Three-Equation Eddy Viscosity Models

v2-f Model

In order to account for the near wall turbulence anisotropy and non local pressure strain effects, Durbin (1995) introduced a velocity scale v2 and the elliptic relaxation function f to the standard k-ε turbulence model.

The velocity scale v2 represents the velocity fluctuation normal to the streamline and represents a proper scaling of the turbulence damping near the wall, while the elliptic relaxation function f is used to model the anisotropic wall effects. Compared to the k-ε turbulence models, the v2-f model produces more accurate predictions of wall-bounded flows dominated by separation but suffers from numerical stability issues.

Transport Equations

Turbulent Kinetic Energy k (1) ∂ ( ρ k ) ∂ t + ∂ ( ρ u j ¯ k ) ∂ x j   = ∂ ∂ x j [ ( μ + μ t σ k ) ∂ k ∂ x j ] + P k + D k
Turbulent Dissipation Rate ε (2) ∂ ( ρ ε ) ∂ t + ∂ ( ρ u j ¯ ε ) ∂ x j   = ∂ ∂ x j [ ( μ + μ t σ ε ) ∂ ε ∂ x j ] + P ε + D ε
Velocity Scale v2 (3) ∂ ( ρ v 2 ¯ ) ∂ t + ∂ ( ρ u j ¯ v 2 ¯ ) ∂ x j   = ∂ ∂ x j [ ( μ + μ t σ v 2 ) ∂ v 2 ¯ ∂ x j ] + P v 2 + D v 2

Elliptic Relation for the relaxation function f

L 2 ∂ 2 f ∂ x j 2 − f   = ( C 1 − 1 ) T ( v 2 k − 2 3 ) − C 2 P k ε

where
  • L = C L   m a x ( k 3 / 2 ε , C η [ ν 3 ε ] 1 / 4   ) : the length scale,
  • T = m a x ( k ε , C T [ ν ε ]   ) : the time scale.

Production Modeling

Turbulent Kinetic Energy k (4) P k = μ t S 2
Turbulent Dissipation Rate ε (5) P ε = C ε 1 ε k μ t S 2 = C ε 1 ε k P k
Velocity Scale v2 (6) P v 2 = ρ k f

Dissipation Modeling

Turbulent Kinetic Energy k (7) D k = − ρ ε
Turbulent Dissipation Rate ε (8) D ε = − C ε 2 ρ ε 2 k
Velocity Scale v2 (9) D v 2 = − ρ ε v 2 ¯ k

Modeling of Turbulent Viscosity μ t

(10) μ t = ρ C μ v 2 ¯ T

Model Coefficients

C ε 1 = 1.44, C ε 2 = 1.92, C μ = 0.22, σ k = 1.0, σ ε = 1.3. C 1 = 1.4, C 2 = 0.45, C T = 6.0, C L = 0.25, C η = 85, σ v 2 = 1.0.

Zeta-F Model

The base model of the zeta-f model is the v2f model described by Durbin (1995).

However, by introducing a normalizing velocity scale, the numerical stability issues found in the v2f model have been improved (Hanjalic et al., 2004; Laurence et al., 2004; Popovac and Hanjalic, 2007).

Transport Equations

Turbulent Kinetic Energy k (11) ∂ ( ρ k ) ∂ t + ∂ ( ρ u j ¯ k ) ∂ x j   = ∂ ∂ x j [ ( μ + μ t σ k ) ∂ k ∂ x j ] + P k + D k
Turbulent Dissipation Rate ε(12) ∂ ( ρ ε ) ∂ t + ∂ ( ρ u j ¯ ε ) ∂ x j   = ∂ ∂ x j [ ( μ + μ t σ ε ) ∂ ε ∂ x j ] + P ε + D ε
Normalized Velocity Scale ς = v 2 ¯ k (13) ∂ ( ρ ς ) ∂ t + ∂ ( ρ u j ¯ ς ) ∂ x j   = ∂ ∂ x j [ ( μ + μ t σ ς ) ∂ ς ∂ x j ] + P ς + D ς

Elliptic Relation for the Relaxation Function f

(14) L 2 ∂ 2 f ∂ x j 2 − f   = 1 T ( C 1 + C ' 2 P k ε − 1 ) ( ς − 2 3 )

where L = C L   m a x ( m i n [ k 3 / 2 ε , k 6 C μ | S | ς ] , C η [ ν 3 ε ] 1 / 4   ) : the length scale, T = m a x ( m i n [ k ε , 0.6 6 C μ | S | ς ] , C T [ ν ε ]   ) : the time scale.

Production Modeling

Turbulent Kinetic Energy k (15) P k = μ t S 2
Turbulent Dissipation Rate ε (16) P ε = C ε 1 ε k μ t S 2 = C ε 1 ε k P k

Velocity Scale ς

P ς = ρ f

Dissipation Modeling

Turbulent Kinetic Energy k (17) D k = − ρ ε
Turbulent Dissipation Rate ε (18) D ε = − C ε 2 ρ ε 2 k
Velocity Scale ς (19) D ζ = − ρ ζ k P k

Modeling of Turbulent Viscosity μ t

(20) μ t = ρ C μ ς k T

Model Coefficients

C ε 1 = 1.44, C ε 2 = 1.92, C μ = 0.22, σ k = 1.0, σ ε = 1.3. C 1 = 1.4, C ' 2 = 0.65, C T = 6.0, C L = 0.36, C η = 85, σ ζ = 1.2.