A gridless boundary condition method for the solution of the Euler equations on embedded Cartesian meshes with multigrid.

*(English)*Zbl 1195.76318Summary: The solution of the Euler equations using a “gridless” boundary condition treatment on a patched, embedded Cartesian field mesh is described. The gridless boundary treatment is implemented by means of a least squares fitting of the conserved flux variables using a cloud of nodes in the vicinity of the body. The method allows for accurate treatment of the surface boundary conditions without the need for excessive refinement of the Cartesian mesh. Additionally, the method does not suffer from problems associated with thin body geometry or extremely fine cut cells near the body. Unlike some methods that consider a gridless (or “meshless”) treatment throughout the entire domain, the use of a Cartesian field mesh allows for effective implementation of multigrid acceleration, and issues associated with global conservation are alleviated. Results are presented for transonic flow about single and dual NACA 0012 airfoil configurations including a convergence study indicating the effectiveness of multigrid acceleration within the construct of the gridless boundary treatment. Where applicable, comparisons to the FLO52 body-fitted Euler code are presented to gauge the accuracy of the method.

##### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76M15 | Boundary element methods applied to problems in fluid mechanics |

##### Keywords:

Cartesian mesh; Gridless; Meshless; Embedded mesh; Euler equations; numerical boundary conditions; Transonic flow
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\textit{D. J. Kirshman} and \textit{F. Liu}, J. Comput. Phys. 201, No. 1, 119--147 (2004; Zbl 1195.76318)

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##### References:

[1] | P.R. Lahur, Y. Nakamura, A New Method for Thin Body Problem in Cartesian Grid Generation, AIAA Paper 99-0919, 1999 |

[2] | Clarke, D.K.; Salas, M.D.; Hassan, H.A., Euler calculations for multielement airfoils using Cartesian grids, AIAA journal, 24, March, 353-358, (1986) · Zbl 0587.76095 |

[3] | R.L. Gaffney, H.A. Hassan, M.D. Salas, Euler Calculations for Wings Using Cartesian Grids, AIAA Paper 87-03563, January 1987 |

[4] | Tidd, D.M.; Strash, D.J.; Epstein, B.; Luntz, A.; Nachshon, A.; Rubin, T., Multigrid Euler calculations over complete aircraft, Journal of aircraft, 29, November-December, 1080-1085, (1992) |

[5] | R.J. Leveque, High Resolution Finite Volume Method on Arbitrary Grids Via Wave Propagation, AIAA 89-1930, 1989 |

[6] | Berger, M.J.; Leveque, R., Stable boundary conditions for Cartesian grid calculations, Computing systems in engineering, 1, 305-311, (1990) |

[7] | R.J. Leveque, M.J. Berger, A rotated difference scheme for Cartesian grids in complex geometries, AIAA Paper CP-91-1602, 1991 |

[8] | H. Forrer, Boundary treatment for a Cartesian grid method, ETH Report 96-04, April 1996 · Zbl 0936.76041 |

[9] | De Zeeuw, D.; Powell, K.G., An adaptively refined Cartesian mesh solver for he Euler equations, Journal of computational physics, 104, 56-68, (1993) · Zbl 0766.76066 |

[10] | Pember, R.B.; Bell, J.B.; Colella, P.; Crutchfield, W.Y.; Welcome, M.L., An adaptive Cartesian grid method for unsteady compressible flow in irregular regions, Journal of computational physics, 1209, 278-304, (1995) · Zbl 0842.76056 |

[11] | J.E. Melton, M.J. Berger, M.J. Aftosmis, M.D. Wong, 3D applications of a Cartesian grid Euler method, AIAA Paper 95-0853, 1995 |

[12] | Coirier, W.J.; Powell, K.G., An accuracy assessment of Cartesian-mesh approaches for the Euler equations, Journal of computational physics, 177, 1995, (1995) |

[13] | Quirk, J.J., An alternative to unstructured grids for computing gas dynamic flows around arbitrarily complex two-dimensional bodies, Computer & fluids, 23, 1, 125-142, (1994) · Zbl 0788.76067 |

[14] | Forrer, H.; Jeltsch, R., A higher order boundary treatment for Cartesian-grid methods, Journal of computational physics, 140, 259-277, (1998) · Zbl 0936.76041 |

[15] | M.J. Aftosmis, M.J. Berger, G. Adomavicius, A parallel multilevel method for adaptively refined Cartesian grids with embedded boundaries, AIAA Paper 2000-0808, 38th Aerospace Sciences Meeting and Exhibit, January 2000 |

[16] | A. Dadone, B. Grossman, An immersed body methodology for inviscid flows on Cartesian grids, AIAA Paper 2002-1059, January 2002 |

[17] | Dadone, A.; Grossman, B., Surface boundary conditions for the numerical solution of the Euler equations, AIAA journal, 32, 285-293, (1995) · Zbl 0800.76323 |

[18] | J.T. Batina, A gridless Euler/Navier-Stokes solution algorithm for complex two-dimensional applications, NASA-TM-107631, June 1992 |

[19] | S.-C. Shih, S.-Y. Lin, A weighted least squares method for Euler and Navier-Stokes equations, AIAA Paper 94-0522, January 1994 |

[20] | J.L. Liu, S.J. Su, A potentially gridless solution method for the compressible Euler/Navier-Stokes equations, AIAA Paper 96-0526, 1996 |

[21] | Vanka, S.P.; Ploplys, N., Meshless methods for navier – stokes equations using radial basis functions, Advances in computational engineering and sciences, 2, (2000) |

[22] | R. Subrata, M. Fleming, Nonlinear subgrid embedded element-free-Galerkin method for monotone CFD solutions, in: Proc. Int. Mechanical Engineering Cong. and Expo., November 2000 |

[23] | Lin, H.; Atluri, S.N., The meshless local petrov – galerkin (MLPG) method for solving incompressible navier – stokes equations, Computer modeling in engineering & sciences, 2, 117-142, (2001) |

[24] | Sridar, D.; Balakrishnan, N., An upwind finite difference scheme for meshless solvers, Journal of computational physics, 189, 1-29, (2003) · Zbl 1023.76034 |

[25] | Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, International journal of numerical methods in engineering, 37, 229-256, (1994) · Zbl 0796.73077 |

[26] | Van Leer, B., Flux-vector splitting for the Euler equations, (), 507-512 |

[27] | A. Jameson, W. Schmidt, E. Turkel, Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping, AIAA Paper 81-1259, AIAA 14th Fluid and Plasma Dynamics Conference, June, 1981 |

[28] | B. Van Leer, C.-H. Tai, K.G. Powell, Design of optimally smoothing multi-stage schemes for the Euler equations, AIAA Paper 89-1933, 1989 |

[29] | Blazek, J., Computational fluid dynamics: principles and applications, (2001), Elsevier Science Ltd Amsterdam · Zbl 0995.76001 |

[30] | Jameson, A., Solution of the Euler equations for two dimensional transonic flow by a multigrid method, Applied mathematics and computation, 13, 327-356, (1983) · Zbl 0545.76065 |

[31] | W.J. Usab, E.M. Murman, Embedded mesh solution of the Euler equations using a multiple-grid-method, AIAA Paper 83-1946, 1983 |

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