A dilating vortex particle method for compressible flow with applications to aeroacoustics

Abstract

Vortex methods have become useful tools for the computation of incompressible fluid flow. In the present work, a vortex particle method for the simulation of unsteady two-dimensional compressible flow is developed and applied to several problems. The method is the first Langrangian simulation method for the full compressible Navier-Stokes equations. By decomposing the velocity into irrotational and solenoidal parts, and using particles that are able to change volume and that carry vorticity, dilation, enthalpy, entropy, and density, the equations of motion are satisfied. A general deterministic treatment of spatial derivatives in particle methods is developed by extending the method of particle strength exchange through the construction of higher-order-accurate, non-dissipative kernels for use in approximating arbitrary differential operators. The application of this technique to wave propagation problems is thoroughly explored. A one-sided operator is developed for approximating derivatives near the periphery of particle coverage; the operator is used to enforce a non-reflecting boundary condition for the absorption of acoustic waves at this periphery. Remeshing of the particles and the smooth interpolation of their strengths are addressed, and a criterion for the frequency of remeshing is developed on the principle axes of the rate-of-strain tensor. The fast multipole method for the fast summation of the velocity field is adapted for use with compressible particles. The new vortex method is applied to co-rotating and leapfrogging vortices in compressible flow, with the acoustic field computed using a two-dimensional Kirchoff surface, and the results agree will with those of previous work or analytical prediction. The method is also applied to the baroclinic generation of vorticity, and to the steepening of waves in the one-dimensional Burgers’ equation, with favorable results in both case