# 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

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### Elliptic Relation for the relaxation function f

${L}^{2}\frac{{\partial}^{2}f}{\partial {x}_{j}^{2}}-f=\frac{\left({C}_{1}-1\right)}{T}\left(\frac{{v}^{2}}{k}-\frac{2}{3}\right)-{C}_{2}\frac{{P}_{k}}{\epsilon}$

- $L={C}_{L}max\left(\frac{{k}^{3/2}}{\epsilon},{C}_{\eta}{\left[\frac{{\nu}^{3}}{\epsilon}\right]}^{1/4}\right)$: the length scale,
- $T=max\left(\frac{k}{\epsilon},{C}_{T}\sqrt{\left[\frac{\nu}{\epsilon}\right]}\right)$: the time scale.

### Production Modeling

^{2}

### Dissipation Modeling

^{2}

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

### 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

### Elliptic Relation for the Relaxation Function f

where $L={C}_{L}max\left(min\left[\frac{{k}^{3/2}}{\epsilon},\frac{\sqrt{k}}{\sqrt{6}{C}_{\mu}\left|S\right|\varsigma}\right],{C}_{\eta}{\left[\frac{{\nu}^{3}}{\epsilon}\right]}^{1/4}\right)$: the length scale, $T=max\left(min\left[\frac{k}{\epsilon},\frac{0.6}{\sqrt{6}{C}_{\mu}\left|S\right|\varsigma}\right],{C}_{T}\sqrt{\left[\frac{\nu}{\epsilon}\right]}\right)$: the time scale.

### Production Modeling

Velocity Scale $\varsigma $

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

### Dissipation Modeling

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

### 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, $C{\text{'}}_{2}$ = 0.65, ${C}_{T}$ = 6.0, ${C}_{L}$ = 0.36, ${C}_{\eta}$ = 85, ${\sigma}_{\zeta}$ = 1.2.