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Improved discontinuity-capturing finite element techniques for reaction effects in turbulence computation. (English) Zbl 1177.76192

Summary: Recent advances in turbulence modeling brought more and more sophisticated turbulence closures (e.g., \(k-\varepsilon , k-\varepsilon -v ^{2}-f\), second moment closures), where the governing equations for the model parameters involve advection, diffusion and reaction terms. Numerical instabilities can be generated by the dominant advection or reaction terms. Classical stabilized formulations such as the Streamline-Upwind/Petrov-Galerkin (SUPG) formulation [A. N. Brook and T. J. R. Hughes, Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982; Zbl 0497.76041); T. J. R. Hughes and T. E. Tezduyar, Comput. Methods Appl. Mech. Eng. 45, 217–284 (1984; Zbl 0542.76093)] are very well suited for preventing the numerical instabilities generated by the dominant advection terms. A different stabilization however is needed for instabilities due to the dominant reaction terms. An additional stabilization term, called the diffusion for reaction-dominated (DRD) term, was introduced by T. E. Tezduyar and Y. J. Park [Comput. Methods Appl. Mech. Eng. 59, 307–325 (1986; Zbl 0593.76096)] for that purpose and improves the SUPG performance. In recent years, a new class of variational multi-scale (VMS) stabilization [T. J. R. Hughes, Comput. Methods Appl. Mech. Eng. 127, No. 1–4, 387–401 (1995; Zbl 0866.76044)] has been introduced, and this approach, in principle, can deal with advection-diffusion-reaction equations. However, it was pointed out in [G. Hauke, Comput. Methods Appl. Mech. Eng. 191, No. 27–28, 2925–2947 (2002; Zbl 1005.76057)] that this class of methods also need some improvement in the presence of high reaction rates.
In this work, we show the benefits of using the DRD operator to enhance the core stabilization techniques such as the SUPG and VMS formulations. We also propose a new operator called the DRDJ (DRD with local variation jump) term, targeting the reduction of numerical oscillations in the presence of both high reaction rates and sharp solution gradients. The methods are evaluated in the context of two stabilized methods: the classical SUPG formulation and a recently-developed VMS formulation called the V-SGS [A. Corsini et al. Comput. Methods Appl. Mech. Eng. 194, No. 45–47, 4797–4823 (2005; Zbl 1093.76032)]. Model problems and industrial test cases are computed to show the potential of the proposed methods in the simulation of turbulent flows.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76V05 Reaction effects in flows
76F99 Turbulence
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