Towards Resurrecting Grid-Free Three Dimensional Vortex Methods

Abstract

We present expressions for approximating the velocity and vortex stretching vectors induced by a far-field collection of point vortices and we propose a set of recurrence relations to properly evaluate these expressions. Expressed as truncated series of spherical harmonics, these approximations are used to develop an error-controlled hybrid adaptive fast solver that combines both O(N) and O(NLOGN) schemes. For a given accuracy, the adaptive solver is used in the context of regularized vortex methods to optimize the speed of the velocity and vortex stretching calculation. This is accomplished by introducing criteria for cell division in building of the tree, conversion of multipole to local expansion coefficients in the downward pass, stopping of the downward pass and choosing between direct and fast summation to compute the vector fields. These criteria are based on key parameters (p; nF ; nT ; d) which take into account the elements distribution, choice of the regularization function, and the computer architecture. The proposed solver automatically adapts to the evolving flow field by periodically updating the optimal values of these parameters to minimize the speed, while meeting the accuracy constraints, by balancing near and far-field calculations. Accuracy of the introduced expressions is investigated by inspecting the convergence of the velocity and the vortex stretching vectors as a function of the expansion order. Performance of the proposed adaptive scheme is investigated in terms of the dependence of cost and accuracy on the various controlling parameters. Evolution of the optimal values of adaptive solver parameters along with the associated computational savings are presented for the case of collision of two vortex rings over a reasonable time span. The adaptive solver, along with the introduced expressions, are used to assess the overall performance of three-dimensional grid-free regularized vortex methods by simulating the collision of two vortex rings, over a long period of time, for different values of Reynolds number covering the range 500-2000. These methods typically rely on operator splitting to handle diffusion and convection separately. In our implementation, the convection step employs a second order Runge-Kutta time integration scheme, where the particles velocities and vortex stretching vectors are computed using the adaptive fast solver that employs the proposed expressions. To model diffusion, we use an extension of the smoothed redistribution scheme to 3D unbounded flows and we compare it with the particle strength exchange diffusion model. We report on the capacity and limitation of the redistribution diffusion model to maintain a divergence free vorticity field without the need to resort to explicit methods and we explore the accuracy of different methods traditionally used to enforce the divergence free condition on the vorticity field.