Rahman-Siikonen-Agarwal Model

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Latest revision as of 19:47, 8 July 2013

Introduction

The Rahman-Agarwal-Siikonen (RAS) Turbulence model is a one-equation eddy viscosity model based on $k-\epsilon$ closure. The R-transport equation along with the Bradshaw and other empirical relations are used to solve for the turbulent viscosity. A damping function, $f_\mu$ , is used to represent the kinematic blocking by the wall. To avoid defining a wall distance, a Helmholtz-type elliptic relaxation equation is used for $f_\mu$ . The model has been validated against a few well-documented flow cases, yielding predictions in good agreement with DNS and experimental data.

RAS Model

The turbulent eddy viscosity is given by

$\mu_t = C_\mu f_\mu \rho \tilde{R} = C_\mu f_\mu \rho k T_t$

The R-transport Equation:

$\begin{matrix} \frac{\partial \rho R}{\partial t} + \frac{\partial \rho u_j R}{\partial x_j} = \frac{\partial}{\partial x_j} \biggl[ \left(\mu +\frac{\mu_t}{\sigma}\right)\frac{\partial R}{\partial x_j} \biggr] +C_1 \rho \sqrt{P \tilde{R}} - C_2 \rho \left(\frac{\partial \tilde{R}}{\partial x_k}\right)^2 \end{matrix}$

Realizable Time Scale:

$T_t = \frac{k}{\epsilon} \sqrt{1+\frac{C_t^2}{Re_T}}, \quad Re_T = \frac{k^2}{\nu \epsilon}$

Coefficient $C_\mu$ :

$C_\mu = \frac{3(1+\eta^2)\alpha_1}{3+\eta^2+6\eta^2\xi^2+6\xi^2}$

$\eta = \alpha_2 T_t S, \quad \xi = \alpha_3 T_t W$

$\alpha_1 = g\left(\frac{1}{4}+\frac{2}{3}\Pi_b^{1/2}\right), \quad \alpha_2 = \frac{3}{8\sqrt{2}}g$

$\alpha_3 = \frac{3}{\sqrt{2}}\alpha_2, \quad g = \left( 1+2\frac{P}{\epsilon} \right) ^{-1}$

$\Pi_b = C_\nu \frac{P}{\epsilon}, \quad \frac{P}{\epsilon} = C_\nu \zeta^2$

$C_\nu = \frac{1}{2(1+T_t S \sqrt{1+\Re^2})}, \quad \zeta = T_t S max(1,\Re)$

$S_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right), \quad S = \sqrt{2S_{ij} S_{ij}}$

$W_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i}\right), \quad W = \sqrt{2S_{ij} S_{ij}}$

Damping Function:

$-L^2 \nabla^2f_\mu + f_\mu = 1$

$L^2 = \zeta(2\zeta+C_\mu Re_T)\sqrt{\frac{\nu^3}{\epsilon}}, \quad (f_\mu)_{wall} = 0$

Other Model Coefficients:

$C_1 = 2C_\mu \zeta (1-C_\mu \zeta), \quad C_2 = min \Biggl[ 2C_\mu,\; \tilde{C}_\mu \sqrt{1+\left( \frac{C_1}{6 \tilde{C}_\mu}\right)^2}\;\Biggr]$

$\sigma = C_\mu + \frac{f_\mu}{C_T}$

$k$ and $\epsilon$ :

$k = \sqrt{\tilde{k}^2 + k_\alpha ^2}$

$\tilde{k} = f_\mu^{0.8}\sqrt{C_\mu}R S_k, \quad k_\alpha = \nu S_\alpha$

$\epsilon = \sqrt{\epsilon_w^2+\tilde{\epsilon}^2+\epsilon_\alpha^2}$

$\tilde{\epsilon} = \frac{k^2}{R+\nu}, \quad \epsilon_\alpha = \nu S_\alpha^2, \quad \epsilon_w=2A_\epsilon\nu(\frac{\partial u}{\partial y})_w^2\approx 2A_\epsilon \nu S_k^2$

$S_k = \sqrt{\tilde{S}^2+S_\alpha^2},$

$\tilde{S} = f_k\left(S-\frac{\vert\eta_1\vert - \eta_1}{C_T}\right), \quad \eta_1 = S-W, \quad f_k=1-\frac{f_\alpha}{C_T}\sqrt{max(1-\Re,0)}$

$S_\alpha = \frac{2C_\alpha f_\alpha}{3\nu}\left(\frac{\sqrt{u_i^2/2}}{1+\mu_T/\mu}\right)^2, \quad C_\alpha=\sqrt{C_\mu^2+\frac{\nu}{R+\nu}}, \quad f_\alpha = 1 - exp\left(-\frac{\mu_T}{36\mu}\right)$

Constants:

$C_T = \sqrt{2}, \quad \tilde{C}_\mu = 0.09$

$0.05\;<\;A_\epsilon\;<\;0.11 \quad Commonly \;A_\epsilon = \tilde{C}_\mu$

References

Rahman, M. M., Siikonen, T., and Agarwal, R. K. (2011), "Improved Low Re-Number One-Equation Turbulence Model", AIAA Vol. 49, No.4, April 2011.

Rahman-Siikonen-Agarwal Model

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RAS Model

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@@ Line 5: / Line 5: @@
 == RAS Model ==
-The turbulent eddy viscosity is given by
+'''The turbulent eddy viscosity is given by'''
 :<math>
@@ Line 11: / Line 12: @@
 </math>
-The R-transport equation is
+'''The R-transport Equation:'''
 :<math>
 \begin{matrix}
-\frac{\partial \rho R}{\partial t} + \frac{\partial \rho u_j R}{\partial x_j} & = & \frac{\partial}{\partial x_j} [ (\mu +\frac{\mu_t}{\sigma})\frac{\partial R}{\partial x_j} ] +C_1 \rho \sqrt{P \tilde{R}} - C_2 \rho (\frac{\partial \tilde{R}}{\partial x_k})^2
+\frac{\partial \rho R}{\partial t} + \frac{\partial \rho u_j R}{\partial x_j} = \frac{\partial}{\partial x_j} \biggl[ \left(\mu +\frac{\mu_t}{\sigma}\right)\frac{\partial R}{\partial x_j} \biggr] +C_1 \rho \sqrt{P \tilde{R}} - C_2 \rho \left(\frac{\partial \tilde{R}}{\partial x_k}\right)^2
 \end{matrix}
 </math>
+'''Realizable Time Scale:'''
+:<math>
+T_t = \frac{k}{\epsilon} \sqrt{1+\frac{C_t^2}{Re_T}}, \quad Re_T = \frac{k^2}{\nu \epsilon}
+</math>
+'''Coefficient <math>C_\mu</math>:'''
+:<math>
+C_\mu = \frac{3(1+\eta^2)\alpha_1}{3+\eta^2+6\eta^2\xi^2+6\xi^2}
+</math>
+:<math>
+\eta = \alpha_2 T_t S, \quad \xi = \alpha_3 T_t W
+</math>
+:<math>
+\alpha_1 = g\left(\frac{1}{4}+\frac{2}{3}\Pi_b^{1/2}\right), \quad \alpha_2 = \frac{3}{8\sqrt{2}}g
+</math>
+:<math>
+\alpha_3 = \frac{3}{\sqrt{2}}\alpha_2, \quad g = \left( 1+2\frac{P}{\epsilon} \right) ^{-1}
+</math>
+:<math>
+\Pi_b = C_\nu \frac{P}{\epsilon}, \quad \frac{P}{\epsilon} = C_\nu \zeta^2
+</math>
+:<math>
+C_\nu = \frac{1}{2(1+T_t S \sqrt{1+\Re^2})}, \quad \zeta = T_t S max(1,\Re)
+</math>
+:<math>
+S_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right), \quad S = \sqrt{2S_{ij} S_{ij}}
+</math>
+:<math>
+W_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i}\right), \quad W = \sqrt{2S_{ij} S_{ij}}
+</math>
+'''Damping Function:'''
+:<math>
+-L^2 \nabla^2f_\mu + f_\mu = 1
+</math>
+:<math>
+L^2 = \zeta(2\zeta+C_\mu Re_T)\sqrt{\frac{\nu^3}{\epsilon}}, \quad (f_\mu)_{wall} = 0
+</math>
+'''Other Model Coefficients:'''
+:<math>
+C_1 = 2C_\mu \zeta (1-C_\mu \zeta), \quad C_2 = min \Biggl[ 2C_\mu,\; \tilde{C}_\mu \sqrt{1+\left( \frac{C_1}{6 \tilde{C}_\mu}\right)^2}\;\Biggr]
+</math>
+:<math>
+\sigma = C_\mu + \frac{f_\mu}{C_T}
+</math>
+'''<math>k</math> and <math>\epsilon</math>:'''
+:<math>
+k = \sqrt{\tilde{k}^2 + k_\alpha ^2}
+</math>
+:<math>
+\tilde{k} = f_\mu^{0.8}\sqrt{C_\mu}R S_k, \quad k_\alpha = \nu S_\alpha
+</math>
+:<math>
+\epsilon = \sqrt{\epsilon_w^2+\tilde{\epsilon}^2+\epsilon_\alpha^2}
+</math>
+:<math>
+\tilde{\epsilon} = \frac{k^2}{R+\nu}, \quad \epsilon_\alpha = \nu S_\alpha^2, \quad \epsilon_w=2A_\epsilon\nu(\frac{\partial u}{\partial y})_w^2\approx 2A_\epsilon \nu S_k^2
+</math>
+:<math>
+S_k = \sqrt{\tilde{S}^2+S_\alpha^2},
+</math>
+:<math>
+\tilde{S} = f_k\left(S-\frac{\vert\eta_1\vert - \eta_1}{C_T}\right), \quad \eta_1 = S-W, \quad f_k=1-\frac{f_\alpha}{C_T}\sqrt{max(1-\Re,0)}
+</math>
+:<math>
+S_\alpha = \frac{2C_\alpha f_\alpha}{3\nu}\left(\frac{\sqrt{u_i^2/2}}{1+\mu_T/\mu}\right)^2, \quad C_\alpha=\sqrt{C_\mu^2+\frac{\nu}{R+\nu}}, \quad f_\alpha = 1 - exp\left(-\frac{\mu_T}{36\mu}\right)
+</math>
+'''Constants:'''
+:<math>
+C_T = \sqrt{2}, \quad \tilde{C}_\mu = 0.09
+</math>
+:<math>
+.05\;<\;A_\epsilon\;<\;0.11 \quad Commonly \;A_\epsilon = \tilde{C}_\mu
+</math>
+==References ==
+* {{reference-paper|author=Rahman, M. M., Siikonen, T., and Agarwal, R. K.|year=2011|title=Improved Low Re-Number One-Equation Turbulence Model|rest=AIAA Vol. 49, No.4, April 2011}}
+[[Category:Turbulence models]]