# Standard k-epsilon model

(Difference between revisions)
 Revision as of 00:14, 14 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 15:16, 21 June 2007 (view source)R.absi (Talk | contribs) (→Transport equations for standard $k$-$\epsilon$ model)Newer edit → (16 intermediate revisions not shown) Line 1: Line 1: - == Transport Equation for standard k-epsilon model == + {{Turbulence modeling}} - For k
+ == Transport equations for standard k-epsilon model == - $\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k$ + + For turbulent kinetic energy $k$
+ :$\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k$

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- $+ :[itex] \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{1 \epsilon}\frac{\epsilon}{k} \left( P_k + C_{3 \epsilon} P_b \right) - C_{2 \epsilon} \rho \frac{\epsilon^2}{k} + S_{\epsilon} \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{1 \epsilon}\frac{\epsilon}{k} \left( P_k + C_{3 \epsilon} P_b \right) - C_{2 \epsilon} \rho \frac{\epsilon^2}{k} + S_{\epsilon} - +$ - [/itex] + == Modeling turbulent viscosity == == Modeling turbulent viscosity == Turbulent viscosity is modelled as:
Turbulent viscosity is modelled as:
- $+ :[itex] \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} +$ + + == Production of k == + + :$+ P_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i}$ [/itex]

+ :$P_k = \mu_t S^2$ + Where $S$ is the modulus of the mean rate-of-strain tensor, defined as :
+ :$+ S \equiv \sqrt{2S_{ij} S_{ij}} +$ - == Model Constants == + == Effect of buoyancy == - $+ :[itex] + P_b = \beta g_i \frac{\mu_t}{{\rm Pr}_t} \frac{\partial T}{\partial x_i} +$ + + +
+ where Prt  is the turbulent [[Prandtl number]] for energy and gi  is the component of the gravitational vector in the ith direction. For the standard and realizable -  models, the default value of Prt  is 0.85. + + The coefficient of thermal expansion, $\beta$ , is defined as
+ :$+ \beta = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_p +$ + + == Model constants == + + :$C_{1 \epsilon} = 1.44, \;\; C_{2 \epsilon} = 1.92, \;\; C_{\mu} = 0.09, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.3 C_{1 \epsilon} = 1.44, \;\; C_{2 \epsilon} = 1.92, \;\; C_{\mu} = 0.09, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.3$ [/itex] + + + [[Category:Turbulence models]]

## Transport equations for standard k-epsilon model

For turbulent kinetic energy $k$

$\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k$

For dissipation $\epsilon$

$\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{1 \epsilon}\frac{\epsilon}{k} \left( P_k + C_{3 \epsilon} P_b \right) - C_{2 \epsilon} \rho \frac{\epsilon^2}{k} + S_{\epsilon}$

## Modeling turbulent viscosity

Turbulent viscosity is modelled as:

$\mu_t = \rho C_{\mu} \frac{k^2}{\epsilon}$

## Production of k

$P_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i}$

$P_k = \mu_t S^2$

Where $S$ is the modulus of the mean rate-of-strain tensor, defined as :

$S \equiv \sqrt{2S_{ij} S_{ij}}$

## Effect of buoyancy

$P_b = \beta g_i \frac{\mu_t}{{\rm Pr}_t} \frac{\partial T}{\partial x_i}$

where Prt is the turbulent Prandtl number for energy and gi is the component of the gravitational vector in the ith direction. For the standard and realizable - models, the default value of Prt is 0.85.

The coefficient of thermal expansion, $\beta$ , is defined as

$\beta = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_p$

## Model constants

$C_{1 \epsilon} = 1.44, \;\; C_{2 \epsilon} = 1.92, \;\; C_{\mu} = 0.09, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.3$