A Two-Dimensional Conservation Laws Scheme for Compressible Flows with Moving Boundaries

The generalized Riemann problem (GRP) scheme for the hydrodynamic conservation laws is extended to a two-dimensional moving boundary tracking (MBT) configuration, aimed at treating time-dependent compressible flows with moving (impermeable) boundary surfaces. A Strang-type operator splitting is employed for the integration of the conservation laws. The boundary motion is also split into Cartesian components that are taken with the respective operator-split integration phases. The conservation laws in boundary cells are integrated by a finite-volume scheme that accounts for changing mesh geometry (cell volume and side area through which mass, momentum, and energy fluxes are taken). The central feature of the scheme is the algorithms for evaluating the changing mesh geometry at boundary cells. These algorithms are based on a “decomposition rule” for evaluating polygon intersection area, which has been used extensively for rezoning in hydrocodes over the past decades. The decomposition rule is combined with the Cartesian splitting of the boundary motion, producing algorithms for the integration of conservation laws in boundary cells that are both consistent and simple. Consistency is taken to mean that the scheme produces an exact solution in the case of a uniform motion of the fluid and the boundary (common velocity). An illustrative example of shock lifting of a light-weight cylinder is presented.

[1]  J. Gillis,et al.  Methods in Computational Physics , 1964 .

[2]  H. Forrer,et al.  Boundary treatment for a Cartesian grid method , 1996 .

[3]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[4]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[5]  John B. Bell,et al.  Cartesian grid method for unsteady compressible flow in irregular regions , 1995 .

[6]  Joseph Falcovitz,et al.  Recent Developments of the GRP Method , 1995 .

[7]  Jean-Antoine Désidéri,et al.  Numerical methods for the Euler equations of fluid dynamics , 1985 .

[8]  Kozo Fujii,et al.  Unified zonal method based on the fortified solution algorithm , 1992 .

[9]  J. Falcovitz,et al.  A second-order Godunov-type scheme for compressible fluid dynamics , 1984 .

[10]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[11]  J. Falcovitz,et al.  An upwind second-order scheme for compressible duct flows , 1986 .

[12]  Bram van Leer,et al.  A simulation technique for 2-D unsteady inviscid flows around arbitrarily moving and deforming bodies of arbitrary geometry , 1993 .

[13]  Joseph Falcovitz,et al.  A high-resolution upwind scheme for quasi 1-D flows , 1985 .

[14]  Numerical simulation of the reflection of a planar shock wave over a double wedge , 1991 .

[15]  Richard Courant,et al.  Supersonic Flow And Shock Waves , 1948 .

[16]  Numerical simulation of the head‐on reflection of a regular reflection , 1993 .

[17]  丹生 慶四郎,et al.  R. Courant and K. O. Friedrichs: Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1976, 464ページ, 23.5×16cm, 7,200円. , 1978 .

[18]  James J. Quirk,et al.  A CARTESIAN GRID APPROACH WITH HIERARCHICAL REFINEMENT FOR COMPRESSIBLE FLOWS , 1994 .