RNG k-epsilon model

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(Difference between revisions)

Revision as of 13:04, 14 September 2005

$\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left(\alpha_k \mu_{\rm eff} \frac{\partial k}{\partial x_j}\right) + P_k + P_b - \rho \epsilon$

$\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left(\alpha_{\epsilon} \mu_{\rm eff} \frac{\partial \epsilon}{\partial x_j}\right) + C_{1 \epsilon}\frac{\epsilon}{k} \left( G_k + C_{3 \epsilon} G_b \right) - C_{2\epsilon} \rho \frac{\epsilon^2}{k} - R_{\epsilon}$

$d \left(\frac{\rho^2 k}{\sqrt{\epsilon \mu}} \right) = 1.72 \frac{\hat{\nu}}{\sqrt{{\hat{\nu}}^3-1+C_\nu}} d{\hat{\nu}}$

$\hat{\nu} = \mu_{\rm eff}/\mu$

and

$C_\nu \approx 100$

$R_{\epsilon} = \frac{C_\mu \rho \eta^3 (1-\eta/\eta_0)}{1+\beta\eta^3} \frac{\epsilon^2}{k}$

$\eta \equiv Sk/\epsilon$

$\eta_0 = 4.38$

$\beta = 0.012$

$\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left(\alpha_{\epsilon} \mu_{\rm eff} \frac{\partial \epsilon}{\partial x_j}\right) + C_{1 \epsilon}\frac{\epsilon}{k} \left( G_k + C_{3 \epsilon} G_b \right) - C_{2\epsilon}^* \rho \frac{\epsilon^2}{k}$

$C_{2\epsilon}^* \equiv C_{2\epsilon} + {C_\mu \eta^3 (1-\eta/\eta_0)\over 1+\beta\eta^3}$

$C_{1\epsilon} = 1.42, \; \; C_{2\epsilon} = 1.68$

RNG k-epsilon model

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Revision as of 13:04, 14 September 2005

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@@ Line 1: / Line 1: @@
-<math>
+:<math>
   \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left(\alpha_k \mu_{\rm eff} \frac{\partial k}{\partial x_j}\right) + P_k + P_b - \rho \epsilon
 </math>
-<math>
+:<math>
   \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left(\alpha_{\epsilon} \mu_{\rm eff} \frac{\partial \epsilon}{\partial x_j}\right) + C_{1 \epsilon}\frac{\epsilon}{k} \left( G_k + C_{3 \epsilon} G_b \right) - C_{2\epsilon} \rho \frac{\epsilon^2}{k} - R_{\epsilon}
 </math>
-<math>
+:<math>
 d \left(\frac{\rho^2 k}{\sqrt{\epsilon \mu}} \right) = 1.72 \frac{\hat{\nu}}{\sqrt{{\hat{\nu}}^3-1+C_\nu}} d{\hat{\nu}}
 </math>
-<math>
+:<math>
 \hat{\nu} = \mu_{\rm eff}/\mu
 </math>
 and
-<math>
+:<math>
 C_\nu  \approx 100
- </math>
+</math>
+:<math>
+R_{\epsilon} = \frac{C_\mu \rho \eta^3 (1-\eta/\eta_0)}{1+\beta\eta^3} \frac{\epsilon^2}{k}
+</math>
+:<math>
+\eta \equiv Sk/\epsilon
+</math>
+:<math>
+\eta_0 = 4.38
+</math>
+:<math>
+\beta = 0.012
+</math>
+:<math>
+\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left(\alpha_{\epsilon} \mu_{\rm eff} \frac{\partial \epsilon}{\partial x_j}\right) + C_{1 \epsilon}\frac{\epsilon}{k} \left( G_k + C_{3 \epsilon} G_b \right) - C_{2\epsilon}^* \rho \frac{\epsilon^2}{k}
+</math>
+:<math>
+C_{2\epsilon}^* \equiv C_{2\epsilon} + {C_\mu \eta^3 (1-\eta/\eta_0)\over 1+\beta\eta^3}
+</math>
+:<math>
+C_{1\epsilon} = 1.42, \; \; C_{2\epsilon} = 1.68
+</math>