Riemann shock tube: 1D normal shocks in air, simulations and experiments

Abstract

This paper presents numerical simulation of the evolution of one-dimensional normal shocks, their propagation, reflection and interaction in air using a single diaphragm Riemann shock tube and validate them using experimental results. Mathematical model is derived for one-dimensional compressible flow of viscous and conducting medium. Dimensionless form of the mathematical model is used to construct space–time finite element processes based on minimization of the space–time residual functional. The space–time local approximation functions for space–time p-version hierarchical finite elements are considered in higher order $H^{k,p}\left({\bar{\Omega }}_{xt}^e\right)$ spaces that permit desired order of global differentiability of local approximations in space and time. The resulting algebraic systems from this approach yield unconditionally positive-definite coefficient matrices, hence ensure unique numerical solution. The evolution is computed for a space–time strip corresponding to a time increment Δt and then time march to obtain the evolution up to any desired value of time. Numerical studies are designed using recently invented hand-driven shock tube (Reddy tube) parameters, high/low side density and pressure values, high- and low-pressure side shock tube lengths, so that numerically computed results can be compared with actual experimental measurements.

Keywords:

• shock tube
• 1D normal shock
• mathematical models
• space–time methods
• time marching

Acknowledgements

The first, third and fourth authors are grateful to Dr. J. Myers, Program Manager of the Mathematical Sciences Division of the Army Research Office (ARO), for the encouragement and support. The second author acknowledges the help of students and colleagues in the Laboratory for Hypersonic and Shock Wave Research at the Indian Institute of Science, Bangalore.

Additional information

Funding

The first, third and fourth authors acknowledge the support of the research reported herein by the Mathematical Sciences Division of the ARO [grant number W-911NF-11-1-0471(FED0061541)] to the University of Kansas and Texas A & M University.