Self-Induced Velocity of a Vortex Ring Using Straight-Line Segmentation

2014 ◽  
Vol 59 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Mahendra J. Bhagwat ◽  
J. Gordon Leishman
Keyword(s):  
Vortex Ring ◽  
Logarithmic Singularity ◽  
The Self ◽  
Numerical Calculations ◽  
Core Radius ◽  
Line Vortex ◽  
The Core ◽  
Analytical Approximations ◽  
Straight Line ◽  
Induced Velocity

The accuracy of discretized induced velocity calculations that can be obtained using straight-line vortex elements has been reexamined, primarily using the velocity field induced by a vortex ring as a reference. The induced velocity of a potential (inviscid) vortex ring is singular at the vortex ring itself. Analytical results were found by using a small azimuthal cutoff in the Biot–Savart integral over the vortex ring and showed that the singularity is logarithmic in the cutoff. Discrete numerical calculations showed the same behavior, with the self-induced velocity exhibiting a logarithmic singularity with respect to the discretization, which introduces an inherent cutoff in the Biot–Savart integral. Core regularization or desingularization can also eliminate the singularity by using an assumed “viscous” core model. Analytical approximations to the self-induced velocity of a thin-cored vortex ring have shown that the self-induced velocity has a logarithmic singularity in the core radius. It is further shown that the numerical calculations require special treatment of the self-induced velocity caused by curvature, which is lost by the inherent cutoff in the straight-line discretization, to correctly recover this logarithmic singularity in the core radius. Numerical solution using straight-line vortex segmentation, augmented with curved vortex elements only for the self-induced velocity calculation, is shown to be second-order accurate in the discretization.

The instability of short waves on a vortex ring

1974 ◽  
Vol 66 (1) ◽  
pp. 35-47 ◽  
Author(s):  
Sheila E. Widnall ◽  
Donald B. Bliss ◽  
Chon-Yin Tsai
Keyword(s):  
Vortex Ring ◽  
Vortex Pair ◽  
Short Wave ◽  
The Self ◽  
Core Size ◽  
The Core ◽  
Bending Waves ◽  
Bending Mode ◽  
Radial Structure ◽  
Bending Modes

A simple model for the experimentally observed instability of the vortex ring to azimuthal bending waves of wavelength comparable with the core size is presented. Short-wave instabilities are discussed for both the vortex ring and the vortex pair. Instability for both the ring and the pair is predicted to occur whenever the self-induced rotation of waves on the filament passes through zero. Although this does not occur for the first radial bending mode of a vortex filament, it is shown to be possible for bending modes with a more complex radial structure with at least one node at some radius within the core. The previous work of Widnall & Sullivan (1973) is discussed and their experimental results are compared with the predictions of the analysis presented here.

A Direct Aerodynamic Proof of the Biot-Savart Law

1951 ◽  
Vol 3 (3) ◽  
pp. 238-239
Author(s):  
J. Lockwood Taylor
Keyword(s):  
Integral Equation ◽  
Direct Proof ◽  
Number Of Solutions ◽  
Line Vortex ◽  
Sources And Sinks ◽  
Straight Line ◽  
Vortex Element ◽  
Electro Magnetic ◽  
Induced Velocity ◽  
Image Position

The expression for the induced velocity of a vortex element,in Glauert's notation (where K is the strength of the vortex, h the length of the normal, and r the distance from the field point to the element, of length ds) which is often known as the Biot-Savart law, and is of fundamental importance in aerodynamics, is frequently left unproved in textbooks. Sometimes reference is made to the electro-magnetic analogy, or to a standard treatise such as Lamb. One well-known work implies that the law is the solution of an integral equation derived from the known induced velocity of an infinite straight-line vortex, but this is misleading, since there is an infinite number of solutions, of which only one is correct. Another textbook gives a lengthy argument starting from the replacement of a line vortex by a sheet of sources and sinks, which is probably not too convincing to the average student. Lamb himself gives a long and rather involved proof based on Helmholtz and Stokes, with appeals to the Theory of Attractions, with which few are likely to be familiar. A simple and direct proof from fluid mechanics seems therefore worth while.

Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity

2000 ◽  
Vol 417 ◽  
pp. 1-45 ◽  
Author(s):  
YASUHIDE FUKUMOTO ◽  
H. K. MOFFATT
Keyword(s):  
Vortex Ring ◽  
Higher Order ◽  
The Self ◽  
Large Reynolds Number ◽  
Arbitrary Distribution ◽  
Inviscid Fluid ◽  
Third Order ◽  
First Order ◽  
The Core ◽  
First Order Theory

A large-Reynolds-number asymptotic solution of the Navier–Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. In order to take account of the influence of elliptical deformation of the core due to the self-induced strain, the method of matched of matched asymptotic expansions is extended to a higher order in a small parameter ε = (v/Γ)1/2, where v is the kinematic viscosity of fluid and Γ is the circulation. Alternatively, ε is regarded as a measure of the ratio of the core radius to the ring radius, and our scheme is applicable also to the steady inviscid dynamics.We establish a general formula for the translation speed of the ring valid up to third order in ε. This is a natural extension of Fraenkel–Saffman's first-order formula, and reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an infinitely thin circular loop of radius R0, that viscosity acts, at third order, to expand the circles of stagnation points of radii Rs(t) and R˜s(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius R˜p(t) as Rs ≈ R0 + [2 log(4R0/√vt) + 1.4743424] vt/R0, R˜s ≈ R0 + 2.5902739 vt/R0, and Rp ≈ R0 + 4.5902739 vt/R0. The growth of the radial centroid of vorticity, linear in time, is also deduced. The results are compatible with the experimental results of Sallet & Widmayer (1974) and Weigand & Gharib (1997).The procedure of pursuing the higher-order asymptotics provides a clear picture of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a line of dipoles along the core centreline, with their axes in the propagating direction, subjected to the self-induced flow field. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specified at the initial instant. This specification removes an indeterminacy of the first-order theory. We derive a new asymptotic development of the Biot-Savart law for an arbitrary distribution of vorticity, which makes the non-local induction velocity from the dipoles calculable at third order.

scholarly journals A New Fueling Process in a Weak Bar

1996 ◽  
Vol 157 ◽  
pp. 439-444
Author(s):  
Keiichi Wada ◽  
Asao Habe
Keyword(s):  
Central Region ◽  
The Self ◽  
Core Radius ◽  
Gaseous Disk ◽  
The Core ◽  
Self Gravity

AbstractA massive gaseous disk in the central region of a galaxy sensitively responds to a weakly distorted potential, and a large amount of gas can be fed into within 1/20 of the core radius of the potential in several 107 yr. The ILRs, the dissipative nature of the gas, and the self-gravity of the gas are essential for triggering this effective fueling. We also found that a counterrotating gaseous core can be formed as a result of the fueling. Our result suggests that the merger of galaxies is not the only way to form the observed counterrotating core in galaxies.

Pathways to Success Through Identity-Based Motivation

2015 ◽  
Author(s):  
Daphna Oyserman
Keyword(s):  
The Self ◽  
The Core ◽  
Very Young Children ◽  
The Future ◽  
Identity Based ◽  
Future Self ◽  
Fidelity Measures ◽  
The Individual ◽  
Taking Action ◽  
Minimal Effort

Everyone can imagine their future self, even very young children, and this future self is usually positive and education-linked. To make progress toward an aspired future or away from a feared future requires people to plan and take action. Unfortunately, most people often start too late and commit minimal effort to ineffective strategies that lead their attention elsewhere. As a result, their high hopes and earnest resolutions often fall short. In Pathways to Success Through Identity-Based Motivation Daphna Oyserman focuses on situational constraints and affordances that trigger or impede taking action. Focusing on when the future-self matters and how to reduce the shortfall between the self that one aspires to become and the outcomes that one actually attains, Oyserman introduces the reader to the core theoretical framework of identity-based motivation (IBM) theory. IBM theory is the prediction that people prefer to act in identity-congruent ways but that the identity-to-behavior link is opaque for a number of reasons (the future feels far away, difficulty of working on goals is misinterpreted, and strategies for attaining goals do not feel identity-congruent). Oyserman's book goes on to also include the stakes and how the importance of education comes into play as it improves the lives of the individual, their family, and their society. The framework of IBM theory and how to achieve it is broken down into three parts: how to translate identity-based motivation into a practical intervention, an outline of the intervention, and empirical evidence that it works. In addition, the book also includes an implementation manual and fidelity measures for educators utilizing this book to intervene for the improvement of academic outcomes.

scholarly journals Influence of Salt on the Self-Organization in Solutions of Star-Shaped Poly-2-alkyl-2-oxazoline and Poly-2-alkyl-2-oxazine on Heating

Polymers ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1152
Author(s):  
Tatyana Kirila ◽  
Anna Smirnova ◽  
Alla Razina ◽  
Andrey Tenkovtsev ◽  
Alexander Filippov
Keyword(s):  
Phase Separation ◽  
Salt Concentration ◽  
Monomer Unit ◽  
Salt Content ◽  
The Self ◽  
Self Organization ◽  
Phase Separation Temperature ◽  
The Core ◽  
Separation Temperature ◽  
Water Salt

The water–salt solutions of star-shaped six-arm poly-2-alkyl-2-oxazines and poly-2-alkyl-2-oxazolines were studied by light scattering and turbidimetry. The core was hexaaza[26]orthoparacyclophane and the arms were poly-2-ethyl-2-oxazine, poly-2-isopropyl-2-oxazine, poly-2-ethyl-2-oxazoline, and poly-2-isopropyl-2-oxazoline. NaCl and N-methylpyridinium p-toluenesulfonate were used as salts. Their concentration varied from 0–0.154 M. On heating, a phase transition was observed in all studied solutions. It was found that the effect of salt on the thermosensitivity of the investigated stars depends on the structure of the salt and polymer and on the salt content in the solution. The phase separation temperature decreased with an increase in the hydrophobicity of the polymers, which is caused by both a growth of the side radical size and an elongation of the monomer unit. For NaCl solutions, the phase separation temperature monotonically decreased with growth of salt concentration. In solutions with methylpyridinium p-toluenesulfonate, the dependence of the phase separation temperature on the salt concentration was non-monotonic with minimum at salt concentration corresponding to one salt molecule per one arm of a polymer star. Poly-2-alkyl-2-oxazine and poly-2-alkyl-2-oxazoline stars with a hexaaza[26]orthoparacyclophane core are more sensitive to the presence of salt in solution than the similar stars with a calix[n]arene branching center.

scholarly journals Dynamical Evolution of Globular Clusters After Core Collapse

1985 ◽  
Vol 113 ◽  
pp. 139-160 ◽  
Author(s):  
Douglas C. Heggie
Keyword(s):  
Surface Density ◽  
Globular Clusters ◽  
Stellar System ◽  
Dynamical Evolution ◽  
Theoretical Models ◽  
Numerical Calculations ◽  
Core Collapse ◽  
Density Profiles ◽  
The Core ◽  
Fokker Planck

This review describes work on the evolution of a stellar system during the phase which starts at the end of core collapse. It begins with an account of the models of Hénon, Goodman, and Inagaki and Lynden-Bell, as well as evaporative models, and modifications to these models which are needed in the core. Next, these models are related to more detailed numerical calculations of gaseous models, Fokker-Planck models, N-body calculations, etc., and some problems for further work in these directions are outlined. The review concludes with a discussion of the relation between theoretical models and observations of the surface density profiles and statistics of actual globular clusters.

Kierkegaard, Metaphysics, and Love

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
George Pattison
Keyword(s):  
Sense Of Self ◽  
The Self ◽  
The Other ◽  
Gabriel Marcel ◽  
The Core ◽  
The Dead ◽  
Alternative Reading ◽  
The Way

AbstractNoting Heidegger’s critique of Kierkegaard’s way of relating time and eternity, the paper offers an alternative reading of Kierkegaard that suggests Heidegger has overlooked crucial elements in the Kierkegaardian account. Gabriel Marcel and Sharon Krishek are used to counter Heidegger’s minimizing of the deaths of others and to show how the deaths of others may become integral to our sense of self. This prepares the way for revisiting Kierkegaard’s discourse on the work of love in remembering the dead. Against the criticism that this reveals the absence of the other in Kierkegaardian love, the paper argues that, on the contrary, it shows how Kierkegaard conceives the self as inseparable from the core relationships of love that, despite of death, constitute it as the self that it is.

Sign in / Sign up

Export Citation Format

Share Document