The terminal velocity of rise of small, distorted gas bubbles in a liquid of small viscosity is calculated. Small viscosity means that the dimensionless group gμ4/ρT3 where g is the acceleration of gravity, μ the viscosity, ρ the density and T the surface tension, is less than 10−8. It is assumed—and the numerical accuracy of the assumption is discussed—that the distorted bubbles are oblate ellipsoids of revolution. The drag coefficient is found by extending the theory given recently (Moore 1963) for the boundary layer on a spherical gas bubble. The results are in reasonable quantitative agreement with the experimental data.

An investigation is made of resonant triads of Tollmien-Schlichting waves in an unstable boundary layer. The triads considered are those comprising a two-dimensional wave and two oblique waves propagating at equal and opposite angles to the flow direction and such that all three waves have the same phase velocity in the downstream direction. For such a resonant triad remarkably powerful wave interations take place, which may cause a continuous and rapid transfer of energy from the primary shear flow to the disturbance. It appears that the oblique waves can grow particularly rapidly and it is suggested that such preferential growth may be responsible for the rapid development of three-dimensionality in unstable boundary layers. The non-linear energy transfer primarily takes place in the vicinity of the critical layer where the downstream propagation velocity of the waves equals the velocity of the primary flow.

The theoretical analysis is initially carried out for a general primary velocity profile; then, in order to demonstrate the essential features of the results, precise interaction equations are derived for a particular profile consisting of a layer of constant shear bounded by a uniform flow. Some exact solutions of the general interaction equations are presented, one of which has the property that the wave amplitudes become indefinitely large at a finite time. The possible relevance of the present theoretical model to the experiments of Klebanoff, Tidstrom & Sargent (1962) is examined.

This article discusses the description of wall-bounded turbulence as a deterministic high-dimensional dynamical system of interacting coherent structures, defined as eddies with enough internal dynamics to behave relatively autonomously from any remaining incoherent part of the flow. The guiding principle is that randomness is not a property, but a methodological choice of what to ignore in the flow, and that a complete understanding of turbulence, including the possibility of control, requires that it be kept to a minimum. After briefly reviewing the underlying low-order statistics of flows at moderate Reynolds numbers, the article examines what two-point statistics imply for the decomposition of the flow into individual eddies. Intense eddies are examined next, including their temporal evolution, and shown to satisfy many of the properties required for coherence. In particular, it is shown that coherent structures larger than the Corrsin scale are a natural consequence of the shear. In wall-bounded turbulence, they can be classified into coherent dispersive waves and transient bursts. The former are found in the viscous layer near the wall, and as very large structures spanning the entire boundary layer. Although they are shear-driven, these waves have enough internal structure to maintain a uniform advection velocity. Conversely, bursts exist at all scales, are characteristic of the logarithmic layer, and interact almost linearly with the shear. While the waves require a wall to determine their length scale, the bursts are essentially independent from it. The article concludes with a brief review of our present theoretical understanding of turbulent structures, and with a list of open problems and future perspectives.

‘Chance is the name we give to what we choose to ignore (Voltaire)’

Using an active grid devised by Makita (1991), shearless decaying turbulence is studied for the Taylor-microscale Reynolds number, Rλ, varying from 50 to 473 in a small (40 × 40 cm2 cross-section) wind tunnel. The turbulence generator consists of grid bars with triangular wings that rotate and flap in a random way. The value of Rλ is determined by the mean speed of the air (varied from 3 to 14 m s–1) as it passes the rotating grid, and to a lesser extent by the randomness and rotation rate of the grid bars. Our main findings are as follows. A weak, not particularly well-defined scaling range (i.e. a power-law dependence of both the longitudinal (u) and transverse (v) spectra, F11(k1) and F22(k1) respectively, on wavenumber k1) first appears at Rλ ∼ 50, with a slope, n1, (for the u spectrum) of approximately 1.3. As Rλ was increased, n1 increased rapidly until Rλ ∼ 200 where n ∼ 1.5. From there on the increase in n1 was slow, and even by Rλ = 473 it was still significantly below the Kolmogorov value of 1.67. Over the entire range, 50 [les ] Rλ [les ] 473, the data were well described by the empirical fit: $n_1 = \frac{5}{3}(1-3.15R_\lambda^{-2/3})$. Using a modified form of the Kolmogorov similarity law: F11(k1) = C1*ε2/3k1–5/3(k1η)5/3–n1 where ε is the turbulence energy dissipation rate and η is the Kolmogorov microscale, we determined a linear dependence between n1 and C1*: C1* = 4.5 – 2.4n1. Thus for n1 = 5/3 (which extrapolation of our results suggests will occur in this flow for Rλ ∼ 104), C1* = 0.5, the accepted high-Reynolds-number value of the Kolmogorov constant. Analysis of the p.d.f. of velocity differences Δu(r) and Δv(r) where r is an inertial subrange interval, conditional dissipation, and other statistics showed that there was a qualitative difference between the turbulence for Rλ < 100 (which we call weak turbulence) and that for Rλ > 200 (strong turbulence). For the latter, the p.d.f.s of Δu(r) and Δv(r) had super Gaussian tails and the dissipation (both of the u and v components) conditioned on Δu(r) and Δv(r) was a strong function of the velocity difference. For Rλ < 100, p.d.f.s of Δu(r) and Δv(r) were Gaussian and conditional dissipation statistics were weak. Our results for Rλ > 200 are consistent with the predictions of the Kolmogorov refined similarity hypothesis (and make a distinction between the dynamical and kinematical contributions to the conditional statistics). They have much in common with similar statistics done in shear flows at much higher Rλ, with which they are compared.

The transient evolution of the bubble-size probability density functions resulting from the breakup of an air bubble injected into a fully developed turbulent water ow has been measured experimentally using phase Doppler particle sizing (PDPA) and image processing techniques. These measurements were used to determine the breakup frequency of the bubbles as a function of their size and of the critical diameter Dc defined as Dc = 1.26 (σ/ρ)3/5ε−2/5, where ε is the rate of dissipation per unit mass and per unit time of the underlying turbulence. A phenomenological model is proposed showing the existence of two distinct bubble size regimes. For bubbles of sizes comparable to Dc, the breakup frequency is shown to increase as (σ/ρ)−2/5ε−3/5 √D/Dc−1, while for large bubbles whose sizes are greater than 1.63Dc, it decreases with the bubble size as ε1/3D−2/3. The model is shown to be in good agreement with measurements performed over a wide range of bubble sizes and turbulence intensities.

Accurate measurements of the density distribution in Ar and N2 shock waves have been made in a shock tube for the Mach number range from 1.55 to 9 and 10 respectively by the absorption of an electron beam. A modified absorption law has been used for data reduction. The density profiles were corrected for the influence of shock curvature and density rise behind the shock wave. The measurements in Ar agree to within 1% with those of Schmidt (1969) in the mean range of Ms but give a slightly smaller density gradient for Ms = 9. Comparison with various theories shows very good agreement with Bird's Monte Carlo simulation in the whole Mach number range for a simple repulsive intermolecular force law. Further, the agreement with the Mott-Smith density profile for the same interaction law is also good, and surprisingly is found to be better for lower than for higher Mach numbers. Qualitative agreement is obtained with the solutions of Hicks & Yen for hard-sphere and Maxwell molecules. The Navier-Stokes and BGK solutions are found to differ significantly from the present experiments even for the lowest measured Mach number (1·55), whereas the Burnett equation gives better agreement, especially with respect to the asymmetry of the profiles.

The measured N2 profiles agree on the whole with the shock-tube measurements of other investigators but show substantial deviations from the low density wind-tunnel experiments of Robben & Talbot (1966b) for higher Mach numbers. Bird's ‘energy sink model’ (1971) is in agreement with the measured density profiles for a realistic interaction law and a suitable rotational collision number. Rotational relaxation in nitrogen is found to be very fast for all Mach numbers. Consequently the coupling between rotational and translational relaxation is very strong.

The motion at low Reynolds number of a drop in a quiescent unbounded fluid is investigated theoretically by means of a singular-perturbation solution of the axisymmetric equations of motion. Special attention is paid to the deformation of the drop. It is shown that for small values of the Weber number We the drop will first deform exactly into an oblate spheroid and then, with a further increase in We, into a geometry approaching that of a spherical cap. These results are quite insensitive to the ratio of the viscosities of the two fluid phases. The first-order effect of the deformation on the drag of the drop is also included in the analysis.

The stability of the horizontal interface between conducting and non-conducting fluids under the influence of an initially uniform vertical electric field is discussed. To produce such a field when the conducting fluid is the heavier it is imagined that a large horizontal electrode is immersed in the non-conducting fluid. As the field increases the part of the interface below the electrode rises till at a voltage V, which depends on the interfacial tension, the height of the electrode above the interface and the density difference, the interface becomes unstable for vertical displacements Z which satisfy the equation $({\frac{\delta^2}{\delta x^2}} + {\frac{\delta^2}{\delta y^2}}+k^2) Z=0.$ The value of k consistent with the lowest value of V is found. When the electrode is situated above the interface at less than a certain distance the lowest value of V is attained when k = 0 so that the horizontal extent of an unstable crest is likely to be great. As the electrode height increases above this critical value k increases and the unstable crests become more closely spaced till an upper limiting value of k is obtained.

Experiments made with several pairs of fluids verify these theoretical conclusions. In some cases sparking occurs before the potential V is reached, but in others, air at atmospheric pressure over water, for instace, the instability occurs first and the jet of water which results permits the passage of a spark which may inhibit further development of the instability. The physical condition which determines the sparking voltage to a fluid may therefore be very different from that which is operative between solid electrodes. This consideration might be relevant to the performance of power-line insulators in wet weather.

Localized arc filament plasma actuators are used to control an axisymmetric Mach 1.3 ideally expanded jet of 2.54 cm exit diameter and a Reynolds number based on the nozzle exit diameter of about 1.1×106. Measurements of growth and decay of perturbations seeded in the flow by the actuators, laser-based planar flow visualizations, and particle imaging velocimetry measurements are used to evaluate the effects of control. Eight actuators distributed azimuthally inside the nozzle, approximately 1 mm upstream of the nozzle exit, are used to force various azimuthal modes over a large frequency range (StDF of 0.13 to 1.3). The jet responded to the forcing over the entire range of frequencies, but the response was optimum (in terms of the development of large coherent structures and mixing enhancement) around the jet preferred Strouhal number of 0.33 (f = 5 kHz), in good agreement with the results in the literature for low-speed and low-Reynolds-number jets. The jet (with a thin boundary layer, D/θ ∼ 250) also responded to forcing with various azimuthal modes (m = 0 to 3 and m = ±1, ±2, ±4), again in agreement with instability analysis and experimental results in the literature for low-speed and low-Reynolds-number jets. Forcing the jet with the azimuthal mode m = ±1 at the jet preferred-mode frequency provided the maximum mixing enhancement, with a significant reduction in the jet potential core length and a significant increase in the jet centreline velocity decay rate beyond the end of the potential core.

A decomposition of turbulent velocity fields into modes that exhibit both localization in wavenumber and physical space is performed. We review some basic properties of such a decomposition, the wavelet transform. The wavelet-transformed Navier—Stokes equations are derived, and we define new quantities such as e(r, x), t(r, x) and π(r, x) which are the kinetic energy, the transfer of kinetic energy and the flux of kinetic energy through scale r at position x. The discrete version of e(r, x) is computed from laboratory one-dimensional velocity signals in a boundary layer and in a turbulent wake behind a circular cylinder. We also compute e(r, x), t(r, x) and π(r, x) from three-dimensional velocity fields obtained from direct numerical simulations. Our findings are that the localized kinetic energies become very intermittent in x at small scales and exhibit multifractal scaling. The transfer and flux of kinetic energy are found to fluctuate greatly in physical space for scales between the energy containing scale and the dissipative scale. These fluctuations have mean values that agree with their traditional counterparts in Fourier space, but have standard deviations that are much larger than their mean values. In space (at each scale r), we find exponential tails for the probability density functions of these quantities. We then study the nonlinear advection terms in more detail and define the transfer T(r|r′, x) between scale r and all scales smaller than r′, at location x. Then we define πsg(r, x), the flux of energy caused only by the scales smaller than r, at x, and find negative values for πsg(r, x), at almost 50% of the physical space at every scale (backscatter). We propose the inclusion of local backscatter in the phenomenological cascade models of intermittency, by allowing some energy to flow from small to large scales in the context of a multiplicative process in the inertial range.

This is an experimental and numerical study of steady free convection in a porous medium, a system dominated by a single non-linear process, the advection of heat. The paper presents results on three topics: (1) a system uniformly heated from below, for which the flow is cellular, as in the analogous Bénard-Rayleigh flows, (ii) the role of end-effects, and (iii) the role of mass discharge. Measurements of heat transfer are used to establish further the validity of the numerical scheme proposed by the author (1966a), while the other flows allow a more extensive study of the numerical scheme under various boundary conditions. The results are very satisfactory even though only moderately non-linear problems can be treated at present.

The main new results are as follows. For the Rayleigh-type flow, above a critical Rayleigh number of about 40, the heat transferred across the layer is proportional to the square of the temperature difference across the layer and is independent of the thermal conductivity of the medium or the depth of the layer. This result is modified when the boundary-layer thickness is comparable to the grain size of the medium. The investigation of end-effects reveals variations in horizontal wave-number and a pronounced hysteresis and suggests an alternative explanation of some observations by Malkus (1954).

Measurements of the relation between mass hold-up and flow rate have been made for glass beads in fully developed flow down an inclined chute, over the whole range of inclinations for which such flows are possible. Velocity profiles in the flowing material have also been measured. For a given inclination it is found that two different flow regimes may exist for each value of the flow rate in a certain interval. One is an ‘energetic’ flow, and is produced when the particles are dropped into the chute from a height, while the other is relatively quiescent and occurs when entry to the chute is regulated by a gate. At some values of the inclination jumps in the flow pattern occur between these branches, and it is even possible for both branches to coexist in the same chute, separated by a shock. A theoretical treatment of chute flow has been based on a rheological model of the material which takes into account both collisional and fractional mechanisms for generating stress. Its predictions include most aspects of the observed behaviour, but quantitative comparison of theory and experiment is difficult because of the uncertain values of some parameters appearing in the theory.

The instability of a partial cavity induced by the development of a re-entrant jet is investigated on the basis of experiments conducted on a diverging step. Detailed visualizations of the cavity behaviour allowed us to identify the domain of the re-entrant jet instability which leads to classical cloud cavitation. The surrounding regimes are also investigated, in particular the special case of thin cavities which do not oscillate in length but surprisingly exhibit a re-entrant jet of periodical behaviour. The velocity of the re-entrant jet is measured from visualizations, in the case of both cloud cavitation and thin cavities. The limits of the domain of the re-entrant jet instability are corroborated by velocity fluctuation measurements. By varying the divergence and the confinement of the channel, it is shown that the extent of the auto-oscillation domain primarily depends upon the average adverse pressure gradient in the channel. This conclusion is corroborated by the determination of the pressure gradient on the basis of LDV measurements which shows a good correlation between the domain of the cloud cavitation instability and the region of high adverse pressure gradient. A simple phenomenological model of the development of the re-entrant jet in an adverse pressure gradient confirms the strong influence of the pressure gradient on the development of the re-entrant jet and particularly on its thickness. An ultrasonic technique is developed to measure the re-entrant jet thickness, which allowed us to compare it with the cavity thickness. By considering an estimate of the characteristic height of the perturbations developing on the interface of the cavity and of the re-entrant jet, it is shown that cloud cavitation requires negligible interaction between both interfaces, i.e. a thick enough cavity. In the case of thin cavities, this interaction becomes predominant; the cavity interface breaks at many points, giving birth to small-scale vapour structures unlike the large-scale clouds which are periodically shed in the case of cloud cavitation. The low-frequency content of the cloud cavitation instability is investigated using spectral analysis of wall pressure signals. It is shown that the characteristic frequency of cloud cavitation corresponds to a Strouhal number of about 0.2 whatever the operating conditions and the cavity length may be, provided the Strouhal number is computed on the basis of the maximum cavity length. For long enough cavities, another peak is observed in the spectra, at lower frequency, which is interpreted as a surge-type instability. The present investigations give insight into the instabilities that a partial cavity may undergo, and particularly the re-entrant jet instability. Two parameters are shown to be of most importance in the analysis of the re-entrant jet instability: the adverse pressure gradient and the cavity thickness compared to the re-entrant jet thickness. The present results allowed us to conduct a qualitative phenomenological analysis of the stability of partial cavities on cavitating hydrofoils. It is conjectured that cloud cavitation should occur for short enough cavities, of the order of half the chordlength, whereas the instability often observed at the limit between partial cavitation and super-cavitation is here interpreted as a cavitation surge-type instability.

Direct numerical simulations are performed of gravity-current fronts in the lock-exchange configuration. The case of small density differences is considered, where the Boussinesq approximations can be adopted. The key objective of the investigation is a detailed analysis of the flow structure at the foremost part of the front, where no previous high-resolution data were available. For the simulations, high-order numerical methods are used, based on spectral and spectral-element discretizations and compact finite differences. A three-dimensional simulation is conducted of a front spreading along a no-slip boundary at a Reynolds number of about 750. The simulation exhibits all features typically observed in experimental flows near the gravity-current head, including the lobe-and-cleft structure at the leading edge. The results reveal that the flow topology at the head differs from what has been assumed previously, in that the foremost point is not a stagnation point in a translating system. Rather, the stagnation point is located below and slightly behind the foremost point in the vicinity of the wall. The relevance of this finding for the mechanism behind the lobe-and-cleft instability is discussed. In order to explore the high-Reynolds-number regime, and to assess potential Reynolds-number effects, two-dimensional simulations are conducted for Reynolds numbers up to about 30 000, for both no-slip and slip (i.e. shear-stress free) boundaries. It is shown that although quantitative Reynolds-number effects persist over the whole range examined, no qualitative changes in the flow structure at the head can be observed. A comparison of the two-dimensional results with laboratory data and the three-dimensional simulation provides evidence that a two-dimensional model is able to capture essential features of the flow at the head. The simulations also show that for the free-slip case the shape of the head agrees closely with the classical inviscid theory of Benjamin.

The present paper is concerned with the conditions of flow in tanks containing stirred fluids. An attempt is made to apply the theoretical concepts of local isotropy to explain the behaviour of liquid in liquid dispersions, subjected to turbulent agitation. Relations describing quantitatively the influence of turbulence on both break-up and coalescence of individual droplets are derived and are compared with experimental evidence. A special type of dispersion is described in which droplet size is controlled by the prevention of coalescence due to turbulence. The dependence of droplet size on energy dissipation per unit mass, as predicted by the theory of local isotropy, is put to an experimental test using geometrically similar vessels of different sizes.

Though the results are not entirely conclusive, experimental evidence suggests that the hypothesis of locally isotropic flow may be applicable to the flow conditions described in the paper, and that statistical theories of turbulence can be of practical value in estimating droplet sizes in agitated dispersions.

The problem of the spreading of a granular mass released at the top of a rough inclined plane was investigated. The evolution of the avalanche was measured experimentally from the initiation up to the deposit using a Moiré image-processing technique. The results are quantitatively compared with the prediction of an hydrodynamic model based on depth-averaged equations. In the model, the interaction between the flowing layer and the rough bottom is described by a non-trivial friction force whose expression is derived from measurements on steady uniform flows. We show that the spreading of the mass is quantitatively predicted by the model when the mass is released on a plane free of particles. When an avalanche is triggered on an initially static layer, the model fails in quantitatively predicting the propagation but qualitatively captures the evolution.

In this paper, we obtain the first-order necessary optimality conditions of an optimal control problem for a distributed parameter system with geometric control, namely, the minimum-drag problem in Stokes flow (flow at a very low Reynolds number). We find that the unit-volume body with smallest drag must be such that the magnitude of the normal derivative of the velocity of the fluid is constant on the boundary of the body. In a three-dimensional uniform flow, this condition implies that the body with minimum drag has the shape of a pointed body similar in general shape to a prolate spheroid but with some differences including conical front and rear ends of angle 120°.

We present the first application of an artificial neural network trained through a deep reinforcement learning agent to perform active flow control. It is shown that, in a two-dimensional simulation of the Kármán vortex street at moderate Reynolds number ($Re=100$), our artificial neural network is able to learn an active control strategy from experimenting with the mass flow rates of two jets on the sides of a cylinder. By interacting with the unsteady wake, the artificial neural network successfully stabilizes the vortex alley and reduces drag by approximately 8 %. This is performed while using small mass flow rates for the actuation, of the order of 0.5 % of the mass flow rate intersecting the cylinder cross-section once a new pseudo-periodic shedding regime is found. This opens the way to a new class of methods for performing active flow control.

The full set of equations of motion for the classical water wave problem in Eulerian co-ordinates is obtained from a Lagrangian function which equals the pressure. This Lagrangian is compared with the more usual expression formed from kinetic minus potential energy.

The viscous sublayer of a turbulent boundary layer on a surface with fine longitudinal ribs (riblets) is investigated theoretically. The mean flow constituent of this viscous flow is considered. Using conformal mapping, the velocity distributions on various surface configurations are calculated. The geometries that were investigated include sawtooth profiles with triangular and trapezoidal grooves as well as profiles with thin blade-shaped ribs, ribs with rounded edges and ribs having sharp ridges and U-shaped grooves. (This latter riblet configuration is also found on the tiny scales of fast sharks.) Our calculations enable us to determine the location of the origin of the velocity profile that lies somewhat below the tips of the ridges. The distance between this origin and the tip of the ridge we call ‘protrusion height’. The upper limit for the protrusion height is found to be 22% of the lateral rib spacing; the coefficient 0.22 being the value of the expression π−1 In 2. This limit is valid for two-dimensional riblet geometries. Analogous experiments with an electrolytic tank are carried out as an additional check on the theoretical calculations. This is also an easy way to determine experimentally the location of the origin of the velocity profile for arbitrary new riblet geometries. A possible connection between protrusion height and drag reduction in a turbulent boundary layer flow is discussed. Finally, the present theory also produces an orthogonal grid pattern above riblet surfaces which may be utilized in future numerical calculations of the whole turbulent boundary layer.