Mads Mølholm Hejlesen

Iterative Brinkman penalization for remeshed vortex methods

We introduce an iterative Brinkman penalization method for the enforcement of the no-slip boundary condition in remeshed vortex methods. In the proposed method, the Brinkman penalization is applied iteratively only in the neighborhood of the body. This allows for using significantly larger time steps, than what is customary in the Brinkman penalization, thus reducing its computational cost while maintaining the capability of the method to handle complex geometries. We demonstrate the accuracy of our method by considering challenging benchmark problems such as flow past an impulsively started cylinder and normal to an impulsively started and accelerated flat plate. We find that the present method enhances significantly the accuracy of the Brinkman penalization technique for the simulations of highly unsteady flows past complex geometries.

A multiresolution method for solving the Poisson equation using high order regularization

We present a novel high order multiresolution Poisson solver based on regularized Greens function solutions to obtain exact free-space boundary conditions while using fast Fourier transforms for computational efficiency. Multiresolution is a achieved through local refinement patches and regularized Greens functions corresponding to the difference in the spatial resolution between the patches. The full solution is obtained utilizing the linearity of the Poisson equation enabling super-position of solutions. We show that the multiresolution Poisson solver produces convergence rates that correspond to the regularization order of the derived Greens functions.

Iterative Brinkman penalization for simulation of impulsively started flow past a sphere and a circular disc

We present a Brinkman penalization method for three-dimensional (3D) flows using particle vortex methods, improving the existing technique by means of an iterative process. We perform simulations to study the impulsively started flow past a sphere at Re=1000 and normal to a circular disc at Re=500. The simulation results obtained for the flow past a sphere are found in qualitative good agreement with previously published results obtained using respectively a 3D vortex penalization method and a 3D vortex method combined with an accurate boundary element method. From the results obtained for the flow normal to a circular disc it is found that the iterative method enables the use of a time step that is one order of magnitude larger than required by the standard non-iterative Brinkman penalization method.

Non-singular Green’s functions for the unbounded Poisson equation in one, two and three dimensions

Abstract In this paper, we derive the non-singular Green’s functions for the unbounded Poisson equation in one, two and three dimensions using a spectral cut-off function approach to impose a minimum length scale in the homogeneous solution. The resulting non-singular Green’s functions are relevant to applications which are restricted to a minimum resolved length scale (e.g. a mesh size h ) and thus cannot handle the singular Green’s function of the continuous Poisson equation. We furthermore derive the gradient vector of the non-singular Green’s function, as this is useful in applications where the Poisson equation represents potential functions of a vector field.

A regularization method for solving the Poisson equation for mixed unbounded-periodic domains

Abstract Regularized Greens functions for mixed unbounded-periodic domains are derived. The regularization of the Greens function removes its singularity by introducing a regularization radius which is related to the discretization length and hence imposes a minimum resolved scale. In this way the regularized unbounded-periodic Greens functions can be implemented in an FFT-based Poisson solver to obtain a convergence rate corresponding to the regularization order of the Greens function. The high order is achieved without any additional computational cost from the conventional FFT-based Poisson solver and enables the calculation of the derivative of the solution to the same high order by direct spectral differentiation. We illustrate an application of the FFT-based Poisson solver by using it with a vortex particle mesh method for the approximation of incompressible flow for a problem with a single periodic and two unbounded directions.