Langley Research Center

Turbulence Modeling Resource

NOTE: This page still needs to be checked for consistency with the original reference

Explicit Algebraic Stress k-kL Turbulence Model

This web page gives detailed information on the equations for a version of an Explicit Algebraic Stress Model (EASM) in k-kL 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. See further discussion on the Explicit Algebraic Stress k-omega page.

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.

Return to: Turbulence Modeling Resource Home Page

Nonlinear ARSM k-kL (2018) Model (k-kL-ARSM2018+J)

The reference for this nonlinear two-equation model is:

Abdol-Hamid, K. S., "Development and Documentation of kL-Based Linear, Nonlinear, and Full Reynolds Stress Turbulence Models," NASA/TM-2018-219820, April 2018, https://ntrs.nasa.gov/citations/20180003199.

The model here is one of several described in the above reference. It includes a jet correction (+J) by default, at the recommendation of the author of the model. Note that the author passed away prior to checking this webpage for consistency; please report any errors or typos that you find to the page curator.

In this model, the turbulent stress relationship is a blend between a linear model and an explicit algebraic stress relationship derived based on a three-basis approximation. It is given by:

$\tau _{ij} = f_2 \tau _{ij}^{(ARSM)} + (1-f_2) \tau _{ij}^{(L)}$

with

$\tau _{ij}^{(ARSM)} = - \rho k\left( {{\beta _1}\;{T_{ij}^{(1)}} + {\beta _2}\;{T_{ij}^{(2)}} + {\beta _4}\;{T_{ij}^{(4)}} + \frac{2}{3}{\delta _{ij}}} \right)$

and $\tau _{ij}^{(L)}$ is the linear version of this model, with $C_{\mu}^*$ = 0.09 = constant and $\beta_2=\beta_4=0$ .

Here,

${T_{ij}^{(1)}} = \left[ {{S_{ij}^*} - \frac{1}{3}tr\left\{ {{S^*}} \right\}} \right]$

${T_{ij}^{(2)}} = \left[ {{S_{ik}^*S_{kj}^*} - \frac{1}{3}tr\left\{ {{S^{*2}}} \right\}} \right]$

${T_{ij}^{(4)}} = \left[ {{S_{ik}^*}{W_{kj}^*} - {W_{ik}^*}{S_{kj}^*}} \right]$

with tr{ } indicating the trace, and

${S_{ij}^*} = \tau S_{ij}$

${W_{ij}^*} = \tau W_{ij}$

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

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

$\tau = \frac{{C_\mu ^{1/4}}}{{{C_\mu }}}\frac{{(kL)}}{{{k^{3/2}}}}$

$\beta_1 = -2 C_{\mu}^* = 2 \alpha$

$\beta_2 = -2 a_4 a_3 \beta_1$

$\beta_4 = a_4 a_2 \beta_1$

The T_ij⁽¹⁾ is the linear part of the model, whereas T_ij⁽²⁾ and T_ij⁽⁴⁾ are nonlinear terms that model the anisotropy.

The two-equation model is given by the following:

$\frac{{\partial \rho k}}{{\partial t}} + \frac{{\partial \rho {u_j}k}}{{\partial {x_j}}} = {{P_k} + \frac{\partial }{{\partial {x_j}}}\left( {\left( {{\mu} + {\sigma _{k}}{\mu _t}} \right)\frac{{\partial \rho k}}{{\partial {x_j}}}} \right) - C_w{\mu}\frac{k}{{{d^2}}}} - {C_{k}}\rho \frac{{{k^{2.5}}}}{{(kL)}}$

$\frac{{\partial \rho (kL)}}{{\partial t}} + \frac{{\partial \rho {u_j}(kL)}}{{\partial {x_j}}} = {{C_{{(kL)}_{\scriptstyle 1}}}\frac{{(kL)}}{k}{P_{kL}} + \frac{\partial }{{\partial {x_j}}}\left( {\left( {{\mu} + {\sigma _{(kL)}}{\mu _t}} \right)\frac{{\partial \rho (kL)}}{{\partial {x_j}}}} \right) - 6{\mu}\frac{{(kL)}}{{{d^2}}}{f_{(kL)}}} - {C_{{(kL)}_{\scriptstyle 2}}}\rho {k^{1.5}}$

where

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

$P_{kL} = \max ({P_k},{\mu _t}^{(L)}{S^2})$

${\mu _t}^{(L)} = C_\mu ^{1/4}\frac{{\rho (kL)}}{{{k^{1/2}}}}$

The eddy viscosity from the model is given by:

${\mu _t} = \frac{{C_\mu ^*}}{{{C_\mu }}}C_\mu ^{1/4}\frac{{\rho (kL)}}{{{k^{1/2}}}}$

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

$\alpha^3 + p \alpha^2 + q \alpha + r = 0$

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 methodology for solving this equation is the same as for the EASMko2003 model. The algorithm for determining this root is 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:

$\tau \equiv 1 / \omega$

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

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

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

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

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

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

The functions are:

$C_{(kL)_1} = \left[ \zeta_1 - \zeta_2 \left( \frac{(kL)}{k L_{vk}} \right)^2 \right]$

$f_{(kL)} = \frac{1+C_{d1} \xi}{1 + \xi^4}$

where

$L_{vk} = \kappa \left| \frac{U'}{U''} \right|$

$U' = \sqrt{2 S_{ij} S_{ij}}$

$U'' = \sqrt{ \left(\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} + \frac{\partial^2u}{\partial z^2} \right)^2 + \left(\frac{\partial^2v}{\partial x^2} + \frac{\partial^2v}{\partial y^2} + \frac{\partial^2v}{\partial z^2} \right)^2 + \left(\frac{\partial^2w}{\partial x^2} + \frac{\partial^2w}{\partial y^2} + \frac{\partial^2w}{\partial z^2} \right)^2 }$

$\xi = \frac{\rho d \sqrt{0.3 k}}{20 \mu}$

and $\rho$ is the density, $\mu$ is the molecular dynamic viscosity, and d is the distance from the field point to the nearest wall. Note that if U' = U'' = 0, then the (kL) production term should be identically zero.