Analytical and Computational Challenges of Incompressible Flows at High Reynolds Number

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Vorticity Deposition and Evolution for Accelerated Inhomogeneous Flows: Experiment, Computation, Analysis and Models

            Norman Zabusky

                                       Department of Complex Physics, Weizmann Institute of Science

Abstract: We present an overview and recent understanding  of accelerated inhomogeneous flows, e.g. shock-accelerated interfaces or Richtmyer-Meshkov flows.^1,2 We use the vortex paradigm¹ and the visiometric approach² and  apply them to the shock-accelerated one-mode, small-amplitude perturbed planar configuration and cylinder in two dimensions and focus on scaling with respect to Atwood number.  We quantify phenomena to validate simulations and create models involving coherent space-time events. We emphasize our recent work,^3-8 including: finite, initial interfacial gradients (which carry the initially deposited circulation) and their subsequent steepening; vortex induced, secondary baroclinic circulation generation which creates unstable vortex bilayers that dominate at intermediate times; vortex bilayer roll-up and the appearance  of “vortex projectiles” (dipolar-like structures) and turbulent domains which drive the mixing of species; possible finite-time ill-posedness; and applicability of a dipolar incompressible point-vortex model for amplitude growth. All these events  occur in an essentially incompressible flow,  if the initial shock has M < 1.5. Finally, we discuss the applicability of these ideas to recent high resolution Rayleigh-Taylor simulations. ^9,10

References 

1. Hawley, J.F. and Zabusky, N.J., Vortex paradigm for shock-accelerated density-stratified interfaces 1989.  Phys. Rev Letters 63, 1241-1244.

2.  Zabusky, N.J., Vortex paradigm for accelerated inhomogeneous flows: Visiometrics for the Rayleigh-Taylor and Richtmyer-Meshkov environments.  Ann. Review of Fluid Mechanics, 1999. 31, 495-535.

3. Zabusky, N.J., Kotelnikov, A.D., Gulak, Y. & Peng, G. Amplitude growth rate of a Richtmyer-Meshkov unstable two-dimensional interface to intermediate times. J. Fluid Mechanics, 475,  147-162. 2003.

4. Peng, G., Zabusky, N.J. and Zhang, S. Vortex-accelerated secondary baroclinic vorticity deposition and late intermediate time dynamics of a two-dimensional  RM interface.  Phys. Fluids 15 (12), 3730-3744, 2003.

5. Zhang, S., Zabusky, N.J.,  Peng, G., Gupta, S.  Shock Gaseous Cylinder Interactions: Dynamically validated initial conditions provide excellent agreement between experiments and Navier-Stokes simulations to late-intermediate time. Phys. Fluids 16(5), 1203-1216, 2004.

6. Zhang, S., Zabusky, N.J.  Peng, G. Vortex dynamics and baroclinically forced inhomogeneous turbulence for shock - planar heavy curtain interactions” J. of Turbulence, 6, 1-27,  2005

7.  Zabusky, N.J. and Peng, G. Vorticity deposition and evolution in shock-accelerated flows: Analysis, Computation and  Experiment”  in Proceedings of the LANL symposium “Modeling and Simulation of Variable Density and Compressible Turbulent Mixing,” Mark A. Christon, Daniel Livescu, and John A. Turner, eds, August 3-5 2005. pp7-9. See   http://www.ccs.lanl.gov/ccs2/docs/05TurbMixSymposium.pdf

8. Lee, D-K., Peng, G. and Zabusky, N. J. Circulation rate of change: A vortex approach for validating and understanding accelerated inhomogeneous flows through intermediate times.  Phys Fluids, to be published  2006.

9. Cook, A., Cabot, W. and Miller, P. The mixing transition in Rayleigh–Taylor instability. J.  Fluid Mechanics, 511: 333-362, 2004

10. Cabot, W.  and Andrew Cook, A. Reynolds number effects on Rayleigh-Taylor instability with possible implications for type 1a supernovae. Nature Physics 2, 562 – 568, 2006.