A numerical study of vortex method schemes for two-dimensional incompressible flows

Vortex methods have been an alternative to Eulerian methods for the calculation of engineering flows. Their inherent adaptivity as well as the fact that the pressure is decoupled from the evolution equations are two of the main reasons that add to the popularity of this class of methods. However there are also some drawbacks in these methodologies. Some of those drawbacks have to do with their efficiency as well as implementation issues. Some times it is not clear when someone is within the mathematical framework of the method or has turned them into a modeling tool. In some cases it seems that the convergence proved by mathematicians completely analytically and with several assumptions regarding the field don't show up in practice especially for more complicated flows or flows involving complex geometries. In this thesis we attempt to answer many of those questions, prove numerical convergence of the various schemes used, investigate the effect of the choice of input parameters and decisions. Furthermore we propose new data-structures and new schemes to alleviate a lot of the implementation problems of the methods. The developed codes are tested in problems with analytic solutions or solutions obtained by other methods. In some cases different alternatives are compared and conclusions are reached as of what is better for what case. Some schemes are included without having been implemented for completeness of text and to inform the reader about more available options.

[1]  D. Shiels,et al.  Simulation of controlled bluff body flow with a viscous vortex method , 1998 .

[2]  J. Christiansen.  Numerical Simulation of Hydrodynamics by the Method of Point Vortices , 1997 .

[3]  Norman J. Zabusky,et al.  Contour Dynamics for the Euler Equations in Two Dimensions , 1997 .

[4]  A. Langdon.  Introduction to “Clouds-in-Clouds, Clouds-in-Cells Physics for Many-Body Simulation” , 1997 .

[5]  Louis F. Rossi,et al.  Merging Computational Elements in Vortex Simulations , 1997, SIAM J. Sci. Comput..

[6]  P. Koumoutsakos,et al.  Simulations of the viscous flow normal to an impulsively started and uniformly accelerated flat plate , 1996, Journal of Fluid Mechanics.

[7]  Georges-Henri Cottet,et al.  Artificial Viscosity Models for Vortex and Particle Methods , 1996 .

[8]  S. Shankar,et al.  A New Diffusion Procedure for Vortex Methods , 1996 .

[9]  C. Williamson.  Vortex Dynamics in the Cylinder Wake , 1996 .

[10]  H. O. Nordmark.  Deterministic high order vortex methods for the 2-d navier - stokes equation with rezoning , 1996 .

[11]  Thomas Westermann,et al.  Particle-in-cell simulations with moving boundaries—adaptive mesh generation , 1994 .

[12]  P. Koumoutsakos,et al.  Boundary Conditions for Viscous Vortex Methods , 1994 .

[13]  Giovanni Russo,et al.  Fast triangulated vortex methods for the 2D Euler equations , 1994 .

[14]  Giovanni Russo,et al.  A deterministic vortex method for the Navier-Stokes equations , 1993 .

[15]  Petros Koumoutsakos,et al.  Direct numerical simulations of unsteady separated flows using vortex methods , 1993 .

[16]  Timothy J. Ross,et al.  Diffusing-vortex numerical scheme for solving incompressible Navier-Stokes equations , 1991 .

[17]  E. Puckett,et al.  A fast vortex method for computing 2D viscous flow , 1990 .

[18]  T. Buttke,et al.  A fast adaptive vortex method for patches of constant vorticity in two dimensions , 1990 .

[19]  Thomas Y. Hou,et al.  Convergence of the point vortex method for the 2-D euler equations , 1990 .

[20]  Dalia Fishelov,et al.  A new vortex scheme for viscous flow , 1990 .

[21]  H. O. Nordmark,et al.  Rezoning for higher order vortex methods , 1991 .

[22]  P. A. Smith,et al.  An efficient surface algorithm for random-particle simulation of vorticity and heat transport , 1989 .

[23]  Elbridge Gerry Puckett,et al.  A Study of the Vortex Sheet Method and its Rate of Convergence , 1989 .

[24]  P. Stansby,et al.  Flow around a cylinder by the random vortex method , 1989 .

[25]  N. Zabusky,et al.  Contour dynamics for the Euler equations: curvature controlled initial node placement and accuracy , 1988 .

[26]  P. A. Smith,et al.  Impulsively started flow around a circular cylinder by the vortex method , 1988, Journal of Fluid Mechanics.

[27]  David G. Dritschel,et al.  Contour surgery: a topological reconnection scheme for extended integrations using contour dynamics , 1988 .

[28]  J. Sethian,et al.  Validation study of vortex methods , 1988 .

[29]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[30]  Ole H. Hald,et al.  Convergence of Vortex methods for Euler's equations, III , 1987 .

[31]  Jonathan Goodman,et al.  Convergence of the Random Vortex Method , 1987 .

[32]  Peter Stansby,et al.  Generalized discrete vortex method for cylinders without sharp edges , 1987 .

[33]  M. Tsutahara,et al.  Flows about a Rotating Circular Cylinder by the Discrete-Vortex Method , 1987 .

[34]  Ole H. Hald,et al.  Convergence of random method with creation of vorticity , 1986 .

[35]  C. Greengard.  Convergence of the Vortex filament method , 1986 .

[36]  Zhen-huan Teng,et al.  Variable-elliptic-vortex method for incompressible flow simulation , 1986 .

[37]  C. R. Anderson.  A method of local corrections for computing the velocity field due to a distribution of vortex blobs , 1986 .

[38]  Andrew J. Majda,et al.  Vorticity and the mathematical theory of incompressible fluid flow , 1986 .

[39]  Ahmed F. Ghoniem,et al.  Grid-free simulation of diffusion using random walk methods , 1985 .

[40]  Christopher R. Anderson,et al.  On Vortex Methods , 1985 .

[41]  Andrew J. Majda,et al.  High order accurate vortex methods with explicit velocity kernels , 1985 .

[42]  S. Roberts.  Accuracy of the random vortex method for a problem with non-smooth initial conditions , 1985 .

[43]  P. Bearman.  VORTEX SHEDDING FROM OSCILLATING BLUFF BODIES , 1984 .

[44]  Marjorie Perlman,et al.  Accuracy of vortex methods , 1983 .

[45]  A. Y. Cheer,et al.  NUMERICAL STUDY OF INCOMPRESSIBLE SLIGHTLY VISCOUS FLOW PAST BLUNT BODIES AND AIRFOILS - eScholarship , 1983 .

[46]  Peter Stansby,et al.  Simulation of flows around cylinders by a Lagrangian vortex scheme , 1983 .

[47]  Andrew J. Majda,et al.  Vortex methods. I. Convergence in three dimensions , 1982 .

[48]  A. Majda,et al.  Vortex methods. II. Higher order accuracy in two and three dimensions , 1982 .

[49]  Z. Teng,et al.  Elliptic-vortex method for incompressible flow at high Reynolds Number☆ , 1982 .

[50]  A. Majda,et al.  Rates of convergence for viscous splitting of the Navier-Stokes equations , 1981 .

[51]  A. Leonard.  Vortex methods for flow simulation , 1980 .

[52]  Alexandre J. Chorin,et al.  Vortex Models and Boundary Layer Instability , 1980 .

[53]  Ole Hald,et al.  Convergence of vortex methods for Euler’s equations , 1978 .

[54]  Alexandre J. Chorin,et al.  Vortex sheet approximation of boundary layers , 1978 .

[55]  Alexandre J. Chorin,et al.  Discretization of a vortex sheet, with an example of roll-up☆ , 1973 .

[56]  A. Chorin.  Numerical study of slightly viscous flow , 1973, Journal of Fluid Mechanics.

[57]  R. Clements,et al.  An inviscid model of two-dimensional vortex shedding , 1973, Journal of Fluid Mechanics.

[58]  G. Pedrizzetti,et al.  Vortex Dynamics , 2011 .

[59]  J. CARRIERt,et al.  A FAST ADAPTIVE MULTIPOLE ALGORITHM FOR PARTICLE SIMULATIONS * , 2022 .