Three-Equation Eddy Viscosity Models

v2-f Model

In order to account for the near wall turbulence anisotropy and non local pressure strain effects, Durbin (1995) introduced a velocity scale v2 and the elliptic relaxation function f to the standard k-ε turbulence model.

The velocity scale v2 represents the velocity fluctuation normal to the streamline and represents a proper scaling of the turbulence damping near the wall, while the elliptic relaxation function f is used to model the anisotropic wall effects. Compared to the k-ε turbulence models, the v2-f model produces more accurate predictions of wall-bounded flows dominated by separation but suffers from numerical stability issues.

Transport Equations

Turbulent Kinetic Energy k

(1)

\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}] + P_{k} + D_{k}

$\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}] + P_{k} + D_{k}$

Turbulent Dissipation Rate ε

(2)

\frac{\partial (ρ ε)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} ε)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ε}}) \frac{\partial ε}{\partial x_{j}}] + P_{ε} + D_{ε}

$\frac{\partial (ρ ε)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} ε)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ε}}) \frac{\partial ε}{\partial x_{j}}] + P_{ε} + D_{ε}$

Velocity Scale v²

(3)

\frac{\partial (ρ ¯ ¯ ¯ v^{2})}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} ¯ ¯ ¯ v^{2})}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{v 2}}) \frac{\partial ¯ ¯ ¯ v^{2}}{\partial x_{j}}] + P_{v 2} + D_{v 2}

$\frac{\partial (ρ \bar{v^{2}})}{\partial t} + \frac{\partial (ρ \bar{u_{j}} \bar{v^{2}})}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{v 2}}) \frac{\partial \bar{v^{2}}}{\partial x_{j}}] + P_{v 2} + D_{v 2}$

Elliptic Relation for the relaxation function f

$L^{2} \frac{\partial^{2} f}{\partial x_{j}^{2}} - f = \frac{(C_{1} - 1)}{T} (\frac{v^{2}}{k} - \frac{2}{3}) - C_{2} \frac{P_{k}}{ε}$ $L^{2} \frac{\partial^{2} f}{\partial x_{j}^{2}} - f = \frac{(C_{1} - 1)}{T} (\frac{v^{2}}{k} - \frac{2}{3}) - C_{2} \frac{P_{k}}{ε}$

where

$L = C_{L} m a x (\frac{k^{3 / 2}}{ε}, C_{η} {[\frac{ν^{3}}{ε}]}^{1 / 4})$ $L = C_{L} m a x (\frac{k^{3 / 2}}{ε}, C_{η} {[\frac{ν^{3}}{ε}]}^{1 / 4})$ : the length scale,
$T = m a x (\frac{k}{ε}, C_{T} \sqrt{[\frac{ν}{ε}]})$ $T = m a x (\frac{k}{ε}, C_{T} \sqrt{[\frac{ν}{ε}]})$ : the time scale.

Production Modeling

Turbulent Kinetic Energy k

(4)

P_{k} = μ_{t} S^{2}

$P_{k} = μ_{t} S^{2}$

Turbulent Dissipation Rate ε

(5)

P_{ε} = C_{ε 1} \frac{ε}{k} μ_{t} S^{2} = C_{ε 1} \frac{ε}{k} P_{k}

$P_{ε} = C_{ε 1} \frac{ε}{k} μ_{t} S^{2} = C_{ε 1} \frac{ε}{k} P_{k}$

Velocity Scale v²

(6)

P_{v 2} = ρ k f

$P_{v 2} = ρ k f$

Dissipation Modeling

Turbulent Kinetic Energy k

(7)

D_{k} = - ρ ε

$D_{k} = - ρ ε$

Turbulent Dissipation Rate ε

(8)

D_{ε} = - C_{ε 2} ρ \frac{ε^{2}}{k}

$D_{ε} = - C_{ε 2} ρ \frac{ε^{2}}{k}$

Velocity Scale v²

(9)

D_{v 2} = - ρ ε \frac{¯ ¯ ¯ v^{2}}{k}

$D_{v 2} = - ρ ε \frac{\bar{v^{2}}}{k}$

Modeling of Turbulent Viscosity $μ_{t}$ $μ_{t}$

(10)

μ_{t} = ρ C_{μ} ¯ ¯ ¯ v^{2} T

$μ_{t} = ρ C_{μ} \bar{v^{2}} T$

Model Coefficients

$C_{ε 1}$ $C_{ε 1}$ = 1.44, $C_{ε 2}$ $C_{ε 2}$ = 1.92, $C_{μ}$ $C_{μ}$ = 0.22, $σ_{k}$ $σ_{k}$ = 1.0, $σ_{ε}$ $σ_{ε}$ = 1.3. $C_{1}$ $C_{1}$ = 1.4, $C_{2}$ $C_{2}$ = 0.45, $C_{T}$ $C_{T}$ = 6.0, $C_{L}$ $C_{L}$ = 0.25, $C_{η}$ $C_{η}$ = 85, $σ_{v 2}$ $σ_{v 2}$ = 1.0.

Zeta-F Model

The base model of the zeta-f model is the v2f model described by Durbin (1995).

However, by introducing a normalizing velocity scale, the numerical stability issues found in the v2f model have been improved (Hanjalic et al., 2004; Laurence et al., 2004; Popovac and Hanjalic, 2007).

Transport Equations

Turbulent Kinetic Energy k

(11)

\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}] + P_{k} + D_{k}

Turbulent Dissipation Rate ε

(12)

\frac{\partial (ρ ε)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} ε)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ε}}) \frac{\partial ε}{\partial x_{j}}] + P_{ε} + D_{ε}

Normalized Velocity Scale

ς = \frac{¯ ¯ ¯ v^{2}}{k}

$ς = \frac{\bar{v^{2}}}{k}$

(13)

\frac{\partial (ρ ς)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} ς)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ς}}) \frac{\partial ς}{\partial x_{j}}] + P_{ς} + D_{ς}

$\frac{\partial (ρ ς)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} ς)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ς}}) \frac{\partial ς}{\partial x_{j}}] + P_{ς} + D_{ς}$

Elliptic Relation for the Relaxation Function f

(14)

L^{2} \frac{\partial^{2} f}{\partial x_{j}^{2}} - f = \frac{1}{T} (C_{1} + C'_{2} \frac{P_{k}}{ε} - 1) (ς - \frac{2}{3})

$L^{2} \frac{\partial^{2} f}{\partial x_{j}^{2}} - f = \frac{1}{T} (C_{1} + C'_{2} \frac{P_{k}}{ε} - 1) (ς - \frac{2}{3})$

where $L = C_{L} m a x (m i n [\frac{k^{3 / 2}}{ε}, \frac{\sqrt{k}}{\sqrt{6} C_{μ} | S | ς}], C_{η} {[\frac{ν^{3}}{ε}]}^{1 / 4})$ $L = C_{L} m a x (m i n [\frac{k^{3 / 2}}{ε}, \frac{\sqrt{k}}{\sqrt{6} C_{μ} | S | ς}], C_{η} {[\frac{ν^{3}}{ε}]}^{1 / 4})$ : the length scale, $T = m a x (m i n [\frac{k}{ε}, \frac{0.6}{\sqrt{6} C_{μ} | S | ς}], C_{T} \sqrt{[\frac{ν}{ε}]})$ : the time scale.

Production Modeling

Turbulent Kinetic Energy k

(15)

$P_{k} = μ_{t} S^{2}$

Turbulent Dissipation Rate ε

(16)

$P_{ε} = C_{ε 1} \frac{ε}{k} μ_{t} S^{2} = C_{ε 1} \frac{ε}{k} P_{k}$

Velocity Scale $ς$

$P_{ς} = ρ f$

Dissipation Modeling

Turbulent Kinetic Energy k

(17)

$D_{k} = - ρ ε$

Turbulent Dissipation Rate ε

(18)

$D_{ε} = - C_{ε 2} ρ \frac{ε^{2}}{k}$

Velocity Scale

$ς$

(19)

$D_{ζ} = - ρ \frac{ζ}{k} P_{k}$

Modeling of Turbulent Viscosity $μ_{t}$

(20)

$μ_{t} = ρ C_{μ} ς k T$

Model Coefficients

$C_{ε 1}$ = 1.44, $C_{ε 2}$ = 1.92, $C_{μ}$ = 0.22, $σ_{k}$ = 1.0, $σ_{ε}$ = 1.3. $C_{1}$ = 1.4, $C'_{2}$ = 0.65, $C_{T}$ = 6.0, $C_{L}$ = 0.36, $C_{η}$ = 85, $σ_{ζ}$ = 1.2.

Three-Equation Eddy Viscosity Models

v2-f Model

Transport Equations

Elliptic Relation for the relaxation function f

Production Modeling

Dissipation Modeling

Modeling of Turbulent Viscosity μtμt<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>μ</mi> <mi>t</mi> </msub> </mrow> </math>

Model Coefficients

Zeta-F Model

Transport Equations

Elliptic Relation for the Relaxation Function f

Production Modeling

Dissipation Modeling

Modeling of Turbulent Viscosity μt<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>μ</mi> <mi>t</mi> </msub> </mrow> </math>

Model Coefficients

Modeling of Turbulent Viscosity $μ_{t}$ $μ_{t}$

Modeling of Turbulent Viscosity $μ_{t}$