# 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)
Turbulent Dissipation Rate ε (2)
Velocity Scale v2 (3)

### Elliptic Relation for the relaxation function f

where
• : the length scale,
• : the time scale.

### Production Modeling

Turbulent Kinetic Energy k (4) ${P}_{k}={\mu }_{t}{S}^{2}$
Turbulent Dissipation Rate ε (5) ${P}_{\epsilon }={C}_{\epsilon 1}\frac{\epsilon }{k}{\mu }_{t}{S}^{2}={C}_{\epsilon 1}\frac{\epsilon }{k}{P}_{k}$
Velocity Scale v2 (6) ${P}_{v2}=\rho kf$

### Dissipation Modeling

Turbulent Kinetic Energy k (7) ${D}_{k}=-\rho \epsilon$
Turbulent Dissipation Rate ε (8) ${D}_{\epsilon }=-{C}_{\epsilon 2}\rho \frac{{\epsilon }^{2}}{k}$
Velocity Scale v2 (9) ${D}_{v2}=-\rho \epsilon \frac{\overline{{v}^{2}}}{k}$

### Modeling of Turbulent Viscosity ${\mu }_{t}$

(10) ${\mu }_{t}=\rho {C}_{\mu }\overline{{v}^{2}}T$

### Model Coefficients

${C}_{\epsilon 1}$ = 1.44, ${C}_{\epsilon 2}$ = 1.92, ${C}_{\mu }$ = 0.22, ${\sigma }_{k}$ = 1.0, ${\sigma }_{\epsilon }$ = 1.3. ${C}_{1}$ = 1.4, ${C}_{2}$ = 0.45, ${C}_{T}$ = 6.0, ${C}_{L}$ = 0.25, ${C}_{\eta }$ = 85, ${\sigma }_{v2}$ = 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)
Turbulent Dissipation Rate ε(12)
Normalized Velocity Scale $\varsigma =\frac{\overline{{v}^{2}}}{k}$(13)

### Elliptic Relation for the Relaxation Function f

(14)

where : the length scale, : the time scale.

### Production Modeling

Turbulent Kinetic Energy k (15) ${P}_{k}={\mu }_{t}{S}^{2}$
Turbulent Dissipation Rate ε (16) ${P}_{\epsilon }={C}_{\epsilon 1}\frac{\epsilon }{k}{\mu }_{t}{S}^{2}={C}_{\epsilon 1}\frac{\epsilon }{k}{P}_{k}$

Velocity Scale $\varsigma$

${P}_{\varsigma }=\rho f$

### Dissipation Modeling

Turbulent Kinetic Energy k (17) ${D}_{k}=-\rho \epsilon$
Turbulent Dissipation Rate ε (18) ${D}_{\epsilon }=-{C}_{\epsilon 2}\rho \frac{{\epsilon }^{2}}{k}$
Velocity Scale $\varsigma$ (19) ${D}_{\zeta }=-\rho \frac{\zeta }{k}{P}_{k}$

### Modeling of Turbulent Viscosity ${\mu }_{t}$

(20) ${\mu }_{t}=\rho {C}_{\mu }\varsigma kT$

### Model Coefficients

${C}_{\epsilon 1}$ = 1.44, ${C}_{\epsilon 2}$ = 1.92, ${C}_{\mu }$ = 0.22, ${\sigma }_{k}$ = 1.0, ${\sigma }_{\epsilon }$ = 1.3. ${C}_{1}$ = 1.4, $C{\text{'}}_{2}$ = 0.65, ${C}_{T}$ = 6.0, ${C}_{L}$ = 0.36, ${C}_{\eta }$ = 85, ${\sigma }_{\zeta }$ = 1.2.