In vortex methods the flow field is recovered at every location of the domain when one considers the collective behavior of all computational elements. The length scales of the flow quantities that are been resolved are characterized by the particle core rather than the interparticle distance. These observations, which stem from the definition itself of vortex methods and are confirmed by its numerical analysis, differentiate particle methods from schemes such as finite differences.The essense of the method is the "communication" of information between the particles, that requires a particle overlap. As a result, a computation is bound to become inaccurate once the particles cease to overlap. Computations involving nonoverlapping finite core particles should be regarded then as modeling and not as direct numerical simulations. Excluding case-specific initial particle distributions (e.g., particles placed on concentric rings to represent an azimuthally invariant vorticity distribution) the loss of overlap (and excessive overlap) is an inherent problem of purely Lagrangian methods.The cause of the problem is the flow strain that may cluster particles in one direction and spread them in another in the neighborhood of hyperbolic points of the flow map, resulting in nonuniform distributions. At the onset of such particle distributions no error is usually manifested in the global quantities of the flow such as the linear and the angular impulse. However, locally the vorticity field becomes distorted and spreading of the particles results in loss of naturally present vortical structures, whereas particle clustering results in the appearance of unphysical ones on the scale of the interparticle separation.A mathematical explanation of this can be found in the numerical analysis of Section 2.6. The equation governing the error involves a right-hand side that is related to the truncation error of the method and that is amplified exponentially in time at a rate given by the first-order derivatives of the flow. These derivatives 206 Cambridge Books Online

International audienceWe present a vortex method for the simulation of the interaction of an incompressible flow with rigid bodies. The method is based on a penalization technique where the system is considered as a single flow, subject to the Navier-Stokes equation with a penalization term that enforces continuity at the solid-fluid interface and rigid motion inside the solid. Level set functions are used to capture interfaces, compute rigid motions inside the solid bodies and model collisions between bodies. A vortex in cell algorithm is built on this method. Numerical comparisons with existing 3D methods on problems of sedimentation and collision of spheres are provided to illustrate the capabilities of the method

This paper is devoted to the design of Vortex-In-Cell (VIC) methods for the direct numerical simulations of wallbounded flows. A first method using body-fitted grid is presented in the particular case of a cylinder wake. This method, which has been used in [Phys. Fluids 14(6) (2002) 2021] to investigate the effect on the wake topology of cylinder rotations, is an extension of the VIC method presented in [J. Comput. Phys. 175 (2002) 702] for periodic geometries. Features of the method that are specific to wall-bounded geometries -interpolation operators, field calculations and vorticity flux formulas to enforce no-slip boundary conditions -are described in details. The accuracy of the method in the calculation of the body forces is investigated by comparisons with experiments and benchmark calculations. A second class of methods is in the spirit of the immersed boundary methods. The paper in particular shows that the noslip conditions are very naturally handled by the vorticity flux formulas, independently of the relative locations of the particles and the body. Numerical experiments on the test-case of a ring impinging on a cylinder suggest that the method is second-order accurate.