where σϵ can be seen as a turbulent-Prandtl number (similar to σk in the TKE equation). Eventually, the modeled equation for the dissipation rate ϵ is obtained.
Remember that the eddy viscosity depended on a characteristic length for the large scales. Similarly to Eq.(5↑), this length scale can be obtained by l ∝ k3 ⁄ 2 ⁄ ϵ, which yields
There are other ways to obtain the eddy viscosity using the turbulent kinetic energy equation and other variables. Some well-known examples are:
the k − l model, where l = k3 ⁄ 2 ⁄ ϵ is the characteristic length scale.
the k − ω model (Wilcox - 1988 [5]), where ω = ϵ ⁄ k is the specific dissipation rate.
the k − τ model, where τ = k ⁄ ϵ is some characteristic time scale.
The second partial differential equations are typically similar, but the modeling differs depending on the method. In addition, the closure coefficients and damping functions vary, as well as the boundary conditions applied to the equations.
Accuracy
Two-equation models perform well for simple flows such as boundary layers in favorable pressure gradients. In this case, the velocity profile and friction coefficient are typically predicted within 5% of the experimental value. The models provide accurate estimates for pipe flows, but are less reliable for free shear flows (in Wilcox (2006), the spreading rate of a far wake is predicted with 30% error by the k − ϵ model [6]). As the flow becomes more complex, the k − ω model seems to provide more accurate predictions than the k − ϵ model. However, the k − ω model suffers from an increased sensitivity to its boundary conditions. In order to tackle this problem, Menter (1992) developed the k − ω SST model (for Shear Stress Transport), which combines the best of each model [3]. A function is implemented to transition from k − ω at the wall (improved predictions) to k − ϵ in the free-stream (reduced boundary conditions sensitivity).
Realizability
It is also interesting to notice that by definition, k and ϵ are positive quantities. An accurate solution of the equations should provide positive values for these parameters. However, these constraints are not explicitly described in the equations. It might therefore happen that with numerical errors, these parameters may be small negative values which have no physical meaning. This introduces the concept of realizability, ie. are the results actually feasible? An additional numerical constraint on realizability such as k ≥ 0 may be imposed during the implementation of these models. The standard models have sometime been improved to account for this problem (k − ϵ Realizable model proposed by Shih (1995) [4]).
References
[1] W.P. Jones and B. Launder. The prediction of laminarization with a two-equation model of turbulence. International Journal of Heat and Mass Transfer, 15(2):301--314, 1972.
[2] A.N. Kolmogorov. Dissipation of energy in locally isotropic turbulence. In Dokl. Akad. Nauk SSSR, volume 32, pages 16--18, 1941.
[3] F.R. Menter. Improved two-equation k-omega turbulence models for aerodynamic flows. NASA STI/Recon Technical Report N, 93:22809, 1992.
[4] T. Shih, W.W. Liou, A. Shabbir, Z. Yang, and J. Zhu. A new k-epsilon eddy viscosity model for high reynolds number turbulent flows. Computers & Fluids, 24(3):227--238, 1995.
[5] D.C. Wilcox. Reassessment of the scale-determining equation for advanced turbulence models. AIAA journal, 26(11):1299--1310, 1988.
[6] D.C. Wilcox. Turbulence modeling for CFD, volume 3. DCW industries La Cañada, 2006.
