Improved discontinuity-capturing finite element techniques for reaction effects in turbulence computation

Recent advances in turbulence modeling brought more and more sophisticated turbulence closures (e.g. k-epsilon, k-epsilon-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 (Brook and Hughes, comput methods Appl Mech Eng 32:199-255, 1982; Hughes and Tezduyar, comput methods Appl Mech Eng 45: 217-284, 1984) 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 Tezduyar and Park (comput methods Appl Mech Eng 59:307-325, 1986) for that purpose and improves the SUPG performance. In recent years a new class of variational multi-scale (VMS) stabilization (Hughes, comput methods Appl Mech Eng 127:387-401, 1995) has been introduced, and this approach, in principle, can deal with advection-diffusion-reaction equations. However, it was pointed out in Hanke (comput methods Appl Mech Eng 191:2925-2947) 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 the 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 (Corsini et al. comput methods Appl Mech Eng 194:4797-4823, 2005). Model problems and industrial test cases are computed to show the potential of the proposed methods in simulation of turbulent flows.