Richtmyer-Meshkov Instabilities : Sinusoidal (fast-slow)
Interfaces
R.M. instabilities at vertical interfaces with sinusoidal perturbations
illustrate the onset of tirbulence and mixing in shock accelerated density
stratified media. The simple geometry and Boussinesq approximations allow
a fundamental insight into the phenomenon.
The contents were presented at ISSW95, Caltech, Pasadena California
The figure shows the initial condition of the numerical shock-tube experiment.
Shock refraction over the density stratification leads to vorticity deposition
and vortex sheet evolution. Later, the vortex sheet rolls up,
displaying a classical Kelvin - Helmholz mushroom structure (see logo).
We present an animation of the density and the
gas-gas interface . Note that the density
animation shows a reflected wave returning because of imperfectly enforced
outflow conditions , but the run is stopped before it affects the interface.
Reduced Modelling
The deposition of the vortex sheet prompts us to investigate its behaviour.
Initially one expects a vortex sheet, later rolling-up to more complicated
structure. This suggested a vortex model for predicting the growth rate of
the instability. We present an amimation of
the vorticity field to elucidate matters.
Vortex models for growth rate
The graph shows a quick deposition of circulation as the shock traverses
the interface; subsequent oscillations and increase are caused by seconadry
baroclinic effects.
The interface is modelled as a sinusoidal sheet with a
sinusoidal strength distribution.
Using a simple Biot-Savart relation we calculate the
difference in the velocity of the foot and head of the interface. At early
times, the numerical (NUM) and the sheet-like model (SMBS) tally very well.
At a later time, as the interface rolls up and is no longer a sheet,
(see late time vorticity map) we approximate the vorticity field by a
single vortex in the centre of the shock tube (LVM).
The initial circulation distribution is obtained analytically, making
the model independent of the simulation. The SMBS and LVM models provide
bounds for the growth rate in these interactions.
Acknowledgements
NCSA, for allowing us CM5 time ; DOE grant DE-FG02-93ER25179.A000
monitored by Dr. Daniel Hitchcock which funded us.
Contacts :
Updated : 31st August, 1995.