Volume 28, Issue 4
High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions

Liang PanKun Xu

Commun. Comput. Phys., 28 (2020), pp. 1321-1351.

Published online: 2020-08

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  • Abstract

The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flows with Cartesian meshes. To study the flow problems in general geometries, such as the flow over a wing-body, the development of HGKS in general curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates from one-dimensional to three-dimensional computations. Based on the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and is transformed back to the physical domain for the update of flow variables inside each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the Jacobian-weighted conservative variables. In the two-stage fourth-order gas-kinetic scheme, the point values as well as the spatial derivatives of conservative variables at Gaussian quadrature points have to be used in the evaluation of the time dependent flux function. The point-wise conservative variables are obtained by ratio of the above reconstructed data, and the spatial derivatives are reconstructed through orthogonalization in physical space and chain rule. A variety of numerical examples from the accuracy tests to the solutions with strong discontinuities are presented to validate the accuracy and robustness of the current scheme for both inviscid and viscous flows. The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is also demonstrated through the numerical example.

  • Keywords

Gas-kinetic scheme, two-stage fourth-order discretization, WENO reconstruction, curvilinear coordinates.

  • AMS Subject Headings

76T10, 76P05, 76N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-1321, author = {Pan , Liang and Xu , Kun}, title = {High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {4}, pages = {1321--1351}, abstract = {

The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flows with Cartesian meshes. To study the flow problems in general geometries, such as the flow over a wing-body, the development of HGKS in general curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates from one-dimensional to three-dimensional computations. Based on the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and is transformed back to the physical domain for the update of flow variables inside each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the Jacobian-weighted conservative variables. In the two-stage fourth-order gas-kinetic scheme, the point values as well as the spatial derivatives of conservative variables at Gaussian quadrature points have to be used in the evaluation of the time dependent flux function. The point-wise conservative variables are obtained by ratio of the above reconstructed data, and the spatial derivatives are reconstructed through orthogonalization in physical space and chain rule. A variety of numerical examples from the accuracy tests to the solutions with strong discontinuities are presented to validate the accuracy and robustness of the current scheme for both inviscid and viscous flows. The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is also demonstrated through the numerical example.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0141}, url = {http://global-sci.org/intro/article_detail/cicp/18102.html} }
TY - JOUR T1 - High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions AU - Pan , Liang AU - Xu , Kun JO - Communications in Computational Physics VL - 4 SP - 1321 EP - 1351 PY - 2020 DA - 2020/08 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2019-0141 UR - https://global-sci.org/intro/article_detail/cicp/18102.html KW - Gas-kinetic scheme, two-stage fourth-order discretization, WENO reconstruction, curvilinear coordinates. AB -

The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flows with Cartesian meshes. To study the flow problems in general geometries, such as the flow over a wing-body, the development of HGKS in general curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates from one-dimensional to three-dimensional computations. Based on the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and is transformed back to the physical domain for the update of flow variables inside each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the Jacobian-weighted conservative variables. In the two-stage fourth-order gas-kinetic scheme, the point values as well as the spatial derivatives of conservative variables at Gaussian quadrature points have to be used in the evaluation of the time dependent flux function. The point-wise conservative variables are obtained by ratio of the above reconstructed data, and the spatial derivatives are reconstructed through orthogonalization in physical space and chain rule. A variety of numerical examples from the accuracy tests to the solutions with strong discontinuities are presented to validate the accuracy and robustness of the current scheme for both inviscid and viscous flows. The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is also demonstrated through the numerical example.

Liang Pan & Kun Xu. (2020). High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions. Communications in Computational Physics. 28 (4). 1321-1351. doi:10.4208/cicp.OA-2019-0141
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