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Volume 32, Issue 3
A High-Order Direct Discontinuous Galerkin Method for Variable Density Incompressible Flows

Fan Zhang & Tiegang Liu

Commun. Comput. Phys., 32 (2022), pp. 850-877.

Published online: 2022-09

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  • Abstract

In this work, we develop a novel high-order discontinuous Galerkin (DG) method for solving the incompressible Navier-Stokes equations with variable density. The incompressibility constraint at cell interfaces is relaxed by an artificial compressibility term. Then, since the hyperbolic nature of the governing equations is recovered, the simple and robust Harten-Lax-van Leer (HLL) flux is applied to discrete the inviscid term of the variable density incompressible Navier-Stokes equations. The viscous term is discretized by the direct DG (DDG) method, the construction of which was initially inspired by the weak solution of a scalar diffusion equation. In addition, in order to eliminate the spurious oscillations around sharp density gradients, a local slope limiting operator is also applied during the highly stratified flow simulations. The convergence property and performance of the present high-order DDG method are well demonstrated by several benchmark and challenging numerical test cases. Due to its advantages of simplicity and robustness in implementation, the present method offers an effective approach for simulating the variable density incompressible flows.

  • AMS Subject Headings

35Lxx, 65Mxx, 65Nxx

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COPYRIGHT: © Global Science Press

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@Article{CiCP-32-850, author = {Zhang , Fan and Liu , Tiegang}, title = {A High-Order Direct Discontinuous Galerkin Method for Variable Density Incompressible Flows}, journal = {Communications in Computational Physics}, year = {2022}, volume = {32}, number = {3}, pages = {850--877}, abstract = {

In this work, we develop a novel high-order discontinuous Galerkin (DG) method for solving the incompressible Navier-Stokes equations with variable density. The incompressibility constraint at cell interfaces is relaxed by an artificial compressibility term. Then, since the hyperbolic nature of the governing equations is recovered, the simple and robust Harten-Lax-van Leer (HLL) flux is applied to discrete the inviscid term of the variable density incompressible Navier-Stokes equations. The viscous term is discretized by the direct DG (DDG) method, the construction of which was initially inspired by the weak solution of a scalar diffusion equation. In addition, in order to eliminate the spurious oscillations around sharp density gradients, a local slope limiting operator is also applied during the highly stratified flow simulations. The convergence property and performance of the present high-order DDG method are well demonstrated by several benchmark and challenging numerical test cases. Due to its advantages of simplicity and robustness in implementation, the present method offers an effective approach for simulating the variable density incompressible flows.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0064}, url = {http://global-sci.org/intro/article_detail/cicp/21048.html} }
TY - JOUR T1 - A High-Order Direct Discontinuous Galerkin Method for Variable Density Incompressible Flows AU - Zhang , Fan AU - Liu , Tiegang JO - Communications in Computational Physics VL - 3 SP - 850 EP - 877 PY - 2022 DA - 2022/09 SN - 32 DO - http://doi.org/10.4208/cicp.OA-2022-0064 UR - https://global-sci.org/intro/article_detail/cicp/21048.html KW - Variable density incompressible flows, direct discontinuous Galerkin method, artificial compressibility, high-order accuracy. AB -

In this work, we develop a novel high-order discontinuous Galerkin (DG) method for solving the incompressible Navier-Stokes equations with variable density. The incompressibility constraint at cell interfaces is relaxed by an artificial compressibility term. Then, since the hyperbolic nature of the governing equations is recovered, the simple and robust Harten-Lax-van Leer (HLL) flux is applied to discrete the inviscid term of the variable density incompressible Navier-Stokes equations. The viscous term is discretized by the direct DG (DDG) method, the construction of which was initially inspired by the weak solution of a scalar diffusion equation. In addition, in order to eliminate the spurious oscillations around sharp density gradients, a local slope limiting operator is also applied during the highly stratified flow simulations. The convergence property and performance of the present high-order DDG method are well demonstrated by several benchmark and challenging numerical test cases. Due to its advantages of simplicity and robustness in implementation, the present method offers an effective approach for simulating the variable density incompressible flows.

Fan Zhang & Tiegang Liu. (2022). A High-Order Direct Discontinuous Galerkin Method for Variable Density Incompressible Flows. Communications in Computational Physics. 32 (3). 850-877. doi:10.4208/cicp.OA-2022-0064
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