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Turbulence Modeling Resource

The Explicit Algebraic Stress k-omega Turbulence Model

This web page gives detailed information on the equations for various versions of Explicit Algebraic Stress Models (EASM) in k-omega form. Note: EASMs are also known as Explicit Algebraic Reynolds Stress Models (EARSM) and Algebraic Reynolds Stress Models (ARSM), but the monikers EASM, EARSM, and ARSM refer to the same thing. EASMs as a class have been developed by several independent groups over the years. As a result, it is difficult to present the many variations completely and cohesively. Currently, only a small subset is given. If any particular variant has been overlooked, please report it to the page curator. It should also be noted that the distinction is drawn between nonlinear EASMs (for which expansion coefficients are extracted directly from Reynolds stress transport equations) and other types of nonlinear eddy-viscosity models (not described on this page). See Phil. Trans. R. Soc. A (2007) 365, pp. 2389-2418.

Nonlinear EASMs are fundamentally different from linear eddy viscosity models in the equation for obtaining the modeled turbulent stresses in the Reynolds-averaged or Favre-averaged Navier-Stokes equations. Linear models use the Boussinesq assumption for the constitutive relation:

$\tau_{ij} = 2 \mu_t \left(S_{ij} - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}$

For nonlinear EASMs, this equation is altered to include additional (nonlinear) terms, as detailed below. Thus, including nonlinear turbulence models like EASM is not simply a matter of computing $\mu_t$ alone. One must also insure that the turbulent stress terms $\tau_{ij}$ are computed appropriately to include the additional nonlinear components in the Navier-Stokes equations.

Unless otherwise stated, for compressible flow with heat transfer this model is implemented as described on the page Implementing Turbulence Models into the Compressible RANS Equations, with perfect gas assumed and Pr = 0.72, Pr_t = 0.90, and Sutherland's law for dynamic viscosity.

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None of the EASM k-omega versions listed here are considered "standard".

Nonlinear EASM k-omega (2005) Model (EARSMko2005)

This model is typically known as the explicit algebraic Reynolds stress model, or EARSM (with an additional "R" in its name). Developed by Hellsten, Wallin, and Johansson, its main references are:

Hellsten, A., "New Advanced k-omega Turbulence Model for High-Lift Aerodynamics," AIAA Journal, Vol. 43, No. 9, 2005, pp. 1857-1869, https://doi.org/10.2514/1.13754.
Wallin, S. and Johansson, A. V., "An Explicit Algebraic Reynolds Stress Model for Incompressible and Compressible Turbulent Flows," J. Fluid Mechanics, Vol. 403, 2000, pp. 89-132, https://doi.org/10.1017/S0022112099007004.

In this model, the turbulent stress relationship can be given by:

$\tau_{ij} = 2 \mu_t \left(S_{ij} - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij} - a_{ij}^{(ex)} \rho k$

where

$a_{ij}^{(ex)} = \beta_3 \left( W_{ik}^*W_{kj}^* - \frac{1}{3} II_{\Omega} \delta_{ij} \right) + \beta_4 \left( S_{ik}^*W_{kj}^* - W_{ik}^*S_{kj}^* \right) + \beta_6 \left( S_{ik}^*W_{kl}^*W_{lj}^* + W_{ik}^*W_{kl}^*S_{lj}^* - II_{\Omega}S_{ij}^* - \frac{2}{3} IV \delta_{ij} \right) + \beta_9 \left( W_{ik}^*S_{kl}^*W_{lm}^*W_{mj}^* - W_{ik}^*W_{kl}^*S_{lm}^*W_{mj}^* \right)$

Note that for 2-D flows, the constitutive model can, if desired (choosing a different set of basis terms) be simplified to:

$a_{ij}^{(ex,2D)} = \beta_4^{(2D)} \left( S_{ik}^*W_{kj}^* - W_{ik}^*S_{kj}^* \right)$

The two-equation model (written in conservation form) is given by the following:

$\frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho u_j k)}{\partial x_j} = \cal P - \beta^* \rho \omega k + \frac{\partial}{\partial x_j} \left[\left(\mu + \sigma_k \mu_t \right)\frac{\partial k}{\partial x_j}\right]$

$\frac{\partial (\rho \omega)}{\partial t} + \frac{\partial (\rho u_j \omega)}{\partial x_j} = \frac{\gamma \omega}{k} \cal P - \beta \rho \omega^2 + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_{\omega} \mu_t \right) \frac{\partial \omega}{\partial x_j} \right] + \sigma_d \frac{\rho}{\omega} {\rm max} \left( \frac{\partial k}{\partial x_k} \frac{\partial \omega}{\partial x_k}, 0 \right)$

where $\rho$ is the density and $\mu$ is the molecular dynamic viscosity, and

$P = \tau_{ij} \frac{\partial u_i}{\partial x_j}$

$S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)$ $S_{ij}^* = \frac{\tau}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)$

$W_{ij}^* = \frac{\tau}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right)$

$\tau = \frac{1}{\beta^* \omega}$

and the turbulent eddy viscosity is computed from:

$\mu_t = \frac{C_{\mu}}{\beta^*} \frac{\rho k}{\omega}$

The variable coefficient $C_{\mu}$ is obtained from:

$C_{\mu} = -\frac{1}{2} \left( \beta_1 + II_{\Omega} \beta_6 \right)$

However, if the 2-D flow (superscript (2D)) constitutive model is used, then

$C_{\mu}^{(2D)} = -\frac{1}{2} \beta_1^{(2D)}$

is used instead. Furthermore,

$\beta_1 = - \frac{N \left( 2 N^2 - 7 II_{\Omega} \right)}{Q}$ $\beta_1^{(2D)} = - \frac{6}{5} \left[\frac{N}{N^2 - 2 II_{\Omega} } \right]$ $\beta_3 = - \frac{12 (IV)}{NQ}$

$\beta_4 = - \frac{2 \left( N^2 - 2 II_{\Omega} \right)}{Q}$ $\beta_4^{(2D)} = - \frac{6}{5} \left[\frac{1}{N^2 - 2 II_{\Omega} } \right]$ $\beta_6 = - \frac{6 N}{Q}$ $\beta_9 = \frac{6}{Q}$

$Q = \frac{5}{6} \left( N^2 - 2 II_{\Omega} \right) \left( 2 N^2 - II_{\Omega} \right)$

$II_{\Omega} = W_{kl}^*W_{lk}^*$ $IV = S_{kl}^*W_{lm}^*W_{mk}^*$

and N is obtained from the solution of a cubic equation. The solution is given by:

$N = \frac{A_3'}{3} + \left( P_1 + \sqrt{P_2} \right)^{1/3} + {\rm sign} \left( P_1 - \sqrt{P_2} \right) \left| P_1 - \sqrt{P_2} \right|^{1/3}$ for $P_2 \geq 0$

$N = \frac{A_3'}{3} + 2\left( P_1^2 - P_2 \right)^{1/6} {\rm cos} \left[ \frac{1}{3} {\rm cos}^{-1} \left( P_1 / \sqrt{ P_1^2 - P_2} \right) \right]$ for $P_2 < 0$

where

$P_1 = \left[ \frac{A_3'^2}{27} + \left( \frac{9}{20} \right) II_{S} - \frac{2}{3} II_{\Omega} \right] A_3'$

$P_2 = P_1^2 - \left[ \frac{A_3'^2}{9} + \left( \frac{9}{10} \right) II_{S} + \frac{2}{3} II_{\Omega} \right]^3$

$A_3' = \frac{9}{5} + \frac{9}{4} C_{diff} \left[ {\rm max} \left( 1 + \beta_1^{(eq)} II_S, 0 \right) \right]$

$II_{S} = S_{kl}^*S_{lk}^*$

$\beta_1^{(eq)} = - \frac{6}{5} \left[ \frac{N^{(eq)}}{\left(N^{(eq)}\right)^2 - 2 II_{\Omega}} \right]$

$N^{(eq)} = \frac{81}{20}$ $C_{diff} = 2.2$

Farfield boundary conditions are not specified for this model. However, the reference states that the model is reasonably insensitive to freestream values of k and $\omega$ , provided that excessively high values are avoided.

Solid wall boundary conditions are:

$\omega_{wall} = \frac{u_{\tau}^2 S_R}{\nu}$

$k_{wall} = 0$

where

$S_R = \left[ \frac{50}{{\rm max} \left(k_s^+, k_{s, min}^+ \right) } \right]^2$ for $k_s^+ < 25$

$S_R = \frac{100}{k_s^+}$ for $k_s^+ \geq 25$

with $k_s^+$ specified for rough walls, and for smooth walls:

$k_{s, min}^+ = {\rm min} \left[ 4.3 \left(d_1^+ \right)^{0.85}, 8 \right]$

and $d_1^+$ is the inner-scaled wall distance of the first solution point next to the solid wall.

The constants in the scale-determining k- $\omega$ model are determined via:

$C = f_{mix} C_1 + \left(1-f_{mix}\right) C_2$

where C represents any of the model coefficients, and

$f_{mix} = {\rm tanh} \left( 1.5 \Gamma^4 \right)$

$\Gamma = {\rm min} \left[ {\rm max} \left( \Gamma_1, \Gamma_2 \right), \Gamma_3 \right]$

$\Gamma_1 = \frac{\sqrt{k}}{\beta^* \omega d}$

$\Gamma_2 = \frac{500 \nu}{\omega d^2}$

$\Gamma_3 = \frac{20 k}{{\rm max} \left[ \frac{d^2}{\omega} \left( \frac{\partial k}{\partial x_k} \frac{\partial \omega}{\partial x_k} \right) , 200 k_{\infty} \right]}$

where d is the distance to the nearest wall. The coefficient values are:

$\gamma_1 = 0.518$ $\gamma_2 = 0.44$

$\beta_1 = 0.0747$ $\beta_2 = 0.0828$

$\sigma_{k1} = 1.1$ $\sigma_{k2} = 1.1$

$\sigma_{\omega 1} = 0.53$ $\sigma_{\omega 2} = 1.0$

$\sigma_{d 1} = 1.0$ $\sigma_{d 2} = 0.4$

$\beta^* = 0.09$

Nonlinear EASM k-omega (2005) Model with Curvature Correction (EARSMko2005-CC)

This model is the same as the (EARSMko2005), with the exception that

$W_{ij}^* = \frac{\tau}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) - \left( \frac{\tau}{A_0} \right) W_{ij}^{*(r)}$

where

$W_{ij}^{*(r)} = -\varepsilon_{ijk} B_{km} S_{pr}^* \frac{D S_{rq}^*}{Dt} \varepsilon_{pqm}$

$B_{km} = \frac{II_S^2 \delta_{km} + 12 III_S S_{km}^* + 6 II_S S_{kl}^* S_{lm}^*}{2 II_S^3 - 12 III_S^2}$

and $A_0 = -0.72$ , $III_S = S_{kl}^* S_{lm}^* S_{mk}^*$ , and $\varepsilon_{ijk}$ is the Levi-Civita symbol, defined as

$\varepsilon_{123} = \varepsilon_{312} = \varepsilon_{231} = 1$

$\varepsilon_{132} = \varepsilon_{213} = \varepsilon_{321} = -1$

with all other $\varepsilon_{ijk}$ zero.

The references for this curvature correction are:

Hellsten, A., "New Advanced k-omega Turbulence Model for High-Lift Aerodynamics," AIAA Journal, Vol. 43, No. 9, 2005, pp. 1857-1869, https://doi.org/10.2514/1.13754.
Wallin, S. and Johansson, A. V., "Modelling Streamline Curvature Effects in Explicit Algebraic Reynolds Stress Turbulence Models," Int. J. Heat and Fluid Flow, Vol. 23, No. 5, 2002, pp. 721-730, https://doi.org/10.1016/S0142-727X(02)00168-6.

Nonlinear EASM k-omega (2005) Model with Better Approximation for 3-D Flows (EARSMko2005a), (EARSMko2005a-CC)

The equations are the same as (EARSMko2005) or (EARSMko2005-CC), with the exception that N gets augmented by an additional term:

$N_{improved} = N + \frac{162 \left[ IV^2 + \left( V - \frac{1}{2} II_S II_{\Omega} \right) N^2 \right]} {20 N^4 \left( N - \frac{1}{2} A_3' \right) - II_{\Omega} \left( 10 N^3 + 15 A_3' N^2 \right) + 10 A_3' II_{\Omega}^2}$

with

$V = S_{kl}^* S_{lm}^* W_{mn}^* W_{nk}^*$

The references for this improved 3-D approximation are:

Hellsten, A., "New Two-Equation Turbulence Model for Aerodynamics Applications," Helsinki University of Technology Laboratory of Aerodynamics, Report A-21, PhD Dissertation, 2004, Espoo, Finland.
Wallin, S. and Johansson, A. V., "An Explicit Algebraic Reynolds Stress Model for Incompressible and Compressible Turbulent Flows," J. Fluid Mechanics, Vol. 403, 2000, pp. 89-132, https://doi.org/10.1017/S0022112099007004.

Nonlinear EASM k-omega (2003) Model (EASMko2003)

The references for this nonlinear two-equation model are:

Rumsey, C. L. and Gatski, T. B., "Summary of EASM Turbulence Models in CFL3D with Validation Test Cases," NASA/TM-2003-212431, June 2003, https://ntrs.nasa.gov/citations/20030064901.
Rumsey, C. L. and Gatski, T. B., "Recent Turbulence Model Advances Applied to Multielement Airfoil Computations," Journal of Aircraft, Vol. 38, No. 5, 2001, pp. 904-910, https://doi.org/10.2514/2.2850.

However, note that the journal reference (EASMko2001) used different values for two of the constants ( $\sigma_k = 0.5$ and $\gamma = 0.575$ ); those listed in the NASA/TM reference (same as below) are considered better, particularly for jet-type flows.

In this model, the turbulent stress relationship is derived based on a three-basis approximation. It is given by:

$\tau_{ij} = 2 \mu_t \left(S_{ij} - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} + \left[a_2 a_4 \left( S_{ik}W_{kj} - W_{ik}S_{kj}\right) - 2 a_3 a_4 \left(S_{ik}S_{kj} - \frac{1}{3}S_{kl}S_{lk}\delta_{ij}\right) \right] \right) - \frac{2}{3} \rho k \delta_{ij}$

The two-equation model (written in conservation form) is given by the following:

$\frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho u_j k)}{\partial x_j} = \cal P - f_{\beta^*} \rho \omega k + \frac{\partial}{\partial x_j} \left[\left(\mu + \sigma_k \mu_t \right)\frac{\partial k}{\partial x_j}\right]$

where

$P = \tau_{ij} \frac{\partial u_i}{\partial x_j}$

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

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

and the turbulent eddy viscosity is computed from:

$\mu_t = C_{\mu}^* \frac{\rho k}{\omega}$

where $\rho$ is the density and $\mu$ is the molecular dynamic viscosity.

The variable coefficient $C_{\mu}^*$ is obtained by solving the cubic equation:

$(-C_{\mu}^*)^3 + p(-C_{\mu}^*)^2 + q(-C_{\mu}^*) + r = 0$

where

$p = - \frac{\gamma_1^*}{\eta^2 \tau^2 \gamma_0^*}$

$q = \frac{1}{(2 \eta^2 \tau^2 \gamma_0^*)^2} \left( \gamma_1^{*2} - 2 \eta^2 \tau^2 \gamma_0^* a_1 -\frac{2}{3} \eta^2 \tau^2 a_3^2 + 2 R^2 \eta^2 \tau^2 a_2^2 \right)$

$r = \frac{\gamma_1^* a_1}{(2 \eta^2 \tau^2 \gamma_0^*)^2}$

The correct root to choose from this cubic equation is the root with the lowest real part. The degenerate case when $\eta \rightarrow 0$ must be avoided. The appendix of Journal of Aircraft, Vol. 38, No. 5, 2001, pp. 904-910, https://doi.org/10.2514/2.2850 provides an algorithm for determining this root, as follows.

If $\eta^2 \tau^2 < 1 \times 10^{-6}$ , then

$C_{\mu}^* = \frac{\gamma_1^* a_1}{\gamma_1^{*2} - 2 \lbrace W^2 \rbrace \tau^2 a_2^2}$

Otherwise, define:

$a \equiv q - \frac{p^2}{3}$

$b \equiv \frac{1}{27} \left( 2 p^3 - 9 p q + 27 r \right)$

$d \equiv \frac{b^2}{4} + \frac{a^3}{27}$

If $d > 0$ , then

$t_1 = \left( -\frac{b}{2} + \sqrt{d} \right)^{1/3}$

$t_2 = \left( -\frac{b}{2} - \sqrt{d} \right)^{1/3}$

$C_{\mu}^* = -{\rm min} \left( -\frac{p}{3} + t_1 + t_2, -\frac{p}{3} -\frac{t_1}{2} - \frac{t_2}{2} \right)$

If $d \leq 0$ , then

$\theta = {\rm cos}^{-1} \left( - \frac{b/2}{\sqrt{-a^3/27} \right)$

$t_1 = -\frac{p}{3} + 2 \sqrt{-\frac{a}{3} } {\rm cos} \left( \frac{\theta}{3} \right)$

$t_2 = -\frac{p}{3} + 2 \sqrt{-\frac{a}{3} } {\rm cos} \left( \frac{2 \pi}{3} + \frac{\theta}{3} \right)$

$t_3 = -\frac{p}{3} + 2 \sqrt{-\frac{a}{3} } {\rm cos} \left( \frac{4 \pi}{3} + \frac{\theta}{3} \right)$

$C_{\mu}^* = -{\rm min} \left( t_1 , t_2 , t_3 \right)$

In this model, $C_{\mu}^*$ is limited to be no smaller than 0.0005.

Other parameters are:

$\tau \equiv 1 / \omega$

$\eta^2 = \lbrace S^2 \rbrace = S_{ij}S_{ij}$

$\lbrace W^2 \rbrace = -W_{ij}W_{ij}$

$R^2 = - \frac{\lbrace W^2 \rbrace}{\lbrace S^2 \rbrace}$ $a_1 = \frac{1}{2} \left( \frac{4}{3} - C_2 \right)$

$a_2 = \frac{1}{2} \left( 2 - C_4 \right)$ $a_3 = \frac{1}{2} \left( 2 - C_3 \right)$

$a_4 = \tau \left[ \gamma_1^* - 2 \gamma_0^* \left( -C_{\mu}^* \right) \eta^2 \tau^2 \right]^{-1}$

$\gamma_0^* = \frac{C_1^1}{2}$ $\gamma_1^* = \frac{1}{2} C_1^0 + \left( \frac{C_{\epsilon 2} - C_{\epsilon 1}}{C_{\epsilon 1} - 1} \right)$

The function $f_{\beta^*}$ is given by:

$f_{\beta^*} = 1$ when $\chi_k \leq 0$

$f_{\beta^*} = \frac{1 + 680 \chi_k^2}{1 + 400 \chi_k^2}$ when $\chi_k > 0$

where

$\chi_k = \frac{C_{\mu}^2}{\omega^3} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$

The P term in the k-equation is limited, replaced by:

${\rm min}(P, 20 f_{\beta^*} \rho \omega k)$

Also, the variables $\tau_{11}$ , $\tau_{22}$ , and $\tau_{33}$ (as defined) are all limited to be negative (this is the realizability constraint).

The farfield boundary conditions given in this reference are:

$\overline k_{farfield} = \frac{k_{farfield}}{a_{ref}^2} = 9 \times 10^{-9}$

$\overline \omega_{farfield} = \frac{\omega_{farfield}}{(\rho_{ref} a_{ref}^2/ \mu_{ref} )} = 9 \times 10^{-8}$

where $\rho_{ref}$ , $a_{ref}$ , and $\mu_{ref}$ are the reference (typically freestream) speed of sound, density, and viscosity, respectively.

The solid wall boundary conditions are the same as those recommended in Menter, F. R., AIAA Journal, Vol. 32, No. 8, August 1994, pp. 1598-1605, https://doi.org/10.2514/3.12149:

$\omega_{wall} = 10 \frac{6 \nu}{\beta (\Delta d_1)^2}$