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Turbulence Modeling Resource

K-e-Rt Turbulence Model

This web page gives detailed information on the equations for the three-equation k-e-Rt turbulence closure. All forms of the model given on this page are linear eddy viscosity models. Linear models use the Boussinesq assumption for the constitutive relation:

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

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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K-e-Rt Model (k-e-Rt)

This model's reference is:

Goldberg, U., Peroomian, O., Batten, P., Chakravarthy, S., "The k-e-Rt Turbulence Closure," Engineering Applications of Computational Fluid Mechanics, Vol. 3, No. 2, 2009, pp. 175-183, https://doi.org/10.1080/19942060.2009.11015263.

Two earlier related references are:

Goldberg, U., Peroomian, O., Chakravarthy, S., "A Wall-Distance-Free k-e Model with Enhanced Near-Wall Treatment," ASME Journal of Fluids Engineering, Vol. 120, No. 3, 1998, pp. 457-462, https://doi.org/10.1115/1.2820684.
Goldberg, U., "Turbulence Closure with a Topography-Parameter-Free Single Equation Model," International Journal of CFD, Vol. 17, No. 1, 2003, pp. 27-38, https://doi.org/10.1080/1061856031000083459.

This three-equation model (written in conservation form) is given by the following:

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

$\frac{\partial (\rho \varepsilon)}{\partial t} + \frac{\partial (\rho u_j \varepsilon)}{\partial x_j} = \frac{1}{T_t} \left( C_{\varepsilon 1} \cal P - C_{\varepsilon 2} \rho \varepsilon + C_{\varepsilon 3} E \right) + \frac{\partial}{\partial x_j} \left[ \left( \mu + \frac{\mu_t}{\sigma_{\varepsilon}} \right) \frac{\partial \varepsilon}{\partial x_j} \right]$

$\frac{\partial (\rho R_t)}{\partial t} + \frac{\partial (\rho u_j R_t)}{\partial x_j} = \left( C_1 - C_2f_2 \right) \sqrt{\rho R_t \cal P} -\rho C_3 D + \frac{\partial}{\partial x_j} \left[ \left( \mu + \frac{\mu_t}{\sigma_{Rt}} \right) \frac{\partial R_t}{\partial x_j} \right]$

where k is the turbulence kinetic energy, $\varepsilon$ is the turbulence kinetic energy dissipation rate, and $R_t$ is an undamped pseudo-eddy viscosity.

The production term is:

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

with

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

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

The extra source term in the $\varepsilon$ equation is:

$E = \rho \sqrt{\varepsilon T_t} {\rm max}\left( \sqrt{k}, (\nu \varepsilon)^{1/4} \right) {\rm max} \left( \frac{\partial k}{\partial x_j} \frac{\partial \tau}{\partial x_j},0 \right)$

The large eddy time scale is:

$\tau = \frac{k}{\varepsilon}$

and the realizable time scale, which reduces to the corresponding Kolmogorov scale for small (dissipative) eddies, is:

$T_t = \tau \left[ {\rm max} \left( 1, \sqrt{\frac{2}{Re_t}} \right) \right]$

The turbulence Reynolds number $Re_t$ (different from the transported pseudo-eddy viscosity $R_t$ ) is:

$Re_t = \frac{k \tau}{\nu}$

The damping function is given by:

$f_2 = \frac{{\rm tanh}\sqrt{\chi}}{{\rm tanh}\left(\sqrt{2} C_{\mu}^{1/4} \chi\right)}$

with

$\chi = \frac{R_t}{C_{\mu} \nu}$

Far from walls, $f_2 = 1$ . Near walls, it grows to very large values.

The sink term D is given by:

$D = \left\{ \begin{array}{l l} \left(\partial R_t / \partial x_j\right) \left(\partial R_t / \partial x_j\right), & \lambda>0 \\ 0, & \lambda \leq 0 \end{array}\right.$

where

$\lambda = \frac{\partial Q}{\partial x_j} \frac{\partial R_t}{\partial x_j}$

$Q = \sqrt{(U-U_0)^2 + (V-V_0)^2 + (W-W_0)^2}$

where (U₀, V₀, W₀) represents the velocity vector of the frame-of-reference. The D term is active only in the immediate vicinity of walls; further away, it vanishes. In the present model the following damping functions are adopted:

$f_{\mu,k-\varepsilon} = \frac{1-{\rm exp}(-A_{\mu}Re_t)}{1-{\rm exp}(-\sqrt{Re_t})} {\rm max}\left( 1, \sqrt{2/Re_t} \right)$

$f_{\mu,Rt} = \frac{{\rm tanh}(\alpha \chi^2)} {{\rm tanh}(\beta \chi^2)}$

The eddy viscosity is given by:

$\mu_{t,k-\varepsilon} = {\rm min} \left( C_{\mu} f_{\mu,k-\varepsilon} \rho k \tau, \frac{\phi \rho k}{|S|} \right)$

$\mu_{t,Rt} = f_{\mu,Rt} \rho R_t$

$\mu_t = {\rm max} \left( \mu_{t,k-\varepsilon},\mu_{t,Rt} \right)$

where $|S|$ is the mean strain magnitude. There are two choices for the $\phi$ parameter: $\phi = 2/3$ (for the Schwartz or "weak" realizability) and $\phi = 0.31$ (for the Bradshaw or "strong" realizability). The former is used in general, and is considered the standard for this model. However, the latter is usually recommended for high speed flows (e.g., transonic and higher) as well as for impinging flows at all speeds. The particular $\phi$ value used should always be reported.

The closure coefficients are:

$\sigma_k = 1.0$ $\sigma_{\varepsilon} = 1.3$

$\sigma_{Rt} = 1.0$ $C_{\varepsilon 1} = 1.44$

$C_{\varepsilon 2} = 1.92$ $C_{\varepsilon 3} = 0.3$

$A_{\mu} = 0.007$ $\alpha = 0.00015$

$\beta = 0.2$ $C_1 = C_2 + \kappa^2(C_3 - 1/\sigma_{Rt}) = 39.918$

$C_2 = (12/11) \sqrt{\beta/\alpha} = 39.834$ $C_3 = 3/(2 \sigma_{Rt}) = 1.5$

There are no specific farfield boundary conditions recommended for this model. Generally they are set based upon the desired freestream turbulence intensity and turbulence length scale. (In this model, the turbulence does not decay in the freestream.)

At the wall, the $k$ and $R_t$ boundary conditions are simple Dirichlet, and $\varepsilon$ is finite:

$k_w = 0$

$R_{t,w} = 0$

$\varepsilon_w = 2 \nu \left(\frac{\partial \sqrt{k}}{\partial n} \right)_w^2$

where n is the wall-normal coordinate direction.

K-e-Rt Model with Rotation/Curvature Correction (k-e-Rt-RC)

This model applies the RC correction of Shur, M. L., Strelets, M. K., Travin, A. K., Spalart, P. R., "Turbulence Modeling in Rotating and Curved Channels: Assessing the Spalart-Shur Correction," AIAA Journal Vol. 38, No. 5, 2000, pp. 784-792, https://doi.org/10.2514/2.1058.

There is no formal reference for this model other than the CFD++ User's Manual. However, the change is simply to multiply the production term of the $R_t$ equation in (k-e-Rt) by the $f_{r1}$ term from the above reference as follows:

$f_{r1} \left( C_1 - C_2f_2 \right) \sqrt{\rho R_t \cal P}$

where $f_{r1}$ is described on the SA-RC page.

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Recent significant updates:
6/30/2015 - mention Pr, Pr_t, and Sutherland's law

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Last Updated: 11/08/2021