Port-Hamiltonian Modeling of Ideal Fluid Flow: Part I. Foundations and Kinetic Energy

TL;DR

This paper develops a systematic port-Hamiltonian modeling framework for ideal fluid flow on Riemannian manifolds, incorporating boundary energy exchanges and modular building blocks for various inviscid fluid systems.

Contribution

It introduces a novel port-Hamiltonian formulation for ideal fluid flow, including boundary energy exchange and a modular interconnection approach for different fluid models.

Findings

Models include compressible and incompressible flows with boundary conditions.

Framework valid on n-dimensional Riemannian manifolds using differential geometry.

Enables geometric description of diverse fluid dynamical systems.

Abstract

In this two-parts paper, we present a systematic procedure to extend the known Hamiltonian model of ideal inviscid fluid flow on Riemannian manifolds in terms of Lie-Poisson structures to a port-Hamiltonian model in terms of Stokes-Dirac structures. The first novelty of the presented model is the inclusion of non-zero energy exchange through, and within, the spatial boundaries of the domain containing the fluid. The second novelty is that the port-Hamiltonian model is constructed as the interconnection of a small set of building blocks of open energetic subsystems. Depending only on the choice of subsystems one composes and their energy-aware interconnection, the geometric description of a wide range of fluid dynamical systems can be achieved. The constructed port-Hamiltonian models include a number of inviscid fluid dynamical systems with variable boundary conditions. Namely, compressible isentropic flow, compressible adiabatic flow, and incompressible flow. Furthermore, all the derived fluid flow models are valid covariantly and globally on n-dimensional Riemannian manifolds using differential geometric tools of exterior calculus.