Menter Shear Stress Transport (SST) k-ω Model

Menter (1994) suggested the SST model to overcome the freestream value sensitivity of the standard k-ω turbulence model by transforming the k-ε model into the k-ω model in the near-wall region, and by utilizing the k-ε model in the turbulent region far from the wall.

The SST model employs a modified source term in the eddy frequency equation. Among the different SST versions, the SST 2003 model (Menter, 2003) will be focused on in this section as it has been shown to be more accurate when predicting flows near stagnation zones by introducing a production limiter to constrain the kinetic energy production. This version also employs the strain rate magnitude in the definition of eddy viscosity rather than the vorticity magnitude employed in the standard SST model (1994). When compared to the k-ε model, the SST 2003 model achieves better accuracy for attached boundary layers and flow separation. The SST 2003 model also overcomes the freestream sensitivity of the k-ω turbulence model. However, this model shares a similar range of weakness with the k-ε equation models for the predictions of flows with strong streamline curvature and/or rotation, as well as flows under extra strains and body forces.

Transport Equations

Turbulent Kinetic Energy k(1)
∂ ( ρ k ) ∂ t + ∂ ( ρ u j ¯ k ) ∂ x j   = ∂ ∂ x j [ ( μ + σ k μ t ) ∂ k ∂ x j ] + P ˜ k + D k
Eddy Frequency (Specific Dissipation Rate) ω (2)
∂ ( ρ ω ) ∂ t + ∂ ( ρ u j ¯ ω ) ∂ x j   = ∂ ∂ x j [ ( μ + σ ω μ t ) ∂ ω ∂ x j ] + P ω + D ω + 2 ( 1 − F 1 ) ρ   σ ω 2 ω ∂ k ∂ x j ∂ ω ∂ x j
where
  • The blending function F 1 = t a n h { ( m i n [ m a x k β * ω d , 500 υ d 2 ω , 4 ρ σ ω 2 k C D k ω d 2 ] ) 4 } ,
  • F 1 { 0   : a c t i v a t i o n   o f   k − ε   m o d e l   f o r   t u r b u l e n t   c o r e   f l o w s   1   : a c t i v a i t i o n   o f   k − ω   f o r   f l o w s   n e a r   w a l l s                                        
  • C D k ω = m a x ( 2 ρ σ ω 2 1 ω ∂ k ∂ x i ∂ ω ∂ x i ,       10 − 10 ) ,
  • d is the distance to the nearest wall.

Production Modeling

Turbulent Kinetic Energy k (3)
P ˜ k = min ( P k ,   10 ρ β * k ω )
where
  • Production P k = μ t S 2
  • Strain rate magnitude S = 2 S i j S i j
  • Strain rate tensor S i j = 1 2 ( ∂ u i ¯ ∂ x j + ∂ u j ¯ ∂ x i )
  • Constant β *
Eddy Frequency (ω) (4)
P ω = α ρ S 2
where
  • Blending function for specifying constants α = α 1 F 1 + α 2 ( 1 − F 1 )
  • α 1 = 5 9 for the inner layer
  • α 2 = 0.44 for the outer layer.

Dissipation Modeling

Turbulent Kinetic Energy k (5)
D k = − ρ β * k ω
Eddy Frequency ω (6)
D ω = − ρ β ω 2

Modeling of Turbulent Viscosity μ t

(7)
μ t = ρ   a 1   k max ( a 1 ω ,   S   F 2 )
where
  • The second blending function F 2 = t a n h ( m a x ( 2 k β * ω   d ,   500 ν d 2 ω ) ) 2 ,
  • Fluid kinematic viscosity ν ,
  • Constant β * .

Model Coefficients

β * = 0.09.

Following constants for SST are computed by a blending function ϕ = ϕ 1 F 1 + ϕ 2 ( 1 − F 1 ) : σ ω 1 = 0.5 , σ ω 2 = 0.856 , σ k 1 = 0.85 , σ k 2 = 1.00 , β 1 = 3 40 , β 2 = 0.0828 .