COMPRESSIBLE TURBULENCE AND INTERFACIAL INSTABILITIES

1
COMPRESSIBLE TURBULENCE AND INTERFACIAL
INSTABILITIES Sreenivas Varadan , Pooya Movahed ,
Prof. Eric Johnsen Department of Mechanical
Engineering, University of Michigan, Ann Arbor

• 
  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.

• 
  NUMERICAL METHODS
• 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/centr
  al 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.

• CODE DESCRIPTION
• 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
Hybrid
Inviscid 10X Viscosity
Inviscid
10X Viscosity

• 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.
• 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.

DECAY OF COMPRESSIBLE ISOTROPIC TURBULENCE WITH
EDDY SHOCKLETS
Dilatation (pseudo-color) Q criterion
(iso-contours) x-Velocity (pseudo-color)
512
x 1024 Weno5

Hybrid Inviscid
10X Viscosity
Inviscid 10X
Viscosity

Turbulent Mach number 0.6 Turbulent kinetic
energy vs. time Velocity
spectra Dilatation
spectra

EVOLUTION OF THE 3D Xe-Ne RT INSTABILITY

• Linear stage
• For small perturbations, linear analysis
• is valid and describes the initial exponential
• growth.

512 x 2048
b) Nonlinear stage Asymmetric structures in
the form of rising bubbles and falling spikes
become apparent. These structures form due to
baroclinic vorticity production.
Weno5

Hybrid Inviscid
10X Viscosity
Inviscid 10X
Viscosity

Turbulent Mach number 1.0 Turbulent kinetic
energy vs time Velocity
spectra Dilatation
spectra

• 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.

WENO COVERAGE FOR THE HYBRID SCHEME (2D RT
INSTABILITY) COARSE
MEDIUM
FINE

• 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.

2D SINGLE-MODE RM INSTABILITY WITH RE-SHOCK
Vortex roll-up
Enstrophy
Euler vs. Navier-Stokes
MUSCL
Before re-shock
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.

• 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.

WENO5
After re-shock 1000X viscosity