Kelvin's circulation theorem

AccessScience ◽  
2015 ◽  
Keyword(s):  
Circulation Theorem

scholarly journals Multisymplectic formulation of fluid dynamics using the inverse map

2007 ◽  
Vol 463 (2086) ◽  
pp. 2671-2687 ◽  
Author(s):  
C.J Cotter ◽  
D.D Holm ◽  
P.E Hydon
Keyword(s):  
Fluid Dynamics ◽  
Conservation Laws ◽  
Fluid Particle ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Numerical Integrators ◽  
Fluid Particles ◽  
Infinite Set ◽  
Circulation Theorem

We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map, the ‘back-to-labels’ map, gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton's principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to Kelvin's circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.

Secondary Flow in Cascades: The Effect of Axial Velocity Ratio

1974 ◽  
Vol 16 (6) ◽  
pp. 402-407 ◽  
Author(s):  
H. Marsh
Keyword(s):  
Secondary Flow ◽  
Incompressible Flow ◽  
Axial Velocity ◽  
Velocity Ratio ◽  
Guide Vanes ◽  
Inlet Guide Vanes ◽  
Blade Row ◽  
Flow Through ◽  
The Difference ◽  
Circulation Theorem

By considering the flow through a many-bladed cascade, a simple theory is developed for the effect of a change in axial velocity on the secondary flow at exit from a cascade. An expression is derived for the difference in the time taken for fluid particles to travel over the two surfaces of the blade and this is used, along with Kelvin's circulation theorem for incompressible flow, to obtain an equation for the distributed secondary vorticity. It is shown that for the row of inlet guide vanes tested by Gregory-Smith (1)†, the change of axial velocity across the blade row has a significant effect on the secondary vorticity.

Some vorticity theorems and conservation laws for non-barotropic fluids

1981 ◽  
Vol 108 ◽  
pp. 475-483 ◽  
Author(s):  
S. D. Mobbs
Keyword(s):  
Thermodynamic Properties ◽  
Conservation Laws ◽  
Potential Vorticity ◽  
Time Dependent ◽  
Dissipative Effects ◽  
Perfect Fluids ◽  
Barotropic Fluids ◽  
Complete Set ◽  
Circulation Theorem

Some theorems concerning the vorticity in barotropic flows of perfect fluids are generalized for non-barotropic flows. The generalization involves replacing the velocity in certain parts of the equations by a time-dependent quantity which is a function of the velocity and thermodynamic properties of the fluid. Results which are generalized include Kelvin's circulation theorem and conservation laws for potential vorticity and helicity. It is shown how the results can be further generalized to include dissipative effects. The possibility of using some of the results in deriving a complete set of Lagrangian conservation laws for perfect fluids is discussed.

A three‐circulation theorem for relativistic plasmas

1996 ◽  
Vol 3 (2) ◽  
pp. 482-489 ◽  
Author(s):  
Klaus Elsässer ◽  
Sergey Popel
Keyword(s):  
Relativistic Plasmas ◽  
Circulation Theorem

scholarly journals The Regulation of Tornado Intensity by Updraft Width

2017 ◽  
Vol 74 (12) ◽  
pp. 4199-4211 ◽  
Author(s):  
Robert J. Trapp ◽  
Geoffrey R. Marion ◽  
Stephen W. Nesbitt
Keyword(s):  
Mathematical Model ◽  
Numerical Simulations ◽  
Simple Mathematical Model ◽  
Simple Hypothesis ◽  
Tornado Damage ◽  
Observational Record ◽  
Circulation Theorem

Abstract Strong to violent tornadoes cause a disproportionate amount of damage, in part because the width and length of a tornado damage track are correlated to tornado intensity (as now estimated through enhanced Fujita scale ratings). The tendency expressed in the observational record is that the most intense tornadoes are often the widest. Herein the authors explore the simple hypothesis that wide intense tornadoes should form more readily out of wide rotating updrafts. This hypothesis is based on an application of Kelvin’s circulation theorem, which is used to argue that the large circulation associated with a wide intense tornado is more plausibly associated with a wide mesocyclone. Because a mesocyclone is, strictly speaking, a rotating updraft, the mesocyclone width should increase with increasing updraft width. A simple mathematical model that is quantified using observations of mesocyclones supports this hypothesis, as do idealized numerical simulations of supercellular thunderstorms.

A Network-Flow Feasibility Theorem and Combinatorial Applications

1959 ◽  
Vol 11 ◽  
pp. 440-451 ◽  
Author(s):  
D. R. Fulkerson
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Network Flow ◽  
Present Author ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Private Communication ◽  
Circulatory Flow ◽  
Necessary And Sufficient ◽  
Circulation Theorem

There are a number of interesting theorems, relative to capacitated networks, that give necessary and sufficient conditions for the existence of flows satisfying constraints of various kinds. Typical of these are the supply-demand theorem due to Gale (4), which states a condition for the existence of a flow satisfying demands at certain nodes from supplies at other nodes, and the Hoffman circulation theorem (received by the present author in private communication), which states a condition for the existence of a circulatory flow in a network in which each arc has associated with it not only an upper bound for the arc flow, but a lower bound as well. If the constraints on flows are integral (for example, if the bounds on arc flows for the circulation theorem are integers), it is also true that integral flows meeting the requirements exist provided any flow does so.

On the fluid dynamics of evolving stars

1981 ◽  
Vol 375 (1761) ◽  
pp. 249-269 ◽  
Keyword(s):  
Fluid Dynamics ◽  
Angular Momentum ◽  
A Priori ◽  
Time Dependent ◽  
Rotating Stars ◽  
Secular Evolution ◽  
Double Stars ◽  
Material Functions ◽  
Model Star ◽  
Circulation Theorem

The problem addressed is that of following the secular evolution of the velocity field and distribution of matter of a model star endowed with an arbitrary amount of angular momentum. A novel feature of the fluid dynamical formulation is the introduction and systematic use of material functions. These functions both facilitate the treatment of the free boun­dary of the star and enable one to use the circulation about certain contours as a priori constants of the motion. The equations governing the evolution of the material functions are adjoined to the Euler equations of fluid dynamics and are to be solved simultaneously with them. No special symmetry assumptions need to be imposed in formulating the equations. This makes it possible to apply them not only in the case of axisymmetric rotating stars, but also in the case of bar-shaped figures that may evolve toward double stars. The formulation is well adapted to the perturbation analysis needed in investigating bifurcation from families of slowly evolving fluid masses. The classes of model stars covered by the formulation include time dependent barotropic models, but are applicable to a significantly wider class of models as well. Even in the context of this wider, non-barotropic class of models, a restricted version of Kelvin’s circulation theorem holds, and plays a major role in rendering determinate the equations of secular evolution.

Nonlinear internal gravity waves in a rotating fluid

1975 ◽  
Vol 71 (3) ◽  
pp. 497-512 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Gravity Waves ◽  
Wave Structure ◽  
Internal Gravity Waves ◽  
Mean Flow ◽  
Mean Velocity ◽  
Main Effect ◽  
Local Wave ◽  
The Mean ◽  
The Waves ◽  
Circulation Theorem

The interaction between internal gravity waves in a rotating frame and the mean flow is discussed for the case when the properties of the mean flow vary slowly on a scale determined by the local wave structure. The principle of conservation of wave action is established. It is shown that the main effect of the waves on the Lagrangian mean velocity is due to an appropriate ‘radiation stress’ tensor. A circulation theorem and a potential-vorticity equation are derived for the mean velocity.

scholarly journals On-shell description of unsteady flames

2008 ◽  
Vol 608 ◽  
pp. 217-242 ◽  
Author(s):  
GUY JOULIN ◽  
HAZEM EL-RABII ◽  
KIRILI A. KAZAKOV
Keyword(s):  
Flame Front ◽  
Gas Velocity ◽  
Flow Equations ◽  
Flame Dynamics ◽  
Front Position ◽  
Potential Component ◽  
Gas Expansion ◽  
Circulation Theorem ◽  
Local Burning ◽  
Front Dynamics

The problem of a non-perturbative description of unsteady premixed flames with arbitrary gas expansion is addressed in the two-dimensional case. Considering the flame as a surface of discontinuity with arbitrary local burning rate and gas velocity jumps, we show that the flame-front dynamics can be determined without having to solve the flow equations in the bulk. On the basis of the Thomson circulation theorem, an implicit integral representation of the downstream gas velocity is constructed. It is then simplified by a successive stripping of the potential contributions to obtain an explicit expression for the rotational component near the flame front. We prove that the unknown potential component is left bounded and divergence-free by this procedure, and hence can be eliminated using the dispersion relation for its on-shell value (i.e. the value along the flame front). The resulting system of integro-differential equations relates the on-shell fresh-gas velocity and the front position. As limiting cases, these equations contain all the theoretical results on flame dynamics established so far, including the linear equation describing the Darrieus–Landau instability of planar flames, and the nonlinear Sivashinsky–Clavin equation for flames with weak gas expansion.

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