On Two-Equation Eddy-Viscosity Models

Internal Report 01/8

On Two-equation Eddy-Viscosity Models

JONAS BREDBERG

Department of Thermo and Fluid Dynamics

HALMERS NIVERSITY OF ECHNOLOGY

C ¢¡ U T £¥¤§¦©¨ £ ¦©¦¥ On Two-equation Eddy-Viscosity Models

JONAS BREDBERG Department of Thermo and Fluid Dynamics Chalmers University of Technology

ABSTRACT This report makes a thoroughly analysis of the two-equation eddy-viscosity models (EVMs). A short presentation of other types of CFD-models is also included. The fundamentals for the eddy-viscosity models are discussed, including the Bous- sinesq hypothesis, the derivation of the ’law-of-the-wall’ and

references to simple algebraic EVMs, such as Prandtls mixing- ¡ length model. Both the exact - and -equations are discerned using DNS-data (Direct Numerical Simulation) for fully developed channel ﬂow. In connection to this a discussion of the

beneﬁts and disadvantegous of the various two-equation mo-

¥¢§¦ £¢©¨ dels ( £¢¤¡ , , and ) is included. Through-out the report special attention is made to the close connection between the various secondary equations. It is de-

monstrated that through some basic transformation rules, it is ¢§¦ possible to re-cast eg. a ¥¢§¡ into a model. The additive terms from such a re-formulation is discussed at length and the possible connections to both the pressure-diffusion process and the Yap-correction are mentioned. The report distinguish between EVMs that are developed with the aid of DNS-data or TSDIA (Two-scale Direct Integration Approximation) and those which are not. Within each group, the turbulence models are compared term by term. The chosen coefficients, damping functions, boundary condition etc. are discussed. The additive terms for the DNS tuned models are compared and criticized. In a lengthy summary some thoughts and advices are given for the development of a two-equation. Apart from a discussion on the type of EVM attention is given to the turbulence model coefficients, Schmidt numbers and the possibilty to include cross-diffusion terms. In the appendices a number of turbulence models are tabula- ted and compared with both DNS-data and experimental data for fully devloped channel flow, a backward-facing step flow and a rib-roughened channel flow. Particularly interest is given to how newer (DNS-tuned) turbulence models compare to

the older, non-optimized EVMs. ¢

Keywords: turbulence model, eddy-viscosity model, ¢ ¡ , ¢©¨ ¦ , , cross-diffusion, TSDIA, DNS

ii Contents 5 The TSDIA and DNS Revolution 16 5.1 Turbulent Time Scales? ...... 16 Nomenclature v 5.2 Turbulent Viscosity ...... 16 5.3 Coupled Gradient and the Pressure- 1 Introduction 1 diffusion Process ...... 17

5.3.1 The -equation ...... 17

2 CFD Models for Turbulence 1 5.3.2 The ¡ -equation ...... 18 2.1 DNS ...... 1 5.3.3 The ¦ -equation ...... 19 5.4 Fine-tuning the Modelled Transport 2.2 Large-Eddy-Simulation (LES) ...... 2 Equations ...... 19 2.3 Reynolds-Averaged-Navier-Stokes (RANS) 2 5.4.1 The -equation ...... 19 2.3.1 Eddy-Viscosity Models ...... 2 5.4.2 The ¡ -equation ...... 20

2.3.2 Reynolds-Stress-Models ...... 2 5.4.3 The ¦ -equation ...... 21

3 Eddy-Viscosity Models (EVMs) 3 6 Enhancing the ¦ -equation? 21

3.1 Boussinesq Hypothesis ...... 3 6.1 Transforming the DNS/TSDIA ¥¢§¡ models 21 3.2 The Mixing-length Model and the Loga- 6.1.1 Turbulent approach, YC- and RS- rithmic Velocity Law ...... 3 models ...... 21 6.1.2 Viscous approach, NS- and HL- 3.2.1 Prandtl’s mixing-length model . . . 3 models ...... 22 3.2.2 Modiﬁed mixing-length models . . . 3 6.2 Additivional Terms in the ¦ -equation . . . . 22

3.2.3 Law-of-the-wall ...... 4

¨§¢¦ © ¦ ¦ §¢¦ ©

6.2.1 Cross-diffusion, ¦ . . . 22

§¦© 3.3 Algebraic or Zero-equation EVMs ...... 4 ¦ 6.2.2 Turbulent diffusion, . . . . 22 3.4 One-equation EVMs ...... 5 6.2.3 Other terms ...... 22

3.5 Turbulent Kinetic Energy Equation . . . . 5 6.3 Back-transforming the ¦ -equation . . . . . 22

3.6 Two-equation EVMs ...... 6 6.4 Cross-diffusion Terms in the ¦ -equation? . 23 3.7 Secondary Quantity ...... 6 7 Summary 24 3.7.1 Wall dependency ...... 6 7.1 The -equation ...... 24 3.7.2 Boundary condition ...... 7 7.2 The Secondary Equation ...... 24 3.8 Dissipation Rate of Turbulent Kinetic 7.3 Schmidt Numbers ...... 25 Energy ...... 8 7.4 Cross-diffusion vs Yap-correction ...... 25 3.9 EVMs Treated in this Report ...... 9 A Turbulence Models 29

4 The Classic Two-equation EVMs 9 A.1 Yang-Shih ¥¢§¡ ...... 29

A.2 Abe-Kondoh-Nagano ¥¢§¡ ...... 29

4.1 Turbulent Viscosity, ¢¡ ...... 9

¡ A.3 Jones-Launder, standard ¢ ...... 30

4.1.1 Near-wall damping ...... 9

¡ A.4 Chien ¥¢ ...... 30

4.1.2 Wall-distance relations ...... 10

£¢¡ £¥¤ A.5 Launder-Sharma + Yap ...... 30

4.1.3 Damping function, ...... 10

A.6 Hwang-Lin ¢ ...... 31

¡ 4.2 Modelling Turbulent Kinetic Energy, - A.7 Rahman-Siikonen ¥¢ ...... 31

equation ...... 11 A.8 Wilcox HRN £¢¤¦ ...... 32

4.2.1 Production ...... 11 A.9 Wilcox LRN £¢¤¦ ...... 32

4.2.2 Dissipation ...... 11 A.10 Peng-Davidson-Holmberg ¥¢§¦ ...... 32

4.2.3 Diffusion ...... 12 A.11 Bredberg-Peng-Davidson £¢¤¦ ...... 33

4.2.4 The modelled -equation ...... 12 B Test-cases and Performance of EVMs 33 4.3 Modelling the Secondary Turbulent Equa- B.1 Fully Developed Channel Flow ...... 33

tion ...... 12 B.2 Backward-facing-step Flow ...... 35 ¡ 4.4 The Jones-Launder ¤¢ model and the B.3 Rib-roughened Channel Flow ...... 36 Dissipation Rate of Turbulent Kinetic

Energy, ¡ -equation ...... 12 C Transformations 38

¦ 4.4.1 Production term ...... 12 C.1 Standard ¢¤¡ model -equation . . . . 38 4.4.2 Destruction term ...... 13 C.1.1 Production term ...... 38 4.4.3 Diffusion ...... 13 C.1.2 Destruction term ...... 38 C.1.3 Viscous diffusion term ...... 38 4.4.4 The modelled ¡ -equation ...... 13 C.1.4 Turbulent diffusion term ...... 38

4.5 Transforming the ¡ -equation ...... 13 C.1.5 Total ...... 39

4.5.1 The ¦ -equation ...... 14

C.2 ¢ ¦ model (with Cross-diffusion Term) ¨

4.5.2 The -equation ...... 14 ¡ -equation ...... 39

4.6 Wilcox and the ¥¢§¦ model ...... 14 C.2.1 Production term ...... 39

4.7 Speziale and the ¥¢¤¨ model ...... 15 C.2.2 Destruction term ...... 39

iii C.2.3 Viscous diffusion term ...... 39 C.2.4 Turbulent diffusion term ...... 39 C.2.5 Cross-diffusion term ...... 40 C.2.6 Total ...... 40

D Secondary equation 41

iv Nomenclature Latin Symbols

Greek Symbols

¡ ¢£¢

¢£¢

Turbulence model constant Turbulence model coefﬁcients ¡

¡ ¢£¢

¢£¢

Various constants Turbulence model coefﬁcients ¡

¤¦¥

¡ ¢£¢ ¢

Length-scale constant Boundary layer thickness ¡

¡ ¢£¢

¡ §/ : ¢

Turbulence model coefﬁcient ¡ Dissipation rate

¤£§

¨ ¨ © § ¡ ¢£¢

¡ S ¡ ¢UT¡ ¡ §/ : ¢

Skin friction coefﬁcient, Reduced dissipation rate

¡ ¢

¡ ¡ §/ : ¢

Hydrualic diameter T Wall dissipation rate

¡ ¢ ¢

Diffusion term Kolmogorov length scale ¡

§ ¡ ¢

¢£¢

Material derivate, Van Karman constant ¡

¦! #" §¦ ¦! #" §¦©

¡ §/H¢

Kinematic viscosity

¡ ¢

¡ YX¨§ Z:-¢

Turbulence model term Density

[

¡ (¢

Rib-size Pressure-diffusion term ¡ ¢

¡ ¢£¢ ¢

Damping function Destruction term ¡

)

¡ ¢

Channel height " Free variable

]

¡ (¢

%^¢

Step height Rotation tensor ¡

+-,.

¡ ¢£¢

¡ %^¢

Tensor indices: ¦ Speciﬁc dissipation rate

¢£¢

streamwise: 1,U,u Turbulent Schmidt number ¡

¡ H¢

wall normal: 2,V,v ¨ Turbulent time scale

¡ § -¢

spanwise: 3,W,w ¨ Shear stress

¡ §/ ¢

H© ¡ §7 -¢

Turbulent kinetic energy ¨ Wall shear,

0 (¢

Integral length scale ¡

1 (¢ Length scale ¡

243 Superscripts

¡ ¢£¢

Nusselt number Viscous value

5 2

¡ § ¢

Static pressure "!` Normalized value using :

¡ (¢

Sb §

Rib-pitch a`

C C B

¡ §7 ¢

S §

Fluctuating pressure `

598

¡ §/;: ¢

` S ¨§

Turbulent production,

3!<=>3!<

¦% §¦©

¡ S ¡ ¨§/ @c

5@?

¡ ¢£¢

Prandtl number Normalized value

A )

§ ¡ ¢£¢

Reynolds number, Alternative value

¡ ¢£¢

¡ Turbulent Reynolds number,

¡ §¢ § ¦ § ¡ , , d

A Subscripts

' B B

¡ ¢£¢

based Reynolds number, Bulk value

Hydraulic diameter D Turbulence model source term [*] *

= Step heigth

¡ ¢

Strain-rate tensor, =

= Quantity based on Kolmogorov scales

§/¨ ¦% §¦© ¦ §¦©

Quantity in -equation F

Turbulent transport term [*] ? Re-attachment poin

¡ Turbulent time scale [s]

= Turbulent quantity

¡ §GH¢

Velocity Wall value

§GH¢

Fluctuating velocity ¡ Normalized using wall distance

3 =>3

< <

¡ §/ ¢ ¡ ¡

Reynolds stresses Quantity in -equation

I ¨ § ¡ §GH¢

Friction velocity, Quantity based on Taylor micro-scale

E;LL/LNM

¦ ¦

KJ § ¢£¢

Normalized value: ¡ Quantity in -equation

¡ (¢

¨ (¢

Wall normal coordinate ¡ Quantity based on the friction velocity

O (¢ Spanwise coordinate ¡ Freestream value

v 1 Introduction 2 CFD Models for Turbulence

In both engineering and academia the most frequent An important question, however less appropriate in this employed turbulence models are the Eddy-Viscosity- paper, is: whether or not a turbulence model should be Models (EVMs). Although the rapidly increasing compu- used at all? With the progress of DNS there is no need ter power in the last decades, the simplistic EVMs still of any modelling of the turbulence ﬁeld or is it?

dominate the CFD community. The question is answered through the study of the dif-

¢ ¡ The landmark model is the model of Jones and ferent approaches used in numerical simulations for tur- Launder [19] which appeared in 1972. This model has bulent ﬂows, via Computation Fluid Dynamics (CFD) been followed by numerous EVMs, most of them based codes. Turbulence models could be conceptionally dis-

on the -equation and an additional transport equation, tinguished from physical accuracy, or computational re-

¢ ¦ ¢ ¨ ¢ ¡ such as the [52], the [47] and the [37] sources point of view. However irrespectively of the basis models. With the emerging Direct Numerical Simula- for the evaluation, DNSs are located on the higher end tions (DNSs), it has now been possible to improve the of the spectrum. DNSs don’t include any modelling at EVMs, especially their near-wall accuracy, to a level not all, apart from numerical approximation and grid reso- achievable using only experimental data. The ﬁrst ac- lution, and hence are treated as accurate, or even more

curate DNS was made by Kim et al. [20] albeit at a low accurate than experiments. They may also be referred

E¡ /L '7B

Reynolds number S and for a simple fully deve- to as numerical experiments made in virtual windtun- loped channel flow testcase. Today however, DNS’s are nels, ie. computers. On the opposite end are the alge- made at both interesting high Reynolds numbers, and braic EVM models, which compute the turbulent visco- of more complex flows, enabling accurate and advanced sity (eddy viscosity) using some algebraic relation1 EVMs to appear. Table 1 lists some of the modelling approaches to tur- This paper will try to explain these newly developed bulence ordered by physical accuracy. EVMs which are based on DNS-data. A number of tur- Substituting physical accuracy with computational bulence models are compared with both DNS-data and demands, the same order is repeated, with DNS consu- experimental data for different flows. Particularly inte- ming most CPU-time, and algebraic models the least. resting is how these newer models compare to the older, Below are the different approaches briefly discussed. non-DNS-tuned EVMs. The majorities of the different ideas when modifying/tuning turbulence models, such as damping functions, boundary conditions, etc. are inclu- 2.1 DNS

ded. A special section deals with the difference between ,

, The main reason why DNS is not used more frequently

¦ ¨ the secondary transported quantities ( ¡ ). Although there is neither any hope nor intention to including all when computing turbulent ﬂows is due to the fact that two-equation EVMs, quite a number of them are tabula- DNS is a VERY resource demanding computation: ted and referenced. ¢ DNS is a time accurate simulation.

ES Time 3D Aniso Trans ¢ DNS’s need to solve the full 3D problem, becuase DNS Y Y Y Y Y turbulence is always 3D. LES Y* Y Y Y Y

RANS-RSM N N N Y Y ¢ All length scales are resolved with a DNS.

RANS-EVM N N N N Y

0 0

V¤£

RANS-Algebraic N N N N N If the length scale ratio is deﬁned as § , where is the integral length scale, and V is the Kolmogorov length Table 1: Turbulence models and physics. ES: The ability scale, then using deﬁnitions it is easy to show that:

to predict the Energy-Spectrum, Time: whether or not

0 §

the computation is time accurate, 3D: if a 3D solution is A

¡ ¦¥ V (1) required, Aniso: if the model predicts anisotropic Rey-

nolds stresses, Trans: if turbulence is a transported or

V : local quantity. Assuming that § is proportional to the number

of § grid points, the computational mesh increases as:

A©¨ c

¡ . Additionally the timestep decreases with increasing Reynolds number and thus the computational time increases rapidly with Reynolds number as seen in Le-

schziner [28]:

L/L L LL/L E L/L LL/L E L ¨

M E L YLNM E L M E;L ENM E;L

N ¨ Time 37h 740h 6.5y 3000y

1Here it is inferred that an algebraic relation, contrary to a differential relation can be explicitly solved, without the need for an iterative solution. It should be noted that real world differential equations rarely have an analytic solution.

1 where N is the number of grid-points, and time is the the description of the eddy-viscosity, with one or more amount of time spent on a 150MFlops machine. terms that involve higher order ﬂow parameters. Non- Evidently from the this table DNS will not be of engi- linear EVMs can, as opposed to the standard EVMs, pre- neering practise in the foreseeable future. The useful- dict anisotropicy, which is of importance for eg. rota- ness of DNSs in respect to research on turbulence, and ting ﬂows. There also exists three- and four-equations as an aid when developing turbulence models, should ho- models, which use the two-equation model concept as

wever be recognized. a basis. The latter class includes models such as the

¢ ¡£¢¡ ¢ ¢ ¡£¢

model of Durbin [13] and the model of Suga et al. [48]. 2.2 Large-Eddy-Simulation (LES) The benefit of LES, and also its drawback, is the model- 2.3.2 Reynolds-Stress-Models ling of the sub-grid scales. The cut-off in wave-number space, enables higher Reynolds number flows to be si- Differential Models: The DSM (Differential-Stress- mulated, however at the expenses of accuracy, especially Models), RSTM (Reynolds-Transport-Stress-Models), or in the near-wall regions. The fundamental principal be- simply RSM solve one equation for each Reynolds stress hind LES is sound, because small scale turbulence is iso- and hence don’t need any modelling of the turbulence to tropic, and thus a simple model could be substitutes for the first order. These model are therefore referred to as the full resolution of a DNS in this region. For bluff-body second-moment closures, since they only model terms in flows this approximation is good as these flows are go- the transport equations for the Reynolds stress (third or ven by large scale turbulent structures. In wall-bounded higher moments). RSMs are numerically more deman- flows, simulations have however shown that the accu- ding, and generally more difficult to converge, compared racy diminish if the cut-off wave-number is not positio- to EVMs. ned in the viscous sub-layer. Thus the requirements for LES is similar to those of DNSs which need to resolve Algebraic Models: The ARSMs (Algebraic-Reynolds- all length scale down to the Kolmogorov wave-number. Stress-Models) or ASMs (Algebraic-Stress-Models) simplify the description of the Reynolds stress transport equations, so that they can be reduced to an algebraic 2.3 Reynolds-Averaged-Navier-Stokes relation. These relations are then solved iteratively, (RANS) due to its implicit construction. ARSMs are even more With a RANS approach the computer demands decrease numerically unstable then the RSM and hence are substantially, however at the expense of excluding the rarely used. multitude of length scales involved in turbulence. Wha- tever the complexity of a RANS-model, it could only com- Explicit Algebraic Models: In the EARSMs pute a single point in the wave-number - energy spect- (Explicit-Algebraic-Reynolds-Stress-Models) or EASMs rum, and thus it is questionable that such a model could (Explicit-Algebraic-Stress-Models) the Reynolds stres- be of much interest. Surprisingly RANS-models still per- ses are described in an explicit formulation, and are form reasonable in many flows – even though its non- thus more easily solved. Several different approaches physical foundations. Consequently these models are have been postulated which involve varies approxi- used extensively in CFD programs. RANS-models can mations on the way to the explicit formulation of the be divided into two major categories, the EVMs and the Reynolds stress, though they are essential similar to the Reynolds Stress Models (RSM). non-linear EVMs, although with more sound theoretical foundations. 2.3.1 Eddy-Viscosity Models Two-equation relations: The eddy-viscosity models

(EVMs) include the commonly used and well-known two-

¢ ¡ equation models, such as the model. The concept behind the eddy-viscosity models are that the unknown

Reynolds stresses, a consequence from the averaging-

= =

D ]

, procedure, are modelled using ﬂow parameters and an eddy-viscosity. EVMs are sub-divided dependent on the way the eddy-viscosity is modelled. Obviously the two-equation models use two equations to describe the eddy-viscosity, while the algebraic or zero-equation models use, normally, a geometrical relation to compute the eddy-viscosity.

Non-linear relations: Based on the popularity of the two-equation models, several different extensions have been proposed, which attempts to improve upon the de- ﬁciencies of these models. The non-linear EVMs extend

2 3 Eddy-Viscosity Models (EVMs) scale is computed using the mixing length and the velocity gradient as:

3.1 Boussinesq Hypothesis

¥ C (7) The eddy-viscosity concept is based on similarity reasoning, with turbulence being a physical concept connected to the viscosity. In the Navier-Stokes equation the The mixing-length model thus becomes:

viscous term is:

¡ C

¥ Mixing-length model

¡ $

¦ ¦% ¦%

=¦¥¨§

S ¤£

(2)

¦© ¦© ¦© with the Reynolds stresses given by the Boussinesq hy- It can be argued that similarly to viscosity, turbulence pothesis, Eq. 5. The mixing-length is closely connected affects the dissipation, diffusion and mixing processes. to the idea of a turbulent eddy or a vortex. Such an eddy Thus it is reasonable to model the Reynolds stresses in a would be restricted by the presence of a wall, and hence fashion closely related to the viscous term. The Reynolds the length scale should be damped close to a wall. The

stress term produced by the Reynolds-averaging is: idea by Prandtl that the turbulent length scale varies li- =

A nearly with the distance to the wall may be used as an

¡© 3 = 3

< <

¦ ¦

S S ¢

(3) initial condition for the turbulent mixing length scale:

¦© ¦©

1 WC

S Prandtl A turbulent ﬂow will, compared to a laminar ﬂow, enhance the above properties, and thus a model for the The proportionality factor, W (the van Karman con-

Reynolds stress could be: stant) is determined through comparison with experi-

L E

¥ W C

3!<= 3 < 8 $

, , , ¦% %

¦ ment, with and is the wall normal distance.

P C O

¢ S © 8 £

(4) The mixing-length model makes the eddy-viscosity

¦© ¦©

local, in the meaning that the turbulence is only directly

8 P

where , is a fourth rank tensor that could have both affected by the surrounding ﬂow, through the local value

§ spatial and temporal variations, as well as directional of . properties (anisotropic). In the EVM concept this coeffi- The accuracy of this model is only reasonable for cient looses the directional properties, and hence turbu- simple flows, and only in the logarithmic region, see Fig. lence becomes isotropic, the spatial variation is modeled 1. using some algebraic relation, while the temporal variation is in most cases dropped. Thus the eddy-viscosity, 3.2.2 Modified mixing-length models

¡ , using the EVM concept is incorporated in the RANS equation as: Attempts to extend the validity of this model have been

= made several times, where the most notable are the

3 3 $

<= <

¦% , , ¦%

C O ¦¥

= damping close to the wall by van Driest [50], the cut-

¢ S ¡ © £

(5)

¦ © ¦© off by Escudier [14], and the freestream modiﬁcation by Klebanoff [21]:

This method was ﬁrst postulated by Boussinesq [4] and

1 1

¢¦ ¢ §G¨

consequently denoted the Boussinesq hypothesis. S van Driest

1 1

,^L L%$ "!#

S Escudier

3.2 The Mixing-length Model and the Lo- 1

Klebanoff

C § garithmic Velocity Law I If one accepts the Boussinesq hypothesis, then it de- The van Driest modiﬁcation is an empirical damping pends on the description of the eddy-viscosity, how the

function that ﬁts experimental data, and also changes C

turbulence model will perform. Naturally the eddy- C

¡ the near-wall asymptotic behaviour of , from to c . viscosity should depend on turbulence quantities so- Although neither of them are correct (DNS-data gives

mehow. The turbulence modelling community still de- C

: ¡

¥ ), the van Driest damping generally improves the bates regarding which parameters, are the most appro- predictions. It has, since its ﬁrst appearance, repeatedly priate. By noting that the eddy-viscosity has the dimen-

been used in turbulence models to introduce viscous ef-

§/H¢ sion of ¡ , the most obvious choice is to model the fects in the near-wall region. eddy-viscosity using a velocity scale, and a length scale, The cut-off to the turbulent length scale is based on si- as:

milarity to the defect layer modiﬁcation by Clauser [10].

¡9S (6) Escudier however found that for boundary-layer flows a different and lower coefficient, as given above, was more appropriate than the one used for wake flows. 3.2.1 Prandtl’s mixing-length model The Klebanoff modification originates from the expe- One of the first turbulence model to appear, the mixing- rimental studies of intermittency, where it was found

length model by Prandtl [39], used the turbulent mix- that the ﬂow approaching the freestream (from within

= ing length scale, 1 , as the length scale. The velocity the boundary layer) is not always turbulent, but rather

0.2 25 0.18 DNS

log-law !

0.16 20 "

0.14

0.12 15 # 0.1

PSfrag replacements 0.08 10

0.06 DNS ¢¤£¦¥¨§ © PSfrag replacements ¡ DNS ¢£¥¨§ 0.04 Prandtl 5 vDriest+Escudier 0.02 vDriest+Klebanoff

0 0

0 0.2 0.4 0.6 0.8 1 −2 −1 0

)

10 $%'&)(+*-,/.1010 10

Figure 1: Turbulent length-scale in fully developed 'B

%$ ' Figure 2: Log-law. DNS-data, Kim [33],

B et al.

$L 'GB

channel ﬂow. DNS-data, Kim et al. [33], S and

%$/L ' B (thin lines) and (thick lines)

S , and mixing-length models.

Then the following is true:

changing from laminar to turbulent intermittently. Kle-

¨;©

< <)2

banoff introduced a factor, , which reduced the eddy- ¢ (9) 43

viscosity to model this effect. Here however for clarity,

3 <

the modiﬁcation is applied to the turbulent length scale, <

Using the mixing-length model for ¢ an expression instead of, as devised, the turbulent viscosity. for the velocity proﬁle could be ﬁnd as:

Invoking the Boussinesq hypothesis, Eq. 5, and the

Prandtl model, then in a fully developed channel ﬂow,

W C

< <

where only and is of any importance, the tur-

bulent mixing-length scale is computed as: E (10)

` `

Sb S #

B W65

< <

C (8)

§ where , based on experimental data. The accuracy of this law, as can be seen in Fig. 2, is – not surprisingly – In Fig. 1, the turbulent length-scale computed using only acceptable in the logarithmic region. The standard

this relation with a priori DNS-data of Kim et al. [33],

A A

$ $L

practise of plotting the velocity in a linear-logarithmic

' '

B B S S and , is compared with the stan- graph (as done here) hides the discrepancies of the log-

dard (Prandtl’s) mixing-length model, and the modiﬁed law very well. In the ﬁgure the validity of the van Kar-

versions. man constant, W , and the constant, , using the DNS-

As notable from the ﬁgure the turbulent length scale

)

L data are also shown.

E;L

W W

S ¨

models are very inaccurate beyond § . The In the ﬁgure is plotted as . The two horizontal

L E ¦

van Driest damping function improves the result for ¨

)

L lines show the commonly accepted values of and

§ . Prandtl’s mixing-length model is only rea-

)

E L

L for and , respectively. As noted is well represented,

C ¨ sonable accurate within § , which is not although the value appears to be on the low side. The

a very attractive situation. Although its limitations this van Karman constant is however not a constant at all,

A A

'GB ' B S

model forms the basis for all eddy-viscosity models, and and for these two DNS-data sets ( S and

87 $/L L E

W C `

also gives the well-known logarithmic law for the velo- ), S is within error only between

$:9 E

city proﬁle. and ` respectively.

3.2.3 Law-of-the-wall 3.3 Algebraic or Zero-equation EVMs Assuming that we have fully developed channel ﬂow Although its erroneous predictions and non-generality, with: the mixing-length model has formed the basis for other turbulence models, even recent one. The Cebeci and i The convective terms are negligible. Smith [8], and Baldwin and Lomax [3] have had some ii The total shear is constant (and equal to the wall success, especially in airfoil design. The accurate pre- shear). diction of these model is however more contributed to the introduced ad hoc (or empirical) functions and con- iii The viscous shear is negligible compared to the tur- stants, rather than any additional physics included in bulent shear. the models. Since the zero equation models don’t have

any transport of turbulence, they cannot be expected to 5 ¤¦¥

accurately predict any flows which have non-local me- § chanisms. The most important of these mechanisms 4 is the history effect, ie. the influence of flow processes downstream the event. Numerical simulations with the 3 zero-length models are thus usually restricted to atta- ched boundary-layer flows, which can be modelled using 2 only local relations. See however Wilcox [54] for a tho- ¨ rough discussion on the algebraic models and their performance. 1

0 3.4 One-equation EVMs PSfrag replacements −1 In order to avoid the local behaviour of the mixing- length turbulence models, a transport equation is nee- −2

0 0.2 0.4 0.6 0.8 1 )

ded for some turbulent quantity. A model that could C §

conserve turbulence should improve the predictions in

¤ C

ﬂows that depends on both the streamwise position and Figure 3: Proﬁle of and its exponential, , -

%$ '/B

the wall-normal (cross-stream) position. dependency. DNS-data, Kim et al. [33], S (thin

$L ' B

A most interesting turbulent quantity is the trace of S =

= lines) and (thick lines).

A 3 3 $ $

L L

< < < < < <

f f

the Reynolds stresses: S ,

which is denoted the turbulent kinetic energy, . It is

reasonable to believe that for increasing normal stres- with, =

ses, the shear stresses would also increase, and hence

598 3 < 3 <

¦%

S ¢

can be used to determine in the Boussinesq relation.

As previously shown, Eq. 6, the eddy-viscosity is gene- ¦©

3 <=

rally described as the product of a velocity scale and a ¦

¡aS

length scale. Using the turbulent kinetic energy as the ¦©

< <

transported quantity, the eddy-viscosity is modelled as: 6

[ 8

S ¢

¦©

¥ (11)

3 F 3 3

= = 8

< < < ¦

S ¢

¨ Such a scheme is used in the model by Wolfshtein [55], ¦©

where the turbulent length scale is pre-described using

¦ ¦

an algebraic expression, based on geometrical condi- £

¦©

tions. Although the soundness of including a transport ¦©

5 [ 8 equation, the one-equation models do not improve the 8

where is the production, ¡ the dissipation, the

8 ¢ predictions greatly compared with the zero-equation mo- pressure-diffusion, the turbulent diffusion and 8 the dels, mainly due to the required a priori knowledge of viscous diffusion term. the length scale. Thus apart from the Wolfshtein model The importance of the individual terms are shown in there exist rather few one-equation models. The recent Fig. 4, where each term is plotted as a fraction of the Spalart-Allmaras model [46] which solves a transport absolute sum of all terms. This was done because the equation for the turbulent viscosity itself, has however sum of the terms is zero for fully developed ﬂow, where had some success. See Wilcox [54] for further informa- the convective term is negligible. tion. In the near-wall region, the dissipation is balanced by the viscous diffusion (barely notable on this scale), while in the off-wall region although still close to the wall (the 3.5 Turbulent Kinetic Energy Equation buffer layer) the dissipation plus the turbulent-diffusion

term balance the production term. In the region from

) )

L E L

C C

S § S The turbulent kinetic energy appears in almost every § to the dominant terms are the pro- EVM and hence it is of interest to study this quantity in duction and dissipation terms. The commonly accepted

depth. The proﬁle of and the wall-distance dependency production equals dissipation in the logarithmic region

C¡

of , (ie. the exponent in the relation, ¥ ), for fully is a good approximation. Towards the centre of the chan- developed channel ﬂow (DNS-data) are shown in Fig. 3. nel the production gradually decreases, while the turbu-

lent diffusion increases. In the middle of the channel

)

The transport equation for , as derived in Bredberg ( § ) the dissipation equals the turbulent diffusion [5] is: term. As noted, nowhere is the pressure-diffusion term

crucial for the balance which is also reﬂected in the in-

$ [ $ F $ £¢ 5

8 8 8

8 cluded turbulence models where this term is seldom mo- ¢§¡ (S (12) deled.

5 boundary condition. 0.5 2. The way the turbulent viscosity is modelled.

3. Modelling of the exact terms in the ¡ -equation.

0.1

3.7 Secondary Quantity 0

−0.1 Four different types of two-equation model were men-

¢ ¡ ¢ ¦ tion above, of these only two, the and the are used frequently. However at this stage, for comple-

teness, all four will be assessed. Noting the dimension of

§/H¢ the eddy-viscosity, ¡ , it is a matter of dimensional analysis to combine the correct powers of the turbulent PSfrag replacements −0.5

kinetic energy and the secondary quantity to establish a

0 0.2 0.4 0.6 0.8 1 ¡

) relation for as:

S ¤£7"

5 F [

8 8 8

, , , ,¢¡ ,

¢ (13)

¡ ¢ ¢ ¢ ¢

Figure 4: ¢ , , , , DNS-data, Kim

A A

%$ $L

' '

B B S et al. [33], S (thin lines) and (thick where the following two expressions need to be fulﬁlled:

lines). d

¡ (¢ ¨KS ¨¦¥ ¡ ¨§/¢

¡ E

H¢ ¢ S ¢ ¨¦¥ ¡ ©§G¢ 3.6 Two-equation EVMs ¡

In this report, as previously stated, the main focus will The different secondary turbulent quantities have the

¡ ¡ ¡

¢ ¡ ¡ §/;: ¢ ¨ ¡ H¢

be on two-equation EVMs, which all use the turbulent following dimensions; ¡ , , , and

¡ / ¢ kinetic energy as one of the solved turbulent quantities. ¦ , respectively, and hence the turbulent viscosity

Apart from the transport equation for , the models add need to be modelled according to:

another transport equation for a second turbulent quan-

1 1

tity. The main difference between these models is the ¢ ¥

choice of this quantity.

¥¢§¡ ¡

The commonly accepted idea is that the eddy-viscosity ¥ ¡

may be expressed as the product of a velocity scale and a ¡

£¢©¨ ¨

length scale. Thus the obvious choice would be , as used

in the Wolfshtein model, Eq. 11, to combine , with

1 ¥¢§¦ ¡ ¥

. Although its logical construction this combination has ¦ not been used with any two-equation turbulence model. Instead of a velocity-length scale model the overwhel- In order to give some idea about the likelihood of suc- mingly majority of the used two-equation EVMs are ba- cess when employing any of the above combinations, the

secondary quantities are discussed below in regards to

¡ ¡ sed on the ¢ concept, which uses the dissipation, , in both their wall dependency and boundary condition. the -equation to construct the eddy-viscosity. The sub-

sequent relation is based on dimensional reasoning, and

1 ¢ as such is no improvement compared to a model. 3.7.1 Wall dependency

However using the ¢©¡ concept one avoids the additional complication of how to model the dissipation rate in The secondary quantities using the above relation for

the -equation. the turbulent viscosity are plotted in fully developed

¢ ¨ Another version is the model, which uses the tur- channel ﬂow, using a priori DNS-data. Fig. 5 gives the bulent time scale to construct the eddy-viscosity. This proﬁles in the channel while Fig. 6 shows a close-up of

type have an advantageous boundary condition as com- the near-wall region.

¢ ¡ pared to the -type. The only model using this con- The behaviour of the secondary quantities are sub- cept, the Speziale, Anderson and Abid, [47], is however divided into three different areas: numerical unstable and have not been a success.

¢©¦ 1. the core of the channel, spanning from roughly

) )

E E

The models originally developed by Kolmogorov L

C C

S § S [23], however more recently promoted by Wilcox [52], § (the logarithmic layer and uses the reciprocal to the time scale or vorticity. This defect layer), secondary quantity is however more commonly referred to as the speciﬁc dissipation rate of turbulent kinetic 2. the near-wall region, although away from the wall energy. (buffer layer), and The major differences and also beneﬁts of using either 3. the immediate wall region (viscous sub-layer). of the above mentioned types of two-equation EVMs are:

1. The used secondary turbulent quantity, and its

1 3

10 ¤£¥

¤ 10

§¡ £¢

§ £¢

§§

§ £¢ ¤

¤ ©

§ £¢

2 ¤

§ ¢

¡ ¡

Lines:

¡ ¡ ¡

Lines:

1 10 0 10

0 10 PSfrag replacements PSfrag replacements

−1 10

−1 10 −2 −1 0 10

10 10 0 1

)

C 10 10

Figure 5: Variation of secondary quantity, whole region.

'GB Figure 6: Variation of secondary quantity, near-wall re-

$L S DNS-data, Kim et al. [33], . 'GB

gion. DNS-data, Kim et al. [33], S .

)

L L L E

C C C

` ` " §

C C

C 3.7.2 Boundary condition

C C %

¡ const The physically correct boundary condition for the re-

C C C

¨ spectively secondary quantities are:

C C C

¦ % %

Table 2: Secondary quantities computed from DNS-data.

¡G©

, ,

, constant

S ¡ § ¡KS §¢ ¡ ¦ S ¨§¢ ¡ ¨ S ¡ §

. L (14)

¨H© S

¦ ©

An approximate wall-dependency is given by table 2. 1

Note that the variables smoothly changes from one re-

© © ¦ wever for both ¡ and there are a number of different gion to another, and not in the discontinuous way indica- choices in the speciﬁcation of the boundary condition. ted by the table. Observe also that the secondary quan-

tities are calculated using the indicated relations with ¡ ¡

¡ Using DNS-data [33] it has been shown that is ﬁ-

DNS-data for ¡ and . The asymptotic behaviour does hence not need to match the physical correct boundary nite and non-zero, at the wall, however it might not be a conditions of Eq. 14. The discrepancies between these constant, since DNS-data appear to indicate a Reynolds two relations are normally corrected using van Driest number dependencies. The two most adopted models for

type of damping functions in turbulence models. the ¡ boundary conditions are:

In order to visualize the accuracy of the different se-

¡G© S ¨¥

condary quantities they are approximated using powers C (15)

C ¡

of in the ﬁgures. The ¢ model is the worst case as

indicated. ¡ in the core region seems to decrease slightly

¡ S ¨¥ C

C (16) %

more than the tendency given by the table, see Fig.

5. In addition it is impossible to curve-ﬁt ¡ in the near-

wall region with either a -line or a constant approach, which becomes identical if ¥ . The dissipation rate, ¡

Fig. 6. ¡ , can either be modelled as the true dissipation rate, ¡

or a reduced version, . The reduced and the true dissi-

¢ ¨ ¢ ¦ The and models are a mirror image of each pation rate are connected according to:

other. These models are best ﬁtted using a linear appro-

¡aS ¡ ¢UT¡ ach in the viscous sub-layer, and then smoothly chan- (17)

ging to a quadratic variation. In the core of the channel

¡ ¡ ¡ the linear wall dependency is only approximative, ho- where T is the boundary value for . If is solved, a

wever on a slightly more accurate level than for the ¡ . zero boundary condition is imposed in the code, and an

The simplest type to curve-ﬁt is the ¥¢ model which additional term is necessary in the -equation.

follows a quadratic behaviour from the wall outwards to

C ¦

a rapid change to around the start of the logarithmic ¦ The boundary condition for is a consequence 1

region. however behave spuriously in the centre of the of both its deﬁnition and the construction of the ¦ - © channel as seen in Fig. 5. equation. Following Wilcox, the ¦ is given through

2.5 it is possible, although a bit tedious, to derive the exact

¢¡ ¡

2 § -equation, see Bredberg [5]:

5 $ 5 $ 5 $ 5 $ F $ [ $ £¢ \

¤ ¤ ¤

¤ ¤ ¤ ¤ ¤

¢ 1.5 S (21) 1

where

3!<

3 =

0.5 <

¦ ¦

S ¢ ¨¥ 8 8

" 0 Mixed production

¦© ¦© ¦©

3 = 3 = <

−0.5 <

¦ ¦ ¦%

8 8

¢ ¨¥

S Mean production

¦© ¦©

−1 ¦©

3!<=

5 3 <

−1.5 ¦ ¤

PSfrag replacements :

S ¢ ¨¥ 8 8

Gradient production

¦© ¦© ¦©

−2 <

= =

3 < 3 <

¦ ¦

8 8 ¢ ¨¥

−2.5 S Turbulent production

¦© ¦© ¦ ©

0 0.2 0.4 0.6 0.8 1

)

F 3

S ¢ ¡

Turbulent diffusion

¦ ©

C ¡

Figure 7: Proﬁle of and its exponential, , - ©

3!<=

[

¦ ¦ ¦

dependency. DNS-data, Kim et al. [33], (thin

%$/L

S ¢ ¨ 8 8 '

B Pressure diffusion

¦© ¦ © ¦©

lines) and (thick lines).

¦ ¡

S 8 8 Viscous diffusion

equating the destruction and viscous term in the ¦ - ¦©

2 ¦ ©

3 =

equation . The resulting boundary condition becomes: <

S ¨ £¨

Destruction

¦© ¦ ©

¦ S

C (18)

Comparing the ¡ -equation with the -equation, Eq. 12,

This value is not entirely compatible with the deﬁnition

the equations include the same type of terms, although

¦ S ¡¥§ ¡ of and the above boundary condition for , the ¡ -equation have several production terms.

although there is only a constant which separates them:

Q Q

8 8 /L

For turbulence modelling purpose, the -equation is a

£ ¤

S § S ¦ S ¨¥§ J S ¨¨

¦ vs . Furthermore

E L severe obstacle as only a single term, as opposed to all Menter [32] included, an additional factor of , hence but one in the -equation, can be implemented exactly. the exact value of ¦ at the wall seems to be of little im- Even employing a RSM type of model the situation does

portance. not improve, as all terms, apart from the viscous dissi- ¢©¦ In the model of Bredberg et al., [7], through the pation term, include ﬂuctuating velocities which are not use of a viscous cross-diffusion term, a slightly different solved using a turbulence model. The other terms are

boundary condition is obtained: not normally modelled using various simpliﬁcations. In

¨ EVMs these terms are lumped together either as pro-

¦ S

C (19) duction, destruction or diffusion terms, without much physical background, apart from that the exact terms

This is identical to the boundary condition of ¡ in the mainly are a source, a sink or enhancing diffusivity to

¢ ¡ ¦

Chien model [9] if the deﬁnition of is used to the equation. ¦ translate ¡ into . Unfortunately the individual terms are not available in the open literature, and hence an importancy control 3.8 Dissipation Rate of Turbulent Kine- of the different terms, similar to Fig. 4 can not be inclu- tic Energy ded here.

Based on the complexity of each term in the ¡ -

, ,

¦ ¨ Of the above secondary quantities ( ¡ and ) there ex- equation, and also the number of unknown involved, it is ists only an exact equation for one of them, namely the quite understandable that their exist a magnitude of dif-

dissipation rate of turbulent kinetic energy, ¡ . Choosing ferent approaches to the modelling of this equation. One another turbulent quantity as a secondary variable, the requirement for these models are that the near-wall be-

resulting modelled equation is necessarily derived on re- haviour of the individual terms in ¡ -equation, given by

asoning from the ¡ -equation. the following table, are truthfully obeyed.

5 5 5 5

C C C

Figure 7 shows the distribution of the dissipation rate C

¤ ¤ ¤ ¤

¥ ¥ ¥ ¥

F [ \

C C¦¥ C§¥ ¦¥

in fully developed channel ﬂow for the two sets of DNS- C

¤ ¤ ¤

¥ ¥ ¥ ¥

data available. Using the deﬁnition of ¡ :

3 = 3 = <

< As the table indicates, in the near-wall region the most

¦ ¦

¡ (20) important terms to model are the pressure-diffusion,

¦© ¦ © viscous diffusion and destruction (dissipation) terms.

2The other terms in the -equation being negligible in the viscous See Rodi and Mansour [43] for further information on

sub-layer. the exact ¡ -equation.

8 3.9 EVMs Treated in this Report 4 The Classic Two-equation There exists a number of two-equation EVMs today, with EVMs an ever increasing number, as these models are fairly easy to modify, tune and improve. It is thus not pos- In the present paper, classic models is defined as mo- sible, not even necessary to include all models in a re- dels which have not used DNS-data to fundamentally port like this. Choosing models with different secon- alter the modelled equations, but rather base the model- dary quantities, and of different complexity (number of ling on comparisons with experiments and through more terms/damping functions etc.) a limited number of mo- or less heuristic theoretical explanations and simplifica- dels is sufficient to exemplify and compare nearly all ex- tions. Into this group falls naturally all models which isting two-equation EVMs, at least at a theoretical level. dates prior to the first DNSs: JL, C, LSY, but also some Thus similar to previous papers on this subject, such of the later models which either have not used DNS data as Patel et al. [36], Sarkar and So [44], and Wilcox [53] bases at all: WHR, or merely used them to tune con- around ten models have been chosen. These are: stants and damping functions: YS, AKN, WLR, SAA.

The other models: YC, NS, HL, RS, PDH, BPD, have ¢

£¢¤¡ models: used either DNS data or TSDIA results as the founda- Yang and Shih, 1993 [56] (YS) tion for their theory. Yoshizawas [59] TSDIA (Two-Scale Direct-Interaction Approximation) analysis is here trea- Abe, Kondoh and Nagano, 1994 [1] (AKN) ted, similarly to DNS, as a numerical advancement rat-

Yoon and Chung, 1995 [58] (YC) her than an improvement in physical understanding.

¡ £¢ models:

Jones and Launder, 1972 [19] (JL) 4.1 Turbulent Viscosity, ¢¡ Chien, 1982 [9] (C) Of the four different types of two-equations EVMs discussed earlier, only three types are used for the turbu-

Launder and Sharma, 1974 [25] 1 ¢ lence models listed above. The version has so far not with Yap-correction, 1987 [57] (LSY) been used as a basis for an EVM. The turbulent viscosity Nagano and Shimada, 1995 [34] (NS) for these three types is computed according to one of the

Hwang and Lin, 1998 [17] (HL) following notations:

Rahman and Siikonen, 2000 [40] (RS) ¡

¥¢§¡ S

¢ ¡

£¢¤¦ models:

¤ ¤

¡ S

Wilcox, 1988 [52] (WHR) ¥¢§¦

¤ ¤

¡ S ¨

Wilcox, 1993 [53] (WLR) £¢©¨ ¤ Peng, Davidson and Holmberg, 1997 [38] (PDH) ¤

in the above formula can be established from equili-

¤ ¤

L L $ Bredberg, Davidson and Peng, 2001 [7] (BDP) brium ﬂow, where in the logarithmic region 2 .

¢ However both DNS-data, Moser et al. [33], and expe-

£¢©¨ ¤ model: ¤ riments indicates a reduction of in the near-wall re- Speziale, Abid and Anderson, 1992 [47] (SAA)

gion.

¤ ¤ ¤ The ’correct’ , or effective ¤ can be computed as:

The models are henceforth referred to with the abbrevi-

< <

ation indicated above. C

¢ § §

¤ (22)

3 "

¨§ ¡ ¨ § ¦ ¢ ¡ ¢ ¨

where " is either , or for the , and

¤ ¤

¢ ¦ type respectively. The effective for the different turbulence models is compared with DNS-data in Fig. 8. Note that for EVMs, which do not compute the Reynolds

stresses, the Boussinesq hypothesis , Eq. 5, is substitu-

S ¡¥§ ¢ ¡ ted into the above equation as: ¡ ( model).

4.1.1 Near-wall damping ¤ The agreement between DNS and the constant ¤ is very poor in the near-wall region. The inclusion of a fun-

ction which reduces (damps) ¢¡ in the near-wall region can signiﬁcantly improve the result as notable. Models which employ damping function, are commonly referred to as Low-Reynolds-Number (LRN) turbulence models, and differs from their High-Reynolds-Number (HRN)

9 0.12 distance functions are frequently used:

0.1

A A

S ¢¤¡ S ¥¢§¦ ¡

0.08 ¡ type type

¡ ¦

PSfrag replacements C

A¡

0.06

¤ A

¤ 0.04 DNS

§ ¡

: c

0.02 C C £¢

LSY A

S ¨ YS 0

AKN C

¢ C

WHR S

¨§ ¡ −0.02 WLR I

SAA C

J −0.04

S S ¡

0 20 40 60 80 100 B7C

I `

S ¨H© § S

¤ Figure 8: ¤ in the near wall region. Fig. 9 shows their near-wall variation using DNS-data for fully developed channel flow. The most appreciated version is the first, because it doesn’t depend on the wall counterparts in that they can (and need to) be resolved distance at all, and hence simplifies the treatment in

down to the wall. complex geometries. The last variant should be avoi- B

ded, because it uses , which becomes zero in separa- B

The HRN-models use wall functions to bridge the near tion and re-attachment points. Employing a -based

wall region, and hence only concern about the value of

¤ ¤ ¤ ¤ damping function for re-circulating ﬂows is thus highly

in the inertial sub-layer, where is fairly accurate ¤

¤ questionable, with spurious results as a consequence,

L L%$

¢ ¡ approximated by . See also the complementary see the predictions using eg. the Chien model for paper on wall boundary condition [6]. the backward-facing-step and the rib-roughened case in Appendix B. Consult also Launder [24].

There are mainly two effects of adding a damping fun-

ction to ¡ : 200

180

¤ ¤ ¢ Reduce near a wall. §©¨

160 § §©

140 §

120

< <

Correct the asymptotic behaviour of . PSfrag replacements 100

¤ ¤

For the LRN-models, as seen in Fig. 8, the effective 60 £¥¤ is reduced, through the damping function, . 40

DNS-data show that the near-wall asymptotic beha- 20

viour of the involved variables in the deﬁnition of ¥¡ are:

C¦¥ C C¦¥

C 0

¡ ¦ ¨

¥ ¥ ¥

¥ , , and . Thus the damping 0 10 20 30 40 50

*¦¥

C C

¢ ¡ ¢ ¦

function should vary either as: ( ) or

¢ ¨

( ) to impose the correct near-wall behaviour of the

: ¡ Figure 9: Wall-distance variation of damping function

turbulent shear stress as ¥ . parameters. The immediate near-wall region is inserted in the upper-right corner of the ﬁgure.

4.1.2 Wall-distance relations

£ ¤ 4.1.3 Damping function, Nearly all LRN EVMs use an exponential function and a variable somehow related to the wall distance, in their The used damping functions for the turbulent viscosity deﬁnition of the damping function. The following wall- by the turbulence models are listed below:

¥¢¤¡ models: 4.2 Modelling Turbulent Kinetic Energy,

-equation

¤ ¤ £¢¤

¡9S

¡ The exact turbulent kinetic energy, Eq. 12, repeated

£¤ ¨

¢ here for clarity, is:

$ A

S ¦

E JL

¦ ©

3 =

3!<

E L L E/E

$ 3 < 3 < $

¦ ¦ ¦%

S ¢ ¦ ¢

C ¥

S ¢ ¢ £ ¢

¦© ¦© ¦©

£¤

$ A

S ¦

GL E

LSY ¤

£¤

A¡ A

E E% M E L NM E L

¡ :

$ S ¡ ¢¦ ¢ ¢ ¢ 3 $ 3 = 3 =

< < <

¦ ¦ ¦

¥ ¥¨§

£ £

A£¢ ¢

EaM E L

¥ ¥¥¤

¦© ¨ ¦© ¦©

¢ ¢

£¤

¥¨§

S ¢¦ £ ¢ ¡

E (23)

E ¡

M Note that the turbulent diffusion ( ) and the pressure-

[@8

¦¨§ ¢

LL £

A AKN

: diffusion ( ) terms are normally combined into one c

¡ term, see Eq. 27.

¢ ¡ In the standard model, the viscous term is the

¥¢¤¦ models: only exact term on the right-hand side, while the other

¤ ¤ £¢¤

are modelled using, in some case rather questionable as-

¡9S

¦ sumptions. Below the terms are treated individually.

£¤ E

S WHR

$ A

L L E;L 9

£¤

¡ §G¨

¨ 4.2.1 Production

$ A

E;L 9

E WLR

§G¨

¡ The production term in an EVM is:

= = =

¥¢©¨

$ 5 3 =>3

8 < <

¦% ¦% ¦% ¦%

model: 2

=¥

S ¢

¡ £

(24)

¤ £

¤ ¤

¦© ¦© ¦© ¦©

S ¨

$ `

E For fully developed channel ﬂow with only one velo-

¥ ¥

# S

£ £

97L A SAA

¡ city gradient component (in the wall normal direction,

§ ) most turbulence models predict an accurate level

with the models abbreviated as deﬁned earlier. ¤ £ of turbulent production. However in ﬂows with a high

The computed with these models are shown in Fig.

¢ ¡

£¢¤ degree of anisotropy, the linear EVMs ( etc.) fails

10. The relevant for DNS-data is computed by nor-

¤ ¤

L%$ L through their use of the isotropic eddy-viscosity, . In

malizing Eq. 22 with the standard value of S . It

£¢¤ order to rectify this an extension to the Boussinesq hy-

should be noted that never reaches unity using DNS-

¤ ¤

L $ L pothesis is needed. In non-linear EVMs the introduction data, hence the value of S is slightly too high. of higher order terms in the relation for the Reynolds stresses improves the predictions in ﬂows involving eg. rotation, curvature and re-circulating regions. 1.2

4.2.2 Dissipation 1

PSfrag replacements The dissipation term is either modeled using its own ¢ ¡

0.8 transport equation, as in the models or via some re-

¢ ¦

lation with and the secondary turbulent quantity (

¥¢¤¨ £ ¤ and models).

DNS 0.6

¢ ¡ In the case of the models there is a choice for

LSY

¡ the dissipation rate: either the true, ¡ , or the reduced, .

C ¡

0.4 If the reduced is used, then there is a modiﬁcation, an 8

SAA & ¡ extra term, , in the -equation, since the true always YS

appear in the -equation. WLR

0.2 The reduced dissipation rate method is employed in & PDH 8 the JL, C and LSY models, with taking two different AKN 0 forms:

WHR 0 20 40 60 80 100

& 8

JL `

¢ ¡aST ¢ ¨¥

S C

£¢¤

Figure 10: in the near wall region. For legend, see

& 8

¢ ¡aST ¢ ¨¥ Fig. 8 S

C JL, LSY

When the true ¡ is solved (YS and AKN models), there The performance of the -equation for the different tur-

is no additional term in the -equation. bulence models in fully developed channel ﬂow are com-

¢ ¦

In the models, the dissipation term is model as pared with DNS-data in Figs. 14 and 15, Appendix B.

S ¦ ¦ ¡ . Dependent on the deﬁnition of , it may be necessary to introduce an additional coefﬁcient. In the

Wilcox models, ¡ is modelled as: 4.3 Modelling the Secondary Turbulent

Q Equation

@S ¦

¡ (25)

¢ ¡ In the case of the numerous models, the standard

with praxis for the modelling of the ¡ -equation is to use the

L L%$

modelled -equation as a basis, and then dimensionally J

S WHR

$b>A

map onto it, with added constants and damping func-

§ §

c ¡

L L%$

$ A

S WLR tions.

¡ §

¨ ¢ ¦ ¢§¨ The ¦ - and -equation in the - and - models,

could similarly be a direct dimensional transformation ¨ In the ¢ model (SAA) the dissipation rate is model-

led as of the -equation. Another possibility would be to use

¡ ¦

the modelled - and -equations to construct either a -

¨ @S ¡ (26) or a -equation. In all cases, the secondary transport ¨ equation would be essentially identical, and could sche- matically be displayed as:

4.2.3 Diffusion

= =

¤ £ ¤ £

$ $

¦% " " " ¦% ¦%

¢ § § ¡ £ ¡

Pressure-diffusion and turbulent-diffusion: In S

¦© ¦© ¦©

¢ ¡

the standard , as well as for other classic models,

the two diffusion terms are lumped together and mo-

$ [ $ & $ $

¦ ¦ " ¡ delled through a gradient model, known as the simple

¥ §

£¨

§ §

gradient diffusion hypothesis (SGDH): _

¦© § ¦©

= =

[ 8 $ F%8 3 $ 3 3

¦ < < <

S ¢

¨ ¦© (30)

(27)

5 \

¦ ¡ ¦

§ §

8 where is the production term, the destruction

[

¦© ¦© §

term, § the pressure-diffusion term, the diffusion

& §

term and an additional term. " is one of the secon-

¦ ¨

Viscous diffusion: dary turbulent quantity, ¡ , or .

£¢

¦ ¦

£ ¤£¦¥

(28)

¦ © ¦© 4.4 The Jones-Launder model and the Dissipation Rate of Turbulent Ki-

Since this term can be implemented exactly there is no ¥ need for a model. netic Energy, -equation

All the classic modeled ¡ -equations are developed by

4.2.4 The modelled -equation simply multiplying each term in the modelled -

¥§ equation by ¡ and adding a number of coefﬁcients.

Summing up, the classic modelled -equation is:

= =

Thus these -equations consists of a production, a de-

$ $ & 8 $

¦% ¦% ¦%

struction term, a viscous diffusion term (which is exact)

=¦¥

¢¤¡ S ¡ £

¦ © ¦© ¦© and a turbulent diffusion term.

¢ (29)

$ $

¦ ¦

¥ §

_ ¦ ©

¦ © 4.4.1 Production term

¢ ¦ ¢ ¨ where ¡ , in the case of the and models is The production terms in the exact dissipation rate equa-

substituted according to Eqs. 25 or 26. 8 & tion, should according to Rodi and Mansour [43], be dis-

The additional term represents the difference tributed into both the production and the destruction

between the true ¡ – always used in the -equation – term in the modelled equation. ¡

and the reduced . If the latter is used in the dissipation 8

& In the classic model, this is of course of less impor- ¡ rate equation, takes the value of T , deﬁned above. tance, because the modelling is more based on dimensio-

Apart from the treatment of the dissipation rate the nality reasoning than actually capture the physics term 5

difference between the ’classic’ turbulence models, is 5

¤ ¤

8 by term. Of the production terms in Eq. 21, and _

restricted to the Schmidt number, , which adopts the 5 ¤ should be included in the destruction term, while c is

following values: modelled as a production term:

= =

¥¢§¡ ¥¢§¦ ¢©¨

¤ £

5 ¡ ¦% $ ¦% ¦%

=¦¥

¡ £

JL LSY YS AKN WHR WLR SAA S (31)

¦© ¦© ¦© 1.0 1.0 1.0 1.4 2.0 2.0 1.36

A A

h ¡

where is a tunable damping function, which has been For the two extremes, ¡ and the following

set to in all of the described models. The constant ¤ relations holds:

£ ¤

E% , L

is set to:

£ ¤

E% ,

JL C LSY YS AKN h (38)

1.55 1.35 1.44 1.44 1.5

¢ ¡ ¤

: The Chien model simulates this behaviour very ac-

The term , see Eq. 21, could be argued to be negligibly £ curately using its relation for . small in comparison to the other production terms, however it could still inﬂuence the results, Rodi and Man- sour [43]. Thus in some models, (JL and LSY), an addi- 4.4.3 Diffusion

tional term is included to simulate its effect: ©

= Pressure and turbulent diffusion terms: Similar

¦ 2

to the -equation a simple gradient term is used to model

¨ S

¡ (32)

¦©

the diffusion effect:

[ $ F ¦ ¦ ¡

¤ ¤

Note however that this term imposes numerical pro- ¥

S £

_ (39)

¦© blems due to its second derivate and is only used in mo- ¦© dels by Launder et al. [19], [25]. The involved Schmidt numbers are:

4.4.2 Destruction term JL C LSY YS AKN

5 5 \

¤ ¤

¤ 1.3 1.3 1.3 1.3 1.4

The destruction term is the sum of the , and ¢ terms, and is modelled as:

Viscous diffusion term: As noted above this term

¤ £

5 $ 5 \

¡¢

¤ ¤

S (33) can be implemented exactly as:

£¢

¦ ¦ ¡

with a damping function, as follows: ¥

S £

(40)

¦© ¦©

E L

S ¢ ¦ ¢ ¢¡

¡ JL

E L ¡

¥ 4.4.4 The modelled -equation

¢ ¨/¨ ¦ £ ¢

S C

E L

The modelled ¡ -equation is the summation of the above

S ¢ ¦ ¢

¡ LSY

£ E terms, with the resulting equation similar to the model-

S YS

led -equation, Eq. 30 as:

= =

¡ E L E J

¥¨§ ¥

¢ ¦ ¢ £ S ¢¦ £ ¢

¤ ¤ £

$ $

¡ ¡ ¦ ¦ ¦ ¡

¤ ¤

=¥

S ¢

¡ £

¦© ¦© ¦©

AKN ¢ (41)

$ & $ $

¦ ¦ ¡

¥ §

¦© ¦©

The constant, ¤ , has the following values:

JL C LSY YS AKN ¤ is an additional term simulating the extra production

2.0 1.8 1.92 1.92 1.9 in the JL and LSY models. The predicted ¡ for fully developed channel ﬂow using the different turbulence models The reason behind the damping function is argued from are displayed in Figs 16 and 17, in Appendix B. the process of decaying turbulence. Townsend [49] gives

the decay as: ¥

4.5 Transforming the -equation

¤£

¥ (34)

E Contrary to the dissipation rate, , there exists no ex-

¥ S ¨ ¥ S ¨ ¨

where initially, but increases to in the ¦

A L act equation for the - and -equations, as previously

ﬁnal period, for which ¡ . In decaying turbulence, disclosed. Thus in order to model these quantities, the the only terms that are of importance are the destruc- equations need to be constructed. This can be done in

tion, and time derivate terms: two different ways are:

S ¢ ¡

(35) 1. Similarly to the modelled ¡ -equation, the -equation

is dimensionally changed through a multiplication

¤ £

¡ ¡

¦ § ¨ §

S ¢

(36) of either or to yield a new secondary trans-

port equation.

¥ ¦£

Solving the equation system for , with ¥ gives:

¦ ¨

E 2. The - or -equation can be constructed from a

¡ £

¤ transformation of the modelled - and -equations,

¥ZS

E (37)

¨ ¡¥§ ¨§ ¡

by substituting ¦ or with either or .

¡ ¨ The ﬁrst method is straight forward and it is not further Using the standard - and -equation the -equation be- discussed here. The second choice is much more compli- comes:

cated and also adds a number of new terms to the secon-

¤ ¤ £

5 $

¨ ¨ 8

E E

¤ ¤

S ¢ ¢

dary equation. Using this technique the new turbulence ¢

¢ ¡

model will be identical to the original model, ho-

$ $ $

¦ ¨ ¦ ¨ ¦ ¨ ¨

¡ ¡

¥ ¥

wever expressed in a new secondary variable with a dif- ¥

£ ¢ £¨ £

_ _

¤ ¤

¦© ¦© ¨ ¦©

ferent, and improved, near-wall behaviour. The trans-

¦ ¨

$ $ $

¡ ¡ ¦ ¦ ¡ ¦ ¨

formation to a - and -equation is treated separately ¨

¥ ¥ §

¢ 8

£ £

_ _ _ ¤

below. ¤

¦ © ¦ © ¦ ©

(45)

Note that there are some peculiar effects in the modelled 4.5.1 The ¦ -equation

¨ - equation:

The transport equation for the speciﬁc dissipation rate ¢ The production and destruction term using stan-

( ¦ ) obtained from the - and -equations reads: dard values for the coefﬁcients, gives that the pro-

E duction term decreases , while the destruction

¦ ¡ ¡ ¡ ¨

¥ term increases .

¤ ¤

8 8

S £ ¢ S S

E The destruction term is constant, and set according

5 \ $ [ $ F $ £¢

¤ ¤ ¤ ¤

S 8 ¡ ¢ ¢

¢ (42) to the used coefﬁcient ( ). This makes the equa-

8 $ [ 8 $ F%8£$

5 tion numerically stiff which is a disadvantagous fe-

¢ ¡ ¢¤¡ ¢

ature of the ¨ -equation.

Compared with the ¦ -equation an additional term ¡ Dependent on the complexity of the modelled - and - appears, namely the square of the derivate of ¨ .

equations there will be different versions of the resulting

¦ ¢ ¡ £

-equation. Using the standard model as a basis ¢

(Eqs. 29 and 41), the resulting transport equation for ¦ 4.6 Wilcox and the model becomes3 (see Appendix C for a thorough derivation):

It can be claimed that the ¦ -equation in the standard

¢ ¦

[52] was derived using the more complex method of

£ ¤ ¤

$ 598

¦ E ¦ E

¢ ¡ ¢ ¦ ¤

¤ transforming a model to a . It is however more

S ¢ ¢ ¢ ¦

likely that the simpler route of dimensionally modify the

$ $ $

¨ ¦ ¡ ¦ ¦ ¦

-equation was used, since the -model only composes of

¦© ¦©

a production term, a destruction term and the two stan-

$ $

$ dard diffusion terms (turbulent- and viscous- respecti-

¦ ¦ ¦ ¦ ¦

¡ ¡ ¡

§ ¥ ¥

£ £¨ ¢

_ _ _

¤ vely).

¦© ¦© ¦©

Thus using method 1, the -equation is multiplied

(43)

§ ¦

with ¦ to yield the -equation as:

£ ¤ ¤

$ 5 8 $

¦ ¦ ¦ ¦ ¦ ¡

The term which includes the second order derivate of

£ £ £

¥ §

¦ ¢ S £¨

¦© ¦ ©

could normally be neglected since the Schmidt numbers

_ _

¤ ¡

of and ( and ) are fairly similar. With identical

¢ ¦ Schmidt numbers (as in the Wilcox model) the term (46) is identical zero.

The Wilcox ¢§¦ models use a slightly different nomenclature for both coefﬁcients and Schmidt numbers, than

shown above; here however, for clarity, a more general

, P 4.5.2 The ¨ -equation nomenclature is used. The Wilcox nomenclature ( ) is

also noted above.

¢ ¦

Performing similar operations on the -equation as for The difference between the ¦ -equation of the few ¦

the -equation gives: models is restricted to, as in the case of the ¡ -equation: _

the values of the Schmidt number ( £ ), coefﬁcients and

¡ ¡

¨ damping functions.

S S ¢ S

¡ ¡

There is however, an additional (unnecessary) com-

5 $ [ $ F $ £¢

8 8 8

¢ ¦

plexity connected to models, which is the deﬁnition

¡ ¢¤¡ ¢ ¢

S (44)

of ¦ . There exist two different versions:

5 \ $ [ $ F $ £¢

¤ ¤ ¤ ¤

¢ ¢ ¡ ¢

Q ¦

¤ Wilcox

3 3

As the transformation only applies to the -equation, the - J

¤¡ ¤¡ ¨

equation in the model identical to original equation in the Q

L L%$ S model. with J .

¤ ¤

B B

_ _ ¤

There is no particular modelling reason for either of ¤

them, and hence the author thinks that the ﬁrst ver- 1.44 1.83 1.36 1.36 sion should be used in order the minimize confusion, as with the damping function deﬁned as:

well as showing the clear similarity between the ¡ - and

¢ ¢

A ¦

-equation. Unfortunately, Wilcox when designing the C

` ¨

E E

£¢¤¦ ¥ § ¥¨§

¦ ¢ S ¢¦ ¢ ¢

$ £ $

standard model, used the second formulation, and £ (50) since then that version has prevailed. There is only a few modelers that use the alternative version, eg. Abid Note that the ﬁrst part of the damping function is inclu- et al. [2]. ¤

ded in ¤ in the Speziale et al. paper. Due to the additional constant in the deﬁnition of ¦ for the Wilcox models the turbulent viscosity is modeled

as:

¡ (47)

E P

with J in order to predict the same level of tur- P bulence. For the WLR J , is damped close to the wall, while in the WHR it is unity, see Section 4.1.

4 ¢ ¦ The Schmidt numbers for the Wilcox models are: WHR WLR

2.0 2.0

¤ ¤ Q

P £

with the coefﬁcients £ and as:

WHR WLR ¤

£ 0.55 0.55

£ 0.075 0.075

As can be seen the ¦ -equation of the WHR and WLR mo-

dels are identical apart from an added damping function ¤ to the production term of the latter. £ is in the WLR-

model multiplied with the following relation:

$ A

L E E;L 9

¡ §G¨

$ A

E L 9

E (48)

§G¨ ¡

The two models are usually referred to as the HRN version (WHR) and LRN version (WLR) of the standard

¥¢¤¦ .

£¡ 4.7 Speziale and the model

The only two-equation EVM that utilizes the ¨ -equation is the model by Speziale et al. [47]. Speziale, contrary to Wilcox, took almost certainly the more complicated road toward the construction of a secondary transport equation, since all the additional terms, as presented in

Eq. 45, are present in the SAA-model:

¤ ¤ £

$ 5

¨ E E ¨

¤ ¤

¢ ¢ (S ¢

$ $

¨ ¦ ¦ ¨

¦© ¦©

(49)

$ $

¨ ¦ ¨

¥ ¥

¢ £¨ £

¨ ¦©

$ ¦ ¨ $ ¦

¥ §

¦ © ¦ ©

¦ ¨§¢¦© with the -term zero due to identical Schmidt numbers. The coefﬁcients used in the model are: 4Note that Wilcox use the reciprocal denotation for the Schmidt number, here however for clarity the standard praxis is used.

S ¨§ ¡ 5 The TSDIA and DNS Revolution time scale is not appropriate. Employing ¨ results

in that the time scale vanishes when approaching a wall, ¡ In this section models which have been developed using where is zero, and non-zero. Instead the common DNS-data or TSDIA, either directly or indirect will be praxis is to invoke the Kolmogorov time scale which is

treated. based on the molecular viscosity and the dissipation as

S § ¡

With the appearance of DNS-data for fully developed ¨ . Near a wall this relation is preferable as

A A

E /L %$

' '

B B S

ﬂow, Kim et al. [20], S , Kim [18], and both quantities are valid and non-zero.

$/L

£ B

¢ ¡ ¢ £¢ Moser et al. [33], , the modelling of turbulence Durbin [13] in his model, was one of models, and especially the dissipation rate equation, has the ﬁrst to introduce a Kolmogorov time scale, in the been improved. Several papers have analyzed the com- deﬁnition of the time scale:

puted DNS-data, eg. Mansour et al. [29], and Rodi and

Mansour [43], which have resulted in new proposals to ,

S ¤£

the modelling of the individual terms in the exact equa- ¨ (53)

¡ ¡

tions.

Yoshizawa [59] used a different approach to im-

§ ¡

with S . This relation is substituted with in prove the turbulence models. Based on the two-scale both the formulation of the turbulent viscosity, and in direct-interaction approximation (TSDIA), he came up the ¡ -equation. with new modelling strategies, which introduces cross-

The idea of a speciﬁc time scale is also used by the YS-

¢ ¡ diffusion terms in both the and -equations. and RS-models, however with slightly different deﬁni- In the paper these diffusion terms are, in the -

equation: tions:

¤ ¤

8 8 8 8

¦ ¦ : ¦ ¦ ¡

¥ ¥

£ £ S

¨ YS

¦ © ¡ ¦© ¦© ¡ ¦©

¡ ¡

(51)

¨S ¨

£ RS

¡ ¡

and in the ¡ -equation:

¤ ¤

$ 8 8 $

¦ ¡ ¦ ¦

¦ The reduced dissipation rate, , in the RS-model is com-

¤ ¤ ¤

¥ ¥

£ £

¡KS ¡ ¢ ¡ T ¡

¦ © ¡ ¦ © ¦© ¦ ©

puted as with the boundary condition for given

¡G© ¡aSb¨T § ¡

by .

¤ ¤ ¤

$ $ $

¦ ¦ ¡ ¦ ¡ ¦

8£¢ ¢

¥ ¥

£ £

¦ © ¡ ¦© ¦ © ¡ ¦© 5.2 Turbulent Viscosity

(52) F

Note that the standard turbulent diffusion terms, 8 and DNS/TSDIA-models use the same structure as the clas- F

¤ , respectively, are included and note also that the de- sic model in the formulation of the turbulent viscosity. notation of the coefﬁcient have been changed compared However the time scale modiﬁed EVMs (YS and RS) uses

with the paper. the following relation:

¤ £ ¤

Using these new ﬁndings a number of models have ¤

S ¨

` ¢

seen the light which mimick DNS-data for both and (54)

` ¢ ¡ proﬁles, and in some cases even their budgets. Rodi and Mansour [43] (RM), Nagano and Shimada [34] (NS) Note that the YS-model is classiﬁed as a classic model and Peng et al. [38] (PDH) used results from DNS ana- here, and is thus described in Section 4. The damping lysis, while Yoon and Chung [58] (YC), Hwang and Lin functions for the other models are: [17] (HL), Rahman and Siikonen [40] (RS) and Bredberg et al. [7] (BPD) used TSDIA results.

£¢¤¡ models:

5.1 Turbulent Time Scales?

¤ ¤£¤

¡9S

Although the idea of a speciﬁc turbulent time scale does ¡

not originate from either DNS-data or TSDIA, it is ques- £¤ ¦¥

S YC

£¤

L L/L

tionable that it would have evolved in such a way wit- E

S ¢ ¢ ¨

hout the aid of DNS-databases. ¡

¢ ¨ ¢¥¡

¡ E

Similar to the model, several models use the M

¥ ©§

¦¨§ ¢ £ L

¨ ¨§ ¡

turbulent time scale, , as a substitute for the ratio, . ¡

E L L E L L/L

¢ C

Little is however gained through only a nomenclature C

¢¦ ¢ ¢

S HL

A ¢ A

change and hence these models also included modiﬁca-

E L L E L L/L

¢¦ ¢ ¢ tions to the turbulent time scale. S RS The standard argument to introduce a speciﬁc time

scale is that near a wall the ﬂow is not turbulent any- Note that the RS-model use the turbulent time scale, ¨

§ ¡ ¢¡ more, and hence using turbulent quantities to deﬁne the instead of in the deﬁnition of .

16 ¥¢¤¦ models: The inclusion of cross-diffusion terms in the secondary

equation could also be the results of the transformation

¤ ¤ £¢¤

¡ ¦ ¨

¡9S

from the -equation to either a - or a -equation. In

¡ ¦

§ both cases there appear cross-diffusion terms (as well as

£¤

E ¡ L

L other terms) in the transformed equation. Through wi-

¢ £ S ¨ E L sely chosen Schmidt numbers (they need to be identical,

¦ see Appendix D) the cross-diffusion terms may be the

L L/L E

¡ M L $:9 ¡

only failing link between a transformed ¥¢ model and

A ¦ § ¢ £

LL PDH

¡ ¨

a ¢¤¦ model. Thus by adding a cross-diffusion term to

¢ ¦

the ¦ -equation, the behaviour of the model would

£¤

$ $

L $E L L%$

£ S

¢ ¡

be similar to the model; further information is given

: ¡

¦ in the next section.

M E ¡

A well documented discrepancy of the ¢¤¦ models is

¢¦¨§ ¢ £ BPD

¨ the sensitivity of the predictions to the freestream va-

lue of ¦ , see eg. Menter [31]. Thus by adding a cross-

¢ ¦

Strangely enough the YC-models damping function was diffusion term to the ¦ -equation the model could

¢ ¦

not described in the paper, and since near-wall results become more universal. models which include a

£ ¤ are presented, it is unlikely that is unity, as for HRN- cross-diffusion term are eg. Menter [32] Peng et al. [38], models. The model is anyhow included here, since it is Kok [22], Wilcox [54], Bredberg et al. [7]. one of few models that uses TSDIA results. The effect Another reason for the inclusion of these additional

of the various damping functions are shown in Fig. 11. terms is numerical. The classic versions of both the

¤ ¡

Note that for the DNS-data, is computed using, Eq. - and -equation are not well balanced close to a wall

L L%$

22, normalized with S . As indicated by the ﬁ- because there is a missing term: the pressure-diffusion. £ gure this value is slightly too high, as ¤ does not reach Since the classic models were mostly based on 1) expe- unity. riments – which are unable to measure the pressure- diffusion, and 2) theory – which shows its relatively 1.2 small importance, the pressure-diffusion term was with good arguments neglected. The appearance of DNSs have however improved the 1 understanding of the near-wall behaviour of turbulent quantities, and also supplied a valuable data base. 0.8 Using both near-wall asymptotic analysis of the individual term and the budgets of the transport equations,

£¤ the importance of the pressure-diffusion term close to a 0.6 wall can be established.

PSfrag replacements This is graphically shown for the -equation in Fig. 0.4 DNS 4, while an asymptotic analysis of the terms in the ¡ - RS equation is given by the table on page 8. Although

PDH hardly discernible in Fig. 4 the dissipation, ¡ , in the - 0.2 (8 BPD equation is balanced by the viscous diffusion, , and

the pressure-diffusion, [@8 in the near-wall region. Si- ¥ 0 C

0 20 40 60 80 100 miarly the non-zero ( ¥ ) (thus important) terms at

¤ `

the wall in the ¡ -equation are the destruction, , the

[

¢ ¤

viscous diffusion, ¤ and the pressure-diffusion, .

B S Figure 11: in the near wall region. Especially the classic ¡ -equation beneﬁts from the addition of a pressure-diffusion term, since in these models

the wall value for the dissipation is set using either a

5.3 Coupled Gradient and the Pressure- & ¡ term, , as in the -models (JL, C, LSY, RS, HL, NS). diffusion Process

Based on the Two-Scale Direct-Interaction Approxima- 5.3.1 The -equation

tion (TSDIA) of Yoshizawa [59] there appear a num-

¡ ber of coupled gradient terms in the - and -equation. The result of the TSDIA operation of the -equation, is These are usually referred to as cross-diffusion terms. two diffusion terms. The ﬁrst term is the standard dif- By cross-diffusion it is meant that the term is a pro- fusion term (SGDH), while the second term is a cross-

duct of gradient of both turbulent quantities, ie. eg. diffusion term.

¦ ¨§¢¦© ¦ ¡ §¢¦©

. Of the newer model included in this paper, three use

It is argued in several models that the cross-diffusion an additional diffusion term in the -equation. These terms are beneﬁcial to the accuracy of the model, and are the NS-, HL- and RS-models. Only Rahman- that they have a similar effect on the equations as the Siikonen cite the TSDIA results as the main origin for pressure-diffusion term in the exact equations. this term. Nagano-Shimada and Hwang-Lin argue the

17 inclusion of an additional term from near-wall asympto- The main argument for the term in the HL-model is

tic analysis. however the need for a well balanced -equation near NS and HL state that this additional diffusion term gi- the wall, for which the above model is well suited.

ves similar effect to the -equation as the exact pressure- The NS-model differs remarkebly from the HL-model,

diffusion term, even though their dissimilarities, and and its formulation is rather questionable. The ﬁrst

[

¦ ¨§¦© hence denote it . Rahman and Siikonen on the other term, , is identical to the viscous diffusion,

hand don’t give any physical reasoning behind the term, while the second term is very similar to the square-root

& 8

and simple denote it : of the boundary condition for ¡ used in the JL- and LSY-

models, see Eq. 15:

& 8

¦ ¡ , L ¦ ¨§ ¡

S ¡ "!#

¨ ¦© ¦©

¡7© S ¨ ¢

A JL, LSY

¦©

[ 8

¦ ¦

S ¡ § ¢ ¦ £ ¢

¦© ¦©

S ¦ ¢ ¨¥ ¡

¦©

, L M ¥

£ NS

¨ ¡ ¡

Note that ¡ has been substituted with in the NS-

[

¦ T

¦ formulation to simplify the comparison. In the limit of

S ¢

£ HL these are identical, and hence it is a valid substi-

¨ ¦© ¡ ¦© tution. The only difference between the two models is the ad-

The main difference between the three models are that

dition of ¡ inside the gradient and the exponential fun- the RS-model is purely a turbulent term, similar to the

TSDIA-result. The two others models include viscous ction in the NS-model. The addition of ¡ is of course effects, through the usage of the molecular viscosity. The necessary to ensure the correct dimension. Assuming a

latter is more plausible, if a comparison is made to the constant gradient of ¡ close to the wall it is fair to state ¤

¢ ¡

pressure-diffusion term, since a turbulent term become that the NS-model adds ¡ to the -equation [ 8 zero close to a wall where the pressure-diffusion term is through the term. most important.

Because there is no reference to the pressure-diffusion 5.3.2 The ¡ -equation process in the RS-model, it is of more interest to compare it with the TSDIA result, Eq. 51. Although the Compared to the -equation, the TSDIA result for the

formulations are similar, they are not identical, which ¡ -equation includes a wealth of different terms. In fact can be shown by expanding them and including the all the possible combination of gradients of ¡ and is re- stated coefﬁcients: presented and of course no EVM utilizes all these forms. Models that explicitly state that the origins for the cross- RS-formulation: diffusion terms are from the TSDIA are the YC- and RS-

models, while the other two models (NS and HL) argue

L L ¦ ¦ ¡ ¦ ¡ :

from a balancing point of view.

¢ £

As for the -equation, the additional term is simply de-

¡ ¦© ¦ © ¡ ¦ ©

: &

noted ¤ for the TSDIA models (YC, RS), while the NS- and HL-models states that they are adding a pressure- TSDIA-formulation (Eq. 51): [

diffusion term, and hence denote the term, ¤ :

L 9 ¦ ¦ ¡ ¦ ¡ : ¦ ¡ :

L EE¡ : ¦ ¡¥§ ¦

¢ ¨ ¢ ¨

¤ ¢

S YC

¡ ¦© ¦ © ¡ ¦© ¡ ¦ ©

¡ ¦© ¦©

[

¦ ¡ ¦

,^L

¥ §

S ¢

As can be seen there is one term missing in the RS- £ NS

¦©

formulation, but worse is that the terms have opposite ¦©

[

¦ ¡ ¦ ¤

sign. The coefﬁcients in the RS-model are also markedly ¥

¢ £

S HL

¦©

lower. ¦ ©

¡ ¨§ ¡ ¦

This latter fact is however of minor importance as co- ¦

¤ ¢

S RS

¨ ¦ © ¦ ©

efﬁcient are normally based more on numerical tuning than theoretical foundations. As a comparison the co-

efﬁcient for the SGDH term is also markedly different: Note that the expression for the YC-model is most pro-

E 9 L L%$

§ ¡ ¨ § ¡

¨ in TSDIA, compared with: , in the stan- bably erroneous, since the dimension is not correct. In

¥¢¤¡

: § ¡

dard model. the following the multiplier has been changed to

§ ¡

Although the HL-model is a viscous term, its construc- to yield the correct dimension.

§ ¡ tion is very similar to the TSDIA model. If in the Note also that in the RS-model, is deﬁned according

TSDIA formulation is exchanged with , the two models to what was stated in the above Section 5.1. becomes identical, with only the coefﬁcients different. The NS and HL models are almost identical, while the Naturally the behaviour of the molecular and turbulent YC (the corrected version) and RS are similar. RS uses viscosity are very different, and hence such a compari- the gradient of the turbulent time, while the YC-model

son is really not appropriate. uses its reciprocal, ¦ . Both models combine this with the

gradient of . The expression in front of the gradients Thus the common praxis, also used here, is to either ¦

are merely to correct the dimension using a combination make a direct substitution of ¡ to or do a proper trans-

¡ ¢§¡ £¢ ¦ of and , for all models. formation of a model to a model, as was done

A noteworthy reﬂection is that similar to the added in Section 4.6. Note however that already in the for- ¡

term in the -equation, the HL- and NS-models use a mulation of the -equation we are using modelled terms.

¢ ¡ viscous term, while the YC- and RS-models use a turbu- Thus dependent on the original model, there will lent term. exist a number of different ¦ -equations. A discussion re- The connection to the full derived expression of Yos- garding this is left to the next Section. Note also that hizawa (Eq. 52) is loose, however at least the YC- and the -equation will always be unaffected by this trans-

RS-model retains some of the terms in the TSDIA ex- formation.

¢ ¡ pression, which can be visualized through a simpliﬁca- The result of transforming the classic model was

tion of the expressions: shown in Eq. 43. The two additional terms in the ¦ - ¦

equation as compared with the standard ¢ model of

¦ ¡¥§ ¦ ¦ ¦ ¡ ¦

Wilcox are the cross-diffusion term:

S ¢

£ YC

¡ ¦© ¦© ¡ ¦© ¦ © ¦©

¨ ¦ ¦ ¦ ¡

(55)

¦ ¦ ¦ ¦ ¡ ¦ ¡ ¡ ¦ ¡

¦© ¦©

¥ ¥

£ S ¢ £ ¢

¦ © ¦© ¦© ¦© ¦ © ¦©

and the turbulent diffusion like term:

HL, NS

¦ ¦

¡ ¡

¦ ¨§ ¡ ¦ ¦ ¦ ¡ ¡ ¦

¡ ¡ ¡

¢ 8

¥ _

_ (56)

S ¢ £

RS ¤

¦©

¦© ¦© ¦© ¦© ¦©

All three models include the cross-diffusion term, The two models which are based on this transforma-

¡¥§¦© ¦ ¨§¢¦© ¦ , and the square of the derivate of . The tion, the PDH- and BPD-models, include only the cross-

NS- and HL-models also include the second derivate of diffusion term, while the second gradient of -term is

. dropped. In the case of the PDH-model only a turbulent Which of the different terms that are most important part is retained, while the BPD-model includes both a is not reported in the Yoshizawa paper. From a numeri- viscous and turbulent part of the cross-diffusion term:

cal standpoint it is however disadvantageous to include

¦ ¦ ¦ ¡

any second derivate in the modelling, similarly to the £ PDH

¦© ¦ ©

NS- and HL-models.

¡ ¦ ¦ ¦

It should also be noted that any term which is con- £

BPD

¦© ¦©

structed with the molecular viscosity may be numerically unstable. In particular when is multiplied with ¤ gradients of quantities which are unrestricted. There is with £ numerically optimized to: always a risk that during the iterative process such a PDH BPD term may become un-proportionally large. A code run- 0.75 1.1 ning with either the NS- and HL-models would thus most probably be less numerical stable than the other It should be noted that the addition of a viscous term

formulations. necessitates a modiﬁcation of the other terms in the ¦ -

¢ ¦ equation, since this term make the standard un- balanced in the near-wall, while a turbulent term only 5.3.3 The ¦ -equation modiﬁes the equation further out, and could be added to

There exists, at least to the best knowledge of the author, existing models.

¢ ¦ no models that are directly based on results from The main argument for this cross-diffusion term va- either TSDIA or DNS-data. ries, however several authors, including Menter [31],

In order to get DNS-data for an ¦ -equation it is neces- [32], Peng et al. [38], Kok [22], Wilcox [54], claim that

sary to derive an exact ¦ -equation, which is a function it improves the prediction of ﬂows with a freestream

< <

¢ ¦ of primitive quantities, such as etc. Using DNS- boundary, for which the Wilcox is less accurate. data it would then be possible to compute the individual In the derivation of the BPD-model [7] the viscous term

terms in the ¦ -equation and then use this as a modelling was however attributed to model a pressure-diffusion ¢¥¦ basis for the model. However as the accuracy of the process. budget in the ¡ -equation is rather questionable, and seldom directly used, the (possible) marginal gains of such a derivation is not in balance with the arduous workload 5.4 Fine-tuning the Modelled Transport needed. Equations In addition it is questionable how to formulate the ex-

5.4.1 The -equation ¦ act ¦ -equation, even to exactly deﬁne . The reciprocal time scale, or the speciﬁc dissipation rate? It is most Apart from the addition of a cross-diffusion, some of the

possible that the exact ¦ -equation will be quite different newer models also incorporate a variable Schmidt num-

dependent on the concept of ¦ . ber, which, if the SGDH-model is used, is necessary to

19 simulate the near-wall reduction of the diffusion. The The NS-model uses, similarly to Launder, only the

NS- and HL-model use such a scheme: ﬁrst term, however with a different coefﬁcient:

¤ £

S ¢

$ A

E NS

E L L/L

¤ (59)

¢ §

S ¢ ¢ ¨

E E% E

S ¢ ¦ ¢ E;L HL

The models based on the TSDIA results (YC and RS) do not include a model for this term. Using a priori tests the HL-model faithfully reproduce

DNS-data, however one should be careful to include too 5 ¤ many ﬂow dependent variables into a model, as those Modelling the production term, c : The fourth pro- are more likely to introduce numerical problems. duction term is the main production and is modelled in the classic way:

The other models (YC, RS, PDH and BPD) use a stan-

¤ £

5 8 5

dard constant Schmidt number: ¡

¤ c

S (60)

£¢¤¡ ¥¢¤¦

where none of the models utilize the damping function,

£ ¤

YC RS PDH BPD E ¤

thus S . The coefﬁcient, , is set to: 1 1 0.8 1 YC NS HL RS Note that the PDH-model has chosen a slightly lower 1.17 1.45 1.44 1.44 Schmidt number than the commonly used. However as

the basis for choosing a unity value is rather vague, as Note that the YS- and RS-models use the turbulent time

¨§ ¡

the models above indicate, there should be nothing to scale, instead of , in the relation for the production 8 deter modellers from change _ through numerical opti- term.

mization. \

In addition the PDH-model uses a variable coefﬁcient Modelling the destruction term, ¤ : As noted above

for the dissipation term, through an additional damping £ 8 the model for the destruction term, is the sum of the

function, : two ﬁrst production terms and the destruction term. In

¤ £ 8

8 these newer turbulence models the destruction term is

aS ¦

¡ modelled as in the classic model:

¡ L 9

E (57)

£ ¤

¡¢

S ¢ ¨/¨ ¦ ¢ £

E;L

¤ ¤

S (61)

, the damping function, is used only in the NS-model:

5.4.2 The -equation ¦

E L ¡

S ¢ ¦ ¢ £ (62)

The sections below follow the presentation in the com-

prehensive paper on the modelling of the ¡ -equation by ¤

Rodi and Mansour, [43]. The coefﬁcient ¤ is given the following value:

5 YC NS HL RS

¤ ¤ Modelling the production terms, and : 1.92 1.9 1.92 1.92

Because these terms give a negative source of ¡ , ie. being destruction terms, they are treated togther with the The turbulent diffu- destruction term, below. Modelling turbulent diffusion: sion is modelled as in the classic model using the stan-

dard gradient hypothesis (SGDH):

5 ¤

Modelling the production term, : : In comparison

¦ ¦ ¡

¥ £

to the other production terms, this term is very small, S

5 $ 5 \

$b5 (63)

¦© ¦ ©

¤ ¤ ¤

¢ however relatively to the sum of c , it becomes signiﬁcant. Rodi and Mansour [43] devised the

5 The difference among the turbulence models is restric- ¤ following model for the : -term:

ted to the Schmidt number:

L 9

¤ ¤

$ 5

¦ ¦% ¦

¦ YC

¤ ¤

E% :

S ¡¨£

C C C

C (58)

¦ ¡ ¦ ¦ ¦

S A $

E NS

§ ¦ ¢

¤ ¤

where and are tunable constants. E

S ¢¦ ¢ E L HL

The ﬁrst term is identical to the term used in the mo-

S ¨ ¤

dels by Launder et al. (JL and LSY), (with ), S

: RS while the second is a numerically cumbersome construc-

tion. The predictions when including both terms in the The NS- and HL-models adopt damping functions simi-

¡ -equation are however improved, at least for channel lar to that for the -equation, see above. The other mo- ﬂows. dels use constant Schmidt numbers.

20 Viscous diffusion: Since this term is exact, it is not 6 Enhancing the -equation? modelled:

Even though the emphasis of this report is general, ie.

¦ ¦ ¡

£ (64) that all types of EVMs are discussed, this section will

¦© ¦ ©

be entirely devoted to the ¢©¦ -type of models. In some

¢ ¦ way this reﬂects the good experience of the models

5.4.3 The ¦ -equation by the author, however equally important is to produce a

¢ ¡

Similar to the models, the main change made to counter-balance to the wealth of information regarding

¢ ¡

the £¢§¦ models, due to DNS-data, has been to improve the -type of models. In addition anyone who desires

¢ ¡ and add damping functions. However for the ¦ -equation, to develop a new model, may consult both DNS-

there exists no rigid analysis as the one made by Rodi data and the TSDIA analysis to enhance the modelling. ¢ ¦ and Mansour, and hence improvements are purely based The basis for improving a model relays more on en- on numerical optimization. hanced treatment of transformations and improved un- The PDH- and BPD-models introduce an additional derstanding of modelling theory, which is the purpose of cross-diffusion term, as noted previously, but are other- the present report. It should however be recognized that

wise similar to the standard ¥¢¤¦ model. in the end all turbulence models are numerically tuned using test-cases, with comparison to experiments and, if available, DNS-data. The performance of the model for Production: The PDH-model adds a damping func- basic ﬂows, included in Appendix B, is a most important tion, in the modelling of the production term, whereas ¤ issue in the validation procedure. the BPD-model only differs through the coefﬁcient, £ : There will be numerous references made to the Bred- PDH BPD berg et al. [7] BPD-model as this report is partly the 0.42 0.49 result of the work done during the development phase of the new model. The main focus in the following analysis

with the damping function in the PDH given as: will be the origin and connection of the cross-diffusion

¦ §

A terms in various recent turbulence models.

S ¢

E (65)

£ ¥ 6.1 Transforming the DNS/TSDIA Destruction: The model for the destruction is the

¤ models

¦ ¦

same as for the standard ¢ model , with only

marginally different coefﬁcients, £ : ¦

In the above sections, the modelling of the -equation in

¢ ¦

PDH BPD the PDH and BPD models was discussed in con-

¢ ¡ 0.075 0.072 nection with the transformation of the standard -

equation, using the relations shown in Eq. 43. It is ho-

¢ ¡ wever unnecessary to use the inaccurate classic

Turbulent diffusion: The SGDH-model is used with model as the basis for such a transformation when there

¢ ¦ different Schmidt numbers compared to the Wilcox exists newer and more accurate versions.

models:

¢ ¡

Below, instead of employing the classic model, ¡ PDH BPD the TSDIA/DNS modiﬁed ¤¢ models which include

1.35 1.8 additional terms will be used. In the formulation of

the ¦ -equation, as shown in Eq. 42, terms in the -

In summation the DNS-data have, similarly to the ¢§¡

¦ §

equation should be multiplied with ¢ and those in

E ¢ ¦

models, been used to improve the predictions for the

§ the ¡ -equation with . models, however for the latter models the change is less fundamental and only apply to improved damping functions and coefﬁcients. The only additional term, the 6.1.1 Turbulent approach, YC- and RS-models cross-diffusion term in the ¦ -equation, appears more

due to the transformation of the ¡ -equation than to DNS- ¡ If the additional terms in the - and -equations are data. modelled using a turbulent approach, as in the YC- and

RS-models, the extra terms appearing in the ¦ -equation are (RS-model):

from the -equation:

& 8

¦ ¡ ¦ ¦ ¦ ¨§ ¡

S ¢ S ¡

¨ ¦© ¦©

¦ (66)

¡ ¦ ¦ ¦ ¦ ¦

S ¢

¨ ¦© ¦© ¦ ¦ ©

¦ §¦©¨ ¦ ¦ §¦©

from the ¡ -equation: 6.2.1 Cross-diffusion,

E E

¦ ¨§ ¡ ¦ ¡

The most frequent term in the resulting -equation, is

¤ ¤

8 S ¢ F 8 S

¦ ¨§¢¦ ©¨ ¦ ¦ §¢¦ ©

¦ © ¦ ©

the cross-diffusion term, . It appears due

¡ ¢

E to the standard terms in the classic formulation, and

¡ ¦ 8 ¦ ¦

§ ©

¤ ¤

S F S S 8 S

§ ¡

8 also due to terms similar to both the RS-models and

& [ [

¦ ¦© ¦© ¦

¤ ¤

¢ , and the HL-models and . Dependent on the

¦ ¡ ¦ ¦

§ magnitude of the term, the back-transformation of a

¦© ¦© ¦

¢ model, see later, will possibly give a more correct

¢ ¡ (67) model, with a pressure-diffusion like term. Thus it is not surprising that the addition of a cross-diffusion The sum of the above terms becomes:

term to the ¦ -equation was one of the ﬁrst major modiﬁ-

cations to the standard ¥¢§¦ model.

¦ ¦ ¦ ¦ ¦

¡ ¡

§ ¥

£ (68)

¨ ¦© ¦ © ¨ ¦ ¦©

¦ §¦© 6.2.2 Turbulent diffusion,

Thus the addition of terms ’a la’ the RS-model gives two

¢ ¡ Transforming the standard model, the second-

terms in the modelled ¦ -equation. The first is similar ¦ to what results from classic transformation, while the gradient of also appears in the -equation. However as repeatedly stated the coefficient in front of this term second is a new construction. could be made small through proper usage of the Sch- midt numbers. In any case any second-gradient term 6.1.2 Viscous approach, NS- and HL-models is unwanted in CFD-codes, due to implementation and If one instead uses the models of NS or HL, which numerical stability reasons. involve viscous terms, then the extra terms appearing in the ¦ -equation is (HL-model): 6.2.3 Other terms The two final terms from the transformation of the

from the -equation: ¢©¡

DNS/TSDIA modiﬁed models are both relations of

[ 8

¦ ¦ ¦ ¦ ¡

only , either using a ﬁrst or second derivate. The NS-

S ¢ ¢ S

¦© ¡ ¦ ©

and HL-model resulted in a viscous diffusion term that

canceled the normal diffusion term, and hence such a

¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦

¢ ¢ ¢ S

£ model is rather questionable. It would be possible to in-

¦ ¦© ¦© ¦ © ¦© ¦©

troduce a term based on the square of the derivate of

¦ ¦ §¢¦ © ¦ ¦ (69) , as to the -equation. It should however be noted that such a term would be very large close to a

from the ¡ -equation: wall, and hence a viscous term based on this idea, as gi- E

E ven by the NS- and HL-models, would be very difﬁcult to

[

¦ ¡ ¦

¤ ¤

8 S 8 S

£ make numerically stable. A model including the turbu-

¦ © ¦©

(70) lent viscosity (RS- and YC-models), could possibly be in-

¦ ¦ ¦ ¦ ¦

troduced with the same arguments as the cross-diffusion

¦ © ¦© ¦ ©

term, however again with a more delicate tuning, due to the tendency of such a term to become inaccessibly large. Summing them up yields the following extra terms in

the ¦ - equation:

6.3 Back-transforming the ¢ -equation

¦ ¦ ¦ ¦ ¦ ¦ ¦

¥ ¥

¢ S ¢ ¦ £

£ (71) From a modeller’s point of view, the critical secondary

¦ ¦© ¦ © ¦© ¦ ¦©

quantity is the dissipation rate, since this is the only variable for which there exists an exact equation. In The ﬁrst term of the LHS is similar to what was found addition the availability of DNS-data for terms in the

in the RS-model, although here it is a viscous term. ¡ ¡ -equation makes it possible to scrutinize an -model to The second term (LHS) cancels out the normal viscous levels unreachable using any other secondary quantity.

diffusion in the ¦ -equation which is indeed very strange!

¢ ¦ Using a model it is thus of the outermost interest It is questionable if the HL-model pressure-strain model

to see the resulting terms in a ¡ -equation when the mo-

is very reliable. Such a model should nonetheless not be

¢ ¡ del is re-casted into a model using the previous used in the modelling of ¦ -equation.

transformation rules. Selecting the BPD ¢¤¦ model as

the basis – since this model includes most terms – the ¢

6.2 Additivional Terms in the -equation ¡ -equation is transformed as:

¤ Although it is possible to add any term with a dimensio- ¤

8 $ 8

¦ ¡ ¦

S S S ¦

¦ ¦

nally correct combination of and , to the -equation,

5 \ $ [ $ F $ ¢ $

it feels however more important to especially note those 8 (72)

£ £ £ £

S ¡ ¢ ¢

$ 8 598 $ [ 8 $ F%8 $ £¢

term appearing from the transformations of the diffe- 8

¢ ¦4¡ ¢¤¡ rent ¥¢§¡ models.

M M

¡ §

Vis: ¨§ Turb: The two other terms are closely connected, and always

¡¥§¦© ¦ ¨§¦© ¦ ¡¥§¦© ¦ ¨§¢¦© ¦ appear together, however here more emphasis is paid to

WHR -2 -1 the cross-diffusion term. ¢£¦ PDH -2 -1.24 Transforming the standard model (WHR) results

BPD -0.9 -0.67 in a negative cross-diffusion in the ¡ -model. The intro-

£ ¤

YC 0 -1.26/ duction of a cross-diffusion term in the ¦ -equation (PDH,

RS 0 1 BPD), could dependent on the magnitude of the coefﬁci- ¤

HL,NS -1 0 ent, , change this according to:

E E

$ [

¦ ¡¥§¦© ¦ §¦© ¡

Table 3: Cross-diffusion term, , in the - ¤

_ _

equation £

E E

$ [

8 S S _

_ (74)

M M

E E

¡¥§ ¡ ¡ §

Vis: Turb:

$ [

¦ ¨§¦© ¦ ¨§¦©

_ _

WHR 2 1 ¡ PDH 2 1.24 Based on the construction of the newer ©¢ models

BPD 0.9 0.67 and DNS-data suggest that the cross-diffusion in the ¡ -

£ ¤ YC 0 1.26/ equation should be negative; however read also the di- RS 0 -1 scussion on the Yap-correction below.

HL,NS 1 0 The term with the ﬁrst derivate of , is similarly af-

fected by the inclusion of a cross-diffusion term in the

Table 4: Terms involving the ﬁrst derivate of , ¦

-equation, with the resulting condition as above, Eq.

¦ ¨§¦© ¡ , in the -equation 74.

The coefﬁcients from the transformed ¡ -equation of

¢ ¦ the different models and the new DNS/TSDIA mo-

The result becomes, see Appendix C: diﬁed ¡ -equation are gathered in tables 3 and 4.

$ 598 $

¡ E ¡ E ¡¢

¢ £ 8 ¢

S 6.4 Cross-diffusion Terms in the -

¢ E

E equation?

¤ ¤

$ $

¦ ¦ ¡

¥ §

¢ ¨ ¢ 8 ¢

£ _

_ So should there be a cross-diffusion term in the model-

¦ © ¦© ¦

¢ led -equation? Based on the above comparison one can

E E

¤ ¤

$ $ $

¦ ¡

¡ not simply due to the effect of the cross-diffusion term

§ ¥ ¥

¨ ¢ £ £ ¢

_ _

¦© ¡

in the ¦ -equation on the transformed -equation include

$ $ $

¡ ¡ ¦ ¦ ¡ ¦ ¡

¡ the term. Transforming the standard -equation, eg.

¥ ¥ §

8 ¢

£ £

_ _ _ £

£ the WHR or WLR model, would also result in a cross-

¦© ¦© ¦©

diffusion term in the ¡ -equation, if this is the appre- (73)

ciated result. However one should bare in mind, that

¤ ¤

The coefﬁcients and are the constants connected it is only through the questionably high Schmidt num-

bers of the Wilcox’s models that the coefﬁcients in-front

to the viscous- and turbulent cross-diffusion terms re-

¦ ¨§¦© ¦ ¡¥§¦© ¦ ¨§¦©

spectively. Because coefﬁcients and damping functions of the -term, and the -term (in

the back-transformed ¡ -equation) that the WHR/WLR-

¢ ¡ are tunable, and also differ considerably among the models they are left-out of the analysis. Table 3 and 4 models becomes similar (actually identical) to those of however list the coefﬁcients for the relevant terms. the newer ¥¢§¡ models. With a reduction of the Schmidt number, as in the

The additional terms in the back-transformed ¡ -

PDH-model, there becomes a necessity to include a

¢ ¡ ¦ ¢ ¦

equation ( model transformed -equation

cross-diffusion term in the ¦ -equation, in order to keep model back-transformed ¡ -equation), compared with

the cross-diffusion term in the transformed ¡ -equation the standard ¡ -equation, are;

reasonable.

¦ ¡¥§¦© ¦ ¨§¦©

The cross-diffusion term: . Stating that a cross-diffusion term should be inclu-

ded in the ¦ -equation because the transformation of the

¢ The term including the square of the ﬁrst derivate

standard modelled -equation indicates this, is not good

¦ ¨§¢¦© of : . enough. Because the modelled ¡ -equation is simply what

¢ it says: a model. There is no reason to be mathemati-

¦ ¨§¦© The second derivate of : .

cally correct when designing the ¦ -equation, especially Based on the coefﬁcients connected to each term, it is when noting the theory behind the construction of the

reasonable to pay less attention to the last term. In the ¡ -equation in the ﬁrst place. case of equal Schmidt number this term is also identical Better arguments are those based on the predicted

zero. Note also that neither of the two models (RS, YC) results using the different ¡ -equations. Dependent on

that add a purely turbulent term to ¡ -equation include which construction gives the overall most correct re- this term. sults, one could declare if cross-diffusion terms should

23 be included or not in the ¦ -equation. If these newer mo- 7 Summary dels (those which include cross-diffusion terms in the -

and ¡ -equations) only improve the predictions in wall- In this section a short theoretical summary of the EVMs

bounded ﬂows, one should consider to still use the stan- are included. The performance of the different models

¢ ¡

dard as the transformation basis, and hence add for three fundamental ﬂows are presented in Appendix

¢ ¦

cross-diffusion terms to the model. Even if the B.

¢ ¡ cross-diffusion modiﬁed models prove to be more general applicable models, it is still possible to construct

7.1 The -equation ¢©¦ a model without any cross-diffusion terms. By ca-

refully tuning the coefﬁcients and Schmidt numbers one The difference between the -equations are generally

could still arrive at the desirable cross-diffusion modi- rather small among the EVMs. It does not matter whet-

¢ ¡

¡ £¢ ¦ ﬁed model, since the transformation will produce her it is a ¥¢ or model, they all use a production

cross-diffusion terms. term, a destruction term, a turbulent- and a viscous-

¢ ¡ Simulations using the BPD-model [7] however indi- diffusion term. Some of the newer models add a

cate that the cross-diffusion terms play an important pressure-diffusion term, which improves the result mar-

¢ ¦ role in reducing the inaccurate behaviour of the ginally, although indisputable enhances the near-wall models for free shear layer ﬂows. It is also considerably behaviour of the turbulent quantities. The gain is ho- easier, although more time consuming, to develop a more wever marginal and cannot deﬁnitely be attributed to general model through adding terms to the standard mo- a pressure-diffusion process and hence such a term was del. not included in the BPD-model. It is believed that wisely constructed damping functions, as in the PDH-model, could perform a similar effect as a pressure-diffusion term, without the additional computational effort and the risk of deteriorate numerical stability. A variable Schmidt number could also be included with the same argument as a pressure-diffusion term, and also sharing some of its drawbacks. A further discussion on the Sch- midt numbers follows below.

7.2 The Secondary Equation The second transport equation in the formulation of

a two-equation EVM differs a lot more than the - equation. First there is the choice of the solved variable. The turbulence modelling community cannot agree, although strong arguments from all sides, which type of

model performs the best. The two main candidates, the

¢ ¡ ¢ ¦ and the , both have their advantage and di-

sadvantage.

¢ ¡ The model is thought as the more general model,

with good predictions for both free shear ﬂows (mixing

¢ ¦ layer, jets and wakes) and wall bounded ﬂows. The model certainly perform better in wall-bounded ﬂows,

especially for adverse pressure ﬂows, although its fre- ¢©¦ estream problem is notorious. The model is credi- ted with a higher numerical stability. The conclusion from this should maybe be that the EVMs are not the all-performing turbulence model which one desires, but rather case-dependent, although certainly less than an algebraic zero-equation model. Within each group the difference between the models

is, similarly to the -equation, restricted to coefﬁcients and damping functions. The magnitude of the coefﬁci-

ents in the ¡ -equation are tuned along the following li-

nes:

¤ ¢

¤ : Backward-facing step and re-attachment

length, since ¤ has a strong inﬂuence on separa-

ted ﬂows.

¤ ¤

S § ¢

: Decaying turbulent which states: ¥

where ¥ is 1.25 to 2.5.

¤ ¤ ¤ ¤

E 9 E

B ¢ W

_ _ _

¤ ¤ ¤ ¤

S S ¢ § ¢ ¨ ¥ S ¢

: Log-law: I and for a model with (SAA): .

£ ¤ The required Schmidt numbers naturally differ depen-

Damping of ¤ : , is introduced to improve the decay-

dent on the used coefﬁcients, and notably is that for the

ing turbulence predictions. Recent studies, [15] also in- W

¢ ¡ S

¤ standard model a value of is needed to

¤ _

dicate that there should be a case-dependency on . ¤ S

give . Dependent on the numerical optimization done, models With a SGDH-model, the varying Schmidt number of use different sets of constants. This apply mainly for the

¤ the HL-model is a very accurate approximation for fully

_ ¤ ¤ and .

developed channel ﬂow DNS-data. However as indi-

¢ ¡

Note that the above discussed was focused to the cated by Launder, different ﬂows, necessities different ¨ model, however similar rules applies for the ¦ - and - constants, and in reﬂection of that, the standard value equation. may very well be as good as any DNS-data optimized values. In addition a variable Schmidt number imposes yet 7.3 Schmidt Numbers another worry for numerical problems, as well as incre- ased implementation efforts. Here only some comments on the magnitude of the Sch-

midt numbers are included. It should be emphasized

E E

_ _

¤ 7.4 Cross-diffusion vs Yap-correction S that the common acceptable values ( and S ) is as little founded as any other combination. Gene- It has been argued [38], [7] that the cross-diffusion

rally the Schmidt numbers should depend on the wall- term appearing in the ¡ -equation could be connected distance, with a reduced value as the wall is approach, to the Yap-correction term of the Launder-Sharma mo- similarly to the models by HL and NS. These number del. Whether the term is due to a cross-diffusion term

can hence not be considered constants at all, and are in the ¦ -equation or due to modelling argument from open for numerical optimization. DNS/TSDIA is of less importance. Jones and Launder [19] simply concluded that Sch- Launder et al. [57], [24] found that when using a LRN midt numbers close to unity accord with expectations. ¢¤¡ model the predicted heat transfer level in the vici- Launder and Spalding [26], through extensive exami- nity of the re-attachment point could be several times nations of free turbulent flows, specified the now well too high. This discrepancy is the result of the excessive accepted standard values. They however stated that levels of near-wall length-scale that is generated in se- slightly different values may be necessary in wall- parated flows. bounded flows, without giving any specific values. Yap [57] devised a term, which in non-equilibrium Indeed Launder in a subsequent paper [16] opts for flows reduces the turbulent length-scale when added to

slightly different values. Although the model is a RSM, the right-hand-side of the ¡ -equation:

and as such does not employ Schmidt numbers, the mo-

§ §

E/E E

§ ¨

L ¡¢ : E :

del, in the limit of thin shear layers, gives a ratio of

¥ ¥ ¥

¤¦¥ ¤¦¥

S £ £ ¢ £ C

C (76)

¡ ¡ for turbulent diffusion of the -equation to that of the

¡ -equation. Thus contrary to the standard values, this

_ _ ¤

RSM uses a higher than . The term has no§ effect in local equilibrium, for which

¤ ¥

: § ¡S Nagano and Tagawa (NT) [35], citing both the near a wall, and hence the term vanishes. Hanjalic-Launder [16] turbulent diffusion ratio as well In re-circulating regions the Yap-correction term ho-

as improved predictions of the -proﬁle in the core re- wever increases the near-wall dissipation, and hence re-

gion, increase the Schmidt number in the -equation to duces the heat transfer to a more acceptable level.

_ _ ¤ S . The NT-model retains the standard value, The question is, whether or not a cross-diffusion term which is connected through the log-law to the other co- affects the ﬂow in a similar manner? Using the LSY-

efﬁcients. Several models have continue the usage of model as the reference model, the wall-normal (the most

_ ¡

higher (AKN, HL). inﬂuential) gradients of and are positive close to the

¢ ¦ In the case of the Wilcox models, one could re- wall, while further out they both become negative. A ally start to question what is a reasonable values for the (positive) cross-diffusion term thus affects most parts of

diffusion coefficient of the turbulent kinetic energy. The a flow with an additional source term to the ¡ -equation, 8 WHR/WLR-models use twice as high _ , as the standard while the effects of a Yap-correction term, is confined to

¥¢¤¡ models. the near-wall region.

In addition should the diffusion coefﬁcient of ¦ be si- A direct comparison between the two terms is difﬁcult

"ZS § £ milar to that of ¡ ? Using the variable, , Laun- to make, due to the influence the modified in the core of der showed (see also Appendix D), that with the usage the flow could have on the near-wall heat transfer. Even of the logarithmic law, the coefficients in the secondary more alarming is that nearly all models (not RS) use a transport equation should satisfy: coefficient in the cross-diffusion term that is negative,

see table 3. The pressure-diffusion term is thus con-

¤ ¤

¥ structed to reduce the dissipation rate, contrary to the

S ¢

§ § W

_ (75)

§ idea behind the Yap-correction term.

The addition of the cross-diffusion term to the ¦ -

The Schmidt number using this equation with standard equation however reduces the negative source the cross-

¥S ¡

coefﬁcients becomes, for a ¥¢§¡ model with (LSY): diffusion in the -equation. In Bredberg et al. [7] it was

E% E 9 E L

_ _

¤ £

¢ ¦ ¥4S Sb¨ ¨ S , for a model with (WHR): , also noted that the Schmidt number has a strong impact

25 on the cross-diffusion term, which should be considered References when comparing the Wilcox-, PDH- and BPD-models in

table 3. Based on the values in this table, the predic- [1] K. Abe, T. Kondoh, and Y. Nagano. A new turbu- ¦ ted heat transfer using the ¢ models should result in lence model for predicting fluid flow and heat trans- levels from low to high: the BPD-, the WHR-, and with fer in separating and reattaching flows - I. flow fi- the PDH-model reaching the highest levels. Not surpri- eld calculations. Int. J. Heat and Mass Transfer, singly the simulations, Figs. 22 and 23 in Appendix B, 37:139–151, 1994. indicates a rather more complex nature for the predicted heat transfer level. The Yap-correction term on the other [2] R. Abid, C. Rumsey, and T. Gatski. Prediction of hand reduces the Nusselt number markedly for the LSY- nonequilibrium turbulent flows with explicit alge- model, which can be noted in the same figures through a braic stress models. AIAA Journal, 33:2026–2031, comparison with the similar, although not Yap corrected, 1995. JL-model. [3] B.S. Baldwin and H. Lomax. Thin-layer approxima- The conclusion from this can thus only serve to show tion and algebraic model for separated turbulent the futility of trying to adopt posterior discussions on the flows. AIAA Paper 78-257, Huntsville, AL, USA, turbulence models, which are too coupled to single out 1978. the effect of a modified term, or coefficient. [4] T.V. Boussinesq. Mem.´ pres Acad. Sci.,3rd ed Paris XXIII p. 46, 1877. Acknowledgments

[5] J. Bredberg. Prediction of ﬂow and heat trans-

¢ ¡

Funding for the present work has been provided by fer inside turbine blades using EARSM,

¢ ¦ STEM, Volvo Aero Corporation and ALSTOM Power via and turbulence models. Thesis for the De- the Swedish Gas Turbine Center. gree of Licentiate of Engineering, Dept. of Thermo and Fluid Dynamics, Chalmers University of Te- chnology, Gothenburg, 1999. Also available at www.tfd.chalmers.se/˜bredberg. [6] J. Bredberg. On the wall boundary condition for turbulence model. Report 00/4, Dept. of Thermo and Fluid Dynamics, Chalmers University of Te- chnology, Gothenburg, 2000. Also available at www.tfd.chalmers.se/˜bredberg.

[7] J. Bredberg, S-H. Peng, and L. Davidson. An impro-

¢ ¦ ved turbulence model applied to recirculating ﬂows. Accepted for publication in Int. J. Heat and Fluid Flow, 2002.

[8] T. Cebeci and A.M. Smith. Analysis of turbulent boundary layers. In Series in Applied Mathematcis and Mechanics, Vol XV, 1974. [9] K.Y. Chien. Predictions of channel and boundary- layer ﬂows with a low-Reynolds-number turbulence model. AIAA Journal, 20:33–38, 1982.

[10] F.H. Clauser. The turbulent boundary layer. Ad- vances in Applied Mechanics, 6:1–51, 1956. [11] L. Davidson and B. Farhanieh. CALC-BFC. Report 95/11, Dept. of Thermo and Fluid Dynamics, Chal- mers University of Technology, Gothenburg, 1995.

[12] F.W. Dittus and L.M.K. Boelter. Heat transfer in automobile radiators of the tubular type. Univ. of Calif. Pubs. Eng., 2:443–461, 1930. [13] P.A. Durbin. Near-wall turbulence closure modeling without damping functions. Theoretical Com- putational Fluid Dynamics, 3:1–13, 1991. [14] M.P. Escudier. The distribution of mixing-length in turbulent ﬂows near walls. Report, twf/tn/12, Im- perical College, Heat Transfer Section, 1966.

26 [15] W.B. George. Lecture notes, turbulence theory. Re- [30] W.H. McAdams. Heat Transmission. McGraw-Hill, port, Dept. Thermo and Fluid Dynamics, Chalmers New York, 2nd edition, 1942. University of Technology, Gothenburg, 2001.

[31] F.R. Menter. Inﬂuence of freestream values on

¢ ¦ [16] K. Hanjalic´ and B.E. Launder. Contribution to- turbulence model prediction. AIAA Journal, wards a Reynolds-stress closure for low-Reynolds- 30:1657–1659, 1992. number turbulence. J. Fluid Mechanics, 74:593– 610, 1976. [32] F.R. Menter. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA

[17] C.B. Hwang and C.A. Lin. Improved low-Reynolds- Journal, 32:1598–1605, 1994.

¢ number model based on direct numerical simulation data. AIAA Journal, 36:38–43, 1998. [33] R.D. Moser, J. Kim, and N.N. Mansour. Direct numerical simulation of turbulent channel flow up to [18] J.Kim. On the structure of pressure fluctuations in Re=590. Physics of Fluids, 11:943–945, 1999. Data simulated turbulent channel flow. J. Fluid Mecha- available at www.tam.uiuc.edu/Faculty/Moser/. nics, 205:421–451, 1989. [34] Y. Nagano and M. Shimada. Rigorous modeling of [19] W.P. Jones and B.E. Launder. The prediction of la- dissipation-rate equation using direct simulation. minarization with a two-equation model of turbu- JSME International Journal, 38:51–59, 1995.

lence. Int. J. Heat and Mass Transfer, 15:301–314,

¢ ¡ [35] Y. Nagano and M. Tagawa. An improved mo- 1972. del for boundary layer flows. J. Fluids Engineering, [20] J. Kim, P. Moin, and R. Moser. Turbulence sta- 112:33–39, 1990. tistics in fully developed channel flow at low Rey- [36] V.C. Patel, W. Rodi, and G. Scheuerer. Turbu- nolds number. J. Fluid Mechanics, 177:133–166, lence models for near-wall and low Reynolds num- 1987. ber flows: A review. AIAA Journal, 23:1308–1319, [21] P.S. Klebanoff. Characteristics of turbulence in a 1985. boundary layer with zero pressure gradient. Re- [37] S-H. Peng and L. Davidson. New two-equation eddy port, tn 3178, NACA, 1954. viscosity transport model for turbulent flow compu-

[22] J.C. Kok. Resolving the dependence on freestream tation. AIAA Journal, 38:1196–1205, 2000.

¢ ¦ values for the turbulence model. AIAA Jour- [38] S-H. Peng, L. Davidson, and S. Holmberg. A mo-

nal, 38:1292–1295, 2000.

¢ ¦ dified low-Reynolds-number model for recir- [23] A.N. Kolomogorov. Equations of turbulent mo- culating flows. J. Fluid Engineering, 119:867–875, tion in incompressible viscous fluids for very large 1997. Reynolds numbers. Izvestia Academy of Sciences, [39] L. Prandtl. Bericht uber¨ die ausgebildete turbu- USSR, Physics, 6:56–58, 1942. lenz. ZAMM, 5:136–139, 1925.

[24] B.E. Launder. On the computation of convective [40] M.M. Rahman and T. Siikonen. Improved low-

heat transfer in complex turbulent ﬂows. ¢ J. Heat Reynolds-number model. AIAA Journal, Transfer, 110:1112–1128, 1988. 38:1298–1300, 2000.

[25] B.E. Launder and B.I. Sharma. Application of the [41] G. Rau, M. Cakan, D. Moeller, and T. Arts. The energy-dissipation model of turbulence to the calcu- effect of periodic ribs on the local aerodynamic lation of ﬂow near a spinning disc. Letters in Heat and heat transfer performance of a straight cooling and Mass Transfer, 1:131–138, 1974. channel. J. Turbomachinery, 120:368–375, 1998.

[26] B.E. Launder and D.B. Spalding. The numeri- [42] C.M. Rhie and W.L. Chow. Numerical study of the cal computation of turbulent ﬂows. Computational turbulent ﬂow past an airfoil with trailing edge se- Methods Appl. Mech. Eng., 3:269–289, 1974. paration. AIAA Journal, 21:1525–1532, 1983.

[27] H. Le, P. Moin, and J. Kim. Direct numerical si- [43] W. Rodi and N.N. Mansour. Low Reynolds number

mulation of turbulent ﬂow over a backward-facing ¢ modelling with the aid of direct numerical si- step. J. Fluid Mechanics, 330:349–374, 1997. mulation. J. Fluid Mechanics, 250:509–529, 1993.

[28] M.A. Leschziner. Turbulence modelling for phy- [44] A. Sarkar and R.M.C So. A critical evaluation of sically complex ﬂows pertinent to turbomachinery bear-wall two-equation models against direct nu- aerodynamics. van Karman Lecture Series 1998- merical simulation. Int. J. Heat and Fluid Flow, 02, 1998. 18:197–207, 1997.

[29] N.N. Mansour, J.Kim, and P. Moin. Reynolds-stress [45] P.R. Spalart. Direct simulation of a turbulent

A¡

E E;L

and dissipation-rate budgets in a turbulent channel boundary layer up to S . J. Fluid Mecha- ﬂow. J. Fluid Mechanics, 194:15–44, 1988. nics, 187:61–98, 1988.

27 [46] P.R. Spalart and S.R. Allmaras. A one-equation turbulence model for aerodynamic ﬂows. AIAA Paper 92-439, Reno, NV, USA, 1992. [47] C.G. Speziale, R. Abid, and E.C. Anderson. Criti- cal evaluation of two-equation models for near-wall turbulence. AIAA Journal, 30:324–331, 1992. [48] K. Suga, M. Nagaoka, N. Horinouchi, K. Abe, and Y. Kondo. Application of a three equation cubic eddy viscosity model to 3-D turbulent ﬂow by the unstructed grid method. In K. Hanjalic´ Y. Nagano and T. Tsuji, editors, 3:rd Int. Symp. on Turbulence, Heat and Mass Transfer, pages 373–381, Nagoya, 2000. Aichi Shuppan.

[49] A.A. Townsend. The structure of turbulent shear ﬂow. Cambridge University Press, Cambridge, 1976.

[50] E.R. van Driest. On turbulent ﬂow near a wall. Ae- ronautical Sciences, 23:1007–, 1956. [51] B. van Leer. Towards the ultimate conservative difference monotonicity and conservation combined in a second-order scheme. J. Computational Physics, 14:361–370, 1974. [52] D.C. Wilcox. Reassessment of the scale- determining equation for advanced turbulence models. AIAA Journal, 26:1299–1310, 1988. [53] D.C. Wilcox. Comparison of two-equation turbulence models for boundary layers with pressure gradient. AIAA Journal, 31:1414–1421, 1993.

[54] D.C. Wilcox. Turbulence Modeling for CFD. DCW Industries, Inc., 1998. [55] M. Wolfshtein. The velocity and temperature distribution in one-dimensional ﬂow with turbulence augmentation and pressure gradient. Int. J. Heat

and Mass Transfer, 12:301–318, 1969. ¢ [56] Z. Yang and T.H. Shih. New time scale based model for near-wall turbulence. AIAA Journal, 31:1191–1198, 1993.

[57] C.R. Yap. Turbulent heat and momentum transfer in recirculation and impinging ﬂows. PhD thesis, Dept. of Mech. Eng., Faculty of Technology, Univ. of Manchester, 1987. [58] B.K. Yoon and M.K. Chung. Computation of com-

pression ramp ﬂow with a cross-diffusion modiﬁed

¥¢ model. AIAA Journal, 33:1518–1520, 1995. [59] A. Yoshizawa. Statistical modeling of a transport equation for the kinetic energy dissipation rate. Physics of Fluids, 30:628–631, 1987.

¤ ¤

5 F $ $

¡ ¦ ¤

A Turbulence Models ¤

S ¢ ¡ § ¨ ¨

¡ ¡

¦©

Nearly all of the treated turbulence model in the report, ¢ (81)

$ $

¦ ¡ ¦ ¡ ¥

listed on page 9, have been used to compute three diffe- §

¦ © ¦ © rent test-cases. Turbulence models not included below, were excluded due to: numerical problems (SAA), insuf- Boundary conditions for ¡ is:

ﬁcient information (YC), or lack of time (NS). The remai-

ning models are: ¦

¡ (82)

¦©

£¢¤¡ models:

Yang and Shih, 1993 [56] (YS) The coefﬁcients are:

¤ ¤ ¤ ¤

_ _

¤ ¤

Abe, Kondoh and Nagano, 1994 [1] (AKN) ¤

0.09 1.44 1.92 1.0 1.3

¡ £¢ models: Jones and Launder, 1972 [19] (JL) Pros/Cons: Chien, 1982 [9] (C) + Numerically stable Launder and Sharma, 1974 [25] – Strange deﬁnition for the time scale.

with Yap-correction, 1987 [57] (LSY) ¡ Hwang and Lin, 1998 [17] (HL) – Awkward damping function for .

Rahman and Siikonen, 2000 [40] (RS) – Second-derivate of velocity in the ¡ -equation is nume-

rically unsatisfying. ¢

£¢¤¦ models:

Boundary condition for ¡ is numerically awkward to Wilcox, 1988 [52] (WHR) – implement. Wilcox, 1993 [53] (WLR) Peng, Davidson and Holmberg, 1997 [38] (PDH) Model Performance: Bredberg, Davidson and Peng, 2001 [7] (BDP) Channel: Good performance.

The models are displayed below with constants tabula- BFS: Underestimated re-attachment length, erroneous ¤£§ ted. In addition an individual summary of their perfor- .

mance in the respectively test-case, is included. More 243 detail information as well as comparisons are found in Rib: Slight overestimation of .

the following section, Appendix B.

£ ¥ The notation § is used for the material derivate, A.2 Abe-Kondoh-Nagano ie. the convective plus time derivate term. The produc-

tion term is deﬁned as:

¤ ¤£¤

= =

¡9S

5 $

¦% ¦% ¦

=¦¥

¡ £

¢ A

(77)

¦© ¦© ¦ ©

£¤

J ¡

E E

¥¨§ ¥

£ £

¢ ¦ ¢ S E LL ¦¨§ ¢

The boundary conditions for and are both zero, while ¦ the boundary condition for ¡ or is displayed along with (83)

each turbulence model.

598 $ $

¡ ¦

The deﬁnition of the variables in the damping function ¦

¥ §

S ¢§¡ £¨ 8

can be found on page 10. _ (84)

¦© ¦ ©

£ ¥

¤ ¤ £

5 $ $

¡ ¡¢ ¦ ¡ ¦ ¡

A.1 Yang-Shih ¡

¤ ¤

¥ §

S ¢

¦© ¦©

¤ ¤£¢¤

¡9S ¡

J ¡

E E L

£¤

A¡ A

E E% NM E L NM E;L

¥¨§ ¥

S ¢ £ ¢ ¢ ¦ § ¢ £ E

¡ :

S ¡ ¢¦ ¢ ¢ ¢

(78)

¢ ¢

EaM E;L

¥ ¥ ¤

¢ ¢ (85)

where a speciﬁc deﬁnition for the time scale is used: Boundary conditions for ¡ is:

¨¥

$ F

¡G© S

C (86) 9S

¡ (79)

¡ ¡

The coefﬁcients are:

¤ ¤ ¤

_ _

¤ ¤ ¤

598 $ ¦ $ ¦

¥ §

S ¢§¡ £¨

_ (80)

¦© ¦© 0.09 1.5 1.9 1.4 1.4

£¡ ¥ Pros/Cons: A.4 Chien

+ Numerically stable ¤ ¤£¤

¡9S ¡

(90)

£¤

E L EE

S ¢¦ ¢

Model Performance: ¡ ¢

Channel: Good performance.

598 $ ¦ ¦

¥ §

S ¢¡ ¢ ¨¥ 8 £ C

_ (91)

¦ © ¦©

BFS: Decent estimate of the re-attachment length, and

¤ §

¤ ¤ £

5 $

¡ ¡ L ¡ ¡¢

¤ ¤

S ¢ ¢ ¨ ¢

Rib: Good performance. C

¡ ¦ ¡

¦ (92)

£¡ ¥

¥ §

¦ © ¦©

A.3 Jones-Launder, standard

E L

¥¤

S ¢ ¨/¨ ¦ ¢ ¡ §

¤ ¤£¢¤

¡9S

The coefﬁcients are:

¤ ¤ ¤

_ _

¤ ¤

(87) ¤

£¤

¢ ¨

$ A

S ¦

/L E

£ 0.09 1.35 1.8 1.0 1.3

§ ¡

Pros/Cons:

598 $ $

¦ ¦ ¦ ¡

¥ §

¢¡ ¢ ¨¥ S 8 ¨

£ + Numerically stable.

¦© ¦© ¦©

C (88) – Damping functions based on ` .

Model Performance:

£ ¤ ¤

5 8 $ $

¡ ¡ ¡ ¦

¤ Overestimation of in the inertial sub-range.

¢ ¨ ¡

S Channel:

¦©

¢ BFS: Underestimated re-attachment length and erro-

$ $

¦ ¦ ¡

¡ (89)

¥ §

£ neous .

¦© ¦©

E Rib: Terrible overestimation of .

S ¢ ¦ ¢

£¡ ¥

The coefﬁcients are: A.5 Launder-Sharma + Yap

¤ ¤ ¤ ¤

_ _

¤ ¤ ¤

¤ £

¤ ¤

0.09 1.55 2.0 1.0 1.3 ¡

(93)

S ¦ $ A

E GL

¡ §

Pros/Cons: A

Damping functions based on ¡

+ ©

$ $ 5

¦ ¦ ¦

§ ¥

S 8 ¢¡ ¢ ¨¥

_ ¡

– Second-derivate of velocity in -equation is numeri-

¦© ¦ © ¦© cally unsatisfying. (94)

– Boundary condition for ¡ is numerically awkward to

implement.

¤ ¤ £

¡ ¡ 598 ¡¢ $ ¦ ; $

¤ ¤

S ¢ ¨ ¡

¦©

– Model is poorly numerically optimized.

§ §

$ $

: L ¡¢ E :

¥ ¥ ¥

¥ ¥

¤ ¤

£ £ £ C

C (95)

¡ ¡

Model Performance:

$ $

¦ ¡ ¦ ¡

¥ §

Channel: Underestimated and ¤

¦ © ¦ ©

E L

S ¢ ¦ ¢ ¡

BFS: Underestimated re-attachment length, and overe-

¤ §

¤ ¤ ¤ ¤¦¥

8 _

stimated _

¤ ¤ ¤

23 Rib: Overestimated 0.09 1.44 1.92 2.55 1.0 1.3

Pros/Cons: Model Performance:

¡ + Damping functions based on . Channel: Excellent estimation of and , and even . + Greatly improved near-wall prediction with the Yap- BFS: Underestimated re-attachment length.

correction. 243

Rib: Surprisingly bad estimation of . ¡ – Second-derivate of velocity in -equation is numerically unsatisfying. Notes: Based on the included test-cases it seems that the HL-model predicts too high level of turbulence for – Boundary condition for ¡ is numerically awkward to implement. non-simple ﬂows which might be a consequence of the variable Schmidt numbers. The theory is however

sound, although one might try to use a different wall- ¢

Model Performance: dependent parameter. It seems that C may be too in-

Channel: Underestimated , sensitive for variating levels of turbulence.

£¡ ¥ BFS: Overestimation of the re-attachment length.

A.7 Rahman-Siikonen 243

Rib: Fairly good estimation of

¤ ¤£¤

¡9S ¡

£¤

A A

L LL E L L E Repeatedly used as a comparative model, thus ¢ (99)

Notes: :

¢ S ¢¦ ¢ a well established model with overall average predic- ¡ tion. Several models however give improved result, with where the turbulent time scale is deﬁned as:

simpler terms, and improved numerical stability.

¡9S ¨§¡ ¨¥ ¨§ ¡

(100)

£¡ ¥

A.6 Hwang-Lin

5 $

¤ £

¤ ¤

S ¢¡ ¢ ¨

¦ © ¡

(96)

£¤

E L LE L L/L

C C

$ $

¦ ¨§¡ ¦ ¡

¡ , L

S ¢ ¢ ¢ ¡

§ (101)

"!#

¨ ¦© ¦©

$ $

¦ ¦

¥ §

£¨

5 © $

8 ¦ ¦ ¡ ¦

¦© ¦©

(S ¢ ¢¡ ¢ ¨

¦© ¨ ¦© ¡ ¦ ©

¤¡

¤ ¤

A 5 F

¡ 8

¢ (97)

¤ ¤

¢ S ¡ ¢ ¨ § § ¢ ¢

£ ¡

$ $

¦ ¦ ¡

¥ §

8 £

¢ ¢

¦ © ¦©

$ $

¦ ¨§¡ ¦ ¡ ¦

¡ ¡

§ ¥ § 8

E E% E E;L

¢ £

S ¢ ¢ §

¦© ¦© ¦ © ¦©

(102)

¤ ¤ ¤ ¤

_ _

¤ ¤

5 $

¡ ¡ ¡¢ ¦

¤ ¤ ¤

¤ ¤

S ¨ ¨ ¢ ¢

¦©

0.09 1.44 1.92 1.0 1.3

$ $

¦ ¡ ¡ ¦ ¦ ¡

¦ (98)

¥ ¥ §

£ £

¦ © ¦© ¦ © ¦ ©

Pros/Cons:

E% E L

S ¢ ¦ ¢

§ + Improved representation of the turbulent time scale,

¤ ¤ ¤

¤ with enhanced near-wall predictions.

¤ ¤

0.09 1.44 1.92 – Unstable model.

Pros/Cons: Model Performance: B

+ Variable Schmidt-numbers improve results for chan- Channel: Severely underestimated . § nel ﬂow. ¤ BFS: Underestimated , with a too slow recovery.

+ Extensively tuned using DNS-data. 243 Rib: Qualitatively erroneous prediction of . – Several terms are awkward to implement.

This is a modiﬁed Chien ¢©¡ model with only – Numerically unstable model. Notes: marginally improved results. In addition this model is – Variable Schmidt numbers are numerically undesi- both more demanding when implementing, and also less rable. numerically stable.

A.8 Wilcox HRN ¢ Pros/Cons:

+ Damping functions based on ¡ .

¡9S

(103) ¦

Model Performance:

5 $ $

¦ ¦

J J

(S ¢ ¦

¡ (104) Channel: Ok.

¦© ¦©

BFS: Overestimated re-attachment length, and undere-

¤ ¢

stimated .

598 $ $

¦ ¦ ¦ ¦ ¦

¢ ¦ S ¡

(105) 243

with the constants, using Wilcox nomenclature as: Q

Q This model is identical to the HRN version,

Notes:

_ _ J

J apart from the added damping functions. Notable is that 0.09 5/9 0.075 0.5 0.5 the WLR-model only improves the prediction for channel ﬂow, with a slight reduction of the agreement for the ot- Boundary condition for ¦ is:

her two cases.

¦ S

C (106)

£ ¢ A.10 Peng-Davidson-Holmberg

Pros/Cons:

¤ £

¤ ¤

S No damping function. ¡

+ ¦

: ©

L L E ¡

S ¨ ¢ ¦ ¢

§ £

Model Performance: E;L

L LLE

. ©

Channel: Underestimated

¡ M L $ 9

A §

¦ ¢ £

L/L ¨

BFS: Overestimated the re-attachment length, and un- ¡ ¤£§

derestimated . (111) 243

Rib: Underestimation of .

¤ £

8 8 $ $ 598

¦ ¡ ¦

§ ¥

¦ S ¢ 8

¦© ¦©

A.9 Wilcox LRN ¦

A (112)

¡ E L 9

S ¢ ¨¨ ¦ ¢ £

E L

$ A

L E L 9

L (107)

¨ ¡ §G¨

$ A

E E;L 9

§G¨

¤ £ ¤ ¤

5 $

¦ ¦ ¦ ¦ ¦

£ £ £ £

¦ S ¢

¦ © ¦©

$ $

Q ¦ ¡ ¦ ¦

$ 598 $ ¦ ¦

¥ §

J J

S ¢ ¦ ¡ _

£ (113)

¦© ¦©

¦© ¦ ©

$b>A

$ E¡

(108) A

§ ¡ §

E ¡

S $ A

E L/L E

¦ S ¢

£ E%

¡ §

c ¢

The boundary conditions for the speciﬁc dissipation rate

$ 5 8 $

¦ ¦ ¦ ¦ ¦

S ¡ ¦

¢ is:

¦© ¦©

$ A

E L E

(109)

¡ §

¦ S ¥

$ A

$ £ E

C (114)

With coefﬁcients as: The constants in this model are:

¤ ¤ ¤ ¤ ¤

8 8

_ _

£ £ £ £

0.075 0.5 0.5 0.09 1.0 0.42 0.075 0.75 0.8 1.35

Boundary condition for ¦ is:

Pros/Cons:

¦ S

C (110) ¡ + Damping functions based on .

Model Performance: B Test-cases and Performance of B

Channel: Ok, however underestimated . EVMs

¤ § BFS: Excellent re-attachment length and . The listed two-equation EVMs on page 29 have been

243 used to compute three different flows: the fully deve- Slight underestimation of . Rib: loped channel flow, the backward-facing-step (BFS) flow

and the rib-roughened channel ﬂow.

£ A.11 Bredberg-Peng-Davidson ¢ The computations were made using the incompres-

sible ﬁnite-volume method code CALC-BFC, [11]. The

¤ ¤£¤

code employs the second order bounded differencing

¡9S

scheme van Leer [51] for the convective derivates, and

A E

¤ the second order central differencing scheme for the ot-

£¤

$ $

L $ E L L%$ E ¡

¥ ¥

A §

£ S ¢ ¦ ¢ £

her terms. The SIMPLE-C algorithm is used to deal

¨ : ¡ with the velocity-pressure coupling. CALC-BFC uses (115) Boundary-Fitted-Coordinate, on a non-staggered grid

with the Rhie-Chow [42] interpolation to smooth non- ¢

physical oscillations.

598 8 $ $

¦ ¡ ¥

§ For all cases only two-dimensional computations were

8 S ¢ ¦ £

_ (116)

¦ © ¦©

made, to reduce computational costs. The difference between the predicted centerline value using a 3D mesh

compared with a 2D mesh is usually negligible. In ad-

¤ ¤

5 $

¦ ¦

£ £

S ¢ ¦

dition as this is a comparative study for different turbu-

¢ lence models any 3D-effects may be excluded.

$ $ $ $

¦ ¦ ¦ ¦ ¦ ¦

¡ ¡

¥ §

¦© ¦© ¦ © ¦© (117) B.1 Fully Developed Channel Flow This flow is a 1D-flow, with a variation only in the wall The boundary conditions for the specific dissipation rate normal direction. Due to its simplicity this was one of is:

the ﬁrst cases computed using DNS, and has since then

¨ become a standard case for turbulence model compari-

¦ S

C (118) £

son. As DNS gives such a wealth of information most designers use these data bases to improve their models.

The constants in the model are given as Thus it is not surprising that the accuracy of post-DNS

¤ ¤ ¤ ¤ ¤

8 8 _

_ models (from 1990 and later) are signiﬁcantly higher

£ £ £ £ than those prior to DNS. The DNS used as a comparison 0.09 1.0 0.49 0.072 1.1 1.0 1.8 here are the data from Moser et al. [33], with a Reynolds

number based on the channel half width and friction ve-

A A

%$ $L

'7B ' B S

Pros/Cons: locity of S and .

S §/ The predicted a` proﬁles are compared in + Damping functions based on ¡ . Figs. 12 and 13. The difference between the models are + Only a single damping function only minor, however some models slightly under-predict

the friction velocity with a resulting over-prediction of

¢ ¦ – Slightly less numerically stable than the other @` . models, due to the added viscous cross-diffusion

term. Model YS AKN JL C LSY WHR WLR ¢¡

¥ 58.1 57.3 54.8 55.0 54.3 57.7 58.5 Model Performance: Nu 40.1 38.9 33.9 35.6 34.9 38.2 40.4

Channel: Good. Table 5: Friction velocity and Nusselt number. DNS,

A 243

%$ ¡ E L/L/L M $ E E

' B B

¤£

KJ S § S S S and .

BFS: Excellent re-attachment length, however a slight

¤ §

overestimation of in the re-development region. 243 Rib: Excellent prediction of .

Model RS HL PDH BPD ¢¡

¥ 49.9 53.4 56.1 54.4 Nu 43.7 50.3 55.3 52.2

Table 6: Friction velocity and Nusselt number. DNS,

A 243

$ $/L ¡ E L/L/L M $

B B

¤£

S KJ S § S S and .

The normalized friction velocity is listed in tables 5

33 25 5 DNS 4.5 JL C 20 4 LSY YS 3.5 PSfrag replacements AKN PSfrag replacements WHR 15 3

WLR ¥

DNS

¢¦¥ JL 2.5 C 10 LSY 2 YS AKN 1.5 SAA WHR 5 SAA WLR 1 0.5

0 0

0 50 100 150 200 250 300 350 400 0 1 2 ¥

* 10 10 10

$%'& ( *¦¥ 0

B '7B

a` S §/ S

' B

Figure 12: . DNS-data, Kim et al. [33], B

` S ¨§/ S

Figure 14: . DNS-data, Kim et al. [33],

%$ . .

25 5

4.5 DNS 20 RS 4 HL PDH 3.5 BPD 15

¦¥ ¢ 2.5 PSfrag replacements 10 PSfragDNSreplacements 2 RS HL 1.5 5 PDH BPD 1 0.5 0

100 200 300 400 500 600 ¥ * 0 0 1 2

10 10 10

$%'& ( *¦¥ 0

'7B B

S S §/

Figure 13: a` . DNS-data, Kim et al. [33],

%$/L

S ¨§/ S

. Figure 15: . DNS-data, Kim et al. [33],

$/L .

and 6 along with the predicted Nusselt number using a

` S

constant Prandtl number model. The Nusselt number is Progressing to the turbulent kinetic energy,

¨§/

not available for the DNS-data and instead the Dittus- , the proﬁles are compared in semi-log plots, Figs.

23 A 5@?

L L

' ¥¥¤ ¡ ¥¥¤

S ¨ Boelter equation [12] is used c , 14 and 15. The spread between the turbulence models is as introduced by McAdams [30]. much larger for this quantity than for the velocity pro- All models predict both the Nusselt number and the ﬁles. Generally the predictions improve the newer the friction velocity fairly accurate. The largest deviation model is. To quantify the models, the predicted max-

for the friction velocity is given by the RS-model which ima are shown in tables 7 and 8. Only the JL, LSY and

9'7

` under-predicts with more than , which is too much WHR-models give that is too much in error. The clo-

for such a simple test-case as the fully developed channel sest agreement is given by the HL-model. The WLR- 7 ﬂow. The PDH-model over-predict DNS-data with . model improves the result substantially compared with

The Nusselt number is always under-predicted, with the the WHR, using the added damping functions. However

largest deviation predicted by the RS-model ( ¨ ). The as noticed later, an improvement in the predicted turbu- 7

WLR-model gives a correct value within ¨ . lent kinetic energy don’t automatically improve results

From these data only the RS-model could be conside- for other quantities.

¡/` S ¡¢ §/

red as inadequate, which is quite surprising since this The predicted c -proﬁles, are shown in Figs.

¡Y` model is rather new and could have used DNS-data to 16 and 17. Note that for the non- ¡ based models, more correctly tune damping functions and extra terms. needs to be computed using the relation expressed in the

-equation since it is not explicitly solved. For models

0.3 Model YS AKN JL C LSY WHR WLR

DNS 4.47 3.97 3.58 4.39 3.11 2.68 4.29

¥ ¢¤£ ¥§¦§¨ JL ¡ 0.18 0.19 0.21 0.18 0.22 0.13 0.25 C

LSY

$ ¡

' B

` ¡G` S

YS Table 7: Maximum value of and . DNS,

89 L

0.2 AKN

` S ¡G` S ¨/¨ PSfrag replacements WHR and .

WLR G`

¡ 0.15 Model RS HL PDH BPD ¥

4.66 4.62 4.25 4.35

¨ ¥

SAA Table 8: Friction velocity and Nusselt number. DNS,

$/L ¡ 9 L

' B

` S ¡G` S ¨ 0.05 S and .

0 0 1 2

10 10 10

C model in fully developed channel, and in eg. separated

5 test-cases is weak. The performance of the HL-model is

3 A '7B

B also considerably reduced for the other two test-cases, as

¡/` S ¡ ¨§ S

Figure 16: c . DNS-data, Kim et al. [33],

$ . noted below.

0.25

DNS 5h RS HL 0.2 PDH BPD h

PSfrag replacements

0.15 10h 30h

¡G` Figure 18: Geometrical conditions, not to scale, BFS- PSfrag replacements 0.1 case.

0.05 B.2 Backward-facing-step Flow

0 0 1 2 For the backward-facing step ﬂow, the ﬂow undergoes se-

10 10 10

C paration, re-circulation and re-attachment followed by a

5 re-developing boundary layer. In addition, this ﬂow in-

3 A '7B

B volves a shear-layer mixing process, as well as an ad-

S ¡ ¨§ c S

Figure 17: ¡ . DNS-data, Kim et al. [33],

%$/L verse pressure, thus the backward-facing step (BFS) is . an attractive ﬂow for comparing turbulence models.

The case used here is the one that has been studied ¡

that solve the reduced dissipation rate , the boundary using DNS by Lee et al. [27], see Fig. 18. This case has

£E L/L

'¤

/© S

value, ¡ , is added in the plots. a relatively low Reynolds number, , based on *

The result for ¡ is considerably worse than any other the step-height, . In the present computation the inﬂow

E L

red by the models. DNS-data give the maximum value of Neumann condition was applied for all variables at the

The other models give the maximum in the buffer-layer. the walls. The overall calculation domain ranges from

* * *

E;L L L

* *

however that only for the YS- and HL-model the compa- The channel height is in the inlet section and after

rison to the DNS-data is of any value, due to the large the step, yielding an expansion ratio of ¨ . discrepancy in the location of the maximum for the ot- The inlet condition in the BFS-case is crucial for a her models. In the viscous sub-layer only the HL-model critical evaluation and comparison. The DNS data of

yields prediction in agreement with DNS-data, while for Spalart [45] for the velocity proﬁle and the -proﬁle, are

¡ ¦ ` all models give reasonable results. Indisputable directly applied at the inlet. The inﬂow or was spe-

it is the variable Schmidt numbers employed by the HL- ciﬁed in such a way that the model prediction matches

§ S ¢

¡ S

- and -profiles. However the importance of capturing The skin friction coefficient, defined as

¨ § these two proﬁles, especially the dissipation rate, is over- ¨ , is shown in Figs. 19 and 20. The predic- rated as the connection between the performance of the ted friction coefﬁcient varies from very good (PDH) to

35 −3 x 10 Model YS AKN JL C LSY WHR WLR

¢¤£¦¥¨§ 4.62 5.65 6.17 5.86 6.83 7.46 7.96

Table 9: Re-attachment point and maximum friction co-

4 *

E;LL/L M L

§ S ¨ S

PSfrag replacements 2 ¤£§ Model RS HL PDH BPD

DNS ¢£¥§

YS Table 10: Re-attachment point and maximum friction

L −2 E;L/LLNM

AKN

WHR £ WLR −4

that the level of skin friction coefﬁcient was highly af-

¤ §

Figure 19: along lower wall. DNS-data, Le et al. [27] fected by the diffusion of turbulent kinetic energy, hence

¤ 8

the value of _ . Using the BPD-model, can be decrea- 8 −3 _ x 10 sed by increasing , which may explain the signiﬁcantly

6 lower values retrieved by the WHR and WLR models.

¢ ¦

5 However the PDH-model, also a model of similar

¤ §

construction, predicts a lower than the BPD-model, 8

4 _ ¡

even though it employs a lower . In addition all ¢

E _

3 models use S and predict markedly different friction coefﬁcients. The coupled behaviour of the different

2 terms in the turbulence models are hence made obvious.

§ ¤ 1 Thus adjusting a coefficient based on only a single test- case, is not advised, since the model may then deterio- PSfrag replacements 0 rate for other flows. Furthermore models purely tuned −1 DNS for simple, as the well documented fully developed chan- RS nel flow test-case, may perform worse than non-tuned −2 HL PDH model. The HL- and the YS-model may possibly be such −3 BPD models.

−4

−10 −5 0 5 * 10 15 20

§ ¤ Figure 20: along lower wall. DNS-data, Le et al. [27]

PSfrag replacements CYCLIC CYCLIC

rather poor (JL,C,RS). Although hardly discernible, the

§ ¤

Chien-model reproduces a highly questionable pro- C

ﬁle around the re-attachment point. The ` -based damping functions is the main reason for this spurious behaviour, which also yields erroneous heat transfer in the Figure 21: Geometry, RR-case. rib-roughened case. One of the commonly used quantities to justify the accuracy of a turbulence model in a BFS-case is the re- B.3 Rib-roughened Channel Flow

attachment length of the main separation. Tables 9 and ¡

The two models which yielded the most favourable va- The Reynolds number, based on the mean velocity and

/L L/L/L

` S

lues for ¡ at the wall in the channel ﬂow case, YS and the channel height was . The height-to-

HL, completely fail to capture the re-attachment point channel hydraulic diameter, § , was 0.1, and the pitch-

in this case. The YS-model under-predicts by as to-rib-height ratio, § , see Fig. 21.

¢ ¦ much as ¨ . The Wilcox models (WHR and WLR) The experiment provides both ﬂow ﬁeld and heat

don’t give very accurate predictions either, with a ove- transfer measurements, however here only the center-

E $:7 9'7

restimation of the re-attachment point of and ¨ line Nusselt number is used in the comparison. The respectively. Notable is that the improved WLR-model measured Nusselt numbers were normalized with the

10 Model YS AKN JL C LSY WHR WLR ¦

243

E 9 7 AKN 2

Table 11: Nusselt number. Experiment: ,

243

$ WHR E

6 WLR S .

PSfrag replacements ¢¡¥¤ £

£ 5 ¢¡

194 379 156 190

§ ¨ 3 158 313 132 163 % -9 +80 -24 -6

243

E 9 2

Table 12: Nusselt number. Experiment: ,

243

E $

S . 0 £

0 1 2 3 4 5 6 7 8 9

' §

243 results are predicted with the JL, C and HL-models, Figure 22: along lower wall. Experiment, Rau et al. especially the C-model, with a large over-prediction of [41] the Nusselt number. The Wilcox models (WHR, WLR) 6 under-predict the Nusselt number by a fair margin, Exp while the PDH-model slightly less. The effect of the Yap- RS correction is apparent, with the LSY-model predicts less 5 HL than half the Nusselt number as compared with the JL- PDH model. The former is also in better agreement with the BPD experimental data. Similarly the RS-model is an im- 4 provement over its predecessor, the C-model, however

¢¡¦¤ it still gives an erroneous predicted heat transfer in the £ 3 re-circulating zones upstream and downstream the rib. ¢¡ Tables 11 and 12 summaries the predicted Nusselt PSfrag replacements number. The tables give both the average and the max- 2 imum Nusselt number for each model. The comparative values of the experiment is less accurate, especially the

1 averaged value, due to the lack of data around the rib.

E/9 S The value of is mostly likely on the higher side. The deviation of turbulence models from the expe- 0

0 1 2 3 4 5 6 7 8 9 riment is also listed.

5 '

§ It is of interest to compare the predicted Nusselt num-

ber for the rib-roughened with the predicted skin friction

¤ 243 Figure 23: along lower wall. Experiment, Rau et al. in the BFS-case. In tables 9 and 10 the maximum [41] in the re-developing zone, ie. after the re-attachment point is listed. Using Reynolds analogy it is reasonable to believe that models which underestimate the maxi- Dittus-Boelter equation, [12] as introduced by McAdams mum friction coefﬁcient will also underestimate the heat [30]:

transfer coefﬁcient. The JL- and C-models that seve-

243 ;A ;5@?

L L

¥ ¤ ¡ ¥¥¤

c rely over-predict the heat transfer in the rib-roughened ¨ S (119) case also give too large skin friction. The HL-model with

The uncertainty in the resulting enhancement factor, a similar large overestimation of the heat transfer, ho-

23 243 ¤

:7 § , was estimated to be for the experiment. wever under-predicts . The models that under-predict The computations were made using periodic boundary 243 (WHR, WLR) also gives low values of skin friction condition at the streamwise boundaries, which was ve- in the backward-facing step case. The RS-model would riﬁed in the experiment to prevail in the test section. also follow this identity if it was not for the erroneous This reduces the uncertainty in the result due to the in- behaviour around the rib, which raises the average heat let boundary condition. No-slip condition and constant transfer. heat ﬂux were applied at the walls. The rib was, as in the experiment, insulated. Figs. 22 and 23 compare the Nusselt numbers predicted from the different turbulence models. Two models (AKN, BPD) give accurate predicted Nusselt number, (+6% and -6%, respectively) although the AKN-model yields too high values downstream the rib. The worst

C Transformations C.1.3 Viscous diffusion term

£ ¥

¦ ¡ ¦ ¦ ¦ ¦ ¦ ¦

S 8 S ¢ S

C.1 Standard model -equation ¢

¦© ¦© ¦ © ¦©

¦ ¦ ¦ ¦ ¦ ¦

Following Wilcox, the speciﬁc dissipation rate of turbu-

¥ §

S S £ ¦ ¢

¦© ¦ © ¦© ¦©

lent kinetic energy, , is deﬁned as

$ $ $

¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦

S ¦ ¢

¦ 8

(120)

¦ ¦

¢ S

¤ Q

L L $

¦©

J S ¦

with S . The -equation is now constructed

$ $

¨ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¡

from the - and -equations through dimensional reaso-

S ¢ S

¦© ¦© ¦© ¦© ¦©

ning, and the above deﬁnition as:

¨ ¦ ¦ ¦ ¦ ¦

S S

¦ ¡ ¦ ¡

¦© ¦© ¦©

¤ ¤

S 8 S 8 ¢

£ (121)

¨ ¦ ¦ ¦ ¦ ¦ ¦

¦© ¦© ¦© ¦©

Inserting the exact equations for , and , yields: (125)

¢ ¢

E E

5 5 \ $

¦ ¦ ¦

¤ ¤ §

§ C.1.4 Turbulent diffusion term

¤ ¤

S 8 ¢ ¢ 8 ¢ ¡

¢¡

¦ ¦ ¡ ¦ ¦ ¡ ¦ ¡

Production Destruction

¥ ¥

¢ ¢

E E

S 8 8 £ ¢ £ S

_ _

[ [ 8 $ $

¦ ¦

¦© ¦ © ¦© ¦©

§ §

¤ ¤

8 8

¢ ¢

¦ ¦ ¦ ¦ ¦ ¡ ¦ ¡

¥ ¥

8 S £ ¢ £ S

_ _

£¡ ¥¤

¦© ¦© ¦© ¦ ©

Pressure diffusion Turbulent diffusion ¦

E (126)

$ $

¦ ¡ ¦ ¦

¦ © ¦©

Assuming constant Schmidt numbers:

¥¦

¡ ¢

Viscous diffusion E

¦ ¦ ¦ ¦ ¦ ¦ ¦

¥¨§ ¥

S ¦ ¢ 8

¡ £ £ ¡ _

(122) _

¦© ¦© ¦© ¦© ¦©

$ $ $

¦ ¦ ¦ ¦ ¡ ¦ ¦

¡ S ¦ ¡ ¦

¢ ¡

When the standard model is used as the transforma- _

tion basis, the different term in the resulting ¦ -equation

$ $ $

¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦

becomes: ¡

¡ ¡

¦ ¦ ¦ ¡ ¦

¢ 8 ¡ S

¦© ¦© ¦ ©

C.1.1 Production term

$ $

¦ ¨¥ ¡ ¦ ¦ ¦ ¡ ¡ ¦

S ¢ 8

_ _ _

¤ ¤

¦© ¦ © ¦ ©

E E

E E E

5 5 5 5 5

¦ 8 ¡ 8 ¦ 8

¦ $ ¦ ¦ $ ¦ ¦ ¦ ¦ $ ¦

¡ ¡ ¡

£ ¤ ¤

¤ ¤

S 8 ¢ S 8 ¢ S

£ ¢ 8

_ _ _ _

¤ ¤ ¤

(123)

E ¦

¤ ¢

S (127)

The standard turbulent diffusion term did not appear in the derivation above, however it is there, which can be shown through contracting the two right most terms as:

C.1.2 Destruction term

¦ ¦ ¡ ¦ ¡ ¦ ¡ ¦ ¦ ¦ ¦

S £

_ _

_ (128)

¤ ¤ ¤

E E

\ \ \£8

¦ ¡¢ ¦

¤ ¤ £

¤ ¤

8 ¢ 8 S ¢ ¡KS

S Note that it is necessary to assume a constant to make

(124)

¤ ¤

E such an operation valid. Furthermore the term with the

S ¢ ¦

gradient of turbulent viscosity in Eq. 127 is expanded

using its deﬁnition: C.2.2 Destruction term

E E

¤ ¤

¦ ¦ ¦

¥ ©

£ ¢ S § ¡9S S

_ _

¦© ¦ © ¦

E E

¦ ¦ ¦

¤ ¤ ¤ ¤

\ 8£ \£8 $ \ 8 8 $

¥ ¥

¢ 8 S S

£ £

¤ £ £

S ¦ S ¦ ¦ ¦ S

_ _

¦© ¦ ¦©

¤ ¤

E E E

£ £

$ $

E ¡¢ E

¦ ¦ ¦ ¦ ¦

¥ ¥

¤ ¤

8 ¦ S 8 S

£ £

¥ ¥

¢ £ S £ ¢ S

_ _

¦ ¦© ¦ ¦© ¦©

E (134)

¦ ¦

¥ ¥

£ ¢ £ ¢ S

_ _

¦ ©

E E

¦ ¦ ¦

¢ ¢ 8

_ _

¦© ¦© (129)

C.2.3 Viscous diffusion term ¡ Note that no damping function was used for as these are normally dependent on the wall-distance. Thus the

turbulent diffusion term in the ¦ -equation is:

E E

$ $ $

¡ ¦ ¦ ¦ ¡ ¡ ¦ ¦

$ 8

¦ ¦ ¦

¥ ¥

S £ 8 £ ¢ 8

S S ¦

_ _ _ _

¤ ¤

¦© ¦© ¦©

¦© ¦©

$ $

¦ ¡ ¡ ¦ ¦ ¡ ¦ ¦

¡ ¦ ¦ ¡

¥ ¥ ¥

¢ 8

£ £ £

S S

_ _ _

¤ ¤

¦© ¦© ¦©

¦© ¦ ©

(130) E

¦ ¦ ¦ ¡ ¦ ¡ ¡

¥¨§

S ¢ S

¦© ¦© ¦© ¦ ©

C.1.5 Total E

$ $

¡ M ¦ ¦ ¦ ¡

S ¢

¦© ¦© ¦©

The -equation now becomes:

= = =

¦ ¡ ¦ ¡ ¦ ¦ ¡

¦ ¦ ¦% ¦ ¦%

S ¢ ¢

¡ £

¦© ¦© ¦© ¦©

¦© ¦© ¦©

¡ ¦ ¦ ¦ ¡ ¦ ¡

¤ ¤

$ $ $

¡ ¡ ¦ ¦ ¦ E

S ¢ ¢

¨¥ 8 ¢ ¢ ¦

_ _

¦© ¦© ¦ © ¦©

¦© ¦©

$ $

¦ ¦ ¦ ¦

¡ ¡ ¡ ¡

¡ ¦ ¡ ¦ ¨ ¡ ¦ ¦

¥ ¥ ¥

8 8 ¢ ¢

£ £ £

¢ ¢ ¢ S

_ _ _ _

¤ ¤

¦© ¦©

¦ © ¦© ¦© ¦©

$ $

¦ ¦ ¦

$ $

¨¥ ¦ ¦ ¡ ¨¥ ¡ ¦ ¦ ¡

¥ §

¤ S ¢ S

¦© ¦ ©

¦ © ¦© ¦ © ¦©

(131)

$ $

¨¥ ¦ ¦ ¡ ¨¥ ¡ ¦ ¦ ¦ ¡

¥ ¥

S ¢

£ £

Term) -equation

The ¡ -equation could be constructed through the same

¢ ¦ method as for the ¦ -equation. Using a model the

¡ -equation is:

¤ ¤ ¤

8 8 $ 8

¡ C.2.4 Turbulent diffusion term

¦ S ¦ S (132)

with the different terms as:

¡ ¡ ¦ ¦ ¦ ¦ ¦

¥ ¥¨§

S S 8 ¦

£ £ _

C.2.1 Production term _

¦ © ¦© ¦© ¦ ©

8 $

¡ ¦ ¡ ¦

¤ ¤ ¤

598 $ 598 5 8£ 5 8 $ 5 8

¦ ¡

S 8 8

¤ £ £

8 S S ¦ S £

¦© ¦ ©

¦ ¡ ¦ ¡

$ 598

¥ ¥ §

£ £

¦ © ¦©

(133) (136)

39 Here again it is necessary to assume constant Schmidt C.2.5 Cross-diffusion term numbers:

Now dealing with the cross-diffusion term from the ¦ -

$ $

¡ ¦ ¡ ¦ ¦

¤ ¤ ¤ ¤

$ 8 8

¡ ¦ ¦ ¦

8 S

¤ £

S S S

¦© ¦© ¦©

¦© ¦©

¦ ¡ ¡ $ ¦ ¦

¤ ¤ ¤

8£ $

¦ ¦ ¡

¥¨§

¢ ¡ S

S ¡ £ S

¦© ¦© ¦©

¦© ¦©

$ $

¦ ¦ ¦ ¡

¡ ¤ ¤

¦ ¡ ¡ ¦ ¦

¡ S

S ¢ S

¡ £

¦© ¦© ¦©

¤ ¤

¦ ¦ ¡ ¡ ¡ ¦

$ $ $

¦ ¡ ¦ ¡ ¡ ¦ ¦ ¡ ¦ ¡ ¡

¢ S

¦© ¦© ¦©

¦© ¦© ¦© ¦© ¦©

(141)

$ $

¡ ¦ ¦ ¦ ¦ ¡

¡ ¡

¢ ¢

¦© ¦ © ¦ © ¦©

¦ ¨ ¡ ¡ ¦ ¡¢ ¡ ¦

¢ S

_ ¡

£ Adding it all up, the -equation from the cross-diffusion

¦© ¦© ¦©

modelled £¢¤¦ becomes:

$ $

¦ ¦ ¡ ¦ ¨ ¡

¡ ¡ ¡

= = =

8 ¢ S ¢

_ _ _

$ $

¡ ¦ ¦% E ¡ ¦%

£ £

¦© ¦© ¦©

S ¡ £ ¢

¦© ¦ © ¦©

E E

$ $ $

¦ ¡ ¦ ¦ ¡ ¨ ¡ ¡

¥ ¥

8 ¢ £ £

E ¡

_ _ _

£ £

¦© ¦© ¦©

¢ ¢ 8

$ $

¦ ¡ ¦ ¦ ¡

¡ ¡

E E

¤ ¤

$ $

¡ ¦ ¦ ¡

_ _

£ £

¥ §

_ _

¦© ¦©

(137) ¢

E E

¤ ¤

$ $ $

¡ ¡ ¦

¥ § ¥

¨ ¢ 8 ¢

£ £

_ _

¦©

Two terms are re-arranged: ¢

$ $ $

¦ ¦ ¡ ¦ ¡

¡ ¡ ¡

¥ ¥ §

£ ¢ £

_ _ _

£ £

¦© ¦ © ¦©

¡ ¦ ¡ ¦ ¡ ¦ ¡ ¦ ¦ ¡ ¡

(142)

_ _

_ (138)

£ £ £

For a model without cross-diffusion term (WHR,WLR)

¤ ¤

(the viscous term) and (the turbulent term) is set

and the term with the gradient of turbulent viscosity is to zero. For the two cross-diffusion ¥¢§¦ models treated

¤ ¤

L L 9

¢¡ S

expanded assuming a HRN-model for : here, the coefﬁcients is S , (PDH) and

¤ ¤

E% E S

S (BPD).

E E

¤ ¤

¡ ¦ ¡ ¦

¥ ©

£ 8 ¢ S S § ¡ S

_ _

¦ © ¦© ¡

E E

¡ ¦ ¦

¥ ¥

S 8 ¢ S

£ £

_ _

¦ © ¡ ¦©

E E

¤ ¤

¡ ¨ ¦ ¦ ¡ ¦

¥ ¥

£ £ ¢ S 8 ¢ S _

_ (139)

¡ ¦© ¡ ¦© ¦ ©

E E

¡ ¡ ¦

¥ ¥

S 8 ¢ ¢

£ £

_ _

¦ ©

E E

¦ ¡ ¦ ¡

¢ 8 ¢

_ _

¦© ¦©

and hence the turbulent diffusion term becomes:

E E

$ $

¦ ¡ ¦ ¦ ¡

¡ ¡ ¡

¥ ¥

8 8

S ¢ ¢ £ £

_ _ _ _

£ £

¦© ¦© ¦©

$ $ $

¡ ¡ ¡ ¦ ¡ ¦ ¡ ¦

¥ ¥ ¥

£ £ £

_ _ _

£ £

¦© ¦© ¦©

(140)

D Secondary equation Continuing to expand the derivate gives:

¤ 1 1

¦ ` ¦ £ ¦

A comparison between the used secondary quantities ¥

£ `

S ¢ £ ¢ ¢

C C C

§ ¦ ¦ ¦ £

could be made by solving a transport equation for the

¤ 1

¤ (149)

S § £

variable, " . In the logarithmic region, the

¥ ¦

¢ C

convective term and the viscous diffusion could be neg- _

§ ¦ lected, hence the transport equation for the secondary £ quantity is reduced to ”production – dissipation equal to Assuming a linear variation of the turbulent length-

the diffusion” as: scale the second derivate is zero, and with the derivate

of equals to zero, the diffusion term becomes:

¤ ¤

¦ `

1 1

¢ S

§ ¡ £ §

§ §

¤ ¤

£ £G`

¥ ` ¥ ` ¦

1 1

S S

¤ ¤ C

_ (150)

/` £G`

Destruction £

§ I

Production

(143)

¦ ¦ §

C C

_ In summation the transport equation for the secondary

¦ § ¦

quantity is:

Diffusion ©

§ §

¤ ¤

` ` ¥

1 1

§ S § ¤

where only gradients normal to the wall are retained for ¤

£G` £G` §

simplicity. Furthermore in order to compare the secon- I

¤ ¤

dary quantities, relations for the involved variable need ¥

¢ S

§ §

to established. Here, similarly to Launder and Spalding

§ [26] the law of the wall for the velocity, turbulent kinetic (151) energy, turbulent length-scale, and the turbulent visco-

sity are used: where ¥ takes the value according to the used secondary

E variable. For the EVM-types referred to in this report

S #

B W

5 the following values for are obtained:

¡ E

¥¢§¡ ¥S

¤ ¤

¡ E

¥4S

(144) ¥¢§¦

W C

¡ E

¥¢¤¨ ¥4S ¢

¡ E

¥¢ ¥S ¢

¤ 1

S ¡ In conclusion all the studied two-equation models thus Using the above relations the production term can be need to fulﬁll the following relation for the Schmidt

simpliﬁed to: number in the logarithmic region:

¤ ¤ ¤ ¤

% ¦

¥ ¥

§ S

1 1

¤ ¤ ¤ ¤ £ £

§ ¡ S § S

(152) C W C

§ ¢ §

£ £#%

1 S

§ Different values may however be applicable through the

¤ ¤

£/` §

tuning of the coefﬁcients, § and . (145)

The same approach is used for the diffusion term:

¦ ¦ § £

C C

¦ ¦ §

1 1 C

C (146)

¦ £ ¦ §¦ ¢ ¦ £¨§¦

¥¨§

¦ §

Assuming that the law of the wall holds, the derivate of

is zero, and hence:

¦ ¦ £

S ¢ £

C C

_ (147)

¦ § ¦

Inserting the relation for the turbulent viscosity with a

constant Schmidt number gives:

§ §

¤ ¤ ¤ ¤

1 1

` ` £

¦ ¥ ¦ ¦ ¦

¥ ¥

1 1

S ¢ S ¢

£ £

C C C C

_ _

¦ ¦ ¦ ¦

§ §

£#% £ (148)