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.  J.P. Mellado and S.Sarkar, Large-eddy simulation of Rayleigh-Taylor turbulence with compressible miscible fluids, Phys. Fluids. 17, 076101 (2005)  B.J Olson and A.W Cook, Rayleigh-Taylor Shock waves, Phys. Fluids. 19, 128108 (2007).  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.