A High-Order Symmetric Interior Penalty Discontinuous Galerkin Scheme
to Simulate Vortex Dominated Incompressible Fluid Flow

Song Lunji; Song Lunji

doi:10.3934/Math.2016.1.43

AIMS Mathematics

2016, Volume 1, Issue 1: 43-63. doi: 10.3934/Math.2016.1.43

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Research article

A High-Order Symmetric Interior Penalty Discontinuous Galerkin Scheme to Simulate Vortex Dominated Incompressible Fluid Flow

Song Lunji ^{1,2
,
,}

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China
2.
Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487, USA
3.
Department of Aerospace Engineering and Mechanics, The University of Alabama, Tuscaloosa, AL 35487, USA

Received: 16 April 2016 Accepted: 25 April 2016 Published: 27 April 2016

A high-order Symmetric Interior Penalty discontinuous Galerkin (SIPG) method has been used for solving the incompressible Navier-Stokes equation. We apply the temporal splitting scheme in time and the SIPG discretization in space with the local Lax-Friedrichs flux for the discretization of nonlinear terms. A fully discrete semi-implicit splitting scheme has been presented and high-order discontinuous Galerkin (DG) finite elements are available. Under a constraint of the CFL condition, two benchmark problems in 2D are investigated: one is a lid-driven cavity flow to verify the high-order discontinuous Galerkin method is accurate and robust; the other is a flow past a circular cylinder, for which we mainly check the Strouhal numbers with the von Kármán vortex street, and also simulate the boundary layers with walls and corresponding dynamical behavior with Neumann conditions on the top and bottom boundaries, respectively. We predict the Strouhal number for the range of Reynolds number 50 ≤ Re ≤ 400, making a comparison between the predicted values by our numerical method and the referenced values from physical experiments.
- Navier-Stokes equations,
- von K´arm´an vortex street,
- discontinuous Galerkin method,
- interior penalty
Citation: Song Lunji. A High-Order Symmetric Interior Penalty Discontinuous Galerkin Schemeto Simulate Vortex Dominated Incompressible Fluid Flow[J]. AIMS Mathematics, 2016, 1(1): 43-63. doi: 10.3934/Math.2016.1.43

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Abstract

A high-order Symmetric Interior Penalty discontinuous Galerkin (SIPG) method has been used for solving the incompressible Navier-Stokes equation. We apply the temporal splitting scheme in time and the SIPG discretization in space with the local Lax-Friedrichs flux for the discretization of nonlinear terms. A fully discrete semi-implicit splitting scheme has been presented and high-order discontinuous Galerkin (DG) finite elements are available. Under a constraint of the CFL condition, two benchmark problems in 2D are investigated: one is a lid-driven cavity flow to verify the high-order discontinuous Galerkin method is accurate and robust; the other is a flow past a circular cylinder, for which we mainly check the Strouhal numbers with the von Kármán vortex street, and also simulate the boundary layers with walls and corresponding dynamical behavior with Neumann conditions on the top and bottom boundaries, respectively. We predict the Strouhal number for the range of Reynolds number 50 ≤ Re ≤ 400, making a comparison between the predicted values by our numerical method and the referenced values from physical experiments.

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