Shock interaction with organized structures: Theory and computation

Unsteady interactions between shocks and turbulence are important phenomena frequently encountered in high-speed flows. In this dissertation the problem of a shock interaction with an entropy spot is studied by means of both theoretical analysis and nonlinear computation. The main objective of the studies is to apply both theoretical and computational approaches to study the physics underlying such shock interaction process.; The theoretical analysis is based on the Fourier decomposition of the upstream disturbance, the interaction of each Fourier mode with the shock, and the reconstruction of the downstream disturbance via the inverse Fourier transform. The theory is linear in that it assumes the principle of superposition and that the Rankine-Hugoniot relations are linearized about the mean position of the shock. The numerical simulation is carried out within the framework of the unsteady and compressible Euler equations, coupled with an equation for the shock motion, solved numerically by a sixth-order accurate spatial scheme and a fourth-order Runge-Kutta time-integration method. Analyses of the results are concentrated on the case of a Mach 2.0 shock interaction with an entropy spot that has a Gaussian density distribution. The theoretical analysis and the numerical simulation are verified with each other for small amplitude disturbances. The roles of the evanescent and the non-evanescent waves and the mechanisms for downstream disturbance generations are explored in details.; In addition, the quasi three-dimensional interaction between a shock and a vortex ring is investigated computationally within the framework of the axisymmetric Euler equations. The vortex ring, which is based on Lamb's formula, has an upstream circulation Γ = 0.01 and its aspect ratio R lies in the range 8 ≤ R ≤ 100. The shock Mach number varies in the range 1.1 ≤ M₁ ≤ 1.8. The interaction results in the streamwise compression of the vortex core and the generation of a toroidal acoustic wave and entropy disturbances. The acoustic wave propagates and interacts with itself on the symmetry axis of the vortex ring, engendering high amplitude rarefaction and compression peaks upstream and downstream of the transmitted vortex core. This results in a significant increase in the centerline sound pressure level, especially near the shock (due to the upstream movement of the rarefaction peak) and in the far downstream (due to the downstream movement of the compression peak). The magnitude of the compression peak increases nonlinearly with M₁. For a given M₁, vortex rings with small aspect ratios (R < 30) generate pressure disturbances whose maximum amplitude scales inversely with R.