The transition of a laminar two-dimensional wake is studied experimentally to establish the role of amplitude and phase modulations in the spectral-broadening and energy-redistribution process. Multiple instability modes fo and fi are triggered by acoustic excitation. The spectrum of the fluctuating velocity field formed by the growing and interacting instabilities shows the development of a complicated side- band structure reminiscent of amplitude- and phase-modulated waves. Digital com- plex demodulation techniques are used to obtain quantitative measurements of local instantaneous amplitude and phase modulations. Measurements of the modulation time traces, their modulation indices, the lag between phase and amplitude modula- tions, and the power spectra of the modulations are presented. Our results show that both phase and amplitude modulation play a role in the transition process. The dominant modulation frequency of both amplitude and phase is that of the difference mode fv = f1−f0 produced by the interaction of the two excited instabilities. Phase modulation becomes progressively more important as transition proceeds down- stream, and seems to play the dominant role in the spectral-broadening and energy- redistribution process. Measurements of the bicoherency spectrum indicate that sideband structures, and accompanying modulations, are produced by nonlinear interactions between the low-frequency difference mode and higher-frequency in- stability modes. Some limited measurements indicate that finite-amplitude induced nonlinear dispersion effects ω(k, a2) may provide a physical mechanism by which amplitude modulations generated by nonlinear interactions can induce simultaneous phase modulations.

The shape of a rising bubble, or of a falling drop, in an incompressible viscous fluid is computed numerically, omitting the condition on the tangential traction at the bubble or drop surface. When the bubble is sufficiently distorted, its top is found to be spherical and its bottom is found to be rather flat. Then the radius of its upper surface is in fair agreement with the formula of Davis & Taylor (1950). This distortion occurs when the effect of gravity is large while that of surface tension is small. When the effect of surface tension is large, the bubble is nearly a sphere.

The shape is found, together with the flow of the surrounding fluid, by assuming that both are steady and axially symmetric, with the Reynolds number being large. The flow is taken to be a potential flow. The boundary condition on the normal component of normal stress, including the viscous stress, is satisfied, but not that on the tangential component. The problem is converted into an integro-differential set of equations, reduced to a set of algebraic equations by a difference method, and solved by Newton's method together with parameter variation.

Energy & ability theory is employed to study the finite-amplitude stability of a viscous incompressible fluid occupying the space between a pair of concentric cylinders when the inner-cylinder angular velocity varies linearly with time. For the case with a fixed outer cylinder and increasing inner-cylinder speed, we find an enhancement of stability, consistent with a linear-theory result due to Eagles. When the inner-cylinder speed decreases, we find an initially decreased stability bound, indicating the possibility of hysteresis, while, if the inner cylinder is allowed to reverse direction and linearly increase in speed, we find significant stability enhancement.

We derive a path-integral representation for the effective diffusion function of a passive scalar field. We use it to calculate the long-time effective diffusivity in Gaussian turbulence in the near-Markovian limit. Our results confirm the negative effect of vorticity predicted by previous discussions. They also demonstrate that the helicity of the turbulence when present may be as important an influence as the vorticity.

We derive a path-integral representation for the effective diffusion function of a passive scalar field. We use it to calculate the long-time effective diffusivity in Gaussian turbulence in the near-Markovian limit. Our results confirm the negative effect of vorticity predicted by previous discussions. They also demonstrate that the helicity of the turbulence when present may be as important an influence as the vorticity.

The effect of periodic two-dimensional excitation on the development of a turbulent mixing region was studied experimentally. Controlled oscillations of variable ampli- tude and frequency were applied at the initiation of mixing between two parallel air streams. The frequency of forcing was at least an order of magnitude lower than the initial instability frequency of the flow in order to test its effect far downstream. The effect of the velocity difference between the streams was also investigated in this experiment. A typical Reynolds number based on the velocity difference and the momentum thickness of the shear layer was l04.

It was determined that the spreading rate of the mixing layer is sensitive to periodic surging even if the latter is so small that it does not contribute to the initial energy of the fluctuations. Oscillations at very small amplitudes tend to increase the spreading rate of the flow by enhancing the amalgamation of neighbouring eddies, but at higher amplitudes the flow resonates with the imposed oscillation. The resonance region can extend over a significant fraction of the test section depending on the Strouhal number and a dimensionless velocity-difference parameter. The flow in the resonance region consists of a single array of large, quasi-two-dimensional vortex lumps, which do not interact with one another. The exponential shape of the mean-velocity distribution is not affected in this region, but the spreading rate of the flow with increasing distance downstream is inhibited. The Reynolds stress in this region changes sign, indicating that energy is extracted from the turbulence to the mean motion; the intensity of the spanwise fluctuations is also reduced, suggesting that the flow tends to become more two-dimensional.

Amalgamation of large coherent eddies is resumed beyond the resonance region, but the flow is not universally similar. There are many indications suggesting that the large eddies in the turbulent mixing layer at fairly large Re are governed by an inviscid instability.

Exact solutions are used to contrast the behaviour of vertical dye streaks and of point discharge in the deep ocean. After a few inertial periods the growth of the contaminant cloud in both cases can be modelled by a diffusion equation with slowly varying coefficients. However, the apparent long-term diffusivity for the dye streak can be as much as three times as large as the horizontal diffusivity for the point discharge, depending upon the precise time of release. This anomalously large rate of spreading for the dye streak can persist for several weeks until sufficient time has elapsed for vertical diffusion over the vertical lengthscale of the dominant inertial- frequency waves.

It is demonstrated experimentally that the influence of viscosity on the transition condition in pseudosteady flow is very significant. A mechanism is proposed for this effect, which explains the features of the observed behaviour. In particular, an experimental method of finding the inviscid transition condition, by extrapolation to infinite Reynolds number, gives excellent agreement with the calcwlatcd inviscid sonic criterion. It is thought that this provides the explanation for the usual persistence of regular reflection beyond the sonic: condition.

It is shown experimentally that, in steady flow, transition to Mach reflection occurs at the von Neumann condition in the strong shock range (Mach numbers from 2.8 to 5). This criterion applies with both increasing and decreasing shock angle, so that the hysteresis effect predicted by Hornung, Oertel & Sandeman (1979) could not be observed. However, evidence of the effect is shown to be displayed in an unsteady experiment of Henderson & Lozzi (1979).

Displacements of wind waves in the laboratory were measured with a laser displacement gauge, a recent,ly developed, optical, non-intrusive sensor, which avoids the meniscus effect's that severely limit the frequency response of conventional thin- wire gauges. The new gauge is a digital device, which has a maximum frequency response of 2.5 kHz. Its spatial resolution, which depends on the field of view, is t,ypically 0.016 cm for a 4 cm field of view. The wind-wave displacements were measured at several fetches for three wind speeds. Wave-variance spectra derived from these measurements indicate the presence of a quasi-equilibrium spectrum in the capillary-wave regime. The quasi-equilibrium spectrum follows an $f^{-\frac{7}{3}}$ power law that has been predicted on dimensional grounds. The spect'ral density increases with increasing wind speed from 4 to 10 m/s but is independent of the fetch from 3 to 5 m. In addition, the capillary-wave spectrum is practically unchanged when a, relatively long but. low-amplitude mechanical wave is superposed onto the wind- generat'cd waves.

Measurements have been made of the Couette flow in the annular space between concentric cylinders with a radius ratio of 1.5, the outer cylinder being held stationary and the inner one rotated at speeds to give Taylor numbers in the range 1.0 × 104−2.3 × l06 times the critical value for first instability of the steady viscous flow. Mean velocities have been measured both with Pitot tubes and with linearized hot-wire anemometers, and turbulent intensities and stresses, frequency spectra and space-time correlations have been obtained using single hot-wire anemometers of X-form and linear arrays of eight single-wire anemometers. For Taylor-number ratios to the critical number less than 3 × l05, the most prominent feature of the flow is a system of toroidal eddies, encircling the inner cylinder and uniformly spaced in the axial direction with nearly the separation of the Taylor vortices of the viscous instability. They are superimposed on a background of irregular motion and, except within the thin wall layers, the toroidal eddies contribute more to the total intensity. With increase of rotation speed, the toroidal eddies lose their regularity, and they cannot be clearly distinguished at Taylor-number ratios beyond 5 × l05.

The change of flow type from quasi-regular toroidal to fully irregular turbulent takes place over an extensive range of Taylor-number ratio centred near 3 × l05, and it may be linked with changes in the thin wall layers that separate the flow boundaries from the central region of nearly constant circulation. For ratios over 5 × l05, an appreciable part of the wall layers is comparatively unaffected by flow curvature and has a logarithmic distribution of mean velocity similar to that found in channel flows. It is suggested that the motion in the wall layers changes from a set of Gortler vortices characteristic of curved-wall flow to the more irregular motion found on plane walls, causing the toroidal eddies to break into sections of length ranging from a considerable fraction of the flow perimeter to nearly the separation of the cylinders. Changes in the frequency spectra of the radial and azimuthal velocit'y fluctuations are consistent with such a change.

The problem of determining both the steady and unsteady axially symmetrical motion of a viscous incompressible fluid outside a fixed sphere when the fluid at large distances rotates as a solid body is considered. It is assumed that the Reynolds number for the motion is so large that the boundary-layer equations may be assumed to hold. The steady-state boundary-layer equations are solved using backward- forward differencing and the terminal solutions at the equator and the pole of the sphere are generatedas part ofthe numerical procedure. To check that this steady-state solution can be approached from an unsteady situation, the case of a sphere that is initially rotating with the same constant angular velocity as the fluid and is then impulsively brought to rest is investigated. I n this case the motion is governed by a coupled set of three nonlinear time-dependent partial differential equations, which are solved by employing the semi-analytical method of series truncation to reduce the number of independent variables by one and then solving by numerical methods a finite set of partial differential equations in one space variable and time. The physical properties of the flow are calculated as functions of the time and compared with the known solution at small times and the steady-state solution.

When light rotating fluid spreads over heavier fluid in the vicinity of a vertical wall (coast) a boundary jet of width Λ forms, the leading edge or nose of which propagates with speed ĉ along the coast. A certain fraction 8 of the boundary transport is not carried by the nose but is deflected backwards (detrained) and left behind the propagating nose. Theoretical and experimental results for Λ,ĉ, and δ are given for a quasi-equilibrium (constant-ĉ) regime. Over longer time intervals the laboratory observations suggest that the nose slows down and stagnates, whereupon the trailing flow separates from the coast and an intermittent boundary current forms. These processes may be relevant to the mixing of oceanic coastal currents and the maintenance of the mean current.

In §1 a brief discussion of the general problem of self-excited acoustic oscillations within fixed boundaries is given. I n §2 a second-order analysis is developed for the special case of rectangular cavities. A nonlinear wave equation is derived for essentially arbitrary boundary conditions. The analysis can be extended to other cavity geometries provided that the first-order solutions can be expressed in closed form. Various applications of the analysis are discussed in §3. It turns out that two-dimensional problems of self-excited oscillations generally lead to nonlinear equations containing terms with a time lag. It is anticipated that the time lag (rather than viscous effects or sound radiation) represents the key to a fundamental understanding of the character of the oscillations and the variety of modes appearing in self-excited resonators.

The analysis by Busse (1975) of preferred patterns of convection in spherical shells is extended to include the case of odd degrees l of spherical harmonics. In the general part of the paper only the property of spherical symmetry of the basic state is used. The results are thus applicable to all bifurcation problems with spherical symmetry. Except in the case 1 = 1 a pattern degeneracy of the linear problem exists, which is partly removed by the solvability conditions that are generated when nonlinear terms are taken into account as perturbations. In each of the cases 1 considered so far, at least 1 physically different solutions have been found. The preferred solution among I existing ones is determined for 1≥2 by a stability analysis. In the case l = 3 emphasized in this paper the axisymmetric solution is found to be always unstable, and the solution of tetrahedronal symmetry appears to be generally preferred. The latter result is rigorously established in the special case of a thin layer with nearly insulating boundaries treated in the second part of the paper.

The flow in a rotating annular cylinder, of finite depth, is examined when the Rossby number Ro is O(E½), where E is the Ekman number, and when there is a topography of height O(E½) on the base of the container. The flow, relative to the rigid axial rotation, is forced by differential rotation of the lid and as it moves over the topography the streamlines are deflected parallel to the bottom surface. This induces O(1) velocity variations near the axial walls of the annulus to which the boundary layers there, of thickness O(E¼), respond. For sufficiently large values of a parameter γ ∝Ro/E½ the skin friction can vanish within these layers, with some similaritits to boundary-layer separation in a non-rotating fluid. In this study the interior flow, with horizontal viscous diffusion neglected, is calculated and used to provide a boundary condition for the, E¼ layer flow. Once λ exceeds a finite critical value a singularity is encountered in the boundary layer corresponding to flow separation from the wall. This demonstrates that E¼ layers in a rotating fluid, which for Ro = 0 have little direct influence on the interior flow, can modify the gross properties of the flow for non-zero Rossby numbers, a conclusion also reached by Walker & Stewartson (1972) in a different context.

A steady streaming displacement of fluid elements is observed to occur during oscillatory flow through a Y-shaped tube bifurcation model at Womersley and Reynolds numbers that can exist in the human bronchial tree. The cause of the displacement is the effect of the asymmetric geometry on the oscillating velocity vector field. The steady streaming displacement is greatest for fluid elements that experience the highest velocity through the bifurcation junction. The maximum displacement observed increases with Re and a up to a Re of about 100 and a of about 5, after which a levelling off and gradual decline occur. Photographs of low-Re and low-a: cxpcrimcnts show the effects of secondary components of the steady-streaming displacement field which contribute to a complex circulation.

Visual examination of simultaneous temperature traces from a rake of cold wires placed across a turbulent boundary layer had enabled the identification of coherent temperature fronts. An X-wirelcold-wire arrangement was used simultaneously with the rake to provide measurements of the velocity fluctuations u (longitudinal) and v (normal) and the temperature fluctuation θ. Conditional averages of u, v, θ and products uv, uθ, vθ were obtained by application of conditional techniques based on the detection of the temperature fronts using information obtained at only one point in space. These averages, obtained at various positions across the layer, have been compared with those obtained when the rake was used to detect the fronts. The comparison has indicated that none of the one-point detection techniques is in good quantitative agreement with the rake detection technique, the largest correspondence between the rake technique and any ofthe other one-point techniques being only 51 yo. With the exception of the hole technique used in conjunction with the quadrant decomposition analysis, conditional averages obtained from one-point techniques are in reasonable qualitative agreement with those deduced using the rake.

Measurements of’ the mean-pressure distribution and the Strouhal number on a smooth circular cylinder, circular cylinders with distributed roughness, and circular cylinders with narrow roughness strips were made over a Reynolds-number range 4.0 × l04 to 1.7 × l06 in a uniform flow. A successful high-Reynolds-number (trans- critical) simulation for a smooth circular cylinder is obtained using a smooth circular cylinder with roughness strips. High-Reynolds-number simulation can only be obtained by roughness strips and not by distributed roughness. A similarity parameter correlating the pressure distributions on circular cylinders with distributed roughness in the supercritical range is presented. The same parameter can also be applicable to the drag coefficients of spheres with distributed roughness.

Experiments on the vortex-shedding frequencies of various rectangular cylinders were conducted in a wind tunnel and in a water tank. The results show how Strouhal number varies with a width-to-height ratio of the cylinders in the range of Reynolds number between 70 and 2 × l04. There is found to exist a certain range of Reynolds number for the cylinders with the width-to-height ratios of 2 and 3 where flow pattern abruptly changes with a sudden discontinuity in Strouhal number. The changes in flow pattern corresponding to the discontinuity of Strouhal number have been confirmed by means of measurements of velocity distribution and flow visualization. These data are compared with those of other investigators. The experimental results have been found to show a good agreement with those of numerical calculations.