Zeta-f model
From CFD-Wiki
The zeta-f model is a robust modification of the elliptic relaxation model. For the incompressible Newtonian fluid the final set of equations constituting the 
model is given below.
Turbulent viscosity 
Turbulent kinetic energy 
![\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]](/W/images/math/4/e/a/4eac951b79fdf4d26cac2b6cca1540bd.png)
Turbulent kinetic energy dissipation rate 
![\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]](/W/images/math/5/b/e/5be411c44d64810a0a4d341ada81137c.png)
Normalized velocity scale 
![\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]](/W/images/math/1/4/3/143979f772a60676f05d24a868fa7f47.png)
Elliptic relaxation function 
Production of the turbulent kinetic energy 
Modulus of the mean rate-of-strain tensor 
Turbulence time scale 
![T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{0.6}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu}{\varepsilon} \right)^{1/2} \right]](/W/images/math/e/0/b/e0b175df5f22a619bce6dbf7ebf46775.png)
Turbulence length scale 
![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]](/W/images/math/7/0/7/70730f002515f1c28758512281d69cd2.png)
Model coefficients
, 
, 
, 
, 
, 
, 
, 
, 
, 
and 
.
References
- Popovac, M., Hanjalic, K. Compound Wall Treatment for RANS Computation of Complex Turbulent Flows and Heat Transfer, Flow Turbulence and Combustion, 78, 177-202, 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.
