scholarly journals The motion of a buoyant vortex filament

2018 ◽  
Vol 857 ◽  
Author(s):  
Ching Chang ◽  
Stefan G. Llewellyn Smith
Keyword(s):  
Analytic Solution ◽  
Axial Flow ◽  
External Pressure ◽  
Transverse Component ◽  
Tangential Component ◽  
Force Balance ◽  
Vortex Filament ◽  
Vortex Rings ◽  
Asymptotic Model ◽  
Vortex Element

We investigate the motion of a thin vortex filament in the presence of buoyancy. The asymptotic model of Moore & Saffman (Phil. Trans. R. Soc. Lond.A, vol. 272, 1972, pp. 403–429) is extended to take account of buoyancy forces in the force balance on a vortex element. The motion of a buoyant vortex is given by the transverse component of force balance, while the tangential component governs the dynamics of the structure in the core. We show that the local acceleration of axial flow is generated by the external pressure gradient due to gravity. The equations are then solved for vortex rings. An analytic solution for a buoyant vortex ring at a small initial inclination is obtained and asymptotically agrees with the literature.

Slender-body analysis of the motion and stability of a vortex filament containing an axial flow

1971 ◽  
Vol 50 (2) ◽  
pp. 335-353 ◽  
Author(s):  
Sheila E. Widnall ◽  
Donald B. Bliss
Keyword(s):  
Travelling Waves ◽  
Stability Boundary ◽  
Axial Flow ◽  
Vortex Pair ◽  
Force Balance ◽  
Slender Body ◽  
Vortex Filament ◽  
Core Radius ◽  
Single Filament ◽  
The Stability

Previous results concerning the effects of axial velocity on the motion of vortex filaments are reviewed. These results suggest that a slender-body force balance between the Kutta–Joukowski lift on the vortex cross-section and the momentum flux within the curved filament will give some insight into the behaviour of the filament. These simple ideas are exploited for both a single vortex filament and a vortex pair, both containing axial flow. The stability of a straight vortex filament containing an axial flow to long wave sinusoidal displacements of its centre-line is investigated and the stability boundary obtained. The effect of axial flow on the stability of a vortex pair is explored. It is shown that to lowest order (in the ratio of vortex core radius to distance between the vortices) the effect of axial flow is to reduce the self-induced rotation of a single filament and that this effect can be considered as a change in effective core radius. To the next order, travelling waves appear in the instability, the instability mode for the vortex pair becomes non-planar but the amplification rate of the instability is not affected.

Dynamics of axisymmetric bodies rising along a zigzag path

2008 ◽  
Vol 606 ◽  
pp. 209-223 ◽  
Author(s):  
PEDRO C. FERNANDES ◽  
PATRICIA ERN ◽  
FRÉDÉRIC RISSO ◽  
JACQUES MAGNAUDET
Keyword(s):  
Aspect Ratio ◽  
Transverse Direction ◽  
Transverse Velocity ◽  
Density Ratio ◽  
Transverse Component ◽  
The Body ◽  
Force Balance ◽  
Body Motion ◽  
Major Influence ◽  
Inertia Force

The forces and torques governing the planar zigzag motion of thick, slightly buoyant disks rising freely in a liquid at rest are determined by applying the generalized Kirchhoff equations to experimental measurements of the body motion performed for a single body-to-fluid density ratio ρs/ρf ≈ 1. The evolution of the amplitude and phase of the various contributions is discussed as a function of the two control parameters, i.e. the body aspect ratio (the diameter-to-thickness ratio χ = d/h ranges from 2 to 10) and the Reynolds number (100 < Re < 330), Re being based on the rise velocity and diameter of the body. The body oscillatory behaviour is found to be governed by the force balance along the transverse direction and the torque balance. In the transverse direction, the wake-induced force is mainly balanced by two forces that depend on the body inclination, i.e. the inertia force generated by the body rotation and the transverse component of the buoyancy force. The torque balance is dominated by the wake-induced torque and the restoring added-mass torque generated by the transverse velocity component. The results show a major influence of the aspect ratio on the relative magnitude and phase of the various contributions to the hydrodynamic loads. The vortical transverse force scales as fo = (ρf − ρs)ghπd2 whereas the vortical torque involves two contributions, one scaling as fod and the other as f1d with f1 = χfo. Using this normalization, the amplitudes and phases of the vortical loads are made independent of the aspect ratio, the amplitudes evolving as (Re/Rec1 − 1)1/2, where Rec1 is the threshold of the first instability of the wake behind the corresponding body held fixed in a uniform stream.

scholarly journals Function of the heterocercal tail in sharks: quantitative wake dynamics during steady horizontal swimming and vertical maneuvering

2002 ◽  
Vol 205 (16) ◽  
pp. 2365-2374 ◽  
Author(s):  
C. D. Wilga ◽  
G. V. Lauder
Keyword(s):  
Body Position ◽  
Classical Model ◽  
Center Of Mass ◽  
Critical Role ◽  
Force Balance ◽  
Body Orientation ◽  
Vortex Rings ◽  
Direct Reaction ◽  
Body Angle ◽  
Tail Function

SUMMARYThe function of the heterocercal tail in sharks has long been debated in the literature. Previous kinematic data have supported the classical theory which proposes that the beating of the heterocercal caudal fin during steady horizontal locomotion pushes posteroventrally on the water, generating a reactive force directed anterodorsally and causing rotation around the center of mass. An alternative model suggests that the heterocercal shark tail functions to direct reaction forces through the center of mass. In this paper,we quantify the function of the tail in two species of shark and compare shark tail function with previous hydrodynamic data on the heterocercal tail of sturgeon Acipenser transmontanus. To address the two models of shark heterocercal tail function, we applied the technique of digital particle image velocimetry (DPIV) to quantify the wake of two species of shark swimming in a flow tank. Both steady horizontal locomotion and vertical maneuvering were analyzed. We used DPIV with both horizontal and vertical light sheet orientations to quantify patterns of wake velocity and vorticity behind the heterocercal tail of leopard sharks (Triakis semifasciata) and bamboo sharks (Chiloscyllium punctatum) swimming at 1.0Ls-1, where L is total body length. Two synchronized high-speed video cameras allowed simultaneous measurement of shark body position and wake structure. We measured the orientation of tail vortices shed into the wake and the orientation of the central jet through the core of these vortices relative to body orientation. Analysis of flow geometry indicates that the tail of both leopard and bamboo shark generates strongly tilted vortex rings with a mean jet angle of approximately 30 ° below horizontal during steady horizontal swimming. The corresponding angle of the reaction force is much greater than body angle (mean 11 °) and the angle of the path of motion of the center of mass (mean approximately 0 °), thus strongly supporting the classical model of heterocercal tail function for steady horizontal locomotion. Vortex jet angle varies significantly with body angle changes during vertical maneuvering, but sharks show no evidence of active reorientation of jet angle relative to body angle, as was seen in a previous study on the function of sturgeon tail. Vortex jet orientation is significantly more inclined than the relatively horizontal jet generated by sturgeon tail vortex rings, demonstrating substantial differences in function in the heterocercal tails of sharks and sturgeon.We present a summary of forces on a swimming shark integrating data obtained here on the tail with previous data on pectoral fin and body function. Body orientation plays a critical role in the overall force balance and compensates for torques generated by the tail. The pectoral fins do not generate lift during steady horizontal locomotion, but play an important hydrodynamic role during vertical maneuvering.

An Asymptotic Model of Viscous Flow Limitation in a Highly Collapsed Channel

1998 ◽  
Vol 120 (4) ◽  
pp. 544-546 ◽  
Author(s):  
O. E. Jensen
Keyword(s):  
Pressure Drop ◽  
Viscous Flow ◽  
External Pressure ◽  
Asymptotic Model ◽  
Flow Limitation ◽  
Two Dimensional ◽  
Inertial Effects ◽  
Channel One ◽  
Narrow Region ◽  
Flow Through

A viscous flow through a long two-dimensional channel, one wall of which is formed by a finite-length membrane, experiences flow limitation when the channel is highly collapsed over a narrow region under high external pressure. Simple approximate relations between flow rate and pressure drop are obtained for this configuration by the use of matched asymptotic expansions. Weak inertial effects are also considered.

The Shell in the Lamellibranchia

1953 ◽  
Vol s3-94 (25) ◽  
pp. 57-70
Author(s):  
G. OWEN
Keyword(s):  
Radial Component ◽  
Transverse Component ◽  
Tangential Component ◽  
The Other ◽  
Transverse Diameter ◽  
Differential Growth ◽  
Anterior Portion ◽  
Normal Zone ◽  
Mantle Edge ◽  
Spiral Angle

1. In the Lamellibranchia, the direction of growth at any region of the valve margins may be resolved into: (a) a radial component radiating from the umbo and acting in the plane of the generating curve; (b) a transverse component acting at right angles to the plane of the generating curve; (c) a tangential component acting tangentially to, and in the plane of, the generating curve. The radial component is always present and affects the form of both valves while the transverse component may be reduced or absent in one valve. 2. The lamellibranch mantle/shell is orientated with reference to the normal axis. This normal axis follows that sector of the shell secreted by the normal zone of the mantle edge (i.e. where the effect of the transverse component is greatest) and passes through the umbo, the normal zone and the point at which the greatest transverse diameter of the shell intersects the surface of the valves. 3. The form of the shell valves should be considered with reference to: (a) the outline of the generating curve; (b) the spiral angle of the normal axis; (c) the form (i.e. planospiral or turbinate spiral) of the normal axis. 4. The ‘deflection’ anteriorly of the umbones and the splitting of the anterior portion of the ligament in many bivalves is a consequence of a tangential component affecting the form of the valves. 5. In both gastropods and lamellibranchs, the turbinate spiral shell is the resultant of two differential growth ratios. In the Gastropoda, however, both ratios act perpendicularly to the plane of the generating curve, while in the Lamellibranchia, one acts perpendicularly to this plane while the other acts in the plane of the generating curve.

Dynamics and Instability of a Vortex Ring Impinging on a Wall

2015 ◽  
Vol 18 (4) ◽  
pp. 1122-1146 ◽  
Author(s):  
Heng Ren ◽  
Xi-Yun Lu
Keyword(s):  
Vortex Ring ◽  
Axial Flow ◽  
Core Region ◽  
Vortical Flow ◽  
Vortex Rings ◽  
Flow Transition ◽  
Azimuthal Component ◽  
Vortical Structures ◽  
The Core ◽  
Eddy Simulation

AbstractDynamics and instability of a vortex ring impinging on a wall were investigated by means of large eddy simulation for two vortex core thicknesses corresponding to thin and thick vortex rings. Various fundamental mechanisms dictating the flow behaviors, such as evolution of vortical structures, formation of vortices wrapping around vortex rings, instability and breakdown of vortex rings, and transition from laminar to turbulent state, have been studied systematically. The evolution of vortical structures is elucidated and the formation of the loop-like and hair-pin vortices wrapping around the vortex rings (called briefly wrapping vortices) is clarified. Analysis of the enstrophy of wrapping vortices and turbulent kinetic energy (TKE) in flow field indicates that the formation and evolution of wrapping vortices are closely associated with the flow transition to turbulent state. It is found that the temporal development of wrapping vortices and the growth rate of axial flow generated around the circumference of the core region for the thin ring are faster than those for the thick ring. The azimuthal instabilities of primary and secondary vortex rings are analyzed and the development of modal energies is investigated to reveal the flow transition to turbulent state. The modal energy decay follows a characteristic –5/3 power law, indicating that the vortical flow has become turbulence. Moreover, it is identified that the TKE with a major contribution of the azimuthal component is mainly distributed in the core region of vortex rings. The results obtained in this study provide physical insight of the mechanisms relevant to the vortical flow evolution from laminar to turbulent state.

On the instantaneous cutting of a columnar vortex with non-zero axial flow

1997 ◽  
Vol 351 ◽  
pp. 41-74 ◽  
Author(s):  
J. S. MARSHALL ◽  
S. KRISHNAMOORTHY
Keyword(s):  
Vortex Core ◽  
Flow Model ◽  
Plug Flow ◽  
Axial Flow ◽  
Vortex Rings ◽  
Core Radius ◽  
Expansion Waves ◽  
Columnar Vortex ◽  
Cutting Surface ◽  
Compression And Expansion

A study of the response of a columnar vortex with non-zero axial flow to impulsive cutting has been performed. The flow evolution is computed based on the vorticity–velocity formulation of the axisymmetric Euler equation using a Lagrangian vorticity collocation method. The vortex response is compared to analytical predictions obtained using the plug-flow model of Lundgren & Ashurst (1989). The plug-flow model indicates that axial motion on a vortex core with variable core area behaves in a manner analogous to one-dimensional gas dynamics in a tube, with the vortex core area playing a role analogous to the gas density. The solution for impulsive cutting of a vortex obtained from the plug-flow model thus resembles the classic problem of impulsive motion of a piston in a tube, with formation of an upstream-propagating vortex ‘shock’ (over which the core radius changes discontinuously) and a downstream-propagating vortex ‘expansion wave’ on opposite sides of the cutting surface. Direct computations of the vortex response from the Euler equation reveal similar upstream- and downstream-propagating waves following impulsive cutting for cases where the initial vortex flow is subcritical. These waves in core radius are produced by a series of vortex rings, embedded within the columnar vortex core, having azimuthal vorticity of alternating sign. The effect of the compression and expansion waves is to bring the axial and radial velocity components to nearly zero behind the propagating vortex rings, in a region on both sides of the cutting surface with ever-increasing length. The change in vortex core radius and the variation in pressure along the cutting surface agree very well with the predictions of the plug-flow model for subcritical flow after the compression and expansion waves have propagated sufficiently far away. For the case where the ambient vortex flow is supercritical, no upstream-propagating wave is possible on the compression side of the vortex, and the vortex axial flow is observed to impact on the cutting surface in a manner similar to that commonly observed for a non-rotating jet impacting on a wall. The flow appears to approach a steady state near the point of impact after a sufficiently long time. The vortex response on the expansion side of the cutting surface exhibits a downstream-propagating vortex expansion wave for both the subcritical and supercritical conditions. The results of the vortex response study are used to formulate and verify predictions for the net normal force exerted by the vortex on the cutting surface. An experimental study of the cutting of a vortex by a thin blade has also been performed in order to verify and assess the limitations of the instantaneous vortex cutting model for application to actual vortex–body interaction problems.

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