The formation of ‘optimal’ vortex rings, and the efficiency of propulsion devices

2001 ◽  
Vol 427 ◽  
pp. 61-72 ◽  
Author(s):  
P. F. LINDEN ◽  
J. S. TURNER
Keyword(s):  
Energy Input ◽  
Physical Mechanism ◽  
Vortex Rings ◽  
The Core ◽  
Aspect Ratios ◽  
Axisymmetric Vortex ◽  
Single Ring ◽  
Spherical Vortex ◽  
Limiting Case ◽  
Limiting Value

The formation of an axisymmetric vortex ring by forcing uid impulsively through a pipe is examined. An idealized model of the circulation, impulse and energy provided by the injected plug is developed, and these quantities are equated to the corresponding properties of the class of rings with finite cores described by Norbury (1973). It is shown that, as the length-to-diameter aspect ratio L/D of the plug increases, the size of the core increases in comparison with all the fluid carried along with the ring, until the limiting case of Hill's spherical vortex is reached. For aspect ratios larger than a certain value it is not possible to produce a single ring while conserving circulation, impulse, volume and energy. This implies that the limiting vortex is ‘optimal’ in the sense that it has maximum impulse, circulation and volume for a given energy input. While this matching calculation makes the physical mechanism clear, the L/D ratio that can be achieved in practice is more appropriately taken from the direct experimental measurements of Gharib et al. (1998) who concluded that the limiting value is L/D = 4. This is close to the value found in our calculation.

A family of steady vortex rings

1973 ◽  
Vol 57 (3) ◽  
pp. 417-431 ◽  
Author(s):  
J. Norbury
Keyword(s):  
Theoretical Work ◽  
Vortex Rings ◽  
Small Cross Section ◽  
The Core ◽  
Axisymmetric Vortex ◽  
Recent Theoretical Work ◽  
The Family ◽  
Core Boundary ◽  
Spherical Vortex ◽  
Axis Of Symmetry

Axisymmetric vortex rings which propagate steadily through an unbounded ideal fluid at rest at infinity are considered. The vorticity in the ring is proportional to the distance from the axis of symmetry. Recent theoretical work suggests the existence of a one-parameter family, [npar ]2 ≥ α ≥ 0 (the parameter α is taken as the non-dimensional mean core radius), of these vortex rings extending from Hill's spherical vortex, which has the parameter value α = [npar ]2, to vortex rings of small cross-section, where α → 0. This paper gives a numerical description of vortex rings in this family. As well as the core boundary, propagation velocity and flux, various other properties of the vortex ring are given, including the circulation, fluid impulse and kinetic energy. This numerical description is then compared with asymptotic descriptions which can be found near both ends of the family, that is, when α → [npar ]2 and α → 0.

scholarly journals Perturbation response and pinch-off of vortex rings and dipoles

2012 ◽  
Vol 704 ◽  
pp. 280-300 ◽  
Author(s):  
Clara O’Farrell ◽  
John O. Dabiri
Keyword(s):  
Vortex Formation ◽  
Three Dimensions ◽  
Vortex Rings ◽  
Two Dimensional ◽  
Vortex Pairs ◽  
The Core ◽  
Axisymmetric Vortex ◽  
Prolate Shape ◽  
The Family ◽  
Spherical Vortex

AbstractThe nonlinear perturbation response of two families of vortices, the Norbury family of axisymmetric vortex rings and the Pierrehumbert family of two-dimensional vortex pairs, is considered. Members of both families are subjected to prolate shape perturbations similar to those previously introduced to Hill’s spherical vortex, and their response is computed using contour dynamics algorithms. The response of the entire Norbury family to this class of perturbations is considered, in order to bridge the gap between past observations of the behaviour of thin-cored members of the family and that of Hill’s spherical vortex. The behaviour of the Norbury family is contrasted with the response of the analogous two-dimensional family of Pierrehumbert vortex pairs. It is found that the Norbury family exhibits a change in perturbation response as members of the family with progressively thicker cores are considered. Thin-cored vortices are found to undergo quasi-periodic deformations of the core shape, but detrain no circulation into their wake. In contrast, thicker-cored Norbury vortices are found to detrain excess rotational fluid into a trailing vortex tail. This behaviour is found to be in agreement with previous results for Hill’s spherical vortex, as well as with observations of pinch-off of experimentally generated vortex rings at long formation times. In contrast, the detrainment of circulation that is characteristic of pinch-off is not observed for Pierrehumbert vortex pairs of any core size. These observations are in agreement with recent studies that contrast the formation of vortices in two and three dimensions. We hypothesize that transitions in vortex formation, such as those occurring between wake shedding modes and in vortex pinch-off more generally, might be understood and possibly predicted based on the observed perturbation responses of forming vortex rings or dipoles.

Some steady axisymmetric vortex flows past a sphere

2001 ◽  
Vol 433 ◽  
pp. 315-328 ◽  
Author(s):  
ALAN ELCRAT ◽  
BENGT FORNBERG ◽  
KENNETH MILLER
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Stream Function ◽  
Vortex Rings ◽  
Vortex Flows ◽  
Partial Differential ◽  
Axisymmetric Vortex ◽  
Spherical Vortex ◽  
Attached Vortex ◽  
Vortex Tubes

Steady, inviscid, axisymmetric vortex flows past a sphere are obtained numerically as solutions of a partial differential equation for the stream function. The solutions found include vortex rings, bounded vortices attached to the sphere and infinite vortex tubes. Four families of attached vortices are described: vortex wakes behind the sphere, spherically annular vortices surrounding the spherical obstacle (which can be given analytically), bands of vorticity around the sphere and symmetric pairs of vortices fore and aft of the sphere. Each attached vortex leads to a one-parameter family of vortex rings, analogous to the connection between Hill's spherical vortex and the vortex rings of Norbury.

scholarly journals Luminous supernovae associated with ultra-long gamma-ray bursts from hydrogen-free progenitors extended by pulsational pair-instability

2020 ◽  
Vol 641 ◽  
pp. L10
Author(s):  
Takashi J. Moriya ◽  
Pablo Marchant ◽  
Sergei I. Blinnikov
Keyword(s):  
Light Curve ◽  
Energy Input ◽  
Gamma Ray ◽  
Gamma Ray Bursts ◽  
Core Collapse ◽  
Decay Energy ◽  
Slow Cooling ◽  
Gamma Ray Burst ◽  
The Core ◽  
Optical Light Curve

We show that the luminous supernovae associated with ultra-long gamma-ray bursts can be related to the slow cooling from the explosions of hydrogen-free progenitors that are extended by pulsational pair-instability. We have recently shown that some rapidly-rotating hydrogen-free gamma-ray burst progenitors that experience pulsational pair-instability can keep an extended structure caused by pulsational pair-instability until the core collapse. These types of progenitors have large radii exceeding 10 R⊙ and they sometimes reach beyond 1000 R⊙ at the time of the core collapse. They are, therefore, promising progenitors of ultra-long gamma-ray bursts. Here, we perform light-curve modeling of the explosions of one extended hydrogen-free progenitor with a radius of 1962 R⊙. The progenitor mass is 50 M⊙ and 5 M⊙ exists in the extended envelope. We use the one-dimensional radiation hydrodynamics code STELLA in which the explosions are initiated artificially by setting given explosion energy and 56Ni mass. Thanks to the large progenitor radius, the ejecta experience slow cooling after the shock breakout and they become rapidly evolving (≲10 days), luminous (≳1043 erg s−1) supernovae in the optical even without energy input from the 56Ni nuclear decay when the explosion energy is more than 1052 erg. The 56Ni decay energy input can affect the light curves after the optical light-curve peak and make the light-curve decay slowly when the 56Ni mass is around 1 M⊙. They also have a fast photospheric velocity above 10 000 km s−1 and a hot photospheric temperature above 10 000 K at around the peak luminosity. We find that the rapid rise and luminous peak found in the optical light curve of SN 2011kl, which is associated with the ultra-long gamma-ray burst GRB 111209A, can be explained as the cooling phase of the extended progenitor. The subsequent slow light-curve decline can be related to the 56Ni decay energy input. The ultra-long gamma-ray burst progenitors we proposed recently can explain both the ultra-long gamma-ray burst duration and the accompanying supernova properties. When the gamma-ray burst jet is off-axis or choked, the luminous supernovae could be observed as fast blue optical transients without accompanying gamma-ray bursts.

scholarly journals Triglobal infinite-wing shock-buffet study

2021 ◽  
Vol 925 ◽  
Author(s):  
Wei He ◽  
Sebastian Timme
Keyword(s):  
Stability Analysis ◽  
Three Dimensional ◽  
Base Flow ◽  
Sweep Angle ◽  
Aspect Ratios ◽  
Spatially Periodic ◽  
Time Marching ◽  
High Reynolds ◽  
Limiting Case ◽  
Base Flows

This article uses triglobal stability analysis to address the question of shock-buffet unsteadiness, and associated modal dominance, on infinite wings at high Reynolds number, expanding upon recent biglobal work, aspiring to elucidate the flow phenomenon's origin and characteristics. Infinite wings are modelled by extruding an aerofoil to finite aspect ratios and imposing a periodic boundary condition without assumptions on spanwise homogeneity. Two distinct steady base flows, spanwise uniform and non-uniform, are analysed herein on straight and swept wings. Stability analysis of straight-wing uniform flow identifies both the oscillatory aerofoil mode, linked to the chordwise shock motion synchronised with a pulsation of its downstream shear layer, and several monotone (non-oscillatory), spatially periodic shock-distortion modes. Those monotone modes become outboard travelling on the swept wing with their respective frequencies and phase speeds correlated with the sweep angle. In the limiting case of very small wavenumbers approaching zero, the effect of sweep creates branches of outboard and inboard travelling modes. Overall, triglobal results for such quasi-three-dimensional base flows agree with previous biglobal studies. On the contrary, cellular patterns form in proper three-dimensional base flow on straight wings, and we present the first triglobal study of such an equilibrium solution to the governing equations. Spanwise-irregular modes are found to be sensitive to the chosen aspect ratio. Nonlinear time-marching simulations reveal the flow evolution and distinct events to confirm the insights gained through dominant modes from routine triglobal stability analysis.

Zur Theorie der Kernrotationen III

1961 ◽  
Vol 16 (10) ◽  
pp. 1083-1089
Author(s):  
Hans Hackenbroich
Keyword(s):  
Moment Of Inertia ◽  
Variation Method ◽  
The Core ◽  
Particle Potential ◽  
Vibrational Energies ◽  
Spherical Nuclei ◽  
The One ◽  
Limiting Case ◽  
Mechanical Rotation ◽  
Closed Shells

The interplay of the quantum mechanical rotation of a core consisting of closed shells and nucleons outside the closed shells is considered. The core is characterized by its moment of inertia and its deformation is taken to be the same as the deformation of the one particle potential. Energies and wave functions of the system are calculated with the help of a variation method. The distortion of the wave function of the outer nucleons due to the rotation is considerably smaller than computed from first order perturbation theory. The INGLIS formula for rotational energies is a limiting case of our energy equation.The core moment of inertia enters the model as a parameter. This parameter can be estimated from the vibrational energies of spherical nuclei and from the rotational-vibrational interaction, but the two values obtained are not in good agreement. In our example (158Gd) only the second value gives the correct moment of inertia of the whole nucleus.

scholarly journals Spherical vortices in rotating fluids

2018 ◽  
Vol 846 ◽  
Author(s):  
M. M. Scase ◽  
H. L. Terry
Keyword(s):  
Ideal Fluid ◽  
Rotation Rate ◽  
Wave Motion ◽  
Infinite Family ◽  
Vortex Rings ◽  
Rotating Fluids ◽  
Critical Rotation ◽  
Spherical Vortex ◽  
Inertial Wave ◽  
Special Case

A popular model for a generic fat-cored vortex ring or eddy is Hill’s spherical vortex (Phil. Trans. R. Soc. A, vol. 185, 1894, pp. 213–245). This well-known solution of the Euler equations may be considered a special case of the doubly infinite family of swirling spherical vortices identified by Moffatt (J. Fluid Mech., vol. 35 (1), 1969, pp. 117–129). Here we find exact solutions for such spherical vortices propagating steadily along the axis of a rotating ideal fluid. The boundary of the spherical vortex swirls in such a way as to exactly cancel out the background rotation of the system. The flow external to the spherical vortex exhibits fully nonlinear inertial wave motion. We show that above a critical rotation rate, closed streamlines may form in this outer fluid region and hence carry fluid along with the spherical vortex. As the rotation rate is further increased, further concentric ‘sibling’ vortex rings are formed.

Navier-Stokes simulations of axisymmetric vortex rings

1988 ◽  
Author(s):  
S. STANAWAY ◽  
B. CANTWELL ◽  
P. SPALART
Keyword(s):  
Navier Stokes ◽  
Vortex Rings ◽  
Axisymmetric Vortex

scholarly journals XV. Researches on the theory of vortex rings.—Part II

1885 ◽  
Vol 176 ◽  
pp. 725-780 ◽  
Keyword(s):  
Outer Boundary ◽  
Dense Core ◽  
Vortex Rings ◽  
Present Communication ◽  
Mathematical Methods ◽  
The Core ◽  
Cyclic Motion ◽  
General Investigation ◽  
Surrounding Fluid ◽  
Atom Theory

The present communication forms a continuation of some researches the first part of which was published in Part I. of the Transactions for 1884. In that paper was considered the case of a circular hollow with cyclic motion round it. In the following pages the more general case is investigated where the core is of different density from that of the surrounding fluid, has a hollow inside it, and circulations additional to that due to the rotational filaments actually present. The investigation is not merely one of mathematical interest, for the vortex atom theory of matter has—so far as it has yet been developed—shown such claims on our consideration that anything throwing light on it will be of value. The supposition of a dense core may possibly be necessary to account for the different masses of the yarious elements. As soon as the existence of a core is postulated the ring at once becomes more complex, depending on the density (or even the arrangement of density) of its core, on its vorticity, and on the presence or absence of additional circulations. In what follows the vorticity has been taken uniform; this not only greatly simplifies the mathematical methods, but is also the case we should naturally choose first to investigate. In the general investigation the density is taken to be different from that of the surrounding fluid. The ring is supposed hollow, with an additional circulation round it, and another additional circulation round the outer boundary of the core. It is evident that the presence of the former circulation necessitates the perpetual existence of the hollow. It is shown that the presence of the latter circulation is necessary to render the ring stable when its density is greater than that of the rest of the fluid.

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