Streamfunction-vorticity formulation for incompressible viscid and inviscid flows on general surfaces

TL;DR

This paper introduces a novel streamfunction-vorticity formulation for incompressible flows on complex surfaces, ensuring exact tangentiality, incompressibility, and pressure robustness with computational efficiency.

Contribution

It develops a scalar-based, finite-dimensional method for Navier--Stokes and Euler equations on general surfaces, including non-simply connected ones, with proven equivalence to traditional formulations.

Findings

Method guarantees divergence-free, tangential velocity fields.

Formulation is pressure robust and computationally efficient.

Numerical examples validate the approach.

Abstract

This paper presents a streamfunction-vorticity formulation for the Navier--Stokes and Euler equations on general surfaces. Notably, this includes non-simply connected surfaces, on which the harmonic components of the velocity field play a fundamental role in the dynamics. By relying only on scalar and finite-dimensional quantities, our formulation ensures that the resulting methods give exactly tangential and incompressible velocity fields, while also being pressure robust. Compared to traditional methods based on velocity-pressure formulations, where one can only guarantee these structural properties by increasing the computational costs, this is a key advantage. We rigorously validate our formulation by proving its equivalence to the well understood velocity-pressure formulation under reasonable regularity assumptions. Furthermore, we demonstrate the applicability of the approach with numerical examples.