Numerical Investigation of the Aerodynamic Droplet Breakup at Mach Numbers Greater Than 1
Publication: Journal of Energy Engineering
Volume 147, Issue 1
Abstract
The present work examines numerically the breakup of water droplets exposed to gas flows at Mach numbers , which resemble the ambient conditions encountered in the injection systems of supersonic combustion ramjet (scramjet) engines. A computational fluid dynamics (CFD) model is utilized that solves the compressible Navier-Stokes equations, the energy equation, and the mass conservation in volume fraction form [volume of fluid (VOF) method] along with two equations of state to model the density variations of the two phases. In addition, a coupled VOF/Lagrange model is employed to capture the appearance of microdroplets, which are smaller than the smallest grid cell. As a first step, a two-dimensional planar simulation (water column) is performed at conditions of in order to validate the numerical model; its results are compared against published experimental and numerical data. Good agreement is observed for the temporal evolution of droplet shape, the streamwise deformation, and the leading-edge displacement, as well as the shock wave reflection. Subsequently, the validated model is utilized to perform a three-dimensional (3D) simulation at , which corresponds to the conditions of previous experimental studies, and its results are compared against the experimental data as well as the results from previous numerical studies, showing good agreement. Furthermore, surface instabilities are observed at the droplet surface initiated by interfacial instabilities due to the shearing effect and the interaction with the shock wave, pertaining to Kelvin-Helmholtz and Rayleigh-Taylor instabilities, despite the stabilizing contribution of surface tension; viscosity effects are found to play an insignificant role.
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Data Availability Statement
All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
Financial support from the MSCA-ITN-ETN of the European Union’s H2020 programme, under REA Grant agreement No. 675676 is acknowledged.
References
Achenbach, E. 1974. “Vortex shedding from spheres.” J. Fluid Mech. 62 (2): 209–221. https://doi.org/10.1017/S0022112074000644.
Allaire, G., S. Clerc, and S. Kokh. 2002. “A five-equation model for the simulation of interfaces between compressible fluids.” J. Comput. Phys. 181 (2): 577–616. https://doi.org/10.1006/jcph.2002.7143.
Barth, T., and D. Jespersen. 1989. “The design and application of upwind schemes on unstructured meshes.” In Proc., 27th Aerospace Sciences Meeting, 366. Reston, VA: American Institute of Aeronautics and Astronautics.
Boiko, V., A. Papyrin, and S. Poplavskii. 1987. “Dynamics of droplet breakup in shock waves.” J. Appl. Mech. Tech. Phys. 28 (2): 263–269. https://doi.org/10.1007/BF00918731.
Boiko, V., and S. Poplavski. 2009. “On the dynamics of drop acceleration at the early stage of velocity relaxation in a shock wave.” Combust. Explosion Shock Waves 45 (2): 198–204. https://doi.org/10.1007/s10573-009-0026-4.
Bo Shen, Q. Y., T. Oliver, and D. Joachim. 2019. “Simulation of the primary breakup of non-Newtonian liquids at a high-speed rotary bell atomizer for spray painting processes using a VOF-Lagrangian hybrid model.” In Proc., 29th European Conf. on Liquid Atomization and Spray Systems. Paris: Sorbonne Univ.
Chang, C. H., X. Deng, and T. G. Theofanous. 2013. “Direct numerical simulation of interfacial instabilities: A consistent, conservative, all-speed, sharp-interface method.” J. Comput. Phys. 242 (Jun): 946–990. https://doi.org/10.1016/j.jcp.2013.01.014.
Chang, C. H., and M. S. Liou. 2007. “A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+-up scheme.” J. Comput. Phys. 225 (1): 840–873. https://doi.org/10.1016/j.jcp.2007.01.007.
Engel, O. G. 1958. “Fragmentation of waterdrops in the zone behind an air shock.” J. Res. Nat. Bur. Stds. 60 (245): 19.
Gor, G. Y., D. W. Siderius, V. K. Shen, and N. Bernstein. 2016. “Modulus–pressure equation for confined fluids.” J. Chem. Phys. 145 (16): 164505. https://doi.org/10.1063/1.4965916.
Green, B. 1995. Fluid vortices. Berlin: Springer.
Guan, B., Y. Liu, C.-Y. Wen, and H. Shen. 2018. “Numerical study on liquid droplet internal flow under shock impact.” AIAA J. 56 (9): 3382–3387. https://doi.org/10.2514/1.J057134.
Guildenbecher, D. R., C. López-Rivera, and P. E. Sojka. 2009. “Secondary atomization.” Exp. Fluids 46 (3): 371–402. https://doi.org/10.1007/s00348-008-0593-2.
Hébert, D., J. L. Rullier, J. M. Chevalier, I. Bertron, E. Lescoute, F. Virot, and H. El-Rabii. 2019. “Investigation of mechanisms leading to water drop breakup at Mach 4.4 and Weber numbers above 105.” SN Appl. Sci. 2 (1): 69. https://doi.org/10.1007/s42452-019-1843-z.
Hirt, C. W., and B. D. Nichols. 1981. “Volume of fluid (VOF) method for the dynamics of free boundaries.” J. Comput. Phys. 39 (1): 201–225. https://doi.org/10.1016/0021-9991(81)90145-5.
Igra, D., and K. Takayama. 2001a. “A study of shock wave loading on a cylindrical water column.” Rep. Inst. Fluid Sci. 13 (3): 19–36.
Igra, D., and K. Takayama. 2001b. “Numerical simulation of shock wave interaction with a water column.” Shock Waves 11 (3): 219–228. https://doi.org/10.1007/PL00004077.
Ivings, M., D. Causon, and E. Toro. 1998. “On Riemann solvers for compressible liquids.” Int. J. Numer. Methods Fluids 28 (3): 395–418. https://doi.org/10.1002/(SICI)1097-0363(19980915)28:3%3C395::AID-FLD718%3E3.0.CO;2-S.
Jain, S. S., N. Tyagi, R. S. Prakash, R. Ravikrishna, and G. Tomar. 2018. “Secondary breakup of drops at moderate Weber numbers: Effect of density ratio and Reynolds number.” Int. J. Multiphase Flow 117: 25–41.
Johnsen, E. 2008. Numerical simulations of non-spherical bubble collapse: With applications to shockwave lithotripsy. Pasadena, CA: California Institute of Technology.
Joseph, D., G. Beavers, and T. Funada. 2002. “Rayleigh–Taylor instability of viscoelastic drops at high Weber numbers.” J. Fluid Mech. 453 (Feb): 109–132. https://doi.org/10.1017/S0022112001006802.
Joseph, D. D., J. Belanger, and G. Beavers. 1999. “Breakup of a liquid drop suddenly exposed to a high-speed airstream.” Int. J. Multiphase Flow 25 (6–7): 1263–1303. https://doi.org/10.1016/S0301-9322(99)00043-9.
Kaiser, J., S. Adami, and N. A. Adams. 2017. “Direct numerical simulation of shock-induced drop breakup with a sharp-interface-method.” In Proc., 10th Int. Symp. on Turbulence and Shear Flow Phenomena. Ypsilanti, MI: LaVision.
Koukouvinis, P., M. Gavaises, O. Supponen, and M. Farhat. 2016. “Numerical simulation of a collapsing bubble subject to gravity.” Phys. Fluids 28 (3): 032110. https://doi.org/10.1063/1.4944561.
Koukouvinis, P., A. V. Roncero, C. Rodriguez, M. Gavaises, and L. Picket. 2020. “High pressure/high temperature multiphase simulations of dodecane injection to nitrogen: Application on ECN Spray-A.” Fuel 275 (Sep): 117871. https://doi.org/10.1016/j.fuel.2020.117871.
Lafaurie, B., C. Nardone, R. Scardovelli, S. Zaleski, and G. Zanetti. 1994. “Modelling merging and fragmentation in multiphase flows with surfer.” J. Comput. Phys. 113 (1): 134–147. https://doi.org/10.1006/jcph.1994.1123.
Lemmon, E. 2013. Reference fluid thermodynamic and transport properties (REFPROP). NIST standard reference database 23. Gaithersburg, MD: NIST.
Liang, C. 2016. “Computational methods for the investigation of liquid drop phenomena in external gas flows.” Ph.D. dissertation, Dept. of Mathematical Sciences, Michigan Technological Univ.
Liou, M. S., C. H. Chang, H. Chen, and J. J. Hu. 2009. “Numerical study of shock-driven deformation of interfaces.” In Shock waves, 919–924. Berlin: Springer.
Liu, N., Z. Wang, M. Sun, H. Wang, and B. Wang. 2018. “Numerical simulation of liquid droplet breakup in supersonic flows.” Acta Astronaut. 145 (Apr): 116–130. https://doi.org/10.1016/j.actaastro.2018.01.010.
Liu, Z., and R. D. Reitz. 1997. “An analysis of the distortion and breakup mechanisms of high speed liquid drops.” Int. J. Multiphase Flow 23 (4): 631–650. https://doi.org/10.1016/S0301-9322(96)00086-9.
Malgarinos, I., N. Nikolopoulos, and M. Gavaises. 2015. “Coupling a local adaptive grid refinement technique with an interface sharpening scheme for the simulation of two-phase flow and free-surface flows using VOF methodology.” J. Comput. Phys. 300 (Nov): 732–753. https://doi.org/10.1016/j.jcp.2015.08.004.
Malgarinos, I., N. Nikolopoulos, and M. Gavaises. 2016. “A numerical study on droplet-particle collision dynamics.” Int. J. Heat Fluid Flow 61 (Oct): 499–509. https://doi.org/10.1016/j.ijheatfluidflow.2016.06.010.
Malgarinos, I., N. Nikolopoulos, and M. Gavaises. 2017a. “Numerical investigation of heavy fuel droplet-particle collisions in the injection zone of a fluid catalytic cracking reactor, Part I: Numerical model and 2D simulations.” Fuel Process. Technol. 156 (Feb): 317–330. https://doi.org/10.1016/j.fuproc.2016.09.014.
Malgarinos, I., N. Nikolopoulos, and M. Gavaises. 2017b. “Numerical investigation of heavy fuel droplet-particle collisions in the injection zone of a fluid catalytic cracking reactor, Part II: 3D simulations.” Fuel Process. Technol. 156 (Feb): 43–53. https://doi.org/10.1016/j.fuproc.2016.09.012.
Malgarinos, I., N. Nikolopoulos, M. Marengo, C. Antonini, and M. Gavaises. 2014. “VOF simulations of the contact angle dynamics during the drop spreading: Standard models and a new wetting force model.” Adv. Colloid Interface Sci. 212 (Oct): 1–20. https://doi.org/10.1016/j.cis.2014.07.004.
Meng, J., and T. Colonius. 2015. “Numerical simulations of the early stages of high-speed droplet breakup.” Shock Waves 25 (4): 399–414. https://doi.org/10.1007/s00193-014-0546-z.
Meng, J. C., and T. Colonius. 2018. “Numerical simulation of the aerobreakup of a water droplet.” J. Fluid Mech. 835 (Jan): 1108–1135. https://doi.org/10.1017/jfm.2017.804.
Meng, J. C. C. 2016. Numerical simulations of droplet aerobreakup. Pasadena, CA: California Institute of Technology.
Menon, E. S. 2014. Transmission pipeline calculations and simulations manual. Houston: Gulf Professional Publishing.
Morsi, S., and A. Alexander. 1972. “An investigation of particle trajectories in two-phase flow systems.” J. Fluid Mech. 55 (2): 193–208. https://doi.org/10.1017/S0022112072001806.
Nicholls, J. A., and A. A. Ranger. 1969. “Aerodynamic shattering of liquid drops.” AIAA J. 7 (2): 285–290. https://doi.org/10.2514/3.5087.
Perry, R. H., and D. W. Green. 1999. Perry’s chemical engineers’ handbook. New York: McGraw-Hill. CD-ROM.
Quirk, J. J., and S. Karni. 1996. “On the dynamics of a shock–bubble interaction.” J. Fluid Mech. 318: 129–163. https://doi.org/10.1017/S0022112096007069.
Sakamoto, H., and H. Haniu. 1990. “A study on vortex shedding from spheres in a uniform flow.” Trans. J. Fluids Eng. 112 (4): 386–392. https://doi.org/10.1115/1.2909415.
Sonntag, R. E., G. J. Van Wylen, and C. Borgnakke. 2008. Fundamentals of thermodynamics. Hoboken, NJ: Wiley.
Stefanitsis, D. 2020. “Direct numerical simulation of secondary droplet breakup in fuel sprays.” Ph.D. thesis, School of Mathematics, Computer Science and Engineering, Univ. of London.
Stefanitsis, D., I. Malgarinos, G. Strotos, N. Nikolopoulos, E. Kakaras, and M. Gavaises. 2017a. “Numerical investigation of the aerodynamic breakup of diesel and heavy fuel oil droplets.” Int. J. Heat Fluid Flow 68: 203–215. https://doi.org/10.1016/j.ijheatfluidflow.2017.10.012.
Stefanitsis, D., I. Malgarinos, G. Strotos, N. Nikolopoulos, E. Kakaras, and M. Gavaises. 2017b. “Numerical investigation of the aerodynamic breakup of diesel droplets under various gas pressures.” In Proc., 28th Conf. on Liquid Atomization and Spray Systems. Paris: Sorbonne Univ.
Stefanitsis, D., I. Malgarinos, G. Strotos, N. Nikolopoulos, E. Kakaras, and M. Gavaises. 2018a. “Numerical investigation of the aerodynamic breakup of droplets in tandem.” Int. J. Multiphase Flow 113 (Apr): 289–303. https://doi.org/10.1016/j.ijmultiphaseflow.2018.10.015.
Stefanitsis, D., G. Strotos, N. Nikolopoulos, and M. Gavaises. 2019a. “Numerical investigation of the aerodynamic breakup of a parallel moving droplet cluster.” Int. J. Multiphase Flow 121 (Dec): 103123. https://doi.org/10.1016/j.ijmultiphaseflow.2019.103123.
Stefanitsis, D., G. Strotos, N. Nikolopoulos, E. Kakaras, and M. Gavaises. 2018b. “Numerical examination of the aerodynamic breakup of droplets in chain formation.” In Proc., 14th Triennial Int. Conf. on Liquid Atomization and Spray Systems. Madison, WI: Institute for Liquid Atomization and Spray Systems International.
Stefanitsis, D., G. Strotos, N. Nikolopoulos, E. Kakaras, and M. Gavaises. 2019b. “Improved droplet breakup models for spray applications.” Int. J. Heat Fluid Flow 76 (Apr): 274–286. https://doi.org/10.1016/j.ijheatfluidflow.2019.02.010.
Strotos, G., I. Malgarinos, N. Nikolopoulos, and M. Gavaises. 2016a. “Aerodynamic breakup of an -decane droplet in a high temperature gas environment.” Fuel 185 (Dec): 370–380. https://doi.org/10.1016/j.fuel.2016.08.014.
Strotos, G., I. Malgarinos, N. Nikolopoulos, and M. Gavaises. 2016b. “Numerical investigation of aerodynamic droplet breakup in a high temperature gas environment.” Fuel 181 (Oct): 450–462. https://doi.org/10.1016/j.fuel.2016.04.126.
Strotos, G., I. Malgarinos, N. Nikolopoulos, and M. Gavaises. 2016c. “Predicting droplet deformation and breakup for moderate Weber numbers.” Int. J. Multiphase Flow 85 (Oct): 96–109. https://doi.org/10.1016/j.ijmultiphaseflow.2016.06.001.
Strotos, G., I. Malgarinos, N. Nikolopoulos, and M. Gavaises. 2016d. “Predicting the evaporation rate of stationary droplets with the VOF methodology for a wide range of ambient temperature conditions.” Int. J. Therm. Sci. 109 (Nov): 253–262. https://doi.org/10.1016/j.ijthermalsci.2016.06.022.
Strotos, G., N. Nikolopoulos, K. Papadopoulos, A. Theodorakakos, and M. Gavaises. 2015. “Performance of VOF methodology in predicting the deformation and breakup of impulsively accelerated droplets.” In Proc., 13th Triennial Int. Conf. on Liquid Atomization and Spray Systems, 23–27. Madison, WI: Institute for Liquid Atomization and Spray Systems International.
Surov, V. 1995. “Numerical modeling of the interaction of a strong shock wave with liquid drops.” J. Appl. Mech. Tech. Phys. 36 (3): 354–359. https://doi.org/10.1007/BF02369771.
Sutherland, W. 1893. “LII: The viscosity of gases and molecular force.” London Edinburgh Dublin Philos. Mag. J. Sci. 36 (223): 507–531. https://doi.org/10.1080/14786449308620508.
Theofanous, T., and G. Li. 2008. “On the physics of aerobreakup.” Phys. Fluids 20 (5): 052103. https://doi.org/10.1063/1.2907989.
Theofanous, T., V. Mitkin, C. Ng, C. Chang, X. Deng, and S. Sushchikh. 2012. “The physics of aerobreakup: II. Viscous liquids.” Phys. Fluids 24 (2): 022104. https://doi.org/10.1063/1.3680867.
Theofanous, T. G., G. J. Li, and T. N. Dinh. 2004. “Aerobreakup in rarefied supersonic gas flows.” J. Fluids Eng. 126 (4): 516–527. https://doi.org/10.1115/1.1777234.
Toro, E. F. 1997. “The Riemann problem for the Euler equations.” In Riemann solvers and numerical methods for fluid dynamics: A practical introduction, edited by E. F. Toro, 115–157. Berlin: Springer.
Wagner, W., and A. Pruß. 2002. “The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use.” J. Phys. Chem. Ref. Data 31 (2): 387–535. https://doi.org/10.1063/1.1461829.
Wierzba, A., and K. Takayama. 1988. “Experimental investigation of the aerodynamic breakup of liquid drops.” AIAA J. 26 (11): 1329–1335. https://doi.org/10.2514/3.10044.
Wu, J.-Z., H.-Y. Ma, and M.-D. Zhou. 2015. Vortical flows. Berlin: Springer.
Xiao, F., Z. Wang, M. Sun, N. Liu, and X. Yang. 2017. “Simulation of drop deformation and breakup in supersonic flow.” Proc. Combust. Inst. 36 (2): 2417–2424. https://doi.org/10.1016/j.proci.2016.09.016.
Yi, X., Y. Zhu, and J. Yang. 2017. “On the early-stage deformation of liquid drop in shock-induced flow.” In Proc., 30th Int. Symp. on Shock Waves 2, 1269–1273. Berlin: Springer.
Yoshida, T., and K. Takayama. 1990. “Interaction of liquid droplets with planar shock waves.” J. Fluids Eng. 112 (4): 481–486. https://doi.org/10.1115/1.2909431.
Zandian, A., W. A. Sirignano, and F. Hussain. 2018. “Understanding liquid-jet atomization cascades via vortex dynamics.” J. Fluid Mech. 843 (12): 293–354. https://doi.org/10.1017/jfm.2018.113.
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Published In
Journal of Energy Engineering
Volume 147 • Issue 1 • February 2021
Copyright
© 2020 American Society of Civil Engineers.
History
Received: Apr 6, 2020
Accepted: Jul 16, 2020
Published online: Oct 19, 2020
Published in print: Feb 1, 2021
Discussion open until: Mar 19, 2021
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