Resolution and implementation of the nonstationary vorticity velocity pressure formulation of the Navier\u2013Stokes equations

Mohamed Abdelwahed,Henda Ouertani,Nejmeddine Chorfi,Ebtisam Alharbi

doi:10.1186/s13661-020-01464-8

Mohamed Abdelwahed, Henda Ouertani + Show 2 more

Open Access

https://doi.org/10.1186/s13661-020-01464-8

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Journal: Boundary Value Problems	Publication Date: Oct 23, 2020
Citations: 1	License type: open-access

Affiliation: King Saud University

Abstract

This paper deals with the iterative algorithm and the implementation of the spectral discretization of time-dependent Navier–Stokes equations in dimensions two and three. We present a variational formulation, which includes three independent unknowns: the vorticity, velocity, and pressure. In dimension two, we establish an optimal error estimate for the three unknowns. The discretization is deduced from the implicit Euler scheme in time and spectral methods in space. We present a matrix linear system and some numerical tests, which are in perfect concordance with the analysis.

Highlights

The nonlinear Navier–Stokes equations model the flow of a viscous and incompressible fluid such as water, air, and oil in stationary or nonstationary states
The modification of any of the parameters associated with these equations leads to new research problems
The equivalent variational formulation of the Navier–Stokes equations provided with these boundary conditions admits three unknowns: the vorticity, velocity, and pressure [2,3,4,5]

Summary

Introduction

The nonlinear Navier–Stokes equations model the flow of a viscous and incompressible fluid such as water, air, and oil in stationary or nonstationary states. Our interest concerns the nonstationary Navier–Stokes equations with boundary conditions on the normal component of the velocity and the tangential components of the potential vector vorticity Such a problem allows us to model, for instance, two fluids separated by a membrane or the flow in a network of pipes. The equivalent variational formulation of the Navier–Stokes equations provided with these boundary conditions admits three unknowns: the vorticity, velocity, and pressure [2,3,4,5] This formulation has been studied in several works for the finite element discretization of the Stokes and Navier–Stokes problems in the stationary case (see [3, 6]). Π ω when is a polygon with the largest angle equal to ω

The time and full discrete problems

Conclusion