A mass-conserving volume-of-fluid method: Volume tracking and droplet surface-tension in incompressible isotropic turbulence

Abstract We have developed a volume of fluid (VoF)/projection method for simulating droplet-laden incompressible turbulent flows with uniform density and viscosity. The method is mass-conserving, wisp-free, and consistent (i.e., the VoF function, C, satisfies the condition 0 ⩽ C ⩽ 1). First, we present the results of the VoF method for tracking volumes of initially spherical shape and with zero surface tension in analytical velocity fields (linear translation, solid-body rotation, single-vortex flow, and Taylor–Green vortex) and in incompressible isotropic turbulence at Reλ0 = 75 and 190. These numerical tests show that (i) our VoF method is mass-conserving, consistent, and wisp-free; (ii) for a CFL number of 0.1, the VoF geometrical error has almost a second-order convergence rate for a mesh resolution with more than 10 grid points per diameter; (iii) in the isotropic turbulence case, a resolution of about 32 grid points per diameter of the sphere is required in order to limit the VoF geometrical error below 1%. Then, in order to simulate droplet-laden flows, we have adopted the continuum surface force (CSF) model to compute the surface tension force. We have modified the sequence of the VoF advection sweeps, and show that, in the case of droplet in a translating reference frame, the r.m.s. of the spurious currents is about 1% of the translating velocity. Finally, we present DNS results of fully-resolved droplet-laden incompressible isotropic turbulence at Re λ 0 = 75 using a computational mesh of 1024 3 grid points and 7000 droplets of Weber number We rms = 0.5 , and initial droplet diameter equal to the Taylor length-scale of turbulence.

[1]  Bernhard Weigand,et al.  Direct numerical simulation of evaporating droplets , 2008, J. Comput. Phys..

[2]  U. Schumann.  Realizability of Reynolds-Stress Turbulence Models , 1977 .

[3]  P. Colella,et al.  A conservative three-dimensional Eulerian method for coupled solid-fluid shock capturing , 2002 .

[4]  L YoungsD,et al.  Time-dependent multi-material flow with large fluid distortion. , 1982 .

[5]  Charles S. Peskin,et al.  Flow patterns around heart valves: a digital computer method for solving the equations of motion , 1973 .

[6]  Julio Hernández,et al.  Analytical and geometrical tools for 3D volume of fluid methods in general grids , 2008, J. Comput. Phys..

[7]  Julio Hernández,et al.  On reducing interface curvature computation errors in the height function technique , 2010, J. Comput. Phys..

[8]  Eugenio Aulisa,et al.  Interface reconstruction with least-squares fit and split advection in three-dimensional Cartesian geometry , 2007, J. Comput. Phys..

[9]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[10]  Ian M. Mitchell,et al.  A hybrid particle level set method for improved interface capturing , 2002 .

[11]  S. Zaleski,et al.  Modelling Merging and Fragmentation in Multiphase Flows with SURFER , 1994 .

[12]  F. Faura,et al.  A new volume of fluid method in three dimensions—Part I: Multidimensional advection method with face‐matched flux polyhedra , 2008 .

[13]  S. Elghobashi,et al.  On the physical mechanisms of two-way coupling in particle-laden isotropic turbulence , 2003 .

[14]  J. Brackbill,et al.  A continuum method for modeling surface tension , 1992 .

[15]  G. Taylor,et al.  LXXV. On the decay of vortices in a viscous fluid , 1923 .

[16]  G. Tryggvason,et al.  A front-tracking method for viscous, incompressible, multi-fluid flows , 1992 .

[17]  D. Fletcher,et al.  A New Volume of Fluid Advection Algorithm , 2000 .

[18]  Matthew W. Williams,et al.  A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework , 2006, J. Comput. Phys..

[19]  S. Cummins,et al.  Estimating curvature from volume fractions , 2005 .

[20]  Stéphane Popinet,et al.  An accurate adaptive solver for surface-tension-driven interfacial flows , 2009, J. Comput. Phys..

[21]  W. Rider,et al.  Reconstructing Volume Tracking , 1998 .

[22]  S. Zaleski,et al.  Interface reconstruction with least‐square fit and split Eulerian–Lagrangian advection , 2003 .

[23]  Heinz Pitsch,et al.  Combination of 3D unsplit forward and backward volume-of-fluid transport and coupling to the level set method , 2013, J. Comput. Phys..