The wake of a circular cylinder is investigated near the oscillation threshold by means of a laser probe. Above the threshold the transient regime is studied and described by a Stuart-Landau law (already found to be relevant in explaining free-oscillating regimes). Below the critical point, impulse and resonant regimes are examined, so the coefficients of the Stuart-Landau equation are determined.

Moreover, in the supercritical regime, the behaviour of the (externally forced) oscillating system is described, varying parameters such as threshold deviation, forcing frequency and amplitude. The different zones of entrainment and desynchronization are given for simple or harmonic frequency.

The onset of periodic behaviour in two-dimensional laminar flow past bodies of various shapes is examined by means of finite-element simulations. The transition from steady to periodic flow is marked by a Hopf bifurcation, which we locate by solving an appropriate extended set of steady-state equations. The bodies considered are a circular cylinder, triangular prisms of various shapes, and flat plates and elliptical cylinders aligned over a range of angles to the direction of flow. Our results for the circular cylinder are in good agreement with experimental observations and with the results of time-dependent calculations.

Two experiments were performed to study the response of turbulent boundary layers to sudden changes in surface curvature and pressure gradient. In the first experiment, the behaviour of a boundary layer negotiating a two-dimensional curved hill was examined. Prior to separating on the leeward side of the hill, the layer experienced a short region of concave surface curvature, followed by a prolonged region of convex surface curvature. The corresponding pressure gradient changed from adverse to favourable, and back to adverse. In the second experiment, the flow over a symmetrical wing was studied. This wing had the same profile as the hill with a very similar pressure distribution. The obvious difference between the two experiments was the use of leading and trailing edge plates in the hill flow. The results show that an internal layer forms in the flow over the curved hill, and that this internal layer displays many similarities to the boundary layer observed on the free wing. The internal layer grows as an independent boundary layer beneath a turbulent free-shear layer, and as it develops it establishes its own wall (inner) and wake (outer) regions. The perturbation responsible for initiating the growth of the internal boundary layer seems to be an abrupt change in surface curvature. Once formed, the internal boundary layer dictates the skin-friction distribution and the process of separation over the hill. The effect of the perturbation in wall curvature appears to be different from that due to prolonged convex curvature in that the former affects the flow in the vicinity of the wall instantly, while the latter affects the flow far away from the wall only after the flow turns through some angle. Physical explanations are offered for the qualitative difference between the effects of mild and strong convex curvature, and for the saturated behaviour observed in strongly curved flows. Finally, the results are compared with the behaviour of wind flow over terrestrial hills. In both cases, the internal layer dominates the flow behaviour, even though the scaling laws for the flows over actual hills are not obeyed in the present case. A qualitative comparison reveals that the present internal layer is thicker than that reported in meteorological flows. This appears to be due to the effect of curvature, which perturbs the wake region of the internal layer in the present hill flow, while in meteorological studies the effect of curvature is generally small enough to be neglected.

Experimental results are presented that show the existence of organized structures in a compressible, turbulent boundary layer. Results were obtained using arrays of hot wires and wall pressure transducers in a Mach-3 zero-pressure-gradient boundary layer. The VITA method of conditional sampling was used to deduce average pressure events at the wall and mass flux events throughout the boundary layer; these results show qualitative similarity to those found in incompressible flows. By conditioning upon the middle hot wire from a three-wire probe, evidence is found suggesting that structures exist of a height comparable with the boundary-layer thickness. Furthermore, two-point conditional sampling was used to show that an average pressure event could be extracted by conditioning upon mass-flux events. From this procedure we found that the structures maintain their shape as they travel downstream and also that their spanwise extent is very limited.

The inferred angle from correlations between two hot wires, and between a hot wire and a wall-pressure transducer, indicate that the average structure is inclined at approximately 45° for a large part of the boundary layer. This result agrees well with structures observed in schlieren photographs of supersonic boundary layers. Measurements of the instantaneous angle show a wide distribution of structure angles, and the general behaviour of the large-scale structures is consistent with the hairpin loop model of wall turbulence.

A linear, stationary internal-wave model is used to determine the vertical propagation of gravity waves produced by submerged moving bodies in a stratified fluid. The turbulent wake generated by the moving object is taken into account parameterically. If the stability of the stratification above the body increases (i.e. the body is under a thermocline) resonance effects occur. Large-amplitude internal waves are generated and they give rise to strong divergence fields at the water surface. These will persist far downstream of the object and are potentially detectable. Limitations of the linear model and comparisons with experimental results available in the literature are discussed.

Wind-tunnel measurements of lift, drag and wake velocity spectra were carried out under (tonal) acoustic excitation for a smooth airfoil in the chord-Reynolds-number (Rec) range of 4 × 104−1.4 × 105. The data are supported by smoke-wire flowvisualization pictures. Small-amplitude excitation in a wide, low-frequency range is found to eliminate laminar separation that otherwise degrades the airfoil performance at low Rec near the design angle of attack. Excitation at high frequencies, scaling as $U_{\infty}^{\frac{3}{2}}$, eliminates a pre-stall, periodic shedding of large-scale vortices; U∞ is the free-stream velocity. Significant improvement in lift is also achieved during post-stall, but with large-amplitude excitation. Wind-tunnel resonances strongly influence the results, especially in cases requiring large amplitudes. It is shown that large transverse velocity fluctuations, induced near the airfoil by specific cross-resonance modes, lead to the most effective separation control; resonances inducing only large-amplitude pressure fluctuations are demonstrated to be less effective.

The problem of calculating nonlinear two-dimensional free-surface potential flow about a circular cylinder rising to a free surface is solved numerically. The deeplysubmerged circular cylinder is accelerated smoothly from rest to a uniform vertical velocity. A boundary/integral-equation method is used to obtain free-surface elevations and streamlines about the rising cylinder for several final speeds. Results, including pressure forces, are compared with a cylinder rising to a rigid wall and a cylinder moving in an infinite fluid.

We present three-dimensional numerical simulations of convection in a low-Prandtlnumber fluid confined between two infinite horizontal bounding surfaces maintained at constant temperatures. We consider the case of free-slip boundary conditions for a fluid of Prandtl number Pr = 0.2 and that of rigid boundary conditions with Pr = 0.025. In the former situation, we observe stationary, periodic, biperiodic and chaotic regimes as the Rayleigh number is increased. In the later situation, the dynamics involves very different characteristic times, and only stationary and time-periodic solutions have been simulated. Convergence to the later regime may occur after a long transient where the amplitude of the oscillation is slowly modulated.

A transformation technique is used to solve the problem of steady free-surface flow of an ideal fluid over a semi-infinite step in the bottom. Application of the exact free-surface condition results in a nonlinear integro-differential equation for the free-surface angle and solutions of this equation are dependent on step height and Froude number. Linearized solutions, based upon small step height are presented and indicate that the nature of the free surface formed depends on whether the upstream flow is subcritical or supercritical. As the step height is increased, solutions to the exact nonlinear equations are obtained using the predictions of the linear theory, or possibly a previous nonlinear solution, as an initial estimate.

The linear stability of the flow of an incompressible viscous fluid through a curved pipe of circular cross-section is considered. There is a sinusoidal pressure gradient, with zero mean, acting down the pipe. The flow is shown to be unstable to a Taylor-Görtler mode of instability, with vortices aligned with the basic flow first appearing at the outer bend of the pipe when a critical value of the Taylor number is exceeded. A WKBJ perturbation solution is constructed and the form of the vortex amplitude is determined. This solution is found to break down in the vicinity of the pipe's outer bend, and an inner solution is presented to overcome this. The solution is determined by identifying a saddle point in the complex plane of the cross-sectional angle coordinate. This leads to an eigenvalue problem for the Taylor number, for fixed wavenumber and cross-sectional angle coordinate, which in turn leads to the determination of the critical Taylor number above which instability sets in.

Buoyancy-driven recirculation in a liquid-filled rectangular cavity cooled from above is shown to be locally modulated by an unstable thermal layer at the surface. Interferometric observations suggest that fluctuations that occur in a plume descending through the upper liquid layers are of the type described by Howard (1964) and by Krishnamurti & Howard (1981). Temperature measurements across the surface layer are in reasonable agreement with the diffusive heat-conduction model, but indicate that near the plume the fluid is cooler than elsewhere. Quasi-steady measurements of the velocity distribution in the upper regions of the cavity were made using multiple-exposure laser speckle velocimetry. Interrogation of the specklegrams with a Young's fringe technique yields a velocity-vector field of about two thousand elements. These data are used to calculate the corresponding velocity components and estimates of the vorticity distribution. The results compare favourably with measurements recorded directly from Fourier filtering methods.

The growth, breakdown, and transition to turbulence of counter-rotating streamwise vortices, generated via a Görtler instability mechanism, was used to experimentally model the eddy structures found in transitional and turbulent flat-plate boundary layers. The naturally occurring vortices have been studied using smoke-wire visualization and multiple-probe hot-wire rakes. Results show that low-speed regions are formed between the vortices as low-momentum fluid is removed away from the wall. The low-speed regions grow in the normal direction faster than a nominally Blasius boundary layer and create strongly inflexional normal and spanwise profiles of the streamwise velocity component. Instability oscillations develop on these unstable profiles that scale with the local shear-layer thickness and velocity difference. Contrary to expectations however, the spatial scales of the temporal velocity fluctuations correlate better with the velocity gradient in the spanwise direction than with the normal velocity gradient. The nonlinear growth of the oscillations is quite rapid and breakdown into turbulence occurs within a short timescale.

Experiments were performed to investigate the supersonic flow of a turbulent boundary layer over a number of compression-corner models. Upstream of each corner, the free-stream Mach number was 2.9 and the incoming boundary layer was typical of a two-dimensional, zero-pressure-gradient, high-Reynolds-number flow. Three different corner angles were used, namely 8°, 16° and 20°, and at the highest angle the interaction was strong enough to cause separation. Each flow was investigated using normal and inclined hot wires, and measurements of the longitudinal mass-flux fluctuations and the mass-weighted turbulent shear stress are presented. The behaviour of the kinematic and Reynolds stresses, deduced from the mass-weighted quantities by applying Morkovin's ‘strong Reynolds analogy’, is also considered. In all three flow cases, the interaction dramatically amplifies the turbulent stresses, and the amplification increases with increasing turning angle. Different stress components are amplified by different amounts, however, and the structure parameter $-\overline{u^{\prime}v^{\prime}}/\overline{u^{\prime 2}}$ changes significantly through the interaction. Perhaps more importantly, the nature of these changes depends on the strength of the interaction. This result is rather unexpected, and it is believed to be due mainly to the unsteadiness of the shock system. It is suggested that the apparently random motion of the shock system affects the normal stresses more than the shearing stresses, and, since the unsteadiness increases with corner angle, the effect on the turbulence structure also becomes more pronounced.

Nearly all theoretical work in geophysical fluid dynamics is based on approximate forms of the equations of motion, but the best ground-rules for deriving such approximate forms are not clear. Traditionally, scale analysis and global energy conservation have been the guiding principles. The existence of analogues of Lagrangian potential vorticity conservation has been seen as at least aesthetically desirable, but consequent improvements in practical accuracy have not often been demonstrated. A simple case study is here offered in order to illuminate these issues. The Type 1 quasi-geostrophic model (QG1) is adopted as a reference formulation and several approximations to it are examined. They are formally accurate to zeroth or first-order in a Burger number B, but may include some higher-order terms and may imply analogues of global energy conservation and Lagrangian potential vorticity conservation. The approximate forms are all characterized by exclusion of the external Rossby mode, and each is related to a certain geostrophic formulation which is familiar in dynamical oceanography. The various approximations are assessed by examining their behaviour in three test problems which may be treated analytically: finite-amplitude internal Rossby wave propagation, zonal flow stability criteria and linearized internal free waves on baroclinic zonal flows. Two of the problems yield support for the hypothesis that the practical accuracy of an approximation may be improved by including higher-order terms in such a way that a potential vorticity conservation analogue is implied. The validity of this hypothesis in the QG1 case could be further investigated by solving more complicated test problems. Its general applicability cannot of course be claimed on the basis of a single case study; but the results obtained here afford evidence in its favour.

In the Czochralski process a single crystal is grown from liquid in a crucible. An axial magnetic field suppresses turbulence in the melt and thus reduces the density of microdefects in the crystal. This paper treats the melt motion due to buoyancy and thermocapillarity. The magnitude of this motion decreases roughly like B−2, as the magnetic field strength B is increased. The separate circulations due to buoyancy and thermocapillarity are roughly equal at an early stage of growing a crystal. However the circulation due to thermocapillarity is nearly independent of the melt depth, while that due to buoyancy is proportional to the square of the depth. Therefore as the crystal grows and the melt depth decreases, thermocapillarity becomes progressively more dominant. When the heat flux into the melt is used to define the characteristic temperature difference and velocity, the stream functions are rather insensitive to changes in the thermal boundary conditions at the free surface and at the crucible bottom, provided the overall heat balance of the system is correctly estimated. This is fortunate because there is considerable uncertainty about these boundary conditions. The exception to this insensitivity is that the melt motion due to thermocapillarity is sensitive to changes in the amount of heat lost through the part of the free surface adjacent to the crystal.

Velocity measurements are presented for turbulent oscillatory flow over rough beds. Two components of velocity were measured with a laser-Doppler anemometer and the rough beds consisted of a single layer of sand, gravel or pebbles on a flat surface. Turbulence intensities showed significant variation during the course of the cycle. Maximum turbulence intensity propagated out from the bed at a more or less constant velocity for all beds. Variation of time-mean turbulence intensity with height was qualitatively similar to that observed in steady flows. Reynolds stress showed several interesting features. Near the bed, maximum Reynolds stress was in phase with one of the two peaks of turbulence intensity but further out it was in phase with the other, i.e. the phase of maximum Reynolds stress showed a 180° phase shift at a certain height above the bed. A related effect was seen in the time-mean eddy viscosity which was negative near the bed but positive further out. It is suggested that these effects are caused by the jets of fluid associated with vortex formation and ejection in oscillatory flow over rough beds. Maximum Reynolds stress was also significantly less than the horizontal force per unit area of bed obtained from the momentum integral. Eddy viscosity and mixing length were found to vary significantly during the course of the cycle. Variation with height of time-mean values of these variables showed similar trends, except in the near-bed region, to those observed in steady flow but derived values of the Kármán constant were significantly lower. Non-dimensional defect velocity appeared to show dependence on a/ks as well as on y/δ in the outer layer away from the bed, even at high Reynolds numbers.

The hydrodynamic problem of a submerged spheroid in waves is analysed based on linearized potential theory. An analytic formulation is derived and demonstrated by considering the problem of a stationary spheroid in head or following seas. Tabulated numerical results are obtained for a spheroid whose major axis is six times the minor axis, submerged at a depth twice the minor axis. Figures for many other cases are also provided. It is suggested that the present method can be extended to the problem of oscillating bodies at forward speed.

This work treats the vertically stratified system of two homogeneous fluid layers confined between horizontal infinite rotating disks which rotate steadily about a common vertical axis. Allowance is made for uniform injection of fluid at either disk. With appropriate restrictions on disk rotational speeds and injection rates a flat interface is possible, and the problem admits similarity solutions to the Navier-Stokes equations of the Kármán-Bödewadt-Batchelor variety. This type of flow allows for a uniformly accessible surface of interphase mass and heat transfer at the two-fluid interface, and, with that as the primary motivation, the present work provides exploratory numerical solutions of the above equations for both corotation and counter-rotation combined with injection.

A linearized theory is given for the case of nearly rigid rotation, with explicit analytical results for the large-Reynolds-number boundary-layer limit. Also, we offer a theoretical discussion of the inviscid limit for arbitrary rotation and injection rates. Based on the type of Euler-cell solutions identified in previous work, we derive the remarkably simple formula \[ \rho_1\omega_1^2\cot^2\frac{\omega_1d_1}{V_1} = \rho_2\omega^2_2\cot^2\frac{\omega_2d_2}{V_2} \] connecting densities ρ, depths d, rotation speeds ω and injection velocities V.

Sample calculations and comparisons are given for property ratios typical of water-kerosene layers. In this case, the linearized theory works exceedingly well for corotation with small injectional Rossby numbers V/ωd. The simple inviscid theory cited above shows excellent agreement with the numerical computations for Reynolds numbers greater than 500 and for Rossby numbers > 1/π, corresponding to strong blowing in the inviscid regime. The larger-wavelength inviscid cell structure appears to provide the kind of stagnation-flow pattern essential to the application envisaged.

It is shown that for a sudden uniform discharge at x = 0, t = 0 in a bounded shear flow, the asymptotic concentration distribution at moderately large times can be well approximated by the tilted Gaussian \[ c=\frac{\overline{q}}{(2\pi\langle\sigma^2\rangle)^{\frac{1}{2}}}\exp\left(-\frac{(x-\overline{u}t-g_0(y,z)+2\alpha D)^2}{2\langle \sigma^2\rangle }\right), \] with \begin{eqnarray*} & \langle \sigma^2\rangle = 2Dt + 2\alpha D(x-\overline{u}t)-3\overline{g^2_0}-4\alpha^2 D^2,\\ & \overline{g_0} = 0,\quad D = \overline{ug_0},\quad \alpha = \frac{(\overline{u-\overline{u}})g^2_0}{2D^2}. \end{eqnarray*} Here u(y, z) is the velocity profile, g0(y, z) the centroid displacement function, and the overbars denote cross-sectional averaging. The tilt parameter α makes the concentration distribution suitably skew. The effectiveness of this simple formula is demonstrated for two-layer flows and for plane Poiseuille flow.

In this paper we use existing perturbation theories for partial differential equations that are approximately integrable. These methods are then used to study the BBM equations, considered as a perturbation of the KdV equations. We show, analytically, that the amplitudes of two solitary waves are altered after interaction. The change in amplitude is also calculated.