Piston driven shock waves in non-homogeneous planar media

TL;DR

This work extends piston-driven planar shock theory to non-homogeneous media with a power-law density profile, introducing self-similar solutions in both Lagrangian and Eulerian coordinates under a time-dependent piston pressure $p_0 t^{\tau}$. By applying dimensional analysis, the authors derive a reduced ODE system for the similarity profiles and characterize the shock via the similarity variable $\xi_s$, with jump conditions linking shocked and unshocked states. The solutions exhibit qualitatively distinct regimes depending on the exponents $\tau$ and $\omega$, including constant-speed, accelerating, and decelerating shocks, as well as finite, vanishing, or diverging density near the piston; energy partition between kinetic and internal forms is shown to be governed by $U(0)$ and remains time-invariant in proportion. Validation against high-resolution hydrodynamic simulations demonstrates very good agreement, supporting the use of these semi-analytic solutions for code verification and for generalizing ablation-driven shock analyses to non-homogeneous density fields.

Abstract

In this work, we analyze in detail the problem of piston driven shock waves in planar media. Similarity solutions to the compressible hydrodynamics equations are developed, for a strong shock wave, generated by a time dependent pressure piston, propagating in a non-homogeneous planar medium consisting of an ideal gas. Power law temporal and spatial dependency is assumed for the piston pressure and initial medium density, respectively. The similarity solutions are written in both Lagrangian and Eulerian coordinates. It is shown that the solutions take various qualitatively different forms according to the value of the pressure and density exponents. We show that there exist different families of solutions for which the shock propagates at constant speed, accelerates or slows down. Similarly, we show that there exist different types of solutions for which the density near the piston is either finite, vanishes or diverges. Finally, we perform a comprehensive comparison between the planar shock solutions and Lagrangian hydrodynamic simulations, by setting proper initial and boundary conditions. A very good agreement is reached, which demonstrates the usefulness of the analytic solutions as a code verification test problem.

Figures (22)

• Figure 1: A schematic description of the hydrodynamics problem: a time dependent pressure piston drives a strong shock wave, into a medium which is initially at rest with a power law density profile.
• Figure 2: A color plot for the mass similarity exponent (the temporal power of the shocked mass, see equation \ref{['eq:shocked_mass']}), as a function of $\tau,\omega$.
• Figure 3: The function $F\left(\xi_{s}\right)=P_{\xi_{s}}\left(0\right)-1$ which has a root at the correct value of the similarity variable at the shock, $\xi_{s}$. The calculation was performed for $\omega=0.5$, $\tau=-0.45$ and $\gamma=1.25$, resulting in $\xi_{s}=2.0189$ (as shown by the black dashed vertical line).
• Figure 4: A color plot for the shock front similarity variable $\xi_{s}$ as a function of the piston pressure temporal power $\tau$ and the initial density spatial power $\omega$ and an adiabatic index $\gamma=1.25$.
• Figure 5: Shock front similarity variable $\xi_{s}$ as a function of $\tau$ for selected values of $\omega$ and as a function of $\tau,\omega$ and $\gamma=1.25$.