scholarly journals On the derivation of the HOMFLYPT polynomial invariant for fluid knots

2015 ◽  
Vol 773 ◽  
pp. 34-48 ◽  
Author(s):  
Xin Liu ◽  
Renzo L. Ricca
Keyword(s):  
Field Strength ◽  
Fluid Mechanics ◽  
Ideal Fluid ◽  
Mechanical Behaviour ◽  
Fluid Flows ◽  
Numerical Implementation ◽  
Rigorous Proof ◽  
Polynomial Invariant ◽  
Real Fluid ◽  
Insight Into

By using and extending earlier results (Liu & Ricca,J. Phys. A, vol. 45, 2012, 205501), we derive the skein relations of the HOMFLYPT polynomial for ideal fluid knots from helicity, thus providing a rigorous proof that the HOMFLYPT polynomial is a new, powerful invariant of topological fluid mechanics. Since this invariant is a two-variable polynomial, the skein relations are derived from two independent equations expressed in terms of writhe and twist contributions. Writhe is given by addition/subtraction of imaginary local paths, and twist by Dehn’s surgery. HOMFLYPT then becomes a function of knot topology and field strength. For illustration we derive explicit expressions for some elementary cases and apply these results to homogeneous vortex tangles. By examining some particular examples we show how numerical implementation of the HOMFLYPT polynomial can provide new insight into fluid-mechanical behaviour of real fluid flows.

A steady vortex ring in Poiseuille flow and rearrangements of a function

2006 ◽  
Vol 462 (2068) ◽  
pp. 1235-1253 ◽  
Author(s):  
Dirar Rebah
Keyword(s):  
Kinetic Energy ◽  
Variational Method ◽  
Fluid Mechanics ◽  
Vortex Ring ◽  
Ideal Fluid ◽  
Poiseuille Flow ◽  
Nonlinear Problems ◽  
Insight Into

This study proves the existence of a steady vortex ring of an ideal fluid in Poiseuille flow. The method that was used is a variational method proposed by Benjamin (Benjamin 1976 The alliance of practical and analytical insight into the nonlinear problems of fluid mechanics , vol. 503, pp. 8–29), in which a steady vortex ring can be obtained as a maximizer of a functional that is related to kinetic energy and the impulse over the set of rearrangements of a prescribed function.

scholarly journals GEORGE KEITH BATCHELOR 8 March 1920–30 March 2000 Founding Editor, Journal of Fluid Mechanics, 1956

2000 ◽  
Vol 421 ◽  
pp. 1-14 ◽  
Author(s):  
HERBERT E. HUPPERT
Keyword(s):  
Fluid Mechanics ◽  
International Organizations ◽  
Solid Mechanics ◽  
Applied Mathematics ◽  
Fluid Flows ◽  
Theoretical Physics ◽  
Founding Editor ◽  
Number Of Factors ◽  
Quantitative Understanding ◽  
The Subject

George Batchelor was one of the giants of fluid mechanics in the second half of the twentieth century. He had a passion for physical and quantitative understanding of fluid flows and a single-minded determination that fluid mechanics should be pursued as a subject in its own right. He once wrote that he ‘spent a lifetime happily within its boundaries’. Six feet tall, thin and youthful in appearance, George's unchanging attire and demeanour contrasted with his ever-evolving scientific insights and contributions. His strongly held and carefully articulated opinions, coupled with his forthright objectivity, shone through everything he undertook.George's pervasive influence sprang from a number of factors. First, he conducted imaginative, ground-breaking research, which was always based on clear physical thinking. Second, he founded a school of fluid mechanics, inspired by his mentor G. I. Taylor, that became part of the world renowned Department of Applied Mathematics and Theoretical Physics (DAMTP) of which he was the Head from its inception in 1959 until he retired from his Professorship in 1983. Third, he established this Journal in 1956 and actively oversaw all its activities for more than forty years, until he relinquished his editorship at the end of 1998. Fourth, he wrote the monumental textbook An Introduction to Fluid Dynamics, which first appeared in 1967, has been translated into four languages and has been relaunched this year, the year of his death. This book, which describes the fundamentals of the subject and discusses many applications, has been closely studied and frequently cited by generations of students and research workers. It has already sold over 45 000 copies. And fifth, but not finally, he helped initiate a number of international organizations (often European), such as the European Mechanics Committee (now Society) and the biennial Polish Fluid Mechanics Meetings, and contributed extensively to the running of IUTAM, the International Union of Theoretical and Applied Mechanics. The aim of all of these associations is to foster fluid (and to some extent solid) mechanics and to encourage the development of the subject.

scholarly journals Energy–Vorticity Theory of Ideal Fluid Mechanics

2009 ◽  
Vol 66 (7) ◽  
pp. 2073-2084 ◽  
Author(s):  
Peter Névir ◽  
Matthias Sommer
Keyword(s):  
Fluid Mechanics ◽  
Shallow Water ◽  
Ideal Fluid ◽  
Characteristic Property ◽  
Incompressible Flows ◽  
Climate Simulations ◽  
Shallow Water Models ◽  
Complete Analogy ◽  
Potential Enstrophy

Abstract Nambu field theory, originated by Névir and Blender for incompressible flows, is generalized to establish a unified energy–vorticity theory of ideal fluid mechanics. Using this approach, the degeneracy of the corresponding noncanonical Poisson bracket—a characteristic property of Hamiltonian fluid mechanics—can be replaced by a nondegenerate bracket. An energy–vorticity representation of the quasigeostrophic theory and of multilayer shallow-water models is given, highlighting the fact that potential enstrophy is just as important as energy. The energy–vorticity representation of the hydrostatic adiabatic system on isentropic surfaces can be written in complete analogy to the shallow-water equations using vorticity, divergence, and pseudodensity as prognostic variables. Furthermore, it is shown that the Eulerian equation of motion, the continuity equation, and the first law of thermodynamics, which describe the nonlinear evolution of a 3D compressible, adiabatic, and nonhydrostatic fluid, can be written in Nambu representation. Here, trilinear energy–helicity, energy–mass, and energy–entropy brackets are introduced. In this model the global conservation of Ertel’s potential enstrophy can be interpreted as a super-Casimir functional in phase space. In conclusion, it is argued that on the basis of the energy–vorticity theory of ideal fluid mechanics, new numerical schemes can be constructed, which might be of importance for modeling coherent structures in long-term integrations and climate simulations.

scholarly journals Energy-Vorticity Theory of Ideal Fluid Mechanics

2009 ◽  
pp. 100802084958071
Author(s):  
Peter Névir ◽  
Matthias Sommer
Keyword(s):  
Fluid Mechanics ◽  
Ideal Fluid

scholarly journals Machine Learning for Fluid Mechanics

2020 ◽  
Vol 52 (1) ◽  
pp. 477-508 ◽  
Author(s):  
Steven L. Brunton ◽  
Bernd R. Noack ◽  
Petros Koumoutsakos
Keyword(s):  
Machine Learning ◽  
Fluid Mechanics ◽  
Domain Knowledge ◽  
Large Scale ◽  
Past History ◽  
Fluid Flows ◽  
Field Measurements ◽  
Industrial Applications ◽  
Current Lines ◽  
Large Scale Simulations

The field of fluid mechanics is rapidly advancing, driven by unprecedented volumes of data from experiments, field measurements, and large-scale simulations at multiple spatiotemporal scales. Machine learning (ML) offers a wealth of techniques to extract information from data that can be translated into knowledge about the underlying fluid mechanics. Moreover, ML algorithms can augment domain knowledge and automate tasks related to flow control and optimization. This article presents an overview of past history, current developments, and emerging opportunities of ML for fluid mechanics. We outline fundamental ML methodologies and discuss their uses for understanding, modeling, optimizing, and controlling fluid flows. The strengths and limitations of these methods are addressed from the perspective of scientific inquiry that considers data as an inherent part of modeling, experiments, and simulations. ML provides a powerful information-processing framework that can augment, and possibly even transform, current lines of fluid mechanics research and industrial applications.

Dynamics of Pipes Conveying Fluid With a Non-Uniform Velocity Profile

2009 ◽  
Author(s):  
Aren M. Hellum ◽  
Ranjan Mukherjee ◽  
Andrew J. Hull
Keyword(s):  
Velocity Profile ◽  
Critical Frequency ◽  
Fluid Flows ◽  
Drop Out ◽  
Total System ◽  
Number Range ◽  
Uniform Velocity ◽  
Real Fluid ◽  
Uniform Case ◽  
Conveying Fluid

Previous work on stability of fluid-conveying cantilever pipes assumed a uniform velocity profile for the conveyed fluid. In real fluid flows, the presence of viscosity leads to a sheared region near the wall. Earlier studies correctly note that viscous forces drop out of the system’s dynamics since the force of fluid shear on the wall is precisely balanced by pressure drop in the conveyed fluid. The effect of shear has therefore not been ignored in these studies. However, a uniform velocity profile assumes that the sheared region is infinitely thin. Prior analysis was extended to account for a fully developed non-uniform profile such as would be encountered in real fluid flows. A modified equation of motion was derived to account for the reduced momentum carried by the sheared fluid. Numerical analysis was carried out to determine a number of velocity profiles over the Reynolds number range of interest and a simple set of curve fits was used when finer discretization was required. Stability analysis of a pipe conveying fluid with these profiles was performed, and the results were compared to a uniform profile. The mass ratio, β, is the ratio of the fluid mass to the total system mass. At β = 0.2, the non-uniform case becomes unstable at a critical velocity, ucr, that is 5.4% lower than the uniform case. The critical frequency, fcr, is 0.36% higher than the uniform case. A more sensitive region exists near β = 0.32. There, the nonuniform velocity ucr is 23% lower than the uniform case and the non-uniform critical frequency fcr is 49% of the uniform case.

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