A robust near-wall elliptic-relaxation eddy-viscosity turbulence model for CFD
Introduction
The –f model of Durbin (1991) appeared as an interesting novelty in engineering turbulence modelling. By introducing an additional (“wall-normal”) velocity scale and an elliptic relaxation concept to sensitize to the inviscid wall blocking effect, the model dispenses with the conventional practice of introducing empirical damping functions. Because of its physical rationale and of its simplicity, it is gaining in popularity and appeal especially among industrial users. Whilst in complex three-dimensional flows, with strong secondary circulation, rotation and swirl, where the evolution of the complete stress field may be essential for proper reproduction of flow features the model remains still inferior to second-moment and advanced non-linear eddy viscosity models, it is certainly a much better option than the conventional near-wall k–ε and similar models.
However, the original –f model possesses some features that impair its computational efficiencies. The main problem is with the wall boundary condition for f, i.e. , which makes the computations sensitive to the near-wall grid clustering and—contrary to most other near-wall models—does not tolerate too small y+ for the first near-wall grid point. The problem can be obviated by solving simultaneously the and f equations, but most commercial as well as in-house codes use more convenient segregated solvers. Alternative formulations of the and f equations have been proposed which permit fw = 0 (Lien et al., 1998), but these perform less satisfactory than the original model and require some re-tuning of the coefficients.
We propose a version of eddy-viscosity model based on Durbin’s elliptic relaxation concept, which solves a transport equation for the velocity scales ratio instead of the equation for . The motivation behind this development originated from the desire to improve the numerical stability of the model, especially when using segregated solvers. Because of a more convenient formulation of the equation for ζ and especially of the wall boundary condition for the elliptic function f, it is more robust and less sensitive to nonuniformities and clustering of the computational grid. Another novelty is the application of a quasi-linear pressure–strain model in the f-equation, based on the formulation of Speziale et al. (1991) (SSG), which brings additional improvements for non-equilibrium wall flows. The computations of flow and heat transfer in a plane channel, behind a backward facing step and in a round impinging jet show in all cases satisfactory agreement with experiments and direct numerical simulations.1
Section snippets
The ζ–f model
The ζ equation can be derived directly from the and k equations of Durbin (1991). The direct transformation yields:where the “cross diffusion” X is a consequence of transformation and can be written in a condensed form as:The solution of the ζ equation (1) instead of should produce the same results. However, from the computational point of view, two advantages can be identified:
- •
instead of ε appearing in the equation, which is difficult
The ζ–f model with quasi-linear pressure–strain term
Instead of using the simple linear IP model for the rapid part of the pressure–strain term as practiced in the conventional –f model, we can adopt the more advanced quasi-linear model of Speziale et al. (1991):which was found to capture better the stress anisotropy in wall boundary layers. Application to the wall normal stress component, with yields the following form of the f equation in conjunction with the ζ
The ζ–f model with fw = 0
One can make further simplifications to satisfy zero wall boundary condition for fw (in analogy with the original Jones and Launder (1972) formulation of the low-Re-number dissipation equation) by solving Eq. (9) but for with and getting f from:which is then used in ζ equation. The second term on the right of Eq. (13) is just an alternative for 2νζ/y2, which follows closer the polynomial expansion of ζ around y = 0.
Some illustrations
As an illustration of performance of the ζ–f model, we present some results of computation of velocity fields and heat transfer in three common test cases: a plane channel flow, a separating flow behind a backward facing step and in a round impinging jet. The temperature field was obtained by solving the RANS energy equation with constant fluid properties, using the isotropic eddy diffusivity νt/σT where νt is given by Eq. (6) and σT = 0.9.
In Fig. 1 profiles of velocity and turbulent quantities
Acknowledgments
This work emerged from the MinNOx project sponsored by the Commission of the European Community, Contract ENK6-CT-2001-00530.
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