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.
Highlights
Fluid mechanics is one of the most fundamental fields that has stimulated many ideas and concepts that are central to modern mathematical sciences
In the classical Hamiltonian theory for fluid dynamical systems, a fundamental difficulty arises in incorporating the spatial boundary conditions of the system, which is the case for general distributed parameter systems
In addition to the system theoretic advantages of modeling ideal fluid flow in the port-Hamiltonian paradigm, the work we present here serves as a stepping stone for modeling fluid–structure interaction in the quest of understanding the flapping-flight of birds within the PORTWINGS project
Summary
Fluid mechanics is one of the most fundamental fields that has stimulated many ideas and concepts that are central to modern mathematical sciences. The procedure to construct the port-Hamiltonian model we are aiming for relies greatly on understanding the underlying geometric structure of the state space of each energetic subsystem This geometric formulation, pioneered by [3] and [6], will allow a systematic derivation of the underlying Hamiltonian dynamical equations and Dirac structures, usually postulated a priori in the literature [2,16,18,19]. While in Part II, we extend the work of [18] by constructing port-Hamiltonian models of adiabatic compressible as well as incompressible flow This first paper is organized as follows: Section 2 introduces the geometric description of ideal fluid flow using differential forms on the spatial domain in the quest of identifying the state space of fluid motion. In Part II, we will present the remaining steps of the port-Hamiltonian procedure which will utilize the distributed stress forces added to the port-Hamiltonian model presented in Section 4 of the present paper to add storage of internal energy for modeling compressible flow, and to add constraint forces to model incompressible flow
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