A new representation of rotational flow fields satisfying Euler's equation of an ideal compressible fluid (original) (raw)
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Variational formulation of ideal fluid flows according to gauge principle
Fluid Dynamics Research, 2008
On the basis of the gauge principle of field theory, a new variational formulation is presented for flows of an ideal fluid. The fluid is defined thermodynamically by mass density and entropy density, and its flow fields are characterized by symmetries of translation and rotation. The rotational transformations are regarded as gauge transformations as well as the translational ones. In addition to the Lagrangians representing the translation symmetry, a structure of rotation symmetry is equipped with a Lagrangian Λ A including the vorticity and a vector potential bilinearly. Euler's equation of motion is derived from variations according to the action principle. In addition, the equations of continuity and entropy are derived from the variations. Equations of conserved currents are deduced as the Noether theorem in the space of Lagrangian coordinate a. Without Λ A , the action principle results in the Clebsch solution with vanishing helicity. The Lagrangian Λ A yields non-vanishing vorticity and provides a source term of non-vanishing helicity. The vorticity equation is derived as an equation of the gauge field, and the Λ A characterizes topology of the field. The present formulation is comprehensive and provides a consistent basis for a unique transformation between the Lagrangian a space and the Eulerian x space. In contrast, with translation symmetry alone, there is an arbitrariness in the transformation between these spaces.
Variational formulation of the motion of an ideal fluid on the basis of gauge principle
Physica D: Nonlinear Phenomena, 2008
On the basis of gauge principle in the field theory, a new variational formulation is presented for flows of an ideal fluid. The fluid is defined thermodynamically by mass density and entropy density, and its flow fields are characterized by symmetries of translation and rotation. A structure of rotation symmetry is equipped with a Lagrangian ΛA including vorticity, in addition to Lagrangians of translation symmetry. From the action principle, Euler's equation of motion is derived. In addition, the equations of continuity and entropy are derived from the variations. Equations of conserved currents are deduced as the Noether theorem in the space of Lagrangian coordinate a. It is shown that, with the translation symmetry alone, there is freedom in the transformation between the Lagrangian a-space and Eulerian x-space. The Lagrangian ΛA provides non-trivial topology of vorticity field and yields a source term of the helicity. The vorticity equation is derived as an equation of the gauge field. Present formulation provides a basis on which the transformation between the a space and the x space is determined uniquely.
Lagrangian and Eulerian Representations of Fluid Flow
This essay considers the two major ways that the motion of a fluid continuum may be described, either by observing or predicting the trajectories of parcels that are carried about with the flow – which yields a Lagrangian or material representation of the flow — or by observing or predicting the fluid velocity at fixed points in space — which yields an Eulerian or field representation of the flow. Lagrangian methods are often the most efficient way to sample a fluid domain and most of the physical conservation laws begin with a Lagrangian perspective. Nevertheless, almost all of the theory in fluid dynamics is developed in Eulerian or field form. The premise of this essay is that it is helpful to understand both systems, and the transformation between systems is the central theme.
Physical Review E, 2015
We consider incompressible Euler flows in terms of the stream function in two dimensions and the vector potential in three dimensions. We pay special attention to the case with singular distributions of the vorticity, e.g., point vortices in two dimensions. An explicit equation governing the velocity potentials is derived in two steps. (i) Starting from the equation for the stream function [Ohkitani, Nonlinearity 21, T255 (2009)], which is valid for smooth flows as well, we derive an equation for the complex velocity potential. (ii) Taking a real part of this equation, we find a dynamical equation for the velocity potential, which may be regarded as a refinement of Bernoulli theorem. In three-dimensional incompressible flows, we first derive dynamical equations for the vector potentials which are valid for smooth fields and then recast them in hypercomplex form. The equation for the velocity potential is identified as its real part and is valid, for example, flows with vortex layers. As an application, the Kelvin-Helmholtz problem has been worked out on the basis the current formalism. A connection to the Navier-Stokes regularity problem is addressed as a physical application of the equations for the vector potentials for smooth fields.
2005
This essay introduces the two methods that are commonly used to describe fluid flow, by observing the trajectories of parcels that are carried along with the flow or by observing the fluid velocity at fixed positions. These yield what are commonly termed Lagrangian and Eulerian descriptions. Lagrangian methods are often the most efficient way to sample a fluid domain and the physical conservation laws are inherently Lagrangian since they apply to specific material parcels rather than points in space. It happens, though, that the Lagrangian equations of motion applied to a continuum are quite difficult, and thus almost all of the theory (forward calculation) in fluid dynamics is developed within the Eulerian system. Eulerian solutions may be used to calculate Lagrangian properties, e.g., parcel trajectories, which is often a valuable step in the description of an Eulerian solution. Transformation to and from Lagrangian and Eulerian systems — the central theme of this essay — is thus ...
Gauge principle and variational formulation for ideal fluids with reference to translation symmetry
Fluid Dynamics Research, 2007
Following the gauge principle in the field theory of physics, a new variational formulation is presented for flows of an ideal fluid. In the present gauge-theoretical analysis, it is assumed that the field of fluid flow is characterized by a translation symmetry (group) and in addition that the fluid itself is a material in motion characterized thermodynamically by mass density and entropy (per unit mass). Local gauge transformation in the present case is local Galilean transformation (without rotation) which is a subgroup of a generalized local Galilean transformation group between non-inertial frames. In complying with the requirement of local gauge invariance of Lagrangians, a gauge-covariant derivative with respect to time is defined by introducing a gauge term. Galilean invariance requires that the covariant derivative should be the convective derivative, i.e. the so-called Lagrange derivative. Using this gauge-covariant operator, a free-field Lagrangian and Lagrangians associated with gauge fields are defined under the gauge symmetry. Euler's equation of motion is derived from the action principle. Simutaneously, the equation of continuity and equation of entropy conservation are derived from the variational principle. It is found that general solution thus obtained is equivalent to the classical Clebsch solution. If entropy of the fluid is non-uniform, the flow will be rotational. However, if the entropy is uniform throughout the space (i.e. homentropic), then the flow field reduces to that of a potential flow. Discussions are given on the issue. From the gauge invariance with respect to translational transformations, a differential conservation law of momentum is deduced as Noether's theorem.
A new formulation of equations of compressible fluids by analogy with Maxwell's equations
Fluid Dynamics Research, 2010
A compressible ideal fluid is governed by Euler's equation of motion and equations of continuity, entropy and vorticity. This system can be reformulated in a form analogous to that of electromagnetism governed by Maxwell's equations with source terms. The vorticity plays the role of magnetic field, while the velocity field plays the part of a vector potential and the enthalpy (of isentropic flows) plays the part of a scalar potential in electromagnetism. The evolution of source terms of fluid Maxwell equations is determined by solving the equations of motion and continuity. The equation of sound waves can be derived from this formulation, where time evolution of the sound source is determined by the equation of motion. The theory of vortex sound of aeroacoustics is included in this formulation. It is remarkable that the forces acting on a point mass moving in a velocity field of an inviscid fluid are analogous in their form to the electric force and Lorentz force in electromagnetism. The significance of the reformulation is interpreted by examples taken from fluid mechanics. This formulation can be extended to viscous fluids without difficulty. The Maxwell-type equations are unchanged by the viscosity effect, although the source terms have additional terms due to viscosities.