The evolution of temperature variance and heat flux in decaying grid turbulence with a linear cross-stream temperature gradient is studied by producing the temperature gradient by means of two different methods: (a) by placing a ‘mandoline’ (Warhaft & Lumley 1978) downstream from the grid but with its wires differentially heated for the present study, and (b) by differentially heating ribbons of nichrome (a ‘toaster’) placed in the plenum chamber of the wind tunnel. For the former method the initial thermal/mechanical lengthscale ratio Lθ/L was varied by changing the mandoline configuration. For this method it is shown that the gradient causes Lθ/L to equilibrate to a value of about 0·9 regardless of its initial value, and that when this value is achieved the temperature variance increases approximately linearly with time. The toaster was used to produce a temperature gradient without the associated initial temperature variance (and initial thermal lengthscale) that is necessarily produced by the mandoline wires; for the toaster the temperature variance was produced solely by the action of turbulence against the temperature gradient. For this experiment too, the thermal variance grew linearly with time, and Lθ/L was approximately the same as the equilibrium value for the mandoline experiments. The equilibrium value of the ratio of temperature-variance production to temperature-variance dissipation was approximately 1·5 for all of the experiments. The ratio of the mechanical-dissipation/thermal-dissipation timescales was also found to equilibrate, but there was considerably more scatter in the data for this parameter. The values of the equilibrium length- and timescale ratios were not affected by the magnitude of the temperature gradient, which was varied for both experiments. Good transverse homogeneity in the thermal field was achieved in all cases, in contrast with previous experiments (using heated grids).

The temperature and velocity fields in a round heated jet were investigated in detail. Both conventional measurements and conditional measurements (zone averages and point averages) were performed. The probability density functions of the lengths of turbulent and non-turbulent durations were also measured. Filtered correlation measurements show that large-scale turbulent motions were responsible for the bulk of momentum and heat transport, and also that small scales were more efficient in transporting heat than in transporting momentum. In no case was heat transported further or more than momentum, however. These results are discussed in detail, particularly with regard to the entrainment. Conservation equations for turbulent-zone variables and the intermittency factor are derived and a model for some of the resulting higher-order correlations is suggested. An exact equation for the intermittency function is presented.

Mathematical methods of control theory are applied to the problem of control of fluid flow with the long-range objective of developing effective methods for the control of turbulent flows. The procedure of how to cast the problem of controlling turbulence into a problem in optimal control theory is presented using model problems through the formalism and language of control theory. Then we present a suboptimal control and feedback procedure for general stationary and time-dependent problems using methods of calculus of variations through the adjoint state and gradient algorithms. This suboptimal feedback control procedure is applied to the stochastic Burgers equation. Two types of controls are investigated: distributed and boundary controls. The control inputs are the momentum forcing for the distributed control and the boundary velocity for the boundary control. Costs to be minimized are defined as the sum of the mean-square velocity gradient inside the domain for the distributed control or the square velocity gradient at the wall for the boundary control; and in both cases a term was added to account for the implementation cost. Several cases of both controls have been numerically simulated to investigate the performances of the control algorithm. Most cases considered show significant reductions of the costs. Another version of the feedback procedure more effective for practical implementation has been considered and implemented, and the application of this algorithm also shows significant reductions of the costs. Finally, dependence of the control algorithm on the time-discretization method is discussed.

The origin of meandering and braiding of alluvial rivers is re-analysed in terms of stability theory. The flow is described by a two-dimensional model, and the transportation of sediment is separated into bed-load transport and transport of suspended sediment, by use of the improved knowledge of sediment transport mechanisms achieved in recent years. The paper explains why it is important to distinguish between the sediment transported as bed load and that in suspension.

The analysis is able to predict whether a river remains stable or tends to meander or braid.

The results of the stability analysis are compared with laboratory experiments and data from natural rivers, and the agreement is satisfactory.

When the vertical boundaries of a container of stratified brine solution are heated, cells are observed to form in the brine with a layered structure. An explanation of this phenomenon has been given in terms of the successive growth of cells from the top and bottom of the container. An alternative explanation is given here which is based on an examination of the stability of the fluid layer in the neighbourhood of the vertical boundaries, a layer in which there exist horizontal gradients in both temperature and salinity, and the explanation is subjected to comparison with a carefully controlled laboratory experiment. The theoretical description is fairly general and it seems possible that the effects of horizontal gradients of salinity and temperature, which approximately compensate each other so as to leave no horizontal density gradient, may be important in the ocean, as suggested by Stern (1967).

The stability of a horizontal layer of Maxwellian fluid heated from below is considered. Critical Rayleigh numbers, wave-numbers, and frequencies for overstability are determined for both free and rigid boundaries. Elasticity is found to destabilize the fluid, and the presence of rigid boundaries is found to be slightly stabilizing.

At any free surface at which the tangential stress τns vanishes there must be a surface vorticity ω = −2κq, where κ is the curvature and q the tangential velocity. In a surface wave on water, this condition produces a (Stokes) boundary layer with thickness of order δ = (2ν/σ)½, where ν is the kinemetic viscosity and σ the radian frequency of the wave. To first order in the wave steepness parameter ak, the vorticity remains within the boundary layer, but at second order some escapes through the Stokes layer. The mean vorticity $\overline{\omega}$ at the outer edge of the Stokes layer is of order 2(ak)2σ, twice the mean vorticity generated at the free surface.

These results are applied to steep capillary waves, particularly the parasitic capillaries often seen on the forward face of short gravity waves. Because of the high value of σ for the capillaries, the vorticity they generate is much larger than that generated by the gravity wave itself. Hence the capillaries contribute significantly to the vortex (roller) often found at the crest of short gravity waves, when capillaries are present. It is argued that the crest roller and the capillaries form a cooperative system, a ‘capillary roller’ in which each supports the other, with the aid of surface tension and viscosity. Energy is supplied by the gravity wave.

A capillary roller is one instance of a more general phenomenon: a ‘capillary bore’, which is a noticeable feature of many disturbed water surfaces.

The linear stability of two superimposed layers of fluid, heated from below and separated by a thin conducting plate, is investigated. It is shown that when the ratio of the depths of the layers is close to 1:2, two distinct modes of convection can occur with preferred horizontal wavenumber in the ratio 1:2. The nonlinear evolution of a disturbance consisting of both modes is considered, and it is shown that travelling waves are the preferred mode of nonlinear convection for a wide range of parameter values. Other possible types of behaviour, including modulated waves and an attracting homoclinic trajectory, are also described in detail.

The flow of an ideal dissociating gas through a nearly conical nozzle is considered. The equations of one-dimensional motion are solved numerically assuming a simple rate equation together with a number of different values for the rate constant. These calculations suggest that deviations from chemical equilibrium will occur in the nozzle if the rate constant lies within a very wide range of values, and that, once such a deviation has begun, the gas will very rapidly ’freeze’. The dissociation fraction will then remain almost constant if the flow is expanded further, or even if it passes through a constant area section. An approximate method of solution, making use of this property of sudden ’freezing’ of the flow, has been developed and applied to the problem of estimating the deviations from equilibrium under a wide range of conditions. If all the assumptions made in this paper are accepted, then lack of chemical equilibrium may be expected in the working sections of hypersonic wind tunnels and shock tubes. The shape of an optimum nozzle is derived in order to minimize this departure from equilibrium.

It is shown that, while the test section conditions are greatly affected by ’freezing’, the flow behind a normal shock wave is only changed slightly. The heat transfer rate and drag of a blunt body are estimated to be reduced by only about 25% even if complete freezing occurs. However, the shock wave shape is shown to be rather more sensitive to departures from equilibrium.

The distribution of vorticity in the wake of a hovering bird or insect is considered. The wake is modelled by a chain of coaxial small-cored circular vortex rings stacked one upon another; each member of the chain is generated by a single wing-stroke. Circulation is determined by the animal's weight and the time for which a single ring must provide lift; ring size is calculated from the circulation distribution on the animal's wing. The theory is equally applicable to birds and insects, although the mechanism of ring formation differs. This approach avoids the use of lift and drag coefficients and is not bound by the constraints of steady-state aerodynamics; it gives a wake configuration in agreement with experimental observations. The classical momentum jet approach has steady momentum flux in the wake, and is difficult to relate to the wing motions of a hovering bird or insect; the vortex wake can be related to the momentum jet, but adjacent vortex elements are disjoint and momentum flux is periodic.

The evolution of the wake starting from rest is considered by releasing vortex rings at appropriate time intervals and allowing them to interact in their own velocity fields. The resulting configuration depends on the feathering parameter f (which depends on the animal's morphology); f increases with body size. At the lower end of the wake rings coalesce to form a single large vortex, which breaks away from the rest of the wake at intervals. Wake contraction depends on f; the minimum areal contraction of one-half (as in momentum-jet theory) occurs only in the limit f → 0, but values calculated for smaller insects of just over one-half suggest that the momentum jet may be a good approximation to the wake when f is small.

Induced power in hovering is calculated as the limit of the mean rate of increase of wake kinetic energy as time progresses. It can be related to the classical momentum-jet induced power by a simple conversion factor. For an insect or hummingbird the usual momentum-jet estimate may be between 10 and 15% too low, but for a bird it may be as much as 50% too low. This suggests that few, if any, birds are able to sustain aerobic hovering, and that as small a value of f as possible would be necessary if the bird were to hover.

Tip losses (energy cost of the vortex-ring wake compared with the equivalent momentum jet) are negligible for insects, but can be in the range 15–20% for birds.

The Ranque–Hilsch effect, observed in swirling flow within a single tube, is a spontaneous separation of total temperature, with the colder stream near the tube centreline and the hotter air near its periphery. Despite its simplicity, the mechanism of the Ranque–Hilsch effect has been a matter of long-standing dispute. Here we demonstrate, through analysis and experiment, that the acoustic streaming, induced by orderly disturbances within the swirling flow is, to a substantial degree, a cause of the Ranque–Hilsch effect. The analysis predicts that the streaming induced by the pure tone, a spinning wave corresponding to the first tangential mode, deforms the base Rankine vortex into a forced vortex, resulting in total temperature separation in the radial direction. This is confirmed by experiments, where, in the Ranque–Hilsch tube of uniflow arrangement, we install acoustic suppressors of organ-pipe type, tuned to the discrete frequency of the first tangential mode, attenuate its amplitude, and show that this does indeed reduce the total temperature separation.

An experimental study is reported of the motion, deformation, and breakup of a synthetic capsule that is freely suspended in Couette flow. The capsule is a liquid drop surrounded by a thin polymeric membrane. The shape and orientation of the capsule are measured in steady flow and following the start-up of Couette flow. Results are compared with predictions of the small-deformation theory of Barthes-Biesel and co-workers. The data suggest that the capsule membrane is viscoelastic, and comparisons with theory yield values of the membrane elastic modulus and the membrane viscosity. The values of the elastic modulus of the capsule membrane deduced from the flow data are compared with independent measurements for the same capsule.

When the flow-induced deformation becomes sufficiently large, the capsules break. Breakup begins at points on the membrane surface near the principal strain axis of the undisturbed flow. By examining the local deformation within the membrane, it is shown that breakup is correlated with local thinning of the membrane and is initiated at points where the thickness is a minimum.

Properties of solitary waves propagating in a two-layer fluid are investigated comparing experiments and theory. In the experiments the velocity field induced by the waves, the propagation speed and the wave shape are quite accurately measured using particle tracking velocimetry (PTV) and image analysis. The experiments are calibrated with a layer of fresh water above a layer of brine. The depth of the brine is 4.13 times the depth of the fresh water. Theoretical results are given for this depth ratio, and, in addition, in a few examples for larger ratios, up to 100[ratio ]1. The wave amplitudes in the experiments range from a small value up to almost maximal amplitude. The thickness of the pycnocline is in the range of approximately 0.13–0.26 times the depth of the thinner layer. Solitary waves are generated by releasing a volume of fresh water trapped behind a gate. By careful adjustment of the length and depth of the initial volume we always generate a single solitary wave, even for very large volumes. The experiments are very repeatable and the recording technique is very accurate. The error in the measured velocities non-dimensionalized by the linear long wave speed is less than about 7–8% in all cases. The experiments are compared with a fully nonlinear interface model and weakly nonlinear Korteweg–de Vries (KdV) theory. The fully nonlinear model compares excellently with the experiments for all quantities measured. This is true for the whole amplitude range, even for a pycnocline which is not very sharp. The KdV theory is relevant for small wave amplitude but exhibit a systematic deviation from the experiments and the fully nonlinear theory for wave amplitudes exceeding about 0.4 times the depth of the thinner layer. In the experiments with the largest waves, rolls develop behind the maximal displacement of the wave due to the Kelvin–Helmholtz instability. The recordings enable evaluation of the local Richardson number due to the flow in the pycnocline. We find that stability or instability of the flow occurs in approximate agreement with the theorem of Miles and Howard.

The suppression of initially isotropic turbulence by the sudden application of a uniform magnetic field is considered. The problem is characterized by three dimensionless numbers, a Reynolds number R, a magnetic Reynolds number Rm and a magnetic interaction parameter N (for definitions, see equations (1.2) and (1.4)). It is supposed that R [Gt ] 1, Rm [Lt ] 1 and N [Gt ] 1. There are two important time scales, ta a time characteristic of magnetic suppression, and t0 = Ntd, the ‘turn-over’ time of the turbulent energy-containing eddies. For 0 < t [Lt ] t0 the response of the energy-containing components of the turbulence to the applied field is linear and the time dependence of the kinetic energy density K(t) and magnetic energy density M(t) are analysed. There are essentially two distinct contributions to each from two domains of wave-number space D1 and D2 (defined in figure 2). In D1 the response is severely anisotropic, while in D2 it is nearly isotropic. The relative importance of the contributions K1(t) (from D1) and K2(t) (from D2) to K(t) depends on the value of the Lundquist number S = (NRm)½. If S [Lt ] 1, then K1(t) dominates for all t [lsim ] t0 and K(t) ∝ t−½ for td [Lt ] t [Lt ] t0. If 1 [Lt ] S [Lt ] R−2m, then a changeover in the dominant contribution occurs when t = O(S½Rm) t0. Analogous results are obtained for the magnetic energy density.

The interaction between a fixed particle and decaying homogeneous isotropic turbulence is studied numerically using an overset grid that provides resolution of all scales of fluid motion. A description of the numerical technique and validation of the solution procedure are presented. An ensemble of 64 simulations with the particle in different regions of the flow is computed. The particle diameter in the simulations is approximately twice the size of the unladen Kolmogorov length scale, and the maximum value of the particle Reynolds numbers due to the turbulent fluctuations is close to 20. Ensemble averages of quantities from the numerical solutions are used to investigate the turbulence modification and the fluid forces on the particle. Volume-averaged profiles of the turbulent kinetic energy and dissipation rate from the overset grid simulations reveal that the displacement of fluid by the particle and the formation of the boundary layer at the particle surface lead to turbulence modification in a local region. Time histories of the force applied to the particle from each overset grid simulation are compared to those predicted by a particle equation of motion. The particle equation of motion is shown to underpredict the root mean square (RMS) force applied to the particle by the turbulence. RMS errors between the forces from the overset grid simulation and those predicted by the particle equation of motion are shown to be between 15% and 30% of the RMS force on the particle. The steady viscous drag force is shown to be the dominant term in the particle equation of motion while the history integral term is negligible.

The adjoint of the perturbed and linearized compressible viscous flow equations is formulated in such a way that its solution can be used to optimize control actuation in order to reduce flow-generated sound. We apply it to a direct numerical simulation of a randomly excited two-dimensional mixing layer, with inflow vorticity-thickness Reynolds number 500 and free-stream Mach numbers 0.9 and 0.2. The control actuation is implemented as general source terms in the flow equations (body forces, mass sources, and internal energy sources) with compact support near the inflow boundary. The noise to be reduced is defined by a space-time integral of the mean-square pressure fluctuations on a line parallel to the mixing layer in the acoustic field of the low-speed stream. Both the adjoint and flow equations are solved numerically and without modelling approximations. The objective is to study the mechanics of the noise generation and its control. All controls reduce targeted noise with very little required input power, with the most effective (the internal energy control) reducing the noise intensity by 11 dB. Numerical tests confirm that the control is not by any simple acoustic cancellation mechanism but instead results from a genuine change of the flow as a source of sound. The comparison of otherwise identical flows with and without control applied shows little change of the flow's gross features: the evolution and pairings of the energetic structures, turbulence kinetic energy, spreading rate, and so on are superficially unchanged. However, decomposition of the flow into empirical eigenfunctions, as surrogates for Fourier modes in the non-periodic streamwise direction, shows that the turbulence structures advect downstream more uniformly. This change appears to be the key to reducing their acoustic efficiency, a perspective that is clarified by comparing the randomly excited mixing layer to a harmonically excited mixing layer, which is relatively quiet because it is highly ordered. Unfortunately, from the perspective of any practical implementation with actuators, the optimized control identified has a complex spatial and temporal structure, but it can be simplified. Two empirical eigenmodes were required to represent it sufficiently to reduce the targeted noise intensity by about 50%. Optimization of a simple single-degree-of-freedom control with an ad hoc spatial structure is less effective.

The axisymmetric creeping motion of a neutrally buoyant deformable drop flowing through a circular tube is analysed with a boundary integral equation method. The fluids are immiscible, incompressible, and the bulk flow rate is constant. The drop to suspending fluid viscosity ratio is arbitrary and the drop radius varies from 0.5 to 1.15 tube radii. The effects of the capillary number, viscosity ratio, and drop size on the deformation, the drop speed, and the additional pressure loss are examined.

Drops with radius ratios less than 0.7 are insensitive to substantial variation in capillary number and viscosity ratio, and computed values of drop speed and extra pressure loss are in excellent agreement with small deformation theories (Hestroni et al. 1970; Hyman & Skalak 1972a). For this drop size range, significant deformation will result only for Ca > 0.25. The onset of a re-entrant cavity is predicted at the trailing end of the drop for Ca ≈ 0.75. Drop speed and meniscus shape become independent of drop size for radius ratios as small as 1.10. The extra pressure loss can be positive or negative depending mainly on the viscosity ratio, however a relatively inviscid drop can cause a positive extra pressure loss when capillary forces are significant. Computed values for extra pressure loss and drop speed are in good agreement with the experimental data of Ho & Leal (1975) for drops of sizes comparable with the tube radius.

The small-gap equations for the stability of Couette flow with respect to non-axisymmetric disturbances are derived. The eigenvalue problem is solved by a direct numerical procedure. It is found that there is a critical value of Ω2/Ω1(Ω1, Ω2 and R1, R2 are the angular velocities and radii of the inner and outer cylinders respectively) of approximately −0·78, above which the critical disturbance is axisymmetric and below which it is non-axisymmetric. In particular for R1/R2 = 0·95, Ω2/Ω1 = −1, the wave-number in the azimuthal direction of the critical disturbance is m = 4. This result is confirmed when the full linear disturbance equations are considered, i.e. the small-gap approximation is not made.

The influence of a transverse sound wave on vortex shedding from a rigid circular cylinder in a duct has been explored at Reynolds numbers from 20000 to 40000. In the absence of sound, the vortex shedding is found to consist of strings of coherent cyclic events which have frequencies that wander randomly about the nominal vortex-shedding frequency. Application of sound at the vortex-shedding frequency eliminates this wander and correlates the shedding along the cylinder axis. The frequency of vortex shedding can be shifted by sound applied either above or below the nominal vortex-shedding frequency. This entrainment is produced by the velocity induced by the sound wave rather than by the sound pressure. These phenomena are also observed in tube rows.

Vortex shedding from flat plates with square leading and trailing edges having chord-to-thickness ratios 3–16 at Reynolds numbers (1–3) × 103 is investigated experimentally in low-speed wind tunnels. It is shown that vortex shedding from flat plates with square leading and trailing edges is characterized by the impinging-shear-layer instability where the separated shear layer becomes unstable in the presence of a sharp trailing edge corner. The Strouhal number which is based on the plate's chord is approximately constant and equal to 0.6 for chord-to-thickness ratios 3–5. With further increase in the ratio it increases stepwise to values that are approximately equal to integral multiples of 0.6.