Construction of Solenoidal Immersed Velocity Vectors Using the Kinematic Velocity--Vorticity Relation

TL;DR

This paper introduces a method to construct incompressible, solenoidal velocity vectors from vorticity fields using kinematic relations, applicable to 2D and 3D problems with various boundary conditions.

Contribution

It presents a novel approach that employs velocity-vorticity relations to generate solenoidal velocity fields satisfying boundary conditions.

Findings

Solenoidality achieved up to machine accuracy for periodic boundaries.

Method improves solenoidality for homogeneous boundary conditions.

Applicable to both 2D and 3D velocity field problems.

Abstract

The present paper suggests a method for obtaining incompressible solenoidal velocity vectors that satisfy approximately the desired immersed velocity boundary conditions. The method employs merely the mutual kinematic relations between the velocity and vorticity fields (i.e, the curl and Laplacian operators). An initial non-solenoidal velocity field is extended to a regular domain via a zero-velocity margin, where an extended vorticity is found. Re-calculation of the velocities (subjected to appropriate boundary conditions), yields the desired solenoidal velocity vector. The method is applied to the two- and three-dimensional problems for the homogeneous Dirichlet, as well as periodic boundary conditions. The results show that the solenoidality is satisfied up to the machine accuracy for the periodic boundary conditions (employing the Fourier--spectral solution method), while an improvement in the solenoidality is achievable for the homogenous boundary conditions.