# Zeta-f model

The zeta-f model is a robust modification of the elliptic relaxation model. The set of equations, for the incompressible Newtonian fluid, constituting the $\zeta-f$ model is given below.

## Turbulent viscosity $\nu_t$

$\nu_t = C_\mu \, \zeta \, k \, T$

## Turbulent kinetic energy $k$

$\frac{\partial k}{\partial t} + U_j \frac{\partial k}{\partial x_j} = P_k - \varepsilon + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{k}} \right) \frac{\partial k}{\partial x_j} \right]$

## Turbulent kinetic energy dissipation rate $\varepsilon$

$\frac{\partial \varepsilon}{\partial t} + U_j \frac{\partial \varepsilon}{\partial x_j} = \frac{C_{\varepsilon 1} P_k - C_{\varepsilon 2} \varepsilon}{T} + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\varepsilon}} \right) \frac{\partial \varepsilon}{\partial x_j} \right]$

## Normalized velocity scale $\zeta$

$\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} P_k + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_j} \right]$

## Elliptic relaxation function $f$

$L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{P_k}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)$

## The production of the turbulent kinetic energy $P_k$

$P_k = - \overline{u_i u_j} \frac{\partial u_j}{\partial x_i}$

$P_k = \nu_t S^2$

## The modulus of the mean rate-of-strain tensor $S$

$S \equiv \sqrt{2S_{ij} S_{ij}}$

## The turbulence time scale $T$

$T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{0.6}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]$

## The turbulence length scale $L$

$L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \, \frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta} \left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]$

## The coefficients

$C_\mu = 0.22$, $\sigma_{k} = 1$, $\sigma_{\varepsilon} = 1.3$, $\sigma_{\zeta} = 1.2$, $C_{\varepsilon 1} = 1.4 (1 + 0.012 / \zeta)$, $C_{\varepsilon 2} = 1.9$, $C_1 = 1.4$, $C_2' = 0.65$, $C_T = 6$, $C_L = 0.36$ and $C_{\eta} = 85$.

## References

• Popovac, M., Hanjalic, K. Compound Wall Treatment for RANS Computation of Complex Turbulent Flows and Heat Transfer, Flow, Turbulence and Combustion, DOI 10.1007/s10494-006-9067-x, 2007.
• Hanjalic, K., Popovac, M., Hadziabdic, M. A robust near-wall elliptic-relaxation eddy-viscosity turbulence model for CFD, Int. J. Heat Fluid Flow, 25, 1047–1051, 2004.