Motion of a vortex near the boundary between two heavy fluids

The motion of a vortex beneath the surface of a heavy fluid has been discussed in both linear [1, 2] and nonlinear [3–5] formulation. The density of the upper medium is neglected, which makes it possible to replace the continuity of pressure during transition through the boundary between the media by constancy of the pressure at the boundary of the heavy fluid. In this paper, the problem is solved in a general nonlinear formulation, including the mutual effects of media motion, and the vortex can be in either the upper or lower medium. Steady-state motion of a vortex of given intensity near the boundary between two heavy fluids is discussed in terms of a model of an ideal and incompressible medium. Approximate expressions are obtained for the boundary.     

Hydrodynamic characteristics of a vortex source performing translational motion in a multilayer heavy fluid

The problem of translational motion of a vortex source in a three-layer fluid bounded by a bottom from below is considered. The fluid in each layer is perfect, incompressible, heavy, and homogeneous. Based on the previously developed method, formulas for disturbed complex velocities of the fluid in each layer and the wave drag and lift force of the vortex source are obtained. The vortex motion is considered near the interface of two semi-infinite fluid media and in a two-layer fluid with different conditions at the boundary. In all cases, the hydrodynamic characteristics of the vortex source are given as functions of the Froude number. In a number of problems, these characteristics have discontinuities at the transition through the critical Froude numbers. The character of these discontinuities is studied analytically. Omsk Department of Sobolev Institute of Mathematics, Siberian Division, Russian Academy of Sciences, Omsk 644099. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 140–146, September–October, 2000.     

Vortex dynamics mechanisms of separated boundary layer in a highly loaded low pressure turbine cascade

This paper investigates the vortex dynamics in the suction-side boundary layer on an aero-engine low pressure turbine blade at two different Reynolds numbers at which short and long laminar separation bubbles occur. Different vortical patterns are observed and investigated through large eddy simulation (LES). The results show that at the higher Reynolds number, streamwise streaks exist upstream of separation line. These streaks initiate spanwise undulation in the form of vortex tubes, which roll-up and shed from the shear layer due to the Kelvin–Helmholtz instability. The vortex tubes alternately pair together and eventually distort and break down to small-scale turbulence structures near the mean reattachment location and convect into a fully turbulent boundary layer. At the lower Reynolds number, streamwise streaks are strong and the separated flow is unable to reattach to the blade surface immediately after transition to turbulence. Therefore, bursting of short bubbles into long bubbles can occur, and vortex tubes have larger diameters and cover a part of the blade span. In this case vortex pairing does not occur and vortex shedding process is promoted mainly by flapping phenomenon. Moreover, the results of dynamic mode decomposition (DMD) analysis show a breathing motion as a source of unsteadiness in the separation location, which is accompanied by the flapping phenomenon.     

Effects of micro-ramps on a shock wave/turbulent boundary layer interaction

Stereoscopic particle image velocimetry is used to investigate the effects of micro-ramp sub-boundary layer vortex generators, on an incident shock wave/boundary layer interaction at Mach 1.84. Single- and double-row arrangements of micro-ramps are considered. The micro-ramps have a height of 20% of the unperturbed boundary layer thickness and the measurement planes are located 0.1 and 0.6 boundary layer thicknesses from the wall. The micro-ramps generate packets of individual vortex pairs downstream of their vertices, which produce counter-rotating longitudinal streamwise vortex pairs in a time-averaged view. These structures induce a pronounced spanwise variation of the flow properties, namely the mixing across the boundary layer interface. The probability of reversed-flow occurrence is decreased by 20 and 30% for the single- and double-row configurations, respectively. Both configurations of micro-ramps stabilize the shock motion by reducing the length of its motion by about 20% in the lower measurement plane. The results are summarized by a conceptual model describing the boundary layer’s and interaction’s flow pattern under the effect of the micro-ramps.     

The evolution of an initially circular vortex near an escarpment. Part I: analytical results

The motion of a quasigeostrophic, equivalent-barotropic, initially circular vortex patch near an infinitely long topographic escarpment is studied using f-plane dynamics. There are two time scales in the problem: the advective time scale associated with the vortex, and the time scale for topographic vortex stretching. Analytical progress is possible when these two time scales are well-separated and results are presented here.If topographic vortex stretching dominates advection by the vortex the vortex is said to be ‘weak’. The vortex patch remains circular on the topographic time scale, and dispersive topographic waves rapidly propagate the initial disturbance away from the vicinity of the vortex. Subsequently cross-isobath motion is inhibited, and the vortex moves as though the escarpment were a plane wall. The same behaviour was observed for the motion of a weak singular vortex near an escarpment by Dunn, McDonald and Johnson [7], who named the phenomenon the ‘pseudoimage’ of the vortex.If advection dominates over topographic effects, the vortex is said to be ‘intense’. The vortex also remains circular to leading order, but the relative vorticity produced by the swirl of the vortex is less able to escape the vicinity of the vortex. The vortex follows a similar curved trajectory to those observed for intense vortices on the β-plane. The dipolar mechanism for this behaviour is described. Large time solutions are inhibited by the form of the escarpment topography, but examination of the equations leads to the conclusion that the leading order solution may be predict the motion for times beyond its formal range of validity.     

Elementary aspects of vortex motion

Several aspects of vortex motion are considered, with a special stress on the present status of idealization, such as point vortices or vortex filament. As an introduction, elements of vortices induced by the transient flow past an obstacle are considered and their role and development are stated.

Following this introduction, a general survey of the issues in this symposium is made sketchily. As an example, the motion of point vortices in the presence of an external flow or a boundary is discussed on the basis of the Hamiltonian formalism. The cases of linear flow and semicircular boundary are taken as examples of regular and chaotic motions. Secular behaviour of a pair of vortices in the flow is remarked.     

Swept-Wing Boundary Layer Receptivity to a Steady Free-Stream Vortex Disturbance

A single trailing vortex developed behind a micro-wing immersed in a free stream was used to study the vortex receptivity of a swept-wing boundary layer. As a result of the interaction, longitudinal-velocity disturbances develop in the boundary layer. On the swept wing, disturbance transformation occurs near the leading edge and is accompanied by the formation of a wave packet consisting of waves typical of cross-flow instability. Disturbances with other characteristics are also detected. These disturbances may be attributable to distributed boundary-layer receptivity to the free-stream vortex disturbance considered.     

Coaxial interactions of two vortex rings or of a ring with a body

Inviscid coaxial interactions of two vortex rings, including head-on collisions and leapfrogging motions, are considered using a contour dynamics technique. Interactions of vortex rings with solid bodies are also investigated by combining the contour dynamics technique with a boundary integral equation method. Numerical results show that a clean, successful passage motion is possible for two vortex rings with not too thick cores. In both cases of head-on collisions and leapfrogging motions, very large core deformations are observed when two vortex rings get close to each other. A head-tail structure is formed in the later stage of a head-on collision of two fat vortices. Numerical results also show that a vortex ring will stretch and slow down when it moves toward a solid boundary, will shrink and speed up when it moves away from a solid boundary, and will either translate steadily or approach an oscillating asymptotic state when it is far away from any boundaries. The project supported by The National Education Commission of China and NASA under cooperative grant agreement #NCC5-34.     

Radiation and scattering of sound from a vortex ring

The mechanisms of generation and scattering of sound by a vortex ring are investigated on the basis of fluid dynamics. The vortex ring can serve as a simple dynamic model of the large-scale structures observed in shear flows. Moreover, it is probably the most easily studied vortex element that can be created experimentally. The sound scattering investigation also served to determine the extent to which the vortex is affected by sound, its selectivity with respect to such parameters as the acoustic frequency, the angle of incidence of the wave, etc. The perturbed motion is considered against the background of the steady-state motion of the ring. The perturbed motion in the vortex core is determined on the basis of linear incompressible fluid dynamics. Two terms of the expansion in the M number of the far acoustic field generated by the perturbations in the core are found in accordance with Lighthill's theory. The acoustic power and directivity of the radiation and the acoustic instability growth rate are calculated. It is shown that the scattering of sound by the vortex ring is a resonance effect, and the scattering amplitude near resonance is determined. The acoustic action on the hydrodynamic structure of the flow in the core of the ring is especially intense near the resonances and extends over a period short as compared with the characteristic time of the acoustic instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 83–95, May–June, 1987.     

Interactions between vortices and flexible walls

We give two fundamental solutions for the motion of a point vortex near a flexible wall, up to first order in wall deflection, using computational methods. For a point vortex near an infinite horizontal wall, the deformation of the wall intensifies the flow at the wall near the vortex, and increases the speed of the vortex. Near a circular wall there is a strong mutual amplification of the deflection of the wall and the pressure force induced by the deflection, as the point vortex approaches the wall. The total force on the wall diverges as the inverse cube of the distance to the point vortex, and the induced speed of the point vortex diverges as the inverse fourth power of distance to the wall.     

Acoustic waves emitted by a vortex ring passing near a wedge-like plate

Acoustic waves emitted by a vortex ring moving near a thin wedge-like plate of finite width have been studied. The experiments are performed for three configurations: the plate (A) is held edgeways to the direction of the vortex motion, (B) is held sideways to the direction, and (C) is held edgeways at an angle of 45° against the vortex motion. The observed sound wave is of dipole radiation type, and the magnitude of the pressure is large in the direction of the normal to the plate plane and small in parallel. The observed pressure is proportional to the third power of the vortex speed. The instantaneous force exerted on the plate by the vortex motion has also been examined. The force vector is mainly normal to the plate plane. The observed profiles agree within a reasonable degree of accuracy with the theoretical ones calculated for the vortex ring interacting with the flat plate of thickness zero.     

Evolution of Two Vortices Near a Solid Surface

A numerical solution is found for the problem of evolution of two oppositely rotating vortices in a viscous incompressible fluid near a solid surface. The mechanism of vortex dissipation is determined. The trajectory of vortex motion is constructed.     

Thrust generation and wake structure of wiggling hydrofoil

Marine animals and micro-machines often use wiggling motion to generate thrust. The wiggling motion can be modeled by a progressive wave where its wavelength describes the flexibility of wiggling animals. In the present study, an immersed boundary method is used to simulate the flows around the wiggling hydrofoil NACA 65-010 at low Reynolds numbers. One can find from the numerical simulations that the thrust generation is largely determined by the wavelength. The thrust coefficients decrease with the increasing wavelength while the propulsive efficiency reaches a maximum at a certain wavelength due to the viscous effects. The thrust generation is associated with two different flow patterns in the wake： the well-known reversed Karman vortex streets and the vortex dipoles. Both are jet-type flows where the thrust coefficients associated with the reversed Karman vortex streets are larger than the ones associated with the vortex diploes.     

Wind-Up of a Spanwise Vortex in Deepening Transition and Stall

A fundamental flow problem of unsteady wind-up of a spanwise vortex is studied in this theoretical work on deepening dynamic stall and transition in a boundary layer, internal layer or related unsteady motion. It examines the nonlinear evolution of the spanwise vortex produced when the local wall pressure develops a maximum or minimum, subsequent to the finite-time break-up of an interacting layer and the impact of normal pressure gradients. The evolution is controlled by an inner–outer interaction between the effects of the normal pressure gradient and the momentum jumps across and outside the vortex, which is situated near the strong inflexion point induced in the mean flow. Although the work concentrates on a particular internal-flow context, many of the flow properties found are generic and in particular apply for a more general case including external flows. Analysis and associated computations point to two main distinct trends in the vortex response, depending to a large extent on a parameter gauging the relative strengths of the above effects. The response is either an explosive one, provoking enhanced wind-up, growth and pressure in the vortex, or it is implosive, causing the vortex to shrink and virtually empty itself through unwinding, leaving little local pressure variation. A further discussion includes the after-effects of this vortex response and some of the connections with experiments and direct computations on deepening stall and transition. Received 22 February 1999 and accepted 28 March 2000     

Vibrational dynamics of a light body in a liquid-filled rotating cylinder

The behavior of a light free cylindrical body in a rapidly rotating horizontal cylinder containing a liquid under vibrational action (the vibration direction is perpendicular to the rotation axis) is investigated. An intense rotation of the body relative to the cavity is detected. Depending on the vibration frequency, the body rotation velocity in the laboratory reference system may be higher or lower than the cavity rotation velocity and in the resonance region they may differ by several times. The mechanism of motion generation is theoretically described. It is shown that the motion is related with the excitation of inertial oscillations of the body: the cause of the motion is an average vibrational force generated due to nonlinear effects in the Stokes boundary layer near the oscillating body. The formation of large-scale axisymmetric vortex structures periodic along the rotation axis, which appear under conditions of inertial oscillation of the body during its motion, both leading and lagging, is detected.     

Effect of a Thin Longitudinal Inviscid Vortex on a Two-Dimensional Pre-Separation Boundary Layer

For large Reynolds numbers, an asymptotic solution of the Navier-Stokes equations describing the effect of a thin longitudinal vortex with a constant circulation on the development of an incompressible steady two-dimensional laminar boundary layer on a flat plate is obtained. It is established that, in a narrow wall region extending along the vortex filament, the viscous flow is described by the 3-D boundary layer equations. A solution of these equations for small values of the vortex circulation is studied. It is found that the solution of the two-dimensional pre-separation boundary layer equations collapses. This is attributable to the singular behavior of the 3-D disturbances near the zero-longitudinal-friction points.     

Meander of a fin trailing vortex and the origin of its turbulence

The low-frequency meander of a trailing vortex shed from a tapered fin installed on a wind tunnel wall has been studied using stereoscopic particle image velocimetry in the near-wake at Mach 0.8. Distributions of the instantaneous vortex position reveal that the meander amplitude increases with downstream distance and decreases with vortex strength, indicating meander is induced external to the vortex. Trends with downstream distance suggest meander begins on the fin surface, prior to vortex shedding. Mean vortex properties are unaltered when considered in the meandering reference frame, apparently because turbulent fluctuations in the vortex shape and strength dominate positional variations. Conversely, a large peak of artificial turbulent kinetic energy is found centered in the vortex core, which almost entirely disappears when corrected for meander, though some turbulence remains near the core radius. Turbulence originating at the wind tunnel wall was shown to contribute to vortex meander by energizing the incoming boundary layer using low-profile vortex generators and observing a substantial increase in the meander amplitude, while greater turbulent kinetic energy penetrates the vortex core. An explanatory mechanism has been hypothesized, in which the vortex initially forms at the apex of the swept leading edge of the fin where it is exposed to turbulent fluctuations within the wind tunnel wall boundary layer, introducing an instability into the incipient vortex core.     

Study of water waves with submerged obstacles using a vortex method with Helmholtz decomposition

The present study develops a 2‐D numerical scheme that combines the vortex method and the boundary integral method by a Helmholtz decomposition to investigate the interaction of water waves with submerged obstacles. Viscous effects and generation of vorticity on the free surface are neglected. The second kind of Fredholm integral equations that govern the strengths of vortex sheets along boundaries are solved iteratively. Vorticity is convected and diffused in the fluid via a Lagrangian vortex (blob) method with varying cores, using the particle strength exchange method for diffusion, with particle redistribution. A grid‐convergence study of the numerical method is reported. The inviscid part of the method and the simulation of the free‐surface motion are tested using two calculations: solitary wave propagation in a uniform channel and a moving line vortex in the fluid. Finally, the full model is verified by simulating periodic waves travelling over a submerged rectangular obstacle using nonuniform vortex blobs with a mapping of the redistribution lattice. Overall, the numerical model predicts the vortices' evolution and the free‐surface motion reasonably well. Copyright © 2008 John Wiley & Sons, Ltd.