Vortex Method

The results of a numerical investigation of Rayleigh-Taylor instability in an inviscid, incompressible flow by vortex methods was presented. The numerical procedure was briefly described. The effect of surface-tension on the smoothing of irregular motion of vortices in the vortex simulation of Reyleigh-Taylor instability is shown. The irregular motion appears as an effect of short-wave disturbances, the source of which is a round-off error. Inclusion of surface tension allows for the observation of the formation of singularities. The singularities make an infinite jump discontinuity in the curvature of the vortex sheet. It is observed that, for a sufficiently small Atwood number and for initial amplitude perturbation large enough, two singularities appear in a half period of the vortex sheet and only one appears for greater Atwood numbers. The results for desingularized equations of motion were also presented. Desingularization was made by introducing the delta-square parameter to t...

This paper presents numerical simulation results for vortex-induced vibration of a circular cylinder in laminar flow. A vortex method is implemented to solve the two-dimensional Navier-Stokes equations in terms of vorticity. In order to validate the numerical code, the flow past a fixed cylinder is first investigated for which enough experimental and numerical results are available. Basic characteristics of the dynamic response and vortex shedding for an elastically mounted circular cylinder are then investigated for 70 < Re < 170. The lock-in phenomenon is captured at certain reduced velocities where the lift coefficient takes a considerable value associated with a high amplitude response. The wake structure exhibits the 2S or C (2S) modes of vortex shedding in this range of Reynolds numbers, as opposed to the 2P mode which is observed in the turbulent flow regime. The numerical results are in acceptable agreement with available experimental and numerical data.

The motion of free surfaces in incompressible, irrotational, inviscid layered flows is studied by evolution equations for the position of the free surfaces and appropriate dipole (vortex) and source strengths. The resulting Fredholm integral equations of the second kind may be solved efficiently in both storage and work by iteration in both two and three dimensions. Applications to breaking water waves over finite-bottom topography and interacting triads of surface and interfacial waves are given.

During the motion of a fluid interface undergoing Rayleigh-Taylor instability, vorticity is generated on the interface baronclinically. This vorticity is then subject to Kelvin-Helmholtz instability. For the related problem of evolution of a nearly flat vortex sheet without density stratification (and with viscosity and surface tension neglected), Kelvin-Helmholtz instability has been shown to lead to development of curvature singularities in the sheet. In this paper, a simple approximate theory is developed for Rayleigh-Taylor instability as a generalization of Moore&#39;s approximation for vortex sheets. For the approximate theory, a family of exact solutions is found for which singularities develop on the fluid interface. The resulting predictions for the time and type of the singularity are directly verified by numerical computation of the full equations. These computations are performed using a point vortex method, and singularities for the numerical solution are detected using...

Vortex methods rely principally on a discretization of the continuous two-dimensional timedependent vorticity field into a large number of vortex &quot;blobs&quot;, whose position and strengthdetermine the underlying velocity field. In this paper, the convergence of the random vortexmethod (RVM) for a complex flow is studied in function of three discretization parameters. Twoof these parameters are related to the spatial discretization of the vorticity, i.e. \Gamma (sheet or blobstrength) and h...

Short Lecture Session 3.5b т эсп 6 Kuwahara, К.: Numerical study of flow past an inclined flat plate by an inviscid model. J. Phys. Soc. Japan 35 (1973). 5. 7 Clements, RR: An inviscid model of two-dimensional vortex shedding. J. Fluid Mech. 57 (1973), 2. 8 Kiya, M.; Arie, M.: A ...

Fast algorithms for the computation of $N$-body problems can be broadly classified into mesh-based interpolation methods, and hierarchical or multiresolution methods. To this last class belongs the well-known fast multipole method (FMM), which offers O(N) complexity. This paper presents an extensible parallel library for $N$-body interactions utilizing the FMM algorithm, built on the framework of PETSc. A prominent feature of

Linear and nonlinear dilational and sinuous capillary waves on thin inviscid infinite and semi-infinite planar liquid sheets in a void are analysed in a unified manner by means of a method that reduces the two-dimensional unsteady problem to a one-dimensional unsteady problem. For nonlinear dilational waves on infinite sheets, the accuracy of the numerical solutions is verified by comparing with an analytical solution. The nonlinear dilational wave maintains a reciprocal relationship between wavelength and wave speed modified from the linear theory prediction by a dependence of the product of wavelength and wave speed on the wave amplitude. For the general dilational case, nonlinear numerical simulations show that the sheet is unstable to superimposed subharmonic disturbances on the infinite sheet. Agreement for both sinuous and dilational waves is demonstrated for the infinite case between nonlinear simulations using the reduced one-dimensional approach, and nonlinear two-dimension...

Linear and nonlinear dilational and sinuous capillary waves on thin inviscid infinite and semi-infinite planar liquid sheets in a void are analysed in a unified manner by means of a method that reduces the two-dimensional unsteady problem to a one-dimensional unsteady problem. For nonlinear dilational waves on infinite sheets, the accuracy of the numerical solutions is verified by comparing with an analytical solution. The nonlinear dilational wave maintains a reciprocal relationship between wavelength and wave speed modified from the linear theory prediction by a dependence of the product of wavelength and wave speed on the wave amplitude. For the general dilational case, nonlinear numerical simulations show that the sheet is unstable to superimposed subharmonic disturbances on the infinite sheet. Agreement for both sinuous and dilational waves is demonstrated for the infinite case between nonlinear simulations using the reduced one-dimensional approach, and nonlinear two-dimension...

ABSTRACT To model incompressible flow over a body of arbitrary geometry when using vortex methods, it is necessary to construct an irrotational field to impose the impermeability condition at the surface of the object. In order to achieve this impermeability, this paper uses a boundary integral equation based on the single-layer representation for the velocity potential. Specifically, we formulate this exterior Neumann problem in terms of a source/sink boundary integral equation. The solution to this integral equation is then coupled with an interpolation procedure which smoothes the transition between near-wall and interior regimes. We describe the numerical scheme embedding this strategy and discuss its accuracy and efficiency. For validation purposes, we consider the potential and vortical flow over a circular cylinder, for which an analytical solution and the commonly used method of images are available.

Abstract. Convergence of the grid-free point vortex method is proved for three-dimensional Euler equations with smooth solutions. Two new techniques are used to obtain consistency and stability of the method. The first one is Strang&amp;amp;#x27;s trick [Numer. Math., 6 (1964), pp. 37-46] which ...

The results of a numerical investigation of Rayleigh-Taylor instability in an inviscid, incompressible flow by vortex methods was presented. The numerical procedure was briefly described. The effect of surface-tension on the smoothing of irregular motion of vortices in the vortex simulation of Reyleigh-Taylor instability is shown. The irregular motion appears as an effect of short-wave disturbances, the source of which is a round-off error. Inclusion of surface tension allows for the observation of the formation of singularities. The singularities make an infinite jump discontinuity in the curvature of the vortex sheet. It is observed that, for a sufficiently small Atwood number and for initial amplitude perturbation large enough, two singularities appear in a half period of the vortex sheet and only one appears for greater Atwood numbers. The results for desingularized equations of motion were also presented. Desingularization was made by introducing the delta-square parameter to the Cauchy type integral; in this way the singularity of the integrand was removed.

Vortex methods using vorticity–velocity formulations have become an increasingly powerful and popular means of studying complex fluid flow systems. The problem of combining an integral equation method and grid-free discrete vortex method (DVM) when studying three-dimensional wall-bounded flows is considered. While the normal boundary condition is satisfied by means of a boundary integral equation (BIE), we also consider the problem of recovering pressure from given vorticity and velocity fields when using Lagrangian DVMs in terms of a BIE. For validation purposes, vortical flow past a sphere and past a flat plate are considered, for which the commonly used method of images is available. Results of near-wall boundary-layer flow simulations are then presented as an illustration of the numerical scheme. The importance of hairpin vortices is highlighted. Finally, results on wall compliance fluid flow are displayed emphasizing the versatility of the numerical method.