Advances in viscous vortex methods?meshless spatial adaption based on radial basis function interpolation

Abstract

SUMMARYVortex methods have a history as old as ÿnite di erences. They have since faced di culties stemming from the numerical complexity of the Biot-Savart law, the inconvenience of adding viscous e ects in a Lagrangian formulation, and the loss of accuracy due to Lagrangian distortion of the computational elements. The ÿrst two issues have been successfully addressed, respectively, by the application of the fast multipole method, and by a variety of viscous schemes which will be brie y reviewed in this article. The standard method to deal with the third problem is the use of remeshing schemes consisting of tensor product interpolation with high-order kernels. In this work, a numerical study of the errors due to remeshing has been performed, as well as of the errors implied in the discretization itself using vortex blobs. In addition, an alternative method of controlling Lagrangian distortion is proposed, based on ideas of radial basis function (RBF) interpolation (brie y reviewed here). This alternative is formulated grid-free, and is shown to be more accurate than standard remeshing. In addition to high-accuracy, RBF interpolation allows core size control, either for correcting the core spreading viscous scheme or for providing a variable resolution in the physical domain. This formulation will allow in theory the application of error estimates to produce a truly adaptive spatial reÿnement technique. Proof-of-concept is provided by calculations of the relaxation of a perturbed monopole to a tripole attractor.

Subgrid-Scale Modeling of Turbulent Flow Around Circular Cylinder by Mesh-Free Vortex Method

“…The vortex method that uses the RWM is currently referred to as the Random Vortex Method (Barba et al, 2005). Hence, the total vortex displacement is given by , ,…”

“…In other words, approximately one out of five vortices is "turbulent" at the last step of the simulation, in the sense that they have significant turbulent activity in their neighborhood. Additionally, we observe that the range of Re t computed in this simulation varies from 1.510 3 to 4.510 4 Figure 10 displays the continuous development of the wake's streamlines as the simulation evolves. The wake is born in a symmetric fashion and gradually looses its symmetry due to the amplification of existing perturbations in the flow, simulated by the random (or stochastic) component of the algorithm for vorticity diffusion.…”

Emergence and evolution of tripole vortices from net-circulation initial conditions

“…Figure 1 shows the evolution of the total vorticity for this case. ͑See Barba et al 23 for a comparison of results with our method to the results of Rossi et al, 9 using similar line-contour plots of total vorticity and perturbation vorticity.͒…”

“…The current limitation is mainly one of memory, as a square matrix of rank equal to the number of particles is built on initialization and upon each process of spatial adaption. As described in detail by Barba 22 and Barba et al, 23 the method utilizes radial basis function interpolation to find the circulation strength of the vortex particles on initialization and after spatial adaption. Radial basis function interpolation is a highly accurate method for interpolation of scattered data, [25][26][27] but it requires the solution of a global influence problem.…”

A self-organizing Lagrangian particle method for adaptive-resolution advection–diffusion simulations

“…Rather, they are conceptually related to the interpolation scheme used in Behrens's adaptive semi-Lagrangian method [5] for radial basis functions. Differential operators can be consistently approximated on the evolving, irregular set of particles using the DC-PSE scheme [39], which can be seen as a generalization of vorticity redistribution schemes [41,4,25] to arbitrary differential operators.…”