Parallel implementation of fast methods for vortex influence computation in vortex methods for 2d incompressible flows simulation

Abstract

Vortex methods are a powerful tool for solving engineering problems of incompressible flow simulation at small subsonic speeds. The main idea is to consider vorticity as a primary computed variable. Vorticity distribution is simulated by a set of elementary vorticity carriers — vortex elements. Their velocity in the flow is a sum of the convective and diffusive ones. The simplest way to compute the convective velocity of each vortex element is to summarize the influences of all the other vortices, it should be done at every time step. Such problem is similar to the N -body gravitational problem, its computational complexity is proportional to N 2 (N is number of vortices). This fact restricts significantly the applicability of vortex methods. Two approximate fast methods of logarithmic (N log N ) computational complexity are implemented and investigated. The first method is analogous of the Barnes — Hut fast method for the gravitational N -body problem; the second one is based on the possibility of convolution integral fast calculation through Fast Fourier Transform (FFT) technique with further results correction, which permits to take into account the influence of closely- spaced vortices. Sequential and parallel implementations of both methods are developed. Numerical experiments show that the FFT-based method is more efficient in comparison to the Barnes — Hut method; it provides the acceleration of about 1000 times for the velocities calculation for N = 500 000 vortex elements (in comparison to the direct “point- to-point” calculation). The number of mesh cells doesn’t effect the method accuracy, however it determines the computational complexity of the algorithm. It is found that the mesh size should be chosen according to the derived estimation of the algorithm’s numerical complexity and available computational resources.