It is shown that, according to the linearized theory of long waves in a rotating, unbounded sea, if there is a discontinuity in depth along a straight line separating two regions each of uniform depth, then wave motions may exist which are propagated along the discontinuity and whose amplitude falls off exponentially to either side. Thus the discontinuity acts as a kind of wave-guide.

The period of the waves is always greater than the inertial period. The wave period also exceeds the period of Kelvin waves in the deeper medium. As the ratio of the depth tends to infinity, the wave period tends to the inertial period or to the Kelvin wave period, whichever is the greater. On the other hand as the wavelength decreases (within the limits of shallow-water theory) so the waves tend to the non-divergent planetary waves found recently by Rhines.

In an infinite ocean of uniform depth free waves with period greater than a pendulum-day cannot normally be propagated without attenuation (if the Coriolis parameter is constant). But non-uniformities of depth provide a means whereby such energy may be channelled over great distances with little attenuation.

It is suggested that a gradually diminishing discontinuity will act as a chromatograph, each position along the discontinuity being marked by waves of a particular period.

A study has been made of the varicose instability of an axisymmetrical jet with a velocity distribution radially uniform at the nozzle mouth except for a laminar boundary layer at the wall. The evolutionary phenomena of instability, such as the rolling up of the cylindrical vortex layer into ring vortices, the coalescence of ring vortex pairs, and the eventual disintegration into turbulent eddies, have been investigated as a function of the Reynolds number using smoke photography, stroboscopic observation, and the light-scatter technique.

Emphasis has been placed on the wavelength with maximum growth rate. The jet is highly sensitive to sound and the effects of several types of acoustic excitation, including pure tones, have been determined.

In this paper it is shown how a previous theory for treating shocks moving into fluid at rest (Whitham 1957, 1959) can be changed to a form suitable for treating propagation into a steady uniform flow. It is necessary only to transform the original theory to a moving frame but the details are not trivial.

An earlier treatment of the diffraction of a shock wave advancing into a region of uniform flow, based on Chisnell's (1965) extension of Whitham's (1957) rule for shock diffraction, is corrected for an algebraic error and then compared with an analogous treatment based on the more recent extension derived by Whitham (1968). The basis for comparison is the pressure just behind a shock wave that is diffracted by a thin wedge travelling at supersonic speed. The approximation provided by Whitham's extension is both simpler than, and typically superior to, that provided by Chisnell's extension (although the numerical differences are small in the Mach-number régime considered).

The laminar flow of a wall jet over a curved surface is considered. A unique similarity solution is obtained for both concave and convex surfaces when the local radius of curvature is proportional to x3/4. This solution satisfies a similar invariant condition to the one derived by Glauert for the wall jet over a plane surface. The variation of the shape of the velocity profile, the skin friction, and the surface pressure as a function of curvature is given.

Air flowing over a liquid surface encounters an increased resistance if waves are present. The relation of this increased resistance to the properties of the waves has been studied. Air and a liquid flowed co-currently in an enclosed channel which is 12 in. wide and 1 in. high and which is long enough so that flow in the air and the liquid and the interfacial structure are fully developed. The drag on interfaces with three-dimensional wave structures was found to increase with the square of the gas velocity and to depend more on the height of the waves than on other parameters characterizing the interface. The ratio of the equivalent sand roughness to the root-mean-square of the fluctuations in the height of the liquid film is approximately equal to 3 √2. The velocity profiles in the gas were found to be different from what has been reported for flows over sand roughened surfaces.

This paper is concerned with the steady, symmetric, two-dimensional flow of a viscous, incompressible fluid issuing from an orifice and falling freely under gravity. A Reynolds number is defined and considered to be small. Due to the apparent intractability of the problem in the neighbourhood of the orifice, interest is confined to the flow region below the orifice, where the jet is bounded by two free streamlines. It is assumed that the influence of the orifice conditions will decay exponentially, and so the asymptotic solutions sought have no dependence upon the nature of the flow at the orifice. In the region just downstream of the orifice, it is expected that the inertia effects will be of secondary importance. Accordingly the Stokes solution is sought and a perturbation scheme is developed from it to take account of the inertia effects. It was found possible only to express the Stokes solution and its perturbations in the form of co-ordinate expansions. This perturbation scheme is found to be singular far downstream due to the increasing importance of the inertia effects. Far downstream the jet is expected to be very thin and the velocity and stress variations across it to be small. These assumptions are used as a basis in deriving an asymptotic expansion for small Reynolds numbers, which is valid far downstream. This expansion also has the appearance of being valid very far downstream, even for Reynolds numbers which are not necessarily small. The method of matched asymptotic expansions is used to link the asymptotic solutions in the two regions. An extension of the method deriving the expansion far downstream, to cover the case of an axially-symmetric jet, is given in an appendix.

The Navier–Stokes equations are used to study the unsteady structure of a weak shock wave reflecting from a plane wall. Both an adiabatic and an isothermal wall are considered. Incident and reflected shock structures are found by expanding the dependent variables in asymptotic series in the shock strength; the first-order terms are shown to satisfy an equation analogous to Burgers equation. The structure of the wave during reflexion is obtained from an expansion in which the first-order terms satisfy the acoustic equations. The isothermal wall boundary condition requires the introduction of a thermal layer adjacent to the wall. In this case viscosity and convection play a role secondary to the wall temperature boundary condition in determining the structure of the reflected wave. The presentation is simplified by introducing a generalized Burgers equation that gives the same first-order results as the Navier–Stokes equations. Correct second-order results are obtained from this equation simply by applying a correction to the result for the temperature.

The rate of damping of perturbations on a shock wave reflected from a perturbed flat wall was measured in a shock tube. Incident shock wave Mach numbers of 1·45 and 1·09 in air together with sinusoidal and Gaussian wall perturbations were employed. These measurements were compared with a modified form of a linearized theory due to Zaidel (1960). The linearization was performed about the basic solution of a plane shock wave reflected normally from a flat wall.

The rate of decay and the frequency and phase of oscillations agreed very well with the theoretical predictions; the amplitudes of the oscillations were some-what larger than predicted. The reflected shock shape was initially in good agreement with theory, but higher frequency perturbations on the reflected shock front caused deviations from the predicted shape after the shock front had travelled about one wall-wavelength away from the wall.

The generally satisfactory agreement between theory and experiment supports the use of linearized analysis in predicting shock wave stability.

Measurements of the distributions of pressure on a bluff flat plate (fence) have been correlated with the characteristics of the smooth-wall boundary layer in which it is immersed. For zero pressure-gradient flows, correlations are obtained for the variation of form drag with plate height h which are analogous in form to the ‘law of the wall’ and the ‘velocity-defect law’ for the boundary-layer velocity profile. The data for adverse pressure-gradient flows is suggestive of a ‘law of the wake’ type correlation. Pressures on the upstream face of the bluff-plate are determined by a wall-similarity law, even for h/δ > 1, and are independent of the pressure-gradient history of the flow; the separation induced upstream is apparently of the Stratford-Townsend type. The effects of the history of the boundary layer are manifested only in the flow in the rear separation bubble, and then only for h/δ > ½. The base pressure is also sensitive to free-stream pressure gradients downstream of the bluff-plate. The relative extent of upstream influence of the bluff-plate on the boundary layer is found to increase rapidly as h/δ decreases. One set of measurements of the mean flow field is also presented.

The injection of a two-dimensional jet into a uniform stream is considered, the fluids being assumed inviscid and incompressible. When the total head of the jet is much larger than that of the uniform flow, the motion is characterized by two disparate length scales, and uniformly valid asymptotic solutions can be found by the method of matched expansions. Inner and outer expansions are developed for the jet and the external flow. The first-order outer solution in the jet is the usual thin jet approximation which fails in the neighbourhood of the jet exit except for 90° injection, when it is uniformly valid. The basic non-linearity introduced by the pressure condition along the vortex sheet separating the jet from the external flow appears as a non-linear boundary condition for the first-order outer solution in the external flow. A novel feature of the analysis is the necessity of imposing a logarithmic singularity as an ‘inner’ boundary condition for the outer solution in the external flow. The first-order fluid speed and streamline deflection angle are shown to be given correctly to O(1) uniformly in the external flow (for all injection angles) by the first-order outer solution.

The stability is considered of the flow with velocity components \[ \{0,\Omega r[1+O(\epsilon^2)],\;2\epsilon\Omega r_0f(r/r_0)\} \] (where f(x) is a function of order one) in cylindrical polar co-ordinates (r, ϕ, z), bounded by the rigid cylinders r/r0 = x1 and r/r0 = 1 (0 [les ] x1 < 1). When ε [Lt ] 1, the flow is shown to be unstable to non-axisymmetric inviscid disturbances of sufficiently large axial wavelength. The case of Poiseuille flow in a rotating pipe is considered in more detail, and the growth rate of the most rapidly growing disturbance is found to be 2εΩ.

An experimental investigation on reverse transition from turbulent to laminar flow in a two-dimensional channel was carried out. The reverse transition occurred when Reynolds number of an initially turbulent flow was reduced below a certain value by widening the duct in the lateral direction. The experiments were conducted at Reynolds numbers of 625, 865, 980 and 1250 based on half the height of the channel and the average of the mean velocity. At all these Reynolds numbers the initially turbulent mean velocity profiles tend to become parabolic. The longitudinal and vertical velocity fluctuations ($\overline{u^{\prime 2}}$ and $\overline{v^{\prime 2}}$) averaged over the height of the channel decrease exponentially with distance downstream, but $\overline{u^{\prime}v^{\prime}} $ tends to become zero at a reasonably well-defined point. During reverse transition $\overline{u^{\prime}}\overline{v^{\prime}}/\sqrt{\overline{u^{\prime 2}}}\sqrt{\overline{v^{\prime 2}}}$ also decreases as the flow moves downstream and Lissajous figures taken with u’ and v’ signals confirm this trend. There is approximate similarly between $\overline{u^{\prime 2}} $ profiles if the value of $\overline{u^{\prime 2}_{\max}} $ and the distance from the wall at which it occurs are taken as the reference scales. The spectrum of $\overline{u^{\prime 2}} $ is almost similar at all stations and the non-dimensional spectrum is exponential in wave-number. All the turbulent quantities, when plotted in appropriate co-ordinates, indicate that there is a definite critical Reynolds number of 1400±50 for reverse transition.