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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 deve- loped 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 discus- sion on the type of EVM attention is given to the turbulence model coefficients, Schmidt numbers and the possibilty to in- clude 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 gi- ven 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 P ¡ ¢£¢ ¢£¢ Turbulence model constant Turbulence model coefficients ¡ ¤ Q ¡ ¢£¢ ¢£¢ Various constants Turbulence model coefficients ¡ ¤¦¥ R ¡ ¢£¢ ¢ 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 V ¡ ¢ ¢ Diffusion term Kolmogorov length scale ¡ W § ¡ ¢ ¢£¢ 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 2 ¡ § -¢ spanwise: 3,W,w ¨ Shear stress 2 B ¡ §/ ¢ H© ¡ §7 -¢ Turbulent kinetic energy ¨ Wall shear, 0 (¢ Integral length scale ¡ 1 (¢ Length scale ¡ 243 Superscripts ¡ ¢£¢ Nusselt number Viscous value 5 2 B ¡ § ¢ Static pressure "!` Normalized value using : 5 B ¡ (¢ Sb § Rib-pitch a` 2 6 C C B ¡ §7 ¢ S § Fluctuating pressure ` 598 B ¡ §/;: ¢ ` S ¨§ Turbulent production, = 3!<=>3!< B ¦% §¦© ` ¡ S ¡ ¨§/ @c 5@? ¡ ¢£¢ Prandtl number Normalized value A ) ' § ¡ ¢£¢ Reynolds number, Alternative value A ¡ ¢£¢ ¡ Turbulent Reynolds number, ¡ §¢ § ¦ § ¡ , , d A Subscripts ' B B ¡ ¢£¢ based Reynolds number, Bulk value C B Hydraulic diameter D Turbulence model source term [*] * = Step heigth D e ¡ ¢ Strain-rate tensor, = = Quantity based on Kolmogorov scales $ E §/¨ ¦% §¦© ¦ §¦© Quantity in -equation F Turbulent transport term [*] ? Re-attachment poin F ¡ Turbulent time scale [s] = Turbulent quantity f ¡ §GH¢ Velocity Wall value = < 3 C §GH¢ Fluctuating velocity ¡ Normalized using wall distance 3 =>3 < < ¡ §/ ¢ ¡ ¡ Reynolds stresses Quantity in -equation g B © I ¨ § ¡ §GH¢ Friction velocity, Quantity based on Taylor micro-scale E;LL/LNM B B ¦ ¦ KJ § ¢£¢ Normalized value: ¡ Quantity in -equation " ¡ (¢ © Streamwise coordinate Free variable C ¨ (¢ Wall normal coordinate ¡ Quantity based on the friction velocity h 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 A E¡ /L '7B Reynolds number S and for a simple fully deve- to as numerical experiments made in virtual windtun- loped channel flow testcase.
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