A symposium on the subject ‘Concentrated vortex motions in fluids’ was held at the University of Michigan, Ann Arbor, Michigan, from 6 to 11 July 1964. The symposium was organized by the International Union for Theoretical and Applied Mechanics, and participation was restricted to about 150 people invited on the basis of their active interest in the subject. The author was the secretary of the international committee which planned the meeting, and has prepared the following account of the scientific developments of the symposium in order to make them widely available.[dagger] There will be no other publication of the proceedings of the symposium.Editors’ note. This novel method of reporting the proceedings of a symposium to the scientific community at large deserves consideration by the organizers of other meetings. There are obvious objections to the common plan of publishing in one volume the papers presented at a meeting: being an isolated volume it is not certain to be acquired by libraries; the papers are published at a time and under conditions not known in advance to authors; and papers are not usually subjected to the scrutiny of referees before publication. However, some kind of widely available report on an important meeting seems to be desirable, particularly when participation is by invitation only. Dr Küchemann's detailed account of the papers presented and the discussion that followed them should enable readers to see whether there were any developments at the Michigan Symposium of direct interest to them, and any such developments can be followed up with the aid of the very complete list of references at the end of the report.The Editors will be glad to consider other similar reports of conferences concerned with some aspect of fluid mechanics for publication in the Journal. Further details of the work described at the symposium, and of related previous work, are given in the references quoted.

The local and total mass flow through a circular orifice and a two-dimensional slit with large upstream to downstream pressure ratio is calculated for large Knudsen number. The solution is obtained by performing one iteration on an integral form of the kinetic equation, starting with the free molecular solution for the distribution function. A relaxation model is used to describe the inter-molecular collisions. There is found to be a variation of approximately 40% in the local mass flow perturbation (from the free molecular value) across the orifice. For the slit the variations are smaller when the Reynolds number is less than 0·3, but are comparable for greater values. This Reynolds number, defined by $Re = 2 R \rho (kT|m)\frac {{1}}{{2}}/\m$ with R the radius of the orifice or half-width of the slit and with the flow field quantities evaluated far upstream, appears to be a less ambiguous quantitative parameter than the inverse Knudsen number. The ratio of total mass flow to the free molecular value is given by [1+0·083 Re+o(Re)] for the orifice and [1−0·057 Re ln Re+0·055 Re+o(Re)] for the slit. There is agreement within experimental error between the theoretical results and available experimental data for the circular orifice when Re < 1.

In a backward boundary layer the fluid has, in the mathematical model, been flowing along a solid wall through an infinite distance. The co-ordinate distance x along the boundary is measured upstream, and the velocity U of the flow outside the boundary layer is taken as negative. The main application is to flow in converging passages.

The existence of similar solutions is considered, with emphasis on the correct asymptotic behaviour for large values of the stretched co-ordinate normal to the wall. This emphasis is shown to be necessary in considering backward boundary layers.

For two-dimensional flow in converging passages the requirement that a boundary layer should be possible for vanishingly small viscosity with a potential core flow is shown to lead directly to Hamel's spirals as the shape of the boundary streamlines.

Flow in axisymmetric converging passages is considered. For flow in a cone there is no limit as the viscosity tends to zero, and no potential core flow with a boundary layer is possible. The nature of a solution of the Navier-Stokes equations for laminar flow is considered.

The steady, axisymmetric, converging motion of a viscous incompressible fluid inside an infinite right circular cone is considered. It is shown that the exact solution of the Navier-Stokes equation for the stream function Ψ is of the form Ψ(r,θ) = AF(rv/A,θ), where (r, θ) are spherical polar co-ordinates chosen so r = 0 is the apex and θ = 0 is the axis of the cone, 2πA is the volumetric flow rate, and v the kinematic viscosity of the fluid. Asymptotic expansions of the stream function are found for large and small rv/A.

For large rv/A, Stokes's method for slow motions is generalized to obtain a complete asymptotic expansion. Except for cones of special angles, all terms in this expansion may theoretically be found.

For small rv/A a solution is constructed in two parts, namely, an inner expansion which starts from boundary-layer type equations as well as the no-slip condition at the wall, and an outer expansion in unstretched variables rv/A and cosθ which satisfies the boundary conditions at the axis of the cone. The condition that the inner solution merge with the outer solution with an exponentially small error requires an outer solution near the apex which is not potential sink flow, as might perhaps have been expected from the solution for two-dimensional flow in a wedge. The simplest outer flow satisfying the requirement is a vortex motion. Complete inner and outer expansions are developed and it is shown that they contain only six undetermined constants which must be determined by joining this solution numerically to the Stokes solution upstream. The inclusion of logarithmic terms in these expansions has not been found necessary.

In addition to the random processes described as horizontal turbulence, there are certain more regular processes which may contribute to horizontal mixing. One of these occurs in a shearing current, where the vertical gradient of velocity combined with vertical turbulent mixing leads to an effective diffusion in the horizontal direction. It is shown that this effect occurs in an alternating flow, such as a tidal current, as well as in a steady flow. In estuaries and coastal waters horizontal mixing by the shear effect may be associated with tidal currents, density currents or wind-driven currents. In each case the effective coefficient of horizontal diffusion, Kx or Ky, is inversely proportional to the coefficient of vertical eddy diffusion Kz. The occurrence of a stable gradient of density therefore increases the effective horizontal mixing very considerably. Results obtained from observations in the Mersey estuary and Irish Sea are compared with theoretical estimates of these effects.

The fluid motion associated with puffs (strongly turbulent masses of fluid moving through surroundings with which they mix readily) is considered, for cases in which any buoyancy force acts in the direction of gross motion, the surroundings are unstratified, the internal and external densities are approximately equal, and the motion of the surroundings is directly associated with the motion of the puff. It is assumed that the size of any one such puff is directly proportional to the distance travelled, i.e. that r = z/n.

It is shown that d{ρra(dz/dt)}/dt = CaMag, where t is time, ρ is the density of the surrounding fluid, g is the gravitational acceleration, Ca is a numerical constant, and a = 3 or 2 in the cases of axial or planar mean symmetry. M3 is the mass excess, and M2 is the mass excess per unit length.

Previous experiments with buoyant puffs having zero initial momentum (i.e. experiments with thermals) confirm these equations and give the values C2 [bumpe ] 0·33 and C3 = 0·27. New experiments with non-buoyant puffs having considerable initial momentum also confirm the equations, with the same values for C2 and C3. These results support the view that the turbulence inside thermals is primarily maintained through their mean motion rather than directly by gravitational instability.

The values reported by Willmarth & Wooldridge (1962) for the root-mean-square wall pressure and the power spectra of the wall pressure are in error. G. M. Corcos and M. K. Bull pointed out to me, in April 1963, that the area beneath the power spectra of figure 4 does not agree with the values of $\sqrt {(p^2)}| q \infty$ and $\sqrt {(p^2)}| \tau_w$ reported in table 3 and in §4·1. The measurements for the wall pressure at high speed, U∞ = 206ft./sec, were fortunately recorded on magnetic tape. It was therefore possible to check the measurements for that case.

A descriptive experimental study was made in both air and water of the temporally periodic motion that occurs in the vortex whistle and cyclone separator. The motion can be described in terms of an oscillator that derives its energy from hydrodynamic instability of the steady swirling flow and whose frequency is determined by an angular velocity characteristic of this steady flow. The relevant dynamical parameters are the Rossby number and Reynolds number for the steady flow with the addition of the Strouhal number for the time-dependent flow. The results of this study were compared with the vortex breakdown phenomenon over swept-back wings. Breakdown can be described in the same terms as for the other two cases and it appears that all three motions are basically the same.

The instability of a two-dimensional jet with respect to three-dimensional disturbances and that of an axially symmetric jet with respect to azimuthally periodic disturbances are studied, for the inviscid flow of a compressible fluid. In both cases the undisturbed velocity is assumed to be uniform in the jet. It is shown analytically that a two-dimensional jet is unstable under small disturbances, either subsonic or supersonic. There is no upper limit in Mach number, as was found for a plane vortex sheet, above which the flow is completely stable. Numerical calculations for the eigenvalues for both the two-dimensional jet and the axially symmetric jet have been made. The results indicate that the increase of Mach number tends to stabilize the flow. For the two-dimensional jet, the larger the angle between the direction of wave propagation and that of the main flow, the more the flow will be destabilized. For the axially symmetric jet, the flow is more unstable under azimuthally periodic disturbances than under rotationally symmetric ones, at small wave number.

A mathematical analysis of the behaviour of small disturbances to Poiseuille flow in a circular pipe can be tackled in two different ways. In the first of these, the disturbance velocity is expressed as a combination of terms, each of which represents a disturbance of a fixed wavelength that decays with time. In the second approach, each term which contributes to the disturbance velocity represents a disturbance of a fixed frequency that decays with downstream distance. The latter method is given more prominence in this paper, since, in experiments in which disturbances are introduced into the flow in a controlled manner, a constant frequency-generating device is commonly used.

In the main, it is assumed not only that the Reynolds number of the Poiseuille flow is large, but that the frequency, f, of the disturbance is large compared with the frequency, v/a2, where a is the radius of the pipe and v the kinematic viscosity of the fluid. The mathematical problem is then of the singular perturbation type, and the disturbances with the smallest damping rates are confined to thin layers. A simple, but crude, analysis shows among other things that the radius a t which the disturbance velocity is a maximum is roughly that at which the velocity of the Poiseuille flow is equal to the frequency, f, times the disturbance wavelength. Eigenfunctions are found precisely for the two limiting cases in which, as fa2/v tends to infinity, the disturbance becomes confined to a thin layer situated (a) near the centre of the pipe, and (b) near the wall. The eigenfunctions are presented graphically in such a way that immediate comparison can be made with some of Leite's experimental results. Good agreement is found. A real disturbance, however, is made up of several modes, each of which is damped at a different rate with increasing downstream distance. Possible changes in the form and apparent damping rate of a disturbance are discussed in terms of particular case.

Next, an asymptotic procedure is carried out, which proves to give a good approximation to the eigenvalues and eigenfunctions over a wide range of conditions. From the eigenvalue equation so obtained it is possible to calculate the wave-speed and damping rate for each mode as a function of the non-dimensional frequency, fa2/v, and the Reynolds number. For simplicity, the calculations are carried out for the case in which the Reynolds number is infinite, so that the eigenvalues depend only on fa2/v. For each mode it is found that the damping rate is an increasing function of the frequency for high frequencies, but the frequency is decreased the damping rate approaches a limiting value. These limiting values can be quite small for the Reynolds numbers, R, at which experiments are normally carried out, and for some low-frequency disturbances the distance required for the disturbance amplitude to be reduced by half is as much as R/1000 diameters.

Consideration is given to the flow of an elastico-viscous liquid in a curved pipe under a pressure gradient. The cross-section of the pipe is an ellipse, the axes of which are in an arbitraty position with respect to the radius of curvature of the pipe. The method of solution is an extension of that used by Dean (1927) and by Thomas & Walters (1963) in their consideration of flow through a curved pipe of circular cross-section.

It is shown that the liquid elements move along the pipe in two sets of spirals. When the axes of the ellipse are in an asymmetrical position the streamline projections on the cross-section of the pipe are strongly dependent on the elasticity in the liquid. This is not so when the axes are in a symmetrical position. However, in this case, the pitch of the spirals is strongly dependent upon the elasticity of the liquid.

It is also shown that the flux through the pipe is independent of the curvature of the pipe to first order in the curvature.

A highly rarefied gas, initially in equilibrium, is compressed by an infinite plane piston. The resulting flow is considered when the gas is unbounded and also when the gas is bounded by a second stationary wall. In the first case, the density ahead of the piston is analysed for specularly and diffusely reflecting surfaces. In the second case, the average temperature of the gas is found as a function of the wall separation for specularly reflecting surfaces. This (non-equilibrium) temperature is compared with that generated in the corresponding continuum flow which involves multiple shock wave reflexion. It is shown that, for a given average density gradient, the free molecule temperatures are very much higher than the continuum temperatures.