Linear interaction of a cylindrical entropy spot with a shock

The interaction of a cylindrical element of hot or cold gas (the "entropy spot") with a shock wave is considered. An exact solution in the limit of weak spot amplitudes is elaborated, using the linear interaction analysis theory and the procedure of decomposition proposed by Ribner (Technical Report No. 1164, NACA, 1953). The method is applied to an entropy spot with a Gaussian profile. Results are presented for a wide range of shock Mach numbers, with a special interest at M₁ = 2. The resulting vorticity field consists of a pair of primary counter-rotating vortices, as well as a pair of secondary vortices of opposite sign and weaker amplitude. An expression for the circulation in half a plane is derived and compared to existing results. The pressure field consists of a cylindrical acoustic wave which propagates away from the transmitted spot and an evanescent nonpropagative field confined behind the shock. For a hot spot, the cylindrical wave is a rarefaction wave 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 discussed. The linear solution is compared with numerical simulation results for M₁ = 2 and M₁ = 4. The comparison shows qualitative and quantitative agreement 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 profile, 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 interaction problem for the three kinds of plane waves (entropy, vorticity, and pressure waves) impinging upon a shock.