Experiments are conducted to generate progressive wave fields in deep water with two-dimensional surface patterns for which two parameters are systematically varied: (i) the aspect ratio of the cells comprising the surface patterns and (ii) a measure of nonlinearity of the input wave field. The goal of these experiments is to determine whether these patterns persist, what their main features are, whether standard models of waves describe these features, and whether there are parameter regimes in which the patterns are stable. We find that in some parameter regimes, surface patterns in deep water do persist with little change of form during the time of the experiment. In other parameter regimes, particularly for large-amplitude experiments, the patterns evolve more significantly. We characterize the patterns and their evolutions with a list of observed features. To describe the patterns and features, we consider two models: ($a$) the standard ($2+1$) nonlinear Schrödinger equation and ($b$) coupled nonlinear Schrödinger equations for two interacting wavetrains. Exact solutions of these models provide qualitative explanations for many of the observed features.

The forces acting on a clean spherical bubble fixed on the axis of a turbulent pipe flow are computed using large-eddy simulation for a bubble Reynolds number $\hbox{\it Re}_{B}\,{=}\,500$ (based on the bubble diameter and centreline velocity in the pipe) and a bulk Reynolds number $\hbox{\it Re}\,{=}\,6000$ (based on the pipe diameter and bulk velocity). The force is found to be influenced by all the length and time scales down to the Kolmogorov microscales. The results show that the lift force experiences much larger fluctuations than the drag force, an effect which is shown to be specific to clean bubbles as compared with solid spheres. Using the instantaneous undisturbed velocity and vorticity at the centre of the bubble, the drag force is well predicted by Moore's drag law while Auton's expression for the shear-induced lift force yields a good description of the transverse force.

Supersonic boundary-layer receptivity to different types of free-stream disturbance is studied for a Mach 4.5 boundary-layer flow over a flat plate by using the approaches of both direct numerical simulation and linear stability theory. This paper is Part 3 of a three-part study of the receptivity of supersonic boundary layers to free-stream disturbances. The present paper investigates receptivity to four types of different free-stream disturbances, i.e. slow and fast acoustic waves, entropy waves, and vorticity waves. A high-order shock-fitting scheme is used in the numerical simulation in order to accurately account for the effects of interactions between free-stream disturbance waves and the oblique shock wave. Numerical results on the generation of fast acoustic waves by free-stream entropy waves or vorticity waves are compared with those of a linear theory. Good agreement is obtained in both wave angles and amplitudes immediately behind the bow shock. It is found that the second-mode receptivity to free-stream slow acoustic waves is several times stronger than that to free-stream fast acoustic waves. This is because free-stream slow acoustic waves can directly induce and interact with the first and second Mack modes, while free-stream fast acoustic waves cannot. Instead, the free-stream fast acoustic waves can only induce and interact with stable mode I waves, which in turn induce unstable Mack modes. In the cases of receptivity to free-stream entropy waves and vorticity waves, it is found that the oblique shock wave created by the displacement of the boundary layer plays an important role because boundary-layer disturbances are mainly induced by fast acoustic waves generated behind the shock by free-stream forcing waves. As a result, mechanisms of the receptivity to free-stream entropy and vorticity waves are very similar to those of the receptivity to free-stream fast acoustic waves.

We consider transport of a passive scalar by an isotropic turbulent velocity field in the presence of a mean scalar gradient. The velocity–scalar cospectrum measures the distribution of the mean scalar flux across scales. An inequality is shown to bound the magnitude of the cospectrum in terms of the shell-summed energy and scalar spectra. At high Schmidt number, this bound limits the possible contribution of the sub-Kolmogorov scales to the scalar flux. At low Schmidt number, we derive an asymptotic result for the cospectrum in the inertial–diffusive range, with a $-11/3$ power law wavenumber dependence, and a comparison is made with results from large-eddy simulation. The sparse direct-interaction perturbation (SDIP) is used to calculate the cospectrum for a range of Schmidt numbers. The Lumley scaling result is recovered in the inertial–convective range and the constant of proportionality was calculated. At high Schmidt numbers, the cospectrum is found to decay exponentially in the viscous–convective range, and at low Schmidt numbers, the $-11/3$ power law is observed in the inertial–diffusive range. Results are reported for the cospectrum from a direct numerical simulation at a Taylor Reynolds number of 265, and a comparison is made at Schmidt number order unity between theory, simulation and experiment.

Direct numerical simulation (DNS) of fully developed turbulent flow in elliptical ducts is performed. The mean cross-stream secondary flows exhibited by two counter-rotating vortices which are symmetrical about the major ellipse's axis are examined. The mean flow characteristics and turbulence statistics are obtained. The variation of the statistical quantities such as the Reynolds stresses and turbulence intensities along the minor axis of the elliptical cross-section are found to be similar to plane channel data. The turbulent statistics along the major axis are found to be inhibited by the secondary flow transferring high-momentum fluid from the duct's centre towards the wall. The instantaneous velocity fields in the near-wall region reveal structures similar to the ‘streaks’ except in the vicinity of the major axis endpoints where significant reduction of the turbulent activity due to the wall transverse curvature effect is found.

Steady Couette and pressure-driven turbulent channel flows have large regions in which the gradients of the viscous and Reynolds stresses are approximately in balance (stress gradient balance regions). In the case of Couette flow, this region occupies the entire channel. Moreover, the relevant features of pressure-driven channel flow throughout the channel can be obtained from those of Couette flow by a simple transformation. It is shown that stress gradient balance regions are characterized by an intrinsic hierarchy of ‘scaling layers’ (analogous to the inner and outer domains), filling out the stress gradient balance region except for locations near the wall. The spatial extent of each scaling layer is found asymptotically to be proportional to its distance from the wall.

There is a rigorous connection between the scaling hierarchy and the mean velocity profile. This connection is through a certain function $A(y^+)$ defined in terms of the hierarchy, which remains $O(1)$ for all $y^+$. The mean velocity satisfies an exact logarithmic growth law in an interval of the hierarchy if and only if $A$ is constant. Although $A$ is generally not constant in any such interval, it is arguably almost constant under certain circumstances in some regions. These results are obtained completely independently of classical inner/outer/overlap scaling arguments, which require more restrictive assumptions.

The possible physical implications of these theoretical results are discussed.

The far-field description of the flow produced by a source of both momentum and mass is given in the form of a coordinate expansion. The first term of the expansion corresponds to the classical self-similar solution first written by Landau. The next two terms, associated with the non-zero rate of mass, are found in the paper.

We examine available data from experiment and recent numerical simulations to explore the supposition that the scalar dissipation rate in turbulence becomes independent of the fluid viscosity when the viscosity is small and of scalar diffusivity when the diffusivity is small. The data are interpreted in the context of semi-empirical spectral theory of Obukhov and Corrsin when the Schmidt number, $\hbox{\it Sc}$, is below unity, and of Batchelor's theory when $\hbox{\it Sc}$ is above unity. Practical limits in terms of the Taylor-microscale Reynolds number, $R_\lambda$, as well as $\hbox{\it Sc}$, are deduced for scalar dissipation to become sensibly independent of molecular properties. In particular, we show that such an asymptotic state is reached if $R_\lambda \hbox{\it Sc}^{1/2}\,{\gg}\,1$ for $\hbox{\it Sc} \,{<}\, 1$, and if $\ln(\hbox{\it Sc})/R_\lambda\,{\ll}\,1$ for $\hbox{\it Sc} \,{>}\,1$.

We consider the countercurrent flow of two incompressible immiscible viscous fluids in an inclined channel. This configuration may lead to the phenomena of ‘flooding’, i.e. the transition from a countercurrent regime to a cocurrent regime. This transition is marked by a variety of transient behaviour, such as the development of large-amplitude waves that impede the flow of one of the fluids to the reversal of the flow of the denser fluid. From a lubrication approximation based on the ratio of the channel height to the downstream disturbance wavelength, we derive a nonlinear system of evolution equations that govern the interfacial shape separating the two fluids and the leading-order pressure. This system, which assumes fluids with disparate density and dynamic viscosity ratios, includes the effects of viscosity stratification, inertia, shear and capillarity. Since the experimental constraints for this effective system are unclear, we consider two ways to drive the flow: either by fixing the volumetric flow rate of the gas phase or by fixing the total pressure drop over a downstream length of the channel. The latter forcing results in a single evolution equation whose dynamics depends non-locally on the interfacial shape. From both of these driven systems, admissible criteria for Lax shocks, undercompressive shocks and rarefaction waves are investigated. These criteria, through a numerical verification, do not depend significantly on the inertial effects within the more dense layer. The choice of the local/non-local constraints appears to play a role in the transient growth of undercompressive shocks, and may relate to the phenomena observed near the onset of flooding.

We re-analyse experiments on a foam sheared in a two-dimensional Couette geometry. We characterize the bubble deformation by a texture tensor. Our measurements are local in time: they show two regimes, one transient and one stationary. They provide both the average and fluctuations of the anisotropy. Measurements are also local in space: they show that both the deformation and the elastic contribution to the stress field do not localize, varying smoothly across the shear gap. We can thus describe the foam as a continuous medium with elastic properties.

Laboratory experiments are conducted to study the rheological behaviour of high-concentration granular–liquid mixtures. Steady uniform free-surface flows are obtained using a recirculating flume. Cases in which a loose deposit forms underneath the flow are contrasted with runs for which basal shear occurs along the flume bottom. The granular motions are observed through the channel sidewall, and analysed with recently developed Voronoï imaging methods. Depth profiles of mean velocity, solid concentration, and granular temperature are obtained, and complemented by stress estimates based on force balance considerations. These measurements are used to probe variations in rheological behaviour over depth, and to clarify the role of the granular temperature. The flows are found to evolve a stratified structure. Distinct sub-layers are characterized by either frictional or collisional behaviour, and transitions between one and the other occur at values of the Stokes number which suggest that viscous effects intervene. The observed frictional behaviour is consistent with shear cell tests conducted at very low shear rates. On the other hand, the collisional data corroborate both the Bagnold description and the more recent kinetic theories of granular flows, provided that one accounts for the inertia of the interstitial liquid.

This paper describes the boundary-layer structure of flow through a porous medium in a two-dimensional rectangular cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated. In the limit of large Darcy–Rayleigh number, the solution involves a horizontal boundary layer near the upper surface where the main thermal gradients occur. For a monotonic temperature distribution at the upper surface, these drive fluid to the colder end of the cavity where it descends within a narrow vertical boundary layer before returning to the horizontal layer. The horizontal and vertical layers form an interactive system which is solved by a combination of asymptotic analysis and numerical computation. A complete solution is obtained for the case of a quadratic temperature distribution at the upper surface. The solution of the interactive boundary-layer system determines the almost constant temperature in the core region below the horizontal and vertical layers, which contains relatively weak variations in both the thermal and velocity fields.

High-resolution laser-Doppler anemometer measurements were acquired in a two-dimensional turbulent boundary layer over a $4^\circ$ ramp. The goals were to provide a detailed data set for an adverse pressure gradient boundary layer far from separation and to examine near-wall behaviour of the Reynolds stresses as compared to flat-plate boundary layers. The flow develops over a flat plate, reaching a momentum thickness Reynolds number of 3350 at an upstream reference location. The boundary layer is then subjected to a varying pressure gradient along the length of the ramp and partially redevelops on a downstream flat plate. Mean velocity measurements show a log law region in all velocity profiles, but the outer layer does not collapse in deficit coordinates indicating that the boundary layer is not in equilibrium. Measurements of non-dimensional stress ratios and quadrant analysis of the two-component data indicate relatively small changes to the turbulence structure. However, the streamwise normal stress has an extended outer layer plateau, and the shear stress and wall-normal stress have outer layer peaks. Near the wall, the streamwise normal stress and shear stress collapse with flat-plate data using standard scaling, but the wall normal stress is substantially larger than flat-plate cases.

The value of the Prandtl number $P$ exerts a strong influence on convection-driven dynamos in rotating spherical shells filled with electrically conducting fluids. Low Prandtl numbers promote dynamo action through the shear provided by differential rotation, while the generation of magnetic fields is more difficult to sustain in high-Prandtl-number fluids where higher values of the magnetic Prandtl number $P_m$ are required. The magnetostrophic approximation often used in dynamo theory appears to be valid only for relatively high values of $P$ and $P_m$. Dynamos with a minimum value of $P_m$ seem to be most readily realizable in the presence of convection columns at moderately low values of $P$. The structure of the magnetic field varies strongly with $P$ in that dynamos with a strong axial dipole field are found for high values of $P$ while the energy of this component is exceeded by that of the axisymmetric toroidal field and by that of the non-axisymmetric components at low values of $P$. Some conclusions are discussed in relation to the problem of the generation of planetary magnetic fields by motions in their electrically conducting liquid cores.

Time-dependent computations of the two-dimensional incompressible uniform-velocity laminar flow past a normal flat plate (of unit half-width) in a channel are presented. Attention is restricted to cases in which the well-known anti-symmetric (von Kármán-type) vortex shedding is suppressed by the imposition of a symmetry plane on the downstream plate centreline. With a further symmetry plane at the channel's upper boundary, the only two governing parameters in the problem are the channel half-width, $H$, and the Reynolds number, $Re$ (based on the body half-width and the upstream velocity, $U$). The former is restricted to the range $3\,{\leq}\, H \,{\leq}\,30$ and the interest lies in determining the nature of the initial instability which occurs in the separated wake as $Re$ is gradually increased. It is found that for sufficiently large $H$ and at a critical $Re$, a long-time-scale global (supercritical) instability is initiated, which in its saturated (limit) state takes the form of ‘lumps’ of vorticity being periodically shed from the tail end of the separated bubble. Stability calculations of corresponding mean flow profiles (typical of those found in the separated wake) are undertaken by examining the impulse response of particular profiles via appropriate solution of the Orr–Sommerfeld equation. The results of this analysis extend those available from related published work and are consistent with the behaviour found from the numerical computations. Taken together, all the results suggest that this type of global instability may be generic to many kinds of separated wakes and, indeed, may provide the fundamental explanation for the very low-frequency oscillations often noticed in fully turbulent wake bubbles.