V2-f models

**Turbulence modeling**
	Turbulence
	RANS-based turbulence models Linear eddy viscosity models Algebraic models Cebeci-Smith model ; Baldwin-Lomax model ; Johnson-King model ; A roughness-dependent model ; ; One equation models Prandtl's one-equation model ; Baldwin-Barth model ; Spalart-Allmaras model ; ; Two equation models k-epsilon models Standard k-epsilon model ; Realisable k-epsilon model ; RNG k-epsilon model ; Near-wall treatment ; ; k-omega models Wilcox's k-omega model ; Wilcox's modified k-omega model ; SST k-omega model ; Near-wall treatment ; ; Realisability issues Kato-Launder modification ; Durbin's realizability constraint ; Yap correction ; Realisability and Schwarz' inequality ; ; ; ; Nonlinear eddy viscosity models Explicit nonlinear constitutive relation Cubic k-epsilon ; EARSM ; ; v2-f models model ; model ; ; ; Reynolds stress model (RSM) ;
	Large eddy simulation (LES) Smagorinsky-Lilly model ; Dynamic subgrid-scale model ; RNG-LES model ; Wall-adapting local eddy-viscosity (WALE) model ; Kinetic energy subgrid-scale model ; Near-wall treatment for LES models ;
	Detached eddy simulation (DES)
	Direct numerical simulation (DNS)
	Turbulence near-wall modeling
	Turbulence free-stream boundary conditions Turbulence intensity ; Turbulence length scale ;

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Introduction

The $\overline{v^2}-f$ 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. Instead of turbulent kinetic energy $k$ , the $\overline{v^2}-f$ model uses a velocity scale $\overline {v^2}$ (hence the name $\overline{v^2}-f$ or the v2-f model) 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 anisotropic wall effects are modelled through the elliptic relaxation function $f$ , by solving a separate elliptic equation of the Helmholtz type.

In order to improve the computational preformances of the $\overline{\upsilon^2}-f$ 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 $\overline{\upsilon^2}$ as the velocity scale, the normalised wall-normal velocity scale $\zeta = \overline{\upsilon^2} / k$ is used (hence the name $\zeta-f$ or the zeta-f model). This turbulence variable can be regarded as the ratio of the two time scales: scalar $k / \varepsilon$ (isotropic), and lateral $\overline{\upsilon^2} / \varepsilon$ (anisotropic). Following the definition of $\zeta$ , the new transport equation is derived from the equations for $\overline{\upsilon^2}$ and $k$ , and solved instead of the transport equation for $\overline{\upsilon^2}$ .

The $\overline{\upsilon^2} - f$ equations

The turbulent viscosity is defined as

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

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

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

and the elliptic equation for the relaxation function $f$

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

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 $T$

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

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

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

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

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

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

where the turbulence time scale $T$

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

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 $\overline{\upsilon^2}-f$ 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.

Laurence, D.R., Uribe J.C., Utyuzhnikov, S.V. A Robust Formulation of the v2-f Model, Flow Turbulence and Combustion, 73, 169-185, 2004.

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.

This article is a stub, a short article which needs to be improved. You can help by expanding it.

@@ Line 1: / Line 1: @@
+{{Template:Turbulence modeling}}
 ==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.
-The <math>\overline{v^2}-f</math> model uses a velocity scale, <math>\overline {v^2}</math>, instead of [[turbulent kinetic energy]], <math>k</math>, 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.
+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.
@@ Line 17: / Line 19: @@
 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 - \varepsilon \frac{\overline{\upsilon^2}}{k} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\overline{\upsilon^2}}} \right) \frac{\partial \overline{\upsilon^2}}{\partial x_k} \right]
+<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{\mathcal{P}}{\varepsilon}</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>
@@ Line 30: / Line 32: @@
 and the turbulence time scale <math>T</math>
-<math>\displaystyle T = \max \left[ \frac{\displaystyle k}{\displaystyle \varepsilon} , C_T \left( \frac{\displaystyle \nu^3}{\displaystyle \varepsilon} \right)^{1/2} \right]</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).
@@ Line 43: / Line 45: @@
 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} \mathcal{P} + \frac{\partial}{\partial x_k} \left[ \left( \nu + \frac{\nu_t}{\sigma_{\zeta}} \right) \frac{\partial \zeta}{\partial x_k} \right]</math>
+<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{\mathcal{P}}{\varepsilon} \right) \left( \zeta - \frac{2}{3} \right)</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{a_T}{\sqrt{6} C_{\mu} |S|\zeta} \right), C_T  \left( \frac{\nu^3}{\varepsilon} \right)^{1/2} \right]</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}, \,
+<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>
@@ Line 62: / Line 64: @@
 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==
@@ Line 73: / Line 74: @@
 *<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.
+*<b>Laurence, D.R., Uribe J.C., Utyuzhnikov, S.V.</b> A Robust Formulation of the v2-f  Model, Flow Turbulence and Combustion, 73, 169-185, 2004.
+*<b>Popovac, M., Hanjalic, K.</b> Compound Wall Treatment for RANS Computation of Complex Turbulent Flows and Heat Transfer, Flow Turbulence and Combustion, 78, 177-202, 2007.
-{{stub}}
 [[Category:Turbulence models]]
+{{stub}}

V2-f models

From CFD-Wiki

Latest revision as of 05:45, 7 February 2014

Contents

Introduction

The $\overline{\upsilon^2} - f$ equations

The $\zeta - f$ equations

Notes

References

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V2-f models

From CFD-Wiki

Latest revision as of 05:45, 7 February 2014

Contents

Introduction

The equations

The equations

Notes

References

Views

My wiki

wiki navigation

wiki search

wiki toolbox

The $\overline{\upsilon^2} - f$ equations

The $\zeta - f$ equations