Adaptive clipping‐and‐redistribution algorithms for bounded and conservative high‐order interpolations applied to discontinuous and reactive flows. (9th December 2022) - Overton‐Katz, Nathaniel; Gao, Xinfeng; Johansen, Hans; Guzik, Stephen M.
Abstract: A new adaptive clipping‐and‐redistribution method is presented which provides bounds‐preservation for multidimensional interpolation in the context of high‐order finite‐volume discretizations with adaptive mesh refinement (AMR). The underlying finite‐volume method (FVM) for the computational fluid dynamics applications is fourth‐order accurate for smooth solutions and utilizes AMR for computational efficiency in solving multiscale problems involving turbulence and combustion. High‐order interpolation between different AMR levels is required. However, this operation often leads to numerical issues because combustion species must have physical bounds preserved. The present study overcomes two major challenges in the development of the high‐order interpolation method. First, the method needs to be bound‐preserving near extrema or discontinuities to prevent the emergence of unphysical oscillations while maintaining fourth‐order accuracy in smooth flows. Second, the method needs to satisfy the conservation requirement in multiple dimensions, particularly in the context of curvilinear coordinate transformations. Additionally, the method is designed to be localized and computationally inexpensive. The new interpolation scheme is demonstrated by solving reacting flows, which are extremely sensitive to unphysical overshoots in conserved quantities. The test problems are shock‐induced H 2 H_2 ‐ O 2 O_2 combustion and a C 3 H 8 C_3H_8 ‐air flame in a practical bluff‐body combustor. Results show the method prevents new extrema near discontinuities while maintaining high‐order accuracy in smooth regions. In particular, the method is extremely beneficial for combustion with stiff chemistry. With the proposed new method, even if flame fronts cross AMR interfaces or new grids are created in the vicinity of the flame, solution stability is retained. Abstract : For conservative finite‐volume methods, adaptive mesh refinement requires high‐degree polynomial interpolation, which is prone to oscillations around nonsmooth features. We develop the HO‐ACR method and demonstrate that it successfully prevents oscillations while maintaining conservation, solution‐dependent bounds‐preservation, and higher‐order accuracy in smooth regions. This can be demonstrated with a "top‐hat" profile of mass fraction, shown after it has been refined and then advected for 100 steps. Although all four methods are conservative, the base AMR and coarse grid methods both have large, persistent overshoots while the HO‐ACR method prevents these effects and closely resembles the more expensive fine grid‐only solution.