SST (Menter’s Shear Stress Transport)

History

The SST two equation turbulence model was introduced in 1994 by F.R. Menter to deal with the strong freestream sensitivity of the k-omega turbulence model and improve the predictions of adverse pressure gradients. The formulation of the SST model is based on physical experiments and attempts to predict solutions to typical engineering problems. Over the last two decades the model has been altered to more accurately reflect certain flow conditions. The Reynold's Averaged Eddy-viscosity is a pseudo-force and not physically present in the system. The two variables calculated are usually interpreted so k is the turbulence kinetic energy and omega is the rate of dissipation of the eddies.

SST (Menter’s Shear Stress Transport) turbulence model

∂ ( ρ k ) ∂ t + ∂ ( ρ u j k ) ∂ x j = ρ P − β ∗ ρ ω k + ∂ ∂ x j [ ( μ + σ k μ t ) ∂ k ∂ x j ]

∂ ( ρ ω ) ∂ t + ∂ ( ρ u j ω ) ∂ x j = γ ν t P − β ρ ω 2 + ∂ ∂ x j [ ( μ + σ ω μ t ) ∂ ω ∂ x j ] + 2 ( 1 − F 1 ) ρ σ ω 2 ω ∂ k ∂ x j ∂ ω ∂ x j

Variable Definition

P = τ i j ∂ u i ∂ x j

τ i j = μ t ( 2 S i j − 2 3 ∂ u k ∂ x k δ i j ) − 2 3 ρ k δ i j

S i j = 1 2 ( ∂ u i ∂ x j + ∂ u j ∂ x i )

μ t = ρ a 1 k m a x ( a 1 ω , Ω F 2 )

ϕ = F 1 ϕ 1 + ( 1 − F 1 ) ϕ 2

F 1 = t a n h ( a r g 1 4 )

a r g 1 = m i n [ m a x ( k β ∗ ω d , 500 ν d 2 ω ) , 4 ρ σ ω 2 k C D k ω d 2 ]

C D k ω = m a x ( 2 ρ σ ω 2 1 ω ∂ k ∂ x j ∂ ω ∂ x j , 10 − 20 )

F 2 = t a n h ( a r g 2 2 )

a r g 2 = m a x ( 2 k β ∗ ω d , 500 ν d 2 ω )

K-W Closure

σ k 1 = 0.85 , σ w 1 = 0.65 , β 1 = 0.075

K-e Closure

σ k 2 = 1.00 , σ w 2 = 0.856 , β 2 = 0.0828

SST Closure Constants

β ∗ = 0.09 a 1 = 0.31

Far Field

U ∞ L < w f a r f i e l d < 10 U ∞ L

10 − 5 U ∞ 2 R e L < k f a r f i e l d < 0.1 U ∞ 2 R e L

Boundary/Wall Conditions

ω w a l l = 10 6 ν β 1 ( Δ d 1 ) 2

k w a l l = 0