Realisable k-epsilon model
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+ | {{Turbulence modeling}} | ||
== Transport Equations == | == Transport Equations == | ||
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:<math> \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_j} (\rho \epsilon u_j) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_{\epsilon}}\right) \frac{\partial \epsilon}{\partial x_j} \right ] + \rho \, C_1 S \epsilon - \rho \, C_2 \frac{{\epsilon}^2} {k + \sqrt{\nu \epsilon}} + C_{1 \epsilon}\frac{\epsilon}{k} C_{3 \epsilon} P_b + S_{\epsilon} </math> | :<math> \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_j} (\rho \epsilon u_j) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_{\epsilon}}\right) \frac{\partial \epsilon}{\partial x_j} \right ] + \rho \, C_1 S \epsilon - \rho \, C_2 \frac{{\epsilon}^2} {k + \sqrt{\nu \epsilon}} + C_{1 \epsilon}\frac{\epsilon}{k} C_{3 \epsilon} P_b + S_{\epsilon} </math> | ||
+ | |||
+ | Where <br> | ||
+ | |||
+ | <math> C_1 = \max\left[0.43, \frac{\eta}{\eta + 5}\right] , \;\;\;\;\; \eta = S \frac{k}{\epsilon}, \;\;\;\;\; S =\sqrt{2 S_{ij} S_{ij}} </math> <br> | ||
+ | |||
+ | In these equations, <math> P_k </math> represents the generation of turbulence kinetic energy due to the mean velocity gradients, calculated in same manner as standard k-epsilon model. <math> P_b </math> is the generation of turbulence kinetic energy due to buoyancy, calculated in same way as standard k-epsilon model. | ||
+ | |||
+ | == Modelling Turbulent Viscosity == | ||
+ | |||
+ | :<math> \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} </math> <br> | ||
+ | where <br> | ||
+ | <math> C_{\mu} = \frac{1}{A_0 + A_s \frac{k U^*}{\epsilon}} </math> <br> | ||
+ | <math> U^* \equiv \sqrt{S_{ij} S_{ij} + \tilde{\Omega}_{ij} \tilde{\Omega}_{ij}} </math> ;<br> | ||
+ | <math> \tilde{\Omega}_{ij} = \Omega_{ij} - 2 \epsilon_{ijk} \omega_k </math> ; <br> | ||
+ | <math> \Omega_{ij} = \overline{\Omega_{ij}} - \epsilon_{ijk} \omega_k </math> <br> | ||
+ | |||
+ | where <math> \overline{\Omega_{ij}} </math> is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity <math> \omega_k </math>. The model constants <math> A_0 </math> and <math> A_s </math> are given by: <br> | ||
+ | <math> A_0 = 4.04, \; \; A_s = \sqrt{6} \cos \phi </math> <br> | ||
+ | |||
+ | <math> \phi = \frac{1}{3} \cos^{-1} (\sqrt{6} W), \; \; W = \frac{S_{ij} S_{jk} S_{ki}}{{\tilde{S}} ^3}, \; \; \tilde{S} = \sqrt{S_{ij} S_{ij}}, \; \; S_{ij} = \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right) </math> | ||
+ | |||
+ | |||
+ | ==Model Constants == | ||
+ | |||
+ | <math> C_{1 \epsilon} = 1.44, \;\; C_2 = 1.9, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.2 </math> | ||
+ | |||
+ | |||
+ | == References == | ||
+ | |||
+ | See section [[K-epsilon_models#References|References]] in the parent page [[K-epsilon models]]. | ||
+ | |||
+ | [[Category:Turbulence models]] |
Latest revision as of 19:57, 16 December 2014
Contents |
Transport Equations
Where
In these equations, represents the generation of turbulence kinetic energy due to the mean velocity gradients, calculated in same manner as standard k-epsilon model. is the generation of turbulence kinetic energy due to buoyancy, calculated in same way as standard k-epsilon model.
Modelling Turbulent Viscosity
where
;
;
where is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity . The model constants and are given by:
Model Constants
References
See section References in the parent page K-epsilon models.