V2-f models

==Introduction==
The <math>\overline{v^2}-f</math> model is similar to the [[Standard k-epsilon model]]. Additionally, it incorporates also some near-wall turbulence anisotropy as well as non-local pressure-strain effects. It is a general turbulence model for low [[Reynolds number|Reynolds-numbers]], that does not need to make use of wall functions because it is valid upto solid walls.
Instead of [[turbulent kinetic energy]] <math>k</math>, the <math>\overline{v^2}-f</math> model uses a velocity scale <math>\overline {v^2}</math> (hence the name <math>\overline{v^2}-f</math> or the ''v2-f model'') for the evaluation of the eddy viscosity. <math>\overline {v^2}</math> can be thought of as the velocity fluctuation normal to the streamlines. It can provide the right scaling for the representation of the damping of turbulent transport close to the wall. The anisotropic wall effects are modelled through the elliptic relaxation function <math>f</math>, by solving a separate elliptic equation of the Helmholtz type.


In order to improve the computational preformances of the <math>\overline{\upsilon^2}-f</math> model, a variant of this eddy-viscosity model is derived when the change of variables is introduced.
Instead of using the wall-normal velocity fluctuation <math>\overline{\upsilon^2}</math> as the velocity scale, the normalised wall-normal velocity scale <math>\zeta = \overline{\upsilon^2} / k</math> is used (hence the name <math>\zeta-f</math> or the [[zeta-f model]]).
This turbulence variable can be regarded as the ratio of the two time
scales: scalar <math>k / \varepsilon</math> (isotropic), and lateral <math>\overline{\upsilon^2} / \varepsilon</math> (anisotropic).
Following the definition of <math>\zeta</math>, the new transport equation is derived from the equations for <math>\overline{\upsilon^2}</math> and <math>k</math>, and solved instead of the transport equation for <math>\overline{\upsilon^2}</math>.

==The <math>\overline{\upsilon^2} - f</math> equations==
The turbulent viscosity is defined as

<math>\nu_t^{\overline{\upsilon^2}} = C_\mu \, \overline{\upsilon^2} \, T</math>

and the turbulent quantities, in addition to standard <math>k</math> and <math>\varepsilon</math>, are obtaned from two more equations: the transport equation for <math>\overline{\upsilon^2}</math>

<math>\frac{\partial \overline{\upsilon^2}}{\partial t} + U_j \frac{\partial \overline{\upsilon^2}}{\partial x_j} = k f - \frac{\overline{\upsilon^2}}{k} \varepsilon + \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_j} \right]
</math>

and the elliptic equation for the relaxation function <math>f</math>

<math>L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\overline{\upsilon^2}}{k} - \frac{2}{3} \right) - C_2 \frac{P_k}{\varepsilon}</math>

where the turbulence length scale <math>L</math>

<math>\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]</math>

and the turbulence time scale <math>T</math>

<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu}{\displaystyle \varepsilon} \right)^{1/2} \right]</math>

are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below).

The coefficients used read: <math>C_\mu = 0.22</math>, <math>\sigma_{\overline{\upsilon^2}} = 1</math>, <math>C_1 = 1.4</math>, <math>C_2 = 0.45</math>, <math>C_T = 6</math>, <math>C_L = 0.25</math> and <math>C_{\eta} = 85</math>.

==The <math>\zeta - f</math> equations==
The turbulent viscosity is defined as

<math>\nu_t^\zeta = C_\mu \, \zeta \, k \, T</math>

and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for <math>\zeta</math> reads

<math>\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]</math>

and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function <math>f</math>

<math>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)</math>

where the turbulence time scale <math>T</math>

<math>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]</math>

and the turbulence length scale <math>L</math>

<math>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]</math>

are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints.

The coefficients used read:  <math>C_\mu = 0.22</math>, <math>\sigma_{\zeta} = 1.2</math>, <math>C_1 = 1.4</math>, <math>C_2' = 0.65</math>, <math>C_T = 6</math>, <math>C_L = 0.36</math> and <math>C_{\eta} = 85</math>.

==Notes==
This model can not be used to solve Eulerian multiphase problems.

Mathematically and physically the <math>\overline{\upsilon^2}-f</math> and the <math>\zeta-f</math> model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the <math>\zeta-f</math> model performs better in the complex flow calculations.

== References ==

*<b>Durbin, P.</b> Separated flow computations with the <math>k-\epsilon-\overline{v^2} </math>model, AIAA Journal, 33, 659-664, 1995.

*<b>Popovac, M., Hanjalic, K.</b> 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.


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[[Category:Turbulence models]]