A mixed variational discretization for non-isothermal compressible flow in pipelines

TL;DR

This paper introduces a variational mixed finite element method for simulating non-isothermal compressible flow in pipelines, ensuring conservation laws and accommodating complex physical effects.

Contribution

It develops a variational formulation and a mixed finite element discretization that inherently preserve physical conservation laws for non-isothermal compressible flow.

Findings

Conservation of mass is exactly maintained in the numerical scheme.

Energy is slightly dissipated, and entropy increases due to implicit time discretization.

The approach is adaptable to more complex flow models and boundary conditions.

Abstract

We consider the non-isothermal flow of a compressible fluid through pipes. Starting from the full set of Euler equations, we propose a variational characterization of solutions that encodes the conservation of mass, energy, and entropy in a very direct manner. This variational principle is suitable for a conforming Galerkin approximation in space which automatically inherits the basic physical conservation laws. Three different spaces are used for approximation of density, mass flux, and temperature, and we consider a mixed finite element method as one possible choice of suitable approximation spaces. We also investigate the subsequent discretization in time by a problem adapted implicit time stepping scheme for which exact conservation of mass as well as a slight dissipation of energy and increase of entropy are proven which are due to the numerical dissipation of the implicit time discretization. The main arguments of our analysis are rather general and allow us to extend the approach with minor modification to more general boundary conditions and flow models taking into account friction, viscosity, heat conduction, and heat exchange with the surrounding medium.