Several years ago Dean (1948) showed how to solve the linearized potential problem of water waves passing over a submerged circular cylinder. He discovered the remarkable fact that there is no reflexion from the cylinder; the transmitted waves have the same amplitude as the incident waves, but they suffer a phase shift in passing the cylinder. Soon after the publication of Dean's paper, Ursell (1950) investigated the problem anew. He placed the solution on a rigorous basis, supplied a uniqueness proof, and developed a form of the solution with which it was reasonable to perform calculations.

An array of surface hot-film shear-stress sensors together with a traversing hot-wire probe is used to identify the conditional structure associated with a large-scale skin-friction event in a high-Reynolds-number turbulent boundary layer. It is found that the large-scale skin-friction events convect at a velocity that is much faster than the local mean in the near-wall region (the convection velocity for large-scale skin-friction fluctuations is found to be close to the local mean at the midpoint of the logarithmic region). Instantaneous shear-stress data indicate the presence of large-scale structures at the wall that are comparable in scale and arrangement to the superstructure events that have been previously observed to populate the logarithmic regions of turbulent boundary layers. Conditional averages of streamwise velocity computed based on a low skin-friction footprint at the wall offer a wider three-dimensional view of the average superstructure event. These events consist of highly elongated forward-leaning low-speed structures, flanked on either side by high-speed events of similar general form. An analysis of small-scale energy associated with these large-scale events reveals that the small-scale velocity fluctuations are attenuated near the wall and upstream of a low skin-friction event, while downstream and above the low skin-friction event, the fluctuations are significantly amplified. In general, it is observed that the attenuation and amplification of the small-scale energy seems to approximately align with large-scale regions of streamwise acceleration and deceleration, respectively. Further conditional averaging based on streamwise skin-friction gradients confirms this observation. A conditioning scheme to detect the presence of meandering large-scale structures is also proposed. The large-scale meandering events are shown to be a possible source of the strong streamwise velocity gradients, and as such play a significant role in modulating the small-scale motions.

Oblique detonation waves are simulated to study the evolution of their morphology as gasdynamic and chemical parameters are varied. Although two kinds of transition pattern have previously been observed, specifically an abrupt transition and a smooth one, the determining factors for the transition pattern are still unclear. Numerical results show that the transition pattern is influenced by the inflow Mach number, chemical activation energy and heat release. Despite the fact that these parameters were known to influence the detonation instability, the transition pattern variation cannot be predicted according to the instability criterion. In this study, the difference in the oblique shock and detonation angles is proposed as the criterion to determine the transition pattern with the aid of shock-polar analysis. It is found that the smooth transition will appear when the angle difference is small, while the abrupt transition will occur when the difference is large. The shift from the smooth transition to the abrupt transition occurs when the angle difference is about $1{5}^{\ensuremath{\circ} } $–$1{8}^{\ensuremath{\circ} } $. The previously proposed criterion using the characteristic time ratio is also examined and compared with the present angle difference criterion, and the latter is proved to provide better results.

Electrochemical techniques have been used to measure the velocity gradients at the surface of a cylinder for Reynolds numbers from 5 × 103 to 105. This is a companion study to that already reported by Dimopoulos & Hanratty (1968) for a Reynolds number range of 60–360. The use of a specially designed sandwich electrode enabled the direction of the velocity gradient as well as its magnitude to be measured. Of particular interest is the region of definite length after separation where the velocity gradient is negative, followed by an ill-defined region where the flow moves in the positive direction. Still farther downstream the direction of flow changes with time in an irregular fashion. The measured velocity gradients prior to separation are described satisfactorily by boundary-layer theory. The presence of a splitter plate in the rear of the cylinder eliminates periodic fluctuations in the wake and has a significant effect on the boundary layer prior to separation.

Periodic oscillations are found in a one-dimensional model of thermal convection. The model consists of a fluid-filled tube bent into rectangular shape and standing in a vertical plane. The fluid is heated at the centre of the lower horizontal segment and cooled at the centre of the upper horizontal segment. When a certain parameter exceeds unity, a periodic motion of the fluid is found in which the flow is always in the same direction but in which the speed varies. Inertia is unimportant for this oscillation, which depends upon the interplay between frictional and buoyancy forces.

We describe a series of laboratory experiments in which aqueous salt solutions were cooled and solidified from above. These solutions serve as model systems of metallic castings, magma chambers and sea ice. As the solutions freeze they form a matrix of ice crystals and interstitial brine, called a mushy layer. The brine initially remains confined to the mushy layer. Convection of brine from the interior of the mushy layer begins abruptly once the depth of the layer exceeds a critical value. The principal path for brine expelled from the mushy layer is through ‘brine channels’, vertical channels of essentially zero solid fraction, which are commonly observed in sea ice and metallic castings. By varying the initial and boundary conditions in the experiments, we have been able to determine the parameters controlling the critical depth of the mushy layer. The results are consistent with the hypothesis that brine expulsion is initially determined by a critical Rayleigh number for the mushy layer. The convection of salty fluid out of the mushy layer allows additional solidification within it, which increases the solid fraction. We present the first measurements of the temporal evolution of the solid fraction within a laboratory simulation of growing sea ice. We show how the additional growth of ice within the layer affects its rate of growth.

A super-hydrophobic array of grooves containing trapped gas (stripes) has the potential to greatly reduce drag and enhance mixing phenomena in microfluidic devices. Recent work has focused on idealized cases of stick-perfect slip stripes. Here, we analyse the experimentally more relevant situation of a pressure-driven flow past striped slip-stick surfaces with arbitrary local slip at the gas sectors. We derive approximate formulas for maximal (longitudinal) and minimal (transverse) directional effective slip lengths that are in a good agreement with the exact numerical solution for any surface slip fraction. By representing eigenvalues of the slip length tensor, we obtain the effective slip for any orientation of stripes with respect to the mean flow. Our results imply that flow past stripes is controlled by the ratio of the local slip length to texture size. In the case of a large (compared to the texture period) slip at the gas areas, surface anisotropy leads to a tensorial effective slip, by attaining the values predicted earlier for a perfect local slip. Both effective slip lengths and anisotropy of the flow decrease when local slip becomes of the order of texture period. In the case of a small slip, we predict simple surface-averaged isotropic flows (independent of orientation).

An analytical model is presented for sound radiation from a semi-infinite unflanged annular duct. The duct carries a jet which issues into a uniform mean flow while an inner cylindrical centre body extends downstream from the duct exit. This geometrical arrangement forms an idealized representation of a turbofan exhaust where noise propagates along the annular bypass duct, refracts through the external bypass stream and radiates to the far field. The instability wave of the vortex sheet and its interaction with the acoustic field are accounted for in an exact way in the current solution. Efficient numerical procedures are presented for evaluating near-field and far-field solutions, and these are used as the basis for a parametric study to illustrate the effect of varying the hub–tip ratio, and the ratio of jet velocity to external flow velocity. Since the ‘Kutta’ condition can be turned on and off in the current solution, this capability is used to assess the effect of vortex shedding on noise radiation. Far-field directivity patterns are presented for single modes and also for a multi-mode ‘broadband’ source model in which all cut-on modes are assumed to be present with equal modal power. Good agreement is found between analytical solutions and experimental data. Near-field pressure maps of the acoustic and instability portions of the solution are generated for selected tones.

The motion of buoyant transient cavities with non-condensible contents is investigated numerically using a boundary-integral method. The bubble contents are described by an adiabatic gas law. Motion is considered in the neighbourhood of a rigid boundary, in an axisymmetric geometry. We investigate whether the non-condensible contents will resist the formation of jets. It is found that jets form upon collapse and, in general, completely penetrate the bubble before it rebounds, but circumstances are identified under which the non-spherical bubble will rebound prior to this occurrence. In these cases the bulk of the jet growth occurs upon rebound. Furthermore, the interaction between the buoyancy force causing jet formation upwards, and the Bjerknes attraction of the rigid boundary causing jet formation towards it, is investigated and general principles discussed which allow the behaviour to be interpreted. The concept of the Kelvin impulse is utilized.

In our previous qualitative paper, it was shown that when a vertical ice surface melts into a stable salinity gradient, the melt water spreads out into the interior in a series of nearly horizontal layers. The experiments reported here are aimed at quantifying this effect, which could be of some importance in the application to melting icebergs. Experiments have also been carried out with heated and cooled vertical walls at larger Rayleigh numbers R than those of previous experiments.

The main result is that for most of our experiments there is no significant difference between these three cases when properly scaled. The layer thickness over a wide range of R is described to within the experimental accuracy by \[ h=0.65 [\rho(T_w,S_{\infty}) - \rho(T_{\infty},S_{\infty})]\left/\frac{d\rho}{dz}\right., \] where the term in brackets is the horizontal buoyancy difference evaluated at the mean salinity and dp/dz is the vertical density gradient due to salinity. In the case of ice melting into warm water the effective wall temperature Tw is approximately 0°C, whereas in colder water the freezing point depression must be taken explicitly into account. A detailed examination of the vertically flowing inner melt water layer in both homogeneous and salinity stratified cases has been made. This layer and the melt water which is mixed outwards from it into the turbulent horizontal layers have little effect on the outer flow. At high R and large external salinity, however, mixing can reduce the effective salinity at the inner edge of the horizontal layers, and thus the layer scale. A puzzling feature is the relatively weak dependence of layer scale on local salinity, though the vigour of convection and the rate of melting are greater where the salinity is high.

The direct application of our results to oceanographic situations predicts layer scales under typical summer conditions of order tens of metres in the Antarctic and of order metres in the Arctic. More measurements will be needed, especially close to icebergs, before the application of these ideas to polar regions can be properly evaluated.

It is shown that a random superposition of inertial waves in a rotating conducting fluid can act as a dynamo, i.e. can systematically transfer energy to a magnetic field which has no source other than electric currents within the fluid. Dynamo action occurs provided the statistical properties of the velocity field lack reflexional symmetry, and this occurs when conditions are such that there is a net energy flux (positive or negative) in the direction of the rotation vector Ω.

If the magnetic field grows from an infinitesimal level, then the mode of maximum growth rate dominates before the back-reaction associated with the Lorentz force becomes significant. This mode is first determined, and then the back-reaction associated with it alone is analysed. It is shown that the magnetic energy grows exponentially during the stage when the Lorentz forces are negligible, then reaches a maximum depending on the values of the parameters \[ R_m = u_0 l/\lambda,\quad Q = \Omega l^2/\lambda, \] (u0 = initial r.m.s. velocity, l = length scale characteristic of the velocity field, λ = magnetic diffusivity) and ultimately decays as t−1 (equation (5.15)). This decay is coupled with a decay of the velocity field due to ohmic dissipation, and it occurs because there is no external source of energy for the fluid motion.

A theory for fully developed turbulent pipe and channel flows is proposed which extends the classical analysis to include the effects of finite Reynolds number. The proper scaling for these flows at finite Reynolds number is developed from dimensional and physical considerations using the Reynolds-averaged Navier–Stokes equations. In the limit of infinite Reynolds number, these reduce to the familiar law of the wall and velocity deficit law respectively.

The fact that both scaled profiles describe the entire flow for finite values of Reynolds number but reduce to inner and outer profiles is used to determine their functional forms in the ‘overlap’ region which both retain in the limit. This overlap region corresponds to the constant, Reynolds shear stress region (30 < y+ < 0.1R+ approximately, where R+ = u*R/v). The profiles in this overlap region are logarithmic, but in the variable y + a where a is an offset. Unlike the classical theory, the additive parameters, Bi, Bo, and log coefficient, 1/κ, depend on R+. They are asymptotically constant, however, and are linked by a constraint equation. The corresponding friction law is also logarithmic and entirely determined by the velocity profile parameters, or vice versa.

It is also argued that there exists a mesolayer near the bottom of the overlap region approximately bounded by 30 < y+ < 300 where there is not the necessary scale separation between the energy and dissipation ranges for inertially dominated turbulence. As a consequence, the Reynolds stress and mean flow retain a Reynolds number dependence, even though the terms explicitly containing the viscosity are negligible in the single-point Reynolds-averaged equations. A simple turbulence model shows that the offset parameter a accounts for the mesolayer, and because of it a logarithmic behaviour in y applies only beyond y+ > 300, well outside where it has commonly been sought.

The experimental data from the superpipe experiment and DNS of channel flow are carefully examined and shown to be in excellent agreement with the new theory over the entire range 1.8 × 102 < R+ < 5.3 × 105. The Reynolds number dependence of all the parameters and the friction law can be determined from the single empirical function, H = A/(ln R+)α for α > 0, just as for boundary layers. The Reynolds number dependence of the parameters diminishes very slowly with increasing Reynolds number, and the asymptotic behaviour is reached only when R+ [Gt ] 105.

The linear instability and breakdown to turbulence induced by an isolated roughness element in a boundary layer at Mach $2. 5$, over an isothermal flat plate with laminar adiabatic wall temperature, have been analysed by means of direct numerical simulations, aided by spatial BiGlobal and three-dimensional parabolized (PSE-3D) stability analyses. It is important to understand transition in this flow regime since the process can be slower than in incompressible flow and is crucial to prediction of local heat loads on next-generation flight vehicles. The results show that the roughness element, with a height of the order of the boundary layer displacement thickness, generates a highly unstable wake, which is composed of a low-velocity streak surrounded by a three-dimensional high-shear layer and is able to sustain the rapid growth of a number of instability modes. The most unstable of these modes are associated with varicose or sinuous deformations of the low-velocity streak; they are a consequence of the instability developing in the three-dimensional shear layer as a whole (the varicose mode) or in the lateral shear layers (the sinuous mode). The most unstable wake mode is of the varicose type and grows on average ${\sim }17\hspace{0.167em} \% $ faster than the most unstable sinuous mode and ${\sim }30$ times faster than the most unstable boundary layer mode occurring in the absence of a roughness element. Due to the high growth-rates registered in the presence of the roughness element, an amplification factor of $N= 9$ is reached within ${\sim }50$ roughness heights from the roughness trailing edge. The independently performed Navier–Stokes, spatial BiGlobal and PSE-3D stability results are in excellent agreement with each other, validating the use of simplified theories for roughness-induced transition involving wake instabilities. Following the linear stages of the laminar–turbulent transition process, the roll-up of the three-dimensional shear layer leads to the formation of a wedge of turbulence, which spreads laterally at a rate similar to that observed in the case of compressible turbulent spots for the same Mach number.

High-Reynolds-number data in turbulent boundary layers are analysed to examine statistical moments of streamwise velocity fluctuations ${u}^{\prime } $. Prior work has shown that the variance of ${u}^{\prime } $ exhibits logarithmic behaviour with distance to the surface, within an inertial sublayer. Here we extend these observations to even-order moments. We show that the $2p$-order moments, raised to the power $1/ p, $ also follow logarithmic behaviour according to $\langle \mathop{({u}^{\prime + } ){}^{2p} \rangle }\nolimits ^{1/ p} = {B}_{p} - {A}_{p} \ln (z/ \delta )$, where ${u}^{\prime + } $ is the velocity fluctuation normalized by the friction velocity, $\delta $ is an outer length scale and ${B}_{p} $ are non-universal constants. The slopes ${A}_{p} $ in the logarithmic region appear quite insensitive to Reynolds number, consistent with universal behaviour for wall-bounded flows. The slopes differ from predictions that assume Gaussian statistics, and instead are consistent with sub-Gaussian behaviour.

We consider a small droplet of water sitting on top of a heated superhydrophobic surface. A toroidal convection pattern develops in which fluid is observed to rise along the surface of the spherical droplet and to accelerate downwards in the interior towards the liquid/solid contact point. The internal dynamics arise due to the presence of a vertical temperature gradient; this leads to a gradient in surface tension which in turn drives fluid away from the contact point along the interface. We develop a solution to this thermocapillary-driven Marangoni flow analytically in terms of streamfunctions. Quantitative comparisons between analytical and experimental results, as well as effective heat transfer coefficients, are presented.

Motivated in part by the problem of large-scale lateral turbulent heat transport in the Earth's atmosphere and oceans, and in part by the problem of turbulent transport itself, we seek to better understand the transport of a passive tracer advected by various types of fully developed two-dimensional turbulence. The types of turbulence considered correspond to various relationships between the streamfunction and the advected field. Each type of turbulence considered possesses two quadratic invariants and each can develop an inverse cascade. These cascades can be modified or halted, for example, by friction, a background vorticity gradient or a mean temperature gradient. We focus on three physically realizable cases: classical two-dimensional turbulence, surface quasi-geostrophic turbulence, and shallow-water quasi-geostrophic turbulence at scales large compared to the radius of deformation. In each model we assume that tracer variance is maintained by a large-scale mean tracer gradient while turbulent energy is produced at small scales via random forcing, and dissipated by linear drag. We predict the spectral shapes, eddy scales and equilibrated energies resulting from the inverse cascades, and use the expected velocity and length scales to predict integrated tracer fluxes.

When linear drag halts the cascade, the resulting diffusivities are decreasing functions of the drag coefficient, but with different dependences for each case. When β is significant, we find a clear distinction between the tracer mixing scale, which depends on β but is nearly independent of drag, and the energy-containing (or jet) scale, set by a combination of the drag coefficient and β. Our predictions are tested via high- resolution spectral simulations. We find in all cases that the passive scalar is diffused down-gradient with a diffusion coefficient that is well-predicted from estimates of mixing length and velocity scale obtained from turbulence phenomenology.

The first Hopf bifurcation of the infinite cylinder wake is analysed theoretically and by direct simulation. It is shown that a decomposition into a series of harmonics is a convenient theoretical and practical tool for this investigation. Two basic properties of the instability allowing the use and truncation of the series of harmonics are identified: the lock-in of frequencies in the flow and separation of the rapid timescale of the periodicity from the slow timescale of the non-periodic behaviour. The Landau model is investigated under weak assumptions allowing strong nonlinearities and transition to saturation of amplitudes. It is found to be rather well satisfied locally at a fixed position of the flow until saturation. It is shown, however, that no truncated expansion into a series of powers of amplitude can account correctly for this fact. The validity of the local Landau model is found to be related to the variation of the form of the unstable mode substantially slower than its amplification. Physically relevant characteristics of the Hopf bifurcation under the assumption of separation of three timescales – those of the periodicity, amplification and deformation of the mode – are suggested.

Critical speeds for the onset of Taylor vortices and for the later development of wavy vortices have been determined from torque measurements and visual observations on concentric cylinders of radius ratios R1/R2 = 0·894–0·954 for a range of values of the clearance c and length L: c/R1 = 0·0478–0·119 and L/c = 1–107. Effectively zero variation of the Taylor critical speed with annulus length was observed. The speed at the onset of wavy vortices was found to increase considerably as the annulus length was reduced and theoretical predictions are realistic only for L/c values exceeding say 40. The results were similar for all four clearance ratios examined. Preliminary measurements on eccentrically positioned cylinders with c/R1 = 0·119 showed corresponding effects.

Structure functions of turbulent velocity fluctuations up to fourth order have been measured at several heights in the atmospheric boundary layer over the open ocean, and the results are compared with theoretical predictions for separations in the inertial subrange. The behaviour of second- and third-order quantities shows substantial agreement with the predictions of Kolmogorov's original theory over a wide range of separations, but the results of a recent modification of the theory, attempting to account for intermittency in the local dissipation rate, are also consistent with the data over somewhat shorter separation intervals. The behaviour of the measured fourth-order structure function disagrees with that predicted from Kolmogorov's original work, but good agreement is found with the results of the modified theory.

A general class of solutions is studied describing three-dimensional steady convection flows in a fluid layer heated from below with boundaries of low thermal conductivity. Non-linear properties of the solutions are analysed and the physically realizable convection flow is determined by a stability analysis with respect to arbitrary three-dimensional disturbances. The most surprising result is that square-pattern convection is preferred in contrast to two-dimensional rolls that represent the only form of stable convection in a symmetric layer with highly conducting boundaries. The analysis is carried out in the limit of infinite Prandtl number and for a particular boundary configuration. But it is shown that the results hold for arbitrary Prandtl number to the order to which they have been derived and that other assumptions about the boundaries require only minor modifications as long as their thermal conductance remains low.