EARSMWallin model
The EARSMWallin (Explicit Algebraic Reynolds Stress Model) is a RANS turbulence model developed by Wallin and Johanson (WALLIN and JOHANSSON 2000). It is based on the Boussinesq approximation but goes beyond the linear eddy-viscosity concept by providing an explicit algebraic formulation for the Reynolds stress tensor. This allows the model to capture anisotropic turbulence effects more accurately than traditional two-equation models. The model uses k-omega TNT by Kok (Kok 2000) as the underlying turbulence model.
Model Equations
The transport equations for the kOmegaTNT model are given by:
\[ \begin{aligned} \frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho k u_j)}{\partial x_j} &= P_k - \beta^* \rho k \omega + \frac{\partial}{\partial x_j} \left[ (\mu + \sigma_k \mu_t) \frac{\partial k}{\partial x_j} \right], \\ \frac{\partial (\rho \omega)}{\partial t} + \frac{\partial (\rho \omega u_j)}{\partial x_j} &= P_\omega - \beta \rho \omega^2 + \frac{\partial}{\partial x_j} \left[ (\mu + \sigma_\omega \mu_t) \frac{\partial \omega}{\partial x_j} \right] + C_D, \end{aligned} \] where: \[ \begin{aligned} P_k &= \tau_{ij} \frac{\partial u_i}{\partial x_j}, \\ P_\omega &= \frac{\alpha_\omega \omega}{k} P_k, \\ C_D &= \frac{\sigma_d \rho}{\omega} \max\left[\frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}, 0\right]. \end{aligned} \]
Reynolds stress tensor is computed as: \[ \tau_{ij} = 2 \mu_t S_{ij}^* - \frac{2}{3} \rho k \delta_{ij} - \rho k a_{ij}^{ex}, \] where \(a_{ij}^{ex}\) is the extra anisotropy tensor and \[ S_{ij}^* = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} - \frac{2}{3} \delta_{ij} \frac{\partial u_k}{\partial x_k} \right) \] and \[ \Omega_{ij}^* = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right). \]
The extra anisotropy tensor \(a_{ij}^{ex}\) is given by: \[ \begin{aligned} a_{ij}^{ex} =& \beta_3\left(\mathbf{\Omega}^2 - \frac{1}{3} I\!I_\Omega\,\mathbf{I}\right) + \beta_4\left(\mathbf{S\Omega} - \mathbf{\Omega S}\right) + \\ &+ \beta_6\left(\mathbf{S\Omega^2} \mathbf{\Omega^2S} - I\!I_S\,\mathbf{S} - \frac{2}{3} I\!V\, \mathbf{I}\right) + \\ &+ \beta_9\left(\mathbf{\Omega S\Omega^2} - \mathbf{\Omega^2 S \Omega}\right). \end{aligned} \]
Here \(\mathbf{S}\) and \(\mathbf{\Omega}\) are the second order tensors. The inner products are defined as \(\mathbf{\Omega}^2_{ij} = \Omega_{ik} \Omega_{kj}\).
Normalized mean strain-rate and rotation-rate tensors are defined as: \[ \mathbf{S} = \tau S_{ij}^*, \quad \mathbf{\Omega} = \tau \Omega_{ij}^*, \] where \[\tau = \max\left(\frac{k}{\epsilon}, C_\tau\sqrt{\frac{\mu}{\rho \epsilon}}\right) \] is the turbulence time scale (\(\epsilon=\beta^* k \omega\)). The invariants of the tensors are given by: \[ I\!I_S = S_{ij} S_{ji}, \quad I\!I_\Omega = \Omega_{ij} \Omega_{ji}, \quad I\!V = S_{ik} \Omega_{kj} \Omega_{ji}. \]
The eddy viscosity is computed as: \[ \mu_t = - \frac{1}{2} (\beta_1 + I\!I_\Omega \beta_6) \rho k \tau. \]
Model coefficients
The model coefficients for TNT model are defined as:
- \(\beta^* = 0.09\)
- \(\beta = 0.075\)
- \(\sigma_k = \frac{2}{3}\)
- \(\sigma_\omega = 0.5\)
- \(\alpha_\omega = 5/9\)
- \(\sigma_d = 0.5\)
Additional coefficients for EARSM model are given as:
- \(C_\tau = 6\)
- \(\beta_1 = - \frac{N(2N^2 - 7 I\!|_\Omega)}{Q}\)
- \(\beta_3 = - \frac{12 I\!V}{N Q}\)
- \(\beta_4 = \frac{2(N^2 - 2I\!|_\Omega}{Q}\)
- \(\beta_6 = - \frac{6N}{Q}\)
- \(\beta_9 = \frac{6}{Q}\)
where
\[ \begin{aligned} Q &= \frac{5}{6}(N^2 - 2 I\!I_\Omega)(2N^2 - I\!I_\Omega), \\ N &= \begin{cases} \frac{C_1'}{3} + (P_1 + \sqrt{P_2})^{1/3} + \text{sign}(P_1 - \sqrt{P_2})|P_1 - \sqrt{P_2}|^{1/3} & \text{if } P_2 \geq 0, \\ \frac{C_1'}{3} + 2 \sqrt{P_1} \cos\left(\frac{1}{3} \arccos\left(\frac{P_1}{\sqrt{P_1^2 - P_2}}\right)\right) & \text{if } P_2 < 0, \end{cases} \\ P_1 &= \left(\frac{C_1'^2}{27} + \frac{9}{20}I\!I_S - \frac{2}{3}I\!I_\Omega\right)C_1', \\ P_2 &= P_1^2 - \left(\frac{C_1'^2}{9} + \frac{9}{10}I\!I_S + \frac{2}{3}I\!I_\Omega\right)^3, \\ C_1' &= \frac{9}{4}\left[c_1 - 1 + C_{diff} \max(1 + \beta_1^{eq} I\!I_S,0)\right], \\ \beta_1^{eq} &= -\frac{6}{5} \frac{N^{eq}}{(N^{eq})^2 - 2I\!I_\Omega}, \\ N^{eq} &= \frac{9c_1}{4} = \frac{81}{20}. \end{aligned} \] Here \(c_1 = 1.8\) and \(C_{diff} = 2.2\).
Notes on OpenFOAM implementation
OpenFOAM implementation uses slightly different notation for some coefficients:
Cmuis used for \(\beta^*\)betais used for \(\beta\)gammais used for \(\alpha_\omega\)alphaKis used for \(\sigma_k\)alphaOmegais used for \(\sigma_\omega\)alphaDis used for \(\sigma_d\)Ctauis used for \(C_\tau\)