Calculations of the growth and collapse of transient vapour cavities near a free surface when buoyancy forces may be important are made using the boundary-integral method described in Part 1. Bubble shapes, particle paths, pressure contours and centroid motion are used to illustrate the calculations. In the absence of buoyancy forces the bubble migrates away from the free surface during the collapse phase, yielding a liquid jet directed away from the free surface. When the bubble is sufficiently close to the free surface, the nonlinear response of the free surface produces a high-speed jet (‘spike’) that moves in the opposite direction to the liquid jet and, in so doing, produces a stagnation point in the fluid between the bubble and the free surface. For sufficiently large bubbles, buoyancy forces may be dominant, so that the bubble migrates towards the free surface with the resulting liquid jet in the same direction. The Kelvin impulse provides a reasonable estimate of the physical parameter space that determines the migratory behaviour of the collapsing bubbles.

The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

As an extension of the authors' work on isotropic vortical turbulence interacting with a shock wave (Lee, Lele & Moin 1993), direct numerical simulation and linear analysis are performed for stronger shock waves to investigate the effects of the upstream shock-normal Mach number (M1). A shock-capturing scheme is developed to accurately simulate the unsteady interaction of turbulence with shock waves. Turbulence kinetic energy is amplified across the shock wave, and this amplification tends to saturate beyond M1 = 3.0. An existing controversy between experiments and theoretical predictions on length scale change is thoroughly investigated through the shock-capturing simulation: most turbulence length scales decrease across the shock, while the dissipation length scale (ρq3/ε) increases slightly for shock waves with M1<1.65. Fluctuations in thermodynamic variables behind the shock wave are nearly isentropic for M1<1.2, and deviate significantly from isentropy for the stronger shock waves, due to the entropy fluctuation generated through the interaction.

Averaged equations governing the motion of equal rigid spheres suspended in a potential flow are derived from the equation for the probability distribution. A distinctive feature of this work is the derivation of the disperse-phase momentum equation by averaging the particle equation of motion directly, rather than the microscopic equation for the particle material. This approach is more flexible than the usual one and leads to a simpler and more fundamental description of the particle phase. The model is closed in a systematic way (i.e. with no ad hoc assumptions) in the dilute limit and in the linear limit. One of the closure quantities is related to the difference between the gradient of the average pressure and the average pressure gradient, a well-known problem in the widely used two-fluid engineering models. The present result for this quantity leads to the introduction of a modified added mass coefficient (related to Wallis's ‘exertia’) that remains very nearly constant with changes in the volume fraction and densities of the phases. Statistics of this coefficient are provided and exhibit a rather strong variability of up to 20% among different numerical simulations. A detailed comparison of the present results with those of other investigators is given in § 10.

As a further illustration of the flexibility of the techniques developed in the paper, in Appendix C they are applied to the calculation of the so-called ‘particle stress’ tensor. This derivation is considerably simpler than others available in the literature.

It is shown that the surface wind drift in the ocean substantially reduces the maximum wave height ξx and wave orbital velocity that can be attained before breaking. If q is the magnitude of the surface drift at the point where the wave profile crosses the mean water level and c is the wave speed, then \[ \zeta_{\max} = \frac{c^2}{2g}\bigg(1-\frac{q}{c}\bigg)^2. \] Incipient breaking in a steady wave train is characterized by the occurrence of stagnation points at wave crests, but not necessarily by discontinuities in slope. After breaking, there is in the mean flow a stagnation point relative to the wave profile near the crest of the broken wave, on one side of which the water tumbles forward and behind which it recedes more smoothly to the rear. Some simple flow visualization studies indicate the general extent of the wake behind the breaking region.

Linear stability of incompressible flow in a circular pipe is considered. Use is made of a vector function formulation involving the radial velocity and radial vorticity only. Asymptotic as well as transient stability are investigated using eigenvalues and ε-pseudoeigenvalues, respectively. Energy stability is probed by establishing a link to the numerical range of the linear stability operator. Substantial transient growth followed by exponential decay has been found and parameter studies revealed that the maximum amplification of initial energy density is experienced by disturbances with no streamwise dependence and azimuthal wavenumber n = 1. It has also been found that the maximum in energy scales with the Reynolds number squared, as for other shear flows. The flow field of the optimal disturbance, exploiting the transient growth mechanism maximally, has been determined and followed in time. Optimal disturbances are in general characterized by a strong shear layer in the centre of the pipe and their overall structure has been found not to change significantly as time evolves. The presented linear transient growth mechanism which has its origin in the non-normality of the linearized Navier–Stokes operator, may provide a viable process for triggering finite-amplitude effects.

An axisymmetric hot-air jet discharging into cold ambient air is investigated experimentally. We consider the transitional regime, that is, Reynolds numbers at which the jet is initially laminar. In the first part of the paper it is demonstrated by several different experiments that, for sufficiently low Reynolds number and a ratio of jet exit to ambient density below approximately 0.7, global oscillations of the ‘jet column’ become self-excited, a behaviour which is related to local absolute instability in the potential core region. The onset of the global oscillations is identified as a Hopf bifurcation and two axisymmetric global modes are observed below the critical density ratio. Finally, it is shown that in the (self-excited) limit-cycle regime the spreading of the hot jet is intermittently quite spectacular, with half-angles in excess of 45°. Using flow visualization, this large spreading of low-density jets is related to the generation of strong ‘side jets’ emanating from the jet column.

In previous computational fluid dynamics studies of breaking waves, there has been a marked tendency to severely over-estimate turbulence levels, both pre- and post-breaking. This problem is most likely related to the previously described (though not sufficiently well recognized) conditional instability of widely used turbulence models when used to close Reynolds-averaged Navier–Stokes (RANS) equations in regions of nearly potential flow with finite strain, resulting in exponential growth of the turbulent kinetic energy and eddy viscosity. While this problem has been known for nearly 20 years, a suitable and fundamentally sound solution has yet to be developed. In this work it is demonstrated that virtually all commonly used two-equation turbulence closure models are unconditionally, rather than conditionally, unstable in such regions. A new formulation of the $k$–$\unicode[STIX]{x1D714}$ closure is developed which elegantly stabilizes the model in nearly potential flow regions, with modifications remaining passive in sheared flow regions, thus solving this long-standing problem. Computed results involving non-breaking waves demonstrate that the new stabilized closure enables nearly constant form wave propagation over long durations, avoiding the exponential growth of the eddy viscosity and inevitable wave decay exhibited by standard closures. Additional applications on breaking waves demonstrate that the new stabilized model avoids the unphysical generation of pre-breaking turbulence which widely plagues existing closures. The new model is demonstrated to be capable of predicting accurate pre- and post-breaking surface elevations, as well as turbulence and undertow velocity profiles, especially during transition from pre-breaking to the outer surf zone. Results in the inner surf zone are similar to standard closures. Similar methods for formally stabilizing other widely used closure models ($k$–$\unicode[STIX]{x1D714}$ and $k$–$\unicode[STIX]{x1D700}$ variants) are likewise developed, and it is recommended that these be utilized in future RANS simulations of surface waves. (In the above $k$ is the turbulent kinetic energy density, $\unicode[STIX]{x1D714}$ is the specific dissipation rate, and $\unicode[STIX]{x1D700}$ is the dissipation.)

Liquid in a slot flows owing to a temperature gradient applied along its free surface. The thermal variation of surface tension induces a steady viscous flow directed on the surface from hot to cold, and recirculating below. For small aspect ratios A, giving flow in thin, two-dimensional slots, an asymptotic theory valid for A → 0 is used to obtain the fluid and thermal fields as well as the interfacial shapes. Solutions are obtained for both fixed lines and fixed angles at the contact between the interface and the solid side walls.

The properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows are explored both experimentally and theoretically. Available high-quality data reveal a dynamically relevant four-layer description that is a departure from the mean profile four-layer description traditionally and nearly universally ascribed to turbulent wall flows. Each of the four layers is characterized by a predominance of two of the three terms in the governing equations, and thus the mean dynamics of these four layers are unambiguously defined. The inner normalized physical extent of three of the layers exhibits significant Reynolds-number dependence. The scaling properties of these layer thicknesses are determined. Particular significance is attached to the viscous/Reynolds-stress-gradient balance layer since its thickness defines a required length scale. Multiscale analysis (necessarily incomplete) substantiates the four-layer structure in developed turbulent channel flow. In particular, the analysis verifies the existence of at least one intermediate layer, with its own characteristic scaling, between the traditional inner and outer layers. Other information is obtained, such as (i) the widths (in order of magnitude) of the four layers, (ii) a flattening of the Reynolds stress profile near its maximum, and (iii) the asymptotic increase rate of the peak value of the Reynolds stress as the Reynolds number approaches infinity. Finally, on the basis of the experimental observation that the velocity increments over two of the four layers are unbounded with increasing Reynolds number and have the same order of magnitude, there is additional theoretical evidence (outside traditional arguments) for the asymptotically logarithmic character of the mean velocity profile in two of the layers; and (in order of magnitude) the mean velocity increments across each of the four layers are determined. All of these results follow from a systematic train of reasoning, using the averaged momentum balance equation together with other minimal assumptions, such as that the mean velocity increases monotonically from the wall.

Measurements of heavy particle dispersion have been made using direct numerical simulations of isotropic turbulence. The parameters affecting the dispersion of solid particles, namely particle inertia and drift due to body forces were investigated separately. In agreement with the theoretical studies of Reeks, and Pismen & Nir, the effect of particle inertia is to increase the eddy diffusivity over that of the fluid (in the absence of particle drift). The increase in the eddy diffusivity of particles over that of the fluid was between 2 and 16%, in reasonable agreement with the increases reported in Reeks, and Pismen & Nir. The effect of a deterministic particle drift is shown to decrease unequally the dispersion in directions normal and parallel to the particle drift direction. Eddy diffusivities normal and parallel to particle drift are shown to be in good agreement with the predictions of Csanady and the experimental measurements of Wells & Stock.

Experimental and theoretical studies of the effect of an ultrasonically absorptive coating (UAC) on hypersonic boundary-layer stability are described. A thin coating of fibrous absorbent material (felt metal) was selected as a prototype of a practical UAC. Experiments were performed in the Mach 6 wind tunnel on a $7^{\circ}$ half-angle sharp cone whose longitudinal half-surface was solid and other half-surface was covered by a porous coating. Hot-wire measurements of ‘natural’ disturbances and artificially excited wave packets were conducted on both solid and porous surfaces. Stability analysis of the UAC effect on two- and three-dimensional disturbances showed that the porous coating strongly stabilizes the second mode and marginally destabilizes the first mode. These results are in qualitative agreement with the experimental data for natural disturbances. The theoretical predictions are in good quantitative agreement with the stability measurements for artificially excited wave packets associated with the second mode. Stability calculations for the cooled wall case showed the feasibility of achieving a dramatic increase of the laminar run using a thin porous coating of random microstructure.

In existing experiments it is known that the slow evolution of nonlinear deep-water waves exhibits certain asymmetric features. For example, an initially symmetric wave packet of sufficiently large wave slope will first lean forward and then split into new groups in an asymmetrical manner, and, in a long wavetrain, unstable sideband disturbances can grow unequally to cause an apparent downshift of carrier-wave frequency. These features lie beyond the realm of applicability of the celebrated cubic Schrödinger equation (CSE), but can be, and to some extent have been, predicted by weakly nonlinear theories that are not limited to slowly modulated waves (i.e. waves with a narrow spectral band). Alternatively, one may employ the fourth-order equations of Dysthe (1979), which are limited to narrow-banded waves but can nevertheless be solved more easily by a pseudospectral numerical method. Here we report the numerical simulation of three cases with a view to comparing with certain recent experiments and to complement the numerical results obtained by others from the more general equations.

A general method for computing the hydrodynamic interactions among an infinite suspension of particles, under the condition of vanishingly small particle Reynolds number, is presented. The method follows the procedure developed by O'Brien (1979) for constructing absolutely convergent expressions for particle interactions. For use in dynamic simulation, the convergence of these expressions is accelerated by application of the Ewald summation technique. The resulting hydrodynamic mobility and/or resistance matrices correctly include all far-field non-convergent interactions. Near-field lubrication interactions are incorporated into the resistance matrix using the technique developed by Durlofsky, Brady & Bossis (1987). The method is rigorous, accurate and computationally efficient, and forms the basis of the Stokesian-dynamics simulation method. The method is completely general and allows such diverse suspension problems as self-diffusion, sedimentation, rheology and flow in porous media to be treated within the same formulation for any microstructural arrangement of particles. The accuracy of the Stokesian-dynamics method is illustrated by comparing with the known exact results for spatially periodic suspensions.

A large-Reynolds-number asymptotic theory is presented for the problem of a vortex tube of finite circulation [Gcy ] subjected to uniform non-axisymmetric irrotational strain, and aligned along an axis of positive rate of strain. It is shown that at leading order the vorticity field is determined by a solvability condition at first-order in ε = 1/R[Gcy ] where R[gcy ] = [gcy ]/ν. The first-order problem is solved completely, and contours of constant rate of energy dissipation are obtained and compared with the family of contour maps obtained in a previous numerical study of the problem. It is found that the region of large dissipation does not overlap the region of large enstrophy; in fact, the dissipation rate is maximal at a distance from the vortex axis at which the enstrophy has fallen to only 2.8% of its maximum value. The correlation between enstrophy and dissipation fields is found to be 0.19 + O(ε2). The solution reveals that the stretched vortex can survive for a long time even when two of the principal rates of strain are positive, provided R[gcy ] is large enough. The manner in which the theory may be extended to higher orders in ε is indicated. The results are discussed in relation to the high-vorticity regions (here described as ‘sinews’) observed in many direct numerical simulations of turbulence.

This paper describes experiments undertaken to study in detail the control of vortex shedding from circular cylinders at low Reynolds numbers by using feedback to stabilize the wake instability. Experiments have been performed both in a wind tunnel and in an open water channel with flow visualization. It has been found that feedback control is able to delay the onset of the wake instability, rendering the wake stable at Reynolds numbers about 20% higher than otherwise. At higher flow rates, however, it was not possible to use single-channel feedback to stabilize the wake - although, deceptively, it was possible to reduce the unsteadiness recorded by a near-wake sensor. When control is applied to a long span only the region near the control sensor is controlled. The results presented in this paper generally support the analytical results of other researchers.

Waves at an unstable horizontal interface between two fluids moving vertically through a saturated porous medium are observed to grow rapidly to become fingers (i.e. the amplitude greatly exceeds the wavelength). For a diffusing interface, in experiments using a Hele-Shaw cell, the mean amplitude taken over many fingers grows approximately as (time)2, followed by a transition to a growth proportional to time. Correspondingly, the mean wave-number decreases approximately as (time)−½. Because of the rapid increase in amplitude, longitudinal dispersion ultimately becomes negligible relative to wave growth. To represent the observed quantities at large time, the transport equation is suitably weighted and averaged over the horizontal plane. Hyperbolic equations result, and the ascending and descending zones containing the fronts of the fingers are replaced by discontinuities. These averaged equations form an unclosed set, but closure is achieved by assuming a law for the mean wave-number based on similarity. It is found that the mean amplitude is fairly insensitive to changes in wave-number. Numerical solutions of the averaged equations give more detailed information about the growth behaviour, in excellent agreement with the similarity results and with the Hele-Shaw experiments.

A systematic procedure for initializing supersonic and hypersonic turbulent boundary layers at controlled Mach number and Reynolds number conditions is described. The initialization is done by locally transforming a true direct numerical simulation flow field, and results in a nearly realistic initial magnitude of turbulent fluctuations, turbulence structure and energy distribution. The time scales necessary to forget the initial condition are studied. The experimental conditions of previous studies are simulated. The magnitude of velocity and temperature fluctuations, as well as the turbulent shear stresses given by the direct numerical simulations are in agreement with the experimental data.

It is shown experimentally that free shear flows can be substantially altered through direct control of the large coherent vortices present in the flow.

First, flow-visualization experiments are conducted in Kalliroscope fluid at Reynolds number 550. A foil is placed in the wake of a D-section cylinder, sufficiently far behind the cylinder so that it does not interfere with the vortex formation process. The foil performs a heaving and pitching oscillation at a frequency close to the Strouhal frequency of the cylinder, while cylinder and foil also move forward at constant speed. By varying the phase of the foil oscillation, three basic interaction modes are identified. (i) Formation of a street of pairs of counter-rotating vortices, each pair consisting of one vortex from the initial street of the cylinder and one vortex shed by the foil. The width of the wake is then substantially increased. (ii) Formation of a street of vortices with reduced or even reverse circulation compared to that of oncoming cylinder vortices, through repositioning of cylinder vortices by the foil and interaction with vorticity of the opposite sign shed from the trailing edge of the foil. (iii) Formation of a street of vortices with circulation increased through merging of cylinder vortices with vortices of the same sign shed by the foil. In modes (ii) and (iii) considerable repositioning of the cylinder vortices takes place immediately behind the foil, resulting in a regular or reverse Kármán street. The formation of these three interaction patterns is achieved only for specific parametric values; for different values of the parameters no dominant stable pattern emerges.

Subsequently, the experiments are repeated in a different facility at larger scale, resulting in Reynolds number 20000, in order to obtain force and torque measurements. The purpose of the second set of experiments is to assess the impact of flow control on the efficiency of the oscillating foil, and hence investigate the possibility of energy extraction. It is found that the efficiency of the foil depends strongly on the phase difference between the oscillation of the foil and the arrival of cylinder vortices. Peaks in foil efficiency are associated with the formation of a street of weakened vortices and energy extraction by the foil from the vortices of the vortex street.

An investigation into the existence of hairpin vortices in turbulent channel flow is conducted using a database generated by the large-eddy simulation technique. It is shown that away from the wall the distribution of the inclination angle of vorticity vector gains its maximum at about 45° to the wall. Two-point correlations of velocity and vorticity fluctuations strongly support a flow model consisting of vortical structures inclined at 45° to the wall. The instantaneous vorticity vectors plotted in planes inclined at 45° show that the flow contains an appreciable number of hairpins. Vortex lines are used to display the three-dimensional structure of hairpins, which are shown to be generated from deformation (or roll-up) of sheets of transverse vorticity.