HIGH-RESOLUTION NUMERICAL METHODS FOR COMPRESSIBLE MULTI-FLUID FLOWS AND THEIR APPLICATIONS IN SIMULATIONS ZHENG JIANGUO (B.S & M.E., University of Science and Technology of China, Hefei, Anhui, P.R China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009 Acknowledgments I would like to express my sincere gratitude to my supervisors, Associate Professors T S Lee and S H Winoto for their advice, support and guidance during my thesis research I am deeply grateful to my parents, my elder sisters as well as other members of my family and my girl friend Hua Yi for their love and constant support Without them, this work would have never been possible I am indebted to Associate Professor Ma Dongjun at University of Science and Technology of China for many good suggestions I also wish to thank Dr Zhang Weiqun at the Center for Cosmology and Particle Physics of New York University for his help in implementation of adaptive mesh refinement I would like to thank all my friends for their friendship and encouragement during my four-year study at National University of Singapore Finally, I am grateful to National University of Singapore for providing me with a scholarship i Contents Acknowledgments i Contents ii Summary vi List of Tables viii List of Figures ix List of Symbols xvi Chapter Introduction and Literature Review 1.1 Background 1.2 Diffuse Interface Methods 1.2.1 Methods for Flows with Stiffened Gas EOS 1.2.2 Multi-uid Flows with Mie-Grăneisen EOS u 1.2.3 Flows Involving Barotropic Components 10 1.3 Applications of Multi-fluid Algorithms in Simulations 11 1.3.1 Numerical Simulations of Richtmyer-Meshkov Instability 11 1.3.2 Shock-bubble Interactions 13 1.4 Objectives and Significance of Study 15 1.5 Outline of Thesis 16 ii Chapter High-resolution Methods for Multi-fluid Flows with Stiffened Gas Equation of State 18 2.1 Governing Equations 19 2.1.1 Inviscid Model 19 2.1.2 Model with Viscous Effect and Gravity 22 Numerical Methods 24 2.2.1 Reconstruction of Variables 25 2.2.2 Unsplit PPM Scheme 32 2.2.3 Dimensional-splitting PPM Scheme 33 2.2.4 HLLC Riemann Solver 37 2.2.5 Two-shock Riemann Solver 38 2.2.6 Oscillation-free Property 41 2.2.7 Solution of the Diffusion Equations 43 2.3 Adaptive Mesh Refinement 44 2.4 Numerical Results 47 2.5 Summary 57 2.2 Chapter Interface-capturing Methods for Flows with General Equation of State 72 3.1 Governing Equations 73 3.2 Method Based on MUSCL-Hancock Scheme 77 3.2.1 Variables Reconstruction 77 3.2.2 HLLC Riemann Solver 79 3.2.3 Oscillation-free Property of the Present Method 82 Piecewise Parabolic Method 84 3.3.1 Unsplit PPM 84 3.3.2 Dimensional-splitting PPM 85 Adaptive Mesh Refinement 87 3.3 3.4 iii 3.5 Numerical Results with MUSCL 88 3.6 Numerical Results with PPM 96 3.7 Summary 97 Chapter High-resolution Methods for Barotropic Two-fluid and Barotropic-nonbarotropic Two-fluid Flows 4.1 112 PPM for Barotropic-nonbarotropic Two-fluid Flows 113 4.1.1 4.1.2 Governing Equations 115 4.1.3 Lagrangian-remapping PPM for Multi-fluid Flows 117 4.1.4 Riemann Solver 121 4.1.5 4.2 Equation of State 113 Numerical Results 123 PPM for Barotropic Two-fluid Flows 127 4.2.1 4.2.2 4.3 Model Equations 127 Results of Numerical Simulations 129 Summary 130 Chapter Numerical Simulations of Richtmyer-Meshkov Instability Driven by Imploding Shock 5.1 139 Richtmyer-Meshkov Instability Driven by Imploding Shock 139 5.1.1 5.1.2 Effects of Shock Strength and Perturbation Amplitude on RMI 146 5.1.3 5.2 Single-mode RMI 140 Random-mode Air-Helium Simulation 146 Summary 150 Chapter Numerical Simulations of Shock-Bubble Interactions 6.1 157 Interactions of Shock with Helium Cylinder 157 6.1.1 Setup for Numerical Simulations 157 6.1.2 Results for Ma = 1.2 159 iv 6.1.3 Results at Higher Mach Numbers 161 6.2 Interactions of Shock with Helium Sphere 162 6.3 Interactions of Shock with Krypton Cylinder 166 6.4 Interactions of Shock with Krypton Sphere 168 6.5 Summary 170 Chapter Conclusions and Future Work 192 Bibliography 195 v Summary This thesis is concerned with the development of high-resolution diffuse interface methods for resolving compressible multi-fluid flows The developed methods are subsequently applied to simulate Richtmyer-Meshkov instability (RMI) driven by cylindrical shock and shock-bubble interactions in two and three dimensions Based on ensemble averaging for multi-component flows, an inviscid compressible multi-fluid model is recovered The viscous effect and gravity can also be introduced into the model The direct Eulerian piecewise parabolic method (PPM) is modified slightly and generalized to integrate numerically the hyperbolic part of governing equations Although the resulting dimensional-splitting and unsplit PPMs are complicated, they prove more accurate in interface capturing The present methods are able to resolve the material interfaces sharply and deal with problems involving high density and pressure ratios as well as large differences in equations of state (EOSs) across an interface The use of adaptive mesh refinement (AMR) allows us to capture flow features at disparate scales MUSCL-Hancock method is extended to resolve the multi-fluid ows with components modeled by Mie-Grăneisen EOS which is referred to as general or complex u EOS By adapting HLLC approximate Riemann solver to the advection equation, the volume fraction is updated properly As a result, the method proves very stable under different situations, which is a remarkable advantage As Mie-Grăneisen EOS u can model a large number of real materials, this method can be applied to many vi problems In addition, PPM is also extended to handle general EOS To simulate flows involving one or two barotropic components, methods based on Lagrangian-remapping (LR) PPM are developed The basic idea is that the mixtures of two fluids are considered to be nonbarotropic The solution procedure is divided into Lagrangian step and separate remapping step The methods can produce sharper profiles for discontinuities, particularly contact discontinuities than other diffuse interface methods They prove quite stable and effective in dealing with the multi-fluid flows LR PPM is applied to numerically study RMI The results with our method for air-SF6 interface driven by a planar shock are found in good agreement with predictions of front tracking and theoretical models The evolution of single-mode air-SF6 interface driven by an imploding shock is highly different from that of the planar case The so-called reshock is observed In addition, random-mode perturbations are imposed on air-helium interface to mimic real problems Random nature of the perturbations significantly alters evolution of the interface The effects of shock strength and perturbation amplitude on RMI are also examined The study also concentrates on the numerical investigation of cylindrical and spherical bubbles in air accelerated by shock with Mach numbers (Ma ) in the range of 1.2 ≤ Ma ≤ The bubbles may be lighter or heavier than the ambient air, forming different configurations It is found that the time evolution of a specific bubble filled with helium or krypton is significantly altered by the shock strength For three-dimensional (3D) bubbles, some new flow features observed in experiments are reproduced numerically Our 3D results agree qualitatively well with the experimental images Some qualitative findings are reported on Mach number effects on the bubbles evolutions vii List of Tables 5.1 Air-SF6 simulation parameters 151 5.2 Air-helium simulation parameters 151 6.1 Properties of gases used in this study 172 viii List of Figures 2.1 Locations of characteristics for supersonic flow (a) and subsonic flow (b) The dashed lines labeled -, and + correspond to characteristics of speeds u − c, u and u + c, respectively Here, fluid is assumed to propagate from left to right 2.2 59 The structure of a grid block In two dimensions, each block has 12 × 12 interior cells bounded by the dash-dot line and has guard cells at each boundary 2.3 59 The sketch of a simple computational domain covered with a set of grid blocks at different refinement levels The solid lines show outlines of these blocks 2.4 60 The flux conservation at a jump in refinement The flux f on the coarse cell interface should be equal to sum of the fluxes f1 , f2 on the fine cell interfaces 2.5 60 Density contours of the square air bubble advection problem at time t = × 10−4 , using dimensional-splitting PPM The blue dashed lines show the bubble outline at initial time ix 61 Chapter Conclusions and Future Work In this thesis, the high-resolution numerical methods are developed to resolve compressible multi-fluid flows with various equations of state which are widely used to model a wide range of materials Then, the methods are applied to simulate Richtmyer-Meshkov instability and shock-bubble interactions in two and three dimensions As compared with some other methods, the dimensional-splitting and unsplit PPMs for the multi-fluid flows can provide more accurate results and have a few remarkable advantages The derivation of the inviscid model is easy to understand and the viscous effect and gravity can also be accounted for With the slight modifications in contact discontinuity detection and flattening, the splitting PPM can resolve discontinuities, particularly contact discontinuities, more sharply than other diffuse interface methods Although the unsplit PPM without detection technique is a little more diffusive than the splitting version, it is easier to implement Both PPM schemes are able to handle problems with high density and pressure ratios, proving very stable and robust In addition, the use of block-structured AMR allows the flow features at disparate scales to be resolved sufficiently To deal with the multi-fluid flows with general EOS, i.e Mie-Grăneisen EOS, u the interface-capturing methods based on MUSCL and PPMs are developed MieGrăneisen EOS can model a large number of materials including gas, liquid as well u as solid under high pressure The HLLC approximate Riemann solver is adapted to the advection equation As update of volume fraction is based on the Riemann 192 Chapter Conclusions and Future Work problems solutions rather than the velocity at previous time step, the volume fraction is advected properly and the resulting schemes are more stable than some other methods, as demonstrated by the numerical results MUSCL-Hancock scheme and PPMs are extended to reconstruct the primitive variables PPMs are much more complex than MUSCL, but are more accurate AMR can sharpen the captured interfaces significantly by increasing grid resolution locally It is found that our methods can be applied to many complex problems Besides, Lagrangian-remapping (LR) PPM is generalized to simulate barotropic two-fluid and barotropic-nonbarotropic two-fluid flows The idea behind the methods for the two types of flows is to derive a nonbarotropic artificial EOS for the two-fluid mixtures The present methods are more accurate than other secondorder schemes and produce sharper representations of the contact discontinuities, indicating that they are suitable for the interface capturing The Riemann problem solution procedure is much simpler than that in the direct Eulerian PPM In addition, LR PPM is easier to extend to the multi-fluid flows as the volume fraction is just passively advected in the separate remapping step Cylindrical RMI is investigated numerically using LR PPM code The code is validated by simulating a RMI driven by a planar shock and good agreement between the present and benchmark solutions is achieved It is found the evolution of air-SF6 interface driven by an imploding shock is much more complex than that in the case of planar shock The reshock and resulting phase inversion are observed The theoretical models are no longer applicable to the cylindrical case For the airhelium interface with random-mode perturbations, the random nature of amplitudes and wave lengths varying with the interface positions greatly complicates and affects the evolution process In addition, the shock strength and perturbation amplitude also have significant effects on RMI development The shock-accelerated bubbles filled with helium or krypton are simulated to 193 Chapter Conclusions and Future Work investigate Mach number effects on the bubbles evolutions For all cases, the higher Mach numbers lead to the stronger compression process, larger interface deformations, and more rapid bubble evolution In the case of 2D helium cylinder, when Ma is increased beyond 2, the evolution process is different from that with weak shock For the 3D helium sphere, the downstream vortex ring and upstream lobe are reproduced numerically Their dimensions and evolutions are affected strongly by the shock strength The 3D results at Ma = 1.5 are comparable to the experimental images There are significant differences in evolutions between krypton cylinder and sphere In the 3D krypton sphere case, the spike as well as hat-shaped structure arising from the collision of the reverse jet with the upstream center of sphere are observed in experiment and reproduced through our simulations The sphere shape is not altered significantly by the shock strength when Mach number is larger than Although the high-resolution numerical methods have been developed, validated and applied to simulations successfully, the work in this thesis can be extended in the following aspects: In numerical models, complex physics on the interfaces such as surface tension, heat transfer, etc, should be considered because the real interfacial problems are complicated To deal with the multi-fluid problems in irregular domains, methods based on unstructured adaptive mesh refinement in multiple dimensions should be developed In addition, problems with moving boundaries can also be considered It is desirable to numerically simulate the evolution of shock-sphere interactions at late times as the flows may become turbulent 194 Bibliography Abarbanel, S and Gottlieb, D (1981) Optimal time splitting for two- and threedimensional Navier-Stokes equations with mixed derivatives Journal of Computational Physics, 41:1–33 Abgrall, R (1996) How to prevent pressure oscillations in multicomponent flow calculations: A quasi conservative approach Journal of Computational Physics, 125:150–160 Allaire, G., Clerc, S., and Kokh, S (2002) A five-equation model for the simulation of interfaces between compressible fluids Journal of Computational Physics, 181:577–616 Ashgriz, N and Poo, J Y (1991) FLAIR: Flux line-segment model for advection and interface reconstruction Journal of Computational Physics, 93:449–468 Bagabir, A and Drikakis, D (2001) Mach number effects on shock-bubble interaction Shock Waves, 11:209–218 Bates, K R., Nikiforakis, N., and Holder, D (2007) Richtmyer-Meshkov instability induced by the interaction of a shock wave with a rectangular block of SF6 Physics of Fluids, 19:036101(1–16) Benjamin, R., Besnard, D., and Haas, J (1993) Shock and reshock of an unstable interface LANL report LA-UR92C1185, Los Alamos National Laboratory 195 Bibliography Berger, M J and Colella, P (1989) Local adaptive mesh refinement for shock hydrodynamics Journal of Computational Physics, 82:64–84 Berger, M J and Oliger, J (1984) Adaptive mesh refinement for hyperbolic partial differential equations Journal of Computational Physics, 53:484–512 Brouillette, M (2002) The Richtmyer-Meshkov instability Annual Review of Fluid Mechanics, 34:445–468 Colella, P and Woodward, P R (1984) The piecewise parabolic method (PPM) for gas-dynamics simulation Journal of Computational Physics, 54:174–201 Drew, D A and Passman, S L (1999) Theory of multicomponent fluids SpringerVerlag, New York Dutta, S., Glimm, J., Grove, J W., Sharp, D H., and Zhang, Y M (2004) Spherical Richtmyer-Meshkov instability for axisymmetric flow Mathematics and Computers in Simulation, 65:417–430 Fedkiw, R P., Aslam, T., Merriman, B., and Osher, S (1999) A non-oscillatory Eulerian approach to interface in multimaterial flows (the ghost fluid method) Journal of Computational Physics, 152:457–492 Fryxell, B., Olson, K., and Ricker, P et al (2000) Flash: an adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes The Astrophysical Journal Supplement Series, 131:273–334 Giordano, J and Burtschell, Y (2006) Richtmyer-Meshkov instability induced by shock-bubble interaction: Numerical and analytical studies with experimental validation Physics of Fluids, 18:036102(1–10) Glimm, J., Grove, J., Zhang, Y M., and Dutta, S (2002) Numerical study of 196 Bibliography axisymmetric Richtmyer-Meshkov instability and azimuthal effect on spherical mixing Journal of Statistical Physics, 107:241–260 Glimm, J., Grove, J W., Li, X L., Oh, W., and Sharp, D H (2001a) A critical analysis of Rayleigh-Taylor growth rates Journal of Computational Physics, 169:652–677 Glimm, J., Grove, J W., Li, X L., and Shyue, K M et al (1998) Three-dimensional front tracking SIAM Journal on Scientific Computing, 19:703–727 Glimm, J., Grove, J W., Li, X L., and Tan, D C (2000) Robust computational algorithms for dynamic interface tracking in three dimensions SIAM Journal on Scientific Computing, 21(6):2240–2256 Glimm, J., Li, X L., Liu, Y J., and Zhao, N (2001b) Conservative front tracking and level set algorithms Proceedings of the National Academy of Sciences, 98(25):14198–14201 Glimm, J., Marchesin, D., and McBryan, O (1981) A numerical-method for two phase flow with an unstable interface Journal of Computational Physics, 39:179– 200 Gueyffier, D., Li, J., Nadim, A., Scardovelli, R., and Zaleski, S (1999) Volumeof-fluid interface tracking with smoothed surface stress methods for threedimensional flows Journal of Computational Physics, 152:423–456 Haas, J F and Sturtevant, B (1987) Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities Journal of Fluid Mechanics, 181:41–76 Hawley, J and Zabusky, N J (1989) Vortex paradigm for shock accelerated density stratified interfaces Physical Review Letters, 63(12):1241–1244 197 Bibliography Henderson, L F., Colella, P., and Puckett, E G (1991) On the refraction of shock waves at a slow-fast gas interface Journal of Fluid Mechanics, 224:1–27 Hirt, C W and Nichols, B D (1981) Volume of fluid(VOF) method for the dynamics of free boundaries Journal of Computational Physics, 39:201–225 Holmes, R (1994) A numerical investigation of the Richtmyer-Meshkov instability using front tracking Ph.D Thesis, State University of New York at Stony Brook Jacobs, J W (1993) The dynamics of shock accelerated light and heavy gas cylinders Physics of Fluids A, 5(9):2239–2247 Johnsen, E and Colonius, T (2006) Implementation of WENO schemes in compressible multicomponent flow problems Journal of Computational Physics, 219:715–732 Karni, S (1994) Multi-component flow calculations by a consistent primitive algorithm Journal of Computational Physics, 112:31–43 Karni, S (1996) Hybrid multifluid algorithms SIAM Journal on Scientific Computing, 17:1019–1039 Kumar, S., Orlicz, G., Tomkins, C., Goodenough, C., Prestridge, K., Vorobieff, P., and Benjamin, R (2005) Stretching of material lines in shock-accelerated gaseous flows Physics of Fluids, 17:082107(1–11) Kumar, S., Vorobieff, P., Orlicz, G., Palekar, A., Tomkins, C., Goodenough, C., Marr-Lyon, M., Prestridge, K P., and Benjamin, R (2007) Complex flow morphologies in shock-accelerated gaseous flows Physica D, 235:21–28 Layes, G., Jourdan, G., and Houas, L (2003) Distortion of a spherical gaseous interface accelerated by a plane shock wave Physical Review Letters, 91(17):174502(1– 4) 198 Bibliography Layes, G., Jourdan, G., and Houas, L (2005) Experimental investigation of the shock wave interaction with a spherical gas inhomogeneity Physics of Fluids, 17:028103(1–4) Layes, G and Le Metayer, O (2007) Quantitative numerical and experimental studies of the shock accelerated heterogeneous bubbles motion Physics of Fluids, 19:042105(1–13) Lee, T S., Zheng, J G., and Winoto, S H (2009) An interface-capturing method for resolving compressible two-fluid flows with general equation of state Communications in Computational Physics, 6(5):1137–1162 LeVeque, R J and Shyue, K M (1996) Two-dimensional front tracking based on high-resolution wave propagation methods Journal of Computational Physics, 123:354–368 Liu, T G., Khoo, B C., and Yeo, K S (2003) Ghost fluid method for strong shock impacting on material interface Journal of Computational Physics, 190:651–681 Lohner, R (1987) An adaptive finite element scheme for transient problems in CFD Computer Methods in Applied Mechanics and Engineering, 61:323–338 Ma, D J (2002) Study of high resolution numerical methods for compress- ible/incompressible interfacial flows Ph.D Thesis, University of Science and Technology of China MacNeice, P., Olson, K M., Mobarry, C., de Fainchtein, R., and Packer, C (2000) PARAMESH: A parallel adaptive mesh refinement community toolkit Computer Physics Communications, 126:330–354 Marquina, A and Mulet, P (2003) A flux-split algorithm applied to conservative models for multicomponent compressible flows Journal of Computational Physics, 185:120–138 199 Bibliography Meshkov, E E (1970) Instability of a shock wave accelerated interface between two gases NASA Tech Trans F-13 Miller, G H and Puckett, E G (1996) A high order Godunov method for multiple condensed phases Journal of Computational Physics, 128:134–164 Mulder, W., Osher, S., and Sethian, J A (1992) Computing interface motion in compressible gas dynamics Journal of Computational Physics, 100:209–228 Murrone, A and Guillard, H (2005) A five equation reduced model for compressible two-phase flow computations Journal of Computational Physics, 202:664–689 Niederhaus, J H J., Greenough, J A., Oakley, J G., Ranjan, D., Anderson, M H., and Bonazza, R (2008) A computational parameter study for the threedimensional shock-bubble interaction Journal of Fluid Mechanics, 594:85–124 Nourgaliev, R R., Dinh, T N., and Theofanous, T G (2006) Adaptive characteristics-based matching for compressible multifluid dynamics Journal of Computational Physics, 213:500–529 Osher, S and Fedkiw, R P (2001) Level set methods: an overview and some recent results Journal of Computational Physics, 169:463–502 Osher, S and Sethian, J A (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations Journal of Computational Physics, 79:12–49 Palekar, A., Vorobieff, P., and Truman, C R (2000) Two-dimensional simulation of Richtmyer-Meshkov instability AIAA Paper, American Institute of Aeronautics and Astronautics Parker, B J and Youngs, D L (1992) Two and three dimensional Eulerian simula- 200 Bibliography tion of fluid flow with material interfaces Technical report, UK Atomic Weapons Establishment Perigaud, G and Saurel, R (2005) A compressible flow model with capillary effects Journal of Computational Physics, 209:139–178 Picone, J M and Boris, J P (1988) Vorticity generation by shock propagation through bubbles in a gas Journal of Fluid Mechanics, 189:23–51 Pilliod Jr, J E and Puckett, E G (1997) Second-order accurate volume-of-fluid algorithms for tracking material interface Technical report LBNL-40744, Lawrence Berkeley National Laboratory Pilliod Jr, J E and Puckett, E G (2004) Second-order accurate volume-of-fluid algorithms for tracking material interfaces Journal of Computational Physics, 1999:465–502 Qiu, J X., Khoo, B C., and Shu, C W (2006) A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes Journal of Computational Physics, 212:540–565 Quirk, J J and Karni, S (1996) On the dynamics of a shock-bubble interaction Journal of Fluid Mechanics, 318:129–164 Ranjan, D., Anderson, M., Oakley, J., and Bonazza, R (2005) Experimental investigation of a strongly shocked gas bubble Physical Review Letters, 94:184507(1–4) Ranjan, D., Niederhaus, J., Motl, B., Anderson, M., Oakley, J., and Bonazza, R (2007) Experimental investigation of primary and secondary features in highmach-number shock-bubble interaction Physical Review Letters, 98:024502(1–4) Ranjan, D., Niederhaus, J H J., Oakley, J G., Anderson, M H., Bonazza, R., and Greenough, J A (2008) Shock-bubble interactions: Features of divergent 201 Bibliography shock-refraction geometry observed in experiments and simulations Physics of Fluids, 20:036101(1–20) Richtmyer, R D (1960) Taylor instability in shock acceleration of compressible fluids Communications on Pure and Applied Mathematics, 13:297–319 Rider, W J and Kothe, D B (1995) Stretching and tearing interface tracking methods AIAA Paper 95-1717, American Institute of Aeronautics and Astronautics Rider, W J and Kothe, D B (1998) Reconstructing volume tracking Journal of Computational Physics, 141:112–152 Rudman, M (1997) Volume-tracking methods for interfacial flow calculations International Journal for Numerical Methods in Fluids, 24:671–691 Rudman, M (1998) A volume-tracking method for incompressible multifluid flows with large density variations International Journal for Numerical Methods in Fluids, 28:357–378 Saltz, D., Graham, M J., Holmes, R L., Zoldi, C A., Weaver, R P., and Gittings, M L (1999) Richtmyer-Meshkov instability in cylindrical geometry Technical report Samtaney, R and Zabusky, N J (1994) Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws Journal of Fluid Mechanics, 269:45–78 Saurel, R and Abgrall, R (1999a) A multiphase Godunov method for compressible multifluid and multiphase flows Journal of Computational Physics, 150:425–467 Saurel, R and Abgrall, R (1999b) A simple method for compressible multifluid flows SIAM Journal on Scientific Computing, 21(3):1115–1145 202 Bibliography Scardovelli, R and Zaleski, S (1999) Direct numerical simulation of free-surface and interfacial flow Annual Review of Fluid Mechanics, 31:567–603 Sethian, J A and Smereka, P (2003) Level set methods for fluid interfaces Annual Review of Fluid Mechanics, 35:341–372 Shi, J., Zhang, Y T., and Shu, C W (2003) Resolution of high order WENO schemes for complicated flow structures Journal of Computational Physics, 186:690–696 Shu, C W (1997) Essentially non-oscillatory and weighted essentially non- oscillatory schemes for hyperbolic conservation laws ICASE Report NASA/CR97-206253 No 97-65, National Aeronautics and Space Administration Shu, C W and Osher, S (1988) Efficient implementation of essentially nonoscillatory shock-capturing scheme Journal of Computational Physics, 77:439– 471 Shyue, K M (1998) An efficient shock-capturing algorithm for compressible multicomponent problems Journal of Computational Physics, 142:208–242 Shyue, K M (1999a) A fluid-mixture type algorithm for compressible multicomponent flow with van der waals equation of state Journal of Computational Physics, 156:43–88 Shyue, K M (1999b) A volume-of-fluid type algorithm for compressible two-phase flows International Series of Numerical Mathematics, 130:895–904 Shyue, K M (2001) A fluid-mixture type algorithm for compressible multicomponent ow with Mie-Grăneisen equation of state Journal of Computational u Physics, 178:678–707 203 Bibliography Shyue, K M (2004) A fluid-mixture type algorithm for barotropic two-fluid flow problems Journal of Computational Physics, 200:718–748 Shyue, K M (2006a) Mathematical models and numerical methods for compressible multicomponent flow Workshop on CFD based on Unified Coordinates– Theory & Applications, Research Center for Applied Sciences Shyue, K M (2006b) A volume-fraction based algorithm for hybrid barotropic and non-barotropic two-fluid flow problems Shock Waves, 15(6):407–423 Shyue, K M (2006c) A wave-propagation based volume tracking method for compressible multicomponent flow in two space dimensions Journal of Computational Physics, 215:219–244 Strang, G (1968) On the construction and comparison of difference schemes SIAM Journal on Numerical Analysis, 5(3):506–517 Sun, M and Takayama, K (1999) Conservative smoothing on an adaptive quadrilateral grid Journal of Computational Physics, 150:143–180 Taylor, G I (1950) The instability of liquid surfaces when accelerated in a direction perpendicular to their planes I Proc R Soc A., 201:192–196 Thompson, P A (1972) Compressible-fluid dynamics McGraw-Hill, New York Tomkins, C., Prestridge, K., Rightley, P., and Marr-Lyon, M (2003) A quantitative study of the interaction of two Richtmyer-Meshkov-unstable gas cylinders Physics of Fluids, 15(4):986–1004 Tomkins, C., Prestridge, K., Rightley, P., Vorobieff, P., and Benjamin, R (2002) Flow morphologies of two shock-accelerated unstable gas cylinders Journal of Visualization, 5(3):273–283 204 Bibliography Toro, E F (1999) Riemann solvers and numerical methods for fluid dynamics: A practical introduction Springer-Verlag, Berlin Heidelberg van Brummelen, E H and Koren, B (2003) A pressure-invariant conservative Godunov-type method for barotropic two-fluid flows Journal of Computational Physics, 185:289–308 van Leer, B (1979) Towards the ultimate conservative difference scheme V A second-order sequel to Godunov’s method Journal of Computational Physics, 32:101–136 van Leer, B (2006) Upwind and high-resolution methods for compressible flow: from donor cell to residual-distribution scheme Communications in computational physics, 1(2):192–206 Wackers, J and Koren, B (2005) A fully conservative model for compressible twofluid flow International Journal for Numerical Methods in Fluids, 47:1337–1343 Woodward, P and Colella, P (1984) The numerical simulation of two-dimensional fluid flow with strong shocks Journal of Computational Physics, 54:115–173 Yang, Y., Zhang, Q., and Sharp, D H (1994) Small amplitude theory of RichtmyerMeshkov instability Physics of Fluids, 6(5):1856–1873 Zabusky, N J (1999) Vortex paradigm for accelerated inhomogeneous flows: Visiometrics for the Rayleigh-Taylor and Richtmyer-Meshkov environments Annual Review of Fluid Mechanics, 31:495–536 Zhang, Q and Graham, M J (1998) A numerical study of Richtmyer-Meshkov instability driven by cylindrical shocks Physics of Fluids, 10(4):974–992 Zhang, S., Zabusky, N J., Peng, G Z., and Gupta, S (2004) Shock gaseous cylinder interactions: Dynamically validated initial conditions provide excellent agreement 205 Bibliography between experiments and numerical simulations to late-intermediate time Physics of Fluids, 16(5):1203–1216 Zhang, W Q and MacFadyen, A I (2006) Ram: a relativistic adaptive mesh refinement hydrodynamics code The Astrophysical Journal Supplement Series, 164:255–279 Zheng, H W., Shu, C., and Chew, Y T (2008a) An object-oriented and quadrilateral-mesh based solution adaptive algorithm for compressible multi-fluid flows Journal of Computational Physics, 227:6895–6921 Zheng, J G., Lee, T S., and Ma, D J (2007) A piecewise parabolic method for barotropic two-fluid flows International Journal of Modern Physics C, 18(3):375– 390 Zheng, J G., Lee, T S., and Winoto, S H (2008b) Numerical simulation of Richtmyer-Meshkov instability driven by imploding shocks Mathematics and Computers in Simulation, 79:749–762 Zheng, J G., Lee, T S., and Winoto, S H (2008c) A piecewise parabolic method for barotropic and nonbarotropic two-fluid flows International Journal of Numerical Methods for Heat & Fluid Flow, 18(6):708–729 Zoldi, C A (2002) A numerical and experimental study of a shock-accelerated heavy gas cylinder Ph.D Thesis, SUNY Stony Brook 206 ... this study and recommendation for future work are given 17 Chapter High- resolution Methods for Multi- fluid Flows with Stiffened Gas Equation of State In this chapter1 , high- resolution methods are... construct high- resolution methods with AMR, which are very robust and stable Chapter 1.2.3 Introduction and Literature Review Flows Involving Barotropic Components Practical applications often involve... material interfaces and can handle flows with more complex EOS 1.3 Applications of Multi- fluid Algorithms in Simulations 1.3.1 Numerical Simulations of Richtmyer-Meshkov Instability When an incident