Realizable k- ε Model

The Standard k−ɛ^[1] is a well-established model capable of resolving through the boundary layer.^[2] The second model is Realizable k−ɛ, an improvement over the standard k−ɛ model.^[3] It is a relatively recent development and differs from the standard k−ɛ model in two ways.The realizable k−ɛ model contains a new formulation for the turbulent viscosity and a new transport equation for the dissipation rate, ɛ, that is derived from an exact equation for the transport of the mean-square vorticity fluctuation. The term "realizable" means that the model satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of turbulent flows. Neither the standard k-ɛ model nor the RNG k-ɛ model is realizable. It introduces a Variable Cμ instead of constant. An immediate benefit of the realizable k-ɛ model is that it provides improved predictions for the spreading rate of both planar and round jets. It also exhibits superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation. In virtually every measure of comparison, Realizable k-ɛ demonstrates a superior ability to capture the mean flow of the complex structures.^[4]

Transport equations

$\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_j} (\rho k u_j) = \frac{\partial}{\partial x_j} \left [ \left(\mu + \frac{\mu_t}{\sigma_k}\right) \frac{\partial k} {\partial x_j} \right ] + P_k + P_b - \rho \epsilon - Y_M + S_k$

$\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_j} (\rho \epsilon u_j) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_{\epsilon}}\right) \frac{\partial \epsilon}{\partial x_j} \right ] + \rho \, C_1 S \epsilon - \rho \, C_2 \frac{{\epsilon}^2} {k + \sqrt{\nu \epsilon}} + C_{1 \epsilon}\frac{\epsilon}{k} C_{3 \epsilon} P_b + S_{\epsilon}$

Where

$C_1 = \max\left[0.43, \frac{\eta}{\eta + 5}\right] , \;\;\;\;\; \eta = S \frac{k}{\epsilon}, \;\;\;\;\; S =\sqrt{2 S_{ij} S_{ij}}$

In these equations, $P_k$ represents the generation of turbulence kinetic energy due to the mean velocity gradients, calculated in same manner as standard k-epsilon model. $P_b$ is the generation of turbulence kinetic energy due to buoyancy, calculated in same way as standard k-epsilon model.

Modelling turbulent viscosity

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

where
$C_{\mu} = \frac{1}{A_0 + A_s \frac{k U^*}{\epsilon}}$
$U^* \equiv \sqrt{S_{ij} S_{ij} + \tilde{\Omega}_{ij} \tilde{\Omega}_{ij}}$ ;
$\tilde{\Omega}_{ij} = \Omega_{ij} - 2 \epsilon_{ijk} \omega_k$ ;
$\Omega_{ij} = \overline{\Omega_{ij}} - \epsilon_{ijk} \omega_k$

where $\overline{\Omega_{ij}}$ is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity $\omega_k$. The model constants $A_0$ and $A_s$ are given by:
$A_0 = 4.04, \; \; A_s = \sqrt{6} \cos \phi$

$\phi = \frac{1}{3} \cos^{-1} (\sqrt{6} W), \; \; W = \frac{S_{ij} S_{jk} S_{ki}}{{\tilde{S}} ^3}, \; \; \tilde{S} = \sqrt{S_{ij} S_{ij}}, \; \; S_{ij} = \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right)$

Model constants

$C_{1 \epsilon} = 1.44, \;\; C_2 = 1.9, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.2$

The realizable model is unable to satisfactorily predict the radial velocity; it is also the most computationally-expensive model.^[5]

References

1. P. L. Davis, A. T. Rinehimer, and M.Uddin
2. D. C. Wilcox. Turbulence Modeling for CFD, 2nd ed. DCW Industries, 2006
3. T. H. Shih, W. W. Liou, A. Shabbir, Z. Yang, and J. Zhu. A New k-ε Eddy Viscosity Model for High Reynolds Number Turbulent Flows—Model Development and Validation. Computers Fluids. 24(3):227-238, 1995
4. http://www.cd-adapco.com/sites/default/files/technical_document/pdf/PRU_2012.pdf
5. Osama A. Marzouk, E. David Huckaby, Simulation of a Swirling Gas-Particle Flow Using Different k-epsilon Models and Particle-Parcel Relationships, Engineering Letters, 18:1, EL_18_1_07