Analytical theory for planar shock focusing through perfect gas lens and shock tube experiment designs

In this paper, we present a generalization of the gas lens technique developed by Dimotakis and Samtaney [“Planar shock cylindrical focusing by a perfect-gas lens,” Phys. Fluids 18, 031705 (2006)]. This technique is devoted to converting a planar shock wave into a cylindrical one through a shaped interface between two gases. We revisit this theory and demonstrate that the shape of the lens is either an ellipse or a hyperbola. A simple formula for its eccentricity is analytically obtained: e=Wt/Wi, where Wt and Wi are the transmitted and incident shock wave velocities, respectively. Furthermore, our theory is valid for fast-slow and slow-fast configurations. It also allows the generation of spherical converging shock waves. We present numerical simulations that successfully validate our lens design. Finally, we use the gas lens technique in order to design shock tube experiments: shock wave and hydrodynamic instabilities are studied and discussed in convergent geometry.