/&

"

$/&%$

N

*

N

^ 4

&

`

"%$/&

N

$

N

& *

N

_ "+ > *

Note that we here use the term incompressible in the sense that density isindependent of pressure (

^ 0 ) , but it does not mean that density isconstant; it can be dependent on for example temperature or concentration.

Inserting Eq. 2.1 into the continuity equation (2.2) and the Navier-Stokesequation (2.4) we obtain the time averaged continuity equation and Navier-Stokes equation

"

-$)& * & ^ 0 "+ 9*

-$/&

-

$/&

-

$

N

N

^ 4

-

&

`

"

-

$/&

N

-

$

N

& *'4

J

&

JN

N

_ "+ 9*

A new term J & JN appears on the right-hand side of Eq. 2.6 which iscalled the Reynolds stress tensor. The tensor is symmetric (for example J 6 J : ^J

:

J

6 ). It represents correlations between fluctuating velocities. It is an ad-ditional stress term due to turbulence (fluctuating velocities) and it is un-known. We need a model for J & J N to close the equation system in Eq. 2.6.

% $ '%

Q

8

8

8

a

8

a a

Q

,

0

This is called the closure problem: the number of unknowns (ten: three veloc-

7 '

ity components, pressure, six stresses) is larger than the number of equa-tions (four: the continuity equation and three components of the Navier-Stokes equations).

For steady, two-dimensional boundary-layer type of flow (i.e. bound-ary layers along a flat plate, channel flow, pipe flow, jet and wake flow, etc.)where

-

P

-$

7

K

Q

"+ 3*

Eq. 2.6 reads

-$ -$

K

-

P

-$

Q

^ 4

-

K

Q

`

-$

Q

4

JZL

_ "+ ,9*

K ^ K

6 denotes streamwise coordinate, and Q ^ K : coordinate normal tothe flow. Often the pressure gradient

-

K is zero.To the viscous shear stress `

-$

Q on the right-hand side of Eq. 2.863ED

> 6

appears an additional turbulent one, a turbulent shear stress. The total shearstress is thus

8

a a

^

`

-

$

Q

4

JL

In the wall region (the viscous sublayer, the buffert layer and the logarith-mic layer) the total shear stress is approximately constant and equal to thewall shear stress 8 , see Fig. 2.1. Note that the total shear stress is constantonly close to the wall; further away from the wall it decreases (in fully de-veloped channel flow it decreases linearly by the distance form the wall).At the wall the turbulent shear stress vanishes as J ^ L ^0 , and the viscousshear stress attains its wall-stress value 8V^

J

:

. As we go away from thewall the viscous stress decreases and turbulent one increases and at Q

,

0

they are approximately equal. In the logarithmic layer the viscous stress isnegligible compared to the turbulent stress.

In boundary-layer type of flow the turbulent shear stress and the ve-locity gradient

-$

Q have nearly always opposite sign (for a wall jet this

;

% $ '%

K

Q

L

0

L

0

Q

6

Q

:

$:"Q.*

4

JZL

is not the case close to the wall). To get a physical picture of this let usstudy the flow in a boundary layer, see Fig. 2.2. A fluid particle is movingdownwards (particle drawn with solid line) from Q : to Q 6 with (the turbu-lent fluctuating) velocity L . At its new location the $ velocity is in averagesmaller than at its old, i.e. -$:"Q 6 * -$:"Q : * . This means that when the par-ticle at Q : (which has streamwise velocity $:"Q : * ) comes down to Q 6 (wherethe streamwise velocity is $:"Q 6 * ) is has an excess of streamwise velocitycompared to its new environment at Q 6 . Thus the streamwise fluctuation ispositive, i.e. J 0 and the correlation between J and L is negative ( JZL 0 ).

If we look at the other particle (dashed line in Fig. 2.2) we reach thesame conclusion. The particle is moving upwards ( L 0 ), and it is bringinga deficit in $ so that J 0 . Thus, again, JL 0 . If we study this flow fora long time and average over time we get JZL 0 . If we change the signof the velocity gradient so that

-$

Q

0 we will find that the sign of JZLalso changes.

Above we have used physical reasoning to show the the signs of JLand

-

$

Q are opposite. This can also be found by looking at the produc-tion term in the transport equation of the Reynolds stresses (see Section 5).In cases where the shear stress and the velocity gradient have the samesign (for example, in a wall jet) this means that there are other terms in thetransport equation which are more important than the production term.

There are different levels of approximations involved when closing theequation system in Eq. 2.6.

' $O%

An algebraic equation is used to compute a tur-bulent viscosity, often called eddy viscosity. The Reynolds stress ten-sor is then computed using an assumption which relates the Reynoldsstress tensor to the velocity gradients and the turbulent viscosity. Thisassumption is called the Boussinesq assumption. Models which arebased on a turbulent (eddy) viscosity are called eddy viscosity mod-

>

6 7

els.

3!5 % $ O

In these models a transport equation is solvedfor a turbulent quantity (usually the turbulent kinetic energy) anda second turbulent quantity (usually a turbulent length scale) is ob-tained from an algebraic expression. The turbulent viscosity is calcu-lated from Boussinesq assumption.

5 $O%

These models fall into the class of eddy vis-cosity models. Two transport equations are derived which describetransport of two scalars, for example the turbulent kinetic energy and its dissipation 2 . The Reynolds stress tensor is then computedusing an assumption which relates the Reynolds stress tensor to thevelocity gradients and an eddy viscosity. The latter is obtained fromthe two transported scalars.

10(

" 3%$ > 6 $ O

Here a transport equation is derived forthe Reynolds tensor J & JN . One transport equation has to be added fordetermining the length scale of the turbulence. Usually an equationfor the dissipation 2 is used.

Above the different types of turbulence models have been listed in in-creasing order of complexity, ability to model the turbulence, and cost interms of computational work (CPU time).

0 4

%Z ' $ %

is equal to zero. In order to make Eq. 2.9 valid upon contraction we add9!;

O\

&

N

to the left-hand side of Eq. 2.9 so that

J

&

JON

^ 4

`ba

" -$)&

N

-$

N

&+*

;

\

&

N

%_ "+ 0 *

Note that contraction of\

&

N gives\

& & ^

\

6 6

\

: :

\

^ ^ ;

V2 W . U "b& 0 W 2[4

In eddy viscosity models we want an expression for the turbulent viscosity`ba

^

ca . The dimension of c'a is

:

(same as c ). A turbulent velocity $ $

%8X

$ O

scale multiplied with a turbulent length scale gives the correct dimension,i.e.

c

a

"+ 3 *

Above we have used and which are characteristic for the large turbulentscales. This is reasonable, because it is these scales which are responsiblefor most of the transport by turbulent diffusion.

In an algebraic turbulence model the velocity gradient is used as a ve-locity scale and some physical length is used as the length scale. In bound-ary layer-type of flow (see Eq. 2.7) we obtain

ca

^

:

&

.

$

Q

"+ 9*

where Q is the coordinate normal to the wall, and where &.

is the mixinglength, and the model is called the mixing length model. It is an old modeland is hardly used any more. One problem with the model is that &

.

isunknown and must be determined.

More modern algebraic models are the Baldwin-Lomax model [2] andthe Cebeci-Smith [6] model which are frequently used in aerodynamicswhen computing the flow around airfoils, aeroplanes, etc. For a presen-tation and discussion of algebraic turbulence models the interested readeris referred to Wilcox [46].

"%$X& 0( 4 0

U

&[(

W

$'&

(

WDU

,

33 # D'

The equation for turbulent kinetic energy ^6

:

J

&

J

& is derived from theNavier-Stokes equation which reads assuming steady, incompressible, con-stant viscosity (cf. Eq. 2.4)

"

$)&%$

N

*

N

^ 4

&

`

$)&

N N

_ "+ ;9*

3D [2E 3

The time averaged Navier-Stokes equation reads (cf. Eq. 2.6)

"

-$)&1-$

N

*

N

^ 4 -

&

`

-$)&

N N

4

"

J

&

JN

*

N

"+ > *

Subtract Eq. 2.14 from Eq. 2.13, multiply by J & and time average and weobtain

$ & $

N

4

-$ & -$

N

N

J

& ^

4

4 -

&

J

&

`

$)& 4 -$)&

N N

J

&

"

J

&

JN

*

N8J

&

"+ 9*

The left-hand side can be rewritten as

" -$ &

J

& * " -$

N

J N

*'4 -$ & -$

N

N

J

& ^

-$ &

J N

J

& -$

N

J

&

J N

N

J

& _ "+ 9*

Using the continuity equation "

J N

*

N

^ 0 , the first term is rewritten as

-$ &

J N

N

J

& ^

J

&

J N

-$ &

N

_ "+ 3*

We obtain the second term (using "

-$

N

*

N

^ 0 ) from

-

$

N

N^

-

$

N

J

&

J

&

N

^

-

$

N J

&

J

&

N

J

&

J

&

N

^

J

&

-

$

N;J

&

N

"+ ,9*

The third term in Eq. 2.16 can be written as (using the same technique as inEq. 2.18)

"

J

N

J

&

J

&

*

N

_ "+ 9*

The first term on the right-hand side of Eq. 2.15 has the form

4 &

J

&3^ 4"

J

&+* & "+ !0 *

The second term on the right-hand side of Eq. 2.15 reads

`

J

&

N N;J

& ^

`

"

J

&

N>J

& *

N

4

J

&

N>J

&

N

"+ 1 *

For the first term we use the same trick as in Eq. 2.18 so that

`

"

J

&

N

J

&

*

N

^

`

"

J

&

J

&

*

N N

^

`

N N

"+ 39*

The last term on the right-hand side of Eq. 2.15 is zero. Now we canassemble the transport equation for the turbulent kinetic energy. Equa-tions 2.17, 2.18, 2.20, 2.21,2.22 give

"

-$

N

.*

N

^ 4

J

&

JON

-$/&

N

4

JON

JN>J

&

J

& 4

`

N

N

4

`

J

&

N;J

&

N

"+ 3;9*

The terms in Eq. 2.23 have the following meaning.

3D [2E 3

%EX

.

$ '

. The large turbulent scales extract energy from the meanflow. This term (including the minus sign) is almost always positive.

The two first terms represent3'

% $ %

by pressure-velocityfluctuations, and velocity fluctuations, respectively. The last term isviscous diffusion.

10( % 6%7D

. This term is responsible for transformation of kineticenergy at small scales to internal energy. The term (including theminus sign) is always negative.

In boundary-layer flow the exact equation read

-$

K

-

P

Q

^ 4

JZL

-$

Q

4

Q

L

LJ

&

J

& 4

`

Q

4

`

J

&

N;J

&

N

"+ ?> *

Note that the dissipation includes all derivatives. This is because the dis-sipation term is at its largest for small, isotropic scales where the usualboundary-layer approximation that

J

&

K

J

&

Q is not valid.

33 3D [Z

!#"%$/&($)&+*

The equation for the instantaneous kinetic energy ^6

:

$/&%$/& is derivedfrom the Navier-Stokes equation. We assume steady, incompressible flowwith constant viscosity, see Eq. 2.13. Multiply Eq. 2.13 by $ & so that

$/& "

$)&%$

N

*

N

^ 4$/&

&

`

$)& $)&

N)N

_ "+ 39*

The term on the left-hand side can be rewritten as

"

$

&

$

&

$

N

*

N

4

$

&

$

N

$

&

N

^

$

N

"%$

&

$

&

*

N

4

$

N

"%$

&

$

&

*

N

^

$

N

"%$/&%$/& *

N

^ "

$

N

7*

N

"+ 39*

where ^ !#"%$)&($)&+* .The first term on the right-hand side of Eq. 2.25 can be written as

4$

&

&

^ 4 "%$

&

*

&

_ "+ 93*

The viscous term in Eq. 2.25 is rewritten in the same way as the viscousterm in Section 2.4.1, see Eqs. 2.21 and 2.22, i.e.

`

$/&%$/&

N N

^

`

N N

4

`

$)&

N

$/&

N

_ "+ 3,9*

Now we can assemble the transport equation for by inserting Eqs. 2.26,2.27 and 2.28 into Eq. 2.25

"

$

N

5*

N

^

`

N)N

4 "%$

&

*

&

4

`

$

&

N

$

&

N

_ "+ 39*

,

3D [2E 3

We recognize the usual transport terms due to convection and viscous dif-fusion. The second term on the right-hand side is responsible for transportof by pressure-velocity interaction. The last term is the dissipation termwhich transforms kinetic energy into internal energy. It is interesting tocompare this term to the dissipation term in Eq. 2.23. Insert the Reynoldsdecomposition so that

`

$/&

N

$)&

N

^

`

-$/&

N

-$)&

N

`

J

&

N;J

&

N

_ "+ ;!0 *

As the scales of -$ is much larger than those of J & , i.e. J & N $/& N we get

`

$/&

N

$)&

N

,

`

J

&

N;J

&

N

_ "+ ;1 *

This shows that the dissipation taking place in the scales larger than thesmallest ones is negligible (see further discussion at the end of Sub-section 2.4.3).

33 3D [Z

!#"1-$

&

-$

&

*

The equation for !#".-$)-$/& * is derived in the same way as that for !#"%$ &($)& * .Assume steady, incompressible flow with constant viscosity and multiplythe time-averaged Navier-Stokes equations (Eq. 2.14) so that

-

$/& "

-

$)&

-

$

N

*

N

^ 4

-

$/&

-

&

`

-

$)&

-

$)&

N)N

4

-

$)&

"

J

&

JN

*

N

_ "+ ;39*

The term on the left-hand side and the two first terms on the right-handside are treated in the same way as in Section 2.4.2, and we can write

"

-

$

N

-

5*

N

^

`

-

N)N

4 "

-

$/&

-

* &4

`

-

$/&

N

-

$)&

N

4

-

$/&

"

J

&

JON

*

N

7

"+ ;3;9*

where - ^ !#" -$/& -$)&+* . The last term is rewritten as

4

-

$/&

"

J

&

JN

*

N

^ 4"

-

$/&

J

&

JON

*

N

"

J

&

JON

*

-

$/&

N

_ "+ ;?> *

Inserted in Eq. 2.33 gives

"

-

$

N

-

5*

N

^

`

-

N)N

4 "

-

$/&

-

* &4

`

-

$/&

N

-

$)&

N

4 "1-$)&

J

&

JN

*

N

J

&

JN

-$)&

N

_

"+ ;39*

On the left-hand side we have the usual convective term. On the right-handside we find: transport by viscous diffusion, transport by pressure-velocityinteraction, viscous dissipation, transport by velocity-stress interaction andloss of energy to the fluctuating velocity field, i.e. to . Note that the lastterm in Eq. 2.35 is the same as the last term in Eq. 2.23 but with oppositesign: here we clearly can see that the main source term in the equation(the production term) appears as a sink term in the - equation.

It is interesting to compare the source terms in the equation for (2.23), (Eq. 2.29) and - (Eq. 2.35). In the equation, the dissipation term is

4

`

$

&

N

$

&

N

^ 4

`

-$

&

N

-$

&

N

4

`

J

&

N

J

&

N

_ "+ ;39*

33 $O $

%

In the - equation the dissipation term and the negative production term(representing loss of kinetic energy to the field) read

4

`

-$/&

N

-$)&

N

J

&

JON

-$)&

N

7

"+ ;93*

and in the equation the production and the dissipation terms read

4

J

&

JON

-$)&

N

4

`

J

&

N;J

&

N

_ "+ ;3,9*

The dissipation terms in Eq. 2.36 appear in Eqs. 2.37 and 2.38. The dissi-pation of the instantaneous velocity field $ &^ -$/& J & is distributed intothe time-averaged field and the fluctuating field. However, as mentionedabove, the dissipation at the fluctuating level is much larger than at thetime-averaged level (see Eqs. 2.30 and 2.31).

W 0 W 2[2 W

"%$'&[03(

In Eq. 2.23 a number of terms are unknown, namely the production term,the turbulent diffusion term and the dissipation term.

In the production term it is the stress tensor which is unknown. Since7 $ '

we have an expression for this which is used in the Navier-Stokes equationwe use the same expression in the production term. Equation 2.10 insertedin the production term (term II) in Eq. 2.23 gives

^ 4

J

&

JN

-

$/&

N

^

`ba

-

$/&

N

-

$

N

&

-

$)&

N

4

;

-

$)& & "+ ;39*

Note that the last term in Eq. 2.39 is zero for incompressible flow due tocontinuity.

The triple correlations in term III in Eq. 2.23 is modeled using a gradient' O%

$& 6

law where we assume that is diffused down the gradient, i.e from region ofhigh to regions of small (cf. Fouriers law for heat flux: heat is diffusedfrom hot to cold regions). We get

J

N

J

&

J

&

^ 4

`ba

N

_ "+ >90 *

where is the turbulent Prandtl number for . There is no model for thepressure diffusion term in Eq. 2.23. It is small (see Figs. 4.1 and 4.3) andthus it is simply neglected.

The dissipation term in Eq. 2.23 is basically estimated as in Eq. 1.12. The$66%7 %

velocity scale is now

^ "+ >. *

so that

2 I

c

J

&

N;J

&

N

^

#

"

"+ > 9*

!0

3 3D $ %

The modelled equation can now be assembled and we get

"

-$

N

.*

N

^

`

`ba

N

N

4

#

"

"+ > ;9*

We have one constant in the turbulent diffusion term and it will be de-termined later. The dissipation term contains another unknown quantity,the turbulent length scale. An additional transport will be derived fromwhich we can compute . In the :452 model, where 2 is obtained from itsown transport, the dissipation term

#

"

in Eq. 2.43 is simply

2 .For boundary-layer flow Eq. 2.43 has the form

-$

K

-

P

Q

^

Q

`

`ba

Q

`ba

-$

Q

:

4

#

"

_ "+ >3> *

( W

"%$X& 0( 0 W 2[4

In one equation models a transport equation is often solved for the turbu-lent kinetic energy. The unknown turbulent length scale must be given,and often an algebraic expression is used [4, 48]. This length scale is, forexample, taken as proportional to the thickness of the boundary layer, thewidth of a jet or a wake. The main disadvantage of this type of model isthat it is not applicable to general flows since it is not possible to find ageneral expression for an algebraic length scale.

However, some proposals have been made where the turbulent lengthscale is computed in a more general way [14, 30]. In [30] a transport equa-tion for turbulent viscosity is used.

1

W 0 W 2[2 W

"%$X& 03(

An exact equation for the dissipation can be derived from the Navier-Stokesequation (see, for instance, Wilcox [46]). However, the number of unknownterms is very large and they involve double correlations of fluctuating ve-locities, and gradients of fluctuating velocities and pressure. It is better toderive an 2 equation based on physical reasoning. In the exact equation for2 the production term includes, as in the equation, turbulent quantitiesand and velocity gradients. If we choose to include J & JN and -$)& N in theproduction term and only turbulent quantities in the dissipation term, wetake, glancing at the equation (Eq. 2.43)

A

^ 4

5 A 6

2

-$/&

N

-$

N

&

-$/&

N

"+; *

_

^ 4

5

A

:

2

:

_

Note that for the production term we have

A

^

5@A

6

" 2!!.*

. Now we canwrite the transport equation for the dissipation as

"

-

$

N

2?*

N

^

`

`ba

ZA

2

N

N

2

"

5

A

6

4

5

A

:

2?* "+; 9*

For boundary-layer flow Eq. 3.2 reads

-$2

K

-

P

2

Q

^

Q

`

`ba

A

2

Q

5

A

6

2

`ba

-$

Q

:

4

5

A

:

2

:

"+; ;9*

"b2[2 ( $'&[03( 4

The natural way to treat wall boundaries is to make the grid sufficiently fineso that the sharp gradients prevailing there are resolved. Often, when com-puting complex three-dimensional flow, that requires too much computerresources. An alternative is to assume that the flow near the wall behaveslike a fully developed turbulent boundary layer and prescribe boundaryconditions employing wall functions. The assumption that the flow nearthe wall has the characteristics of a that in a boundary layer if often nottrue at all. However, given a maximum number of nodes that we can af-ford to use in a computation, it is often preferable to use wall functionswhich allows us to use fine grid in other regions where the gradients of theflow variables are large.

In a fully turbulent boundary layer the production term and the dissi-pation term in the log-law region ( ;!0 Q 030 ) are much larger than the

3

Z 3' % Z

0 200 400 600 800 1000 1200

20

10

0

10

20

!#"$"%'&(

!*)+"#

,

.-0/1

; 2

43

65

17

> ;

)#*98,

>3>9030

J

$

,

0O_ 0!> ;

other terms, see Fig. 3.1. The log-law we use can be written as

$

J

^

]

G

J

Q

c

"+; > *

G

^ _ 0 "+; 9*

Comparing this with the standard form of the log-law

$

J

^;:

J

Q

c

=< "+; 9*

we see that

: ^

]

"+; 3*

< ^

]

1G

_

In the log-layer we can write the modelled equation (see Eq. 2.44) as

0 ^

`ba

-$

Q

:

4

2_ "+; ,9*

where we have replaced the dissipation term

#

"

by

2 . In the log-lawregion the shear stress 4

JZL is equal to the wall shear stress 8 , see Fig. 2.1.The Boussinesq assumption for the shear stress reads (see Eq. 2.10)

8 ^ 4

JZL

^

`ba

-$

Q

"+; 9*

Using the definition of the wall shear stress 8 ^

J

:

, and inserting Eqs. 3.9,3.15 into Eq. 3.8 we get

5@F

^

J

:

:

"+; 0 *

3;

Z 3' %

From experiments we have that in the log-law region of a boundary layerJ

:

!

,

0O_ ; so that 5;F ^ 0O_ 09 . 5

> Z%

When we are using wall functions and 2 are not solved at the nodesadjacent to the walls. Instead they are fixed according to the theory pre-sented above. The turbulent kinetic energy is set from Eq. 3.10, i.e.

' [

^

5

6 :

F

J

:

"+; 3 *

where the friction velocity J is obtained, iteratively, from the log-law (Eq. 3.4).Index

denotes the first interior node (adjacent to the wall).The dissipation 2 is obtained from observing that production and dis-

sipation are in balance (see Eq. 3.8). The dissipation can thus be writtenas

' )Z

2 ^

^

J

]

Q

"+; 9*

where the velocity gradient in the production term 4 JZL

$

Q has beencomputed from the log-law in Eq. 3.4, i.e.

$

Q

^

J

]

Q

_ "+; ;9*

For the velocity component parallel to the wall the wall shear stress is'

[

O

b!

used as a flux boundary condition (cf. prescribing heat flux in the temper-ature equation).

When the wall is not parallel to any velocity component, it is more con-venient to prescribe the turbulent viscosity. The wall shear stress 8 is ob-tained by calculating the viscosity at the node adjacent to the wall from thelog-law. The viscosity used in momentum equations is prescribed at thenodes adjacent to the wall (index P) as follows. The shear stress at the wallcan be expressed as

8

^

`

a

-

$

`

a

-

$

where -$

denotes the velocity parallel to the wall and is the normaldistance to the wall. Using the definition of the friction velocity J

8 ^

J

:

we obtain

`ba

$

^

J

:

`ba

^

J

$

J

Substituting J -$

with the log-law (Eq. 3.4) we finally can write

`ba

^

J

]

"

G

*

where ^ J . c .

?>

33

:452

$

W 0 W 2

In the :452 model the modelled transport equations for and 2 (Eqs. 2.43,3.2) are solved. The turbulent length scale is obtained from (see Eq. 1.12,2.42)

^

:

2

_ "+; > *

The turbulent viscosity is computed from (see Eqs. 2.11, 2.41, 1.12)

c a

^

5 F

6 :

#^

5 F

:

2

_ "+; 9*

We have five unknown constants 58FZ795 A 6>795 A :F constant was determined above (Sub-section 3.2). The equation in the logarithmic part of a turbulent boundarylayer was studied where the convection and the diffusion term could beneglected.

In a similar way we can find a value for the 5 A 6 constant . We look at the 5

> Z%

2 equation for the logarithmic part of a turbulent boundary layer, where theconvection term is negligible, and utilizing that production and dissipationare in balance

^

2 , we can write Eq. 3.3 as

0 ^

Q

`

a

DA

2

Q

+

"

5

A

6

4

5

A

:

*

2

:

"+; 9*

The dissipation and production term can be estimated as (see Sub-section 3.2)

2 ^

:

"+; 3*

^

J

]

Q

7

which together with

^

2 gives

^

]D5

F

Q%_ "+; ,9*

In the logarithmic layer we have that

.

Q ^0 , but from Eqs. 3.17, 3.18 wefind that

2?

Q

^ 0 . Instead the diffusion term in Eq. 3.16 can be rewrittenusing Eqs. 3.17, 3.18, 3.15 as

A

^

Q

`ba

A

Q

:

]D5

F

Q

^

:

]

:

A

:

5

6 :

F

"+; 9*

3

33

:476

$O

Inserting Eq. 3.19 and Eq. 3.17 into Eq. 3.16 gives

5 A 6

^

5 A :

4

]

:

5

6 :

F

DA

"+; !0 *

The flow behind a turbulence generating grid is a simple flow whichallows us to determine the 5 A : constant. Far behind the grid the velocity

5

> Z%

gradients are very small which means that

^0 . FurthermoreP

^0 andthe diffusion terms are negligible so that the modelled and 2 equations(Eqs. 2.43, 3.2) read

-$

XK

^ 4

2 "+; 1 *

-$

2

XK

^ 4

5@A :

2

:

"+; 39*

Assuming that the decay of is exponential K

, Eq. 3.21 gives 4 K

6

. Insert this in Eq. 3.21, derivate to find 2! XK and insert it intoEq. 3.22 yields

5

A

:

^

"+; 3;9*

Experimental data give ^ *

"

-$

N

6 *

N

^

`

`ba

C

6

N

N

6

"

5BCD6

4

5BCE:

6 * "+; 39*

`ba

^

6

7

2 ^

6%_

3

33

:458

$O

The constants are determined as in Sub-section 3.3:

^ 0O_ 09 , 5 CD6 ^ 9! ,5BCE:

^ ;9 >90 , C

^ and C ^ .When wall functions are used and 6 are prescribed as (cf. Sub-section 3.2):

%^ "

*

6 :

J

:

7

6 Z^ "

*

6 :

J

]

Q

_ "+; 39*

In regions of low turbulence when both and 2 go to zero, large numer-ical problems for the 4=2 model appear in the 2 equation as becomeszero. The destruction term in the 2 equation includes 2

:

! , and this causesproblems as 0 even if 2 also goes to zero; they must both go to zeroat a correct rate to avoid problems, and this is often not the case. On thecontrary, no such problems appear in the 6 equation. If 0 in the 6equation in Eq. 3.24, the turbulent diffusion term simply goes to zero. Notethat the production term in the 6 equation does not include since

6

5BCD6

^

6

5BCD69`ba

-$

&

K

N

-$

N

K &

-$

&

K

N

^

5BCD6

-$

&

K

N

-$

N

K &

-$

&

K

N

_

In Ref. [35] the 4 6 model was used to predict transitional, recirculatingflow.

W

0

W

2

One of the most recent proposals is the 4 8 model of Speziale

[39]where the transport equation for the turbulent time scale 8 is derived. Theexact equation for 8 ^ .?2 is derived from the exact and 2 equations. Themodelled and 8 equations read

"

-$

N

.*

N

^

`

`ba

N

N

4

8

"+; 93*

"

-

$

N

8 *

N

^

`

`ba

:

8

N

N

8

"( 4

5

A

6

*

"

5

A

:

4 *

8

"+; 3,9*

`

`ba

6

N

8

N

4

8

`

`ba

:

8

N

8

N

`ba

^

5BF

18

7

2 ^ .?8

The constants are: 5;F , 5 A 6 and 5 A : are taken from the :4=2 model, and

^

6

^

:

^

In the previous section we discussed wall functions which are used in orderto reduce the number of cells. However, we must be aware that this isan approximation which, if the flow near the boundary is important, canbe rather crude. In many internal flows where all boundaries are eitherwalls, symmetry planes, inlet or outlets the boundary layer may not bethat important, as the flow field is often pressure-determined. For externalflows (for example flow around cars, ships, aeroplanes etc.), however, theflow conditions in the boundaries are almost invariably important. Whenwe are predicting heat transfer it is in general no good idea to use wallfunctions, because the heat transfer at the walls are very important for thetemperature field in the whole domain.

When we chose not to use wall functions we thus insert sufficientlymany grid lines near solid boundaries so that the boundary layer can beadequately resolved. However, when the wall is approached the viscouseffects become more important and for Q the flow is viscous dom-inating, i.e. the viscous diffusion is much larger that the turbulent one(see Fig. 4.1). Thus, the turbulence models presented so far may not becorrect since fully turbulent conditions have been assumed; this type ofmodels are often referred to as high-

)#*

number models. In this sectionwe will discuss modifications of high-

)#*

number models so that they canbe used all the way down to the wall. These modified models are termedlow Reynolds number models. Please note that high Reynolds numberand low Reynolds number do not refer to the global Reynolds number(for example

)#*

,)#*

.

,)#*

.

etc.) but here we are talking about the localturbulent Reynolds number

)#*

^

c formed by a turbulent fluctuationand turbulent length scale. This Reynolds number varies throughout thecomputational domain and is proportional to the ratio of the turbulent andphysical viscosity c'a c , i.e.

)#*

ca

c . This ratio is of the order of 100 orlarger in fully turbulent flow and it goes to zero when a wall is approached.

We start by studying how various quantities behave close to the wallwhen Q 0 . Taylor expansion of the fluctuating velocities J & (also validfor the mean velocities -$/& ) gives

J

^

6

Q

:

Q

:

_>_>_

L

^

6

Q#

:

Q

:

_>_>_

M

^

5

5 6

Q#

5;:

Q

:

_>_>_

" > *

where _>_>_ 5;: are functions of space and time. At the wall we have no-slip, i.e. J ^ L ^ M ^ 0 which gives ^ ^ 5 . Furthermore, at the wall

J

K ^

M

S ^ 0 , and the continuity equation gives

L

Q^ 0 so that

3,

5 "

:452

$ %

0 5 10 15 20 25 30

0.2

0.1

0

0.1

0.2

0.3

Q

2

>

1

)#*

^ $

\

c

^ !,3!0

J

$ ^ 0O_ 09!0 5

17

2

"

*

J

c

6

^0 . Equation 4.1 can now be written

J

^

6

Q

:

Q

:

_>_>_

L

^

:

Q

:

_>_>_

M

^

56

Q

5;:

Q

:

_>_>_

" > 9*

From Eq. 4.2 we immediately get

J

:

^

:

6

Q

:

_>_>_ ^ "Q

:

*

L

:

^

:

:

Q

_>_>_ ^ "Q

*

M

:

^

5

:

6

Q

:

_>_>_ ^ "Q

:

*

JL

^

6

:

Q _>_>_ ^ "Q *

^ "

:

6

5

:

6

*)Q

:

_>_>_ ^ "Q

:

*

-$

Q ^

6

_>_>_ ^ "Q

*

" > ;9*

In Fig. 4.2 DNS-data for the fully developed flow in a channel is pre-sented.

!#0 "!$#

W

0

W

2[4

There exist a number of Low-Re number :452 models [32, 7, 10, 1, 27].When deriving low-Re models it is common to study the behavior of theterms when Q 0 in the exact equations and require that the correspond-ing terms in the modelled equations behave in the same way. Let us study

3

5 "

:452

$ %

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

Q

J

?

J

L

?

J

M

?

J

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Q.

\

J

?

-$

L

?

-$

M

?

-$

>

1

)#*

^ $

\

c

^ !,3!0

J

$ ^ 0O_ 09!0

J

?

&

^

J

:

&

the exact equation near the wall (see Eq. 2.24).

-$

K

-

P

Q

^ 4

JL

-$

Q

"Q

*

4

L

Q

4

Q

LJ

&

J

&

"Q

*

`

:

Q

:

4

`

J

&

N>J

&

N

"Q

*

" > > *

The pressure diffusion

L

Q term is usually neglected, partly because itis not measurable, and partly because close to the wall it is not important,see Fig. 4.3 (see also [28]). The modelled equation reads

-

$

K

-

P

Q

^

`ba

-

$

Q

:

"Q

*

Q

`ba

Q

"Q

*

`

:

Q

:

4

2

"Q

*

" > 9*

When arriving at that the production term is "Q

* we have used

ca

^

5@F

:

2

^

"Q

*

"Q

*

^ "Q

* " > 9*

Comparing Eqs. 4.4 and 4.5 we find that the dissipation term in the mod-elled equation behaves in the same way as in the exact equation when

;!0

5 "

:452

$ %

0 5 10 15 20 25 300.1

0.05

0

0.05

0.1

0.15

0.2

0.25

Q

> ;

1

)#*

^ $

\

c

^ !,3!0

J

$ ^ 0O_ 09!0 5

17

J

c

Q

0 . However, both the modelled production and the diffusion termare of "Q

* whereas the exact terms are of "Q * . This inconsistency ofthe modelled terms can be removed by replacing the 5 F constant by 5;F

H

F

whereH

F is a damping functionH

F so thatH

F

^ "Q

6

* when Q 0 andH

F

when Q !0 . Please note that the term damping term in thiscase is not correct since

H

F actually is augmenting `3a when Q 0 ratherthan damping. However, it is common to call all low-Re number functionsfor damping functions.

Instead of introducing a damping functionH

F , we can choose to solvefor a modified dissipation which is denoted

33 $65 3D ,9*

The equation needs to be modified since the diffusion term cannot balancethe destruction term when Q 0 .

W !#" ( WDU ! *

" U - " ! 0 !$# W 0 W 2)4

There are at least a dozen different low Re 452 models presented in theliterature. Most of them can be cast in the form [32] (in boundary-layerform, for convenience)

-$

K

-

P

Q

^

Q

`

`ba

Q

` a

-$

Q

:

4

2 " > 9*

-$ 3 *

2 ^

$ D %$ [Z

2

$

> *

From this equation we get immediately a boundary condition for 2 as

2

%^

c

:

Q

:

_ " > 9*

From Eq. 4.14 we can derive alternative boundary conditions. The exactform of the dissipation term close to the wall reads (see Eq. 2.24)

2 ^

c

J

Q

:

M

Q

:

" > 9*

where

Q

K

,

S and J,

M

L have been assumed. UsingTaylor expansion in Eq. 4.1 gives

2 ^

c

:

6

5

:

6

_>_>_ " > 3*

In the same way we get an expression for the turbulent kinetic energy

^

:

6

5

:

6

Q

:

_>_>_ " > ,9*

so that

Q

:

^

:

6

5

:

6

_>_>_ " > 9*

Comparing Eqs. 4.17 and 4.19 we find

2

%^

c

Q

:

_ " > !0 *

;3;

33 5

:452

$O

In the Sharma-Launder model this is exactly the expression for

in Eqs. 4.12and 4.13, which means that the boundary condition for 39*

which gives Eq. 4.21.

W

0 !;! "D, WDU 0 W 2

Near the walls the one-equation model by Wolfshtein [48], modified byChen and Patel [7], is used. In this model the standard equation is solved;the diffusion term in the -equation is modelled using the eddy viscosityassumption. The turbulent length scales are prescribed as [15, 11]

F

^

5

4

"(4

)

:

F

*

7

A

^

5

4

"(4

)

:

A

*

( is the normal distance from the wall) so that the dissipation term in the -equation and the turbulent viscosity are obtained as:

2 ^

:

A

7 `ba

^

5@F

F

" > 3;9*

The Reynolds number)

and the constants are defined as

)

^

c

7D5@F

^0O_ 09

7 5

^

]D5

F

7

:

F

^ ?0

7

:

A

^

5

The one-equation model is used near the walls (for)

3!0 ), and thestandard high-

)#*

:452 in the remaining part of the flow. The matching linecould either be chosen along a pre-selected grid line, or it could be definedas the cell where the damping function

4

"(4

)

:

F

*

takes, e.g., the value 0O_ 3 . The matching of the one-equation model and the:4=2 model does not pose any problems but gives a smooth distribution of`ba and 2 across the matching line.

;?>

33

5 "

:476

$O

W 2[0 ! # W 0 W 2

A model which is being used more and more is the 4 6 model of Wilcox [45].The standard 46 model can actually be used all the way to the wall with-out any modifications [45, 29, 34]. One problem is the boundary conditionfor 6 at walls since (see Eq. 3.25)

6^

2

^ "Q

:

* " > ?> *

tends to infinity. In Sub-section 4.3 we derived boundary conditions for2 by studying the equation close to the wall. In the same way we canhere use the 6 equation (Eq. 3.25) close to the wall to derive a boundarycondition for 6 . The largest terms in Eq. 3.25 are the viscous diffusion termand the destruction term, i.e.

0 ^

`

:

6

Q

:

4

5

CE:

6

:

_ " > 39*

The solution to this equation is

6^

c

5BCE:

Q

:

" > 39*

The 6 equation is normally not solved close to the wall but for Q _ 6 iscomputed from Eq. 4.26, and thus no boundary condition actually needed.This works well in finite volume methods but when finite element methodsare used 6 is needed at the wall. A slightly different approach must thenbe used [16].

Wilcox has also proposed a :4=6 model [47] which is modified for vis-cous effects, i.e. a true low-Re model with damping function. He demon-strates that this model can predict transition and claims that it can be usedfor taking the effect of surface roughness into account which later has beenconfirmed [33]. A modification of this model has been proposed in [36].

;3

33

5 "

:476

$O

33

5 "

:476

$ O

The :476 model of Peng et al. reads [36]

K

N

" -$

N

.* ^

K

N

c

ca

K

N

4

5

H

6

6

K

N

" -$

N

6 * ^

K

N

c

ca

C

6

K

N

6

"

5BCD6

H

C

4

5BCE:

6 *

5BC

ca

K

N

6

K

N

ca

^

H

F

6

H

^ 450O_ !3

4

)

a

0

H

F

^ 0O_ 093

4

4

)

a

0

0O_ 9!

0O_ 030#

)

a

4

)

a

!030

:

H

C

^ >_ ;

4

)

a

_ ;

4

)

a

7 5BCE:

^0O_ 0 !

5BC

^ 0O_ !

7

^0O_ ,

7

C

^ 39*

;3

33

5 "

:476

$O

with the turbulent Reynolds number defined as)

a

^ .1" 6

c

* . The constantsin the model are given as

^ 0O_ 09

7

F

^

7

C

^

7

CE:

^0O_ 0 !

7

^

7

C

^

: %

In Reynolds Stress Models the Boussinesq assumption (Eq. 2.10) is not usedbut a partial differential equation (transport equation) for the stress tensoris derived from the Navier-Stokes equation. This is done in the same wayas for the equation (Eq. 2.23).

Take the Navier-Stokes equation for the instantaneous velocity $ & (Eq. 2.4).Subtract the momentum equation for the time averaged velocity -$ & ((Eq. 2.6)and multiply by JN . Derive the same equation with indices ! and " inter-changed. Add the two equations and time average. The resulting equationreads

" -$

J

&

J N

*

&

N

^ 4

J

&

J

-$

N

4

J N J

-$ &

&

N

"

J

&

N

J N

& *

&

N

4

J

&

JN8J

JN

\

&

J

&

\

N

4

c

"

J

&

JON

*

&

N

4=

c

J

&

J

N

2&

N

"+ *

where

&

N is the production of J & JN [note that

& & ^

6

:

(

& & ^

6 6

: :

)];

&

N is the pressure-strain term, which promotes isotropy of the turbu-lence;

2

&

N is the dissipation (i.e. transformation of mechanical energy intoheat in the small-scale turbulence) of J & JON ;

&

N and

&

N are the convection and diffusion, respectively, of J & JN .

Note that if we take the trace of Eq. 5.1 and divide by two we get theequation for the turbulent kinetic energy (Eq. 2.23). When taking the tracethe pressure-strain term vanishes since

& &b^

J

& & ^0 "+ 9*

due to continuity. Thus the pressure-strain term in the Reynolds stressequation does not add or destruct any turbulent kinetic energy it merelyredistributes the energy between the normal components ( J

:

, L:

and M:

).Furthermore, it can be shown using physical reasoning [19] that & N acts toreduce the large normal stress component(s) and distributes this energy tothe other normal component(s).

;3,

" !3%$& $ %

# W ,( 032 4 *$ UXW 4

" !3%$& $ % $ $ 0 X 6 $O%

two derivative J&

and J N

are not correlated for !

^ " . This is the same asassuming that for small scales J & and JON are not correlated for !

^ " which isa good approximation since the turbulence at these small scale is isotropic,see Section 1.4.

We have given models for all unknown term in Eq. 5.1 and the modelledReynolds equation reads [24, 17]

" -$

J

&

JON

*

^ 4

J

&

J

-$

N

4

JN;J

-$ &

`

"

J

&

JON

*

5;=

J J

2

"

J

&

JN

*

&

N

4

;

\

&

N

2

"+ 9*

where & N should be taken from Eq. 5.3. For a review on RSMs (ReynoldsStress Models), see [18, 22, 26, 38].

# W ,( 032 4 *$ UXW 490

63 '6

0 0

A A

B BU (r)r

3

^

5

!

$ ^

$3"

*

7

$

^0

i) complex and difficult to implement;ii) numerically unstable because small stabilizing second-order deriva-

tives in the momentum equations (only laminar diffusion);iii) CPU consuming.

U

"%$

UXW

W

$X4

Curvature effects, related either to curvature of the wall or streamline cur-vature, are known to have significant effects on the turbulence [3]. Bothtypes of curvature are present in attached flows on curved surfaces, andin separation regions. The entire Reynolds stress tensor is active in theinteraction process between shear stresses, normal stresses and mean ve-locity strains. When predicting flows where curvature effects are impor-tant, it is thus necessary to use turbulence models that accurately predictall Reynolds stresses, not only the shear stresses. For a discussion of curva-ture effects, see Refs. [12, 13].

When the streamlines in boundary layer type of flow have a convex(concave) curvature, the turbulence is stabilized (destabilized), which damp-ens (augments) the turbulence [3, 37], especially the shear stress and theReynolds stress normal to the wall. Thus convex streamline curvature de-creases the stress levels. It can be shown that it is the exact modelling of theproduction terms in the RSM which allows the RSM to respond correctly tothis effect. The 452 model, in contrast, is not able to respond to streamlinecurvature.

The ratio of boundary layer thickness\

to curvature radius)

is a com-mon parameter for quantifying the curvature effects on the turbulence. Thework reviewed by Bradshaw demonstrates that even such small amountsof convex curvature as

\

)

^ 0O_ 0# can have a significant effect on the tur-bulence. Thompson and Whitelaw [42] carried out an experimental inves-

>.

63 '6

x

ystreamline

3

K

3

"

*

tigation on a configuration simulating the flow near a trailing edge of anairfoil, where they measured

\

) ,

0O_ 09; . They reported a 50 percent de-crease of

L

:

(Reynolds stress in the normal direction to the wall) owing tocurvature. The reduction of

J

:

and 4

JZL was also substantial. In additionthey reported significant damping of the turbulence in the shear layer inthe outer part of the separation region.

An illustrative model case is curved boundary layer flow. A polar coor-dinate system 4 (see Fig. 5.1)) with

locally aligned with the streamlineis introduced. As $ ^ $!" * (with

$

0 and $ ^ 0 ), the radialinviscid momentum equation degenerates to

$

:

4

^0 "+ 3*

Here the variables are instantaneous or laminar. The centrifugal force ex-erts a force in the normal direction (outward) on a fluid following the stream-line, which is balanced by the pressure gradient. If the fluid is displaced bysome disturbance (e.g. turbulent fluctuation) outwards to level A, it en-counters a pressure gradient larger than that to which it was accustomedat ^ , as "($ * " $ * , which from Eq. 5.7 gives "

*

"

*

.Hence the fluid is forced back to ^ . Similarly, if the fluid is displacedinwards to level B, the pressure gradient is smaller here than at ^ andcannot keep the fluid at level B. Instead the centrifugal force drives it backto its original level.

It is clear from the model problem above that convex curvature, when

$

0 , has a stabilizing effect on (turbulent) fluctuations, at least inthe radial direction. It is discussed below how the Reynolds stress modelresponds to streamline curvature.

Assume that there is a flat-plate boundary layer flow. The ratio of thenormal stresses

J

:

and

L

:

is typically 5. At one K station, the flow isdeflected upwards, see Fig. 5.2. How will this affect the turbulence? Let usstudy the effect of concave streamline curvature. The production terms

&

N

owing to rotational strains can be written as

>

63 '6 %

X% $# 63

XXVOD3

"!

7

J

:

4

17

_

6 6

^ 4

JZL

-$

Q

"+ ,9*

"!

7

JZL

4

17

_

6[:

^ 4

J

:

-

P

K

4

L

:

-$

Q

"+ 9*

"!

7

L

:

4

17

_

: :

^ 4

JZL

-

P

K

"+ 0 *

:452

^

`

a

-

$

Q

-

P

K

:

"+ 3 *

As long as the streamlines in Fig. 5.2 are parallel to the wall, all pro-duction is a result of

-$

Q . However as soon as the streamlines are de-flected, there are more terms resulting from

-

P

K . Even if

-

P

K is muchsmaller that

-$

Q it will still contribute non-negligibly to

6[: as

J

:

ismuch larger than

L

:

. Thus the magnitude of

6[: will increase (

6[: is nega-tive) as

-

P

K

0 . An increase in the magnitude of

6[: will increase 4 JL ,which in turn will increase

6 6 and

: : . This means that

J

:

and

L

:

willbe larger and the magnitude of

6[: will be further increased, and so on. Itis seen that there is a positive feedback, which continuously increases theReynolds stresses. It can be said that the turbulence is #%$&('*)"+,-,./$(# owingto concave curvature of the streamlines.

However, the 452 model is not very sensitive to streamline curvature(neither convex nor concave), as the two rotational strains are multipliedby the same coefficient (the turbulent viscosity).

If the flow (concave curvature) in Fig. 5.2 is a wall jet flow where

-

$

Q

0 , the situation will be reversed: the turbulence will be &0'*)"+1,-,./$(# . If thestreamline (and the wall) in Fig. 5.2 is deflected downwards, the situationwill be as follows: the turbulence is stabilizing when

-

$

Q

0 , and desta-bilizing for

-$

Q

0 .The stabilizing or destabilizing effect of streamline curvature is thus

dependent on the type of curvature (convex or concave), and whether thereis an increase or decrease in momentum in the tangential direction withradial distance from its origin (i.e. the sign of

-$

). For convenience,these cases are summarized in Table 5.1.

> ;

XXO

E Z $ " E D$D

x1

x2x1

x2

x

y

;

3

W 2 WDU "%$X& 0( "b( # W $'" U "$X&[03(

When the flow accelerates and/or decelerate the irrotational strains (

-$

K ,

-

P

Q and

-

R

S ) become important.In boundary layer flow, the only term which contributes to the pro-

duction term in the equation is 4

JZL

$

Q ( K denotes streamwise direc-tion). Thompson and Whitelaw [42] found that, near the separation pointas well as in the separation zone, the production term 4

"

J

:

4

L

:

*

$

K

is of equal importance. This was confirmed in prediction of separated flowusing RSM [12, 13].

In pure boundary layer flow the only term which contributes to theproduction term in the and 2 -equations is 4

JL

-$

Q . Thompson andWhitelaw [42] found that near the separation point, as well as in the sepa-ration zone, the production term 4

"

J

:

4

L

:

*

-$

K is of equal importance.As the exact form of the production terms are used in second-moment clo-sures, the production due to irrotational strains is correctly accounted for.

In the case of stagnation-like flow (see Fig. 5.3), where J:

,

L

:

the pro-duction due to normal stresses is zero, which is also the results given bysecond-moment closure, whereas :4=2 models give a large production. Inorder to illustrate this, let us write the production due to the irrotationalstrains

-

$

K and

-

P

Q for RSM and :452 :

)

0O_ "

6 6

: :

*^ 4

J

:

-

$

K

4

L

:

-

P

Q

452

^

`ba

-

$

K

:

-

P

Q

:

If J:

,

L

:

we get

6 6

: :

,

0 since

-

$

K ^ 4

-

P

Q due to continuity.The production term

in 452 model, however, will be large, since it willbe & of the two strains.

>3>

" ! " 5

[1] A new turbulence modelfor predicting fluid flow and heat transfer in separating and reattach-ing flows - 1. Flow field calculations. Int. J. Heat Mass Transfer 37 (1994),139151.

[2] !"#$&%'( (*)+,-/.0 Thin-layer approximation and alge-braic model for separated turbulent flows. AIAA 78-257, Huntsville,AL, 1978.

[3] 21345#$687 Effects of streamline curvature on turbulent flow.Agardograph progress no. 169, AGARD, 1973.

[4] 21345#$697';:/

" ! " 5

[13] ?P#Q%Z4=)E Reynolds stress transport modelling of shock-inducedseparated flow. Computers & Fluids 24 (1995), 253268.

[14] ?P#Q%Z4=W)E

" 4 4= Calculation of some parabolic andelliptic flows using a new one-equation turbulence model. In 5th Int.Conf. on Numerical Methods in Laminar and Turbulent Flow (1987), C. Tay-lor, W. Habashi, and M. Hafez, Eds., Pineridge Press, pp. 411422.

[15] ?P#Q%Z4=E)E P% %TEA Navier-Stokes stall predictions usingan algebraic stress model. J. Spacecraft and Rockets 29 (1992), 794800.

[16] ,+EQ%Z7 The Full Multigrid Method Applied to Turbulent Flow in Ven-tilated Enclosures Using Structured and Unstructured Grids. PhD thesis,Dept. of Thermo and Fluid Dynamics, Chalmers University of Tech-nology, Goteborg, 1997.

[17] %Z34 _)I1 Ground effects on pressure fluctua-tions in the atmospheric boundary layer. Journal of Fluid Mechanics 86(1978), 491511.

[18] . ">%SL/ Advanced turbulence closure models: A view of cur-rent status and future prospects. Int. J. Heat and Fluid Flow 15 (1994),178203.

[19] .%' J Turbulence, second ed. McGraw-Hill, New York, 1975.

[20] JFI4> ;)

" ! " 5

[26] )I4 S %Z

" ! " 5

[38] KG)% '

L.0 Second-order near-wall turbulenceclosures: a review. AIAA Journal 29 (1991), 18191835.

[39] KH$6GJ Characteristics of a trailing-edge flow with turbulent boundary-layer separation. Journal of FluidMechanics 157 (1985), 305326.

[43] $G4