Shock Waves

Shocks are non-isentropic phenomena, i.e. entropy is not conserved, and necessitates a special formulation.

The missing energy is generated by an artificial bulk viscosity

$q$ as derived by von Neumann and Richtmeyer. ¹ This value is added to the pressure and is computed by:(1)

$q = q_{a}^{2} ρ l^{2} {(\frac{\partial ε_{k k}}{\partial t})}^{2} - q_{b} ρ l c \frac{\partial ε_{k k}}{\partial t}$

Where,

l: Is equal to $\sqrt[3]{Ω}$ or to the characteristic length
$Ω$: Volume
$\frac{\partial ε_{k k}}{\partial t}$: Volumetric compression strain rate tensor
$c$: Speed of sound in the medium

The values of

$q_{a}$ and

$q_{b}$ are adimensional scalar factors defined as:

$q_{a}$ is a scalar factor on the quadratic viscosity to be adjusted so that the Hugoniot equations are verified. This value is defined by the user. The default value is 1.10.
$q_{b}$ is a scalar factor on the linear viscosity that damps out the oscillations behind the shock. This is user specified. The default value is 0.05.

Default values are adapted for velocities lower than Mach 2. However, for viscoelastic materials (LAW34, LAW35, LAW38) or honeycomb (LAW28), very small values are recommended, that is, 10^-20.

Von Neumann J. and Richtmeyer R., “A method for the numerical calculation of hydrodynamical shocks”, Journal of applied physics, 1950.