A Hybrid Discrete Exterior Calculus Discretization and Fourier Transform of the Incompressible Navier-Stokes Equations in 3D

TL;DR

This paper introduces a hybrid discretization method combining discrete exterior calculus and Fourier transform to efficiently simulate 3D incompressible Navier-Stokes equations, maintaining geometric fidelity and reducing computational costs.

Contribution

It develops a novel hybrid DEC and Fourier transform approach for 3D Navier-Stokes equations, enhancing simulation accuracy and efficiency over traditional methods.

Findings

Results are comparable to existing literature for 3D flow cases

The hybrid method preserves geometric properties of the flow

Demonstrates potential for efficient 3D fluid simulations

Abstract

The simulation of fluid flow problems, specifically incompressible flows governed by the Navier-Stokes equations (NSE), holds fundamental significance in a range of scientific and engineering applications. Traditional numerical methods employed for solving these equations on three-dimensional (3D) meshes are commonly known for their moderate conservation properties, high computational intensity and substantial resource demands. Relying on its ability to capture the intrinsic geometric and topological properties of simplicial meshes, discrete exterior calculus (DEC) provides a discrete analog to differential forms and enables the discretization of partial differential equations (PDEs) on meshes.We present a hybrid discretization approach for the 3D incompressible Navier-Stokes equations based on DEC and Fourier transform (FT). An existing conservative primitive variable DEC discretization of incompressible Navier-Stokes equations over surface simplicial meshes developed by Jagad et al. [1] is considered in the planar dimension while the Fourier expansion is applied in the third dimension. The test cases of three-dimensional lid-driven cavity and viscous Taylor-Green three-dimensional vortex (TGV) flows show that the simulation results using this hybrid approach are comparable to literature.