Abstract
In this work, we investigate the growth of interface perturbations following the interaction of a shock wave with successive layers of fluids. Using the Discontinuous Galerkin method, we solve the two-dimensional multifluid Euler equations. In our setup, a shock impacts up to four adjacent fluids with perturbed interfaces. At each interface, the incoming shock generates reflected and transmitted shocks and rarefactions, which further interact with the interfaces. By monitoring perturbation growth, we characterize the influence these instabilities have on each other and the fluid mixing as a function of time in different configurations. If the third gas is lighter than the second, the reflected rarefaction at the second interface amplifies the growth at the first interface. If the third gas is heavier, the reflected shock decreases the growth and tends to reverse the Richtmyer–Meshkov instability as the thickness of the second gas is increased. We further investigate the effect of the reflected waves on the dynamics of the small scales and show how a phase difference between the perturbations or an additional fluid layer can enhance growth. This study supports the idea that shocks and rarefactions can be used to control the instability growth.
Similar content being viewed by others
Notes
Since the present simulations are two-dimensional, they cannot represent vortex stretching, and thus turbulence. By TKE, our intent is to describe the energy contained in the small scales.
References
Abgrall, R.: How to prevent pressure oscillations in multicomponent flow calculations: a Quasi conservative approach. J. Comput. Phys. 125(1), 150–160 (1996)
Adjerid, S., Devine, K.D., Flaherty, J.E., Krivodonova, L.: A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191(11–12), 1097–1112 (2002)
Adjerid, S., Massey, T.C.: Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem. Comput. Methods Appl. Mech. Eng. 195(25–28), 3331–3346 (2006)
Balakumar, B.J., Orlicz, G.C., Tomkins, C.D., Prestridge, K.P.: Simultaneous particle-image velocimetryplanar laser-induced fluorescence measurements of Richtmyer-Meshkov instability growth in a gas curtain with and without reshock. Phys. Fluids 20(12), 124103 (2008)
Brouillette, M.: The Richtmyer-Meshkov instability. Ann. Rev. Fluid Mech. 34(1), 445–468 (2002)
Cockburn, B., Hou, S., Shu, C.W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54(190), 545–581 (1990)
Cockburn, B., Lin, G., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)
Cockburn, B., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52(186), 411–435 (1989)
Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1997)
Cockburn, B., Shu, C.W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1997)
Collins, B.D., Jacobs, J.W.: PLIF flow visualization and measurements of the Richtmyer-Meshkov instability of an air/SF6 interface. J. Fluid Mech. 464, 113–136 (2002)
Cook, A.W., Dimotakis, P.E.: Transition stages of RayleighTaylor instability between miscible fluids. J. Fluid Mech. 443, 69–99 (2001)
Di Stefano, C.A., Malamud, G., Henry de Frahan, M.T., Kuranz, C.C., Shimony, A., Klein, S.R., Drake, R.P., Johnsen, E., Shvarts, D., Smalyuk, V.A., Martinez, D.: Observation and modeling of mixing-layer development in high-energy-density, blast-wave-driven shear flow. Phys. Plasmas 21(5), 056306 (2014)
Drake, R.P.: High-Energy-Density Physics. Springer, Berlin (2006)
Geuzaine, C., Remacle, J.F.: Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009)
Goncharov, V.N., McKenty, P., Skupsky, S., Betti, R., McCrory, R.L., Cherfils-Clerouin, C.: Modeling hydrodynamic instabilities in inertial confinement fusion targets. Phys. Plasmas 7(12), 5118–5139 (2000)
Hahn, M., Drikakis, D., Youngs, D.L., Williams, R.J.R.: RichtmyerMeshkov turbulent mixing arising from an inclined material interface with realistic surface perturbations and reshocked flow. Phys. Fluids 23(4), 046101 (2011)
Henry de Frahan, M.T., Johnsen, E.: Discontinuous Galerkin method for multifluid Euler equations. In: 21st AIAA Comput. Fluid Dyn. Conf., 2013–2595, pp. 1–12. American Institute of Aeronautics and Astronautics, Reston, Virginia (2013)
Henry de Frahan, M.T., Johnsen, E.: A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces. J. Comput. Phys. 280, 489–509 (2014). doi:10.1016/j.jcp.2014.09.030
Hill, D.J., Pantano, C., Pullin, D.I.: Large-eddy simulation and multiscale modelling of a Richtmyer-Meshkov instability with reshock. J. Fluid Mech. 557(2006), 29–61 (2006)
Holmes, R.L., Dimonte, G., Fryxell, B., Gittings, M.L., Grove, J.W., Schneider, M., Sharp, D.H., Velikovich, A.L., Weaver, R.P., Zhang, Q.: Richtmyer-Meshkov instability growth: experiment, simulation and theory. J. Fluid Mech. 389, 55–79 (1999)
Houim, R.W., Kuo, K.K.: A low-dissipation and time-accurate method for compressible multi-component flow with variable specific heat ratios. J. Comput. Phys. 230(23), 8527–8553 (2011)
Kifonidis, K., Plewa, T., Scheck, L., Janka, H.T., Müller, E.: Non-spherical core collapse supernovae. Astron. Astrophys. 453(2), 661–678 (2006)
Kutta, W.: Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Zeitschr. fr Math. u. Phys. 46, 435–453 (1901)
Landen, O.L., Benedetti, R., Bleuel, D., Boehly, T.R., Bradley, D.K., Caggiano, J.A., Callahan, D.A., Celliers, P.M., Cerjan, C.J., Clark, D., Collins, G.W., Dewald, E.L., Dixit, S.N., Doeppner, T., Edgell, D., Eggert, J., Farley, D., Frenje, J.A., Glebov, V., Glenn, S.M., Glenzer, S.H., Haan, S.W., Hamza, A., Hammel, B.A., Haynam, C.A., Hammer, J.H., Heeter, R.F., Herrmann, H.W., Hicks, D.G., Hinkel, D.E., Izumi, N., Gatu Johnson, M., Jones, O.S., Kalantar, D.H., Kauffman, R.L., Kilkenny, J.D., Kline, J.L., Knauer, J.P., Koch, J.A., Kyrala, G.A., LaFortune, K., Ma, T., Mackinnon, A.J., Macphee, A.J., Mapoles, E., Milovich, J.L., Moody, J.D., Meezan, N.B., Michel, P., Moore, A.S., Munro, D.H., Nikroo, A., Olson, R.E., Opachich, K., Pak, A., Parham, T., Patel, P., Park, H.S., Petrasso, R.P., Ralph, J., Regan, S.P., Remington, B.A., Rinderknecht, H.G., Robey, H.F., Rosen, M.D., Ross, J.S., Salmonson, J.D., Sangster, T.C., Schneider, M.B., Smalyuk, V., Spears, B.K., Springer, P.T., Suter, L.J., Thomas, C.A., Town, R.P.J., Weber, S.V., Wegner, P.J., Wilson, D.C., Widmann, K., Yeamans, C., Zylstra, A., Edwards, M.J., Lindl, J.D., Atherton, L.J., Hsing, W.W., MacGowan, B.J., Van Wonterghem, B.M., Moses, E.I.: Progress in the indirect-drive National Ignition Campaign. Plasma Phys. Control. Fusion 54(12), 124026 (2012)
Latini, M., Schilling, O., Don, W.S.: Effects of WENO flux reconstruction order and spatial resolution on reshocked two-dimensional Richtmyer-Meshkov instability. J. Comput. Phys. 221(2), 805–836 (2007)
Lindl, J.: Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain. Phys. Plasmas 2(11), 3933–4024 (1995)
Mikaelian, K.O.: Numerical simulations of Richtmyer-Meshkov instabilities in finite-thickness fluid layers. Phys. Fluids 8(5), 1269–1292 (1996)
Motl, B., Oakley, J., Ranjan, D., Weber, C., Anderson, M., Bonazza, R.: Experimental validation of a Richtmyer-Meshkov scaling law over large density ratio and shock strength ranges. Phys. Fluids 21(12), 126102 (2009)
Movahed, P., Johnsen, E.: Numerical simulations of the Richtmyer-Meshkov instability with reshock. In: 20th AIAA Comput. Fluid Dyn. Conf., 2011–3689, pp. 1–12. American Institute of Aeronautics and Astronautics, Reston, Virigina (2011)
Movahed, P., Johnsen, E.: A solution-adaptive method for efficient compressible multifluid simulations, with application to the Richtmyer-Meshkov instability. J. Comput. Phys. 239, 166–186 (2013)
Richtmyer, R.D.: Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Math. 13(2), 297–319 (1960)
Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981)
Schilling, O., Latini, M., Don, W.: Physics of reshock and mixing in single-mode Richtmyer-Meshkov instability. Phys. Rev. E 76(2), 026319 (2007)
Shankar, S.K., Lele, S.K.: Numerical investigation of turbulence in reshocked RichtmyerMeshkov unstable curtain of dense gas. Shock Waves 24(1), 79–95 (2013)
Taylor, G.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. A Math. Phys. Eng. Sci. 201(1065), 192–196 (1950)
Vetter, M., Sturtevant, B.: Experiments on the Richtmyer-Meshkov instability of an air/SF6 interface. Shock Waves 4(5), 247–252 (1995)
Weber, C., Haehn, N., Oakley, J., Rothamer, D., Bonazza, R.: Turbulent mixing measurements in the Richtmyer-Meshkov instability. Phys. Fluids 24(7), 074105 (2012)
Acknowledgments
This research was supported in part by the DOE NNSA/ASC under the predictive Science Academic Alliance Program by Grant No. DEFC52-08NA28616, by ONR grant N00014-12-1-0751 under Dr. Ki-Han Kim, by NSF grant CBET 1253157, and through computational resources and services provided by Advanced Research Computing at the University of Michigan, Ann Arbor.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by D. Ranjan.
This paper is based on work that was presented at the 29th International Symposium on Shock Waves, Madison, Wisconsin, USA, July 14–19, 2013.
Rights and permissions
About this article
Cite this article
Henry de Frahan, M.T., Movahed, P. & Johnsen, E. Numerical simulations of a shock interacting with successive interfaces using the Discontinuous Galerkin method: the multilayered Richtmyer–Meshkov and Rayleigh–Taylor instabilities. Shock Waves 25, 329–345 (2015). https://doi.org/10.1007/s00193-014-0539-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00193-014-0539-y