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Report
COMPRESSIBLE TURBULENCE AND INTERFACIAL INSTABILITIES
Sreenivas Varadan , Pooya Movahed , Prof. Eric Johnsen
Department of Mechanical Engineering, University of Michigan, Ann Arbor
NUMERICAL METHODS
INTRODUCTION AND MOTIVATION
• In high-energy-density physics (HEDP), strong shocks, large density variations and highly compressible turbulence are often present.
• Hydrodynamic instabilities play an important role in inertial confinement fusion and astrophysics.
• Rayleigh-Taylor (RT) instability: heavy fluid on top of a light fluid in a downward accelerating field.
• Richtmyer-Meshkov (RM) instability: shock-interface interaction.
• RT and RM instabilities often evolve into turbulent mixing regions.
• Numerical methods for shock waves perform poorly in turbulence problems.
DECAY OF COMPRESSIBLE ISOTROPIC TURBULENCE WITH EDDY SHOCKLETS
Dilatation (pseudo-color)
Q criterion (iso-contours)
x-Velocity (pseudo-color)
CODE DESCRIPTION
• Requirements for numerical schemes for shock waves (adding numerical dissipation to
stabilize the solution) are contradictory with methods for turbulence (prevent numerical
dissipation from overwhelming the small scales).
• High-order accurate hybrid shock-capturing/central difference methods.
• Development of a novel physics-based discontinuity sensor that can handle strong
shocks and contact discontinuities.
• A multi-dimensional geometric approach is used for the discontinuity sensor.
INITIALIZATION OF THE PROBLEMS
Rayleigh-Taylor
• Initialization approach of Mellado et al1 and Cook2 .
• A field of random numbers is initialized, transformed to Fourier space and Gaussian filtered
with the peak mode being 14.
• Conjugate symmetry is enforced before the field is transformed to physical space and
scaled so that the rms fluctuation is 10% of the wavelength corresponding to the peak mode.
• Hydrostatic equilibrium is assumed and fluid initially has a constant temperature.
• The heavier fluid (Xe) sits above the lighter fluid (Ne) so that the system is buoyancy stable
but RT unstable.
• BCs are periodic in the horizontal directions and non-reflecting in the vertical direction.
• 3-D domain decomposition with MPI.
• Scales well on up to 160 processors.
• HDF5 libraries for parallel I/O and
visualization.
COMPARISON OF WENO5 WITH HYBRID FOR THE 2D RT INSTABILITY
512 x 512
Weno5
Inviscid
Hybrid
10X Viscosity
Inviscid
10X Viscosity
Isotropic Turbulence
• Initialization approach of Johnsen et al3.
• The velocity field, in Fourier space, is initialized a model spectrum and transformed.
• The initial Reynolds number, based on the Taylor micro scale, is 100 for all the runs.
• The initial velocity field is only generated on a 2563 grid which is then filtered onto the
coarse grid (643) .
• Triple periodic boundary conditions are used.
Turbulent Mach number: 0.6
Turbulent kinetic energy vs. time
Velocity spectra
512 x 1024
Weno5
Dilatation spectra
EVOLUTION OF THE 3D Xe-Ne RT INSTABILITY
Inviscid
Hybrid
10X Viscosity
Inviscid
10X Viscosity
a) Linear stage:
For small perturbations, linear analysis
is valid and describes the initial exponential
growth.
Turbulent Mach number: 1.0
Turbulent kinetic energy vs time
Velocity spectra
Dilatation spectra
b) Nonlinear stage:
Asymmetric structures in the form of rising
bubbles and falling spikes become apparent.
These structures form due to baroclinic
vorticity production.
512 x 2048
Weno5
Inviscid
10X Viscosity
Hybrid
Inviscid
10X Viscosity
c) Nonlinear interaction:
Strong nonlinear interactions lead
to the break up of the coherent
structures. Larger structures form due to
the amalgamation of the smaller ones.
2D SINGLE-MODE RM INSTABILITY WITH RE-SHOCK
Vortex roll-up
Enstrophy
Euler vs. Navier-Stokes
MUSCL
Before re-shock
d) Transition to turbulence:
Kelvin-Helmholtz (shear) instabilities
occur, thereby increasing the dynamic
range of scales in the problem. This leads
to turbulence at later times.
WENO COVERAGE FOR THE HYBRID SCHEME (2D RT INSTABILITY)
COARSE
MEDIUM
FINE
100X viscosity
COMPRESSIBILITY EFFECTS IN THE 3D RT INSTABILITY
• RT in HEDP problems is not necessarily incompressible: acoustic waves emerge
from the mixing layer and merge into a shock .
• There are significant 3D effects2.
• The central scheme stabilizes weak acoustic waves while WENO is triggered when
the shock forms.
WENO5
After re-shock
1000X viscosity
CONCLUSIONS AND REFERENCES
• Although numerical dissipation stabilizes the solution, it also destroys the small scale turbulent features.
• Pure WENO excessively damps the solution and, as a result, there is no difference as the physical viscosity is increased.
• As the grid is refined, the Hybrid method uses central differences, which have nominally no dissipation and converges more rapidly. Dispersion
and aliasing errors are also minimized.
• Future work includes refinement of the present methods for HEDP and studying problems with stronger shocks and more intense turbulence.
[1] J.P. Mellado and S.Sarkar, Large-eddy simulation of Rayleigh-Taylor turbulence with compressible miscible fluids, Phys. Fluids. 17, 076101
(2005)
[2] B.J Olson and A.W Cook, Rayleigh-Taylor Shock waves, Phys. Fluids. 19, 128108 (2007).
[3] E.Johnsen et al, Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves, J. Comput.
Phys. 228, 1213 (2010)
This research was supported in part by DOE NNSA/ASC under the Predictive Science Academic Alliance Program by
Grant No. DEFC52-08NA28616.

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