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A New Immersed Boundary Method for Stable Two-Phase Flow Simulations of a Deforming Droplet

王, 家瑞 東京大学 DOI:10.15083/0002001892

2021.10.04

概要

The cloud microphysical processes are parameterized in a numerical weather pre- diction model by empirical formulae and the accuracy of forecasts is said to be sen- sitive to the microphysics parameterizations to a considerable extent. The empirical formulae to predict the terminal velocity, shape, and collision efficiency of a free-falling water droplet are extensively used in the parameterizations. However, these formulae are determined by experiments which are conducted at room temperature and standard atmospheric pressure or linear theories in an ideal condition. The validity of the param- eterizations is therefore questionable under general conditions. By building a two-phase flow direct numerical simulation (DNS) model of water droplets with deforming inter- face, it is expected that the empirical formulae can be verified systematically and the microphysical processes can be studied in a more efficient way. This work aims to build such a numerical model with the Immersed Boundary Method (IBM).

The IBM is a popular one-fluid Eulerian-Lagrangian model for two-phase flow sim- ulations. In the IBM, an interface is represented by a series of connected Lagrangian markers and an IB forcing term is added to the Navier-Stokes equations to enforce the interfacial boundary condition. Despite great success in simulations of droplets and bubbles in previous works, it suffers from two instability problems when the capillarity is dominant: the spurious parasitic currents and surface fluctuation. These problems need to be solved before it can be applied to the study of microphysical processes.

It is found in this study that the parasitic currents occur because the discrete delta function does not preserve the irrotational property; while the surface fluctuation is due to the discretization error and sensitivity of the direct polynomial fitting to marker locations. To enable stable water droplet simulations by the IBM, a new irrotational spreading operator is proposed to eliminate the parasitic currents and the global B-spline surface reconstruction by least squares is adopted to suppress the surface fluctuation.

The conventional and new IBM are implemented into a finite volume flow solver of two dimensional (2D) or axisymmetric incompressible Navier-Stokes equations. The Chorin’s projection method is used in the discretization of the Navier-Stokes equations and the multigrid method is adopted to solve the Poisson’s equation for pressure. Two standard test cases have been conducted and the results show that the new IBM has a proper first-order convergence. The simulations of a free-falling axisymmetric droplet and rising bubble of small density ratio are performed to demonstrate the robustness of the new IBM for different degrees of shape deformation.

The new IBM is applied to simulations of axisymmetric and 2D free-falling rain- drops to investigate the terminal velocity, drag coefficient, and shape under different conditions. The terminal velocity and drag coefficient obtained from the axisymmetric raindrop simulation at standard pressure and temperature fit well with the experimental data. Simulation results of an axisymmetric free-falling raindrop of diameter 0.5 mm at different altitudes assuming the lapse rate of 6.5 °C km−1 show that there is a significant discrepancy between the numerical results and some of the widely used empirical for- mulae. The discrepancy is larger at higher altitudes. A new parameterization with better accuracy for terminal velocity aloft of raindrops with diameter smaller than 0.5 mm is proposed. Further simulations of a 0.5 mm free-falling water droplet (2D and ax- isymmetric) with different surface tension coefficients suggest that the change in the shape and terminal velocity is negligible with respect to aerosols; the surface tension coefficient depends on the concentration and type of aerosols. For a free-falling 2D raindrop, the critical Reynolds number for the onset of vortex shedding motion and drag are investigated for theoretical purpose. Moreover, the motion of two simultaneous 2D free-falling water droplets is simulated to study the performance of the new IBM in the merging process and the collision efficiency.

The last part of this thesis is devoted to the localization of smooth surface recon- struction. Although the global B-spline fitting can be easily applicable to 1D interface, it is complicating to directly extend it to a 2D interface. This restricts the application of the new IBM to only small free-falling raindrop or water droplets with low Reynolds number. A new iterative local planar Bézier reconstruction algorithm is developed to replace the global B-spline fitting to enable a more straightforward extension of the new IBM for three dimensional (3D) simulations. The results of the 2D free-oscillating droplet test case show good agreement with the analytic solution. Together with the new irrotational spreading operator, these new schemes will form the foundation for the development of a general model for 3D droplet simulations in the future.

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参考文献

Agrawal, D. C. and V. J. Menon, 1992: Surface tension and evaporation: An empirical relation for water. Phys. Rev. A, 46, 2166–2169.

Alfeld, P., 1984: A bivariate C2 Clough-Tocher scheme. Comput. Aided Geom. D., 1, 257–267.

Alfeld, P., M. Neamtu, and L. L. Schumaker, 1996a: Bernstein-Bézier polynomials on spheres and sphere-like surfaces. Comput. Aided Geom. D., 13, 333–349.

Alfeld, P., M. Neamtu, and L. L. Schumaker, 1996b: Fitting scattered data on sphere- like surfaces using spherical splines. J. Comput. Appl. Math., 73, 5–43.

Angot, P., C. H. Bruneau, and P. Fabrie, 1999: A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math., 81, 497–520.

Arquis, E., and J. P. Caltagirone, 1984: Sur les condtions hydrodynamiques au voisi- nage d’une interface milieu fluide - milieux poreux: application à la convection naturelle. C. R. Acad. Sci. Paris II, 299, 1–4.

Bao, Y., J. Kaye, and C. S. Peskin, 2016: A Gaussian-like immersed-boundary ker- nel with three continuous derivatives and improved translational invariance. J. Comput. Phys., 316, 139–144.

Bashforth, F., and J. C. Adams, 1883: An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of fluid. With an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops. Cambridge, 80.

Beard, K. V., and H. R. Pruppacher, 1968: An experimental test of theoretically calcu- lated collision efficiencies of cloud drops. J. Geophys. Res., 73, 6407–6414.

Beard, K. V., and H. R. Pruppacher, 1969: A determination of the terminal velocity and drag of small water drops by means of a wind tunnel. J. Atmos. Sci., 26, 1066–1072.

Beard, K. V., and H. R. Pruppacher, 1971: A wind tunnel investigation of collection kernels for small water drops in air. Q. J. Roy. Meteor. Soc., 97, 242–248.

Beard, K. V., 1984: Raindrop oscillations: Evaluation of a potential flow model with gravity. J. Atmos. Sci., 41, 1765–1774.

Beard, K. V., and C. Chuang, 1987: A new model for the equilibrium shape of rain- drops. J. Atmos. Sci., 44, 1509–1524.

Beard, K. V., H. T. Ochs III, and R. J. Kubesh, 1989: Natural oscillations of small raindrops. Nature, 342, 408–410.

Beard, K. V., and R. J. Kubesh, 1991: Laboratory measurements of small raindrop distortion. Part 2: Oscillation frequencies and modes. J. Atmos. Sci., 48, 2245– 2264.

Beyer, R. P., and R. J. LeVeque, 1992: Analysis of a one-dimensional model for the immersed boundary method. SIAM J. Numer. Anal., 29, 332–364.

Brackbill, J. U., D. B. Kothe, and C. Zemach, 1992: A continuum method for modeling surface tension. J. Comput. Phys., 100, 335–354.

Briggs, W. L., V. E. Henson, and S. F. McCormick, 2000: A Multigrid Tutorial: Second Edition. Society for Industrial and Applied Mathematics, 187.

Brook, M., and D. J. Latham, 1968: Fluctuating radar echo: Modulation by vibrating drops. J. Geophys. Res., 73, 7137–7144.

Cahn, J. W., J. E. Hilliard, 1958: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys., 28, 258-–267.

Chen, L., H. Wei, and M. Wen, 2017: An interface-fitted mesh generator and virtual element methods for elliptic interface problems. J. Comput. Phys., 334, 327–348.

Chorin, A. J., 1967: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys., 2, 12–26.

Chowdhury, M. N., F. Y. Testik, M. C. Hornack, and A. A. Khan, 2016: Free fall of water drops in laboratory rainfall simulations. Atmos. Res. , 168, 158–168.

Clift, R., J. R. Grace, and M. E. Weber, 1978: Bubbles, drops, and particles. New York; London : Academic Press, 380.

Clough, R. W., and J. L. Tocher, 1965: Finite Element Stiffness Matrices for Analy- sis of Plates in Bending. Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, 515–545.

Dahmen, W., C. A. Micchelli, and H. P. Seidel, 1992: Blossoming begets B-spline bases built better by B-patches. Math. Comput., 59, 97–115.

de Boor, C., 1978: A practical guide to splines. Springer, 348.

Esmaeeli, A., and G. Tryggvason, 2004: Computations of film boiling. Part I: numeri- cal method. Int. J. Heat Mass Tran., 47, 5451–5461.

Esmaeeli, A., and G. Tryggvason, 2005: A direct numerical simulation study of the buoyant rise of bubbles at O(100) Reynolds number. Phys. Fluids, 17, 093303.

Fadlun, E. A., R. Verzicco, P. Orlandi, and J. Mohd-Yusof, 2000: Combined immersed- boundary finite-difference methods for three-dimensional complex flow simula- tions. J. Comput. Phys., 161, 35–60.

Farin, G., 1986: Triangular Bernstein-Bézier patches. Comput. Aided Geom. D., 3, 83–127

Farin, G., 1992: Curves and Surfaces for Computer-Aided Geometric Design (Third Edition). Academic Press, 473.

Fedkiw, R., T. Aslam, B. Merriman, and S. Osher, 1999: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys., 152, 457–492.

Feng, J. Q., and K. V. Beard, 1991: A perturbation model of raindrop oscillation char- acteristics with aerodynamic effects. J. Atmos. Sci., 48, 1856–1868.

Fontes, D., R. Duarte, C. Antonio, and F. Souza, 2018: Numerical simulation of a water droplet splash: Effects of density interpolation schemes. Mech. Res. Comm., 90.

Foote, G. B., and P. S. du Toit, 1969: Terminal velocity of raindrops aloft. J. Appl.Meteorol., 8, 249–253.

Francois, M. M., S. J. Cummins, E. D. Dendy, D. B. Kothe, J. M. Sicilian, and M. W. Williams, 2006: A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J. Comput. Phys., 213, 141–173.

Fyfe, D. E., E. S. Oran, and M. J. Fritts, 1988: Surface tension and viscosity with lagrangian hydrodynamics on a triangular mesh. J. Comput. Phys., 76, 349–384.

Gittens, G. J., 1969: Variation of surface tension of water with temperature. J. Colloid Interf. Sci., 30, 406–412.

Goldstein, D., R. Handler, and L. Sirovich, 1993: Modeling a no-slip flow boundary with an external force field. J. Comput. Phys., 105, 354–366.

Grabowski, W. W., 1998: Toward cloud resolving modeling of large-scale tropical circulations: a simple cloud microphysics parameterization J. Atmos. Sci., 55, 3283–3298.

Greiner, G., 1994: Variational design and fairing of spline surfaces. Comput. Graph.Forum, 13, 143–154.

Gros, E., G. R. Anjos, and J. R. Thome, 2018: Interface-fitted moving mesh method for axisymmetric two-phase flow in microchannels. Int. J. Numer. Meth. Fl., 86, 201–217.

Gu, X., Y. He, and H. Qin, 2006: Manifold Splines. Graph. Models, 68, 237–254.

Gunn, R., and G. D. Kinzer, 1949: The terminal velocity of fall for water droplets in stagnant air. J. Meteorol., 6, 243–248.

Hadamard, J., 1911: Mouvement permanent lent d’une sphere liquid et visqueuse dans un liquid visqueux. Compt. Rend., 152, 1735–1738.

Hagos, S., L. R. Leung, C. Zhao, Z. Feng, and K. Sakaguchi, 2018: How do micro- physical processes influence large-scale precipitation variability and extremes? Geophys. Res. Lett., 45, 1661–1667.

Harlow, F. H., and J. E. Welch, 1965: Numerical calculation of time-dependent viscous incompressible flow of fluid with free surfaces. Phys. Fluids, 8, 2182–2189.

Hirt, C. W., and B. D. Nichols, 1981: Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys., 39, 201–225.

Hoschek, J., and D. Lasser, 1993: Fundamentals of Computer Aided Geometric De- sign. A. K. Peters, Ltd., 727.

Iaccarino, G., and R. Verzicco, 2003: Immersed boundary technique for turbulent flow simulations. Appl. Mech. Rev, 56, 331–347.

Irfan, M., and M. Muradoglu, 2017: A front tracking method for direct numerical simulation of evaporation process in a multiphase system. J. Comput. Phys., 337, 132–153.

Jacqmin, D., 1996: An energy approach to the continuum surface tension method. Proc. 34th Aerosp. Sci. Meet. Exh. AIAA 96-0858.Reno: Am. Inst. Aeron. Astron.

Jacqmin, D., 1999: Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys., 155, 96–127.

Jamet, D., O. Lebaigue, N. Coutris, and J. M. Delhaye, 1995: A Numerical Description of a Liquid-Vapor Interface Based on the Second Gradient Theory. Int. J. Fluid Mech. Res., 22, 1–14.

Jamet, D., D. Torres, and J. U. Brackbill, 2002: On the theory and computation of surface tension: The elimination of parasitic currents through energy conservation in the second-gradient method. J. Comput. Phys., 182, 262–276.

Juric, D., and G. Tryggvason, 1996: A front-tracking method for dendritic solidifica- tion. J. Comput. Phys., 123, 127–148.

Juric, D., and G. Tryggvason, 1998: Computations of boiling flows. Int. J. Multiphas.Flow, 24, 387–410.

Kaplun, S., 1957: Low Reynolds number flow past a circular cylinder. J. Math. Mech.,6, 595–603.

Kessler, E., 1969: On the distribution and continuity of water substance in atmospheric circulations. Meteorol. Monogr., 10, 88.

Kim, J., and P. Moin, 1985: Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys., 59, 308–323.

Komrakova, A. E., D. Eskin, and J. J. Derksen, 2013: Lattice Boltzmann simulations of a single n-butanol drop rising in water. Phys. Fluids, 25, 042102.

Kothe, D., W. Rider, S. Mosso, J. Brock, and J. Hochstein, 1996: Volume tracking of interfaces having surface tension in two and three dimensions. 34th Aerospace Sciences Meeting and Exhibit, Reno, NV, U.S.A..

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, 134–147.

Lamb, H., 1932: Hydrodynamics. Cambridge University Press, 738.

LeClair, B. P., A. E. Hamielec, H. R. Pruppacher, and W. D. Hall, 1972: A Theoreti- cal and Experimental Study of the Internal Circulation in Water Drops Falling at Terminal Velocity in Air. J. Atmos. Sci., 29, 728–740.

Lee, L., and R. J. LeVeque, 2003: An immersed interface method for incompressible Navier-Stokes equations. SIAM J. Sci. Comput., 25, 832–856.

Lima E Silva A. L. F., A. Silveira-Neto, and J. J. R. Damasceno, 2003: Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method. J. Comput. Phys., 189, 351–370.

Lin, Y. L., R. D. Farley, and H. D. Orville, 1983: Bulk parameterization of the snow field in a cloud model. J. Clim. Appl. Meteorol., 22, 1065–1092.

Liu, Y., and Y. Mori, 2012: Properties of discrete delta functions and local convergence of the immersed boundary method. SIAM J. Numer. Anal., 50, 2986–3015.

Liu, S., A. Jacobson, and Y. Gingold, 2014: Skinning cubic Bézier splines and Catmull- Clark subdivision surfaces. ACM Trans. Graph., 33, 190:1–190:9.

Marshall, J. S., and W. Mc K. Palmer, 1948: The distribution of raindrops with size. J. Meteorol., 5, 165–166.

McFiggans, G., P. Artaxo, U. Baltensperger, H. Coe, M. C. Facchini, G. Feingold,S. Fuzzi, M. Gysel, A. Laaksonen, U. Lohmann, T. F. Mentel, D. M. Murphy,C. D. O’Dowd, J. R. Snider, and E. Weingartner, 2006: The effect of physical and chemical aerosol properties on warm cloud droplet activation. Atmos. Chem. Phys., 6, 2593–2649.

Mittal, R., and G. Iaccarino, 2005: Immersed boundary methods. Annu. Rev. Fluid Mech., 37, 239–261.

Mohd Yusof, J., 1997: Combined immersed-boundary/B-spline methods for simula- tions of flow in complex geometries. Annual Research Briefs, Center for Turbu- lence Research, 317–328.

Mori, Y., 2008: Convergence proof of the velocity field for a stokes flow immersed boundary method. Commun. Pur. Appl. Math., 61, 1213–1263.

Morrison, H., and A. Gettelman, 2008: A new two-moment bulk stratiform cloud microphysics scheme in the community atmosphere model, version 3 (CAM3). Part I: Description and numerical tests. J. Climate, 21, 3642–3659.

Müller, S., M. Szakáll, S. K. Mitra, K. Diehl, and S. Borrmann, 2013: Shapes and os- cillations of raindrops with reduced surface tensions: Measurements at the Mainz vertical wind tunnel. Atmos. Res., 119, 38–45.

Noh, W. F., and P. R. Woodward, 1976: SLIC (Simple Line Interface Calculation). In proceedings of 5th international conference of fluid dynamics, edited by A. I. van de Vooren and P.J. Zandbergen. Lecture Notes in Physics, 59, 330–340.

Ong, C. R., and H. Miura, 2018a: An immersed boundary method with irrotational discrete delta vector for droplet simulations of large density ratio. J. Comput. Phys., in revision.

Ong, C. R., and H. Miura, 2018b: Iterative Bézier reconstruction algorithm of smooth droplet surface for the immersed boundary method. SOLA, 14, 170–173.

Osher, S., and J. A. Sethian, 1988: Fronts propagating with curvature-dependent speed: Algorithms based on hamilton-jacobi formulations. J. Comput. Phys., 79, 12–49.

Parodi, A., and K. Emanuel, 2009: A theory for buoyancy and velocity scales in deep moist convection. J. Atmos. Sci., 66, 3449–3463.

Peskin, C. S., 1972: Flow patterns around heart valves: a digital computer method for solving the equations of motion. PhD thesis, physiology, Albert Einstein College of Medicine.

Peskin, C. S., 2002: The immersed boundary method. Acta Numer., 11, 479–517.

Picknett, R. G., 1960: Collection efficiencies for water drops in air. Int. J. Air Pollut.,3, 160–167.

Pinsky, M., A. Khain, M. Shapiro, 2001: Collision efficiency of drops in a wide range of Reynolds numbers: Effects of pressure on spectrum evolution. J. Atmos. Sci., 58, 742–764.

Proudman, I., and J. R. A. Pearson, 1957: Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech., 2, 237–262.

Pruppacher, H. R., and R. L. Pitter, 1971: A semi-empirical determination of the shape of cloud and rain drops. J. Atmos. Sci., 28, 86–94.

Pruppacher, H. R., and K. V. Beard, 1987: A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air. Q. J. Roy. Meteor. Soc., 96, 247–256.

Pruppacher, H. R., and J. D. Klett, 1997: Microphysics of clouds and precipitation.Kluwer Academic Publishers, 954.

Rosen, M. J., and J. T. Kunjappu, 2012: Surfactants and Interfacial Phenomena. Wi- ley, 616.

Rutledge, S. A., and P. V. Hobbs, 1983: The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. VIII: A model for the “seeder-feeder” process in warm-frontal rainbands. J. Atmos. Sci., 40, 1185–1206.

Rybczinski, W., 1911: Über die fortschreitende Bewegung einer flüssigen Kugel in einem zähen Medium. Bull. Acad. Sci. Cracovie, A., 40–46.

Ryskin, G., and L. Leal, 1984: Numerical solution of free-boundary problems in fluid mechanics. Part 2. Buoyancy-driven motion of a gas bubble through a quiescent liquid. J. Fluid Mech., 148, 19–35.

Sartor, J. D., and C. E. Abbott, 1975: Prediction and measurament of the accelerated motion of water drops in air. J. Appl. Meteor., 14, 232–239.

Satoh, M., T. Matsuno, H. Tomita, H. Miura, T. Nasuno, and S. Iga, 2008: Non- hydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations. J. Comput. Phys., 227, 3486–3514.

Scardovelli, R., and S. Zaleski, 1999: Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech., 31, 567–603.

Schwarz, S., T. Kempe and J. Fröhlich, 2016: An immersed boundary method for the simulation of bubbles with varying shape. J. Comput. Phys., 315, 124–149.

Seidl, W., 1983: Surface-active substances on rainwater and atmospheric particles.Pure Appl. Geophys., 121, 1077–1093.

Seidl, W., 2000: Model for a surface film of fatty acids on rain water and aerosol particles. Atmos. Environ., 34, 4917–4932.

Singh, R., and W. Shyy, 2007: Three-dimensional Adaptive Cartesian Grid Method with Conservative Interface Restructuring and Reconstruction. J. Comput. Phys., 224, 150–167.

Spilhaus, A. F., 1948: Raindrop size, shape, and falling speed. J. Meteorol., 5, 108– 110.

Sugioka, M., and S. Komori, 2007: Drag and lift forces acting on a spherical water droplet in homogeneous linear shear air flow. J. Fluid Mech., 570, 155–175.

Sussman, M., P. Smereka, and S. Osher, 1994: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys., 114, 146–159.

Sussman, M., and E. G. Puckett, 2000: A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. J. Comput. Phys., 162, 301–337.

Szakáll, M., K. Diehl, S. K. Mitra, and S. Borrmann, 2009: A Wind Tunnel Study on the Shape, Oscillation, and Internal Circulation of Large Raindrops with Sizes between 2.5 and 7.5 mm. J. Atmos. Sci., 66, 755–765.

Taraniuk, I., A. B. Kostinski, and Y. Rudich, 2008: Enrichment of surface-active com- pounds in coalescing cloud drops. Geophys. Res. Lett., 35.

Taubin, G., 1995: A signal processing approach to fair surface design. Proc. of the 22Nd Annual Conference on Computer Graphics and Interactive Techniques, 351–358.

Thurai, M., V. N. Bringi, A. B. Manic´, N. J. S˘ ekeljic´, and B. M. Notaros˘, 2014: In- vestigating raindrop shapes, oscillation modes, and implications for radio wave propagation. Radio Sci., 49, 921–932.

Tomita, H., 2008: New microphysical schemes with five and six categories by diag- nostic generation of cloud ice. J. Meteor. Soc. Japan, 86A, 121–142.

Tomita, H., and M. Satoh, 2004: A new dynamical framework of nonhydrostatic global model using the icosahedral grid. Fluid Dyn. Res., 34, 357–400.

Tornberg, A. K., and E. Björn, 2004: Numerical approximations of singular source terms in differential equations. J. Comput. Phys., 200, 462–488.

Torres, D. J., and J. U. Brackbill, 2000: The point-set method: Front-tracking without connectivity. J. Comput. Phys., 165, 620–644.

Udaykumar, H. S., H. C. Kan, S. Wei, and R. Tran-Son-Tay, 1997: Multiphase dy- namics in arbitrary geometries on fixed Cartesian grids. J. Comput. Phys., 137, 366–405.

Udaykumar, H. S., R. Mittal, P. Rampunggoon, A. Khanna, 2001: A sharp interface Cartesian grid method for simulating flows with complex moving boundaries. J. Comput. Phys., 174, 345–380.

Unverdi, S. O., and G. Tryggvason, 1992: A front-tracking method for viscous, incom- pressible, multi-fluid flows. J. Comput. Phys., 100, 25–37.

Wang, P. K., and H. R. Pruppacher, 1977: Acceleration to terminal velocity of cloud and raindrops. J. Appl. Meteor., 16, 275–280.

Woods, J. D., and B. J. Mason, 1964: Experimental determination of collection effi- ciencies for small water droplets in air. Q. J. Roy. Meteor. Soc., 90, 373–381.

Yang, X., X. Zhang, Z. Li, and G. W. He, 2009: A smoothing technique for discrete delta functions with application to immersed boundary method in moving bound- ary simulations. J. Comput. Phys., 228, 7821–7836.

Yokoi, K., 2013: A practical numerical framework for free surface flows based on {CLSVOF} method, multi-moment methods and density-scaled {CSF} model: Numerical simulations of droplet splashing. J. Comput. Phys., 232, 252–271.

Youngs, D. L., 1982: Time-dependent multi-material flow with large fluid distortion.Numer. Meth. Fluid D., 24, 273–285.

Zheng, X., 2016: An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension. Adv. Comput. Math., 42, 1225–1257.

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