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Foppl’s stability analysis (1913) of the equilibrium positions of a line vortex pair in the downstream neighbourhood of a circular cylinder in the two-dimensional potential flow of an incompressible fluid moving with uniform speed at infinity has been corrected recently by Smith (1973). Smith shows that it is only for small symmetric displacements from any one pair of points on the locus of possible rest positions which is known as the Foppl curve there is neutral spatial stability, all other displacements being unstable. This correction now makes the results of the stability analysis consistent with the work of Dolaptchiev and Sendov (1959) who determined the trajectories followed by such a vortex pair when constrained so as to move in a wholly symmetric motion. Dolaptchiev and Sendov obtained the analytical solution of the differential equations for the completely symmetric motion, but for motion other than symmetric no simple solutions of this sort appear to exist although the vortex trajectories for any case can be deduced numerically without difficulty.

We consider the motion of the mass of fluid ejected through a sharp-edged orifice by the motion of a piston. The vorticity formed by viscous forces within the separated flow at the sharp edge rolls up to form a concentrated vortex which, after a development period, consists of a core of very fine scale turbulence surrounded by a co-moving bubble of much larger scale turbulence. This bubble entrains outer fluid, mixes with it, and deposits the majority into a wake together with some small fraction of the total vorticity of the ring. Enough fluid is retained to account for the slow growth of the whole fluid mass. A theory which takes account of both the growth process and the loss of vorticity is proposed. By comparison with experimental measurements we have determined that the entrainment coefficient for turbulent vortex rings has a value equal to 0.011 ± 0.001, while their effective drag coefficient is 0.09 ± 0.01. These results seem to be independent of Reynolds number to within experimental accuracy.

The results of an experimental investigation of a turbulent vortex street in the range 103 [lsim ] Re [lsim ] 2 × 104 are presented. The vortex street was created by the motion of a circular cylinder in a motionless fluid (mercury). Photographs obtained showed that the turbulent street, created by the vortex shedding behind the cylinder, persisted at longer downstream distances and higher Reynolds numbers than previously reported in the literature. A theory was developed to account for the experimental measurements pertaining to the change of the geometrical characteristics, h (distance between the two rows of vortices) and α (longitudinal distance between two consecutive vortices on the same row), of the street in the downstream direction. The implications of the structure of the vortex street on the entrainment mechanism of the turbulent wake are discussed. Some observations of the flow in the formation region of the vortices are discussed in relation to existing work.

The point vortex approximation of a vortex sheet in two space dimensions is examined and a remedy for some of its shortcomings is suggested. The approximation is then applied to the study of the roll-up of a vortex sheet induced by an elliptically loaded wing.

A series of measurements of trailing vortices behind lifting hydrofoils is described. These measurements were made in the Caltech Free-Surface Water Tunnel, using a laser-Doppler velocimeter to measure two components of velocity in the vortex wake. Two different model planforms were tested, and measurements were made at several free-stream velocities and angles of attack for each. Velocity profiles were measured at distances downstream of the model of from five to sixty chord lengths. These measurements are the first results of a continuing experimental programme.In § 3 of this paper, the theory of trailing vortices is discussed. The effects of ‘vortex wandering’ upon the measurements are computed, and the corrected results are seen to be in reasonable agreement with the theory.

The intensity of the polar vortex at 10 mb is used to calculate theoretical values of mean total ozone north of 40° latitude. A satisfactory fit is attained between the development in time of the theoretical ozone and that of the mean of the observed total ozone. The results lead to the conclusion, that a one-cell mean meridional motion relative to the polar night vortex is important for the transport of heat and ozone.

The similarity parameters that can be used to relate erosive particle trajectories in hot gas and equivalent cold gas flows are derived from the equations of motion. The flow in a radial turbine is used as the basis for studying the range of applicability of these similarity parameters. The study includes the vortex and rotor regions of the radial turbine and indicates the ranges of Reynolds number within which these parameters can be used. In addition, the study shows that when general trends of particle trajectories are to be considered, and precise trajectories are not necessary, one of the similarity parameters can be used for Reynolds numbers that are not within the limits indicated for precisely similar trajectories.

The method of parametric differentiation has been applied to predict the farfield noise propagation in both a lossless and a dissipative medium. The classical nonlinear acoustical problems of calculating the sound field of two rectilinear vortices rotating about an axis between them in a lossless medium and of calculating the sound field generated by a rotating cylinder in a viscous medium have been investigated. Solutions to these classical problems valid for asymptotically small Mach number have been extended to larger values of the Mach number parameter. It is shown that in the lossless medium, the governing equation transformed to parameter space reduces to a wave equation in the farfield. In the dissipative medium, the system of nonlinear partial differential equations transformed to parameter space reduces to a linear partial differential equation of the propagating type which contains a third‐order derivative as well as the wave operator. Pressure distributions in the farfield are obtained in the l...

We have measured the critical heat current in thermal counterflow in He II as a function of temperature. The results are analyzed in terms of a slightly modified Vinen theory. The agreement between theory and experiment is excellent, providing an independent determination of the parameters of the theory, as well as as one of the few examples of a successful theory of critical-velocity phenomena in liquid He II. (auth)

We have made computational experiments to study the stability and long-time evolution of two-dimensional wakes. We have used the VORTEX code, a finite-difference realization of two-dimensional motions in incompressible inviscid fluids. In the first experiment an initial shear-unstable triangular velocity profile evolves into a non-homogeneous, finite-area, asymmetric vortex array and like-signed regions attract andfuse(or coalesce). Enhanced transport across the profile is due to ‘capture’ and convection of small-scale vortex regions by larger opposite-signed vortex regions. In the following experiments we study the stability of an asymmetric four-vortexfinite-areasystem corresponding to a von Kármán street of point vortices. Here the critical parameter isb/a, the initial transverse-to-longitudinal separation ratio of vortex centres. At\[ b/a = 0.281 \]the four-vortex system is stable and we observe that large-area vortex regions develop elliptical (m= 2), triangular (m= 3), etc. surface modes owing to mutual interactions. Atb/a= 0 the measured growth rate is smaller than that for the corresponding von Kármán system and atb/a= 0·6 the measured growth rate is larger. Atb/a= 0 one vortex undergoes fission in the high-shear field produced by two nearest-neighbour opposite-signed vortex regions. Heuristic comparisons are made with the two-dimensional tunnel experiments of Taneda and others.

The spatial positions of discrete quantized vortex lines in rotating superfluid helium have been directly visualized by a photographic technique. The positions of the lines in the apparatus do not form a regular array.

SummaryThe trailing vortex wake system generated by a lifting wing of finite span is modelled with a pair of laminar, viscous vortex models of a new type. The embryonic form of this viscous wake model is matched, at several interior collocation points, to the inviscid Betz model of the wake convolution process that transforms the wing trailing-edge vortex sheet into a pair of axisymmetric trailing vortices. In the near-wake and intermediate-wake regions downstream of the wing, the viscous wake model indicates that the trailing vortex characteristics (particularly the peak swirl velocity and the circulation profile) are strongly influenced by details of the wing span load distribution. Using simple eddy viscosity concepts, the viscous wake model achieves good qualitative agreement with test results from a limited selection of published experimental wake investigations. The present laminar theory appears to offer a useful basis for developing an engineering prediction method of aeroplane wake characteristics.

A method is developed for the calculation of the initial inviscid form of rolled-up wake vortices behind a wing having arbitrary lift distribution. The method makes use of the Betz assumptions of conservation of wake vorticity and moments of vorticity. It is found that a simple relationship exists between the radial distribution of vorticity in the rolled-up wake and the spanwise lift distribution. Computed tangential velocity profiles for DC-7, DC-9, and C-141 aircraft are shown to compare favorably with profiles measured by the FAA during tower flyby tests of these aircraft in both flapped and unflapped configurations.

A three-dimensional, 11-level, primitive equation model has been constructed for a simulation study of tropical cyclones. The model has four levels in the boundary layer and its 70×70 variable grid mesh encloses a 4000-km square domain with a 20-km resolution near the center. Details of the model, including the parameterization scheme for the subgrid-scale diffusion and convection processes, are described. A weak vortex in the conditionally unstable tropical atmosphere is given as the initial state for a numerical integration from which a tropical cyclone develops in the model. During the integration period of one week, the sea surface temperature is fixed at 302K. The central surface pressure drops to about 940 mb, while a warm moist core is established. The azimuthal component of mean horizontal wind is maximum at about 60 km from the center at all levels. A strong in-flow is observed in the boundary layer. At upper levels, a secondary radial-vertical circulation develops in and around the regi...

Natural convection along a heated inclined wall in a salinity gradient has been studied experimentally. The experiments were carried out in a tank with one wall inclined at 45°. The dimensions of the tank were 20 cm in height and depth, 30 cm wide at the top, and 10 cm wide at the bottom. The inclined wall and the opposite vertical wall were made of aluminum and they were kept at different constant temperatures. The other two walls were of plexiglass. Shadowgraph technique was used to visualize the fluid motion. The onset of celular motion was quite similar to the case of heating along a vertical wall. In the subcritical regime, the fluid moved along the heated wall almost imperceptibly. In the transition regime, convection cells were generated successfully from the bottom boundary. In the supercritical regime, simultaneous occurrence of the convection cells were observed. These cells then grew clear across the tank. The motions in the fully developed cells were quite different from those observed in tanks with vertical walls. The governing parameter which characterizes the fluid motion is the Rayleigh number defined in terms of the height of rise of a heated fluid particle and the reduced gravity. It was found that the Rayleigh number range for the transitional regime is somewhat higher than that for the vertical heated wall. This was due to the stabilizing effect of the additional hydrostatic pressure normal to the wall. Longitudinal vortices along the inclined wall were observed only in a large bottom cell 6 days into the experiment. At that time, the fluid within the cell was of constant salinity. In no other case longitudinal vortices were observed.

Similarity equations, using an assumed transformation which reduces the partial differential equations to sets of ordinary differential equations, are obtained from the boundary layer and the complete Navier-Stokes equations for the interaction of vortex flows with free stream sink flows and a stationary surface. Solutions to the boundary layer equations for the case of the potential vortex that satisfy the prescribed boundary conditions are shown to be nonexistent using the assumed transformation. Direct numerical integration is used to obtain solutions to the complete Navier-Stokes equations under a potential vortex with equal values of tangential and radial free stream velocities. Solutions are found for Reynolds numbers up to 2.0.

It is shown that a quantized vortex ring in superfluid He4, moving freely towards a flat plate, gives to it an impulse equal to the fluid impulse of the ring, provided only that the radius of the plate is larger than the initial vortex radius but smaller than the distance to the source of the ring. This is in agreement with experiment, in contradiction to two previous theoretical analyses.

The expectation of quantized vortices around the nodes of wavefunctions is formulated in terms of Schroedinger quantum mechanics, and the derivation is shown to depend only on the requirement that the wavefunction be single-valued and continuous. The implications of circulation for the topology of the nodes are discussed, and consideration is given to the nodes and circulation of a smooth wavefunction in D dimensions and to the significance of exceptional nodal points. The theoretical results are illustrated by an example of an infinite field of regularly spaced vortices.

Neutron diffraction experiments on vortex single crystals reveal various information on properties of flux-line lattices. In this paper we present the data obtained from a pure single-crystal niobium rod with its axis parallel to a 〈111〉 direction and a Ginzburg–Landau parameter of 0.9. For the largest achievable lattice parameter in this sample of 1940 A at 4.2°K we found a hexagonal flux-line lattice with an azimuthal mosaic spread of 60′ and a radial one of less than 1.2′. From the integrated reflectivity we determined the absolute values of the form factors at the reciprocal lattice points 10, 11, 20, 21, 30 and 31 From these form factors the local magnetic field distribution was calculated.

A quasi-continuous method is developed for solving thin-wing problems. For the purpose of satisfying the wing boundary conditions, the spanwise vortex distribution is assumed to be stepwise-constant, while the chordwise vortex integral is reduced to a finite sum through a modified trapezoidal rule and the theory of Chebyshev polynomials. Wing-edge and Cauchy singularities are acounted for. The total aerodynamic characteristics are obtained by an appropriate quadrature integration. The two-dimensional results for airfoils without flap deflection reproduce the exact solutions in lift and pitching moment coefficients, the leading edge suction, and the pressure difference at a finite number of points. For a flapped airfoil, the present results are more accurate than those given by the vortex-lattice method. The three-dimensional results also show an improvement over the results of the vortex-lattice method. Extension to nonplanar applications is discussed.

General circulation parameters in the northern hemisphere are calculated using atmospheric thermal structure obtained from Nimbus III SIRS multi-channel radiance information. The thermal structure up to 10 mb is obtained by using a regression technique with thickness between pressure levels as the dependent variable. General circulation parameters calculated on a daily basis include zonal and eddy available potential energy, and zonal and eddy kinetic energy. A second set of calculations is performed using National Meteorological Center (NMC) grid data. A comparison of the two sets of calculations indicates that, although the energies calculated from the SIRS-derived structure underestimate the actual energies, maxima, minima and trends are well identified. An example of mid-stratospheric energy changes during a breakdown of the polar-night vortex is also given.

Axially symmetric thermal convection in a rotating environment with a friction layer is investigated by numerically integrating an appropriate system of equations. The study examines the influence of the friction layer upon dry adiabatic thermal convection and the resulting vortex formation process by making comparative computer runs, with and without addition of the friction layer, for four values of ambient-vertical vorticity. The analysis shows that the addition of a friction layer has a significant influence on the resulting velocity distributions. Friction-induced radial convergence adds to the central updraft of the rising thermal and increases its rate of rise. This friction layer enhancement increases with larger rotation rates and consequently overcomes the correspondingly increased rotational suppression. The vortex height is observed to be greater for the friction layer runs. Additionally, the vortex is observed to have a smoother but less intense profile of tangential velocity with th...

The question of modifying Landau' s equations for a rotating superfluid is considered. Bekarevich and Khalataikov considered the problem from the phenomenological viewpoint and derived these extra terms without recourse to vortex lines, assumiring that the internal energy depends uporin the magnitude of the superfluid vorticity. Within the context of Bekarevich and Khalatnkov's theory the energy equation being invariant is equivalent to saying that their dissipation function R should be invariant. Here, it is proposed to show that this is not the case. (MHR)

A rotating plasma experiment has been built which produces a stable, reproducible, super-magnetosonic flow in highly ionized argon. The rotation is produced by a radial electrical discharge across a vertical magnetic field. A preheat discharge gives uniform starting conditions for the flow. The flow velocity exceeds the limit predicted for argon by the Alfven critical velocity theory. Magnetosonic mach numbers up to 2.0 and flow times of 0.5 msec are obtained. The VORTEX plasma is used to model some aspects of the solar wind flow past bodies in the solar system.

Previously developed numerical computational techniques for obtaining unsteady, nonuniform, viscous plane conduit flows are applied to the calculation of oscillatory flow past rectangular humps placed normal and parallel to the flow direction at a single combination of the frequency and pressure-gradient parameters. These results are compared with previously obtained square hump results under the same conditions. The velocity fields and accompanying vortex development and shedding during the oscillatory cycle are described. Quantitative descriptions of such important occurrences as pressure drops, shear stress variation, and energy loss factors are presented. The quantity most likely to be related to blood cell damage, i.e., the maximum shear stress, reached peak values at the sharp angularity of the hump on the upstream side at times associated with those of peak oscillatory flow. The rectangular hump placed normal to the flow and blocking three-fourths of the cross section caused pressure drops and shear stresses many times larger than the square hump blocking one-half the section.

Stability and bifurcation of Couette flow between concentric rotating cylinders are investigated for the case when the ratios of their radii R and angular velocities Ω are nearly equal to unity. The limiting problem in the linear theory when R → 1 and Ω → 1 is the problem of convection stability in the layer [1]. We find that this is also correct in the case of a nonlinear problem. Below we show that solution of the problem of free convection yields the principal term of the expansion of the secondary flow (Taylor vortex) in the powers of a small parameter δ = R − 1. Therefore the results of [2, 3] can be used to provide, in the present case, a strict justification for the use of the Liapunov-Schmidt method to compute the Taylor vortices. The numerical results obtained for the critical Reynolds' number and the amplitude of the secondary flow provide a good illustration of the asymptotic passage as δ → 0.

The mathematical analysis of the induced velocity components of a screw propeller has occupied the attention of fluid dynamicists for many years. Early approaches to the problem concentrated mainly on momentum and blade element theories, neither of which are capable of providing a detailed knowledge of the flow field, except at the propeller plane or the boundary of the fluid. The early vortex theories also suffered from equally significant, but less severe, limitations and such methods were found to be completely unsatisfactory for many applications. In particular, the development of the ducted propeller for the propulsion of supertankers and VTOL aircraft has demanded a highly detailed knowledge of the flow field, especially around the propeller tip.

We present analytic solutions of the Eilenberger equations for the low-frequency Green's functions corresponding to those quasiparticles whose trajectories pass near the center of an isolated vortex. Using these results we find that for type II superconductors in the temperature rangeT c ≫T≫T c 2/eF the order parameter and the supercurrent near the vortex center increase over a lengthξ 1∼ξ BCS T/T c (ξ BCS=BCS coherence length) and that the density of states at the Fermi surface isN 0 2π 3 ξ BCS 2 /3 In (ξ BCS/ξ 1). The results can be reproduced with the Bogoliubov equations for the elementary excitations. It is shown that this peculiar behavior is connected with the low-lying bound states in the vortex core. ForTť c 2/e F one hasξ 1∼k F −1.

A systematic procedure is presented for determining an equilibrium periodic vortex flow of finite amplitude in a viscous fluid between concentric circular cylinders. The equations of motion for the vortex flow is solved by expanding the solution in power series of the amplitude. The stability of the vortex flow thus determined for a wide gap case is studied to axisymmetric small disturbances. The amplification factor of the disturbance is calculated up to the fifth order approximation. The domain of the wave number of stable mode of perturbation is predicted fairly narrower than by the existing third order theory. This is in more quantitative agreement with experiments on the non-uniqueness of vortex flows.

A method based on the Navier-Stokes equations was applied for analyzing the three-dimensional unsteady flow around oscillating wings. These equations were treated in the complete three-dimensional form considering also the irregular geometry of the wing. The vorticity transport equations were discretized using three-dimensional finite elements from a variational formulation and integrated numerically. At each time step of the numerical integration, the velocity field was determined from a representation of the three-dimensional wing by a system of optimized distribution of discrete vortices superimposed on the flow space. The time-dependent boundary conditions on the oscillating wing were specified as external constraint conditions at each step of the numerical integration. Examples of obtained results describing the three-dimensional unsteady flow around a wing were presented.

The von Karman vortex streets formed in the wakes of vibrating, flexible cables were studied using a hot-wire anemometer. All the experiments took place in the flow regime where the vibration and vortex-shedding frequencies lock together, or synchronize, to control the wake formation. Detailed measurements were made of the vortex formation flow for Reynolds numbers between 230 and 650. As in the case of vibrating cylinders, the formation-region length is dependent on a shedding parameter St* related to the natural Strouhal number and the vibrational conditions. Furthermore, the near wake configuration is found to be dependent on the local amplitude of vibration suggesting that the vibrating cylinder rseults are directly applicable in that region.

The turbulent entrainment mechanism of a swirling plume surrounded by a potential vortex and a swirling jet and a wake in a uniform stream has been discussed. With the assumption of the stream-wise similarity of various flow variables, the integrated conservation equations yield a rate of entrainment function which explicitly shows that the external potential vortex retards the rate of entrainment whereas the positive density gradients across the plume interact with the surrounding vortex to weakly increase the rate of entrainment. In a uniform stream, the swirling jets and wakes show distinctly opposite behavior. An increasing initial swirl delays the dissipation of the swirling wake whereas it promotes the dissipation of a swirling jet. The coaxial uniform free stream always retards the rate of entrainment across the swirling jet.

Since 1970, semi-submerged containers have been investigated in the Ship Research Institute. The knowledge of the hydrodynamic characteristics of a shallow-running flat submerged body has become essential for the hull form design of a large semi-submerged ship. Through the study on separation of the resistance components of three flat submerged bodies and an elliptic body of revolution, following problems are discussed.a) Unexpected large resistance of shallow-running submerged bodies (Fn>0. 25) results from generation of the vortices behind ship.b) The resistance due to the vortices is obtained as viscous resistance depend on Froude number because of the vortices behind submerged bodies mainly originated in the current of free waves.c) Wave making resistance of submerged bodies is seriously affected by the vortices.d) Flow symmetrization between the up and down side of the shallow-running flat submerged bodies significantly contributes to reduce the resistance.

The image method is used to examine the upper-surface-blowing jet-airfoil interaction with Mach number nonuniformity. The formulation represents an extension of the classical incompressible results (Ting and Liu, 1969; Koning, 1963). Some characteristics of the interaction are discussed. The main assumptions are (1) inviscid linear theory, (2) two-dimensional jet, (3) no turbulent mixing, and (4) no airfoil thickness effect. A plane jet with Mach number M sub 2 is assumed to be imbedded in a freestream of Mach number M sub 1. A thin airfoil is placed at a distance h below the lower jet surface. For h = 0, this may represent an idealized configuration with an upper-surface blowing jet.

The authors have used the TDGL equations, derived by Eliashberg (1969) for a superconductor with a low concentration of magnetic impurity ( tau sT>>1) near Tc, to calculate the entropy and energy currents and the flux flow conductivity associated with a slowly moving isolated vortex. The effect of the anomalous term in the TDGLE is important. The transported entropy (Sd) and the conductivity ( sigma f) depend on kappa , tau s and tau 1 (transport collision time). Sd is decreased by the addition of relatively small amounts of magnetic impurity. The agreement between measurement on dirty nonmagnetic alloys and theory (with tau sT= infinity ) is good. sigma f is calculated for tau 1/ tau s > tau s/ tau 1, sigma f/ sigma approximately=Cbeta Hc2/(B); Cbeta approximately=(4 kappa 2)-1 for large kappa , approximately=0.1 for kappa approximately=1/ square root 2.

We present data on the flow of superfluid helium through channels of diameters 2.5 and 5.0 µm. Three modes of flow are observed: (1) flow that can be identified with the thermal nucleation of vorticity, (2) flow described by Δz ∝ (t − t0), where Δz is the gravitational head andt the time, and (3) flow described by (Δz)1/2 ∝ (t − t0). Associated with the third mode, flows appear with a critical velocity close to that of the Feynman prediction for small channels.

A new linked mode of operation has been used to obtain virtually simultaneous measurements by a fully instrumented aircraft and a ground-based high power radar within a train of large amplitude Kelvin-Helmholtz billows in the optically clear atmosphere. Both sets of data were analysed to show the detailed distribution of air motion and turbulence within the billows. The resulting pattern was consistent with a train of vortices each of which was characterized by a vertical wind shear of 6 m s−1 over 100 m, with downward motion of about 1 m s−1 on the downshear side and upward motion of about 1 m s−1 on the upshear side. Between the vortices the vertical shear decreased to almost half the maximum value. The most intense radar echoes occurred within inclined layers connecting the top of one region of maximum vorticity to the bottom of the next. The aircraft, which flew through the upper parts of the billows, encountered turbulence of up to moderate intensity as it penetrated the crests of the billows. Between successive penetrations of the billow crests the aircraft flew close to but just above the region of intense radar echo and encountered smooth air with a slowly varying vertical component of motion.

The frictional forces at a vortex line in rotating liquid helium and the generalizations of Landau's hydrodynamical equations are re-examined in the light of recent experimental evidence. There are no unbalanced forces, contrary to a previous suggestion. The mutual friction data are reinterpreted in terms of these new results.

The magnetic field structure of vortices in the mixed state of a lead-indium alloy and of niobium was probed by stopped ${\ensuremath{\mu}}^{+}$ particles. Fourier transforms of time-dependent spin-precession data contain a peak at the average internal field, which is also the applied field. For the alloy only there is a second peak at the saddle point of the internal field distribution. Motional averaging due to quantum diffusion of the muons at low temperatures can account for these results.

The stability of natural convection flow along inclined plates has been investigated using the methods of linear perturbation theory. Attention has been focussed on three dimensional spatially growing disturbances, which take the form of parallel rows of streamwise oriented vortices. Results are presented of the critical Rayleigh numbers versus plate inclination angle. The theory is in qualitative agreement with experiment. The angle at which the three dimensional instability crosses over into the two dimensional wave instability is fairly well predicted by the theory.

Two dimensional vortex shedding behind an inclined flat plate is simulated by point vortex arrays separating from the both edges. The strengths of point vortices are determined from the Kutta condition. The flow round the body is impulsively started from rest with the angle of attack 89{°}, 75{°}, 60{°}, 45{°} or 30{°}. The calculated values of drag coefficients in the case of the angle of attack 89{°} have their marked peak initially and decrease gradually to the values of about one and remain at this stage for some time and then suddenly increase to the values of 4 or 5 and finally oscillate between about 2 and 4. This final stage corresponds to the flow with vortex shedding of Karman's type. Similar results are obtained in other cases.