A high order regularisation method for solving the Poisson equation and selected applications using vortex methods

A regularisation method for solving the Poisson equation using Green’s functions is presented.The method is shown to obtain a convergence rate which corresponds to the design of the regularised Green’s function and a spectral-like convergence rate is obtained using a spectrally ideal regularisation. It is shown that the regularised Poisson solver can be extended to handle mixed periodic and free-space boundary conditions. This is done by solving the equation spectrally in the periodic directions which yields a modified Helmholtz equation for the free-space directions which in turn is solved by deriving the appropriate regularised Green’s functions. Using an analogy to the particle-particle particle-mesh method, a framework for calculating multi-resolution solutions using local refinement patches is presented. The regularised Poisson solver is shown to maintain a high order converging solution for different configurations of the refinement patches.The regularised Poisson solver has been implemented in a high order particle-mesh based vortex method for simulating incompressible fluid flow. A re-meshing of the vortex particlesis used to ensure the convergence of the method and a re-projection of the vorticity field is included to explicitly fulfil the kinematic constraints of the flow field. The high order, unbounded particle-mesh based vortex method is used to simulate the instability, transition to turbulence and eventual destruction of a single vortex ring. From the simulation data, a novel analysis on the vortex ring dynamics is presented based on the alignment of the vorticity vector with the principal axis of the strain rate tensor.A novel iterative implementation of the Brinkman penalisation method is introduced for the enforcement of a fluid-solid interface in re-meshed vortex methods. The iterative scheme is shown to improve the enforcement of the interface and also allow the simulation to perform significantly larger time steps, than what is customary for the method. The improved accuracy ofthe iterative implementation is demonstrated by considering challenging benchmark problems such as the impulsively started flow past a cylinder and a flat plate normal or inclined to theflow. The iterative implementation is shown to enhance the quality of the solution by Brinkman penalisation significantly for simulations of highly unsteady flows past complex geometries. A stochastic method of generating a synthetic turbulent flow field is combined with a 2Dmesh-free vortex method to simulate the effect of an oncoming turbulent flow on a bridge deck cross-section within the atmospheric boundary layer. The mesh-free vortex method is foundto be capable of preserving the a priori specified statistics as well as anisotropic characteristics of the synthesized turbulent flow field. From the simulation, the aerodynamic admittance is estimated and the instantaneous effect of a time varying angle of attack is briefly investigated.The obtained aerodynamic admittance of four aerodynamically different bridge sections are compared to available wind tunnel data, showing good agreement between the two. A vorticity formulated stochastic turbulence generator is presented which improves the kinetic properties of the generated turbulent field compared to present methods. Additional measures, such as explicit high order smoothing of the flow field, is introduced to insure that the generated field can be introduced into numerical simulations without an excessive loss of energydue to numerical dissipation.