davidson, l. - an introduction to turbulence models

8/2/2019 Davidson, L. - An Introduction to Turbulence Models

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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


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"

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 Exact0

Equation . . . . . . . . . . . . . . . . . . .' $

2.4.2 The Equation for' 1 ( 4 6 8 6 8 B

. . . . . . . . . . . . . . .' D

2.4.3 The Equation for' 1 ( 4 E6 8 E6 8 B

. . . . . . . . . . . . . . .' &

2.5 The Modelled0

Equation . . . . . . . . . . . . . . . . . . . .( H

2.6 One Equation Models . . . . . . . . . . . . . . . . . . . . . . .( '

3 Two-Equation Turbulence Models 223.1 The Modelled

PEquation . . . . . . . . . . . . . . . . . . . . .

( (

3.2 Wall Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .( (

3.3 The 0 R P Model . . . . . . . . . . . . . . . . . . . . . . . . . . ( #3.4 The

0 R T

Model . . . . . . . . . . . . . . . . . . . . . . . . .( $

3.5 The0 R V

Model . . . . . . . . . . . . . . . . . . . . . . . . . .( %

4 Low-Re Number Turbulence Models 284.1 Low-Re

0 R P

Models . . . . . . . . . . . . . . . . . . . . . . .( &

4.2 The Launder-Sharma Low-Re0 R P

Models . . . . . . . . . .Y (

4.3 Boundary Condition forP

and`P

. . . . . . . . . . . . . . . . .Y Y

4.4 The Two-Layer0 R P

Model . . . . . . . . . . . . . . . . . . .Y b

4.5 The low-Re0 R T

Model . . . . . . . . . . . . . . . . . . . . .Y #

4.5.1 The low-Re0 R T

Model of Peng et al. . . . . . . . . .Y $

4.5.2 The low-Re 0 R T Model of Bredberg et al. . . . . . . . Y $

5 Reynolds Stress Models 385.1 Reynolds Stress Models . . . . . . . . . . . . . . . . . . . . .

Y &

5.2 Reynolds Stress Models vs. Eddy Viscosity Models . . . . . . b H5.3 Curvature Effects . . . . . . . . . . . . . . . . . . . . . . . . .

b '

5.4 Acceleration and Retardation . . . . . . . . . . . . . . . . . .b b

(


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! " $ & ( 0 2

3 4 6 8 @ B D E F H P R

S T U S X U S a constants in the Reynolds stress modelS c

T

U S c

X constants in the Reynolds stress modelSd e T U S e X constants in the modelled

P

equationS g T U S g X constants in the modelled

Tequation

S p constant in turbulence modelq

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

damping function in pressure strain tensor0

turbulent kinetic energy (s

T

X t

8

t

8

)6

instantaneous (or laminar) velocity inu

-direction6 8

instantaneous (or laminar) velocity inu 8

-directionE6

time-averaged velocity inu

-direction

E6 8

time-averaged velocity inu 8

-directiont v

U

t w shear stressest fluctuating velocity in

u-direction

t

X

normal stress in theu

-directiont

8fluctuating velocity in

u 8-direction

t

8

t x Reynolds stress tensor

instantaneous (or laminar) velocity in

-direction

E

time-averaged velocity in

-directionv fluctuating (or laminar) velocity in

-direction

v

X

normal stress in the

-directionv w shear stress

instantaneous (or laminar) velocity in

-directionE

time-averaged velocity in

-directionw fluctuating velocity in

-directionw

X

normal stress in the

-direction

B D E F H P R

boundary layer thickness; half channel heightP

dissipation wave number; von Karman constant (

H b '

) dynamic viscosity dynamic turbulent viscosity

kinematic viscosity kinematic turbulent viscosity

8instantaneous (or laminar) vorticity component in

u 8-direction

Y


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E

8time-averaged vorticity component in

u8-direction

T specific dissipation ( P 1 0 )T 8

fluctuating vorticity component inu 8

-direction turbulent Prandtl number for variable

V

laminar shear stressV

total shear stressV

turbulent shear stress

B F R "

8 $ 6

&

centerlinew wall

b


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0 2 0 & $

@ 6

H " 6 8 H @

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

E6, which is independent of time (when the mean flow is steady), and

one fluctuating part t so that6 E6

t .There is no definition on turbulent flow, but it has a number of charac-

teristic features (see Tennekes & Lumley [41]) such as:

. 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 flow

becomes turbulent. The turbulence increases the exchange of momentumin e.g. boundary layers and reduces or delays thereby separation at bluff

bodies such as cylinders, airfoils and cars. The increased diffusivity alsoincreases the resistance (wall friction) in internal flows such as in channelsand pipes.

2 4 6 & 9 &

. Turbulent flow occurs at high Reynoldsnumber. For example, the transition to turbulent flow in pipes occurs that

A B C D

( Y H H, and in boundary layers at

A B F D

' H H H H H.

H Q S

# &

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

H # & & U

. 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.

#


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9 &

H

. Even though we have small turbulent scales in the

flow they are much larger than the molecular scale and we can treat theflow as a continuum.

F P

@ 6 B " 4 P

R

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.

E6

4

T

B 4 1 B

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

Pwhich is energy per unit time and unit mass

(P

X

1 ! #). The dissipation is proportional to the kinematic viscosity

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 are

larger 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

V. We

assume that these scales are determined by viscosity and dissipationP

.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 is

P . 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

V

in andP

using dimensional analysis. We write

%

P

1 #

X

1 #

X

1 ! #

4 ' 0 ' B

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,

$


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H

H

6

one for meters #

' ( (

U

4 ' 0 ( B

and one for seconds #

R ' R R Y

U

4 ' 0 Y B

which gives ' 1 b

. In the same way we obtain the expressions for&

andV

so that

% 4

P B

T

U

&

!

P

T

U

V

P

T X

4 ' 0 b B

" $

H

6 8 " 8 6 D %

$

P H " 8 6 D

4 8

@ 6

@ 6

4 " 6 8 H @

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 (

8 E

8 T 8) reads [41, 31, 44]

6

x

8 '

x

x

6 8 '

x

8 '

x x

8 ) 8

x 1

6

1

'

x

4 ' 0 # B

where)

8

x 1 is the Levi-Civita tensor (it is '

if3,

4 0are in cyclic order,

R 'if

3,

4 0are in anti-cyclic order, and

Hif any two of

3,

4 0are equal), and where

4 B'

x denotes derivation with respect tou

x . 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 of

the vorticity lines. If we write it term-by-term it reads

T

6

T

'

T

X

6

T

'

X

!

6

T

'

!

4 ' 0 $ B

T

6

X

'

T

X

6

X

'

X

!

6

X

'

!

T

6

!

'

T

X

6

!

'

X

!

6

!

'

!

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

5

& Q

slender, cylindrical fluid element which vorticity7

. We introduce a cylin-drical coordinate system with the

u

T -axis as the cylinder axis andu

X as the

%


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H

H

6

T

T

' 0 '

" % & )

0

radial coordinate (see Fig. 1.1) so that7

4

T U

H

U

H B

. A positive6

T

'

T willstretch the cylinder. From the continuity equation

6

T

'

T

'

1

4

1

6

X

B '

X

H

we find that the radial derivative of the radial velocity6

X 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 the

turbulent 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

1

X

4 ' 0 % B

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

2

1

X

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 affect

the vorticity components in the other two coordinate directions.The off-diagonal terms in Eq. 1.6 represent vortex tilting. Again, take aH

5

slender fluid element with its axis aligned with theu

X axis, Fig. 1.2. The ve-locity gradient

6

T

'

X will tilt the fluid element so that it rotates in clock-wisedirection. The second term

X

6

T

'

X in line one in Eq. 1.6 gives a contributionto

T .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 interacts

with 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

6

!

s Hand all

derivatives with respect tou

!

are zero. If6

!

s H

the third line in Eq. 1.6vanishes, and if

68 '

!

s H

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

D


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& U

X

X

6

T

4 u

X

B

' 0 (

)

0

remaining terms in Eq. 1.6 will also be zero since

T

6

!

'

X

R 6

X

'

!

s H

X

6

T

'

!

R 6

!

'

T

s H

The interaction between vorticity and velocity gradients will, on av-

erage, create smaller and smaller scales. Whereas the large scales whichinteract with the mean flow have an orientation imposed by the mean flowthe small scales will not remember the structure and orientation of the largescales. Thus the small scales will be isotropic, i.e independent of direction.

@

D R $

" 6

E

The turbulent scales are distributed over a range of scales which extendsfrom the largest scales which interact with the mean flow to the smallestscales where dissipation occurs. In wave number space the energy of ed-dies from to

can be expressed asq

4

B

4 ' 0 D B

where Eq. 1.8 expresses the contribution from the scales with wave numberbetween and

to the turbulent kinetic energy0

. The dimensionof wave number is one over length; thus we can think of wave numberas proportional to the inverse of an eddys radius, i.e

' 1

1 . The totalturbulent kinetic energy is obtained by integrating over the whole wavenumber space i.e.

0

q

4

B

4 ' 0 & B

&


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& U

q

4

B

' 0 Y

%

00

)

&

) %

)

) ) "

0

& )

" #

)

0

)

"

"

% " %

"

0

The kinetic energy is the sum of the kinetic energy of the three fluctuat-ing velocity components, i.e.

0

'

(

t

X

v

X

w

X

'

(

t

8

t

8 4 ' 0 ' H B

The spectrum ofq

is shown in Fig. 1.3. We find region I, II and III whichcorrespond to:

I. In the region we have the large eddies which carry most of the energy.

These eddies interact with the mean flow and extract energy from themean flow. Their energy is past on to slightly smaller scales. Theeddies velocity and length scales are and , respectively.

III. Dissipation range. The eddies are small and isotropic and it is herethat the dissipation occurs. The scales of the eddies are described bythe Kolmogorov scales (see Eq. 1.4)

II. Inertial subrange. The existence of this region requires that the Reynoldsnumber is high (fully turbulent flow). The eddies in this region repre-sent the mid-region. This region is a transport region in the cascadeprocess. Energy per time unit (

P) is coming from the large eddies at

the lower part of this range and is given off to the dissipation rangeat the higher part. The eddies in this region are independent of boththe large, energy containing eddies and the eddies in the dissipationrange. One can argue that the eddies in this region should be char-acterized by the flow of energy (

P) and the size of the eddies

' 1

.Dimensional reasoning gives

q

4

B

S ( 0

2 P

4

5

6

5

4 ' 0 ' ' B

This is a very important law (Kolmogorov spectrum law or theR # 1 Y

law) which states that, if the flow is fully turbulent (high Reynolds

' H


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& U

number), the energy spectra should exhibit aR # 1 Y

-decay. This of-

ten used in experiment and Large Eddy Simulations (LES) and DirectNumerical Simulations (DNS) to verify that the flow is fully turbu-lent.

As explained on p. 6 (cascade process) the energy dissipated at the smallscales can be estimated using the large scales and . The energy at thelarge scales lose their energy during a time proportional to

1 , which gives

P

X

1

!

4 ' 0 ' ( B

' '


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0 2 0 & $ & "

@ 6

H " 6 8 H @

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.

6 8 E

6 8

t

8

E

4 (0

' B

One reason why we decompose the variables is that when we measure flowquantities we are usually interested in the mean values rather that the time

histories. 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

2

4

68

B' 8

H 4 (0

( B

6 8

2

4

6 8 6

x

B

'

x

R

' 8

6 8 '

x

6

x

' 8 R

(

Y

8

x

6

1

'

1

'

x

4 (0

Y B

where4 B

'

x denotes derivation with respect tou

x . 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

6 8

2

4

6 8 6

x

B

'

x

R

' 8

4 6 8 '

x

6

x

' 8 B#

'

x

4 (0

b B

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

1

H) , but it does not mean that density is

constant; it can be dependent on for example temperature or concentration.Inserting Eq. 2.1 into the continuity equation (2.2) and the Navier-Stokes

equation (2.4) we obtain the time averaged continuity equation and Navier-Stokes equation

2

4

E6 8 B ' 8 H 4 (0

# B

E6 8

2

E6 8

E6

x

'

x R

E

' 8

4E

6 8 '

x

E

6

x

' 8 B R

t

8

t x

'

x 4 (

0$ B

A new term t8

t x appears on the right-hand side of Eq. 2.6 which iscalled the Reynolds stress tensor. The tensor is symmetric (for example t T t X

t

X

t

T ). 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 t

8

tx to close the equation system in Eq. 2.6.

' (


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6 7

V

V

V

V

D

' H

(0

'

&

" " " )

0

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

U 9

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

E

E6

U

u

4 (0

% B

Eq. 2.6 reads

E6 E6

u

E

E6

R

E

u

E6

R

t v

4 (0

D B

u u

T denotes streamwise coordinate, and u

X coordinate normal tothe flow. Often the pressure gradient

E

1

uis zero.

To the viscous shear stress

E6 1

on the right-hand side of Eq. 2.8

& Q

& & &

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

V

E6

R

t v

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 the

wall shear stress V , 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 t

v

H

, and the viscousshear stress attains its wall-stress value

V

t

X

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

D

' H

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

E6 1

have nearly always opposite sign (for a wall jet this

' Y


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6 7

u

v

H

v

H

T

X

6 4 B

(0

(

)

& # ) " &

" " "

R

t v

)

#

)

0

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

X to

T with (the turbu-lent fluctuating) velocity v . At its new location the

6velocity is in average

smaller than at its old, i.e.E

6 4

T

B

E6 4

X

B. This means that when the par-

ticle at

X (which has streamwise velocity6 4

X

B

) comes down to

T (wherethe streamwise velocity is

6 4

T

B

) is has an excess of streamwise velocitycompared to its new environment at

T . Thus the streamwise fluctuation ispositive, i.e. t

Hand the correlation between t and v is negative ( t v

H).

If we look at the other particle (dashed line in Fig. 2.2) we reach thesame conclusion. The particle is moving upwards (v

H

), and it is bringinga deficit in

6so that t

H. Thus, again, t v

H. If we study this flow for

a long time and average over time we get t v H

. If we change the signof the velocity gradient so that

E6 1

Hwe will find that the sign of t v

also changes.Above we have used physical reasoning to show the the signs of t v

and

E6 1

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 the

transport equation which are more important than the production term.There are different levels of approximations involved when closing the

equation system in Eq. 2.6.

9 6 &

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 are

based on a turbulent (eddy) viscosity are called eddy viscosity mod-

' b


15/48

% & & &

& & U

els.

S 7 6 &

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.

S 6 &

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

0

and its dissipationP

. The Reynolds stress tensor is then computedusing an assumption which relates the Reynolds stress tensor to the

velocity gradients and an eddy viscosity. The latter is obtained fromthe two transported scalars.

H 4 7 6 & & & & 6 &

Here a transport equation is derived forthe Reynolds tensor t

8

t x . One transport equation has to be added fordetermining the length scale of the turbulence. Usually an equationfor the dissipation

P

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).

H R R 8 @

R R R E $ 6 8 H @

In eddy viscosity turbulence models the Reynolds stresses are linked tothe velocity gradients via the turbulent viscosity: this relation is called theBoussinesq assumption, where the Reynolds stress tensor in the time aver-aged Navier-Stokes equation is replaced by the turbulent viscosity multi-plied by the velocity gradients. To show this we introduce this assumptionfor the diffusion term at the right-hand side of Eq. 2.6 and make an identi-fication

4 E6 8 '

x

E6

x

' 8 B R

t

8

t x

'

x 4

B 4 E6 8 '

x

E6

x

' 8 B

'

x

which gives

t

8

t x

R

4 E6 8 '

x

E6

x

' 8 B 4 (0

& B

If we in Eq. 2.9 do a contraction (i.e. setting indices3 4

) the right-handside gives

t

8

t

8 s ( 0

where0

is the turbulent kinetic energy (see Eq. 1.10). On the other handthe continuity equation (Eq. 2.5) gives that the right-hand side of Eq. 2.9

' #


16/48

9 6 &

is equal to zero. In order to make Eq. 2.9 valid upon contraction we add( 1 Y

8

x

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

t

8

t x

R

4E

6 8 '

x

E

6

x

' 8 B

(

Y

8

x

0 4 (0

' H B

Note that contraction of

8

x gives

8 8

T T

X X

! !

' ' ' Y

"

P

F

4 8 " H

P R

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

. The dimension of is

X

1

"

# (same as ). A turbulent velocity 6 6

( & &

6

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

4 (0

' ' B

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

X

8

F

6

4 (0

' ( B

where

is the coordinate normal to the wall, and where

8

F

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

8

F

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 reader

is referred to Wilcox [46].

4 6 8 H @ R H

8 @

6 8 "

@

D

Q 5

0

The equation for turbulent kinetic energy0

T

X

t

8

t

8is derived from the

Navier-Stokes equation which reads assuming steady, incompressible, con-stant viscosity (cf. Eq. 2.4)

4

6 8 6

x

B '

x

R

' 8

6 8 '

x x

4 (0

' Y B

' $


17/48

&

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

4

E6 8 E6

x

B '

x

R E

' 8

E6 8 '

x x

R

4

t

8

t x

B '

x

4 (0

' b B

Subtract Eq. 2.14 from Eq. 2.13, multiply by t8

and time average and weobtain

68

6

x

R

E68

E6

x

'

x

t

8

R

R E

' 8

t

8

6A 8 R E6 8

'

x x

t

8

4

t

8

t x

B '

x t

8

4 (0

' # B

The left-hand side can be rewritten as

4 E6 8

t

8 B 4 E6

x

tx

B R E6 8 E6

x

'

x

t

8

E6 8

tx

t

8 E6

x

t

8

tx

'

x

t

8 4 (0

' $ B

Using the continuity equation4

t x

B'

x

H

, the first term is rewritten as

E6 8

tx

'

x

t

8

t

8

tx

E6 8 '

x

4 (0

' % B

We obtain the second term (using4

E6

x

B '

x

H) from

E6

x

0

'

x E

6

x

'

(

t

8

t

8

'

x

'

(

E6

x t

8

t

8 '

x

t

8

t

8 '

x

t

8

E6

x t

8

'

x

4 (0

' D B

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

'

(

4

t x t

8

t

8B

'

x

4 (0

' & B

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

R ' 8

t

8 R 4

t

8 B ' 8 4 (0

( H B

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

t

8 '

x x t

8

4

t

8 '

x t

8 B

'

x

R

t

8 '

x t

8 '

x

4 (0

( ' B

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

4

t

8 '

x t

8B

'

x

'

(

4

t

8

t

8B

'

x x

0'

x x

4 (0

( ( B

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

4

E6

x

0 B '

x

R

t

8

t x

E6 8 '

x

R

t x

'

(

t x t

8

t

8 R

0 '

x

'

x

R

t

8 '

x t

8 '

x

4 (0

( Y B

The terms in Eq. 2.23 have the following meaning.

' %


18/48

&

(

.

6

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

The two first terms represent 9 6 $ &

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

H@ # & & U

. 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 exact0

equation read

E6 0

u

E

0

R

t v

E6

R

v

'

(

v t

8

t

8 R

0

R

t

8 '

x t

8 '

x

4 (0

( b B

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

boundary-layer approximation that

t

8 1

u

t

8 1

is not valid.

Q

' 1 ( 4 6 8 6 8 B

The equation for the instantaneous kinetic energy

T

X

6 8 6 8is derived

from the Navier-Stokes equation. We assume steady, incompressible flow

with constant viscosity, see Eq. 2.13. Multiply Eq. 2.13 by 6 8 so that

6 8 4

6 8 6

x

B '

x

R 6 8

' 8

6 8 6 8 '

x x

4 (0

( # B

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

4

68

68

6

x

B'

x

R

68

6

x

68 '

x

6

x

4 68

68

B'

x

R

'

(

6

x

4 68

68

B'

x

'

(

6

x

4 6 8 6 8 B '

x

4

6

x

B '

x

4 (0

( $ B

where ' 1 ( 4 6 8 6 8 B

.The first term on the right-hand side of Eq. 2.25 can be written as

R 68

' 8 R 4 6

8

B' 8

4 (0

( % B

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.

6 8 6 8 '

x x

'

x x

R

6 8 '

x

6 8 '

x

4 (0

( D B

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

4

6

x

B

'

x

'

x x

R 4 68

B' 8

R

68 '

x

68 '

x

4 (0

( & B

' D


19/48

&

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

6 8 '

x

6 8 '

x

E6 8 '

x

E6 8 '

x

t

8 '

x t

8 '

x

4 (0

Y H B

As the scales ofE6

is much larger than those of t8, i.e. t

8 '

x

6 8 '

x we get

6 8 '

x

6 8 '

x

D

t

8 '

x t

8 '

x

4 (0

Y ' B

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).

Q

' 1 ( 4 E68

E68

B

The equation for' 1 ( 4 E6 8 E6 8 B

is derived in the same way as that for' 1 ( 4 6 8 6 8 B

.Assume steady, incompressible flow with constant viscosity and multiplythe time-averaged Navier-Stokes equations (Eq. 2.14) so that

E6 8 4

E6 8

E6

x

B '

x

RE

6 8E

' 8

E6 8

E6 8 '

x x

RE

6 8

4

t

8

t x

B '

x

4 (0

Y ( B

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

4

E6

x

E

B '

x

E

'

x x

R 4E

6 8E

B ' 8 R

E6 8 '

x

E6 8 '

x

RE

6 8

4

t

8

t x

B '

x

U

4 (0

Y Y B

whereE

' 1 ( 4

E6 8

E6 8 B

. The last term is rewritten as

R E6 8

4

t

8

t x

B '

x

R 4 E6 8

t

8

t x

B '

x

4

t

8

t x

B E6 8 '

x

4 (0

Y b B

Inserted in Eq. 2.33 gives

4

E6

x

E

B '

x

E

'

x x

R 4E

6 8E

B ' 8 R

E6 8 '

x

E6 8 '

x

R 4 E6 8

t

8

t x

B '

x

t

8

t x

E6 8 '

x

4 (0

Y # B

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

0. Note that the last

term 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

0equation

(the production term) appears as a sink term in theE

equation.It is interesting to compare the source terms in the equation for

0(2.23),

(Eq. 2.29) andE

(Eq. 2.35). In the equation, the dissipation term is

R

68 '

x

68 '

x

R

E68 '

x

E68 '

x

R

t

8 '

xt

8 '

x

4 (0

Y $ B

' &


20/48

Q

6 y 6

0

In theE

equation the dissipation term and the negative production term

(representing loss of kinetic energy to the 0 field) read

R

E6 8 '

x

E6 8 '

x

t

8

t x

E6 8 '

x

U

4 (0

Y % B

and in the0

equation the production and the dissipation terms read

R

t

8

t x

E6 8 '

x

R

t

8 '

x t

8 '

x

4 (0

Y D B

The dissipation terms in Eq. 2.36 appear in Eqs. 2.37 and 2.38. The dissi-pation of the instantaneous velocity field

6 8 E6 8

t

8is distributed into

the time-averaged field and the fluctuating field. However, as mentionedabove, the dissipation at the fluctuating level is much larger than at the

time-averaged level (see Eqs. 2.30 and 2.31).

H

P P

4 6 8 H @

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. SinceU 6

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

1

R

t

8

t x

E6 8 '

x

E6 8 '

x

E

6

x

' 8

E6 8 '

x

R

(

Y

0E

6 8 ' 8 4 (0

Y & B

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 9

6 $ &

law where we assume that0

is diffused down the gradient, i.e from region ofhigh

0to regions of small

0(cf. Fouriers law for heat flux: heat is diffused

from hot to cold regions). We get

'

(

tx

t

8

t

8 R

1

0'

x

4 (0

b H B

where

1

is the turbulent Prandtl number for0

. 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. The6 & & U

velocity scale is now

0 4 (

0b ' B

so that

P s

t

8 '

x t

8 '

x

0

5

4

4 (0

b ( B

( H


21/48

6 &

The modelled0

equation can now be assembled and we get

4

E6

x

0 B'

x

1

0'

x

'

x

1

R

0

5

4

4 (0

b Y B

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

0 R Pmodel, where

Pis obtained from its

own transport, the dissipation term

0

5

4

1 in Eq. 2.43 is simply

P.

For boundary-layer flow Eq. 2.43 has the form

E6 0

u

E

0

1

0

E6

X

R

0

5

4

4 (0

b b B

@

4 6 8 H @

H

P R

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.

( '


22/48

0 (

0 2 0 & $ & "

"

H

P P

4 6 8 H @

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

Pequation based on physical reasoning. In the exact equation for

Pthe production term includes, as in the

0equation, turbulent quantities

and and velocity gradients. If we choose to include t8

t x andE

6 8 '

x in theproduction term and only turbulent quantities in the dissipation term, wetake, glancing at the

0equation (Eq. 2.43)

e

R

Se

T

P

0

E

6 8 '

x

E

6

x

' 8

E6 8 '

x

4 Y0

' B

" "

R

Se

X

P

X

0

Note that for the production term we have

e

S eT

4 P 1 0 B

1 . Now we canwrite the transport equation for the dissipation as

4

E

6

x

P B '

x

e

P '

x

'

x

P

0

4

S e T

1

R

S e X

P B 4 Y0

( B

For boundary-layer flow Eq. 3.2 reads

E6 P

u

E

P

e

P

Se

T

P

0

E6

X

R

Se

X

P

X

0

4 Y0

Y B

"

4 P P @ " 6 8 H @ R

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 boundary

conditions 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 (

Y H

' H H) are much larger than the

( (


23/48

&

0 200 400 600 800 1000 1200

20

10

0

10

20

! # % & ( 0

23 ( 4 4 ( 7 8 & ( 0

23 ( A # 4 ( 0

D 0 E G % & ( 0

Y0

' P

)

)

Q

0S

) #

) % )

0

U

)

b Y # 0

A B V D

b b H H

t

1 6

D

H H b Y0

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

6

t

'

)

q

t

4 Y0

b B

q

& H 4 Y0

# B

Comparing this with the standard form of the log-law

6

t

X

)

t

` 4 Y0

$ B

we see that

X

'

4 Y0

% B

`

'

) q

In the log-layer we can write the modelled0

equation (see Eq. 2.44) as

H

E6

X

R

P 4 Y0

D B

where we have replaced the dissipation term

0

5

4

1

by

P

. In the log-lawregion the shear stress

R

t v is equal to the wall shear stressV

, see Fig. 2.1.The Boussinesq assumption for the shear stress reads (see Eq. 2.10)

V R

t v

E6

4 Y0

& B

Using the definition of the wall shear stressV

t

X

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

S p

t

X

0

X

4 Y0

' H B

( Y


24/48

&

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

X

1 0

D

H Y so thatS p

H H & . S

&

When we are using wall functions0

andP

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.

9

0

S

T X

p

t

X

4 Y0

' ' B

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

denotes the first interior node (adjacent to the wall).The dissipation

Pis obtained from observing that production and dis-

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

9

P

1

t

!

4 Y0

' ( B

where the velocity gradient in the production termR

t v

6 1

has been

computed from the log-law in Eq. 3.4, i.e.

6

t

4 Y0

' Y B

For the velocity component parallel to the wall the wall shear stress is9

(

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

V 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

V

'

E6

&

'

E6

'

&

whereE

6

'

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

V

t

X

we obtain

'

6

'

&

t

X

'

t

6

'

t

&

Substituting t 1

E6

'

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

'

t

&

)

4

q

&

B

where&

t

&1

.

( b


25/48

Q

0 R P

6

" "

H

P

In the0 R P

model the modelled transport equations for0

andP

(Eqs. 2.43,3.2) are solved. The turbulent length scale is obtained from (see Eq. 1.12,2.42)

0!

X

P

4 Y0

' b B

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

Sp

0

T X

Sp

0

X

P

4 Y0

' # B

We have five unknown constants S p U S e T U S e X U 1 and e , which we hope

should be universal i.e same for all types of flows. Simple flows are chosenwhere the equation can be simplified and where experimental data are usedto determine the constants. The S p constant was determined above (Sub-section 3.2). The

0equation in the logarithmic part of a turbulent boundary

layer was studied where the convection and the diffusion term could beneglected.

In a similar way we can find a value for the S e T constant . We look at the

S

&

P

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

1

P

, we can write Eq. 3.3 as

H

e

P

C

4

Se

T

R

Se

X

B

P

X

0

4 Y0

' $ B

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

P

0 !

X

4 Y0

' % B

1

t

!

U

which together with

1

P

gives

S

!

p

4 Y0

' D B

In the logarithmic layer we have that

0 1

H, but from Eqs. 3.17, 3.18 we

find that

P 1

H. Instead the diffusion term in Eq. 3.16 can be rewritten

using Eqs. 3.17, 3.18, 3.15 as

e

e

0!

X

S

!

p

0

X

X

e

X

S

T X

p

4 Y0

' & B

( #


26/48

Q

0 R T

6

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

Se

T

Se

X

R

X

S

T X

p

e

4 Y0

( H B

The flow behind a turbulence generating grid is a simple flow whichallows us to determine the S e X constant. Far behind the grid the velocity

S

&

gradients are very small which means that

1

H. Furthermore

Hand

the diffusion terms are negligible so that the modelled0

andP

equations(Eqs. 2.43, 3.2) read

E6

0

u

R

P 4 Y0

( ' B

E6

P

u

R

S eX

P

X

0

4 Y0

( ( B

Assuming that the decay of0

is exponential0 u

, Eq. 3.21 gives0

R

u

T

. Insert this in Eq. 3.21, derivate to find P 1 u

and insert it intoEq. 3.22 yields

Se

X

'

4 Y0

( Y B

Experimental data give

' ( #

H H $

[43], and S e X ' & (

is chosen.

We have found three relations (Eqs. 3.10, 3.20, 3.23) to determine threeof the five unknown constants. The last two constants, 1 and e , are opti-mized by applying the model to various fundamental flows such as flow inchannel, pipes, jets, wakes, etc. The five constants are given the followingvalues: S p

H H &, S e T

' b b, S e X

' & (, 1

' H, e

' Y '.

"

H

P

The0 R T

model is gaining in popularity. The model was proposed byWilcox [45, 46, 36]. In this model the standard

0equation is solved, but as a

length determining equationT

is used. This quantity is often called specificdissipation from its definition

T P 1 0

. The modelled0

andT

equation read

4

E6

x

0 B'

x

g

1

0'

x

'

x

1

R

T 0 4 Y0

( b B

4

E6

x

T B '

x

g

T '

x

'

x

T

0

4

S g T

1

R

S g X

0 T B 4 Y0

( # B

0

T

U

P

T 0

( $


27/48

Q

0 R V

6

The constants are determined as in Sub-section 3.3:

H H &

, S g T # 1 &

,S g X

Y 1 b H ,

g

1

( and g

( .When wall functions are used

0and

Tare prescribed as (cf. Sub-section 3.2):

0 4

B

T X

t

X

U

T 4

B

T X

t

4 Y0

( $ B

In regions of low turbulence when both0

andP

go to zero, large numer-ical problems for the

0 R Pmodel appear in the

Pequation as

0becomes

zero. The destruction term in theP

equation includesP

X

1 0, and this causes

problems as0

H

even ifP

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

Tequation. If

0

Hin the

T

equation in Eq. 3.24, the turbulent diffusion term simply goes to zero. Notethat the production term in the

Tequation does not include

0since

T

0

S g T

1

T

0

S g T

E68

u

x

E6

x

u 8

E68

u

x

S g T

E68

u

x

E6

x

u 8

E68

u

x

In Ref. [35] the0 R T

model was used to predict transitional, recirculatingflow.

"

H P

One of the most recent proposals is the0 R V

model of Speziale

[39]

where the transport equation for the turbulent time scale V is derived. Theexact equation for

V 0 1 P

is derived from the exact0

andP

equations. Themodelled

0

andV

equations read

4

E6

x

0 B'

x

1

0'

x

'

x

1

R

0

V

4 Y0

( % B

4

E

6

x

V B '

x

X

V '

x

'

x

V

0

4 ' R

S e T

B

1

4

S e X

R ' B

0

V

4 Y0

( D B

(

0

T

0'

x

V'

x

R

(

V

X

V'

x

V'

x

S p

0 V

U

P 0 1 V

The constants are: S p , S e T and S e X are taken from the0 R P

model, and

1

T

X

' Y $.

( %


28/48

!

0 " 2

0 2 0 & $ & "

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, can

be 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 the

temperature field in the whole domain.When we chose not to use wall functions we thus insert sufficiently

many 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

#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-

A B

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

A B

number models so that they canbe used all the way down to the wall. These modified models are termed

low Reynolds number models. Please note that high Reynolds numberand low Reynolds number do not refer to the global Reynolds number(for example

A B

,A B F

,A B F

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

A B

1

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

1

, i.e.A B

1

. 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

H. Taylor expansion of the fluctuating velocities t

8(also valid

for the mean velocitiesE6 8

) gives

t

T

X

X

v

T

X

X

w

S

S T

S X

X

4 b0

' B

where

S X are functions of space and time. At the wall we have no-slip, i.e. t

v

w

Hwhich gives

S

. Furthermore, at the wall

t

1

u

w

1

H, and the continuity equation gives

v

1

Hso that

( D


29/48

2

S 4

0 R P

6 &

0 5 10 15 20 25 30

0.2

0.1

0

0.1

0.2

0.3

1

P

b0

'

#

$ )

"

0

% ) %

"

) "

( ' # 0

A B

6

1

% D & H0

t

1 6 H H # H0

S

) #

) % )

0

U

)

0!

%

)

1

" "

)

P

# ) # "

)

4

# %

%

%

) "

)

" "

B

) % " %

" # "

)

0)

" &

% # $ ) " %

&

t

1

0

T

H. Equation 4.1 can now be written

t

T

X

X

v

X

X

w

S T

S X

X

4 b0

( B

From Eq. 4.2 we immediately get

t

X

X

T

X

4

X

B

v

X

X

X

4

B

w

X

S

X

T

X

4

X

B

t v

T

X

!

4 !

B

0 4

X

T

S

X

T

B

X

4

X

B

E6 1

T

4

B

4 b0

Y B

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

sented.

3 H 1 3 5

H

P R

There exist a number of Low-Re number0 R P

models [32, 7, 10, 1, 27].When deriving low-Re models it is common to study the behavior of theterms when

H

in the exact equations and require that the correspond-ing terms in the modelled equations behave in the same way. Let us study

( &


30/48

2

S 4

0 R P

6 &

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

t

c

1

t

v

c

1

t

w

c

1

t

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

1

t

c

1 E6

v

c

1E

6

w

c

1 E6

b0

(

#

$ )

"

0

% ) %

"

) "

( ' # 0

A B

6

1

% D & H0

t

1 6 H H # H0

%

) %

%

%

)( ) "

t

c

8

t

X

80

the exact0

equation near the wall (see Eq. 2.24).

E6 0

u

E

0

R

t v

E6

4

!

B

R

v

R

'

(

v t

8

t

8

4

!

B

X

0

X

R

t

8 '

x t

8 '

x

4

B

4 b0

b B

The pressure diffusion

v

1

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

E6 0

u

E

0

E6

X

4

B

1

0

4

B

X

0

X

R

P

4

B

4 b0

# B

When arriving at that the production term is 4

Bwe have used

S p

0

X

P

4

B

4

B

4

B 4 b0

$ B

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

Y H


31/48

2

S 4

0 R P

6 &

0 5 10 15 20 25 300.1

0.05

0

0.05

0.1

0.15

0.2

0.25

b0

Y

#

$ )

"

0

% ) %

"

) "

( ' # 0

A B

6

1

% D & H0

t

1 6 H H # H0

S

) #

) % )

0

U

)

0

# ) # "

) # %

%

%

) "

# )

# "

) #

" "

) % " %

" # "

)

0)

" &

% # )

" %

&

t

1

0

H. However, both the modelled production and the diffusion term

are of 4

Bwhereas the exact terms are of

4 !

B. This inconsistency of

the modelled terms can be removed by replacing the S p constant by S pr

p

wherer

p is a damping functionr

p so thatr

p

4

T

B

when

H

andr

p

'when

# H. Please note that the term damping term in this

case is not correct sincer

p

actually is augmenting

when

H

ratherthan damping. However, it is common to call all low-Re number functionsfor damping functions.

Instead of introducing a damping functionr

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

`P, see Ref. [25] and Section 4.2.

It is possible to proceed in the same way when deriving damping func-tions for the

Pequation [39]. An alternative way is to study the modelled

P

equation near the wall and keep only the terms which do not tend to zero.From Eq. 3.3 we get

E6 P

u

4

T

B

E

P

4

T

B

Se

T

P

0

1

4

T

B

e

P

4

T

B

X

P

X

4

B

R

Se

X

P

X

0

4

X

B

4 b0

% B

where it has been assumed that the production term

1 has been suitablemodified so that

1

4 ! B. We find that the only term which do not

vanish at the wall are the viscous diffusion term and the dissipation term

Y '


32/48

Q 2

6 S Q 2

S 4

0 R P

6 &

so that close to the wall the dissipation equation reads

H

X

P

X

R

Se

X

P

X

0

4 b0

D B

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

H.

3 4 @ 3 B

4 E 4 3 H 1 3 5

H P R

There are at least a dozen different low Re0 R P

models presented in theliterature. Most of them can be cast in the form [32] (in boundary-layerform, for convenience)

E6 0

u

E

0

1

0

E6

X

R

P 4 b0

& B

E6 `P

u

E

`P

e

`P

S Te

r

T

`P

0

E6

X

R

S e X

r

X

`P

X

0

q

4 b0

' H B

Sp

r

p

0

X

`P

4 b0

' ' B

P `P

4 b0

' ( B

Different models use different damping functions (r

p

,r

T

andr

X

) anddifferent extra terms (

andq

). Many models solve for`P

rather than forP

where

is equal to the wall value ofP

which gives an easy boundarycondition

`P H(see Sub-section 4.3). Other models which solve for

Puse

no extra source in the0

equation, i.e.

H.

Below we give some details for one of the most popular low-Re0 R P

models, the Launder-Sharma model [25] which is based on the model of2

6 S

Q

Jones & Launder [20]. The model is given by Eqs. 4.9, 4.10, 4.11 and 4.12where

r

p

R Y b

4 '

A

1 # H B

X

r

T

'

r

X

' R H Y

R

A

X

(

0

X

q

(

X

E6

X

X

A

0

X

`P

4 b0

' Y B

Y (


33/48

% 6 6

P

6

`P

The termq

was added to match the experimental peak in0

around

D

( H [20]. Ther

X

term is introduced to mimic the final stage of decay of turbu-lence behind a turbulence generating grid when the exponent in

0 u

changes from ' ( #

to ( #

.

"

H @ 4

D H @ 8 6 8 H @

H

4 @

In many low-Re0 R P

models`P

is the dependent variable rather thanP

.The main reason is that the boundary condition for

P

is rather complicated.The largest term in the

0

equation (see Eq. 4.4) close to the wall, are thedissipation term and the viscous diffusion term which both are of

4

Bso

that

H

X

0

X

R

P 4 b0

' b B

From this equation we get immediately a boundary condition forP

as

P

X

0

X

4 b0

' # B

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)

P

t

X

w

X

4 b0

' $ B

where

1

1

u

D

1

and t

D

w

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

P

X

T

S

X

T

4 b0

' % B

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

0

'

(

X

T

S

X

T

X

4 b0

' D B

so that

0

X

'

(

X

T

S

X

T

4 b0

' & B

Comparing Eqs. 4.17 and 4.19 we find

P

(

0

X

4 b0

( H B

Y Y


34/48

Q S2

0 R P

6

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

in Eqs. 4.12

and 4.13, which means that the boundary condition for `P is zero, i.e. `P H .In the model of Chien [8], the following boundary condition is used

P

(

0

X

4 b0

( ' B

This is obtained by assuming

T

S T in Eqs. 4.17 and 4.18 so that

P (

X

T

0

X

T

X

4 b0

( ( B

which gives Eq. 4.21.

1 H 3 3 4 D

H

P

Near the walls the one-equation model by Wolfshtein [48], modified byChen and Patel [7], is used. In this model the standard

0

equation is solved;the diffusion term in the

0

-equation is modelled using the eddy viscosityassumption. The turbulent length scales are prescribed as [15, 11]

p

S

0

' R

4 R

A

1 X

p

B#

U

e

S

0

' R

4 R

A

1 X

e

B#

( 0 is the normal distance from the wall) so that the dissipation term in the

0 -equation and the turbulent viscosity are obtained as:

P

0!

X

e

U

S p

0

p

4 b0

( Y B

The Reynolds numberA

and the constants are defined as

A

0

0

U S p

H H &

U S

S

!

p

U

X

p

% H

U

X

e

(

S

The one-equation model is used near the walls (forA

( # H), and the

standard high-A B

0 R Pin the remaining part of the flow. The matching line

could either be chosen along a pre-selected grid line, or it could be definedas the cell where the damping function

' R

4 R

A

1 X

p

B

takes, e.g., the valueH & #

. The matching of the one-equation model and the0 R P

model does not pose any problems but gives a smooth distribution of and

Pacross the matching line.

Y b


35/48

Q S 4

0 R T

6

P H2 1 3 5

H

P

A model which is being usedmore and more is the0 R T

model of Wilcox [45].The standard

0 R T

model can actually be used all the way to the wall with-out any modifications [45, 29, 34]. One problem is the boundary conditionfor

T

at walls since (see Eq. 3.25)

T

P

0

4

X

B 4 b0

( b B

tends to infinity. In Sub-section 4.3 we derived boundary conditions forP

by studying the0

equation close to the wall. In the same way we canhere use the

T

equation (Eq. 3.25) close to the wall to derive a boundary

condition forT

. The largest terms in Eq. 3.25 are the viscous diffusion termand the destruction term, i.e.

H

X

T

X

R

Sg X

T

X

4 b0

( # B

The solution to this equation is

T

$

S g X

X

4 b0

( $ B

TheT

equation is normally not solved close to the wall but for

( # Tis

computed from Eq. 4.26, and thus no boundary condition actually needed.This works well in finite volume methods but when finite element methodsare used

T

is needed at the wall. A slightly different approach must thenbe used [16].

Wilcox has also proposed a0 R T

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].

Y #


36/48

Q S 4

0 R T

6

Q S 4

0 R T

6

The0 R T

model of Peng et al. reads [36]

0

2

u

x

4 E6

x

0 B

u

x

1

0

u

x

1

R

S

1

r

1

T 0

T

2

u

x

4E

6

x

T B

u

x

g

T

u

x

T

0

4

S g T

r

g

1

R

S g X

0 T B

S g

0

0

u

x

T

u

x

r

p

0

T

r

1

' R H % ( (

R

A

' H

r

p

H H ( #

' R

R

A

' H

!

H & % #

H H H '

A

R

A

( H H

X

r

g

' b Y

R

A

' #

T X

U

r

g

' b Y

R

A

' #

T X

S

1

H H &

U S g T

H b (

U S g X

H H % #

S g

H % #

U

1

H D

U g

' Y #

4 b0

( % B

Q S 4

0 R T

6 6 9

A new0 R T

model was recently proposed by Bredberg et al. [5] which reads

0

2

u

x

4 E6

x

0 B

1

R

&

1

0 T

u

x

1

0

u

x

T

2

u

x

4E

6

x

T B

&

g T

T

0

1

R

&

g X

T

X

&

g

0

0

0

u

x

T

u

x

u

x

g

T

u

x

4 b0

( D B

The turbulent viscosity is given by

&

p

r

p

0

T

r

p

H H &

H & '

'

A

!

' R

R

A

( #

X

4 b0

( & B

Y $


37/48

Q S 4

0 R T

6

with the turbulent Reynolds number defined asA

0 1 4 T

B

. The constants

in the model are given as&

1

H H &

U

&

p

'

U

&

g

' '

U

&

g T

H b &

U

&

g X

H H % (

U

1

'

U

g

' D

4 b0

Y H B

Y %


38/48

& " 2 " " & "

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

0

equation (Eq. 2.23).Take the Navier-Stokes equation for the instantaneous velocity

6 8(Eq. 2.4).

Subtract the momentum equation for the time averaged velocityE6

8((Eq. 2.6)

and multiply by t x . Derive the same equation with indices3

and4

inter-changed. Add the two equations and time average. The resulting equationreads

4 E6

1t

8

t x

B'

1

&

8

x

R

t

8

t1

E6

x

'

1

R

t x t1

E68 '

1

8

x

4

t

8 '

x

t x

' 8B

8

x

R

t

8

t x t1

t x

8

1

t

8

x 1

R

4

t

8

t x

B '

1

'

1

8

x

R (

t

8 '

1t

x

'

1

P 8

x

4 #0

' B

where

8

x

is the production oft

8

t x

[note that

8 8

T

X

1

(

8 8

T T

X X

! !

)];

8

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

P8

x is the dissipation (i.e. transformation of mechanical energy intoheat in the small-scale turbulence) of t

8

t x ;&

8

x and

8

x are the convection and diffusion, respectively, of t8

t x .

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

8 8 (

t

8 ' 8 H 4 #0

( B

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 ( t

X

, vX

and wX

).Furthermore, it can be shown using physical reasoning [19] that

8

x acts toreduce the large normal stress component(s) and distributes this energy tothe other normal component(s).

Y D


39/48

4 6 & & &

6 &

5

D @ H P R B 6

R R H

P R

We find that there are terms which are unknown in Eq. 5.1, such as thetriple correlations t

8

tx

t1 , the pressure diffusion

4

tx

8

1

t

8

x 1

B 1

andthe pressure strain

8

x , and the dissipation tensorP 8

x . From Navier-Stokesequation we could derive transport equations for this unknown quantities

but this would add further unknowns to the equation system (the closureproblem, see p. 12). Instead we supply models for the unknown terms.

The pressure strain term, which is an important term since its contribu-

tion is significant, is modelled as [24, 17]

8

x

8

x

'

T

8

x

'

X

c

8

x

'

T

c

8

x

'

X

8

x

'

T

R

S T

P

0

t

8

t x

R

(

Y

8

x

0

8

x

'

X

R

S X

8

x

R

(

Y

8

x

1

c

'

T

R (

S

c

T

P

0

t

X

r

c

a a

'

T

S

c

T

P

0

t

X

r

r

0

5

4

( # # u

P

4 #0

Y B

The object of the wall correction terms

c

'

T and

c

a a

'

T is to take the effect of

the wall into account. Here we have introduced a R0

coordinate system,with along the wall and 0 normal to the wall. Near a wall (the termnear may well extend to

( H H) the normal stress normal to the wall is

damped (for a wall located at, for example,u H

this mean that the normalstress v

X

is damped), and the other two are augmented (see Fig. 4.2).In the literature there are many proposals for better (and more compli-

cated) pressure strain models [40, 23].The triple correlation in the diffusion term is often modelled as [9]

U

S

8

x

S a

t1

t

0

P

4

t

8

t x

B'

1

'

4 #0

b B

The pressure diffusion term is for two reasons commonly neglected. First, itis not possible to measure this term and before DNS-data (Direct NumericalSimulations) were available it was thus not possible to model this term.Second, from DNS-data is has indeed been found to be small (see Fig. 4.3).

The dissipation tensorP 8

x is assumed to be isotropic, i.e.

P 8

x

(

Y

8

x

P 4 #0

# B

From the definition ofP 8

x (see 5.1) we find that the assumption in Eq. 5.5 isequivalent to assuming that for small scales (where dissipation occurs) the

Y &


40/48

4 6 & & &

6 & 6 6

H & &

6 &

two derivative t8

'

1 and t x'

1 are not correlated for3

4

. This is the same as

assuming that for small scalest

8 andt x

are not correlated for 3

4 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]

4 E6

1t

8

t x

B '

1

R

t

8

t1

E6

x

'

1

R

t x t1

E6 8 '

1

4

t

8

t x

B'

1

S a

t1

t

0

P

4

t

8

t x

B '

'

1

8

x

R

(

Y

8

x

P

4 #0

$ B

where 8

x should be taken from Eq. 5.3. For a review on RSMs (Reynolds

Stress Models), see [18, 22, 26, 38].

5 D @ H P R B 6 R R H P R

D

$

8 R " H R 8 6 D

H P R

Whenever non-isotropic effects are important we should consider using RSMs.Note that in a turbulent boundary layer the turbulence is always non-isotropic,

but isotropic eddy viscosity models handle this type of flow excellent as faras mean flow quantities are concerned. Of course a

0 R P

model give verypoor representation of the normal stresses. Examples where non-isotropiceffects often are important are flows with strong curvature, swirling flows,flows with strong acceleration/retardation. Below we present list some ad-

vantages and disadvantages with RSMs and eddy viscosity models.Advantages with eddy viscosity models:

i) simple due to the use of an isotropic eddy (turbulent) viscosity;ii) stable via stability-promoting second-order gradients in the mean-

flow equations;iii) work reasonably well for a large number of engineering flows.

Disadvantages with eddy viscosity models:

i) isotropic, and thus not good in predicting normal stresses ( tX

U

v

X

U

w

X

);ii) as a consequence of i) it is unable to account for curvature effects;

iii) as a consequence of i) it is unable to account for irrotational strains.

Advantages with RSMs:

i) the production terms need not to be modelled;ii) thanks to i) it can selectively augment or damp the stresses due to

curvature effects,

davidson, l. - an introduction to turbulence models

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