~~

"

-. ,.

INTRODUCTION TO

TURBULENCE MODELLING

by

Helge I. Andersson

Lecture Notes in Subject 76572 Turbulent FlowDivision of Applied Mechanics

Department of Physics and MathematicsNorwegian Institute of Technology

Trondheim, October 1988

TABLE OF CONTENTS page

1. Introduction 1

2. The generalized eddy-viscosity concept 2

3. .rurbulent conductivity 5

4. Zero-equation models 7

4.1 Constant eddy-viscosity models 94.2 Mixing-length models 104.3 Characterizing features 14

5. One-equation models 17

5.1 Transport equation for k 195.2 Transport equation for the eddy viscosity 255.3 Transport equation for the shear stress 255.4 Characteristic features 27

6. Two-equation models 29

6.1 The k-~ model of turbulence 306.2 Numerical examples 336. 3 ~_dvantages and disadvantages 40

7. Reynolds stress models 42 .'

:

References for supplementary reading 46 :,~i

;;

- 1 -

1. INTRODUCTION

It is generally accepted that turbulent flows are exactly "

represented by three-dimensional time-dependent Navier-Stokes equ-

ations, which are the second-order Chapman-Enskog approximation

to the Boltzmann equation for molecular motion. Althoug existing

computer algorithms arid programs are capable of solving the full

Navier-Stokes equations, the storage capacity of present day compu-

ters is too small to allow the resolution of the tiniest small-

scale fluctuations of the turbulence.

Most flows of engineering importance are turbulent, and the

desire to make practical calculations useful in design and planning

has led to the development of approximate methods which make computer

simulations of turbulent flows feasible. The first step towards

fairly general prediction methods is the Reynolds-averaging of

the governing conservation equations. However, while the process

of carefully averaging the momentum and thermal-energy equations

can be considered an exact procedure, the resulting averaged equ-

ations do not contain enough information about the turbulence to

form a soluble set of equations. For example, unknown averages

of products of fluctuating quantities:

T.. = - P ~ (1)1J 1 J

QJ. = pc u-:--e (2)P J

appear in the Reynolds averaged momentum and thermal-energy equation,

respectively. Here, eq. (1) represents the apparent turbulent

stresses (Reynolds' stresses), while eq. (2) represents an apparent

turbulent heat conduction in the j'th direction. It should be

remembered, however, that the turbulent stresses and heat flux

account for the physical mechanisms of turbulent transport (convec-

tion) of momentum and heat, respectively.

The process of replacing the unknown averages of products

of fluctuating quantities with equations for (or functions of)

variables which can be considered as dependent variables in the

problem is called modelling. For instance, an equation that suppos-

edly expresses a relation between the unknown Reynolds stresses(1) and the mean velocity components is a turbulence model.

- - 2 -

A turbulence model can be either an algebraic relation or

a set of transport equations for some characteristic turbulent

quantities. The various models are often classified according

to the number of extra partial differential equations which are

considered in addition to the Reynolds-averaged Navier-Stokes equ-

ations. By selecting a proper turbulence model, the total number

of equations involved equals the number of unknown variables, and

a closed set of governing equations is obtained.

The objective of these lecture notes is to outline the essen-

tial ingredients involved in modern turbulence modelling, and to

demonstrate the effectiveness and capabilities of some representative

models at each level (according to the above classification scheme).

We are focussing on the various models required for the solution

of the Reynolds averaged momentum equation, from which the basic

ideas and principles are illuminated. The modelling required for

the solution of the thermal-energy equation follows basically the

same philosophy and strategy. Thus, only a brief account on the

modelling of the turbulent heat flux is given in section 3.

2. THE GENERALIZED EDDY-VISCOSITY CONCEPT

The convenience of the eddy-viscosity concept is coupled

with the mathematical convenience of retaining the same form of

the governing equations for laminar and turbulent flow, and there-

by allowing for the use of the same solution procedure in both

cases. Considering, for instance, the free shear flows like the

turbulent jet, wake and mixing-layer, the averaged streamwise moment-

um equation become: i..

i

"

u ~ + V ~ = - a~ (uv) (3) ~,

The orginal idea of Boussinesq was to define a turbulent or eddy-

viscosity by the relation

- au- PUV = pvT ay (4)

:r;:,~;;;i;:;;i~*;,?~:i;

- 3 -

If we substitute eq. (4) into eq. (3), the momentum equation become'

au au a ( au)u ax + V ay =ay vT -ay (5)

The eddy-viscosity defined by eq. (4) is not a physical property

like the molecular viscosity, but varies with local flow condi-

tions and the geometry of the problem. If the eddy-viscosity is

assumed to be constant, as suggested by Boussinesq in 1877, equ-

ation (5) becomes mathematically equivalent with the streamwise

momentum equations for the corresponding laminar free shear flows.

The value of the eddy viscosity, which can be obtained from experi-

ments, will generally be significantly greater than the molecular

viscosity.

For the free shear flows, only one of the Reynolds stresses

(1) appear in the momentum equation (3). In the general case,

however, relations like eq. (4) are required for all nine compo-

nents of the Reynolds stresses. In analogy with the molecular

stresses

cri j = 2 1.1 5 i j ( 6 )

the turbulent stresses can be modelled according to

T" = - p~ = 2pv 51.' )' - -32pko 1.' )' it.". (7)1.) 1. ) -r ,-~,

~~'\Here, the symbol k denotes the~kinetic energy of the turbulent

motion, defined as

k =: , ~ (8)1. 1.

which is a measure of the normal Reynolds stresses. The latter

term in the model (7) has been included to assure that the iden-

tity

: - 4 -

. ."- 2 ko = - 2k ,- uiui = 2\),Sii -"3 ii ' (9)

is recovered for i=j.

An important feature of eq. (7) is that the turbulent shear

stress

- ( aU av )- p uv = P\)! ay + ~ (10)

reduces to the simpler relation (4) for shear flows in which the

dominating velocity gradient is the cross-stream gradient of thestreamwise velocity component u. Thus, eq. (7) represents a gen-

eralized eddy-viscosity hypothesis.

Now, if we introduce the eddy-viscosity defined by eq. (7)into the Reynolds averaged momentum equations

aui aui 1 a- + u - = - - T (11)at j ax. pax. ij

) )

where

T.. = - Po.. + 2~S.. - p u-:-i::i-:-(12)1.) 1.). 1.) 1. )

is the total average stress tensor, the resulting equations become:

au. aui a [ p 2 ] a [ ( ) 1 -.:!::. + u. - = - '\- - + _3 k + '\- 2 \) + \) 5.. J (13) at J ax. ox. Pox. ,1.J) 1. J

- 5 -

"

These fundamental equations reveal~ that the effective viscosity

is the sum of the molecular and turbulent (eddy) viscosities. Fur-'

thermore, the turbulent kinetic energy k is the turbulent equivalent

to the averaged static pressure. Thus, if the kinetic energy is

not treated as a dependent variable of the model equations, k is

conveniently absorbed in the unknown kinematic pressure.

3. TURBULENT CONDUCTIVITY

Following the idea behind the eddy viscosity concept, the

turbulent heat flux (2) is related to the gradient of the mean

temperature field according to

- ae- pcp Uj6 = KT"a"'Xj (14)

where Kt is the turbulent conductivity (or, alternatively, the eddy

conductivity). This relation is analogous to Fourier's law for

heat flux due to molecular agitation.

Now, if the assumption (14) is introduced in the averaged

thermal energy equation

[ ae ae ) - - - 2.9-Pc - + U. -;;-- - ~<t> ax. (15)p at J OXj J

where

ae -Q : - K- + pc u.e (16)

ax. p JJ

denotes the total mean heat flux, the resulting equation becomes

~-~- ~-~ ~~

...~ , - 6 -

! \.

j

[ ae ae ] - a[ ( ) ae ]pcp a-t + U j ax; = ~<I> - axj - \ K + ~ ~ (17)

This equation demonstrates that the turbulent conductivity adds

to the molecular conductivity to enhance the total heat fluxes.

In analogy with the thermal diffusivity caused by molecular

action, it may often be convenient to introduce the turbulent or

eddy diffusivity:

YT :: KT/pcp (18)

for the turbulent heat transfer. It should be emphasized, however,

that the turbulent diffusivity as well as the turbulent conductivity

are not fluid properties but depend on the mean flow and the turbu-

lence. These scalar quantities may therefore vary significantly

from point to point within a flow field.

The eddy-diffusivity hypothesis (14) is analogous to the

eddy-viscosity hypothesis (4), i.e. the turbulent transport of

a certain property is related to the gradient of that property.

Thus, eq. (14) is based on the assumption that the turbulent tran-

sport of heat is provided by similar processes as the turbulent

transport of momentum. The ratio of the respective coefficients

of diffusion is defined as the turbulent Prandtl number:

'0" = \) /Y (19)T T T

In spite of the fact that the eddy viscosity and the eddy diffu-

sivity may vary considerably throughout a flow field, it is experi-mentally verified that the ratio (19) is approximately constant

within a flow. Furthermore, the maguitude of the turbulent Prandtlnumber is approximately constant for a wide range a flow situations.

1

- 7 - -

This is the famous Reynolds analogy from 1874. Today, engi- .

nee ring heat and fluid flow calculations take advantage of this

analogy and take the turbulent Prandtl number to be a constant

close to 1.

4. ZERO-EQUATION MODELS

Turbulence models using only partial differential equations

for the mean velocity field, and no differential equation for the

turbulence are classified as zero-equation models. All models

belonging to this class use the generalized eddy-viscosity concept

(7). The eddy-viscosity is furthermore related to the mean flow

field via an algebraic relation. Therefore, these models are also

called algebraic models. Because of their simplicity, zero-equ-

ation models have received considerable interest over the years,

and have been in common use for sophisticated engineering appli-

cations during the past decades.

The very simplest approximation of the turbulent effects

on the mean flow can be achieved by assuming that the eddy-visco-

sity is proportional to the molecular viscosity, i.e.

vT = f . v (20)

throughout the flow field. The dimensionless constant f is typi-

cally of the order 1000. From a numerical point of view, this

approach does not require any modifications of computer codes des-

ignated for laminar flow calculations. The only implication of

the turbulence is the increased effective diffusivity, which in

general quenches numerical instabilities arising from diffusion-

like truncation errors.

It is interesting to notice that some useful information

about the flow field may be obtained by this simple approach. For

instance, Hanson, Summers and Wilson (1984,1986) simulated the

wind flow over buildings numerically using the eddy-viscosity model

(20). Figure 1 a shows the predicted wind-flow environment for

~",~

"c - 8 - ..,

.

.; iji~; / I :: '::!.!:~~~~f~:.; 'Jl1

,/

./-;

Fig.l Predicted wind-flow environment for a two-building config-

uration in 3D. Hand h denote the heigths of the downstream and

upstream buildings, respectively. L is the distance of separation.

(Figures taken from Hanson, Summers and Wilson 1986).

a) Streakline pattern for a vertical plane one meter from the

plane of symmetry. L/H = 1.0 and H/h = 5.0.

.8

.7

00.6 0 0

-I-~ .5

0ttJ .4a-U')b .3Z 0- 0~ .2 L/H = .5

.1

.0 .5 1.0 1.5 2.0

W/H

b) Maximum predicted reverse vortex speed at ground level (2m

above the ground) for the case L/H = 0.5. The symbols denote

wind-tunnel measurements by Penwarden and Wise 1975.

-

- 9 -

,a two-building configuration. The recirculation of the flow

downstream of the taller building and the reverse-vortex formed

between the buildings are clearily observed. Figure lb shows the

predicted maximum reverse-vortex speed at the ground level compared

with wind-tunnel measurements.

It should be emphasized, however, that eq.(20) cannot give

satisfactory results in the immediate vicinity of solid boundaries,

simply because the turbulence quenches close to a wall. Then,

according to eq. (7), the eddy viscosity should vanish at the boun-

daries.

4.1 Constant eddy-viscosity models

An example of the constant eddy-viscosity models is the simple

relation

vT = K . ~(x) . Us(x) (21)

suggested by Prandtl in 1942 for free shear flows. While K is

a dimensionless constant, the length scale and velocity scale in

eq.(2l) may be functions of the longitudinal distance x. Accor-

dingly, the eddy-viscosity is allowed to vary in the streamwise

direction but is supposed to be constant over any cross section

of the flow. This is certainly a rough assumption, in particular

in the vicinity of the edges of the shear layers. Due to the inter-

mittent nature of the flow in these regions, a fixed point in space

will alternately be within and outside the turbulent domain. Never-

theless, reasonable results for free shear flows are obtained with

the model (21).

Figure 2 shows that the solution obtained with the model

(21) is in quite good agreement with experimental data for the

plane jet. Near the edge of the jet, however, the experimentalresults indicate a somewhat faster reduction- of the velocity than

obtained by the constant eddy-viscosity model.

~

- 10 -1.0 ~_.- --- -

Ii ~ T~rv:.. "" - Goerllrrl1942J

V (x y) IV! '~'"' I, s 1- ' --- Tollmlrn 119261

~ II . Co Irs I 19561

! Lawollhr.wakr!I I

05 ~ -. -- T--,-., I

I

IDJla: Ii 0 Retchardt (19421 "r' . Foerthmann (19361 \\\.

,L '"\~ ~ .'. ~ ~,".. ." '

'... ~-oL-L-- J ...0 0.5 1.0 1.5 2.0 25

C= ~((

Fig.2 Comparison of theoretical velocity profile with experimental

data for a plane turbulent jet. The solid line is based on the

constant eddy-viscosity model (21), while Tollmien used a mixing-

length type model (22). (Figure taken from White 1974).

It should be remembered that in the particular case of a

plane wake, the streamwise variation of the eddy viscosity (21)

vanishes, and the eddy viscosity becomes a constant througout the

wake.

4.2 Mixing-length models

In order to account for a cross-stream variation of the eddy-

viscosity, for instance in the vicinity of a wall or near the edge

of a jet, Prandtl suggested in 1925 that

2 \dU \vT = £. ay (22)

where 1 is the mixing-length.

The model (22) was orginally based on the mixing-length hypo-

thesis, in which it was assumed that the turbulent eddies interact

by collisions in the same way as the molecules in a gas. Thus,the mixing-length became the turbulent equivalent of the molecular

-

- 11 -

~ ,

mean free path. However, it is today generally accepted that this

physical equivalence is completely erroneous. The turbulent eddies

are not small compared to the width of the mean flow, and they

interact continuously rather than collide instantaneously.

Nevertheless, the model (22) has proved successful in several

flow situations. This is probably due to the observation that

the eddy-viscosity from dimensional argumentss should be propor-

tional to a length scale and a velocity scale. Now, by takingthe velocity scale which characterizes the larger eddies as the

cross-stream velocity gradient times the mixing-length, the formula

(22) is obtained. Equation (22) can thus be considered as the

definition of the mixing-length, like eq. (7) defines the eddy-viscosity. The success of these concepts is achieved because the

latter quantities are more easily correlated with experimental

data than are the Reynolds stresses themselves.

By adopting the mixing-length hypothesis (22), an empirical

model for the mixing-length rather than for the eddy-viscosity

must be selected. prandtl orginally suggested that the mixing-

length should be proportional to the distance from the wall. This

is a reasonable assumption in the inner part of a wall flow, but

outside the viscous sublayer.

In the outer part of the wall layer, Fig.3 shows that the

mixing-length becomes approximately constant. This behaviour sugg-

ests the model:

I ry y ~ YiR. -- alo Yi .5 y

where (23)

r = 0.40 ; al = 0.075

- 12 -

008

0.07

006

OO~PI!

004

0.03

0.02

0.01

00 0.1 0.2 03 04 O~ 06 07 08 0.9 10

rIa

Fig.3 Dimensionless mixing-length distribution across a turbulent

boundary layer at zero pressure gradient, according to data of

Klebanoff 1954. (Figure taken from Cebeci & Bradshaw 1977).

007

006

OO~

004'""T

~00)

, , ,,

002 "

001

00 01 02 03 04 05 06 07 08 09 10

y/8

Fig.4 Dimensionless eddy-viscosity distribution across a turbu-

lent boundary layer at zero pressure gradient, according to data

of Klebanoff 1954. (Figure taken from Cebeci & Bradshaw 1977).

~~i\~~;:~¥~"

- 13 -

are dimensionless constants in accordance with experimental obser-

vations. The position Yi is determined so that the inner region

model should match the outer region model by the requirement of

continuity in the mixing-length.

By analogy with laminar flow over an oscillating flat plate

(Stokes' second problem), the modified mixing-length formula

t = ry [1 - exP(-yu*/VA+)] (24)

was proposed by van Driest in 1956 for the inner region. The term

in the square brackets accounts for the damping effect of the thin

viscous sub layer close to the wall, so that the mixing-length out-

side the viscous shear layer is effectively modified near the wall.

In accordance with experimental results, the dimensionless para-meter A+ = 26. Thus, the term in the square brackets increases

from zero at the wall to about 0.63 at y+ = 26, and tends to 1

as the normal distance from the wall is further increased.

Today, the most popular eddy-viscosity model based on the

mixing-length concept is the two-layer model

vT = [ (ry) 2 [1 - :XP(-YU*/VA+)]2 . 1i¥1 y ~ Yi (25a)

a2 . UE . 0 Yi ~ Y (25b)

proposed by Cebeci and Smith in 1970. Here

UE = inviscid velocity external to boundary layer

*0 = boundary layer displacement thickness

a2 = 0.0168 = dimensionless parameter

- 14 -

The inner region model is obtained by combining van Driest's modi-

fied mixing-length formula (24) with Prandtl's model (22). The

outer region model, on the other hand, is a simple constant eddy-

viscosity model like (21). With a different choise for velocity

and length scale, the outer model (25b) can be replaced by

"T = 0.3 . u* . <5 (26)

where

u* = friction velocity

<5 : boundary layer thickness

a3 : 0.060 - 0.075 : dimensionless parameter

It can easily be verified that the inner-region expression

(25a) yields a linearly increasing eddy-viscosity in the logar-

ithmic region, which is in accordance with the experimental data

plotted in Fig.4. In the outer half of the boundary layer, however,

the eddy-viscosity is observed to decrease. This is obviously

due to the intermittent nature of the outermost part of wall boundary

layers. This behaviour is sometimes accounted for by a dimensionless

function which reduces from 1 to 0 as y goes from 0.5 IS to 1.0 <5

In order to account for the intermittency, the eddy-viscosity model

(25) should be multiplied with this function or intermittency factor.

4.3 Characterizing features

Several eddy-viscosity and mixing-length models have been

used over the years, and some of them still are. Empirical corr-

ections have been made to some of the models to account for the

effects of pressure gradient, low Reynolds number, wall suction

or injection, and curvature effects.

Some typical numerical solutions for boundary layer flows

obtained with the Cebeci & Smith model (25) are shown in Figures

5 and 6. While Fig. 5 shows results for a zero-pressure gradient

flow, the boundary layer in Fig. 6 is in a moderate positive pres-sure gradient. Results for the entrance-region turbulent flow

- 15 -I , ~

"" 5

4

C, . 10']

Z

110' 106 10' In" Int

R~.

Fig.5 Calculated variation of local sk'in-friction coefficientfor zero pressure gradient turbulent boundary layer. The solidline denotes numerical solution of Cebeci & Smith 1974 using a

two-layer model like eq. (25). The symbols denote experimental

data.1.0 = 6.92 ft

0.8

0.6

!!-Ue

0.4

0.2 Re

0R x 10-" A A Ae

C x 10] c,f

0 00 Z 4 6 8

x (ft)

00 1 2 J 4 5

y (in.)

Fig.6 Numerical results of Cebeci & Smith 1974 and experimental

data of Bradshaw & Ferriss 1965 for an equilibrium boundary layerin moderate adverse pressure gradient. The symbols denote the

experimental data, and the solid lines denote calculations using

a two-layer model like eq. (25).

.

- 16 -

..3 i . 9 i . 33 i. 57 i . 80.5I. 0 0

0.8

0

0.6

I:-ro

0.4

0.2

O~b 0'.11 ".0 I ~20',0 d,o '~O 1,2

u/u C.b U~8 1',0 1,20 0~6, 0',8 1:0 1~2

Fig.7 Comparison of calculated and experimental velocity profiles

in the entrance region of turbulent pipe flow. The symbols denote

the experimental data of Barbin & Jones 1963, and the solid lines

denote the numerical solutions of Cebeci & Chang 1977 using an

algebraic turbulence model.

i0.4 !

o.

0.2

Ap.

0.1

0 10 20 30 40 50 60 70 80 90

x

Fig.8 Comparison of calculated and experimental pressure dropin the entrance region of turbulent pipe flow. The symbols de-

note the experimental data of Barbin & Jones 1963, and the solid

line denote the numerical solution of Cebeci & Chang 1977 using

an algebraic turbulence model.

,

Ij

~

- 17-

in a pipe with uniform entry velocity are shown in Figs. 7 and

8.

The advantages of the algebraic models can be summarizedas follows:

* They produce good results for simpler shear flows (free

shear flows as well as external and inernal boundary lay-

ers).* They are easy to use (i.e. include in existing computer codes).

* The model parameters are fairly constant.

* They can provide starting values for iterative calculation

procedures.

The most important limitations of the algebraic models are

that: ,

* Self-preservation is assumed; i.e. the mean flow and the

turbulence should depend only on local conditions.* They produce inaccurate results for separating boundary layers.

* They are not capable of predicting the flow in recirculat

ing zones.

5. ONE - EQUATION MODELS

Later in this section it will be demonstrated that the

mixing-length models are implicitly based on the assumption

that the turbulence is in a state of local equilibrium. There-

fore, the mixing-length models, and other algebraic models,

are unable to account for the transport of characteristic

turbulence quantities.

For instance, in pipe flow the turbulence is produced

near the walls and diffused towards the axis. The mixing-length

model (22), neglecting this transport process, predicts zero

turbulence at the centerline (due to the vanishing mean velocity

I

i

- 18-.

gradient;. Nevertheless, reasonable results for the mean flow

can be obtained (Figures 7 and 8). The algebraic models further-

more neglect the convective transport of turbulence. Accord-

ingly, they are not capable of predicting the downstream decay

of turbulence generated by a grid.

In the present section, models which account for the

convective and diffusive transport of turbulence are considered.

While the zero-equation models considered in the preceding

section were represented by some algebraic expressions, the

one-equation models typically consist of ~ partial different-

ial equation like:

2! + u If- = DIFF + PROD - DISS (27)at j aXj

Here, f is a characteristic scalar property of the turbulence.

The left side of eq. (27) represents the rate of change of

f within a fluid element, while the terms on the right side

account for the various mechanisms which may contribute to

the change of f; i.e. diffusion, production and dissipation.

Equation (27) is thus a differential tran~port equation for

the fluid property f.

The thermal-energy equation for laminar flow of an in-

compressibel fluid

[ ae ae ] a2e pcp at + Uj ~ = K ~~ + ~~ (28)

J J J

is a well-knovn transport equation which exhibits essentially

the same form as equation (27). Like in the model equation

(27), the first term on the right represents diffusion (due

to molecular transport)' and the second term represents produ-

ction of internal energy or heat (due to viscous dissipation).

Notice that no dissipation term occurs in the energy equation

(28) .

- 19 -

5.1 Transport equation for k

The kinetic energy of the turbulent motion k defined

in equation (8) is a physically realizable (i.e. measurable)

quantity which is frequently considered as a dependent variable

in the field of turbulence modelling. From the Navier-Stokes

equations of motion the exact differential equation for q:

- - aua ( 1 2 a ( 1 2 i

~ t -2q ) + u.~ -2q ) = -u.u. ~ - 2vs. .s..0 JoX. 1 J oX. 1J 1J

J J (29)a CE. 1 2 a r 1 .

- ~[u. + -2q )] +~- L 2V u.s. .JoXi 1 p oXi - J 1.J

can be derived, provided that the density is constant. Using

the relation

_1 1-=zk = "2 uiui = "2 q (30)

the differential equation (29) can be rewritten in the quasi-

symbolic form:

ak ak- + U. - = D + P - e: (31)at J ax. k k

J

where

Dk =: -;?- [ u . (E. + ~ '11. )] + ':I ~ rL 2 v u. s . . J' ( 32 )oXi 1 p .. J, oXi . J 1.J

au.P = -'U-:-u-: ~ (33)k - 1. J ox.

J

e: =: 2vs. .s.. (34)

1.J 1.J

Exhibiting the same form as eq. (27), equation (31) is an

exact differential transport equation for the mean turbulent

kinetic energy k.

~;~';['.!i~,: "~:":';"

- 20 -

Having established the additional equation (31), one

should not be led to the conclusion that the closure problem

(i.e. closing the set of equations) has been solved. Unfortu-nately, the terms (32) - (34) involve unknown correlations

of fluctuating quantities, and should therefore be subject

to some approximations (modelling).

The production ~ .Pk represents the production of

turbulent kinetic energy by work done by the interaction of

the mean flow and the turbulent stresses. It can easily be

demonstrated that Pk is positive under typical shear flow

conditions. It should also be remembered that the same term,:

but. with opposite sign, appears in the transport equation)

for the kinetic energy of the mean flow. Thus, Pk represents

the rate of energy transfer from the mean motion to the turbu-

lence.

Since the unknown Reynolds stresses are involved in

the production term, Pk kan be modelled by using the general-

ized eddy-viscosity hypothesis (7), i.e.

( 2 ) au. Pk = 2VT Sij - 3 °ij k ~)

(35)au. 2 au.

J. J.= 2vT S. . -;;-- - -3 k -;;--J.) ox. ox.

) J.

The mean flow is incompressible and the final model for the

production term becomes

au.- 2 J.Pk - vT Sij ax-:- (36)

)

This is an exact expression for the production term in terms

of the eddy-viscosity and the mean flow field.

- 21 -

The diffusive ~ Dk represents the diffusion of turbu-

lent kinetic energy due to turbulent and molecular transport.

Integrating the diffusive term over a volume which completely

encloses the turbulence, application of Gauss' theorem reveals

that the integral of Dk exactly vanishes. Accordingly, this

term neither creates nor destroys turbulent kinetic energy,

but merely promotes a spatial redistribution of it.

The diffusive transport due to molecular action, i.e.

the latter term in eq. (32), is negligible for high local

Reynolds numbers, and is therefore neglected (except in the

immediate vicinity of solid boundaries where viscous effects

always become significant). The turbulent contribution to

the transport of k is assumed to be analogous to the turbulent

transport of heat and momentum. Thus, in analogy with eq.(14),

we assume that

~-~?:---~2. E ::-T:; a k- u. ;; + + Rj - u. - q1Rj 'Y k ~ (37 )J. -- P J.1.t aX.

J.

where the proportionality factor is the turbulent diffusivity

of k. Following the Reynolds analogy between different tur-

bulent transport processes (Section 3), the diffusivity des-

cribing the turbulent transport of k is linearly related to

the eddy-diffusivity. Thus, the dimensionless diffusion number

\IT0=-k Yk (38)

should be (approximately) constant across a flow. Note the

analogy between eq. (38) and (19).

With equations (37) - (38), the modelled expression

for the diffusive term becomes:

a ( \IT ak )D = - - -- (39)k ax. Gk ax.

J. J.

where the dimensionless diffusion number should be determined

from experimental data.

~

..' .

- 22 -

The dissipative ~ (34) represents the viscous dissi-

pation rate of turbulent kinetic energy. This term is always

positive, and energy is therefore always extracted from the

turbulence by the action of molecular (viscous) stresses.

Unlike the viscous diffusion, the viscous dissipation cannot

be neglected. The reason is that the latter term comprises

correlations of fluctuating velocity derivatives, and the

smallest scales of motion is characterized by large velocityderivatives. In fact, one may thought that the smallest length

scale adjusts itself so as to ensure sufficient dissipation

of energy.

It is assumed that energy is transferred from the larger

eddies down to the smallest eddies and then dissipated. Since

the viscous effects are negligible at the larger length scales,

the energy transfer from the large-eddy motion becomes inde-

pendent of the viscosity. The energy dissipation is therefore

solely determined by the length scale 1 and the velocity scale

u of the large scale turbulence. The velocity scale of the

larger eddies can be taken as:

u = k1/2 (40)

Thus, from dimensional analysis, the viscous dissipation (34)

can be estimated as:

U3 k3/2e:~_. ~C - (41)R. D L

where C D is an empirical constant.

A final form of the transport equation (31) for k becomes

~ + U ~ - a ( VT ak ) aui k3/2at j ax. - ~ Ok ax: + 2vT Sij ax-:- - CD L (42)

J J J J

- 23 - "

Here, the exact terms (32) - (34) have been replaced by the

modelled (approximate) terms (36), (39) and (41).The partial

differential equation (42) is a transport equation for the

turbulent kinetic energy (8). However, the use of the generali-

zed eddy viscosity hypothesis (7) introduces ~ unknown turbu-

lent quantities in the Reynolds averaged momentum equations:

the eddy viscosity and the turbulent kinetic energy. Therefore,

we have to establish a relation between these unknown quantities

which can used in connection with equation (42).

The first step in Prandtl's mixing-length hypotesis,

which ultimately led to eq. (22), is the assumption that the

eddy viscosity is proportional to a length scale and a velo-

city scale. An example of a model of this form is the simple

constant eddy-viscosity model (21) for free shear layers.

In his mixing-length approach Prandtl related the velocity

scale to the gradient of the mean velocity field. It can be

anticipated, however, that an improved expression for the

eddy viscosity is obtained if the velocity scale is taken

as a characteristic velocity for the turbulence rather than

for the mean flow.

Prandtl (in 1945) and Kolmogorov (in 1942) independ-

ently suggested that the velocity scale should be proportional

to the square root of the turbulent kinetic energy. Thus,

the Prandtl-Kolmogorov relation between the eddy-viscosity

and the turbulent kinetic energy becomes:

vT = c~ v'k L (43)

In this expression the eddy viscosity is related not only

to the mean flow, like in section 4, but it has also been

associated with the turbulence via k.

Now, the modelled transport equation (42) for k and

the relation (43) still involve the unspecified length scale

L. While k is obtained from the solution of the PDE (42),

--;-7 ;~,~~!:~~~

- 24 -

,

L is taken from an empirical algebraic expression. L, being

a meassure of the size of the larger turbulent eddies, can

be specified in a similar manner as the mixing-length in section

4.2. Thus, the accuracy of the complete model depends on the

quality of the selected expression for L.

The model equations (42), (43) furthermore involve three

dimensionless parameters, which should be determined from

available experimental information. In order to have a useful

turbulence model, it is required that these parameters are

approximately constant within a particular flow field, and

that they do not change from one flow situation to another.

Thus', it is desirable that the model parameters can be con-

sidered as being nearly universal constants.

Finally, in order to solve the PDE (42) boundary condi-

tions for k should be imposed. Outside a shear layer (a wall

boundary layer or a free shear layer) the turbulent kinetic

energy should vanish, i.e. k=O. At a line of symmetry (like

the axis in a pipe) the cross-stream variation of k should

vanish. Near solid boundaries the turbulent fluctuations are

damped so that k=O at the walls. However, we have neglected

the viscous part of the diffusion term (32), and the modelled

equation (42) cannot apply straight to the wall. Accordingly,

a condition on k is imposed at a certain position in the inner

part of the logarithmic region. The relevant value of k is

obtained by assuming that the production term is approximately

equal to the dissipation at this position. A detailed discussion

on boundary conditions for turbulent transport equations requi-

res some knowledge about computational fluid dynamics, and

is beyond the scope of this lectures.

In the case of thin shear layer flow a special case

of the k-equation (42) can be considered. In the near wall

region both the convective and diffusive transport of k are

negligible. Thus, the production term balances the dissipation.

I

- 25 - :

With this particular simplification the length scale L becomes: '.

L = £ (CD/C~) l/~ (44)

i.e. L is proportional to the mixing-length. This observation

demonstrates that the mixing-length models are applicable'

only when the turbulence is in a state of local equilibrium,

i.e. when the production and dissipation of turbulence energy

are in balance.

5.2 Transport equation for the eddy viscosity

A differential transport equation for the eddy viscosity

was proposed by Nee and Kovasznay in 1969. Like the k-mode1

(42), the Nee & Kovasznay model r<quires the specification

of a length scale via an emplrical expression.

Reasonable agreement with experimental results has been

obtained with the resulting model. Nevertheless, the model

has received only modest attention. This is probably because

its dependent variable is a non-physical quantity. While,

for instance, the mean turbulent kinetic energy and the Reynolds

stresses can be measured directly, the eddy-viscosity can

only be derived from experimental data via its "definition"

( 7 ) .

5.3 Transport equation for the shear stress

In 1967 Bradshaw, Ferriss and Atwell developed a one-

equation model which does not employ the generalized eddy

viscosity concept. While the eddy viscosity hypothesis (7)

relates the turbulent shear stresses to local mean flow condi-

tions only, the "structural" assumption

:;;;';;',ic;,:i::r;:~~:~~~~~

. . - 26 -

. '.

T.. = - p~ = apk ilj (45)1.J 1. J

relates the shear stresses to the turbulent kinetic energy

(i.e. the turbulent normal stresses). The assumption (45)

has been used earlier by Nevzglijadov (in 1945) and Dryden

(in 1948), and experimental investigations of wall boundary

layers suggest that the dimensionless constant a is approxi-

mately equal to 0.3.

The differential equation (31) with the right hand side

terms (32)-(34) is an exact transport equation for the mean

turbulent kinetic energy k. Bradshaw and his co-workers used

the "structural" assumption (45) and converted equation (31)into a transport equation for the dominating shear stress

component in wall boundary layers.

Since the shear Reynolds stress becomes a dependent

variable in this formulation, the production term should be

kept in its exact form (33), while the dissipative term (34)

is modelled according to the converted form of equation (41).

Bradshaw and co-workers did not employ the Reynolds analogy

for the diffusive term, but assumed that the diffusion of

the shear stress is proportional to a characteristic velocity

for the large eddies in the outer part of the wall boundary

layer.

The resulting model equation for the transport of the

shear stress involves two empirical functions. The dimension-

less "constant" a in the structural assumption (45) may also

be considered a function of the distance from the wall.

The Bradshaw, Ferris & Atwell model has received consider-

able attention, and is perhaps the most popular one-equation

model. It has been applied to a variety of thin-shear-layer

flows (e.g. steady and unsteady wall boundary layers and free

shear layers), and the agreement with experimental data is

generally quite good.

'.; ~.t~~~,~ -,

- 27 - :

~

Nevertheless, a severe shortcoming of this model is

the limited validity of the "structural" assumption (45).The

turbulent kinetic energy cannot attain negative values. Accor-

dingly, the shear Reynolds stress has to be positive throughout

the flow, and the assumption (45) is not directly applicableto flow problems in which the shear stress changes sign. Fur- ~

thermore, on a line of symmetry like, for instance, on thei

axis of a pipe or along the centerline of a circular jet,

the turbulent shear stress should vanish. The assumption (45)

then implies that k=O, while it has been experimentally verified

that k does not vanish at the line of s~etry.

5.4 Characteristic features

The one-equation models are the first step in the devel-

opment of transport equations for characteristic turbulence

quantities. Unlike the zero-equation models the one-equation

models account for diffusive and convective transport mecha-

nisms, and the latter models are therefore superior to the

former when this transport cannot be neglected.

The main deficiencies of the one-equations models are

that:

* The specification of a length scale is required to complete

the model. No universal prescription is available. The

length scale itself may under certain conditions be subject

to transport processes, and cannot be obtained from local

conditions only.

* The one-equation models require the solution of an addi-

tional PDE. Somewhat more computational effort should

be expected, but this is not a serious limitation today.

* Only modest attention has been paid to the one-equation

models as compared with algebraic models and two-equation

models.

"'

- 28 -

!~,. j

\"i

'1, ~

~Iculatlons and ~i.~x~rlm~ts ot !]

~pu_6z/IJ.' 12400 '~

c,~

';.;i"J0 ' !~-y/6a~ :j

'~~".,,

'"e

;l~ ,;,

1.

E. ~~ and"- calculation sc~~

dillef"

0 y/Oa .

1.

0 .,/00",

Fig. 9 Calculations by Bradshaw, Ferriss and Atwell (1967)for some different equilibrium boundary layers. (Figurestaken from Markatos 1986)

a) Flow past a flat plate

b) Equilibrium boundary layer with UE ~ x-0.15

c) Equilibrium boundary layer in favourable pressure gradient.°l/Twall. dp/dx = -0.35

- 29 -.

6. TWO-EQUATION MODELS

In many situations the length scale L (characterizing the

larger eddies) is not only dependent on local flow cond-

itions. Considering, for instance, the turbulent flow down-

stream of a grid. The length scale of the turbulence,

which is generated by the grid, is mainly determined by

the dimensions of the grid itself, but is nevertheless

convected downstream along with the fluid. We may therefore

anticipate that a more realistic turbulence model can be

obtained if the length scale L is calculated from a transport

equation (PDE) rather than from some (local) algebraic

expression. Alternatively, a transport equation for the

product kmLn may be considered, where k is still the mean

kinetic energy of the turbulence.

In order to obtain a dependent variable which really re-

flects the physics of the turbulence, several combinations

of k and L have been considered over the years. Some of

the more important proposals are listed below:

Authors Year m n

-

Kolmogorov 1942 ~ -1

Rotta 1951 0 I

Rodi & Spalding 1970 1 1

Rotta 1971 1 1

Ng & Spalding 1972 1 1

Saffman 1970 1 -2

Spalding 1971 1 -2

Davidov 1961 3/2 -1

Harlow & Nakayama 1967 3/2 -1

Hanja1ic & Launder 1972 3/2 -1

Launder & Spalding 1974 3/2 -1

Tabe11 Suggested transport equations for kmLn.

.

- 30 -." -

Some of the combinations of k and L has an obvious physical

interpretation. While the combination considered by Kolmogorov(m = ~ and n = -I) represents a turbulent or eddy-frequence,

the combination with m = 1 and n = -2 represents turbulent

vorticity. The far most popular combination today is m= 3/2 and n = -1, by which the dependent variable is identi-

fied as the dissipation rate of turbulent energy.

6.1 The k-£ model of turbulence

The famous k-£ model is a two-equation model for turbulence,

in which both k and £ are governed by partial differential

equations. Instead of modelling the viscous dissipation

term (34) in the transport equation (3l) for k, the dissi-

pation is obtained from a new transport equation which

exhibits essentially the same form as eq. (3l). The exact

transport equation for £ was derived by Davidov in 1961

and Harlow and Nakayama in 1967. However, the exact terms

on the right-hand side, which contribute to the change

of £ for a fluid element, must be subject to crude modelling.

The derivation of the exact form of the £-equaiton will

therefore be omitted in this text, and only its modelled

form will be presented.

It is important to emphasize that the k-£ model is based

on the generalized eddy-viscosity hypothesis (7), which

relates the unknown Reynolds stresses to the mean flow

field via the eddy viscosity. The momentum equations for

the mean flow (13), which are to be solved along with the

turbulence model, are strongly influenced by the eddy viscos-

ity. Therefore, in order to close the system of equations,

an algebraic expression which relates the eddy viscosity

to k and £ is required.

-- 31 -

~

The Prandtl-Kolmogorov relation (43) between the eddy-vis-

cosity and the turbulent kinetic energy is expressed in

terms of the unknown length scale L. However, L can be

eliminated by invoking eq. (41) which arised from simple

dimensional analysis, and the eddy-viscosity can thus be

expressed as:

2v = c /k L = (c c )k IE: (46)T ~ ~ D

where the product of the constants becomes a new dimension-

less constant. The most common form of the k-E: model thus

become:v~ + u.~ = -.:?-(-!~) + p - E: (47)

at J ax. ax. Ok ax. kJ J J

, vaE: aE: a T aE: E: ) ( )at + Uj~ = ax:(cr-~) + k(cE:1Pk - CE:2E: 48

J J E: J

2vT = c~ k IE: (49)

au.1Pk = 2vT S.. ~ (50)

1) ox.J

and the five empirical constants in the model are

c' = c CD = 0.09 (51 a)~ ~

ok = 1.0 °E: = 1.3 (51b)

, :, "

cE1 = 1.44 cE:2 = 1.92 (51c)

The two-equation model (47-51) is the standard version

of the k-E model for high Reynolds number turbulence (Launder

and Spalding 1974). Equation (51) defines the standard

values of the 5 empirical parameters in the k-E: model.

These parameters are constants in the sence that they are

not changed in anyone calculation. It is furthermore

;;

- 32 -." "

to be anticipated that they will not change much from one

flow configuration to another. Modifications of the "cons-

tants" (51) have been proposed in order to account for

low Reynolds number turbulence and such effects as curvature,

rotation and swirl.

The effective Prandtl numbers (51b) are estimated by computer

optimization, while the empirical parameters (51a) and

(51c) are deduced from empirical knowledge about some simple

turbulent flows; i.e. from decay of grid turbulence and

from equilibrium shear layers.

For example, convective transport and diffusion can be

neglected in equilibrium shear layers. Accordingly, the

production of turbulent kinetic energy approximately balances

the dissipation. While the production is given by eq.

(50), the dissipation E can be obtained from eq. (49):

au 2 k2\i (-) = c' - (52 )T ay ~ \iT

Elimination of the eddy viscosity by using eq. (4) gives-2

, ( uv)c~ = k (53)

where both the Reynolds shear stress uv and the turbulent

kinetic energy k are easily measured.

The turbulence model (47-51) together with the continuity

equation and the momentum equation (13) form a closed set

of governing equations which can be solved numerically.

However, in order to solve the model equations we need

a set of boundary conditions. Close to a wall the local

Reynolds number is small and molecular transport mechanism

becomes important. In the standard k-E model, however,'

viscous effects have been neglected and it has become standard

practice to impose boundary conditions at some point near

the wall, but still far enough from it for the direct effects

of viscosity to be negligible. If the dependent variables

.

.:.

- 33 -$

;

in the first computational point near the wall are denoted

by a subscript P, the resulting boundary conditions are:

u u.yUp = ~ R.n(-.:;?!) (54a)

Vp = 0 (54b)

i- 2k = IC' . u (54c)p J.l .

3e:p = U*/KYp (54d)

where Yp is the distance from the wall and u. is the friction

velocity. The boundary conditions (54) are derived by

assuming that the grid point P is located in the fully

turbulent log-law region, where the generation and dissi-

pation of turbulent energy are equal, and where the length

scale is proportional to the distance from the wall.

6.2 Numerical examples

In order to demonstrate the capabilities of the k~ model,

numerical results for some different flow configurations

are presented in this section. First we consider the calcul-

ations by Hackman, Raithby and Strong (1984) for turbulent

flow over a backward-facing step, as shown schematically

in Figure 10. Predicted mean velocity profiles for a laminar

and a turbulent case are shown in Figs. 11 and 12, respectiv-

ely. The predictions are compared with measured data of

Denham and Patric (1974) in the laminar case and experimental

findings of Kim (1978) in the turbulent case. The reattach-

ment length is about 9 step heigths in the laminar case,

and about 7h in the turbulent case. The predicted variation

of the turbulent kinetic energy k and the shear stress

component -uv are shown in Figs. 13 and 14, respectively,

while the variation of the pressure coefficient is shown

in Fig. 15.

~

..

.- 3~ -. ..., AVERAGE INrl;.OW PARAMETERS rOR

VELOCITY IS U KIM'S PROBLEM

H . 0.0762 m

h'0.03BlmH

[ y' 0.1524m - I LZ' 2.338Bm

: --- p . 1.88553 ag/m)I _5

~ . 1.836981 10 ag/m I

h :::::""'- U . 11.B mI.

L I X ,XR I~LI -!. LZ ~

Fig. 10 Backward-facing step flow configuration.

- -- ... 0. !

I

I

2

.c">-

I

0O. '"

U/O

Fig. 11 Mean velocity profiles. Laminar case.

30 KIM

- CARTESIAN- \SHUDS 1421421

\--- CURVILINEAR- IUWDS (421421 ,

2 IIX/h = 1.33 2.67 5.3310 6.22 0 7.11 8.00

.c 0" 0>-

U/U

Fig. 12 Mean velocity profiles. Turbulent case.

~

3 3~---'--- -- ---, '-

I : ,

c. .~r'v""-- Cll~VI..lr;r/.RSHUDS2

2.33 5.89 6.78

0 0.C;

>-

I0 . 0 0.025 . .

k/U2

Fig. 13 Turbulent kinetic energy profiles.

2.33 .~ ~;~~~"':~B9 -- =-cl.C;"->-

. . .

~/U2

Fig. 14 Reynolds shear stress profiles.

o. 0 KIM

-CARTESIAN -SHUDS~ - - KWON a PLETCHERz1../U O.""-"" Z1../ -0 ~ p

u uc. 0. Cp . .-=r, -puc. 2

I../u1%:-:> ."""""""""",'/In -In O. -~ ~a. "L~"" ~ """""

xO. 12

X /h

Fig. 15 Variation of pressure coefficient along the lowerwall downstream of the step.

~-~~-~ ~c",

- 36 -

While the predictions for laminar flow compares favourably

with the experimental data in Fig. 11, some important discre-

pancies are observed in the turbulent case. First, the

calculations underpredict the reattachment length by about

one step height h, depending on the discretization scheme

used for the convective terms. The pred~cted variation

of the pressure coefficient in Fig. 15 is therefore shifted

to the left as compared with the experimental data. Moreover,

the experiments show that the turbulent kinetic energy,

which is generated just downstream of the step corner,

spreads more effectively upwards than indicated by the

predictions in Fig. 13. Finally, the turbulent shear stress

is significantly overpredicted by the turbulence model,

as shown in Fig. 14.

The turbulent flow through a sudden pipe expansion is similar

to the backward-facing step test case, but is obviously

of greater practical relevance. Fig. 16, which together

with Figs. 17-20 have been taken from Rodi's 1980 review,

compares numerical results obtained with ,a one-equation

model and the standard k-E model with experimental data.

It is observed that the k-E model predictions are generally

in better agreement with experimental data than are the

one-equation results.

The k-E model calculations by Durst and Rastogi in Fig.

17 show that the predicted mean velocity profile downstream

of the square obstacle slightly deviates from the measured

velocity profiles. The predictions are otherwise quite

reasonably, and reproduce the two small recirculation zones

in front and top of the block, as well as the large recir-

culation region downstream of the obstacle.

Another example, calculated by Gosman and Young, is the

turbulent flow in a square cavity. Fig. 18 shows the U

- 37 - -~ x 0.5 U/UtT 0 ~~~~"~"O T ' r.'Ts~~ hB:6B~;;;"1;;-~~ '

1EQN.MODEL Ro 6.6 6>. . 0 $ ~Ro 0 PITOT , / ~' " "

6 HOT WIRE /0 0; I" f-1

;::~p"O ~ ,,~ ,,'LiO:4 I~~-o ~~ ~ :6 - , ,,~(f- ,"~60 ,,' ~

0" ,/ ,',' l:,.. 0 ,'f I

0' 0 " 6-0 I A ,0 .0 ' , , ~,. , , 0 , ,n " ,'" , ' ,-

Fig. 16 Mean velocity profiles in sudden pipe expansion.

0 Data=~~=::::~,~~~~~~==!:=:!::::o - C Q I cui a t ion0

0 o i ::::::::::::;;;~~;;;;;;;;;;~;;;;;~;~== c-===-

-2 -1 0 1 2 3 I. 5 6 7 B 9 10) I" x IH

0 stream Ines

y0,

Q

Q

0-3 b) velocity profiles ';iH

Fig. 17 Streamlines and mean velocity profiles in channelobstructed by a square block.

. 0.5

0 0

0 0 V-:11Jr U~ U~ 0

d~

-0.5 0U/U~

0.5 0 0.5 1.0

Fig. 18 Velocity profiles in a siuare cavity.

.

- 38 -

and V velocity profiles predicted along a vertical and

horizontal line, respectively. It can be observed that

the model fails to predict the flow behaviour in the region

where the high-velocity fluid meets the righthand vertical

cavity-wall. Nevertheless, the main features of the flow

are reasonably reproduced.

Finally, we consider some free shear flows, or more speci-

fically, a plane and an axisymmetric wake. Fig. 19 shows

that the k-E model describes the plane wake behind a flat

plate in an excellent manner (also in the upstream region

where the flow is not selfpreserving). In the axisymmetric

wake flow in Figure 20, the ratio of production of turbulent

kinetic energy Pk to its dissipation E is quite low

(~0.15). It is observed that the standard k-E model,

as well as a one-equation k-model and an algebraic mixing-

length model, fail considerably to reproduce the flow develop-

ment. However, if the "constant" c~ in the k-E model is

given as a function of Pk/E, the predictions become more

resonable.

,..-i ~

,

-~C~

r. -~

- 39 - -; .£

. . ~ "

"

- I

- - Predictions Roo. 1'+"2

0 ExperIments- Chevray and Kovasznay (151)

-1

0 xlcmJ 150- O.2.L ,6 11 U/Ue

L

5

x:O x:20cm x:50cm x=150cm7

Initial profileat trollingedge of plate

y(cml

.Fig. 19 Velocity profiles in a plane wake behind a flatplate.

120

( Uo)-3/2 k£model, 8 Expts [152] ~- c =O.09~ ' UE ~ ~80 k£model 8 8

-' 8c~=f(P/E) ---

. . ~ 8 mIxing length -4 . hypothesIs 8 ! - 1- Eq~~-=-

-8_8 - -~:--.-=~ -:- .

0 8 16 40 48 56 64

x/D

Fig. 20 Streamwise variation of maximum velocity defect. in an axisymmetric wake.

- ~~ - - 40 -,

_6.3 Advantaqes and disadvantages (M~katos 1986)

- tt The Kolmogorov energy-frequency model represents a major

advance in turbulence modelling; for it permits the length

scale to be predicted rathef than presumed. This is true also

for all other models that followed Kolmogorov's.

4t The kw model has been successfully employed for the prediction

of- numerous turbulent flows. The chief disadvantage was that,

in order to fit the logarithmic wall law, one constant must

vary with distance from the wall. Because of this, preference

was given to other second-equation variables. This disadvan-

tage has now been removed and the model deserves more attention

than it has received to-date.

4t The k-kl model has been extensively and successfully employed

for the prediction of free turbulent flows and those near

walls. Its main defect is the necessity to introduce addit-

ional quantities to handle the region close to the wall. For

this reason it is not used for elliptic flows, where walls

often play an important part.

~ The k-c model is the currently most popular two-equation model,

mainly for two reasons: La) the c equation may be derived

from the NS equations (but this is also true of the wequation)

(b) the "Prandtl II number for e: has a reasonable value which

fits the experimental data for the spread of the various

entities at locations far from walls. The model still requires

modifications for various effects ~uch as buoyancy, rapid

compression, etc) and still requires further validation in

elliptic 2D and 3D flows.* There is no reason to suppose that,

if an equal amount of attention were given to it, another

two-equation model would not perform as well, or even better.

*It is pointed out that care must be taken to sort out discrepancies

between prediction and experiment that arise separately from turbulence

modelling and from numerical schemes. Therefore, for these flows

turbulence-modelling research is likely to be couple'd with researchdirected towards developing more accurate numerical prediction schemes.

..,

- 41 -

.4t In general,predictions of current transport models (k-~, k-w)

agree fairly well with experimental data for:

. 2D boundary layers and jets along plane walls;

~ . 2D jets, wakes, mixing layers, plumes; ~

. 2D flow in tubes, channels, diffusers and annuli;

. many 3D flows without strong swirl, density variations

or chemical reaction;

. some flows influenced by buoyan.cy and 10w-Reynolds-

number effects.

Ad hoc corrections must be made to the models or to the

"constants" in order to procure agreement with experiments on:-

. boundary layers on convex and concave walls;

. strongly swirling and recirculating flows;

. axi-symmetrical jets in stagnant surroundings;

. 3D wall jets;

. gravity-stratified flows;

. flows involving chemical reactions;

. two-phase flows.

.. Current transport models neglect intermittency (i.e. the

ragged edges of jets and boundary layers), periodicity (i.e.

the eddy-shedding propensity of wakes) and postulate, in

general, gradient-induced diffusion, whereas other diffusion

mechanisms exist. Furthermore, ~e absence of a direct means

of establishing the constants delays progress.

The above disadvantages lead researchers towards more "physical"

models like large-eddy simulation and "two-fluid" models.

.;- "!'i¥':;i;;;~";!

,.

- 42 -

7. REYNOLDS STRESS MODELS

While the algebraic models are based on the assumption of

turbulence in equilibrium, the one- and two-equation models

allow for transport of Qllg turbulent velocity scale. But

models including the turbulent kinetic energy k ~e i~i~ly, fl 1 , ,2 22based on the assumptl.on 0 oca l.sotro~~, l..e. ul = u2 = u3.

In real flows, however, the turbulent energy is usually

produced in one normal stress component, and subsequently

transferred into the other normal stress components.

Anisotropic turbulence can be modelled by separate transportequations for the individual Reynolds-stress components:

r 2 - _1-UI -UlU2 -UlU3

- 1 - 2 _I-UiUj = -U2UI -U2 -U2U3

l - - -:-::T

J-U3UI -U3U2 -U3

therby allowing for different velocity scales for the various

stress components.

Keller and Friedmann suggested in 1924 that the Reynolds

stresses -~. could be obtained from trans port equations,1 Jwhile Chou was the first to derive and publish (in 1945) exact

transport equations for these stress components. The following

procedure may be used:

1) Consider the xi-component of the Navier-Stokes equations.

2) Subtract the xi-component of the Reynolds-averaged N-S.

3) Multiply by Uj.

4) Change indices i and j in the equation from step 3).

5) Add results from step 3) and 4).

6) The result is then Reynolds-averaged, and can be written

as:

."-

- 43 -, .'"

D U"-:u.---1--1 = p.. + <1>.. + D.. - E..

Dt 1J 1J 1J 1J

where the left-hand-side represents change of ~ for a fluid

element. The terms on the right-hand-side are:

aU. au.= --:.L - 1Pij - - Uiuk ax - UjUk -a;:-k k

* production of ~ due to mean strain1 J

P au. au.<1>.. ::: -(--1 + -::l.)

1J P ax. ax.J 1

* pressure-strain correlation

- a 1 -r 1 -rD.. = - -[U.U.U k + - pU.u.. + - Pu.u. kJ1J ax. 1 J P J 1J P 1 J

J

* diffusive term

au. au.E .. ::: 2v --1 --.:::.l

1J ax axk. k.

* dissipative term. The spatial derviatives of the fluctuating

velocity components are so large that the dissipation cannot

be neglected.

It can be demonstrated that the transport equation for

the Reynolds stresses contracts (~.e. summation of the

equations for the 3 normal stresses) to the exact transport

equation for the mean turbulent kinetic energy. In this case

the contributions from the pressure-strain correlations

disappear. These terms do not change the turbulent kinetic

energy, but tend to redistribute energy between the different

normal stress components. Note that <1>11' <1>22' and <1>33 all can benonzero, while the sum <1>.. = o.

11

The production terms Pij are made up of Reynolds stress

components and gradients of the mean velocity field. No

...

- 44 -

approximations are required for these important terms at thislevel of modelling. The other right-hand-side terms, however,

must be subject to modelling, and a set of partial

differential equations are obtained. This kind of turbulence

models is called Differential Stress Models (DSM). In a

general 3-dimensional case, 6 PDE's for the Reynolds stresses ~

are solved together with a PDE for the energy dissipation

rate.

A somewhat simpler class of Reynolds stress models, which

still retains some of the important features of the DSM-

models, is the Algebraic Stress Models (ASM). In this

approach, alternative modelling assumptions are used for all

terms in the DSM model which involve spatial derivatives of

the Reynolds stresses. Thus, an algebraic set of relationships

between the stresses can be derived, which subsequently is

used together with PDE's for the mean turbulent kinetic energy

(k) and its dissipation rate (E).

The accompanying figure shows numerical calculations by

Sultanian, Neitzel & Metzer (1985) of the flow through an

axisymmetric expansion (i.e. a pipe with sudden increase in

diameter). Streamwise mean velocity profiles are shown at

different cross-sections downstream of the expansion for the

case Re=60000, and it is observed that the results obtained

with an ASM model (solid line) compare more favourably with

the experimental points than the calculations (broken lines)

with the standard k-E model.

The Reynolds stress models, DSM and ASM, are today the

most complete and accurate representation of turbulence which

are used in technological applications. Their major advantage

is the exact evaluation of the crucial production terms. The

most general models can be gradually reduced to decrease the

computational time without significantly loss of accuracy.

ASMs, for instance, represent a powerful compromise between

full DSMs and the k-E model in complexity and applicability.

~:"-?;:';,c,;,;,,-::;r:i~i!~]7"'"

,- 45 - - ...

- " ;!

1.O.

4x/D1 = 2 x/D1 = 3 x/D1 =O.

O.

O.

r/R2 O.

O.

O.

O.

O.

0 . .U/U,

1.

O. (U1 = 1.13m/s)

O. x/D1 = 6 x/D, = 8 x/D1 = 10 x/D1 = 12

O.

O.

r/R2 O.

O.

0

0

0..

. 1.0 0 0

U/U,

Fig. 21 Mean axial velocity profiles downstream of an

axisymmetric expansion.