© 2020 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
In this paper, we present the hydrodynamic mechanisms which occur between a low-hypersonic shock wave and a millimetric water droplet. To do so, two numerical models, based respectively on the Volume of Fluid (VOF) and Diffuse Interfaces (DI) approaches, are developed. The goal is to compare the results obtained with the models in order to evaluate which is the most accurate to describe the evolution of the physical phenomena. The studied Mach number and initial droplet diameter are 4.25 and 1.135 mm, respectively. Each model allows the compressible Euler equations to be solved in a 2D-axi-symmetric configuration. The evolution of both air and liquid phases is modelled by a stiffened gas equation of state. For qualitative validation, the numerical results are compared to experimental data recently presented in the literature. In this work, the authors used a shock tube test facility and a shadowgraph visualization technique to observe the phenomenology over a long time. Their investigation shows that the droplet deformation, detached bow shock and recompression waves are well captured by the two models until a Rayleigh dimensionless time of 1.5. Beyond this critical time, and up to 3, some differences appear between the two numerical approaches, especially on the droplet deformation. Globally, the droplet deformation is better described with the VOF model, while the DI model appears to be more accurate when it comes to the evaluation of the position of the bow shock. In the discussion section, some ideas are proposed to improve the models.
Diffuse Interfaces, shock wave, stiffened gas equations, Volume of Fluid, water droplet
[1] Jenkins, D.C. & Bowden, F.P., Practical aspects of rain erosion of aircraft and missiles.Philosophical Transactions of the Royal Society of London Series A, Mathematicaland Physical Sciences, 260(1110), pp. 153–160, 1966. https://doi.org/10.1098/rsta.1966.0040
[2] Meng, J.C. & Colonius, T., Numerical simulations of the early stages of high-speeddroplet breakup. Shock Waves, 25(4), pp. 399–414, 2015. https://doi.org/10.1007/s00193-014-0546-z
[3] Theofanous, T.G. & Li, G.J., On the physics of aerobreakup. Physics of Fluids, 20(5),p. 052103, 2008. https://doi.org/10.1063/1.2907989
[4] Kuhnke, D., Spray wall interaction modeling by dimensionless data analysis. PhD Thesis,2004.
[5] Mundo, C., Droplet-wall collisions: Experimental studies of the deformation andbreakup process. International Journal of Multiphase Flow, 21(2), pp. 151–173, 1995.https://doi.org/10.1016/0301-9322(94)00069-v
[6] Engel, O., Fragmentation of waterdrops in the zone behind an air shock. Journalof Research of the National Bureau of Standards, 60(3), p. 245, 1958. https://doi.org/10.6028/jres.060.029
[7] Reinecke, W.G. & McKay, W.L., Experiments on water drop breakup behind mach 3 to12 shocks. Report AVATD-0172-69-RR, 1969.
[8] Ranger & Nicholls, Water droplet breakup in high speed airstreams. 3rd Inter Conf RainErosion and Allied Phenomena, 1970.
[9] Boiko, V.M., Papyrin, A.N. & Poplavskii, S.V., Dynamics of droplet breakup in shockwaves. Journal of Applied Mechanics and Technical Physics, 28(2), pp. 263–269, 1987.https://doi.org/10.1007/bf00918731
[10] Wierzba, A. & Takayama, K., Experimental investigation of the aerodynamicbreakup of liquid drops. AIAA Journal, 26(11), pp. 1329–1335, 1988. https://doi.org/10.2514/3.10044
[11] Yoshida, T. & Takayama, K., Interaction of liquid droplets with planar shockwaves. Journal of Fluids Engineering, 112(4), pp. 481–486, 1990. https://doi.org/10.1115/1.2909431
[12] Chou, W.H., Hsiang, L.P. & Faeth, G., Temporal properties of drop breakup in the shearbreakup regime. International Journal of Multiphase Flow, 23(4), pp. 651–669, 1997.https://doi.org/10.1016/s0301-9322(97)00006-2
[13] Joseph, D., Belanger, J. & Beavers, G., Breakup of a liquid drop suddenly exposed toa high-speed airstream. Int Journal of Multiphase Flow, 25(6), pp. 1263–1303, 1999.https://doi.org/10.1016/s0301-9322(99)00043-9
[14] Theofanous, T., Li, G. & Dinh, N., Rayleigh-taylor and kelvin-helmholz instabilities inaerobreakup. American Physical Society, 2003.
[15] Igra, D. & Takayama, K., Experimental and numerical study of the initial stages in theinteraction process between a planar shock wave and a water column. Proceedings of23rd International Symposium on Shock Waves, p. 8, 2001.
[16] Nourgaliev, R., Dinh, N. & Theofanous, T., Direct numerical simulation of compressiblemultiphase flows. International Conference on Multiphase Flow, ICMF04, p. 18,2004.
[17] Chen, H., Two-dimensional simulation of stripping breakup of a water droplet. AIAAJournal, 46, pp. 1135–1143, 2008. https://doi.org/10.2514/1.31286
[18] Sanada, T., Ando, K. & Colonius, T., A computational study of high-speed dropletimpact. Fluid Dynamics and Materials Processing, p. 12, 2011. https://doi.org/10.4028/www.scientific.net/ssp.187.137
[19] Sembian, S., Liverts, M., Tillmark, N. & Apazidis, N., Plane shock wave interactionwith a cylindrical water column. Physics of Fluids, 28, p. 056102, 2016. https://doi.org/10.1063/1.4948274
[20] Wang, T., Liu, N., Yi, X., Lu, X. & Wang, P., Numerical study on shock/droplet interactionbefore a standing wall. CiCP, 23(4), 2018. https://doi.org/10.4208/cicp.oa-2016-0254
[21] He´bert, D., Rullier, J.L., Chevalier, J.M., Bertron, I., Lescoute, E., Virot, F. & El-Rabii,H., Investigation of mechanisms leading to water drop breakup at mach 4.4 and webernumbers above 105. SN Applied Sciences, 2(1), p. 69, 2019. https://doi.org/10.1007/s42452-019-1843-z
[22] Tymen, G., Allanic, N., Sarda, A., Mousseau, P., Plot, C., Madec, Y. & Caltagirone, J.,Temperature mapping in a two-phase water-steam horizontal flow. Experimental HeatTransfer, 31(4), pp. 317–333, 2018. https://doi.org/10.1080/08916152.2017.1410505
[23] Cano-Lozano, The use of volume of fluid technique to analyze multiphase flows: Specificcase of bubble rising in still liquids. Applied Mathematical Modelling, 2014.
[24] Bernard-Champmartin, A. & Vuyst, F.D., A low diffusive lagrange-remap scheme forthe simulation of violent air–water free-surface flows. Journal of Computational Physics,274, pp. 19–49, 2014. https://doi.org/10.1016/j.jcp.2014.05.032
[25] Gasc, T., Improving numerical methods on recent multi-core processors. Application toLagrange-Plus-Remap hydrodynamics solver. Thesis, 2016.
[26] Chinnayya, A., Daniel, E. & Saurel, R., Modelling detonation waves in heterogeneousenergetic materials. Journal of Computational Physics, 196(2), pp. 490–538, 2004.https://doi.org/10.1016/j.jcp.2003.11.015
[27] Champmartin, A., Mode´lisation et e´tude nume´rique d’e´coulements diphasiques.Thesis, 2011.
[28] Shu, C.W. & Osher, S., Efficient implementation of essentially non-oscillatory shockcapturingschemes. Journal of Computational Physics, 77(2), pp. 439–471, 1988.
[29] Williams, R., Sub-grid properties and artificial viscous stresses in staggered-meshschemes. Journal of Computational Physics, 374, pp. 413–443, 2018. https://doi.org/10.1016/j.ijimpeng.2010.07.003
[30] Dawes, A., Parallel multi-dimensional and multi-material eulerian staggered meshschemes using localised patched based adaptive mesh refinement (amr) for strong shockwave phenomena. Adaptative Mesh Refinement – Theory and Applications, 41, 2005.
[31] Wilkins, M.L., Use of artificial viscosity in multidimensional fluid dynamic calculations.Journal of Computational Physics, 36(3), pp. 281–303, 1980. https://doi.org/10.1016/0021-9991(80)90161-8
[32] Richard, S., Petitpas, F. & Berry, R.A., Simple and efficient relaxation methods forinterfaces separating compressible fluids, cavitating flows and shocks in multiphasemixtures. Journal of Computational Physics, 228(5), pp. 1678–1712, 2009. https://doi.org/10.1016/j.jcp.2008.11.002
[33] Mirjalili, S., Jain, S.S. & Dodd, M.S., Interface-capturing methods for two-phaseflows: An overview and recent developments. Center for Turbulence Research - AnnualResearch Brief, p. 19, 2017.
[34] Mosso, S. & Clancy, S., Geometrically derived priority system for youngs’ interfacereconstruction. Tech report, Los Alamos National Laboratory, 1995.
[35] Leboucher, J.C., Re´alisation et performances d’un tube a` choc a` deux diaphragmes.Thesis, p. 111, 1973.
[36] Wang, E. & Shukla, A., Analytical and experimental evaluation of energies duringshock wave loading. International Journal of Impact Engineering, 37(12), pp. 1188–1196, 2010. https://doi.org/10.1016/j.ijimpeng.2010.07.003
[37] Meng, J.C. & Colonius, T., Numerical simulation of the aerobreakup of a water droplet.Journal of Fluid Mechanics, 835, p. 1108–1135, 2018. https://doi.org/10.1017/jfm.2017.804
[38] Pilch, M. & Erdman, C., Use of breakup time data and velocity history data to predictthe maximum size of stable fragments for acceleration-induced breakup of a liquiddrop. International Journal of Multiphase Flow, 13(6), pp. 741–757, 1987. https://doi.org/10.1016/0301-9322(87)90063-2
[39] Vallerani, E., An “ideal equivalent gas method” for the study of shock waves in supersonicreal gas flows. Meccanica, 4(3), pp. 234–249, 1969. https://doi.org/10.1007/bf02133438
[40] Mirjalili, S., Ivey, C.B. & Mani, A., Comparison between the diffuse interface andvolume of fluid methods for simulating two-phase flows. International Journal ofMultiphase Flow, 116, pp. 221–238, 2019. https://doi.org/10.1016/j.ijmultiphaseflow.2019.04.019
[41] Baals, D.D. & Corliss, W.R., Wind tunnels of nasa. Technical Report – Research andsupport facilities, 1981.
[42] Sharma, P. & Hammett, G.W., Preserving monotonicity in anisotropic diffusion. Journalof Computational Physics, 227(1), pp. 123–142, 2007. https://doi.org/10.1016/j.jcp.2007.07.026
[43] Pudasaini, S. & Kro¨ner, C., Shock waves in rapid flows of dense granular materials:Theoretical predictions and experimental results. Physical Review E, Statistical,Nonlinear, and Soft Matter Physics, 78, p. 041308, 2008. https://doi.org/10.1103/physreve.78.041308
[44] Kokh, S. & Lagoutie`re, F., An anti-diffusive numerical scheme for the simulationof interfaces between compressible fluids by means of a five-equation model. Journalof Computational Physics, 229, pp. 2773–2809, 2010. https://doi.org/10.1016/j.jcp.2009.12.003
[45] Umemura, A. & Shinjo, J., Detailed sgs atomization model and its implementationto two-phase flow les. Combustion and Flame, 195, pp. 232–252, 2018. https://doi.org/10.1016/j.combustflame.2018.01.026
[46] Vajda, B., Lesˇnik, L., Bombek, G., Bilusˇ, I., Zˇ unicˇ, Z., Sˇkerget, L., Hocˇevar,M., Sˇirok, B. & Kegl, B., The numerical simulation of biofuels spray. Fuel, 144, pp.71–79, 2015. https://doi.org/10.1016/j.fuel.2014.11.090
[47] Pogorevc, P., Kegl, B. & Sˇkerget, L., Diesel and biodiesel fuel spray simulations.Energy and Fuels – ENERG FUEL, 22, 2008.
[48] Hrebtov, M., Mixed lagrangian-eulerian simulation of interaction between a shockwaveand a cloud of water droplets. Journal of Engineering Thermophysics, 29, 2020.