×

zbMATH — the first resource for mathematics

Direct numerical simulation of interfacial instabilities: A consistent, conservative, all-speed, sharp-interface method. (English) Zbl 1299.76097
Summary: We present a conservative and consistent numerical method for solving the Navier-Stokes equations in flow domains that may be separated by any number of material interfaces, at arbitrarily-high density/viscosity ratios and acoustic-impedance mismatches, subjected to strong shock waves and flow speeds that can range from highly supersonic to near-zero Mach numbers. A principal aim is prediction of interfacial instabilities under superposition of multiple potentially-active modes (Rayleigh-Taylor, Kelvin-Helmholtz, Richtmyer-Meshkov) as found for example with shock-driven, immersed fluid bodies (locally oblique shocks) – accordingly we emphasize fidelity supported by physics-based validation, including experiments. Consistency is achieved by satisfying the jump discontinuities at the interface within a conservative 2nd-order scheme that is coupled, in a conservative manner, to the bulk-fluid motions. The jump conditions are embedded into a Riemann problem, solved exactly to provide the pressures and velocities along the interface, which is tracked by a level set function to accuracy of \(O(\Delta x^5,\Delta t^4)\). Subgrid representation of the interface is achieved by allowing curvature of its constituent interfacial elements to obtain \(O(\Delta x^3)\) accuracy in cut-cell volume, with attendant benefits in calculating cell- geometric features and interface curvature \((O(\Delta x^3))\). Overall the computation converges at near-theoretical \(O(\Delta x^2)\). Spurious-currents are down to machine error and there is no time-step restriction due to surface tension. Our method is built upon a quadtree-like adaptive mesh refinement infrastructure. When necessary, this is supplemented by body-fitted grids to enhance resolution of the gas dynamics, including flow separation, shear layers, slip lines, and critical layers. Comprehensive comparisons with exact solutions for the linearized Rayleigh-Taylor and Kelvin-Helmholtz problems demonstrate excellent performance. Sample simulations of liquid drops subjected to shock waves demonstrate for the first time ab initio numerical prediction of the key interfacial features and phenomena found in recent experimental and theoretical studies of this class of problems [T.G. Theofanous, Ann. Rev. Fluid Mech. 43, 661–690 (2011; Zbl 1299.76217)].

MSC:
76F65 Direct numerical and large eddy simulation of turbulence
76M25 Other numerical methods (fluid mechanics) (MSC2010)
Software:
Gerris; AUSM
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abgrall, R., How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach, J. Comput. Phys., 125, 150-160, (1996) · Zbl 0847.76060
[2] R. Babarsky, T.G. Theofanous, An assessment of the state-of-the-art on aerodynamic/explosive dissemination of chemical agents with perspectives for future work, Natl. Ground Intel. Center Rep. WF-67774 (2010).
[3] Bailey, A. B.; Hiatt, J., Sphere drag coefficients for a broad range of Mach and Reynolds number, AIAA J., 10, 11, 1436-1440, (1972)
[4] Bell, J. B.; Colella, P.; Glaz, H. M., A second-order projection method for the incompressible navier – stokes equations, J. Comput. Phys., 85, 257-283, (1989) · Zbl 0681.76030
[5] Bell, J. B.; Marcus, D. L., A second-order projection method for variable-density flows, J. Comput. Phys., 101, 334-348, (1992) · Zbl 0759.76045
[6] P.G. Bidone, Expériences sur la Forme et sur la Direction des Yeines et des Courans d’Eau lancés par diverses Ouvertures, Cited by Rayleigh [71].
[7] Bo, W.; Liu, X.; Glimm, J.; Li, X. L., A robust front tracking method: verification and application to simulation of the primary breakup of a liquid jet, SIAM J. Sci. Comput., 33, 4, 1505-1524, (2011) · Zbl 1465.76084
[8] Boeck, T.; Zaleski, S., Viscous versus inviscid instability of two-phase mixing layers with continuous velocity profile, Phys. Fluids, 17, 032106, (2005) · Zbl 1187.76057
[9] Boeck, T.; Li, J.; López-Pagés, E.; Yecko, P.; Zaleski, S., Ligament formation in sheared liquid – gas layers, Theor. Comput. Fluid Dyn., 21, 59-76, (2007) · Zbl 1170.76360
[10] Brackbill, J. U.; Kothe, D. B.; Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys., 100, 335-354, (1992) · Zbl 0775.76110
[11] S.R. Chakravarthy and S. Osher, High resolution applications of the Osher upwind scheme for the Euler equations, AIAA Paper 83-1943, in: Proceedings of AIAA 6th Computational Fluid Dynamics Conference, 1983, pp. 363-373.
[12] Chandrasekhar, S., Hydrodynamic and hydromagnetic stability, (1961), Oxford University Press · Zbl 0142.44103
[13] Chang, C.-H.; Liou, M.-S., A robust and accurate approach to computing compressible multiphase flow: stratified flow model and AUSM^{+}-up scheme, J. Comput. Phys., 225, 840-873, (2007) · Zbl 1192.76030
[14] Chern, I.-L.; Glimm, J.; McBryan, O.; Plohr, B.; Yaniv, S., Front tracking for gas dynamics, J. Comput. Phys., 62, 83-110, (1986) · Zbl 0577.76068
[15] Chorin, A. J., The numerical solution of the navier – stokes equations for an incompressible fluid, AEC research and development report, NYO-1480-82, (1967), New York University New York
[16] P. Colella, An interview with Peter D. Lax, conducted by Society for Industrial and Applied Mathematics. <http://history.siam.org/oralhistories/lax.htm>, 2006.
[17] Davis, S. F., An interface tracking method for hyperbolic systems of conservation laws, Appl. Numer. Math., 10, 447-472, (1992) · Zbl 0766.65067
[18] Drazin, P. G.; Ried, W. H., Hydrodynamic stability, (2004), Cambridge University Press
[19] Degrez, G.; Boccadoro, C. H.; Wendt, J. F., The interaction of an oblique shock with a laminar boundary revisited. an experiment and numerical study, J. Fluid Mech., 177, 247-263, (1987)
[20] Fedkiw, R. P.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory approach to interface in multimaterial flows (the ghost flow method), J. Comput. Phys., 152, 457-492, (1999) · Zbl 0957.76052
[21] Francois, M. M.; Cummins, S. J.; Dendy, E. D.; Kothe, D. B.; Sicilian, J. M.; Williams, M. W., A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework, J. Comput. Phys., 213, 1, 141-173, (2006) · Zbl 1137.76465
[22] Fuster, D.; Bagué, A.; Boeck, T; Le Moyne, L.; Leboissetier, A.; Popinet, S.; Ray, P.; Scardovelli, R.; Zaleski, S., Int. J. Multiphase Flow, 35, 550-565, (2009)
[23] Fyfe, D. E.; Oran, E. S.; Fritts, M. J., Surface tension and viscosity with Lagrangian hydrodynamics on a triangular mesh, J. Comput. Phys., 76, 349-384, (1988) · Zbl 0639.76043
[24] Gatti-Bono, C.; Colella, P.; Trebotich, D., A second-order accurate conservative front-tracking method in one dimension, SIAM J. Sci. Comput., 31, 6, 4795-4813, (2010) · Zbl 1323.76073
[25] Gardner, C. L.; Glimm, J.; McBryan, O.; Menikoff, R.; Sharp, D. H.; Zhang, Q., The dynamics of bubble growth for rayleigh – taylor unstable interfaces, Phys. Fluids, 31, 3, 447-465, (1988) · Zbl 0641.76099
[26] Godunov, S. K.; Zabrodin, A. V.; Prokopov, G. P., A computational scheme for two-dimensional non stationary problems of gas dynamics and calculation of the flow from a shock wave approaching a stationary state, USSR, Comput. Math. Math. Phys., 1, 1962, 1187-1219, (1962) · Zbl 0146.23004
[27] Glimm, J.; Grove, J. W.; Lin, X. L.; Shyue, K.-M.; Zeng, Y.; Zhang, Q., Three-dimensional front tracking, SIAM J. Sci. Comput., 19, 3, 703-727, (1998) · Zbl 0912.65075
[28] J. Glimm, X. Li, Y. Liu, N. Zhao, Conservative front tracking and level set algorithms, in: Proceedings of the National Academy of Sciences of the United States of America, vol. 98, no. 25, 2001, pp. 14198-14201. · Zbl 1005.65091
[29] Glimm, J.; Grove, J. W.; Li, X. L.; Oh, W.; Sharp, D. H., A critical analysis of rayleigh – taylor growth rates, J. Comput. Phys., 169, 652-677, (2001) · Zbl 1011.76057
[30] Glimm, J.; Grove, J. W.; Zhang, Y., Interface tracking for axisymmetric flows, SIAM J. Sci. Comput., 24, 1, 208-236, (2002) · Zbl 1062.76034
[31] Glimm, J.; Li, X.; Liu, Y.; Xu, Z.; Zhao, N., Conservative front tracking with improved accuracy, SIAM J. Numer. Anal., 41, 5, 1926-1947, (2003) · Zbl 1053.35093
[32] Grove, J., The interaction of shock waves with fluid interfaces, Adv. Appl. Math., 10, 201-227, (1989) · Zbl 0669.76085
[33] Harper, E. Y.; Grube, G. W.; Chang, I., On the breakup of accelerating liquid drops, J. Fluid Mech., 52, 565-591, (1972) · Zbl 0245.76040
[34] Harten, A.; Hyman, J., Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys., 50, 235-269, (1983) · Zbl 0565.65049
[35] Harten, A., ENO schemes with subcell resolution, J. Comput. Phys., 83, 148-184, (1989) · Zbl 0696.65078
[36] Hou, T. Y.; Le Floch, P. G., Why nonservative schemes converge to wrong solutions: error analysis, 62, 206, 497-530, (1994) · Zbl 0809.65102
[37] Hu, X. Y.; Khoo, B. C.; Adams, N. A.; Huang, F. L., A conservative interface method for compressible flow, J. Comput. Phys., 219, 553-578, (2006) · Zbl 1102.76038
[38] Jameson, A., Solution of the Euler equations for two dimensional transonic flow by a multigrid method, Appl. Math. Comput., 13, 327-355, (1983) · Zbl 0545.76065
[39] Karni, S., Multicomponent flow calculations by a consistent primitive algorithm, J. Comput. Phys., 112, 31-43, (1994) · Zbl 0811.76044
[40] Lafaurie, B.; Zaleski, S.; Zanetti, G., Modelling merging and fragmentation in multiphase flows with SURFER, J. Comput. Phys., 113, 134-147, (1994) · Zbl 0809.76064
[41] Lax, P. D., Weak solution of nonlinear hyperbolic equations and their numerical computation, Commun. Pure Appl. Math., 7, 159-193, (1954) · Zbl 0055.19404
[42] Lax, P. D.; Richtmyer, S. D., Survey of the stability of linear finite difference equations, Commun. Pure Appl. Math., 9, 267-293, (1956) · Zbl 0072.08903
[43] Lax, P. D., Hyperbolic system of conservation laws, Commun. Pure Appl. Math., 10, 537-566, (1957) · Zbl 0081.08803
[44] Lax, P. D.; Wendroff, B., Systems of conservation laws, Commun. Pure Appl. Math., 13, 217-237, (1960) · Zbl 0152.44802
[45] LeVeque, R. J.; Shyue, K.-M., One-dimensional front tracking based on high resolution wave propagation methods, SIAM J. Sci. Comput., 16, 2, 348-377, (1995) · Zbl 0824.65094
[46] LeVeque, R. J.; Shyue, K.-M., Two-dimensional front tracking based on high resolution wave propagation methods, J. Comput. Phys., 123, 354-368, (1996) · Zbl 0849.65063
[47] Liepmann, H. W.; Roshko, A., Elements of gasdynamics, (1957), John Wiley and Sons · Zbl 0078.39901
[48] Liou, M.-S., A sequel to AUSM: AUSM^{+}, J. Comput. Phys., 129, 364-382, (1996) · Zbl 0870.76049
[49] Liou, M.-S., A sequel to AUSM: AUSM^{+}-up for all speed, ^{+}, J. Comput. Phys., 214, 137-170, (2006) · Zbl 1137.76344
[50] Liu, T. G.; Khoo, B. C.; Yeo, K. S., Ghost fluid method for strong shock impacting on material interface, J. Comput. Phys., 190, 651-681, (2003) · Zbl 1076.76592
[51] Liu, T. G.; Khoo, B. C.; Wang, C. W., The ghost fluid method for compressible gas – water simulation, J. Comput. Phys., 204, 193-221, (2005) · Zbl 1190.76160
[52] Liu, T. G.; Khoo, B. C., The accuracy of the modified ghost fluid method for gas-gas Riemann problem, Appl. Numer. Math., 57, 721-733, (2007) · Zbl 1190.76155
[53] Liu, J.; Lim, H.-K.; Glimm, J.; Li, X. L., A conservative front tracking method in N-dimensions, J. Sci. Comput., 31, 213-236, (2007) · Zbl 1115.76357
[54] Mao, D.-K., A shock tracking technique based on conservation in one space dimension, SIAM J. Numer. Anal., 32, 5, 1603-1677, (1995) · Zbl 0836.65098
[55] Mao, D.-K., Towards front-tracking based on conservation in two space dimensions II, tracking discontinuities in capturing fashion, J. Comput. Phys., 226, 1550-1588, (2007) · Zbl 1128.65066
[56] Marmottant, P.; Villermaux, E., On spray formation, J. Fluid Mech., 498, 73-111, (2004) · Zbl 1067.76512
[57] Meshkov, E. E., Instability of the interface of two gases accelerated by a shock wave, Sov. Fluid Dyn., 4, 101-104, (1969)
[58] Mikaelian, K. O., Oblique shocks and combined rayleigh – taylor, kelvin – helmholtz, and richtmyer – meshkov instabilities, Phys. Fluids, 6, 6, 1943-1945, (1994) · Zbl 0828.76044
[59] Mulder, W.; Osher, S.; Sethian, Computing interface motion in compressible gas dynamics, J. Comput. Phys., 100, 209-228, (1992) · Zbl 0758.76044
[60] C.-L. Ng, T.G. Theofanous, Modes of aero-Breakup with visco-elastic liquid, in: AIP Conference of Proceedings, vol. 1027, 2008, p. 183.
[61] Nourgaliev, R. R.; Dinh, T. N.; Theofanous, T. G., A pseudo-compressibility method for the simulation of multiphase incompressible flows, Int. J. Multiph. Flow, 30, 7-8, 901-937, (2004) · Zbl 1136.76593
[62] Nourgaliev, R. R.; Dinh, T. N.; Theofanous, T. G., Adaptive characteristics-based matching for compressible multifluid dynamics, J. Comput. Phys., 213, 500-529, (2006) · Zbl 1136.76396
[63] Nourgaliev, R. R.; Theofanous, T. G., High-fidelity interface tracking in compressible flows: unlimited anchored adaptive level set, J. Comput. Phys., 224, 836-866, (2007) · Zbl 1124.76043
[64] Nourgaliev, R. R.; Liou, M.-S.; Theofanous, T. G., Numerical prediction of interfacial instabilities: sharp interface method (SIM), J. Comput. Phys., 227, 3940-3970, (2008) · Zbl 1275.76164
[65] Nourgaliev, R. R.; Kadioglu, S.; Mousseau, V., Marker re-distancing/level set method for high-fidelity implicit interface tracking”, SIAM J. Sci. Comput., 32, 1, 320-348, (2010) · Zbl 1209.35081
[66] Osher, S.; Sethian, J. A., Fronts propagating with curvature-dependent speed: algorithms based on hamilton – jacobi formulations, J. Comput. Phys., 79, 12-49, (1988) · Zbl 0659.65132
[67] Pearson, C. E., A computational method for viscous flow problems, J. Fluid Mech., 21, 4, 611-622, (1965) · Zbl 0131.41204
[68] Popinet, S.; Zaleski, S., A front-tracking algorithm for accurate representation of surface tension, Int. J. Numer. Methods Fluids, 30, 775-793, (1999) · Zbl 0940.76047
[69] Popinet, S., An accurate adaptive solver for surface-tension-driven interfacial flows, J. Comput. Phys., 228, 5838-5866, (2009) · Zbl 1280.76020
[70] Puckett, E. G.; Almgren, A. S.; Bell, J. B.; Marcus, D. L.; Rider, W. J., A high-order projection method for tracking fluid interfaces in variable density incompressible flows, J. Comput. Phys., 130, 269-282, (1997) · Zbl 0872.76065
[71] Rayleigh, F. R.S., On the capillary phenomena of jets, Proc. R. Soc. Lond., 29, 71-97, (1879)
[72] Richtmyer, R. D., Taylor instability in a shock acceleration of compressible fluids, Commun. Pure Appl. Math., 13, 297-319, (1960)
[73] Rider, W. J.; Kothe, D. B., Reconstructing volume tracking, J. Comput. Phys., 141, 112-152, (1998) · Zbl 0933.76069
[74] Sambasivan, S. K.; Udaykumar, H. S., Sharp interface simulations with local mesh refinement for multi-material dynamics in strong shocked flows, Comput. Fluids, 39, 1456-1479, (2010) · Zbl 1245.76111
[75] Schroeder, C.; Zheng, W.; Fedkiw, R., Semi-implicit surface tension formulation with a Lagrangian surface mesh on an Eulerian simulation grid, J. Comput. Phys., 231, 2092-2115, (2012) · Zbl 1382.76197
[76] A.H. Shapiro, Dynamics and Thermodynamics of Compressible Fluid Flow, vol. 2, Wiley, 1953.
[77] Shi, J.; Zhang, Y.-T.; Shu, C.-W., Resolution of high order WENO schemes for complicated flow structures, J. Comput. Phys., 186, 690-696, (2003) · Zbl 1047.76081
[78] J.L. Steger, F.C. Dougherty, J.A. Benek, A Chimera grid scheme, Advances in Grid Generation, in: K. N. Ghia, U. Ghia (eds.), ASME FED, vol. 5, June, 1983.
[79] Shin, S.; Abdel-Khalik, S. I.; Daru, V.; Juric, D., Accurate representation of surface tension using the level contour reconstruction method, J. Comput. Phys., 203, 493-516, (2005) · Zbl 1143.76561
[80] Stengel, K. C.; Oliver, D. S.; Booker, J. R., Onset of convection in a variable-viscosity fluid, J. Fluid Mech., 120, 411-431, (1982) · Zbl 0534.76093
[81] Sommeijer, B. P.; van der Houwen, P. J.; Kok, J., Time integration of three-dimensional numerical transport models, Appl. Numer. Math., 16, 201-225, (1994) · Zbl 0819.65122
[82] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146-159, (1994) · Zbl 0808.76077
[83] Sussman, M.; Almgren, A. S.; Bell, J. B.; Colella, P.; Howell, L. H.; Welcome, M. L., An adaptive level set approach for incompressible two-phase flows, J. Comput. Phys., 148, 81-124, (1999) · Zbl 0930.76068
[84] Swartz, B. K.; Wendroff, B., AZTEC: a front tracking code based on godunov’s method, Appl. Numer. Math., 2, 385-397, (1986) · Zbl 0601.76088
[85] Tauber, W.; Unverdi, S. O.; Tryggvason, G., The nonlinear behavior of a shear immiscible fluid interface, Phys. Fluids, 14, 8, 2871-2885, (2002) · Zbl 1185.76364
[86] Taylor, G. I., On the decay of vortices in a viscous fluid, Philos. Mag., 46, 671-674, (1923) · JFM 49.0607.02
[87] Terashima, H.; Tryggvason, G., A front-tracking/ghost-fluid method for fluid interfaces in compressible flows, J. Comput. Phys., 228, 4012-4037, (2009) · Zbl 1171.76046
[88] T.G. Theofanous, S. Sushckikh, R.R. Nourgaliev, Linear stability of sharp and diffuse interfaces under shear, in: FEDSM07 5th Joint ASME/JSME Fluids Engineering Summer Conference, San Diego, California, USA, July 30-August 2, 2007.
[89] T.G. Theofanous, R.R. Nourgaliev, B. Khomami, Short communication: An experimental/theoretical investigation of interfacial instabilities in superposed pressure-driven channel flow of Newtonian and well characterized viscoelastic fluids - Part I. Linear stability and encapsulation effects, by B. Khomami, K. C. Su, J. Non-Newtonian Fluid Mech. 143 (2007) 131.
[90] T.G. Theofanous, R.R. Nourgaliev, S. Wiri, Brief Communication: Direct numerical simulations of two-layer viscosity-stratified flow, by Qing Cao, Kausik Sarkar, Ajay K. Prasad, Int. J. Multiphase Flow 30 (2004), 1485-1508, Int. J. Multiphase Flow 33 (2007) 789.
[91] Theofanous, T. G.; Li, G. J., On the physics of aerobreakup, Phys. Fluids, 20, 052103, (2008) · Zbl 1182.76756
[92] Theofanous, T. G.; Mitkin, V. V.; Ng, C. L.; Chang, C.-H.; Deng, X.; Sushchikh, S., The physics of aerobreakup part II: viscous liquids, Phys. Fluids, 24, 022104, (2011)
[93] Theofanous, T. G., Aerobreakup of Newtonian and viscoelastic liquids, Annu. Rev. Fluid Mech., 43, 661-690, (2011) · Zbl 1299.76217
[94] T.G. Theofanous, V.V. Mitkin, The physics of aerobreakup Part III: Viscoelastic liquid, Phys. Fluids, in press.
[95] Torres, D. J.; Brackbill, J. U., The point-set method: front-tracking without connectivity, J. Comput. Phys., 165, 620-644, (2000) · Zbl 0998.76070
[96] Tryggvason, G.; Bunner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S.; Jan, Y.-J., A front-tracking method for the computations of multiphase flow, J. Comput. Phys., 169, 708-759, (2001) · Zbl 1047.76574
[97] Tryggvason, G.; Scardovelli, R.; Zaleski, S., Direct numerical simulations of gas – liquid multiphase flows, (2011), Cambridge University Press · Zbl 1226.76001
[98] M. Vargas, Current experimental basis for modeling ice accretions on sweep wings, AIAA-2005-5188, in: Fourth Theoretical Fluid Mechanics Meeting, Toronto, Canada, June 6-9, 2005.
[99] Wang, C. W.; Liu, T. G.; Khoo, B. C., A real ghost fluid method for simulation of multimedium compressible flow, SIAM J. Sci. Comput., 28, 278-302, (2006) · Zbl 1114.35119
[100] P.F. Waters, J.C. Trippe, New concepts in octane boosting of fuels for internal combustion engines, in: 220th National Meeting, American Chemical Society, August 22-24 2000, Washington, DC.
[101] Xu, L.; Liu, T. G., Accuracy and conservation error of various ghost fluid methods for multi-medium Riemann problem, J. Comput. Phys., 230, 4975-4990, (2011) · Zbl 1416.76233
[102] Yan, L.; Mao, D.-K., Further development of a conservative front-tracking method for systems of conservation laws in on space dimension, J. Sci. Comput., 28, 1, 85-119, (2005)
[103] Yih, C. S., Instability due to viscosity stratification, J. Fluid Mech., 27, 2, 337-352, (1967) · Zbl 0144.47102
[104] Zalesak, S., Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31, 335-362, (1979) · Zbl 0416.76002
[105] Zhao, H.-K.; Chan, T.; Merriman, B.; Osher, S., A variational level set approach to multiphase motion, J. Comput. Phys., 127, 179-195, (1996) · Zbl 0860.65050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.