Session: 05-02-02 CFD Methods

Paper Number: 86742

86742 - A Method for Numerical Evaluation of Singular Integrals in Curved Hexahedra and With High-Order Source Functions 

We have developed a new compact high-order discontinuous finite difference method for discretization of the unsteady vorticity transport equations (VTE) for 3-D incompressible flow. In this approach, the velocity field consists of (1) a vortical component induced by the distributed vorticity, and (2) a "corrective" potential flow added to the latter so as to enforce the proper flux boundary conditions. The detailed formulation of this new high-order method and the corresponding benchmark results will be presented in an accompanying FEDSM2022 paper (A High-Order Flux Reconstruction Method for 3-D Vorticity Transport; by Mohammad Haji Mohammadi, Mark Stock, and Adrin Gharakhani). The focus of this paper is on the accurate numerical evaluation of the vortical velocity field. Specifically, a method is developed for evaluating the Biot-Savart singular integral for the velocity field induced by arbitrarily high-order vorticity in arbitrarily high-order curved hexahedral elements. Surprisingly, whereas numerous methods have been introduced in the literature for the evaluation of singular integrals on curved surface elements, corresponding methods for volume elements have not been reported as rigorously. To this end, our method uses Duffy's coordinate transformation and singularity removal strategy which, through a series of transformations, accommodates accurate evaluation of the transformed integrals using standard adaptive cubature techniques. In this paper, the new method will be presented in full detail followed by a series of benchmark tests demonstrating the convergence properties of the singular volume integral as a function of the accuracy order of the vorticity (source) field.

Presenting Author: Adrin Gharakhani Applied Scientific Research