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Research Article

A coupled immersed interface and level set method for simulation of interfacial flows steered by surface tension

H. V. R. Mittal1( )Rajendra K. Ray1Hermes Gadêlha2Dhiraj V. Patil3
School of Basic Sciences, Indian Institute of Technology Mandi, P.O. Box 175005, Kamand, Mandi, India
Faculty of Engineering, University of Bristol, Clifton BS8 1UB, United Kingdom
Department of Mechanical Engineering, Indian Institute of Technology Dharwad, Dharwad, Karnataka, 580011, India
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This work presents a methodology to address the problems of bubbles and drops evolving in incompressible, viscous flows due to the effect of surface tension. It is based on the combination of a recently developed immersed interface method to resolve discontinuities, with the level set (LS) method to reproduce the evolving interfaces. The paramount feature of this immersed interface method is the use of Lagrange interpolation enclosing grid points positioned in the vicinity of the interface and few exceptional grid points positioned on the interface. Different problems are considered to assert the accurateness of the proposed methodology, involving both simple and complex interface geometries. Precisely, the following problems are addressed: circular flow with a fixed interface, the dispersion of capillary waves, initially circular, ellipse, star shaped bubbles oscillating to an equilibrium state, and circular drops deforming in shear flows. The transient evolution of bubbles/drops in terms of their shapes, pressure profiles, velocity vectors, deformation ratios of major and minor axis is analyzed to observe the effect of surface tension. The proposed methodology is seen to recover the exact numerical equilibrium between the surface tension and pressure gradient in the vicinity of complex interface geometries as well, while recreating the flow physics with an adequate level of accuracy with well representation of overall trends. Moreover, the numerical results yield a good level of agreement with the reference data.


D. A. Anderson,, J. C. Tannehil,, R. H. Pletcher, 1984. Computational Fluid mechanics and Heat Transfer. New York: Hemisphere Publishing Corporation.
N. Balcázar Arciniega, 2014. Numerical simulation of multiphase flows: Level-set techniques. Doctoral Thesis. UPC, Departament de Màquines i Motors Tèrmics. Available at
N. Balcázar,, O. Lehmkuhl,, L. Jofre,, J. Rigola,, A. Oliva, 2016. A coupled volume-of-fluid/level-set method for simulation of two- phase flows on unstructured meshes. Comput Fluids, 124: 12-29.
I. Chakraborty,, G. Biswas,, P. S. Ghoshdastidar, 2013. A coupled level-set and volume-of-fluid method for the buoyant rise of gas bubbles in liquids. Int J Heat Mass Tran, 58: 240-259.
Y. C. Chang,, T. Y. Hou,, B. Merriman,, S. Osher, 1996. A level set formulation of Eulerian interface capturing methods for incompressible fluid flows. J Comput Phys, 124: 449-464.
A. J. Chorin, 1967. A numerical method for solving incompressible viscous flow problems. J Comput Phys, 2: 12-26.
K. Connington,, T. Lee, 2012. A review of spurious currents in the lattice Boltzmann method for multiphase flows. J Mech Sci Tech, 26: 3857-3863.
F. Denner,, F. Evrard,, R. Serfaty,, B. G. M. van Wachem, 2017. Artificial viscosity model to mitigate numerical artefacts at fluid interfaces with surface tension. Comput Fluids, 143: 59-72.
S. Farokhirad,, T. Lee,, J. F. Morris, 2013. Effects of inertia and viscosity on single droplet deformation in confined shear flow. Commun Comput Phys, 13: 706-724.
M. M. Francois,, S. J. Cummins,, E. D. Dendy,, D. B. Kothe,, J. M. Sicilian,, 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.
M. Francois,, W. Shyy, 2003. Computations of drop dynamics with the immersed boundary method, part 1: Numerical algorithm and buoyancy-induced effect. Numer Heat Tr B: Fund, 44: 101-118.
Z. Ge,, J. C. Loiseau,, O. Tammisola,, L. Brandt, 2018. An efficient mass-preserving interface-correction level set/ghost fluid method for droplet suspensions under depletion forces. J Comput Phys, 353: 435-459.
H. P. Greenspan, 1977a. On the deformation of a viscous droplet caused by variable surface tension. Stud Appl Math, 57: 45-58.
H. P. Greenspan, 1977b. On the dynamics of cell cleavage. J Theor Biol, 65: 79-99.
E. Gutiérrez,, F. Favre,, N. Balcázar,, A. Amani,, J. Rigola, 2018. Numerical approach to study bubbles and drops evolving through complex geometries by using a level set-moving mesh-immersed boundary method. Chem Eng J, 349: 662-682.
J. Huang,, H. Huang,, S. Wang, 2015. Phase-field-based simulation of axisymmetric binary fluids by using vorticity-streamfunction formulation. Prog Comput Fluid Dy, 15: 352-371.
S. Hysing, 2012. Mixed element FEM level set method for numerical simulation of immiscible fluids. J Comput Phys, 231: 2449-2465.
S. Ketterl,, M. Reißmann,, M. Klein, 2019. Large eddy simulation of multiphase flows using the volume of fluid method: Part 2—A-posteriori analysis of liquid jet atomization. Exp Comput Multiphase Flow, 1: 201-211.
M. Klein,, S. Ketterl,, J. Hasslberger, 2019. Large eddy simulation of multiphase flows using the volume of fluid method: Part 1—Governing equations and a priori analysis. Exp Comput Multiphase Flow, 1: 130-144.
R. J. LeVeque,, Z. Li, 1997. Immersed interface methods for stokes flow with elastic boundaries or surface tension. SIAM J Sci Comput, 18: 709-735.
Z. Li,, K. Ito,, M. C. Lai, 2007. An augmented approach for Stokes equations with a discontinuous viscosity and singular forces. Comput Fluids, 36: 622-635.
Z. Li,, M. C. Lai, 2001. The immersed interface method for the Navier-Stokes equations with singular forces. J Comput Phys, 171: 822-842.
Z. Li,, S. R. Lubkin, 2001. Numerical analysis of interfacial two-dimensional Stokes flow with discontinuous viscosity and variable surface tension. Int J Numer M Fl, 37: 525-540.
M. D. Mier-Torrecilla,, S. R. Idelsohn,, E. Oñate, 2011. Advances in the simulation of multi-fluid flows with the particle finite element method. Application to bubble dynamics. Int J Numer M Fl, 67: 1516-1539.
H. V. R. Mittal, 2016. A class of higher order accurate schemes for fluid interface problems. Doctoral Thesis. Indian Institute of Technology, Mandi, India.
H. V. R. Mittal,, Q. M. Al-Mdallal, 2018. A numerical study of forced convection from an isothermal cylinder performing rotational oscillations in a uniform stream. Int J Heat Mass Tran, 127: 357-374.
H. V. R. Mittal,, Q. M. Al-Mdallal,, R. K. Ray, 2017b. Locked-on vortex shedding modes from a rotationally oscillating circular cylinder. Ocean Eng, 146: 324-338.
H. V. R. Mittal,, J. C. Kalita,, R. K. Ray, 2016. A class of finite difference schemes for interface problems with an HOC approach. Int J Numer M Fl, 82: 567-606.
H. V. R. Mittal,, R. K. Ray, 2018. Solving immersed interface problems using a new interfacial points-based finite difference approach. SIAM J Sci Comput, 40: A1860-A1883.
H. V. R. Mittal,, R. K. Ray,, Q. M. Al-Mdallal, 2017a. A numerical study of initial flow past an impulsively started rotationally oscillating circular cylinder using a transformation-free HOC scheme. Phys Fluids, 29: 093603.
S. Osher,, R. Fedkiw, 2006. Level Set Methods and Dynamic Implicit Surfaces (Vol. 153). Springer Science & Business Media.
S. Osher,, J. A. Sethian, 1988. Fronts propagating with curvature- dependent speed: Algorithms based on Hamilton-Jacobi formulations. J Comput Phys, 79: 12-49.
C. S. Peskin, 1977. Numerical analysis of blood flow in the heart. J Comput Phys, 25: 220-252.
S. Popinet, 2009. An accurate adaptive solver for surface-tension- driven interfacial flows. J Comput Phys, 228: 5838-5866.
A. Prosperetti, 1981. Motion of two superposed viscous fluids. Phys Fluids, 24: 1217-1223.
M. Raessi,, J. Mostaghimi,, M. Bussmann, 2010. A volume-of-fluid interfacial flow solver with advected normals. Comput Fluids, 39: 1401-1410.
R. Scardovelli,, S. Zaleski, 1999. Direct numerical simulation of free- surface and interfacial flow. Ann Rev Fluid Mech, 31: 567-603.
K. S. Sheth,, C. Pozrikidis, 1995. Effects of inertia on the deformation of liquid drops in simple shear flow. Comput Fluids, 24: 101-119.
H. A. Stone,, A. D. Stroock,, A. Ajdari, 2004. Engineering flows in small devices: Microfluidics toward a lab-on-a-chip. Ann Rev Fluid Mech, 36: 381-411.
M. Sussman,, P. Smereka,, S. Osher, 1994. A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys, 114: 146-159.
Z. Tan,, D. V. Le,, Z. Li,, K. M. Lim,, B. C. Khoo, 2008. An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane. J Comput Phys, 227: 9955-9983.
A. Y. Tong,, Z. Wang, 2007. A numerical method for capillarity- dominant free surface flows. J Comput Phys, 221: 506-523.
M. Uh,, S. Xu, 2012. The immersed interface method for two-fluid problems. Available at
S. O. Unverdi,, G. Tryggvason, 1992. A front-tracking method for viscous, incompressible, multi-fluid flows. J Comput Phys, 100: 25-37.
S. Xu,, Z. J. Wang, 2006. An immersed interface method for simulating the interaction of a fluid with moving boundaries. J Comput Phys, 216: 454-493.
H. Z. Yuan,, C. Shu,, Y. Wang,, S. Shu, 2018. A simple mass-conserved level set method for simulation of multiphase flows. Phys Fluids, 30: 040908.
Experimental and Computational Multiphase Flow
Pages 21-37
Cite this article:
Mittal HVR, Ray RK, Gadêlha H, et al. A coupled immersed interface and level set method for simulation of interfacial flows steered by surface tension. Experimental and Computational Multiphase Flow, 2021, 3(1): 21-37.






Web of Science




Received: 13 July 2019
Revised: 18 September 2019
Accepted: 10 October 2019
Published: 06 March 2020
© Tsinghua University Press 2019