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Turbulence Modelling1 Spring 2021 8. TURBULENCE MODELLING1 SPRING 2021 8.1 Eddy-viscosity models 8.2 Advanced turbulence models 8.3 Wall boundary conditions Summary References Appendix: Derivation of the turbulent kinetic energy equation Examples The Reynolds-averaged Navier-Stokes (RANS) equations are transport equations for the mean variables in a turbulent flow. These equations contain net fluxes due to turbulent fluctuations. Turbulence models are needed to specify these fluxes. 8.1 Eddy-Viscosity Models y 8.1.1 The Eddy-Viscosity Hypothesis The mean shear stress has both viscous and turbulent parts. In simple shear (i.e. where 휕푈/휕푦 is the only non-zero mean gradient): ∂푈 τ = μ −⏟ ρ푢푣 ⏟∂푦 (1) turbulent viscous U The most popular type of turbulence model is an eddy-viscosity model (EVM) which assumes that turbulent stress is proportional to mean-velocity gradient in a manner similar to viscous stress. In simple shear (see later for the general case): ∂푈 − ρ푢푣 = μ (2) 푡 ∂푦 푡 is called an eddy viscosity or turbulent viscosity. The overall mean shear stress is then ∂푈 τ = μ (3) eff ∂푦 where the total effective viscosity μeff = μ + μ푡 (4) 1 More advanced descriptions of turbulence and its modelling can be found in: Leschziner, M.A., 2015, Statistical turbulence modelling for fluid dynamics - demystified: an introductory text for graduate engineering students, World Scientific. Wilcox, D.C., 2006, Turbulence Modelling for CFD, 3rd Edition, DCW Industries. Pope, S.B., 2000, Turbulent flows, Cambridge University Press. Schlichting, H. and Gersten, K., 1999, Boundary layer theory, 8th English Edition, Springer-Verlag. CFD 8 – 1 David Apsley Note: (1) This is a model! (2) μ is a physical property of the fluid and can be measured; μ푡 is a hypothetical property of the flow and must be modelled. (3) μ푡 varies with position. (4) At high Reynolds numbers, μ푡 ≫ μ throughout much of the flow. Eddy-viscosity models are widely used and popular because: • they are easy to implement in existing viscous solvers; • extra viscosity aids stability; • they have some theoretical foundation in simple shear flows (see below). However: • there is little theoretical foundation in complex flows; • modelling turbulent transport is reduced to a single scalar, μ푡, and, hence, at most one Reynolds stress can be represented accurately. 8.1.2 The Eddy Viscosity in the Log-Law Region In the log-law region of a turbulent boundary layer it is assumed that: (a) (i) total stress is constant (and equal to that at the wall); (ii) viscous stress is negligible compared to turbulent stress: (turb) 2 τ = τ푤 ≡ ρ푢τ (b) the mean velocity profile is logarithmic; ∂푈 푢 = τ ∂푦 κ푦 The eddy viscosity is then (turb) 2 τ ρ푢τ μ푡 ≡ = = ρ(κ푢τ푦) ∂푈/ ∂푦 푢τ/κ푦 Hence, in the log-law region, with ν푡 = μ푡/ρ as the kinematic eddy viscosity, ν 푡 = κ푢τ푦 (5) In particular, the eddy viscosity is proportional to distance from the boundary. 8.1.3 General Stress-Strain Relationship The stress-strain relationship (2) applies only in simple shear and cannot hold in general because the LHS is symmetric in 푥 and 푦 components but the RHS is not. The appropriate generalisation gives representative shear and normal stresses (from which others can be obtained by “pattern-matching”): CFD 8 – 2 David Apsley ∂푈 ∂푉 − ρ푢푣 = μ ( + ) (6) 푡 ∂푦 ∂푥 ∂푈 2 − ρ푢2 = 2μ − ρ푘 (7) 푡 ∂푥 3 2 The − ρ푘 part (푘 is turbulent kinetic energy) in (7) ensures the correct sum of normal stresses: 3 ρ(푢2 + 푣2 + 푤2) = −2ρ푘 because the mean-velocity terms from this sum would be zero in incompressible flow: ∂푈 ∂푉 ∂푊 + + = 0 ∂푥 ∂푦 ∂푧 Using suffix notation both shear and normal stresses can be summarised in the single formula ∂푈푖 ∂푈푗 2 −ρ푢푖푢푗 = μ푡( + ) − ρ푘δ푖푗 (8) ∂푥푗 ∂푥푖 3 8.1.4 Other Turbulent Fluxes According to Reynolds’ analogy it is common to assume a gradient-diffusion relationship between any turbulent flux and the gradient of the corresponding mean quantity; i.e. ∂Φ − ρ푣ϕ = Γ (9) 푡 ∂푦 The turbulent diffusivity Γ푡 is proportional to the eddy viscosity: μ푡 Γ푡 = (10) σ푡 σ푡 is called a turbulent Prandtl number. Since the same turbulent eddies are responsible for transporting momentum and other scalars, σ푡 is approximately 1.0 for most variables. 8.1.5 Specifying the Eddy Viscosity The kinematic eddy viscosity has dimensions of [velocity] [length], which suggests that it be modelled as ν 푡 = 푢0푙0 (11) Physically, 푢0 should reflect the magnitude of velocity fluctuations and 푙0 the size of turbulent eddies. For example, in the log-law region, ν푡 = κ푢τ푦, or ν푡 = velocity (푢τ) × length (κ푦) For wall-bounded flows a candidate for 푢0 is the friction velocity, 푢τ = √τ푤/ρ. However, this CFD 8 – 3 David Apsley is not a local scale, since it depends on where the nearest wall is. A more appropriate velocity scale in general is 푘1/2, where 푘 is the turbulent kinetic energy. For simple wall-bounded flows, 푙0 is proportional to distance from the boundary (e.g. 푙0 = κ푦). For free shear flows (e.g. jet, wake, mixing layer) 푙0 is proportional to the width of the shear layer. However, both of these are geometry-dependent. For greater generality, we need to relate 푙0 to local turbulence properties. Common practice is to solve transport equations for one or more turbulent quantities (usually 푘 + one other), from which μ푡 can be derived on dimensional grounds. The following classification of eddy-viscosity models is based on the number of transport equations. zero-equation models: – constant-eddy-viscosity models; – mixing-length models: 푙0 specified geometrically; 푢0 from mean flow gradients. one-equation models: – 푙0 specified geometrically; transport equation to derive 푢0; two-equation models: – transport equations for quantities from which 푢0 and 푙0 can be derived. Of these, the most popular in general-purpose CFD are two-equation models: in particular, the 푘 − ε and 푘 − ω models. 8.1.6 Mixing-Length Models (Prandtl, 1925). Eddy viscosity: μ푡 = ρν푡, where ν푡 = 푢0푙푚 (12) The mixing length 푙푚 is specified geometrically and the velocity scale 푢0 is then determined from the mean-velocity gradient. In simple shear: ∂푈 푢 = 푙 | | (13) 0 푚 ∂푦 The model is based on the premise that if a turbulent eddy displaces a y fluid particle by distance 푙푚 its velocity will differ from its surrounds by an amount 푙푚|휕푈/휕푦|. (Any constant of proportionality can be absorbed into the definition of 푙푚.) The resulting turbulent shear stress is (assuming positive velocity lm gradient): l dU ∂푈 ∂푈 ∂푈 2 m dy τ(turb) = μ = ρ푢 푙 ( ) = ρ푙2 ( ) (14) 푡 ∂푦 0 푚 ∂푦 푚 ∂푦 U The mixing length 푙푚 depends on the type of flow. CFD 8 – 4 David Apsley Log Layer In the log layer, ∂푈 푢τ τ(turb) = ρ푢2 and = τ ∂푦 κ푦 Equation (14) then implies that 푙푚 = κ푦 y y General Wall-bounded flows 0.09 In general, 푙푚 is limited to a certain fraction of the boundary-layer depth . Cebeci and Smith (1974) suggest: 푙 = min(κ푦, 0.09δ) (15) 푚 y U lm Free shear flows y For free shear layers (no wall boundary), 푙푚 is assumed proportional to the local shear-layer half-width . Wilcox (2006) suggests: 0.071 (mixing layer) 푙 0.098 (plane jet) 푚 = (16) δ 0.080 (round jet) U { 0.180 (plane wake) U Mixing-length models work well in near-equilibrium boundary layers or very simple free-shear flows. However, although generalisations of the stress-strain relationship (13) exist for arbitrary velocity fields, it is difficult to specify the mixing length 푙푚 for complex flows which do not fit tidily into one of the above geometrically-simple categories. 8.1.7 The 풌 − Model This is probably the most common type of turbulence model in use today. It is a two-equation eddy-viscosity model with the following specification: 푘2 μ푡 = ρν푡, ν = 퐶 (17) 푡 μ ε 푘 is the turbulent kinetic energy and ε is its rate of dissipation. In the standard model 퐶μ is a constant (with value 0.09). 푘 and ε are determined by solving transport equations. For the record (i.e. you don’t have to learn them) they are given here in conservative differential form, including implied summation over repeated index 푖. CFD 8 – 5 David Apsley ∂ ∂ (푘) ∂푘 (푘) (ρ푘) + ( ρ푈푖푘 −Γ ) = ρ(푃 − ε) ∂푡 ∂푥푖 ∂푥푖 ∂ ∂ (ε) ∂ε (푘) ε (ρε) + ( ρ푈푖ε −Γ ) = ρ(퐶ε1푃 − 퐶ε2ε) (18) ∂푡 ∂푥푖 ∂푥푖 푘 rate of advection diffusion source change 푃푘 is the rate of production of turbulent kinetic energy 푘 (see below). The diffusivities of 푘 and ε are related to the molecular and turbulent viscosities: μ μ = μ + 푡 , Γ(ε) = μ + 푡 σ푘 σε and, in the standard model (Launder and Spalding, 1974), model constants are: 퐶μ = 0.09, 퐶ε1 = 1.44, 퐶ε2 = 1.92, 휎푘 = 1, σ = 1.3 (19) Notes. (1) The 푘 − ε model is not a single model but a class of different schemes. Variants have different coefficients, some including dependence on molecular-viscosity effects near boundaries (“low-Reynolds-number 푘 − ε models”) and/or mean velocity gradients (e.g. “realisable” 푘 − ε models). Others have a different ε equation. (2) Apart from the diffusion term, the 푘 equation is that derived from the Navier-Stokes equation. The ε equation is, however, heavily modelled. (3) Although 푘 is a logical choice (as it has a physical definition and can be measured), use of ε as a second scale is not universal and other combinations such as 푘 − ω (ω is a frequency), 푘 − τ (τ is a timescale) or 푘 − 푙 (푙 is a length) may be encountered.
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