Linear interaction of a cylindrical entropy spot with a shock

The interaction of a cylindrical element of hot or cold gas (the "entropy s pot") with a shock wave is considered. An exact solution in the limit of we ak spot amplitudes is elaborated, using the linear interaction analysis the ory and the procedure of decomposition proposed by Ribner (Technical Report No. 1164, NACA, 1953). The method is applied to an entropy spot with a Gau ssian profile. Results are presented for a wide range of shock Mach numbers , with a special interest at M-1=2. The resulting vorticity field consists of a pair of primary counter-rotating vortices, as well as a pair of second ary vortices of opposite sign and weaker amplitude. An expression for the c irculation in half a plane is derived and compared to existing results. The pressure field consists of a cylindrical acoustic wave which propagates aw ay from the transmitted spot and an evanescent nonpropagative field confine d behind the shock. For a hot spot, the cylindrical wave is a rarefaction w ave on its forward front and a compression wave on its upstream propagating parts, and the nonpropagative field corresponds to a pressure deficit. The structure of the transmitted spot and the shock deformation are also discu ssed. The linear solution is compared with numerical simulation results for M-1=2 and M-1=4. The comparison shows qualitative and quantitative agreeme nt when linear as well as nonlinear spot amplitudes are considered. Finally , the method is applied to the case of a constant spot with a tophat profil e, and the results are compared to the case of a Gaussian spot. This paper also contains in the Appendix a general formulation of the linear interacti on problem for the three kinds of plane waves (entropy, vorticity, and pres sure waves) impinging upon a shock. (C) 2001 American Institute of Physics.