EARSM are a class of non-linear EVM derived by a systematic approximation to the second moment closure. Unlike other, more empirical approaches to non-linear modelling, EARSM are rational approximations to RSTM and thus yield clear prospects with respect to model validity.

The Hellsten EARSM \(k\)- \(\omega\) model [19] is an explicit algebraic Reynolds stress model based on the self-consistent approach of Wallin & Johannson [61], optionally including a curvature correction [62] . This EARSM is based on a recalibrated Launder, Reece and Rodi (LRR) RSTM [29], from which a fully explicit and self-consistent algebraic relation is derived. Other topics addressed by the approach are the model behaviour at turbulent-laminar edges, its sensitivity to pressure gradients and the model coefficient calibration.

The approach replaces the eddy-viscosity assumption by a more general constitutive relation for the Reynolds stress anisotropy in terms of the strain- and rotation-rate tensors

\begin{eqnarray} S_{ij}^* & \equiv & \frac{1}{2}\left(\parfrac{U_i}{x_j} +\parfrac{U_j}{x_i}-\frac{2}{3}\parfrac{U_k}{x_k}\delta_{ij}\right) \\ \Omega_{ij}^* & \equiv & \frac{1}{2}\left(\parfrac{U_i}{x_j} -\parfrac{U_j}{x_i}\right). \end{eqnarray}

The model is formulated based on an effective turbulent viscosity \(\mu_t\) and an extra anisotropy \(a^{\left(\rm ex\right)}_{ij}\):

\begin{eqnarray}\label{eq:rstr} -\Emean{{\rho}u_iu_j} = 2\mu_t S^*_{ij} -\frac{2}{3}\rho{k}\delta_{ij} - {\rho}ka^{\left(\rm ex\right)}_{ij} \end{eqnarray}

The effective turbulent viscosity and the extra anisotropy are given by

\begin{eqnarray}\label{eq:eddyv} \mu_t = -\frac{1}{2}\left(\beta_1+\IIw\beta_6\right){\rho}k\tau \end{eqnarray}

and

\begin{eqnarray} \a^{\left(\rm ex\right)} = \beta_3\left(\W^2-\frac{1}{3}\IIw\ \I \right) + \beta_4\left(\S\W-\W\S \right) + \beta_6\left(\S\W^2+\W^2\S-\IIw\S-\frac{2}{3}\IV\ \I \right) + \beta_9\left(\W\S\W^2-\W^2\S\W \right) \end{eqnarray}

To this end, the strain and vorticity tensors are normalized by the turbulent time scale \(\tau={k}/{\eps}={1}/{(\beta^*\omega)}\) with \(\eps\) being the dissipation rate of \(k\):

\begin{eqnarray} \S = S_{ij} = \tau S_{ij}^* \quad \mbox{and} \quad \W = \Omega_{ij}= \tau \Omega_{ij}^* \end{eqnarray}

The invariants are

\begin{eqnarray} \IIs \equiv \tr{\S^2} , \IIw \equiv \tr{\W^2} , \IV \equiv \tr{\S\W^2} \end{eqnarray}

The \(\beta\) coefficients are given by

\begin{equation}\label{eq:beta} \beta_1 = -\frac{N\left(2N^2-7\IIw\right)}{Q} , \beta_3 = -\frac{12N^{-1}\IV}{Q} , \beta_4 = -\frac{2\left(N^2-2\IIw\right)}{Q} , \beta_6 = -\frac{6N}{Q} , \beta_9 = \frac{6}{Q} \end{equation}

where the denominator \(Q\) reads

\begin{eqnarray} Q = \frac{5}{6}\left(N^2-2\IIw\right)\left(2N^2-\IIw\right) \end{eqnarray}

The function \(N\) is given by

\begin{eqnarray} N = \left\{ \begin{array}{ll} \frac{C'_1}{3}+\left(P_1+\sqrt{P_2}\right)^{1/3} +\left(P_1-\sqrt{P_2}\right)^{1/3} & , P_2 \ge 0 \\ \frac{C'_1}{3}+2\left(P_1^2-P_2\right)^{1/6} \cdot {\rm cos}\left(\frac{1}{3}{\rm arccos} \left(\frac{P_1}{\sqrt{P_1^2-P_2}}\right)\right) & ,P_2 < 0 \end{array}\right. \end{eqnarray}

where

\begin{eqnarray} \begin{array}{l} P_1 = \left(\frac{{C'_1}^2}{27}+\frac{9}{20}\IIs-\frac{2}{3}\IIw\right)C'_1 \\ P_2 = P_1^2-\left(\frac{{C'_1}^2}{9}+\frac{9}{10}\IIs+\frac{2}{3}\IIw\right)^3 \end{array} \end{eqnarray}

and

\begin{eqnarray} C'_1 = \frac{9}{5}+\frac{9}{4}C_{\mathrm{diff}}\max\left(1+\beta^{(eq)}_1\IIs,0\right) \end{eqnarray}

\begin{eqnarray} \beta^{(eq)} = -\frac{6}{5}\frac{N^{(eq)}}{\left((N^{(eq)})^2-2\IIw\right)} \qquad N^{(eq)} = 81/20 \qquad C_{\mathrm{diff}} = 2.2 \end{eqnarray}

It should be remarked that Fint* 3D mean flows the relation for \(N\) does not exactly imply self consistency, which is only strictly fulfilled in the 2D limit. In fully three-dimensional cases, \(N\) is represented by a sixth-order equation, to which no explicit solution can be found.

Optionally, for flows in which streamline curvature plays an important role, the use of a curvature correction [62] can be beneficial. In this case, \(\W\) is replaced by

\begin{eqnarray} \widetilde{\W} = \W -\left(\frac{\tau}{A_0}\right)\W^{(r)} \end{eqnarray}

where the correction term is defined as follows:

\begin{eqnarray}\label{eq:cc} \Omega^{(r)}_{ij} = -\epsilon_{ijk}B_{km}S_{pr}\frac{\mbox{D}S_{rq}}{\mbox{D}t}\epsilon_{pqm} \end{eqnarray}

with

\begin{eqnarray} B_{km} = \frac{\IIs^2\delta_{km}+12\IIIs S_{km}+6\IIs S_{kl}S_{lm}} {2\IIs^3-12\IIIs^2} \end{eqnarray}

\begin{eqnarray} \begin{array}{ll} \IIIs \equiv \tr{\S^3} \qquad A_0 = -0.72 \end{array} \end{eqnarray}

The two-equation model base for the Hellsten EARSM \(k\)- \(\omega\) is a modified Menter BSL approach [35] specifically recalibrated to be used in conjunction with the nonlinear constitutive relation. The transport equations are given by

\begin{eqnarray} \parfrac{{\rho}k}{t} + \parfrac{{\rho}kU_j}{x_j} &=& \Prod{k} - \beta^*\rho\omega k + \parfrac{}{x_j}\left[\left(\mu+\sigma_k\mu_t\right) \parfrac{k}{x_j}\right] \label{eq:k}\\ \parfrac{\rho\omega}{t} + \parfrac{\rho\omega{U_j}}{x_j} &=& \Prod{\omega} - \beta\rho\omega^2 + \parfrac{}{x_j}\left[\left(\mu+\sigma_\omega\mu_t\right) \parfrac{\omega}{x_j}\right] + \sigma_d \frac{\rho}{\omega} \max \left(\parfrac{k}{x_j}\parfrac{\omega}{x_j},0 \right)\label{eq:w} \end{eqnarray}

The model coefficients \(\alpha,\beta,\sigma_k,\sigma_\omega,\sigma_d\) are blended in space according to

\begin{eqnarray} C = f_{\mr{mix}} C_1 + (1-f_{\mr{mix}}) C_2, \end{eqnarray}

where the blending function is constructed as

\begin{eqnarray} f_{\mr{mix}} = \tanh(1.5\Gamma^4) \qquad \Gamma = \min \left( \max \left(\Gamma_1,\Gamma_2 \right), \Gamma_3 \right) \end{eqnarray}

with

\begin{eqnarray} \Gamma_1 = \frac{\sqrt{k}}{\beta^* \omega y} \qquad \Gamma_2 = \frac{500 \mu}{\rho \omega y^{2}} \qquad \Gamma_3 = \frac{20 k}{\max \left( y^{2} (\nabla k \nabla \omega) / \omega, 200 k_\infty \right)} \end{eqnarray}

The \(k\)- \(\omega\) transport equations of the background model are extended with the EARSM relations for the eddy viscosity and Reynolds stresses. When using the \(k\)- \(\omega\) base, the turbulent dissipation rate is given by

\begin{equation} \eps = \beta^*\omega k \end{equation}

The production of \(k\) and \(\omega\) are given by

\begin{equation}\label{eq:Prod} \Prod{k} = -\Emean{{\rho}u_iu_j}\parfrac{U_i}{x_j} \quad \mbox{and} \quad \Prod{\omega} = \alpha\frac{\omega}{k}\Prod{k} \end{equation}

For the Reynolds stresses and the turbulent viscosity, the expressions given above are used.

The model coefficients are:

Constant	\(\alpha\)	\(\beta\)	\(\beta^*\)	\(\sigma_k\)	\(\sigma_\omega\)	\(\sigma_d\)
Fint*ner (1)	0.518	0.0747	0.09	1.1	0.53	1.0
outer (2)	0.44	0.0828	0.09	1.1	1.0	0.4

Implementation detail: In line with [9], the expression

\begin{eqnarray} C_{\mu}=-\frac{1}{2}\left(\beta_1+\IIw\beta_6\right) \end{eqnarray}

is replaced by

\begin{eqnarray} C_{\mu}=\frac{3}{5}\frac{N}{N^2-\IIw} \end{eqnarray}

which is mathematically equivalent and valid for both the 3D (which is implemented in TRACE) and the 2D (which could easily be included) version of the model. Furthermore, a limiter ensuring \(C_{\mu}\leq{\beta^*}\) is active by default, it can be switched off using -NoEARSMLimitation ON (see Manual). To enhance convergence in the initial phase of the computation, when the robust switch is on, \(C_{\mu}={\beta^*}\) and \(\a^{\left(\rm ex\right)}=0\).

The model initialisation function is tmHellstenEARSMinitModel() or tmHellstenEARSMccinitModel() respectively, if curvature correction is desired.