Wavelet-based Direct Numerical Simulation of Rayleigh-Taylor Instability

1. Wavelet-based Direct Numerical Simulation of Rayleigh-Taylor Instability Scott Reckinger Advisors: Dr. Oleg V. Vasilyev (CU-Boulder) Dr. Daniel Livescu (LANL)

2. Outline • Motivation • Method – the AWCM • Problem Description • Benchmark Problem – homogeneous compressible turbulent mixing • Preliminary Results • Future Work Group Presentation - Fall 2007

3. Motivation • Variable density flows appear in almost all fluid systems • Fluid species of differing molar masses • Turbulence assisted molecular mixing • Detailed understanding of mixing processes has important consequences in many scientific and engineering fields Group Presentation - Fall 2007

4. Motivation • Rayleigh-Taylor instability • Buoyantly driven instability • Light pushing heavy fluid • Gravity, accelerating fronts, etc. • Mixing layer develops • Mainly quiescent flow • Occurs in ICF, supernovae explosions , oceans and atmospheres, etc. Group Presentation - Fall 2007

5. Hydrodynamics simulation of the Rayleigh-Taylor instability. (Li, Shengtai and Hui Li. Parallel AMR Code for Compressible MHD or HD Equations. Los Alamos National Laboratory.) Motivation • Wide range of temporal and spatial scales at large Re • Relatively few regions with sharp transitions or localized structures • Intermittency is highly localized • Promising approach: Adaptive Mesh for DNS Group Presentation - Fall 2007

6. Method • Adaptive Wavelet Collocation Method (AWCM) • Wavelets are localized in physical and wavenumber space • Wavelet decomposition • Dynamically adapt the mesh by keeping only significant wavelets Group Presentation - Fall 2007

7. Method • Small scale resolution in regions with small scale structures • Coarse resolution in quiescent regions • Retain effective error control Group Presentation - Fall 2007

8. Problem Description • Long-term goal: DNS of Rayleigh-Taylor instability using the AWCM • Initial Application: compressible homogeneous turbulent mixing • Prove the AWCM’s effectiveness in the DNS of complicated turbulent flows • Non-linear terms of Navier Stokes eqns. • Two-fluid variable density setup • Gain insight into the physics behind the complicated mixing processes • Variable density effects • Mass diffisivity (Sc number) • compressibility Group Presentation - Fall 2007

9. Benchmark Problem • Governing equations: compressible Navier-Stokes equations with additional scalar equation (heavy fluid mass fraction) • To test whether PDF skew is due to gravitational or inertial effects, no gravity Group Presentation - Fall 2007

10. Benchmark Problem • Triply-Periodic domain • Scalar field initialization: • Random blobs of pure fluid • Gaussian random distribution with top hat spectrum, corresponding to large scale structures • Double delta PDF • High gradient regions smoothed by filter, followed by pure diffusion, resulting in thin diffusion layer Group Presentation - Fall 2007

11. Benchmark Problem • Velocity field initialization • Gaussian random distribution with following spectrum • Non-dilatational • Thermodynamic variables • Constant temperature • Mean density = 1 • Density fluctuations given by modified pressure Poisson equation (p=ρRT) • Where, Group Presentation - Fall 2007

12. Benchmark Problem • Important parameters: Group Presentation - Fall 2007

13. Ma = 0.1 Sc = 1.0 Preliminary Results Group Presentation - Fall 2007

14. Ma = 0.1 Sc = 1.0 Preliminary Results Group Presentation - Fall 2007

15. Preliminary Results • Skewed PDF toward heavy fluid • Due to inertial effects • Models for variable density turbulent mixing must take this effect into consideration • More interesting to look at density PDF, so next step is to initialize density with symmetric double delta PDF Group Presentation - Fall 2007

16. Future Work • Continue homogeneous compressible turbulent mixing problem • Symmetric density initialization • Test compressibility and mass diffusivity effects • Fully test the AWCM for accuracy and speed up Group Presentation - Fall 2007

17. Future Work • Rayleigh-Taylor instability problem • Influence of parameters on growth rate • compressibility • Schmidt number • stratification of the fluids (Froude number) • initial conditions (steepness of the density profile, perturbation amplitude, etc.) • Mixing processes • Pure fluid amounts – skewed PDF? • Supersonic mixing layer edge? • Arbitrary angle between interface and body force (ICF) Group Presentation - Fall 2007

18. RT fingers evident in the Crab Nebula Picture taken for Flow Visualization course at CU by Laurel Swift. Questions? Group Presentation - Fall 2007

19. References Oleg V. Vasilyev and Christopher Bowman, Second-generation wavelet collocation method for the solution of partial differential equations, Journal of Computational Physics 165 (2000), 660-693. Oleg V. Vasilyev, Solving multi-dimensional evolution problems with localized structures using second generation wavelets, Int. J. Comp. Fluid Dyn., special issue on high-resolution methods in computational fluid dynamics, 17(2), 151-168, 2003. Daniel Livescu and J.R. Ristorcelli, Variable density mixing in buoyancy driven turbulence, submitted to J. Fluid Mech., 2007, LA-UR-07-3399. Group Presentation - Fall 2007