EARSM models
The EARSM (Explicit Algebraic Reynolds Stress) models is a class of RANS turbulence models developed by Wallin and Johanson (Wallin and Johanson 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 EARSM formulation consists of non-linear formulation of the Reynolds stress tensor, which is expressed in terms of the mean strain-rate and rotation-rate tensors. The model is closed by an two-equation k-ω turbulence model.The model is particularly suitable for flows with strong streamline curvature, rotation, and separation, where traditional linear eddy-viscosity models may fail to provide accurate predictions.
Common constitutive relations for EARSM models
The general form of the constitutive relation for the Reynolds stress tensor in EARSM models is given by:
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: \[ \mu_t = - \frac{1}{2} (\beta_1 + I\!I_\Omega \beta_6) \rho k \tau. \]
where
\[ \begin{aligned} Q &= \frac{5}{6}(N^2 - 2 I\!I_\Omega)(2N^2 - I\!I_\Omega), \\ N &= \begin{cases} \frac{C_1'}{3} + (P_1 + \sqrt{P_2})^{1/3} + \text{sign}(P_1 - \sqrt{P_2})|P_1 - \sqrt{P_2}|^{1/3} & \text{if } P_2 \geq 0, \\ \frac{C_1'}{3} + 2 \sqrt{P_1} \cos\left(\frac{1}{3} \arccos\left(\frac{P_1}{\sqrt{P_1^2 - P_2}}\right)\right) & \text{if } P_2 < 0, \end{cases} \\ P_1 &= \left(\frac{C_1'^2}{27} + \frac{9}{20}I\!I_S - \frac{2}{3}I\!I_\Omega\right)C_1', \\ P_2 &= P_1^2 - \left(\frac{C_1'^2}{9} + \frac{9}{10}I\!I_S + \frac{2}{3}I\!I_\Omega\right)^3, \\ C_1' &= \frac{9}{4}\left[c_1 - 1 + C_{diff} \max(1 + \beta_1^{eq} I\!I_S,0)\right], \\ \beta_1^{eq} &= -\frac{6}{5} \frac{N^{eq}}{(N^{eq})^2 - 2I\!I_\Omega}, \\ N^{eq} &= \frac{9c_1}{4} = \frac{81}{20}. \end{aligned} \]
Model coefficients
Model coefficients for EARSM model are given as:
- \(C_\tau = 6\)
- \(c_1 = 1.8\)
- \(C_{diff} = 2.2\)
- \(\beta_1 = - \frac{N(2N^2 - 7 I\!|_\Omega)}{Q}\)
- \(\beta_3 = - \frac{12 I\!V}{N Q}\)
- \(\beta_4 = \frac{2(N^2 - 2I\!|_\Omega}{Q}\)
- \(\beta_6 = - \frac{6N}{Q}\)
- \(\beta_9 = \frac{6}{Q}\)
EARSMko2005
The EARSMko2005 proposed by Hellsten (Hellsten 2005) uses a modified version of k-ω SST turbulence model (Menter et al. 2003) as the base model. The current implementation uses the kOmegaSST model from the OpenFOAM library with the following modificatiions:
- the Reynolds stress tensor is computed using the EARSM formulation described above,
- the turbulent viscosity is computed using the EARSM formulation described above,
- the damping function \(F_1\) is given as \[ F_1 = \tanh\left(\min\left[\max\left(\frac{\sqrt{k}}{\beta^* \omega y}, \frac{500 \nu}{y^2 \omega}\right), \frac{20 k}{\max\left(\frac{y^2}{\omega} \frac{\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j}, 200k_\infty\right)}\right]^4\right), \]
Moreover, some coeeficients of the base k-ω SST model are modified as follows:
alphaK1= 1.1alphaK2= 1.1alphaOmega1= 0.53alphaOmega2= 1.0beta1= 0.0747beta2= 0.0828gamma1= 0.518gamma2= 0.44
EARSMtnt
The EARSMtnt model proposed by Wallin (Wallin 2000) uses a Kok’s k-ω TNT turbulence model (Kok 2000) as the base model. The scale model equations are described in kOmegaTNT section. The original k-ω TNT model is modified by using the above mentioned EARSM formula for the Reynolds stress tensor and the eddy viscosity.
EARSMlm2009
The EARSMlm2009 is a transition and turbulence model based on the EARSM formulation and the Langtry-Menter γ-Reθt transition model (Langtry and Menter 2009). The model uses the same \(F_1\) damping function and coefficient set as the EARSMko2005 model.
Validation of the model is provided at the EARSMlm2009 validation page.
EARSMgammaSST
The EARSMgammaSST model still requires some testing and calibration. Current form is provided for research purposes and is not suitable for production use.
The EARSMgammaSST is an attempt to combine the EARSM formulation with the three-equation gamma model by Menter, Smirnov et al. (Menter et al. 2015).