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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} G_b + S_{\epsilon} </math>
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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>
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Where <br>
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<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>
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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.
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== Modelling Turbulent Viscosity ==
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:<math> \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} </math> <br>
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where <br>
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<math> C_{\mu} = \frac{1}{A_0 + A_s \frac{k U^*}{\epsilon}} </math> <br>
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<math>  U^* \equiv \sqrt{S_{ij} S_{ij} + \tilde{\Omega}_{ij} \tilde{\Omega}_{ij}}  </math>  ;<br>
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<math> \tilde{\Omega}_{ij}  =  \Omega_{ij} - 2 \epsilon_{ijk} \omega_k    </math> ; <br>
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<math>  \Omega_{ij}  =  \overline{\Omega_{ij}} - \epsilon_{ijk} \omega_k    </math> <br>
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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>
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<math> A_0 = 4.04, \; \;  A_s = \sqrt{6}  \cos \phi </math> <br>
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<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>
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==Model Constants ==
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<math> C_{1 \epsilon} = 1.44,  \;\; C_2 = 1.9, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.2  </math>
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[[Category:Turbulence models]]

Latest revision as of 18:20, 22 August 2013

Turbulence modeling
Turbulence
RANS-based turbulence models
  1. Linear eddy viscosity models
    1. Algebraic models
      1. Cebeci-Smith model
      2. Baldwin-Lomax model
      3. Johnson-King model
      4. A roughness-dependent model
    2. One equation models
      1. Prandtl's one-equation model
      2. Baldwin-Barth model
      3. Spalart-Allmaras model
    3. Two equation models
      1. k-epsilon models
        1. Standard k-epsilon model
        2. Realisable k-epsilon model
        3. RNG k-epsilon model
        4. Near-wall treatment
      2. k-omega models
        1. Wilcox's k-omega model
        2. Wilcox's modified k-omega model
        3. SST k-omega model
        4. Near-wall treatment
      3. Realisability issues
        1. Kato-Launder modification
        2. Durbin's realizability constraint
        3. Yap correction
        4. Realisability and Schwarz' inequality
  2. Nonlinear eddy viscosity models
    1. Explicit nonlinear constitutive relation
      1. Cubic k-epsilon
      2. EARSM
    2. v2-f models
      1. \overline{\upsilon^2}-f model
      2. \zeta-f model
  3. Reynolds stress model (RSM)
Large eddy simulation (LES)
  1. Smagorinsky-Lilly model
  2. Dynamic subgrid-scale model
  3. RNG-LES model
  4. Wall-adapting local eddy-viscosity (WALE) model
  5. Kinetic energy subgrid-scale model
  6. Near-wall treatment for LES models
Detached eddy simulation (DES)
Direct numerical simulation (DNS)
Turbulence near-wall modeling
Turbulence free-stream boundary conditions
  1. Turbulence intensity
  2. Turbulence length scale

Transport Equations

 \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_j} (\rho k u_j) = \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


 \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}

Where

 C_1  =  \max\left[0.43, \frac{\eta}{\eta + 5}\right] , \;\;\;\;\; \eta  =  S \frac{k}{\epsilon}, \;\;\;\;\; S =\sqrt{2 S_{ij} S_{ij}}

In these equations,  P_k represents the generation of turbulence kinetic energy due to the mean velocity gradients, calculated in same manner as standard k-epsilon model.  P_b is the generation of turbulence kinetic energy due to buoyancy, calculated in same way as standard k-epsilon model.

Modelling Turbulent Viscosity

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

where
 C_{\mu} = \frac{1}{A_0 + A_s \frac{k U^*}{\epsilon}}
  U^* \equiv \sqrt{S_{ij} S_{ij} + \tilde{\Omega}_{ij} \tilde{\Omega}_{ij}}   ;
 \tilde{\Omega}_{ij}  =  \Omega_{ij} - 2 \epsilon_{ijk} \omega_k      ;
  \Omega_{ij}  =  \overline{\Omega_{ij}} - \epsilon_{ijk} \omega_k

where  \overline{\Omega_{ij}} is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity  \omega_k  . The model constants  A_0  and  A_s  are given by:
 A_0 = 4.04, \; \;  A_s = \sqrt{6}  \cos \phi

  \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)


Model Constants

 C_{1 \epsilon} = 1.44,  \;\; C_2 = 1.9, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.2

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