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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaaaa@36EC@ 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 ( ε k k t ) 2 q b ρ l c ε k k t
Is equal to Ω 3 or to the characteristic length
ε k k t
Volumetric compression strain rate tensor
c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGlbaaaa@39A7@
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.

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