Compact Schemes for Multiscale Flows with Cell-Centered Finite Difference Method
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Compact Schemes for Multiscale Flows with Cell-Centered Finite Difference Method Yao Jin1 · Fei Liao1
· Jinsheng Cai1
Received: 24 November 2019 / Revised: 11 July 2020 / Accepted: 11 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract High-order compact interpolation schemes appropriate for multiscale flows are studied within a cell-centered finite difference method (CCFDM) framework where the robustness of highorder schemes on curvilinear grids can be greatly enhanced due to the satisfaction of geometric conservation law. Two types of compact interpolations are mainly developed in this paper for shock-free flows and shock-embedded flows respectively. The present compact schemes are verified to be superior over the explicit counterparts with same orders in terms of the spectral characteristics. Regarding the shock-free flows, low-dissipation low-dispersion properties are achieved by the spectral optimization. Three optimized compact schemes (Opt4, Opt6 and Opt8) are further validated to be attractive for shock-free problems by carrying out benchmarks from computational aeroacoustics workshops and two typical turbulence cases: Tayler–Green vortex and decaying isotropic turbulence. Regarding high-speed flows in the presence of shock waves, the shock-capturing capability is realized by extending the weighting technique to the compact interpolations. The criteria to choose optimally compact nonlinear sub-stencils on a most general compact global stencil are presented. Interestingly, the explicit WENO-type schemes can be reverted within the proposed compact framework. Three nonlinear compact schemes (UI5, CI6 and CI8) on two practical stencils are analyzed and further compared with their explicit counterparts by a series of numerical experiments. The compact ones are superior to explicit ones in resolving rich flow structures as well as discontinuities. Keywords Multiscale flows · Cell-centered finite difference method · Compact scheme · Spectral optimization · Multi-stencil weighting technique · WENO interpolation
1 Introduction For flows characterized by a broad range of temporal and spatial scales, such as the broadband turbulence noise involving a broad range of energy-containing turbulent vortices and acoustic waves, the highly accurate numerical methods are in demand to resolve adequate scales of
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Fei Liao [email protected] School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China 0123456789().: V,-vol
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Journal of Scientific Computing
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flow structures. Over the past decades, there has been an intensive effort on the development of high-order methods in computational fluid dynamic community, referring the review articles [1–3]. Among all, finite difference method (FDM) on structured grid is widely used due to its high efficiency in retaining high-order accuracy for multidimensional problems. Among the available high-order finite difference schemes, compact schemes [4] possess several superiorities over
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