Turbulence Modeling Resource

The Generalized K-Omega (GEKO) Turbulence Model

This web page gives detailed information on the equations for various forms of the Generalized $k$-$\omega$ (GEKO) turbulence model. The reference for the standard implementation of the GEKO model by Menter and Matyushenko is:

Menter, F. R., & Matyushenko, A., "Generalized k-$\omega$ (GEKO) Two-Equation Turbulence Model," AIAA Journal Vol. 63, No. 11, 2025, pp. 4590-4606. https://doi.org/10.2514/1.J065393

Several options are discussed for the constitutive relationship in the paper. The standard implementation uses the Boussinesq hypothesis, also known as a linear eddy-viscosity model. Including all terms, the Boussinesq hypothesis gives the following constitutive relationship:

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

Menter and Matyushenko recommend a simplified constitutive relationship for compressible flows:

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

And the following version is suggested for incompressible flows:

$$ \tau_{ij} = 2 \mu_t S_{ij} $$

The term $ 1/3 \partial u_k / \partial x_k \delta_{ij} $ should be zero for incompressible flows, and is omitted. The standard version of GEKO omits the isotropic portion, $2/3 \rho k \delta_{ij}$, for both compressible and incompressible flows.

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.

Generalized K-Omega Model (GEKO)

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

$$ \begin{aligned} \frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho u_j k)}{\partial x_j} &= \tilde{P}_k - C_\mu \rho k \omega + \frac{\partial}{\partial x_j}\left[ \left(\mu + \frac{\tilde \mu_t}{\sigma_k}\right)\frac{\partial k}{\partial x_j}\right] \\ \frac{\partial (\rho \omega)}{\partial t} + \frac{\partial (\rho u_j \omega)}{\partial x_j} &= C_{\omega 1}P_\omega - C_{\omega2} F_\mathrm{MIX}\rho \omega^2 + F_\mathrm{NW} CD + \frac{\partial}{\partial x_j}\left[ \left(\mu + \frac{\tilde \mu_t}{\sigma_k}\right)\frac{\partial \omega}{\partial x_j}\right] \\ \end{aligned} $$

where:

$$ P_k = \mu_t S^2 $$ $$ \tilde P_k = \min(P_k, C_{PK} C_\mu \rho k \omega) $$ $$ P_\omega = \frac{\rho \tilde{P}_k}{\tilde{\mu}_t + 0.001\mu} $$ $$ CD = \rho \frac{2}{\sigma_\omega}\frac{1}{\omega}\frac{\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j} $$

and $ S = \sqrt{2 S_{ij} S_{ij}} $, with the rate-of-strain tensor defined as:

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

and the turbulent eddy viscosity is computed from:

$$ \mu_t = \tilde{\mu}_t = \frac{\rho k}{\max(\omega, S/C_R)} $$

The maximum of $S / C_R $ is a realizability limit. Both $\mu_t$ and $\tilde{\mu}_t$ are used to denote the eddy viscosity and are identical in the standard model. These two eddy-viscosities only differ when submodels are used that alter the eddy viscosity definition, such as explicit algebraic Reynolds stress models (EARSM).

Menter and Matyushenko also recommend a production limiter similar to the production limiter in SST. This is accomplished by replacing the unlimited production, $ P_k $, with a limited version, $ \tilde{P}_k $, in the k- and omega-equations. While the original reference does not mention production limiters on $ P_\omega $, the authors have clarified that the same limited production $ \tilde{P}_k $ was used in both equations, as shown in the equations here (ref: private communication with the authors). The standard version of GEKO uses both the realizability limit on the eddy viscosity and the production limiter.

The GEKO model relies on several functions, including $F_{MIX}$, $F_{NW}$, $F_{JET}$, $F_{SEP}$, and $F_{Blend}$.

Similar to SST, there is a blending function $F_{Blend} $ which is equal to 1 close to the wall but equal to 0 in shear layers away from the walls:

$$ \begin{aligned} L_T &= \frac{\sqrt{\tilde k}}{C_\mu \omega} \\ \tilde k &= \max(k, CFb_{Lam}\nu\omega) \\ x_{blend} &= CFb_{Turb} L_T / y \\ F_{Blend} &= \tanh(x_{blend}^4) \end{aligned} $$

The $F_{SEP}$ function is defined as:

$$ F_{SEP} = 1 + f_D \psi$$ $$ \psi = C_\psi (C_{SEP} - 1)F_{Blend}$$ $$f_D = \frac{1}{1 + (\tilde y^+ / A^+)^2}$$ $$C_\psi = 0.2454 C_{sep}^{-0.803}$$ $$\tilde y^+ = \frac{1}{\kappa}\frac{k/\omega}{\nu}$$

The $F_{NW} $ function is defined as:

$$F_{NW} = \begin{cases} 1 &\text{ if }CD > 0,\\ [C_{NW} - (C_{c1} + C_{c2}C_{NW})f_D]F_{Lim} &\text{else} \end{cases}$$

where the $F_{Lim} $ function is defined as:

$$ F_{Lim} = \frac{D_\omega}{\max(D_\omega, 2CD)}$$ $$ D_\omega = C_{\omega 2} F_{MIX} \rho \omega^2 $$

The $F_{MIX} $ function is defined as:

$$ F_{MIX} = F_{SEP} + [C_{MIX} + 0.13 C_{JET}(F_{JET} - 1)](1 - F_{Blend})$$

The $F_{JET} $ function is defined as: $$x_{jet} = 4\left(\frac{Soo - 0.9}{0.15}-\frac{1}{2}\right)$$ $$ Soo = \frac{\min(S,\Omega,\omega)}{0.3\omega}$$ $$ F_{JET} = 1/2(1 + \tanh(x_{jet}))$$ where $\Omega = \sqrt{2 \Omega_{ij} \Omega_{ij}}$ is the voriticity magnitude and $\Omega_{ij}$ is the rate-of-rotation tensor: $$ \Omega_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i}\right) $$ The diffusion constants $\sigma_k$ and $\sigma_\omega$ are also blended using the $F_{Blend}$ function:

$$ \begin{aligned} \sigma_k &= \tilde \sigma_k \left[ 1 + 0.25(C_{SEP} - 1)F_{Blend}\right] \\ \sigma_\omega &= \tilde \sigma_\omega \left[1 + (C_{SEP} - 1)F_{Blend}\right] \end{aligned} $$

Note that the $C_{\omega1}$ coefficient was chosen in order to yield a log-law velocity profile via the expression:

$$ C_{\omega1} = \frac{1}{C_\mu}\left(C_{\omega 2} - \frac{\kappa^2 C_\mu^{1/2}}{\sigma_\omega}\right) $$ The other non-tunable coefficients are prescribed as: $$ \begin{aligned} C_\mu &= 0.09 &C_{\omega2} &= 0.083 \\ \tilde \sigma_k &= 1.0 &\tilde \sigma_\omega &= 1.17 \\ C_{c1} &= 1.7 &C_{c2} &= 1.4 \\ CFb_{Turb} &= 2.0 &CFb_{Lam} &= 1.0 \\ \end{aligned} $$ $$ \begin{aligned} C_{PK} &= 10 \\ C_R &= 1 / \sqrt{3} \approx 0.557 \\ \kappa &= 0.41 \\ A^+ &= 15 \end{aligned} $$

Menter and Matyushenko recommend $CFb_{Lam}=1$ but state that $CFb_{Lam}$ can be increased to 25 in transition simulations to shield a laminar boundary layer.

The main difference between GEKO and other k-omega models such as SST is the introduction of free parameters, which can be tuned by the user. These parameters and their sensible ranges are:

$$ \begin{aligned} 0.7 &\le C_{SEP}&\le 2.5 \\ -2.0 &\le C_{NW}&\le 2.0 \\ -0.2 &\le C_{MIX}&\le 1.0 \\ 0.0 &\le C_{JET}&\le 1.2 \\ \end{aligned} $$

Menter and Matyushenko provide the following default values:

Parameter	$C_{SEP}$	$C_{NW}$	$C_{MIX}$	$C_{JET}$
Default	1.75	0.5	0.0	1.0
GEKO $k-\varepsilon$	1.00	1.0	0.0	0.0

To avoid confusion and facilitate comparisons, the default values should be used unless systematic differences are observed relative to reliable data. There is extensive discussion in the original reference on the purpose and the proper calibration of each tunable coefficient.

Menter and Matyushenko prescribe a naming convention to facilitate sharing of results. The model name should be followed by the constants $(C_{SEP}, C_{NW}, C_{MIX}, C_{JET})$. Under this convention, the standard model is then GEKO (1.75, 0.5, 0.0, 1.0). The "exact" $k$$\varepsilon$ model would be GEKO (1.0, 1.0, 0.0, 0.0). Since many applications only change the $C_{SEP}$ coefficient, a simplified notation can be used with GEKO-($C_{SEP}$), where $C_{SEP}$ is the specified value. For example, the standard model would be GEKO-1.75 while a model with $C_{SEP} = 1$ and all other constants untouched would be GEKO-1.0.

The GEKO model can be used with the same boundary conditions as the SST model. The authors also state that for meshes with a first cell-height of $y^+ < 2$, the value of $\omega$ at the first cell center can be imposed as:

$$ \omega_1 = C_{\mathrm{wall}} \frac{6 \nu}{C_{\omega2} y_1^2} $$

where $C_{\mathrm{wall}}$ is a constant that is dependent on the code and numerics. The SST model used a similar boundary condition with a value of $ C_{\mathrm{wall}} = 10$. There are several other boundary conditions which are described in more detail in Appendix B of the paper by Menter and Matyushenko.

Generalized K-Omega Model with Kato-Launder Source Term (GEKO-KL)

Menter and Matyushenko state that the Kato-Launder correction can be used to suppress spurious production; this modification is not the default because it can have a noticeable effect on flows with rotation and swirl relative to the original model calibration. This correction was originally published in:

Kato, M. and Launder, B. E., "The Modelling of Turbulent Flow Around Stationary and Vibrating Square Cylinders," 9th Symposium on Turbulent Shear Flows, Kyoto, Japan, August 1993, paper 10-4

The Kato-Launder correction models production as $P_k = \mu_t S \Omega$ instead of $P_k = \mu_t S^2$ or a more complete production model. Implementation with the Kato-Launder correction should be called (GEKO-KL).

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Page Curators: Ethan Vogel, Clark Pederson
Last Updated: 2/26/2026