Abstract
Part I of this paper presented a systematic derivation of the Stokes–Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the group of diffeomorphisms as a configuration space for the fluid, the Stokes–Dirac structure is derived by Poisson reduction and then augmented by boundary ports and distributed ports. The additional boundary ports have been shown to appear naturally as surface terms in the pairings of dual maps, always neglected in standard Hamiltonian theory. The port-Hamiltonian model presented in Part I corresponded only to the kinetic energy of the fluid and how its energy variables evolve such that the energy is conserved.In Part II, we utilize the distributed port of the kinetic energy port-Hamiltonian system for representing a number of fluid-dynamical systems. By adding internal energy we model compressible flow, both adiabatic and isentropic, and by adding constraint forces we model incompressible flow. The key tools used are the interconnection maps relating the dynamics of fluid motion to the dynamics of advected quantities.
Highlights
In Part II of this paper, we present the port-Hamiltonian models of a number of fluid dynamical systems on general Riemannian manifolds
The variational derivatives δv Hk ∈ Ωn−1(M) and δμHk ∈ Ω0(M) with respect to the states v ∈ g∗ = Ω1(M) and μ ∈ V ∗ = Ωn(M), respectively, are given by δv Hk = (∗μ) ∗ v = ιvμ, δμHk = 2 ιvv. It was shown in Part I, that the port-Hamiltonian system (3) can be represented by a kinetic energy storage port in addition to two open ports that can be interconnected to other systems
The standard Hamiltonian reduction theorems can be used to derive the Lie–Poisson structure as in [2,3,7]. In this two-parts paper, a systematic procedure to model a variety of fluid dynamical systems on general Riemannian manifolds was presented
Summary
In Part II of this paper, we present the port-Hamiltonian models of a number of fluid dynamical systems on general Riemannian manifolds. The variational derivatives δv Hk ∈ Ωn−1(M) and δμHk ∈ Ω0(M) with respect to the states v ∈ g∗ = Ω1(M) and μ ∈ V ∗ = Ωn(M), respectively, are given by δv Hk = (∗μ) ∗ v = ιvμ, δμHk = 2 ιvv It was shown in Part I, that the port-Hamiltonian system (3) can be represented by a kinetic energy storage port in addition to two open ports that can be interconnected to other systems.
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