V European Conference on Computational Fluid Dynamics Eccomas Cfd 2010 Equivalence Conditions for Finite Volume / Element Discretizations in Cylindrical Coordinates

Abstract. A novel numerical technique for the solution of the compressible flow equations over unstructured grids in a cylindrical reference is presented. The proposed approach is based on a mixed finite volume / finite element method. Equivalence conditions relating the finite volume and the finite element metrics in cylindrical coordinates are derived. Numerical simulations of the explosion and implosion problems for inviscid compressible flows are carried out to evaluate the correctness of the numerical scheme. For the implosion problem, numerical simulations include also the effect of the presence of cylindrical obstacles in the flow field, which have been recently proposed as a mean to modify the shape of a cylindrical converging shock to increase the shock front stability in experimental studies on the sonoluminescence effect. Spherical shock waves are also considered and the modification to the shock geometry due to the presence of a spherical obstacle is evaluated numerically and compared to its cylindrical counterpart.

[1]  Malte T. Kjellander,et al.  Regular versus Mach reflection for converging polygonal shocks , 2007 .

[2]  V. Venkatakrishnan,et al.  IMPLICIT METHOD FOR THE COMPUTATION OF UNSTEADY FLOWS ON UNSTRUCTURED GRIDS , 1995 .

[3]  V. Selmin,et al.  UNIFIED CONSTRUCTION OF FINITE ELEMENT AND FINITE VOLUME DISCRETIZATIONS FOR COMPRESSIBLE FLOWS , 1996 .

[4]  V. Eliasson,et al.  Light emission during shock wave focusing in air and argon , 2007 .

[5]  L. Sedov.  Similarity and Dimensional Methods in Mechanics , 1960 .

[6]  V. Selmin,et al.  The node-centred finite volume approach: bridge between finite differences and finite elements , 1993 .

[7]  Nils Tillmark,et al.  Controlling the form of strong converging shocks by means of disturbances , 2007 .

[8]  Alberto Guardone,et al.  Finite element/volume solution to axisymmetric conservation laws , 2007, J. Comput. Phys..

[9]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .

[10]  L. Fezoui,et al.  A class of implicit upwind schemes for Euler simulations with unstructured meshes , 1989 .

[11]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[12]  Emilio Panarella,et al.  Stability of imploding spherical shock waves , 1995 .

[13]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[14]  Shi Honghui,et al.  The interaction between shock waves and solid spheres arrays in a shock tube , 2004 .

[15]  Randall J. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Nonlinear Systems of Conservation Laws , 2002 .