Turbulence Modeling Cuong Nguyen November 05, 2005 1 Incompressible Case 1.1 Reynolds-averaged Navier-Stokes equations The incompressible Navier-Stokes equations in conservation form are u i x i = 0 (1) ρ u i t + ρ (u u i ) = p + (2µs i ) (2) x x i x where the strain-rate tensor s i is given by s i = 1 ( ui + u ) 2 x x i. (3) By the application of Eq. (1), the equations of motion can be written as ρ u i t + ρu u i = p + µ 2 u i. (4) x x i x i x In turbulent flows, the field properties become random functions of space and time. Hence, the field variables u i and p must be expressed as the sum of mean and fluctuating parts as u i = U i + u i, p = P + p. (5) where the mean and fluctuating parts satisfy u i = U i, u i = 0 (6) p = P, p = 0 (7) with the bar denoting the time average. We insert Eq. (5) into (1)-(2) and take the time average to arrive at the Reynoldsaveraged Navier-Stokes (RANS) equations U i x i = 0 (8) ρ U i t + ρ (U i U ) = P + (2µS i ρu i x x i x u ), (9) 1
where S i is the mean strain-rate tensor S i = 1 ( Ui + U ) 2 x x i. (10) The quantity τ i = u i u is known as the Reynolds stress tensor which is symmetric and thus has six components. By the application of (8) Eq. (9) can then be expressed as U i t + U U i = P + ν 2 U i u i u. (11) x x i x i x x By decomposing the instantaneous properties into mean and fluctuating parts, we have introduced 3 unknown quantities. Unfortunately, we have gain no additional equations. This means our system is not yet closed. To close the system, we must find enough equations to solve for our unknowns. In what follows, we describe several approaches (turbulence models) for solving the RANS equations. 1.2 Turbulence Models 1.2.1 Boussinesq Approximation The Reynolds-averaged approach to turbulence modeling requires that the Reynolds stresses in Equation (11) be appropriately modeled. A common method employs the Boussinesq hypothesis to relate the Reynolds stresses to the mean velocity gradients where the turbulence kinetic energy, k, is defined as u i u = 2ν T S i 2 3 kδ i (12) k = 1 2 u i u i, (13) and ν T is the kinetic eddy viscosity assumed as an isotropic scalar quantity which is not strictly true so that the term approximation is appropriate. 1.2.2 Spalart-Allmaras Model In Spalart-Allmaras Model, the turbulence kinetic energy is not calculated, the last term in Equation (12) is ignored when estimating the Reynolds stresses u i u = 2ν T S i. (14) The model includes eight closure coefficients and three closure functions. Its defining equations are as follows: ν T = νf v1, f v1 = χ 3, χ = ν χ 3 + c 3 v1 ν ν is the molecular viscosity and ν obeys the transport equations ν t + U ν 1 = c b1 S ν + (ν + ν) ν ] + c b2 x σ x k x k σ 2 ν x k ν x k c w1 f w (15) ] 2 ν (16) d
where c b1 = 0.1355, c b2 = 0.622, c v1 = 7.1, σ = 2/3 (17) c w1 = c b1 κ + (1 + c b2), c 2 w2 = 0.3, c w3 = 2, κ = 0.41 (18) σ ] χ 1 + c 6 1/6 f v2 = 1, f w = g w3, g = r + c w2(r 6 r) (19) 1 + χf v1 g 6 + c 6 w3 r = ν Sκ 2 d, ν S = S + 2 κ 2 d f v2, S = 2Ω 2 i Ω i. (20) The tensor Ω i = 1 2 (U i/x U /x i ) is the rotation tensor and d is distance from the closest surface. 1.2.3 The k ω Model The k ω model has been modified over the years, production terms have been added to both the k and ω equations, which have improved the accuracy of the model for predicting free shear flows. The following version of the k ω model is presented Kinetic eddy viscosity: ν T = k/ω (21) Turbulence Kinetic Energy: k t + U k U i = τ i β kω + x x x (ν + σ ν T ) k ] x (22) Specific Dissipation rate: ω t + U ω = α ω x k τ U i i βω 2 + x x (ν + σν T ) ω ] x (23) Closure Coefficients and Relations: α = 13 25, β = β of β, β = βof β, σ = 1 2, σ = 1 (24) 2 β o = 9 125, f β = 1 + 70χ ω, χ ω, = Ω i Ω k S ki 1 + 80χ ω (βoω) 3 (25) { βo = 9 1, χk 100, f β = 0 1+680χ 2 k, χ 1+400χ 2 k > 0, χ k = 1 k ω (26) ω 3 x x k ɛ = β ωk, l = k/ω (27) 3
1.2.4 The Standard k ɛ Model The standard k ɛ model is a semi-empirical model based on model transport equations for the turbulence kinetic energy k and its dissipation rate ɛ. The model transport equation for k is derived from the exact equation, while the model transport equation for ɛ was obtained using physical reasoning and bears little resemblance to its mathematically exact counterpart. In the derivation of the k ɛ model, it was assumed that the flow is fully turbulent, and the effects of molecular viscosity are negligible. The standard k ɛ model is therefore valid only for fully turbulent flows. Kinetic eddy viscosity: ν T = C µ k 2 /ɛ (28) Turbulence Kinetic Energy: k t + U k U i = τ i ɛ + x x x (ν + ν T /σ k ) k ] x (29) Specific Dissipation rate: ɛ t + U ɛ ɛ = C ɛ1 x k τ U i ɛ 2 i C ɛ2 x k + x Closure Coefficients and Relations: (ν + ν T /σ ɛ ) ɛ ] x (30) C ɛ1 = 1.44, C ɛ2 = 1.92, C ɛµ = 0.09, σ k = 1.0, σ ɛ = 1.3 (31) ω = ɛ/(c µ k), l = C µ k 3/2 /ɛ (32) 2 Compressible Case 2.1 Favre-averaged Equations 2.1.1 Gorverning equations The compressible Navier-Stokes equations in conservation form are ρ t + (ρu i ) = 0 (33) x i t (ρu i) + (ρu u i ) = p + t i (34) x x i x ρ (e + 12 )] t u iu i + ρu (h + 12 )] x u iu i = (u t i ) q (35) x x where e is specific internal energy, h = e+p/ρ is specific enthalpy, t i is the viscous stress tensor, and q is the heat flux vector. For gases, the classical ideal gas law is p = ρrt = (γ 1)ρe. (36) 4
For the compressible flow, t i, is given by ( t i = 2µ s i 1 ) u k δ i 3 x k. (37) where the strain-rate tensor s i is given by s i = 1 ( ui + u ) 2 x x i. (38) The convective heat flux q is defined as q = κ T x (39) where κ is thermal conductivity. Furthermore, the specific internal energy and specific enthalpy are given by e = c v T, h = c p T (40) where c v and c p are the specific-heat coefficients (note that γ = c p /c v and R = c p c v ). Then, we have h q = κ/c p = µ h (41) x P r x where P r is the Prandtl number defined by 2.1.2 Mass Averaging P r = c pµ κ. (42) Let Φ(x, t) be any dependent variable. We recall the time (Reynolds) average of P hi(x, t) defined by Φ(x, t) = 1 Φ(x, t)dt (43) T and the Reynolds decomposition defined as We now introduce the density weighted time (Favre) average T Φ = Φ + Φ. (44) Φ = ρφ ρ (45) and define the Favre decomposition as It should be noted that Φ = 0 but Φ 0. Φ = Φ + Φ. (46) 5
2.1.3 Turbulent Equations We first introduce the Reynolds decomposition for ρ and p and the Favre decomposition for u, e, and h as ρ = ρ + ρ, p = p + p, q = q + q (47) u i = ũ i + u i, e = ẽ + e, h = h + h. (48) We next insert them into the governing equations and the take the time average for the governing equations to obtain ρ t + (ρũ i ) = 0 (49) x i t (ρũ i) + (ρũ i ũ ) = p + t i x x i x x (ρu i u ) (50) ρũ ( h + 12ũiũ i ) ρ (ẽ + 12ũiũ ) i + 1 ] t 2 ρu i u i + x ( ) ũ i t i ρu i x u ] ρu i + ũ u i = 2 q ρu h + t i u 1 i ρu 2 u i u i ] (51) p = (γ 1)ρẽ (52) 2.1.4 Approximations Reynolds-stress tensor: ρτ i = ρu i u = 2µ T 2.2 Turbulence Models ( S i 1 ) ũ k δ i 2 3 x k 3 ρkδ i (53) 6