# V2-f models

(Difference between revisions)
 Revision as of 12:09, 4 October 2006 (view source) (Short explanation)← Older edit Revision as of 16:40, 20 January 2007 (view source) (→Limitations)Newer edit → Line 3: Line 3: The $v^2-f$ model uses a velocity scale, $\overline {v^2}$, instead of [[turbulent kinetic energy]], $k$, for the evaluation of the eddy viscosity. $\overline {v^2}$ 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 $v^2-f$ model uses a velocity scale, $\overline {v^2}$, instead of [[turbulent kinetic energy]], $k$, for the evaluation of the eddy viscosity. $\overline {v^2}$ 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. - ==Limitations== + ==The $\upsilon^2 - f$ equations== - Can not be used to solve Eulerian multiphase problems. + The turbulent viscosity is defined as + + $\nu_t^{\upsilon^2} = C_\mu \, \upsilon^2 \, T$ + + and the turbulent quantities, in addition to standard $k$ and $\varepsilon$, are obtaned from two more equations: the transport equation for $\upsilon^2$ + + $\frac{\partial \upsilon^2}{\partial t} + U_j \frac{\partial \upsilon^2}{\partial x_j} = k f - \varepsilon \frac{\upsilon^2}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\upsilon^2}} \right) \frac{\partial \upsilon^2}{\partial x_k} \right] +$ + + and the elliptic equation for the relaxation function $f$ + + $L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\upsilon^2}{k} - \frac{2}{3} \right) - C_2 \frac{\mathcal{P}}{\varepsilon}$ + + where the turbulence length scale $L$ + + $\displaystyle L = C_L \max \left[ \frac{\displaystyle k^{3/2}}{\displaystyle \varepsilon} , C_{\eta} \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/4} \right]$ + + and the turbulence time scale $T$ + + $\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]$ + + are bounded with their respective the Kolmogorov definitions (realisability constraints can also be applied, as given below). + + The coefficients used read: $C_\mu = 0.22$, $\sigma_{\upsilon^2} = 1$, $C_1 = 1.4$, $C_2 = 0.45$, $C_T = 6$, $C_L = 0.25$ and $C_{\eta} = 85$. + + + ==The $\zeta - f$ equations== + The turbulent viscosity is defined as + + $\nu_t^\zeta = C_\mu \, \zeta \, k \, T$ + + and the turbulent quantities are obtained from another set of equations. When the change of variables is done, the obtained transport equation for $\zeta$ reads + + $\frac{\partial \zeta}{\partial t} + U_j \frac{\partial \zeta}{\partial x_j} = f - \frac{\zeta}{k} \mathcal{P} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]$ + + and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function $f$ + + $L^2 \nabla^2 f - f = \frac{1}{T} \left( C_1 - 1 + C'_2 \frac{\mathcal{P}}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)$ + + where the turbulence time scale $T$ + + $T = max \left[ min \left( \frac{k}{\varepsilon},\, \frac{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]$ + + and the turbulence length scale $L$ + + $L = C_L \, max \left[ min \left( \frac{k^{3/2}}{\varepsilon}, \, + \frac{k^{1/2}}{\sqrt{6} C_{\mu} |S| \zeta} \right), C_{\eta} + \left( \frac{\nu^3}{\varepsilon} \right)^{1/4} \right]$ + + are limited with their Kolmogorov values as the lower bounds, and Durbin's realisability constraints. + + The coefficients used read:  $C_\mu = 0.22$, $\sigma_{\zeta} = 1.2$, $C_1 = 1.4$, $C_2' = 0.65$, $C_T = 6$, $C_L = 0.36$ and $C_{\eta} = 85$. + + + ==Notes== + This model can not be used to solve Eulerian multiphase problems. + + Mathematically and physically the $\upsilon^2-f$ and the $\zeta-f$ model are the same, but due to better numerical properties (stability and near-wall mesh requirements) the $\zeta-f$ model performs better in the complex flow calculations. + == References == == References ==

## Introduction

The $v^2-f$ 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-numbers, that does not need to make use of wall functions because it is valid upto solid walls. The $v^2-f$ model uses a velocity scale, $\overline {v^2}$, instead of turbulent kinetic energy, $k$, for the evaluation of the eddy viscosity. $\overline {v^2}$ 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 $\upsilon^2 - f$ equations

The turbulent viscosity is defined as

$\nu_t^{\upsilon^2} = C_\mu \, \upsilon^2 \, T$

and the turbulent quantities, in addition to standard $k$ and $\varepsilon$, are obtaned from two more equations: the transport equation for $\upsilon^2$

$\frac{\partial \upsilon^2}{\partial t} + U_j \frac{\partial \upsilon^2}{\partial x_j} = k f - \varepsilon \frac{\upsilon^2}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\upsilon^2}} \right) \frac{\partial \upsilon^2}{\partial x_k} \right]$

and the elliptic equation for the relaxation function $f$

$L^2 \nabla^2 f - f = \frac{C_1 - 1}{T} \left( \frac{\upsilon^2}{k} - \frac{2}{3} \right) - C_2 \frac{\mathcal{P}}{\varepsilon}$

where the turbulence length scale $L$

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

and the turbulence time scale $T$

$\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]$

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

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

## The $\zeta - f$ equations

The turbulent viscosity is defined as

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

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

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

and, in conjunction with the quasi-linear SSG pressure-strain model, the elliptic equation for the relaxation function $f$

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

where the turbulence time scale $T$

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

and the turbulence length scale $L$

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

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

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

## Notes

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

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

## References

• Durbin, P. Separated flow computations with the $k-\epsilon-\overline{v^2}$model, AIAA Journal, 33, 659-664, 1995.