k-ε model, ASM model

The Reynolds stress transport models probably correspond to the optimal level of closure for practical applications because they seem to embody potentials providing a sufficient level of universality to be applied to a wide range of turbulent shear flows. However, they have not always been sufficiently tested, their development is still in progress and they are sometimes cumbersome for numerical treatment.

Two equation models

The ε equation usually does not need secondary diffusion sources and this is one of the reasons for which the k-εmodel may be preferred.

k-epsilon model

The numerical values of constants recommended by Launder B The numerical values of constants recommended by Launder B.E are the following:

The k-ε model generally gives good results in simple flows as far as the mean velocities and energies are concerned, but it cannot predict satisfactorily the specific characteristics of complex flows (recirculation regions, secondary flows, etc.). The round jet anomaly: In two-dimensional thin shear flows, the shear stress is the only dominating term in the mean momentum equations and consequently the gradient hypothesis with eddy viscosity may be acceptable even if the normal stresses are poorly predicted. But difficulties may remain also for relatively apparently simple flows. So, using the numerical values of the model constants given previously, the spreading rate of a turbulent plane jet is correctly predicted but the spreading rate of a circular jet is overestimated by about 30%. The same problem appears, in the case of Reynolds stress transport models, even more acute. The reason for this seems to come from the ε equation, and Rodi W has suggested an empirical modification.

Rodi’s Modification

Pope’s modification

Hanjalic and Launder’s modification

Non-equilibrium problem The constant cµ has been determined by reference to equilibrium flows in which P = ε. When this is not the case (for example in flows with weak production of turbulence kinetic energy such as wakes), this constant must be modified in order to get a good agreement between prediction and experiments. Rodi W. proposes an empirical relation cµ (P /ε ) deduced from experimental data, in which P /ε is the mean value of the ratio production/dissipation in a cross-section of the flow.

This modification improves the numerical prediction of turbulent wakes.

Realizable k-epsilon model

This formulation also includes the effect of solid rotation.

In this equation, the sink term has been chosen in order to recover at high Reynolds numbers

The k-ε RNG models This method also allows us to complement them on the one hand by explicitly calculating the numerical constants and on the other hand by introducing new terms.

Wilcox model It is a two equation model based on a variable different from ε for calculating the length scale

Transport of a passive scalar in two equation models

Algebraic Stress Modeling These are economical methods for the numerical prediction of the Reynolds stresses and the turbulent fluxes of a passive scalar. If the transport terms (convection and diffusion) in the modeled transport equations of the Reynolds stresses are expressed without involving the Reynolds stress gradients, we then get algebraic relations which can be used to determine the stress components Rij . The same is true for the Fγj equations.

Algebraic modeling of passive scalars In general, the Reynolds stresses and the turbulent heat fluxes are thus obtained by solving a linear system numerically. This implicit system is coupled with the transport equations for k and 􀁈. These models are also called implicit algebraic models.