Study of the long-time dynamics of a viscous vortex sheet with a fully adaptive nonstiff method

A numerical investigation of the long-time dynamics of two immiscible two-dimensional fluids shearing past one another is presented. The fluids are incompressible and the interface between the bulk phases is subjected to surface tension. The simple case of density and viscosity matched fluids is considered. The two-dimensional Navier–Stokes equations are solved numerically with a fully adaptive nonstiff strategy based on the immersed boundary method. Dynamically adaptive mesh refinements are used to cover at all times the separately tracked fluid interface at the finest grid level. In addition, by combining adaptive front tracking, in the form of continuous interface marker equidistribution, with a predictor–corrector discretization an efficient method is introduced to successfully treat the well-known numerical difficulties associated with surface tension. The resulting numerical method can be used to compute stably and with high resolution the flow for wide-ranging Weber numbers but this study focuses o...

[1]  Héctor D. Ceniceros,et al.  The effects of surfactants on the formation and evolution of capillary waves , 2003 .

[2]  Thomas Y. Hou,et al.  The long-time motion of vortex sheets with surface tension , 1997 .

[3]  M. Nitsche.  Singularity formation in a cylindrical and a spherical vortex sheet , 2001 .

[4]  D. Juric,et al.  A front-tracking method for the computations of multiphase flow , 2001 .

[5]  Qiang Zhang,et al.  Three-Dimensional Front Tracking , 1998, SIAM J. Sci. Comput..

[6]  Stephen J. Cowley,et al.  On the formation of Moore curvature singularities in vortex sheets , 1999, Journal of Fluid Mechanics.

[7]  Charles S. Peskin,et al.  Modeling Arteriolar Flow and Mass Transport Using the Immersed Boundary Method , 1998 .

[8]  Russel E. Caflisch,et al.  Singular solutions and ill-posedness for the evolution of vortex sheets , 1989 .

[9]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[10]  M. Minion.  A Projection Method for Locally Refined Grids , 1996 .

[11]  Michael Selwyn Longuet-Higgins,et al.  The deformation of steep surface waves on water - I. A numerical method of computation , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[12]  Isidore Rigoutsos,et al.  An algorithm for point clustering and grid generation , 1991, IEEE Trans. Syst. Man Cybern..

[13]  Daniel D. Joseph,et al.  Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel , 2001, Journal of Fluid Mechanics.

[14]  SheetsThomas,et al.  Singularity Formation in 3-D Vortex , 2001 .

[15]  J. Glimm,et al.  A critical analysis of Rayleigh-Taylor growth rates , 2001 .

[16]  T. Hou,et al.  Singularity formation in three-dimensional vortex sheets , 2001 .

[17]  Thomas Y. Hou,et al.  Convergence of a non-stiff boundary integral method for interfacial flows with surface tension , 1998, Math. Comput..

[18]  T. Taylor,et al.  Computational methods for fluid flow , 1982 .

[19]  Xiaolin Li,et al.  Robust Computational Algorithms for Dynamic Interface Tracking in Three Dimensions , 1999, SIAM J. Sci. Comput..

[20]  Steven A. Orszag,et al.  Analytic structure of vortex sheet dynamics. Part 1. Kelvin–Helmholtz instability , 1982, Journal of Fluid Mechanics.

[21]  M. Berger,et al.  An Adaptive Version of the Immersed Boundary Method , 1999 .

[22]  Steven J. Ruuth,et al.  Implicit-Explicit Methods for Time-Dependent PDE''s , 1993 .

[23]  D. W. Moore,et al.  The spontaneous appearance of a singularity in the shape of an evolving vortex sheet , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[24]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[25]  Samuel L. Manzello,et al.  The influence of liquid pool temperature on the critical impact Weber number for splashing , 2003 .

[26]  G. Baker,et al.  Application of Adaptive Quadrature to Axi-symmetric Vortex Sheet Motion , 1998 .

[27]  B. Wetton,et al.  Analysis of Stiffness in the Immersed Boundary Method and Implications for Time-Stepping Schemes , 1999 .

[28]  Morteza Gharib,et al.  Structures in stratified plane mixing layers and the effects of cross-shear , 1997, Journal of Fluid Mechanics.

[29]  Michael Shelley,et al.  A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method , 1992, Journal of Fluid Mechanics.

[30]  T. Hou,et al.  Removing the stiffness from interfacial flows with surface tension , 1994 .

[31]  Gretar Tryggvason,et al.  The nonlinear behavior of a sheared immiscible fluid interface , 2002 .

[32]  T. Sakajo.  Formation of curvature singularity along vortex line in an axisymmetric, swirling vortex sheet , 2002 .

[33]  R. Krasny.  A study of singularity formation in a vortex sheet by the point-vortex approximation , 1986, Journal of Fluid Mechanics.

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

[35]  Anita Mayo,et al.  An implicit numerical method for fluid dynamics problems with immersed elastic boundaries , 1993 .

[36]  Steven J. Ruuth,et al.  Implicit-explicit methods for time-dependent partial differential equations , 1995 .