Reynolds Stress ARSM

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Helsinki University of Technology. Laboratory of Aerodynamics.

Series B

Teknillinen korkeakoulu. Aerodynamiikan laboratorio.

Sarja B

Espoo 2001, FINLAND

IMPLEMENTING AN EXPLICIT ALGEBRAIC REYNOLDS STRESS

MODEL INTO THE THREE-DIMENSIONAL FINFLO FLOW SOLVER

Report B-52

Ville Hmlinen

TEKNILLINEN KORKEAKOULU

TEKNISKA HGSKOLAN HELSINKI UNIVERSITY OF TECHNOLOGY


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Helsinki University of Technology. Laboratory of Aerodynamics.

Series B

Teknillinen korkeakoulu. Aerodynamiikan laboratorio.

Sarja B

Espoo 2001, FINLAND

IMPLEMENTING AN EXPLICIT ALGEBRAIC REYNOLDS STRESS

MODEL INTO THE THREE-DIMENSIONAL FINFLO FLOW SOLVER

Report B-52

Ville Hmlinen

Approved by Seppo Laine

Helsinki University of Technology

Department of Mechanical Engineering

Laboratory of Aerodynamics

Teknillinen korkeakoulu

Konetekniikan osastoAerodynamiikan laboratorio


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Distribution:Helsinki University of TechnologyLaboratory of AerodynamicsP.O.Box 4400FIN-02015 HUTTel. +358-9-451 3421Fax +358-9-451 3418

Ville Hmlinen

ISBN 951-22-5605-3ISBN 951-22-6824-8 (PDF)ISSN 1456-6990

Printed in OtamediaEspoo 2001, FINLAND


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iii

Abstract

This thesis describes the implementing of an explicit algebraic Reynolds

stress model (EARSM) into the three-dimensional FINFLO flow solver. The

EARSM replaces the Boussinesq eddy-viscosity assumption by a more

general constitutive relation for the second-order correlation in the Rey-

nolds averaged Navier-Stokes equations.

The thesis begins by describing the Navier-Stokes equations, in both their

time-accurate and Reynolds averaged form. Some general aspects of tur-

bulence are also discussed and a differential Reynolds stress model is pre-

sented. Then an algebraic Reynolds stress model (ARSM) is formulated

and simplified to its explicit form. After that some programming aspects of

the model are discussed. Finally, four test cases are selected, calculated

and analysed to verify the correct implementation of the model.

As the first test case a two-dimensional boundary layer over a flat plate is

calculated to verify the connection between the EARSM and the rest of the

FINFLO. The second order effects of the model are studied with the secondtest case, a fully developed flow inside a rectangular duct. Thirdly, a fully

developed flow inside a rotating pipe is calculated to examine the third or-

der effects of the model. As the last test case an axial flow over a cylinder,

part of which rotates, is calculated to further study the behaviour of the

EARSM when a sudden strain component is applied to the flow field. From

the results of the test cases it is then concluded that the EARSM has been

correctly implemented. The model gives significant improvements in most

cases compared to standard eddy-viscosity models without a large in-

crease in the computational effort.


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v

Contents

Abstract .................................................................................................... iii

Contents .................................................................................................... v

Nomenclature.......................................................................................... vii

1 Introduction......................................................................................1

2 Governing equations and eddy-viscosity turbulence models ......3

2.1 Navier-Stokes equations ........................................................3

2.2 Reynolds averaging................................................................5

2.3 General facts of turbulence.....................................................62.4 Eddy-viscosity turbulence models...........................................8

2.4.1 Algebraic models.........................................................9

2.4.2 One-equation models................................................10

2.4.3 Two-equation models................................................11

3 Differential Reynolds stress models ............................................15

3.1 Reasons for using the Reynolds stress models ....................15

3.2 Exact equation for the Reynolds stress tensor......................16

3.3 Modelling the Reynolds stress tensor ...................................18

4 Formulation of an explicit algebraic Reynolds stress model .....21

4.1 Algebraic Reynolds stress model..........................................21

4.2 Explicit algebraic Reynolds stress model..............................26

4.3 Simplified EARSM for two-dimensional mean flow................29

4.4 Simplified EARSM for three-dimensional mean flow.............31

4.5 Solution of the general quasi-linear ARSM equation.............33

5 Programming approach and written subroutines........................37

5.1 Subroutine EARSM1 ............................................................37

5.2 Subroutine TTIMES..............................................................41

5.3 Subroutine ANISFLUX..........................................................42

5.4 Subroutine FLUXP................................................................43

5.4 Subroutine PEEKOO............................................................43

5.5 Subroutine VELGRAD ..........................................................44

5.6 Boundary routines ................................................................44


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Contents

vi

6 Test cases ......................................................................................47

6.1 Flow over a flat plate.............................................................48

6.1.1 Grid ...........................................................................48

6.1.2 Boundary conditions..................................................50

6.1.3 Results ......................................................................51

6.2 Flow inside a rectangular duct ..............................................55

6.2.1 Grid ...........................................................................55


6.2.3 Results ......................................................................57

6.3 Flow inside a rotating pipe ....................................................63

6.3.1 Grid ...........................................................................64

6.3.2 Boundary conditions..................................................656.3.3 Results ......................................................................65

6.4 Flow over a rotating cylinder .................................................69

6.4.1 Grid ...........................................................................70


6.4.3 Results ......................................................................71

7 Summary and conclusions............................................................79

Bibliography ............................................................................................81

Appendix A, Subroutine EARSM1..........................................................87

Appendix B, Subroutine TTIMES ...........................................................91

Appendix C, Subroutine ANISFLUX.......................................................93

Appendix D, Subroutine FLUXP .............................................................95

Appendix E, Subroutine PEEKOO .........................................................97

Appendix F, Subroutine VELGRAD........................................................99


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vii

Nomenclature

Roman alphabet

A Model coefficient

ija Reynolds stress anisotropy tensor

C Model coefficient

fC Skin friction coefficient( )aijC Coriolis tensoreffC Effective eddy-viscosity coefficient

E Total energye Specific internal energy

if Body force vector

H Matrix defined by the equation (4.44)

J Matrix defined by the equation (4.45)

k Turbulent kinetic energy, Heat transfer coefficient

l Turbulent length scale

N Symbol defined bytheequation(4.25), Non-dimensional rotation rate

in Cell face unit normal vector

P Production of turbulence

ijP Production of turbulence tensor

p Pressure

iq Heat flux vector

Re Reynolds number

S Cell face area

ijS Mean strain tensor

T Temperature

t Time

iu Velocity vector

u Friction velocity

u Fluctuating component of the u

V Cell volume

ix Cartesian coordinate vector

+y Dimensionless wall distance


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Nomenclature

viii

Greek alphabet

Coefficient for the terms of the ija

ij Kroneckers delta

Dissipation rate

ij Dissipation rate tensor

ijk Permutation tensor

ij Redistribution tensor

Molecular viscosity

Density

Turbulent time-scale

w Wall shear stress

ij Viscous stress tensorT Kinematic eddy-viscosity

ij Mean vorticity tensor

*ij Absolute mean vorticity tensor

Rij Effective mean vorticity tensor

Sij System rotation tensor

kspecific dissipation rateSk Constant system rotation rate vector

Invariants

SII Second invariant of the ijS

II Second invariant of the ij

III Third invariant of the ijS

IV Third invariant defined as kijkijS

V Fourth invariant defined as likljkijSS


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1

Chapter 1

Introduction

In earlier days fluid dynamics, like other physical sciences, was divided into

theoretical and experimental branches. The equipment and vehicles in-

volving fluid flow were designed and analysed by these two methods. With

the evolution of the digital computer, a third method called ComputationalFluid Dynamics (CFD) has become available. In this computational ap-

proach the equations that govern a process of interest are solved numeri-

cally at certain discrete points of space.

The evolution of numerical methods for solving ordinary and partial differ-

ential equations began approximately at the beginning of the twentieth

century. The automatic digital computer was invented in the early 1940s

and was used from nearly the beginning to solve problems in fluid dynam-

ics. The explosion in computational activity did not, however, begin until the

high-speed digital computers began to be generally available in the 1960s

(Ref. [1]).

The three key elements of CFD are algorithm development, grid generation

and turbulence modelling. In this study only the third element is scrutinised.

Turbulence is inherently three-dimensional and time dependent, and an

enormous amount of information is thus required to completely describe a


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Introduction

2

turbulent flow. This is beyond the capability of the existing computers for

virtually all practical flows. Thus, some kind of approximate and statistical

method, called a turbulence model, is needed.

Helsinki University of Technology has quite a long history in the field of

CFD. The development of the parallel multi-block Navier-Stokes flow solver

called FINFLO was initiated in 1987. The original two-man development

group has grown along with new applications. Nowadays, in 1995 estab-

lished CFD group consists of about 15 researchers from the Laboratory of

Aerodynamics, the Laboratory of Applied Thermodynamics and the Labo-

ratory of Ship Hydrodynamics, all utilising the FINFLO code. The FINFLO is

nowadays suitable for compressible, time dependent, laminar and turbulent

flows and has been applied to dozens of demanding research and devel-

opment projects. The code is able to handle structured multi-block grids

and the equations are solved by an implicit pseudo-time integration scheme

using Roes flux splitting.

The CFD-group has always tried to keep up with the most recent turbu-

lence modelling. Prior to this work the FINFLO included the following tur-

bulence models; Baldwin-Lomax [2], Cebeci-Smith [3], Chiens low Rey-

nolds number

k model [4] and different variants of Menters

k

model (BSL), (SST) [5] [6] and (RCSST) [7]. The aim of this thesis has

been to implement into the FINFLO a new turbulence model called the Ex-

plicit Algebraic Reynolds Stress Model (EARSM) developed by Stefan Wal-

lin and Arne V. Johansson from Sweden. The formulation of the model is

presented briefly in this thesis, but the reader is encouraged to find the

complete description in reference [8]. The EARSM was programmed using

Fortran 77 and 90 as they are used throughout the main code. After the

programming was completed, four test cases were selected, calculated and

analysed to verify the correct implementation of the model.


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Chapter 2

Governing equations and eddy-viscosity turbu-lence models

In this chapter the equations governing the flow field are presented in their

time-accurate and Reynolds averaged form. Then some general remarks of

turbulence are made and importance for the turbulence modelling is em-phasised. At the end of the chapter the basic aspects of the so-called eddy-

viscosity turbulence models are discussed.

2.1 Navier-Stokes equations

The equations of viscous flow have been known for more than 100 years.

The exact number of basic equations depends upon personal preference,

but some relations are, however, more basic than others. Usually the basicsystem of equations is considered to be the three laws of conservation for

physical systems:

- Conservation of mass (i.e. the continuity equation)

- Conservation of momentum (i.e. the Newtons second law)

- Conservation of energy (i.e. the first law of thermodynamics)


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Governing equations and eddy-viscosity turbulence models

4

The continuity equation simply states that the mass must be conserved. In

the Cartesian coordinates ix this equation can be written as

( ) 0=

+

i

i

xu

t (2.1)

where is the density of the fluid, ttime and iu the velocity vector. Ten-

sor notation will be used throughout the text whenever practical. The sec-

ond equation, conservation of momentum, states that momentum must be

conserved. It can be written in the Cartesian coordinates as

( )

j

ij

ii

i

jii

xx

p

fx

uu

t

u

+

=

+

(2.2)

where if is a body force, p the pressure. The viscous stress tensor ij is

+

=k

kij

i

j

j

iij

x

u

x

u

x

u

3

2(2.3)

where is the molecular viscosity and ij the Kroneckers delta. The third

equation, conservation of energy, states that the energy must be con-served. It can be written as

( )[ ] ( ) 0=

+

+

jiji

j

j

j

qux

pEuxt

E (2.4)

whereEis defined as the total energy and can be written as

+= iiuueE2

1 (2.5)

where e is the specific internal energy. Assuming Fouriers heat transfer

law the heat flux vector can be written as

i

ix

Tkq

= (2.6)

where kin the heat transfer coefficient and T the temperature.


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5

2.2 Reynolds averaging

In a turbulent flow the local pressure, density, velocity components and

temperature vary randomly with time. A reasonable approach is to separate

the flow quantities into stationary and random parts. The quantities are thus

usually presented as a sum of the mean flow value and the fluctuating part

as ii uu + . This formulation is inserted into the equations (2.1), (2.2) and(2.4); a process called Reynolds averaging is performed. There are three

forms of averaging present in the turbulence-model research, which are a

time average, a spatial average and an ensemble average. The general

term used to describe these averaging processes is mean. The most

commonly used averaging is the first, time averaging. It can be used onlyfor stationary turbulence, i.e. for a turbulent flow that does not vary with

time on the average. For such a flow the mean flow value is defined as

( )+

=

Tt

t

iT

i dttuT

u 1lim (2.7)

In practice taking the limit to infinity means that the integration time T

needs to be long enough relative to the maximum period of the assumed

velocity fluctuations.

The Reynolds averaged Navier-Stokes equations (RANS) are obtained

from the continuity and momentum equations, (2.1) and (2.2), by taking the

time average of all the terms in the equations. The continuity equation does

not change since it is linear in terms of the velocity. However, the momen-

tum equation is non-linear, which means that all the fluctuating components

do not vanish. An extra term, called Reynolds stress jiuu , appears in the

momentum equation (2.2). The result can be written as

( ) ( ) ( )j

jiij

i

i

i

jii

x

uu

x

pf

x

uu

t

u

+

=

+

(2.8)

where the overbar has been dropped from the mean values. This conven-

tion will be used throughout the text whenever obvious. The equation (2.8)

presents the fundamental problem of turbulence. In order to compute all the


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mean-flow properties of the turbulent flow we need a reasonably accurate

way to compute the Reynolds stress jiuu . This is the fundamental reasonfor the need of the turbulence models.

The complete set of the Reynolds averaged Navier-Stokes equations are

not presented in this text, because they can be found from many refer-

ences, for example [9]. Scalar transport equations are also needed, for ex-

ample to describe the transport of the concentration of species or the mass

fraction of species. Their exact formulation can be found from reference [9]

or [10]. The presentation of the exact formulation of the FINFLO along with

the solution methods used is beyond the scope of this text. The reader is

encouraged to consult references [11], [12] and [13].

2.3 General facts of turbulence

Many basic definitions of turbulence have been written, but probably the

most accurate was formed by Bradshaw in 1974:

"Turbulent fluid motion is irregular condition of flow in which the various

quantities show a random variation with time and space co-ordinates, so

that statistically distinct average values can be discerned. Turbulence has a

wide range of scales."(Ref. [14])

Analysis of the solutions to the Navier-Stokes equations shows that turbu-

lence develops as an instability of the laminar flow. At low Reynolds num-

bers, fluid layers slide smoothly past each other and the molecular viscosity

dampens the randomly occurring small-scale, high frequency disturbances.

The flow is laminar and exactly predictable. When Reynolds number in-

creases, the restraining effects of viscosity are too weak to prevent such

disturbances from amplifying. The disturbances grow gradually and be-

come non-linear. They also interact with neighbouring disturbances. At high

Reynolds number, the flow reaches a chaotic and non-repeating, i.e. tur-

bulent, state. Turbulent motions are characterised by the following proper-

ties: three dimensional, unsteady, random, strongly vorticial, strongly dissi-

pative and strongly diffusive. Turbulence increases friction, heat transfer


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and mixing and spreading rate. It also reduces the separation tendency of

the flow by energising the boundary layer (Ref.[15]).

The non-linearity of the Navier-Stokes equations leads to interactions be-tween turbulent fluctuations of different wavelengths and directions. Ac-

cording to reference [14] the wavelengths of motion usually extend all the

way from a maximum comparable to the width of the flow to a minimum

fixed by viscous dissipation of energy. The main physical process that

spreads the motion over a wide range of wavelengths is called vortex

stretching. The turbulence gains energy if the vortex elements are primarily

oriented in a direction in which the mean velocity gradients can stretch

them. This is called production of turbulence. The turbulent kinetic energy is

then convected, diffused and dissipated.

The larger-scale turbulent motion carries most of the energy and is mainly

responsible for the enhanced diffusivity and attending stresses. The large

eddies have memory and their orientation is sensitive to the mean flow.

The large eddies randomly stretch the smaller eddies, cascading energy to

them. Small eddies lose their orientation preference and become statisti-

cally isotropic. Energy is dissipated by viscosity in the shortest wave-

lengths, although the long-wavelength motion at the start of the cascadesets the rate of dissipation of energy. The shortest wavelengths simply ad-

just accordingly. This kind of turbulence is called equilibrium turbulence.

The ratio of largest to smallest scales increases rapidly as the Reynolds

number increases (Ref. [14]).

If the unsteady Navier-Stokes equations were calculated, a vast range of

time and length scales would have to be computed. In other words, a very

fine grid and very small time steps would be required. This is possible with

todays computers for only very simple problems with small Reynolds num-

bers. Thus a turbulence model is needed to account at least some of the

fluctuating motion by a statistical approach. The model should be, in princi-

ple, simple, broadly general and applicable, based on physics rather than

intuition, computationally stable and coordinately invariable. This is, of

course, an impossible list of requirements. The next sections of this chapter

describe traditional ways to model the turbulence using a so-called Boussi-

nesq eddy-viscosity approximation.


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2.4 Eddy-viscosity turbulence models

In this section the so-called eddy-viscosity turbulence models are briefly

discussed. The models can be divided in three categories; zero-, one- and

two-equation models. These models use the Boussinesq eddy-viscosity

approximation. The word approximation is a little misleading here because

it is really more of a hypothesis than an approximation. It can be written for

one- and two-equation models as

kx

u

x

u

x

uuu ij

k

kij

j

i

i

j

Tji 3

2

3

2

+

= (2.9)

For the zero-equation models, which do not make use of the turbulent ki-

netic energy 2iiuuk , the last term of the equation is zero. The Boussi-nesq approximation is used to compute the Reynolds stress tensor as the

product of the kinematic eddy-viscosity T and the mean strain-rate tensor.

The kinematic eddy-viscosity will be referred hereafter as the eddy-viscosity

for simplicity. The Reynolds stress components are calculated from the

eddy-viscosity for an incompressible flow as

kSuu ijijTji 3

22 = (2.10)

where the mean strain-rate tensor is defined as

+

=i

j

j

iij

x

u

x

uS

2

1(2.11)

The eddy-viscosity is a scalar field variable and the different Reynolds

stress components of a certain point of the flow field thus scale with the

mean strain-rate tensor, i.e. the Reynolds stress components are linearly

proportional to the mean strain-rate tensor. Different ways to calculate the

eddy-viscosity are described in the next subsections.

The equations (2.10) and (2.11) as well as all the following equations in this

text are presented for an incompressible flow. The FINFLO calculation rou-

tine calculates the flow as compressible, of course, while all of the turbu-


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lence models approximate the flow as incompressible. This approximation

is a so-called Morkovins hypothesis meaning that the compressibility af-

fects the turbulence quantities only at hypersonic speeds.

2.4.1 Algebraic models

Models where the eddy-viscosity is completely determined in terms of the

local mean flow variables are referred to as zero-equation or algebraic

models. Algebraic models are the simplest of all turbulence models. They

often use some form of Prandtls mixing length hypothesis in order to com-

pute the eddy-viscosity using terms of mixing length in analogy to the mo-

lecular mixing phenomenon. Whereas the molecular viscosity is an intrinsic

property of the fluid, the eddy-viscosity (and hence the mixing length) de-

pends upon the flow. Because of this, the eddy-viscosity and mixing length

must be specified in advance by an algebraic relation between eddy-

viscosity and length scales of the mean flow. Thus, algebraic models are,

by definition, incomplete models of turbulence. (Ref. [14]).

The eddy-viscosity is usually calculated as

j

i

mixT

dy

dul 2= (2.12)

where mixl is the mixing length analogous with the molecular mean free

path in molecular mixing.

Well-known and much used algebraic models are the Baldwin-Lomax [2]

and the Cebeci-Smith [3] models, both of which are two-layer models with

separate expressions for the computation of the eddy-viscosity in eachlayer. The Cebeci-Smith model is simple and easy to implement. Most

computational effort goes to the calculation of the boundary layer velocity

thickness. The Baldwin-Lomax model was formulated for use in computa-

tions where boundary layer properties such as the boundary layer and the

boundary layer velocity thickness are difficult to determine.

Algebraic models are conceptually very simple, economical in terms of

computer resources and rarely cause any unexpected numerical difficulties.


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The user of the algebraic models must always be aware of the issue of in-

completeness. Both the Cebeci-Smith and Baldwin-Lomax models work

well only for the boundary layer flows for which they have been fine-tuned.

They reproduce skin friction and velocity profiles well for incompressible

turbulent boundary layers provided the pressure gradient is not too strong.

However, they can not be applied to other flow cases, such as wake flows

or free jets.

2.4.2 One-equation models

In one-equation models, one of the two turbulence scales or a combination

of both is determined from a transport equation. The transport equation for

the turbulence kinetic energy can be written as

+

= jjiijjj

iij upuuu

x

k

xx

u

Dt

Dk

1

2

1 (2.13)

where the tensor notation jj xutDtD + is used to denote the rateof change following the mean flow. The term jiij xu is the production

of the turbulence, P , the term jxk is the molecular diffusion, the term2jii uuu is the turbulent flux of the turbulent kinetic energy, k, and the last

term jup is the so-called pressure diffusion term. The last term is usu-ally neglected because its contribution is very small.

The is the dissipation rate per unit mass, defined as

k

i

k

i

x

u

x

u

= (2.14)

The eddy-viscosity is usually calculated as

klC mixT = (2.15)

where the C is a model coefficient. As for algebraic models, additional

information from the mean flow field or geometrical measures is needed.

Most commonly used one-equation models are the Spalart-Allmaras [16]


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and Baldwin-Barth [17] models. Actually, in the models the transported

quantity is 2k , which follows from the definition of the turbulent Reynolds

number TTT luRe , since 21~kuT and 23~klT .

According to reference [15], the one-equation models give usually more

accurate results than algebraic models because more realism is obtained

as a result of deducing the turbulent velocity scale from a turbulence quan-

tity k rather than from gradients of mean velocity. This also implies that,

by obtaining the solution for the turbulent velocity scale from a differential

transport equation, some account is taken of the history effects, which be-

come important in non-equilibrium flows. The differential transport equation

also removes the problems associated to the flow field points where the

local mean velocity gradient is zero.

The main drawbacks with one-equation models are largely the same as for

the mixing-length models. The history effects are not accounted for the

length scale l, which is still prescribed as an algebraic function of local

quantities. According to reference [18] the Spalart-Allmaras model performs

better than the Chiens k model in a decelerating flow with adversepressure gradient because it is specially designed for aerodynamical pur-

poses. It is one of the most popular turbulence models in the field of aero-nautical applications, especially in the U.S.A.

2.4.3 Two-equation models

Two-equation models have served as the foundation for much of the tur-

bulence model research until very recent years. The most significant differ-

ence between the other eddy-viscosity models and the two-equation mod-

els is that the latter are complete, i.e. can be used to predict properties of agiven turbulent flow with no prior knowledge of the turbulence structure or

flow geometry. These models provide an equation not only for computation

of k, but also for the turbulence length scale or equivalent.

The starting point for virtually all linear two-equation eddy-viscosity models

is the Boussinesq approximation (2.9) and the turbulence kinetic energy

equation (2.13). The choice of the second variable is arbitrary and many


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proposals have been presented. By far two of the most popular dependent

variables for the second variable have been the dissipation rate and the

k-specific dissipation rate . Nowadays there is also increasing interest in

the use of the turbulent time-scale . The most common proposals are

presented in the table 2.1

Table 2.1. Proposals for the second variable (Ref. [15]).

Proposer Year Variable Symbol Physical meaning

Kolmogorov 1942 121

lk Frequency

Rotta 1951 l l Length scale

Rotta 1968 kl kl k times the length scale

Harlow & Nakayama 1968 123

lk Energy dissipation rate

Spalding 1969 2

kl Vorticity fluctuations squared

Nee & Kovasznay 1969 21

lkT Eddy viscosity

Speziale 1992

21

lk Time-scale

The k model is the best-known two-equation turbulence model be-cause it is simple to understand and use and relatively easy to program.

The model dates back to the late 1960s but the most used formulation, re-

ferred as the standard k model, was presented in 1972 by Jones andLaunder [19]. The eddy-viscosity used with the Boussinesq approximation

is calculated as

2k

CT= (2.16)

where C is a model coefficient. The k model is used widely in practi-cal engineering calculations even though the standard model can be used

only with attached flows with thin shear layers. The model fails to reproduce

the correct flow behaviour in many important flow situations, such as: ad-


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13

verse pressure gradients, bluff-body flows, separation, streamline curva-

ture, swirl, buoyancy, turbulence driven secondary motion, compressibility

and unsteadiness.

The k model is based on the choice of the specific dissipation rate asthe second variable. The eddy viscosity is calculated as

kT= (2.17)

Even though the wall boundary condition for the is more difficult to for-

mulate and program than for the , the k model has found its way to

many engineering, especially aerodynamic, applications. This is mainly dueto the fact that it can also reproduce the flow field behaviour with adverse

pressure gradient. It usually also predicts separation better than the other

linear two-equation eddy-viscosity models. Actually, the k SST model[7] predicts the separation fairly well, but it is no longer really a linear

model.


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Chapter 3

Differential Reynolds stress models

In this chapter a different approach to model the turbulence is reviewed.

The Boussinesq approximation is abandoned and the transport equation for

the Reynolds stresses is presented instead. In the end of the chapter mod-

elling propositions of the different terms of the transport equation are dis-cussed.

3.1 Reasons for using the Reynolds stress models

Eddy-viscosity models perform reasonably well in attached boundary layer

flows as long as only one component of the Reynolds stress tensor is of

significant importance. In these cases the eddy-viscosity can be thought as

a representative for the significant Reynolds stress component. However, ifthe flow becomes more complicated the eddy-viscosity assumption fails.

There is thus not much hope for a more general validity of the eddy-

viscosity approach (Ref. [8]).

The development of the so-called differential Reynolds stress models (also

called Reynolds stress transport, second-order closure, second-

moment closure and second-order modelling) started officially in 1968,


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when C. Donaldson lectured in Stanford of the need to model the Reynolds

stress terms. The Boussinesq eddy-viscosity hypothesis is abandoned and

the unknown Reynolds stress components are obtained directly from the

solution of differential transport equations in which they are the dependent

variables. The Reynolds stress tensor is symmetric meaning that four

equations need to be solved in two-dimensional and six equations in three-

dimensional flows. An additional equation for the length scale or equivalent

is also solved in each case.

The Reynolds stress models are thus more complicated than the eddy-

viscosity models. Still, they are attractive because they provide a more ac-

curate representation of the turbulence and are valid over a wider range of

flows. They can capture many of the complex effects encountered in nature

and in engineering practice without recourse to the ad-hoc modifications

found necessary in lower-order models. According to reference [15], exam-

ples of flow situations where Reynolds stress models were found to give

accurate predictions include flows with streamline curvature, rotation, swirl

and buoyancy.

3.2 Exact equation for the Reynolds stress tensor

The exact transport equation for the Reynolds-stress tensor jiuu is ob-tained from the momentum equation (2.2) by multiplying the instantaneous

iu component equation by ju , the ju component equation by iu , addingthe two and then time-averaging the result. Following reference [20], the

result can be written for constant-density flows with no body forces after

some rearrangement as

=

t

uuD ji

+

k

ikj

k

j

kix

uuu

x

uuu Terms I and II

( )

++

k

ji

ikjjkikji

k x

uuupupuuu

x

1Term III

+

+i

j

j

i

x

u

x

up

k

j

k

i

x

u

x

u2 Terms IV and V

(3.1)


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Term I is the convection term. It represents the rate of change of jiuu

along a streamline. In steady flows this is equal to the rate at which Rey-

nolds stresses are convected by the mean fluid motion.

Term IIis the production term. It represents the rate of production of jiuu

by mean shear, which is the reason for turbulence as discussed earlier.

The shear stress is generated by interaction of transverse normal stress

and shear strain. The production term is both large and exact, which is a

partial reason for the success of the Reynolds stress models. It will be

hereafter referred to as ijP .

Term III is the diffusion term. It represents the rate of spatial transport of

jiuu by the action of turbulent fluctuations, pressure fluctuations and mo-lecular diffusion.

Term IV is the redistribution term, which is also called the pressure-strain

term. It represents the redistribution of the available turbulent kinetic energy

amongst the fluctuating velocity components. The mean flow direction be-

ing 1x , only 11uu is generated in the shear layer, meaning that it is usuallymuch larger than the other Reynolds stress components. The redistribution

term shares 11uu s energy out to 22uu and 33uu . It has no effect on theoverall level of turbulent kinetic energy since it has zero trace. The redistri-

bution term just drives turbulence towards isotropy by redistributing energy.

This term will be hereafter referred to as ij .

Usually 21uu has the opposite sign to the boundary layer shear strain. Theredistribution term 12 reduces also 21uu since isotropic turbulence mustbe shear-free. The same applies to the possible shear strains to other di-

rections.

Term V is the dissipation term. It represents the dissipation rate of jiuu

due to molecular viscous action. The dissipation term will be hereafter re-

ferred to as ij .


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3.3 Modelling the Reynolds stress tensor

With the exception of the convection and production terms of the equation

for the Reynolds stress tensor, all the other terms introduce new unknown

correlations that must be modelled in terms of known or knowable quanti-

ties in order to close the equations.

The diffusion term is a sum of three parts, which are called the viscous,

pressure and turbulent diffusion. The contribution of the viscous diffusion

term to the total rate of transport of jiuu is small at high values of the tur-bulence Reynolds number. This term is thus often neglected in high Rey-

nolds number applications even though it could be included in the numeri-

cal simulations without any difficulty.

Little is known about the pressure diffusion term since direct measurements

can not be conducted. Estimates of its magnitude are done by indirect

methods, which contain all the errors made in the measurements of the

other terms. The consensus of several experiments, however, suggests

that this term is relatively unimportant and may therefore be neglected (Ref.

[15]).

Daly and Harlow [21] were the first to model the turbulent diffusion term

using the gradient transport hypothesis, meaning that the diffusion of a

quantity is assumed to be proportional to the spatial gradient of the same

quantity. According to reference [15] the turbulent diffusion can be written

as

l

ji

lkSkjix

uuuu

kCuuu

=

(3.2)

where SC is an empirical model coefficient typically set equal to 0.22,

which is obtained by computer optimisation. The ratio k represents a

characteristic time scale of the energy containing eddies.

Alternative models for the turbulent diffusion exist, but they all are consid-

erably more complex than Daly & Harlows model without necessarily pro-


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ducing better overall performance. Examples of such models are Hanjalic &

Launder [22] and Lumley & Khajeh-Nouri [23].

The redistribution term modelling is based on the Poisson equation forthe instantaneous pressure. It is obtained by differentiating the Navier-

Stokes equations in a manner that is beyond the scope of this text. Then,

by subtracting the mean, the following equation for the fluctuating pressure

is obtained

( )

+

=

j

i

j

i

ij

jiji

i x

u

x

u

xx

uuuu

x

p2

1 2

2

2

(3.3)

where a sum of two very distinct terms is identified in the parentheses. The

first term of the sum contains only turbulence quantities and second term

contains mean-velocity gradients. When the equation is solved for homo-

geneous turbulence, the so-called Chous integral for the pressure-strain

correlation is obtained

+

+

=

wij

jl

im

m

l

jml

iml

j

i

r

dvol

rr

uu

x

u

rrr

uuu

x

up,

*2**3

24

1

(3.4)

where * denotes quantities evaluated at position *x and xxr = * . wij ,is a surface integral, which is important only when the typical size of the

energy containing eddies is of the same order as distance from a wall (Ref.

[15]).

Launder, Reece and Rodi (LRR) proposed that the first and the second

term in the parentheses of the equation (3.4) are modelled separately [24].

This is the traditional approach to modelling Chous integral and the great

majority of Reynolds-stress model developers have followed it.

Speziale, Sarkar and Gatski (SSG) proposed that the Chous integral could

be modelled as a whole [25]. This approach is rapidly becoming more

popular than (LLR) as it does not appear to require a specific model for the

surface integral wij , .


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Formulation of an explicit algebraic Reynolds stress model

22

Also a transport equation for the Reynolds stress anisotropy tensor can be

construed in a similar was as the transport equation (3.1) for the Reynolds

stress component tensor. According to reference [8] it can be presented in

a rotating Cartesian coordinate system as

( )aij

ijijijji

l

lji

l

ljiijC

PP

k

uu

x

ku

k

uu

x

uuu

Dt

Dak+++

=

1

1

(4.2)

where the dissipation rate tensor ij and the redistribution tensor ij need

to be modelled. On the other hand, the production terms ijP and 2iiPP=and the Coriolis term ( )aijC do not need any modelling since they can be

calculated directly from the Reynolds stress tensor.

In flows where the Reynolds stress anisotropy varies slowly in time and

space, the transport equation for the Reynolds stress anisotropy tensor is

reduced to an implicit algebraic relation. Many inhomogeneous flows of en-

gineering interest consist of a steady flow and equilibrium turbulence as-

sumption can thus be made. Therefore the convection and diffusion of the

Reynolds stress anisotropy may be neglected. This is equivalent to the as-

sumption made in reference [27] that the convection and diffusion of each

Reynolds stress component scale with the convection and diffusion of the

turbulent kinetic energy. This is indeed the traditional ARSM idea, to ne-

glect convection and diffusion terms in the exact transport equation for the

Reynolds stress anisotropy. This means that the left-side terms in the

equation (4.2) can be neglected and set to zero

01

=

l

lji

l

ljiij

x

ku

k

uu

x

uuu

Dt

Dak

(4.3)

The convection term DtDaij is exactly zero for all stationary parallel mean

flows, such as fully developed channel and pipe flows. For inhomogeneous

flows the assumption of negligible diffusion effects can cause problems,

particularly in regions where the production term is small or where the in-

homogeneity is strong. However, the ARSM assumption incorporates in a

natural way the effects of rotation, effects of streamline curvature and

three-dimensionality of the flow. The assumption results in an implicit alge-


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23

braic equation for the Reynolds stress anisotropy tensor, which can be

written as

( )aij

ijijijjiCPP

kuu ++=

1 (4.4)

The production terms ijP and 2iiPP= and the Coriolis term ( )aijC do notneed any modelling since they can be calculated directly from the Reynolds

stress tensor. However, the dissipation rate tensor ij and the redistribution

ij need to be modelled. In a non-rotating coordinate system, the produc-

tion term is normally written as implied in the equation (3.1). It is rewritten

here for the sake of completeness as

k

ikj

k

j

kiijx

uuu

x

uuuP

= (4.5)

To illustrate the natural way in which rotational effects enter in this type of

formalism it is convenient to split the mean velocity gradient tensor into a

mean strain and a mean vorticity tensor. Symbols ijS and ij are usedhere to represent these tensors, normalised with the turbulent time-scale as

=

+

=

i

j

j

iij

i

j

j

iij

x

u

x

u

x

u

x

uS

2

2

(4.6)

where the turbulent time-scale is defined as

k

(4.7)

A consistent formulation of the equation (4.4) can then be obtained by re-

placing the mean vorticity tensor by the absolute vorticity tensor. According

to reference [8], this can formally be done introducing the absolute mean

vorticity tensor

Sijijij +=* (4.8)


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where the system rotation tensor Sij is defined as

Skjik

Sij = (4.9)

where ijk is the permutation tensor and Sk is the constant rotation rate

vector of the system. The permutation tensor is unity when the indexes are

123, 231 or 312. It is minus unity when the indexes are 213, 321 or 132. In

other cases it is zero. Now, it is possible to rewrite the production term (4.5)

using the equations (4.1), (4.6) and (4.8). It is presented here normalised

by the dissipation rate as

( ) ( )kjikkjikkjikkjikij

ij

aaaSSaS

P**

3

4

+++= (4.10)

The Coriolis term ( )aijC arises from the transformation of the convection

term. It is presented in this text for completeness, even though it was not

programmed into the FINFLO, as the EARSM was only implemented into a

non-rotating coordinate system. For rotational purposes, FINFLO uses a

semi-rotational formulation, which is presented in reference [28]. The

Coriolis term can be written according to reference [8] as

( )kj

S

ik

S

kjik

a

ij aaC = (4.11)

For the present modelling purpose the dissipation rate tensor is assumed to

be isotropic. The equation is written here normalised by the dissipation rate

as

ijij

3

2= (4.12)

The redistribution term is modelled in two subparts, slow and fast redistri-

bution, as proposed in reference [8]. According to reference [29], the slow

redistribution rate can be though to be linear in terms of the anisotropy ten-

sor as

( )

ij

sij

aC1=

(4.13)


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where 1C is a model constant. For the rapid redistribution rate the general

linear Launder, Reece and Rodi (LRR) model [24] is chosen. It is normally

written for a non-rotating system as

( )

+

+=

PDC

x

u

x

uk

CPP

C

ijij

i

j

j

iijij

rij

3

2

11

28

55

230

3

2

11

8

2

22

(4.14)

where the 2C is a model constant and ijD is calculated as

i

kkj

j

kkiij

x

uuu

x

uuuD

= (4.15)

A simple way of obtaining a consistent, frame-independent formulation of

the rapid pressure-strain rate model is to apply the same methodology as

for the production term, i.e. to use the mean strain and vorticity tensors de-

fined in equation (4.4.) and to normalise by . The result can be written as

( )

( )kjikkjik

ijmkkmkjikkjikij

rij

aaC

SaaSSaC

S

**2

2

11

107

3

2

11

69

5

4

+

++

+=

(4.16)

When the equations for the production term (4.5), the Coriolis term (4.11),

the dissipation term (4.12), the slow redistribution term (4.13) and the fast

redistribution terms (4.16) are inserted to the equation (4.4), the implicit al-

gebraic equation for the Reynolds stress anisotropy tensor is then obtained

as

( )

+

+

+=

+

kiikijkjikkjik

kjRik

Rkjikijij

SaaSSaC

aaC

SaP

C

3

2

11

95

11

17

15

81

2

2

1

(4.17)

Here the effective mean vorticity tensor Rij is dependent on the modelconstant 2C and therefore on the choice of the redistribution model.


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The effective mean vorticity tensor Rij of the equation (4.17) can be writtenas

Sijij

Sijij

Rij

CC

C

+++=

++=

17127

1711

2

2

2

* (4.18)

It should be noted, however, that in the present FINFLO formulation the

effective mean rotation rate tensor is the same as the vorticity tensor

ijRij = , for the reasons discussed earlier. The dissipation rate must be

modelled by a transport equation similar to the one used by the two-

equation eddy-viscosity models. It should be noted that equation (4.17) rep-

resents a non-linear relation, since the production to dissipation ratio is de-

fined as

kiikSaP

(4.19)

The algebraic Reynolds stress model has been formulated in the form of

equation (4.17). In the next section this is further simplified.

4.2 Explicit algebraic Reynolds stress model

The implicit ARSM relation for the Reynolds stress anisotropy tensor has

been found to be numerically and computationally cumbersome since there

is no diffusion or damping present in the equations. This usually means

convergence problems. In many applications the computational effort has

been found to be excessively large and the benefits of using ARSM instead

of the full Reynolds stress transport form are then successively lost, as

Wallin discusses [8]. Therefore a more simple and straightforward way to

calculate the Reynolds stress anisotropy is needed. A model, where the

Reynolds stresses are explicitly related to the mean flow field, is called an

explicit algebraic Reynolds stress model. It is much more numerically ro-

bust and has been found to have almost a negligible effect on the compu-

tational effort as compared to the linear k or the linear k model.

The procedure used here to obtain an explicit form is to avoid the non-

linearity by considering the production to dissipation ratio P as an extra


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unknown. The resulting linear equation system can then be formally written

as

( ) 0,,, = PSa RklklklijL (4.20)

and may, in principal, be solved directly. However, by the inspection of the

equation (4.17) it is realised that the anisotropy tensor is dependent on only

two other tensors, ijS andRij , which can be used to form a complete base

for the anisotropy. The most general form for ija in terms of ijS andRij

consists of ten tensorially independent groups to which all higher-order ten-

sor combinations can be reduced with the aid of the Caley-Hamilton theo-

rem. Various techniques have been developed with different level of simpli-

fications in references [26], [30], [31], [32] and [33], for example. The Rey-

nolds stress anisotropy tensor is, however, written here following reference

[8] as

( )SSIISSa +

+

+= 423221

3

1

3

1 IIIIS

( )

++

+++ ISSISSSS VIV

3

2

3

2 22227

22

6

22

5 (4.21)

( ) ( ) ( )+++ 222210

22

9

22

8 SSSSSSSS

For simplicity, boldface has been used to denote second-rank tensors as

ija=a , ijS=S and Rij= . The inner product of two matrices is definedas ( ) ( ) kjikijij SS

2SSS and I represents the identity matrix. The

coefficients may be functions of the five independent invariants of S

and , which can be written as

( )( )

( )

( )

( ) likljkij

kijkij

kijkij

jiij

jiijS

SSV

SIV

SSSIII

II

SSII

====

====

==

22

2

3

2

2

trace

trace

trace

trace

trace

S

S

S

S

(4.22)

Other scalar parameters may also be involved.


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The procedure to solve this equation system is the following:

1) The general form of the anisotropy, the tensor polynomial (4.21), is in-

serted into the simplified ARSM equation (4.24) where N is not yet

determined. Now N is considered to be a known parameter. This re-

sults in a linear equation system for the coefficients.

2) The linear equation system can be solved using the fact that higher-

order tensor groups can be reduced with the aid of the Cayley-

Hamilton theorem. The solution is unequivocal because the ten groups

in the general form (4.21) form a complete basis for the system. The

solution consist of many pages of complicated tensor algebra and it is

not essential to reproduce it here. The -coefficients are now functions

of N , or the production to dissipation ratio.

3) The next step is to formulate and solve the non-linear scalar equation

for N . This is done by inserting the solution of the coefficients into

the tensor polynomial (4.21) for the Reynolds stress anisotropy tensor.

4) The final step is to insert the resulting equation to the simplified ARSM

equation (4.24). Then a non-linear scalar equation for N is obtained,

which for a general three-dimensional flow field is of sixth order.

The solution for the simplified ARSM equation (4.24) is presented in the

following section 4.3 for two- and 4.4 for three-dimensional mean flows. In

section 4.5, the solution for the model without the simplification made from

the equation (4.17) to (4.23) is also presented.

4.3 Simplified EARSM for two-dimensional mean flow

For two-dimensional mean flows the simplified solution of the ARSM equa-

tion is reduced to only two non-zero coefficients, which can be expressed

according to reference [8] as

=

=

IIN

IIN

N

2

1

5

6

25

6

24

21

(4.27)


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In the equation (4.27) it is clearly seen that the denominator cannot become

singular since II is always negative. The non-linear equation for Nin two-

dimensional mean flow can be derived by inserting the tensor polynomial

(4.21) for the Reynolds stress anisotropy tensor with the coefficients from

the equation (4.27), into the definition of N, equation (4.24). The resulting

equation is cubic and can be written following reference [8] as

02210

271

21

3 =+

+ IICNIIIINCN S (4.28)

The equation can be solved in a closed form with the solution for the posi-

tive root being

( ) ( )

( )

Reynolds Stress ARSM

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