Standard k-epsilon model
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(Someone added that C3 was -0.33, without any reference. This can be confusing to OpenFOAM users, because their k-epsilon has another C3, for another reason.) |
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- | + | {{Turbulence modeling}} | |
- | For k <br> | + | == Transport equations for standard k-epsilon model == |
- | <math> \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 </math> | + | |
+ | For turbulent kinetic energy <math> k </math> <br> | ||
+ | :<math> \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 </math> | ||
<br> | <br> | ||
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<br> | <br> | ||
- | <math> | + | :<math> |
\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} | ||
- | + | </math> | |
- | + | ||
== Modeling turbulent viscosity == | == Modeling turbulent viscosity == | ||
Turbulent viscosity is modelled as: <br> | Turbulent viscosity is modelled as: <br> | ||
- | <math> | + | :<math> |
\mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} | \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} | ||
</math> | </math> | ||
- | |||
- | |||
- | |||
- | |||
== Production of k == | == Production of k == | ||
- | <math> | + | :<math> |
P_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i} | P_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i} | ||
</math> | </math> | ||
<br> | <br> | ||
- | <math> P_k = \mu_t S^2 </math> | + | :<math> P_k = \mu_t S^2 </math> |
Where <math> S </math> is the modulus of the mean rate-of-strain tensor, defined as : <br> | Where <math> S </math> is the modulus of the mean rate-of-strain tensor, defined as : <br> | ||
- | <math> | + | :<math> |
S \equiv \sqrt{2S_{ij} S_{ij}} | S \equiv \sqrt{2S_{ij} S_{ij}} | ||
</math> | </math> | ||
+ | == Effect of buoyancy == | ||
- | = | + | :<math> |
+ | P_b = \beta g_i \frac{\mu_t}{{\rm Pr}_t} \frac{\partial T}{\partial x_i} | ||
+ | </math> | ||
- | < | + | |
- | + | <br> | |
+ | where Pr<sub>t</sub> is the turbulent [[Prandtl number]] for energy and g<sub>i</sub> is the component of the gravitational vector in the ith direction. For the standard and realizable - models, the default value of Pr<sub>t</sub> is 0.85. | ||
+ | |||
+ | The coefficient of thermal expansion, <math> \beta </math> , is defined as <br> | ||
+ | :<math> | ||
+ | \beta = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_p | ||
</math> | </math> | ||
+ | |||
+ | == Model constants == | ||
+ | |||
+ | :<math> | ||
+ | C_{1 \epsilon} = 1.44, \;\;\; C_{2 \epsilon} = 1.92,\;\; \; C_{\mu} = 0.09, \;\;\; \sigma_k = 1.0, \;\;\; \sigma_{\epsilon} = 1.3 | ||
+ | </math> | ||
+ | |||
+ | |||
+ | '''Note''': <math>C_{3 \epsilon}</math> depends on the literature being followed and is meant to be used only with the <math>P_b</math> term. Possible values, depending on literature reference: | ||
+ | |||
+ | {| class="wikitable" | ||
+ | |- align="center" | ||
+ | ! Reference !! Constant !! Comments | ||
+ | |- align="center" | ||
+ | | ''unknown'' || <math>C_{3 \epsilon} = -0.33</math> || Note to OpenFOAM users: do not confuse this constant with the one used in their implementations of the k-epsilon turbulence models. Their implementation is different. | ||
+ | |} | ||
+ | |||
+ | |||
+ | == References == | ||
+ | |||
+ | See section [[K-epsilon_models#References|References]] in the parent page [[K-epsilon models]]. | ||
+ | |||
+ | |||
+ | [[Category:Turbulence models]] |
Latest revision as of 20:15, 16 December 2014
Contents |
Transport equations for standard k-epsilon model
For turbulent kinetic energy
For dissipation
Modeling turbulent viscosity
Turbulent viscosity is modelled as:
Production of k
Where is the modulus of the mean rate-of-strain tensor, defined as :
Effect of buoyancy
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, , is defined as
Model constants
Note: depends on the literature being followed and is meant to be used only with the term. Possible values, depending on literature reference:
Reference | Constant | Comments |
---|---|---|
unknown | Note to OpenFOAM users: do not confuse this constant with the one used in their implementations of the k-epsilon turbulence models. Their implementation is different. |
References
See section References in the parent page K-epsilon models.