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Turbulence Modeling Resource

The Chien k-epsilon Turbulence Model

This web page gives detailed information on the equations for various forms of the Chien k-epsilon turbulence model. 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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The first version listed (KE-Chien) is the original published version of this model.

Chien k-epsilon Two-Equation Model (KE-Chien)

The reference for this model is:

Chien, K.-Y., "Predictions of Channel and Boundary-Layer Flows with a Low-Reynolds-Number Turbulence Model," AIAA Journal, Vol. 20, No. 1, 1982, pp. 33-38, https://doi.org/10.2514/3.51043.

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

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

$\frac{\partial (\rho \epsilon)}{\partial t} + \frac{\partial (\rho u_j \epsilon)}{\partial x_j} = C_{\epsilon 1} f_1 \frac{\epsilon}{k} \cal P - C_{\epsilon 2} f_2 \frac{\rho \epsilon^2}{k} + \frac{\partial}{\partial x_j} \left[ \left( \mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + \rho L_{\epsilon}$

where

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

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

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

and the turbulent eddy viscosity is computed from:

$\mu_t = C_{\mu} f_{\mu} \frac{\rho k^2}{\epsilon}$

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

There are no specific farfield boundary conditions recommended for this model. At solid walls, the boundary conditions are:

$k_{wall} = 0$ $\epsilon_{wall} = 0$

The constants and auxiliary functions are:

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

$C_{\mu} = 0.09$ $C_{\epsilon 1} = 1.35$ $C_{\epsilon 2} = 1.80$

$f_1 = 1$ $f_2 = 1 - \frac{0.4}{1.8} e^{-Re_T^2/36}$

$Re_T = \frac{\rho k^2}{\mu \epsilon}$

$f_{\mu} = 1 - e^{-0.0115 d^+}$

with $d^+$ a non-local function of distance to the wall (in wall variables), and thus dependent on properties at the nearest wall location.

$d+ = d \rho u_{\tau} / \mu$

$u_{\tau} = \sqrt{\tau_w/\rho_w}$

$\tau_w = \mu_w \left( \frac{\partial U}{\partial n} \right)_w$

with U the velocity parallel to the wall, n the direction normal to the wall, and d the minimum distance to the wall.

$L_k = -2 \frac{\mu k}{\rho d^2}$

$L_{\epsilon} = -2 \frac{\mu \epsilon}{\rho d^2} e^{-d^+/2}$

Chien k-epsilon Two-Equation Model with Compressibility Correction (KE-Chien-comp)

A reference for this version of the model is:

Kaul, U. K., "Effect of Inflow Boundary Conditions on the Turbulence Solution in Internal Flows," AIAA Journal, Vol. 49, No. 2, pp. 426-432, https://doi.org/10.2514/1.J050532.
Kaul, U. K., "Effect of Inflow Boundary Conditions on the Solution of Transport Equations for Internal Flows," AIAA Paper 2010-4743, June-July 2010, https://doi.org/10.2514/6.2010-4743.

This version of the Chien k-epsilon model is the same as for (KE-Chien) with the exception that the destruction term in the k-equation has an additional compressibility correction:

$\frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho u_j k)}{\partial x_j} = \cal P - \rho \epsilon (1 + M_t^2) + \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_k} \right)\frac{\partial k}{\partial x_j}\right] + \rho L_k$

where $M_t = \sqrt{1.5 k / a^2}$ and a is the local speed of sound $a = sqrt{\gamma p / \rho}$ .

Other versions of the Chien k-epsilon model have also made use of a non-local curvature correction term dependent on the rotational Richardson number, which modifies the $C_{\epsilon 2}$ term in the epsilon equation. But because it is not possible to implement this particular curvature correction for general flows, it is not included here. The interested reader is referred to: Kaul, NASA CR-4141, 1989.

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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