Interaction of fan–jump–fan composite waves in a two-dimensional steady jet for van der Waals gases

We consider a two-dimensional (2D) steady jet generated by gas streaming in parallel supersonic flow out of a duct and into the atmosphere. When the gas is a van der Waals gas, at the corners at the exit of the duct, the parallel supersonic flow expands symmetrically into centered simple waves, jump–fan (JF) composite waves, fan–jump composite waves, or fan–jump–fan (FJF) composite waves. This paper studies the interaction of the symmetric centered FJF composite waves in the jet. The interaction of the FJF composite waves is more involved in comparison to our earlier work on the interaction of the symmetric centered JF composite waves, since the flow in the FJF composite wave interaction zone is rotational. To construct the flow in the interaction zone, we consider a Goursat-type boundary value problem with discontinuous boundary data associated with the 2D and isentropic steady Euler equations. The existence of a global piecewise smooth solution to this Goursat problem is obtained by a new method based on characteristics.

ON THE EXPONENTIAL CONVERGENCE OF THE METHOD OF FUNDAMENTAL SOLUTIONS

It is well known that the method of fundamental solutions (MFS) is a numerical method of exponential convergence. In other words, the logarithmic error is proportional to the node number of spatial discretization. In this study, the exponential convergence of the MFS is demonstrated by solving the Laplace equation in domains of rectangles, ellipses, amoeba-like shapes, and rectangular cuboids. In the solution procedure, the sources of the MFS are located as far as possible and the instability resulted from the ill-conditioning of system matrix is avoided by using the multiple precision floating-point reliable (MPFR) library. The results converge faster for the cases of smoother boundary conditions and larger area/perimeter ratios. For problems with discontinuous boundary data, the exponential convergence is also accomplished using the enriched method of fundamental solutions (EMFS), which is constructed by the fundamental solutions and the local singular solutions. The computation is scalable in the sense that the required time increases only algebraically.