Numerical simulation of radially converging shock waves in the cavity of~a~bubble

A. A. Aganin ; T. F. Khalitova ; N. A. Khismatullina

Matematičeskoe modelirovanie, Tome 26 (2014) no. 4, pp. 3-20.

Résumé

A numerical technique of investigating of the final stage of focusing of radially converging non-spherical shock wave in the neighborhood of the center of an axially symmetric cavitation bubble subjected to strong compression is presented. In the hydrodynamic model used compressibility of the liquid, heat conductivity of the vapor and liquid, evaporation and condensation on the interphase surface are taken into account, realistic wide-range equations of state are utilized. Moving meshes with explicit tracing of the bubble surface are applied. The technique is based on the TVD-modification of the Godunov scheme of the second order of accuracy in space and time. Its efficiency is gained due to allowing for the features of the problem in the final stage of focusing of the non-spherical shock wave in the central area of the bubble. After the value of deformation of the shock wave becomes greater than some threshold (i.e., when the shock wave becomes largely non-spherical) the curvilinear radially diverging mesh in the central area of the bubble is changed by the rectilinear oblique-angled mesh close to the Cartesian one. At the same moment the non-moving system of spherical co-ordinates is replaced by the cylindrical ones. Computation of parameters of cells from one mesh to the other is performed by a method of conservative interpolation. Some results of computation of a test problem and an example illustrating the working capacity of the presented approach are given.

Détail

Keywords: bubble collapse, strong compression of the bubble, radial convergence of shock waves.

Bibliographie

[1] Wu C. C., Roberts P. H., “Shock-wave propagation in a sonoluminescing gas bubble”, Phys. Rev. Lett., 70 (1993), 3424–3427 | DOI

[2] Wu C. C., Roberts P. H., “A model of sonoluminescence”, Proc. R. Soc. Lond. A, 445 (1994), 323–349 | DOI

[3] Moss W. C., Clarke D. B., White J. W., Young D. A., “Hydrodynamic simulations of bubble collapse and picosecond somoluminescence”, Phys. Fluids, 6:9 (1994), 2979–2985 | DOI | MR

[4] Moss W. C., Clarke D. B., Young D. A., “Calculated Pulse Widths and Spectra of a Single Sonoluminescencing bubble”, Science, 276 (1997), 1398–1401 | DOI

[5] Taleyarkhan R. P., West C. D., Cho J. S., Lahey R. T. (Jr), Nigmatulin R. I., Block R. C., “Evidence for nuclear emissions during acoustic cavitation”, Science, 295 (2002), 1868–1873 | DOI

[6] Taleyarkhan R. P., West C. D., Lahey R. T. (Jr), Nigmatulin R. I., Block R. C., Xu Y., “Nuclear Emissions During Self-Nucleated Acoustic Cavitation”, Phys. Review Let., 96 (2006), 034301 | DOI

[7] Nigmatulin R. I., Taleiarkhan R. P., Lekhi R. T. (ml)., “Termoyadernyi sintez na osnove deiteriya pri akusticheskoi kavitatsii”, Vestnik Akademii nauk Respubliki Bashkortostan, 7:4 (2002), 3–25 | MR

[8] Nigmatulin R. I., Taleyarkhan R. P., Lahey R. T. (Jr), “The evidence for nuclear emissions during acoustic cavitation revisited”, J. Power and Energy, 218-A (2004), 345–364 | DOI

[9] Nigmatulin R. I., “Nano-scale thermonuclear fusion in imploding vapor bubbles”, Nuclear Eng and Design, 235 (2005), 1079–1091 | DOI

[10] Nigmatulin R. I., Akhatov I. Sh., Topolnikov A. S., Bolotnova R. Kh., Vakhitova N. K., Lahey R. T. (Jr), Taleyarkhan R. P., “The Theory of Supercompression of Vapor Bubbles and Nano-Scale Thermonuclear Fusion”, Physics of Fluid, 17 (2005), 107106 | DOI | Zbl

[11] Taleyarkhan R. P., Lapinskas J., Xu Y., Cho J. S., Block R. C., Lahey R. T. (Jr), Nigmatulin R. I., “Modeling, analysis and prediction of neutron emission spectra from acoustic cavitation bubble fusion experiments”, Nuclear Engng and Design, 238 (2008), 2779–2791 | DOI

[12] Aganin A. A., Ilgamov M. A., “Kolebaniya sfericheskogo puzyrka gaza v zhidkosti s obrazovaniem udarnykh voln”, Izv. AN. MZhG, 1999 126–133, no. 6

[13] Aganin A. A., Ilgamov M. A., “Chislennoe modelirovanie dinamiki gaza v puzyrke pri skhlopyvanii s obrazovaniem udarnykh voln”, PMTF, 40:2 (1999), 101–110 | Zbl

[14] Aganin A. A., Nigmatulin R. I., Ilgamov M. A., Akhatov I. Sh., “Dinamika puzyrka gaza v tsentre sfericheskogo ob'ema zhidkosti”, Dokl. AN, 369:2 (1999), 182–185 | Zbl

[15] Aganin A. A., “Dynamics of a small bubble in a compressible fluid”, Int. J. Numer. Meth. Fluids, 33 (2000), 157–174 | 3.0.CO;2-A class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | Zbl

[16] Aganin A. A., Ilgamov M. A., “Dinamika puzyrka gaza v tsentre sfericheskogo ob'ema zhidkosti”, Mat. modelirovanie, 13:1 (2001), 26–40 | MR | Zbl

[17] Lin H., Storey B. D., Szeri A. J., “Rayleigh–Taylor instability in violently collapsing bubbles”, Physics of Fluid, 14 (2002), 2925–2928 | DOI

[18] Lin H., Storey B. D., Szeri A. J., “Inertially driven inhomogeneities in violently collapsing bubbles: the validity of the Rayleigh–Plesset equation”, J. Fluid Mech., 452 (2002), 145–162 | DOI | MR | Zbl

[19] Aganin A. A., Khalitova T. F., Khismatullina N. A., “Raschet silnogo szhatiya sfericheskogo parogazovogo puzyrka v zhidkosti”, Vychislitelnye tekhnologii, 13:6 (2008), 17–27 | MR | Zbl

[20] Gaitan D. F., Crum L. A., “Observation of sonoluminescence from a single, stable cavitation bubble in a water/glycerine mixture”, 12th Intern. Symp. On Nonl. Acoustics, Elsevier, New York, 1990, 459–463

[21] Xu Y., Butt A., “Confirmatory experiments for nuclear emissions during acoustic cavitation”, Festschrift Edition Celebrating the 65th Birthday of Prof. Richard T. Lahey, Jr. (20–24 September 2004), Nuclear Engineering and Design, 235, no. 12, eds. Ruis P. Taleyarkan, Paolo Di' Marco, Gunther Lohnert, 2005, 1317–1324

[22] Galimov E. M., Kudin A. M., Skorobogatskii V. N., Plotnichenko V. G., Bondarev O. L., Zarubin B. G., Strazdovskii V. V., Aronin A. S., Fisenko A. V., Bykov I. V., Barinov A. Yu., “Eksperimentalnoe podtverzhdenie sinteza almaza v protsesse kavitatsii”, DAN, 395:2 (2004), 187–191

[23] Dnestrovskii A. Yu., Voropaev S. A., Ponomareva E. A., “Modelirovanie uslovii obrazovaniya almaza pri kavitatsii v benzole”, DAN, 416:5 (2011), 611–614

[24] Plesset M. S., Mitchell T. P., “On the stability of the spherical shape of a vapor cavity in a liquid”, Quart. Appl. Math., 13:4 (1956), 419–430 | MR | Zbl

[25] Birkhoff G., “Note on Taylor Instability”, Quart. Appl. Math., 12:3 (1954), 306–309 | MR | Zbl

[26] Birkhoff G., “Stability of spherical bubbles”, Quart. Appl. Math., 13 (1956), 451–453 | MR | Zbl

[27] Kull H. J., “Theory of the Rayleigh–Taylor instability”, Phys. Rep., 206 (1991), 197–325 | DOI

[28] Aganin A. A., Ilgamov M. A., Nigmatulin R. I., Toporkov D. Yu., “Evolyutsiya iskazhenii sferichnosti kavitatsionnogo puzyrka pri akusticheskom sverkhszhatii”, MZhG, 2010, no. 1, 57–69 | Zbl

[29] Nigmatulin R. I., Aganin A. A., Ilgamov M. A., Toporkov D. Yu., “Iskazhenie sferichnosti parovogo puzyrka v deiterirovannom atsetone”, DAN, 408:6 (2006), 767–771

[30] Ilgamov M. A., “Kachestvennyi analiz razvitiya otklonenii ot sfericheskoi formy pri skhlopyvanii polosti v zhidkosti”, DAN, 401:1 (2005), 37–40

[31] Ilgamov M. A., “Kachestvennaya teoriya ustoichivosti sfericheskoi formy polosti pri szhatii v zhidkosti”, Aktualnye problemy mekhaniki sploshnoi sredy, Izd-vo KGU, Kazan, 2006, 8–35

[32] Ilgamov M. A., “Otklonenie ot sferichnosti parovoi polosti v moment ee kollapsa”, DAN, 440:1 (2011), 35–38 | MR

[33] Ilgamov M. A., “Rasshirenie-szhatie i ustoichivost polosti v zhidkosti pri silnom akusticheskom vozdeistvii”, DAN, 433:2 (2010), 178–181

[34] Aganin A. A., Ilgamov M. A., Khalitova T. F., “Modelirovanie silnogo szhatiya gazovoi polosti v zhidkosti”, Matematicheskoe modelirovanie, 20:11 (2008), 89–103 | MR | Zbl

[35] Aganin A. A., Khalitova T. F., Khismatullina N. A., “Metod chislennogo resheniya zadach silnogo szhatiya nesfericheskogo kavitatsionnogo puzyrka”, Vychislitelnye tekhnologii, 15:1 (2010), 14–32 | Zbl

[36] Aganin A. A., Toporkov D. Yu., Khalitova T. F., Khismatullina N. A., “Evolyutsiya malykh iskazhenii parovogo puzyrka pri ego sverkhszhatii”, Matematicheskoe modelirovanie, 23:10 (2011), 82–96 | MR | Zbl

[37] Evans A. K., “Instability of converging shock waves and sonoluminescence”, Phys. Rev. E, 54:5 (1996), 5004–5011 | DOI

[38] Ramsey S. D., Kamm J. R., Bolstad J. H., “The Guderley problem revisited”, Int. J. CFD, 26:2 (2012), 79–99 | MR

[39] Nigmatulin R. I., Bolotnova R. Kh., “Shirokodiapazonnoe uravnenie sostoyaniya organicheskikh zhidkostei na primere atsetona”, DAN, 415:5 (2007), 617–621 | Zbl

[40] Aganin A. A., Kuznetsov V. B., “Metod konservativnoi interpolyatsii integralnykh parametrov yacheek proizvolnykh setok”, Trudy seminara, Dinamika obolochek v potoke, 18, KFTI KFAN SSSR, Kazan, 1985, 144–160