Numerical study on the interaction of a weak shock wave with an elliptic gas cylinder

The interaction of a weak shock wave with a heavy elliptic gas cylinder is investigated by solving the Eulerian equations in two-dimensional Cartesian coordinates. An interface-capturing algorithm based on the $$\gamma $$γ-model and the finite volume weighed essential non-oscillatory scheme is employed to trace the motion of the discontinuous interface. Three gas pairs with different Atwood numbers ranging from 0.21 to 0.91 are considered, including carbon dioxide cylinder in air (air–$$\hbox {CO}_2$$CO2), sulfur hexafluoride cylinder in air (air–$$\hbox {SF}_6$$SF6), and krypton cylinder in helium (He–Kr). For each gas pair, the elliptic cylinder aspect ratio ranging from 1/4 to 4 is defined as the ratio of streamwise axis length to spanwise axis length. Special attention is given to the aspect ratio effects on wave patterns and circulation. With decreasing aspect ratio, the wave patterns in the interaction are summarized as transmitted shock reflection, regular interaction, and transmitted shock splitting. Based on the scaling law model of Samtaney and Zabusky (J Fluid Mech 269:45–78, 1994), a theoretical approach is developed for predicting the circulation at the time when the fastest shock wave reaches the leeward pole of the gas cylinder (i.e., the primary deposited circulation). For both prolate (i.e., the minor axis of the ellipse is along the streamwise direction) and oblate (i.e., the minor axis of the ellipse is along the spanwise direction) cases, the proposed approach is found to estimate the primary deposited circulation favorably.