The equilibrium configurations of a liquid spreading on a rough solid surface are derived by making expansions in terms of the characteristic slope ε of the surface roughness, which is assumed to be very small. It is also assumed that the microscopic contact angle is a constant and that the liquid–air interface is planar at large distances from the contact line. Expressions for the value of the macroscopic contact angle and a discussion of the existence of contact-angle hysteresis and of stick-jump behaviour of the contact line are given for (i) surfaces with parallel grooves, (ii) surfaces with periodicity in two perpendicular directions and (iii) general non-period surfaces.

Nonstationary oblique shock-wave reflections over compressive wedges in air and argon were investigated using infinite-fringe interferometric techniques. These allowed direct, continuous and accurate observations of the isopycnics (lines of constant density) of the flow field. The initial pressures for these experiments were made as high as possible (15 to 250 torr) in order to increase the number of isopycnics and to enhance their details and distribution along the wedge surface over a shock-Mach-number range 2.0 < Ms [les ] 8.7. Included in the study were two cases of regular reflection (RR) and one of each single Mach reflection (SMR), complex Mach reflection (CMR) and double-Mach reflection (DMR) for air, and one RR, SMR, CMR and DMR for argon. These particular cases, which we investigated previously in N2 and Ar using a finite-fringe technique, have been used by computational fluid dynamicists to check their finite-difference results against our experimental data. It will be shown that the isopycnic structure previously reported by us differs in detail, in most cases, from that of the present study. The major difference arises from the fact that it was only possible previously to obtain discrete points on isopycnics and along the wedge surface. Consequently, the results obtained before were not as accurate. Comparisons were made of actual wall-density distributions with numerical simulations of the density contours of the various flows obtained by a number of authors. Each numerical method displays its advantages and disadvantages in describing the details of the flow fields. The present experimental results for air are new. They are of great interest from a practical viewpoint. The experiments in argon were redone to provide better data for a gas free from real-gas effects in the range of initial conditions considered, in order to simplify the computations in the numerical simulations. Although the recent numerical simulations are better than those reported previously, additional efforts are required to improve the predictions of the shape, location and values of the isopycnics and other flow isolines in the various regions and along the wall, and to render the predictions free of computer ‘noise’. It is worth noting that real-gas effects did not play any significant role in determining the various wave systems in RR, SMR, CMR and DMR; a different claim was made in our previous work. Relaxation of nitrogen in air can be seen however, at the highest shock Mach numbers (Ms = 7.19 and 8.70), with relaxation lengths in good agreement with accepted predictions.

The weak reflection of monochromatic water waves is studied for the cases of slowly varying water depth. A coupled system of equations for the forward-scattering (transmitted) and the backward-scattering (reflected) wavefields are derived from the mild-slope equation (Smith & Sprinks 1975). Parabolic approximation is then used to simplify the equations for the diffraction factor. An iterative numerical scheme is proposed to compute the resulting equations. The scheme converges very quickly for the cases of weak reflection. The accuracy of the present approach is shown by comparing with numerical results obtained by a hybrid finite-element formulation.

A closed-loop multilayer salt-stratified water channel allowing independent adjustment of velocity and density profiles has been developed for the study of stably stratified shear flows. Each of ten gravity-driven layers are adjustable for velocities from 0 to 30 cm/s and densities from 1.0 to 1.1 g/cm3. A turbulence-management section located at the inlet manifold successfully suppresses undesired turbulence due to the ten mixing layers generated at this inlet. This management section also smooths the inherent step structure of the initial velocity and density profiles. Velocity profiles can be maintained throughout an experiment, whereas density profiles change slowly in time. An inner core region of linear stable density gradient will remain stationary for a time dependent on the initial stratification, velocity profile and mixing conditions. Temperature is observed to increase at a rate near 1 °C per hour due to internal heating in the system. A description of calibration techniques and temperature-correction methods for the velocity and conductivity instrumentation is described.

The evolution of unsheared grid-generated turbulence in a stably stratified fluid was investigated in a closed-loop salt-stratified water channel. Simultaneous single-point measurements of the horizontal and vertical velocity and density fluctuations were obtained, including turbulent mass fluxes central in understanding the energetics of the fluctuating motion. When the buoyancy lengthscale was initially substantially larger than the largest turbulent scales, the initial behaviour of the velocity and density fields was similar to that in the non-stratified case. With further downstream development, the buoyancy lengthscale decreased while the turbulence scale grew. Deviations from neutral behaviour occurred when these two lengthscales became of the same order, after the initially larger inertial forces associated with the initial kinetic energy had become weaker and buoyancy forces became important.

Buoyancy forces produced anisotropy in the largest scales first, preventing them from overturning, while smaller-scale isotropic turbulent motions remained embedded within the larger-scale wave motions. These small-scale motions exhibited classical turbulent behaviour and scaled universally with Kolmogorov length and velocity scales. Eventually even the smallest scales of the decaying turbulence were affected by buoyancy, all isotropic motions disappeared, and Kolmogorov scaling failed. The turbulent vertical mass flux decreased to zero for this condition, indicating that the original turbulent field had been completely converted to random internal wave motions.

The transition from a fully turbulent state to one of internal waves occurred rapidly in a time less than the characteristic time of the turbulence based on the largest-scale eddies found in the flow at transition. The dissipation rate for complete transition to a wave field was found to be of the order of εt = 24.5νN2, where ν is the kinematic viscosity and N the Brunt-Väisälä frequency. This is in fairly good agreement with the value 30νN2 predicted by Gibson (1980, 1981).

The present experiments have determined quantitative limits on the range of active turbulent scales in homogeneous stratified turbulence, in terms of an upper limit near the buoyancy lengthscale and a lower limit determined by viscosity in the usual way. This description has been used here to help explain and assimilate the results from the earlier stratified grid-turbulence experiments of Lin & Veenhuizen (1975) and Dickey & Mellor (1980). While some of the features of the present observations may be qualitatively seen in the numerical simulations of the problem of Riley, Metcalfe & Weissman (1981), there are fundamental differences, probably due in part to large differences in initial lengthscale ratios and in the limited range of scales attainable in numerical simulations. The present experiments may serve as a useful test case for future modelling and interpretation of the behaviour of turbulence in stratified flows observed in the oceans and atmosphere.

This experimental study extends our earlier work (Hsu, Hsu & Street 1981) on U∞/c = 1.54 to U∞/c = 0.88, 1.10, 1.36 and 1.87, where U∞ is the mean-free-stream wind velocity and c is the celerity of the water wave. This was accomplished by changing the speed of the turbulent wind, while the water wave was maintained at a frequency of 1.0 Hz and wave slope of 0.1. The consistency between the results of the present and earlier experiments is established. The experimental results indicate that the mean velocity of the typical log-linear profile basically follows the waveform. However, the surface condition for the wind is regarded as supersmooth because the mean turbulent shear stress supported by the current is relatively lower than that supported by a smooth flat plate. The structure of the wave-induced velocity fields is found to be very sensitive to the height of the critical layer. When the critical height is high enough that most of the wave-induced flow in the turbulent boundary layer is below the critical layer, the structure of the wave-induced velocity field is strongly affected by the Stokes layer, which under the influence of the turbulence can have thickness comparable to the boundary-layer thickness. When the critical height is low enough that most of the wave-induced flow in the boundary layer is above the critical layer, the structure of the wave-induced velocity fields is then strongly affected by the critical layer. The structure of the critical layer is found to be nonlinear and turbulently diffusive. This implies that the inclusion of both the nonlinear and the turbulent terms in the wave-perturbed momentum equations is essential to success in the numerical modelling. The response of the turbulent Reynolds stresses to the wave is found to depend on the flow regimes near the interface or in the boundary layer. Near the interface, the wave-induced turbulent Reynolds stresses are found to be produced mainly from the stretching and changing in the direction of the turbulent velocity fluctuations due to the surface displacements. In the boundary layer, the eddy-viscosity-type relation between the wave-induced turbulent Reynolds stresses and the wave-induced velocities as found in Hsu et al. (1981) for U∞/c = 1.54 is also found to hold for the other U∞/c values of this study.

When m different species of small particles are dispersed in fluid the existence of a (small) spatial gradient of concentration of particles of type j is accompanied, as a consequence of Brownian motion of the particles, by a flux of particles of type i. The flux and the gradient are linearly related, and the tensor diffusivity Dij is the proportionality constant. When the total volume fraction of the particles is small, Dij is approximately a linear function of the volume fractions ϕ1, ϕ2, …s, ϕm, with coefficients which depend on the interactions between pairs of particles. The complete analytical expressions for these coefficients given here for the case of spherical particles are a linear combination of the second virial coefficient for the osmotic pressure of the dispersion (measuring the effective force acting on particles when there is a unit concentration gradient) and an analogous virial coefficient for the bulk mobility of the particles. Extensive calculations of the average velocities of the different species of spherical particles in a sedimenting polydisperse system have recently been published (Batchelor & Wen 1982) and some of the results given there (viz. those for small Péclet number of the relative motion of particles) refer in effect to the bulk mobilities wanted for the diffusion problem. It is thus possible to obtain numerical values of the coefficient of ϕk in the expression for Dij, as a function of the ratios of the radii of the spherical particles of types i, j and k. The numerical values for ‘hard’ spheres are found to be fitted closely by simple analytical expressions for the diffusivity; see (4.6) and (4.7). The dependence of the diffusivity on an interparticle force representing the combined action of van der Waals attraction and Coulomb repulsion in a simplified way is also investigated numerically for two species of particles of the same size. The diffusivity of a tracer particle in a dispersion of different particles is one of the many special cases for which numerical results are given; and the result for a tracer ‘hard’ sphere of the same size as the other particles is compared with that found by Jones & Burfield (1982) using a quite different approach.

Recently, Williams (1982) carried out a study of the initial development of the viscous flow in the vicinity of a sharp trailing edge on a symmetrical body impulsively set into motion. The numerical results of that study indicate that, for small or moderate trailing-edge angles, a moving singularity occurs in the solution fairly early in the flow development and that the flow in the vicinity of this singularity exhibits the characteristics of unsteady separation. In the present study, this problem is re-examined with the objective of providing convincing evidence for the existence of such a singularity and describing its structure.

A detailed asymptotic theory is developed for the structure of the boundary-layer solution in the vicinity of the moving singularity. The major features of this theory are then tested by comparison with careful numerical solutions carried as closely as possible to the singularity. The agreement between the asymptotic theory and the numerical integration of a the boundary-layer equations is favourable, and it is concluded that the proposed structure of the singularity is correct for unsteady flow past a sharp trailing edge that is impulsively set into motion.

The nonlinear time-dependent adjustment of a homogeneous rotating-channel flow to the sudden obtrusion of an obstacle is studied. Solutions are obtained using a Lax–Wendroff numerical scheme which allows rotating breaking bores and jumps to form and be maintained. The flow upstream of the obstacle is found to be completely blocked, partially blocked (and hydraulically controlled), or unobstructed depending upon the height of the obstacle. Partial blockage is accomplished through the excitation of a combination of nonlinear Kelvin waves, some of which steepen into interfacial shocks. Riemann invariants for the Kelvin waves are found, and jump conditions on mass, momentum and potential vorticity for the shocks are discussed. The shocks are surrounded by dispersive regions of Rossby deformation scale, and the potential vorticity of passing fluid is altered at a rate proportional to the differential rate of energy dissipation along the line of breakage. For the special case of initially uniform potential vorticity the asymptotic state is found as a function of the initial conditions.

The central question addressed here concerns the occurrence of laminar separation near a non-symmetric trailing edge, on one surface only of an airfoil, and whether or not such an event heralds a ‘catastrophic stall’ in the sense that the flow structure changes significantly from the triple-deck or interactive-boundary-layer form holding for attached flow. Virtually all previous works have conjectured, assumed or argued that there is such a catastrophic stall. The present work, however, points (strongly, we believe) to the opposite view, based on a combination of analytical and numerical grounds. First, the argument for a catastrophic stall, although tempting, is shown to contain a fundamental flaw. Secondly, the present numerical work deliberately aims at including the most important separated-flow features, the acknowledgement of the discontinuities at the trailing-edge station and the effects of reversed flow, in a systematic fashion. This appears to be the first such attempt. As a result the trailing-edge requirements are found to be swept upstream, forcing any flow reversal on just one surface to be followed by a reattachment, however abruptly, just before the trailing-edge point. Thirdly, an analysis of the nearly separated and the just-separated regimes confirms the natural emergence of the reattachment phenomenon and ties in closely with the observed numerical features. In particular, the distance of the reattachment point from the trailing edge is found to be of the tiny order $\overline{\triangle^4} $ or less, where $\overline{\triangle}$ is the small upstream separation distance. Finally, there is shown to be a logical tie-in also with trailing-edge flows involving two-sided separation where no catastrophic stall arises. It is concluded that there is no catastrophic stall and that inter alia the triple-deck/interactive-boundary-layer approach can continue to be used with one-sided separation present.

The study implies some fairly striking features associated with one-sided separating flows, but these do bear a firm resemblance to recent laminar and even turbulent flow computations and experiments. This indicates that, contrary to previous proposals, such computations and experiments are explicable within the realms of interactive-boundary-layer theory.

This paper reports a fundamental study of the fluid dynamics inside a triangular (attic-shaped) enclosure with cold upper wall and warm horizontal bottom wall. The study was undertaken in three distinct parts. In the first part, the flow and temperature fields in the cavity are determined theoretically on the basis of an asymptotic analysis valid for shallow spaces (H/L → 0, where H and L are the attic height and length). It is shown that in the H/L → 0 limit the circulation consists of a single elongated cell driven by the cold upper wall. The net heat transfer in this limit is dominated by pure conduction. In the second part of the study, the transient behaviour of the attic fluid is examined, based on a scaling analysis. The transient phenomenon begins with the sudden cooling of the upper sloped wall. It is shown that both walls develop thermal and viscous layers whose thicknesses increase towards steady-state values. The criterion for the existence of distinct thermal layers in the steady state is (H/L)½RaH¼ > 1, where RaH is the Rayleigh number based on attic height. The corresponding criterion for distinct viscous wall jets is (H/L)½RaH¼ Pr−½ > 1, where Pr is the Prandtl number. The third phase of this study focused on a complete sequence of transient numerical simulations covering the ranges H/L = 0.2, 0.4, 1; RaH/Pr = 10, 103, 105; Pr = 0.72, 6. The numerical experiments verify the flow features described theoretically in the first two parts of the study. The effect of thermal convection on the net heat transfer between the bottom and top walls is illustrated numerically. Finally, the transient numerical experiments show that in the present parametric domain the single-cell circulation pattern is stable with respect to the Bénard instability expected in fluid layers heated from below.

The principle and basic physics of a new type of liquid-in-air jet are described. This jet is generated by a primary fluid jet that emerges from a nozzle below the surface of a stationary (secondary) fluid. After breaking the surface, the jet consists of the central primary jet surrounded by a sheath of secondary fluid which has been entrained by the primary jet during its passage through the secondary fluid. Normally the flow in this compound jet is laminar, and it breaks up into drops due to capillary instability.

In this paper the generation of compound jets is discussed, and the flow pattern in the jet is studied experimentally both in stable and unstable conditions. Three different types of instabilities can be elicited on the jet. The jet mechanism is described quantitatively, and expressions for the jet velocity and some other parameters are derived.

Approximations in statistical turbulence theory often rely on modelling the decay in time of velocity correlations with a simple exponential decay. The decay rate is viewed as a renormalized viscosity. The three simplest implementations of this approximation scheme were originally given independently by Kraichnan, Edwards and Leslie. Each of these investigators used a different formalism and each achieved different renormalization prescriptions. These three different results are reexamined here entirely in terms of direct-interaction theory. The difference in the prescriptions of Kraichnan and Leslie is shown to be the product of different definitions of renormalized viscosity. Edwards’ prescription is shown to result from an inconsistent identification of the non-stationary energy-spectrum relaxation rate with the viscosity. An assessment of the validity of the Markovian closure approximation, and a prescription for non-stationary renormalized viscosity are provided.

A viscous–inviscid interaction procedure is presented for computing incompressible separation bubbles in two-dimensional flows. The analysis consists of the solution of the inviscid-flow equations with a conformal-mapping method and the solution of the boundary-layer equations with an inverse procedure. The boundary-layer equations employ the Cebeci–Smith algebraic eddy-viscosity formulation. The coupling between the inviscid and boundary-layer equations is established through the Hilbert integral by using Veldman's suggestion. An empirical method is used to calculate the location of transition, which is found to play a key role in predicting the behaviour of separating flows. Numerical solutions are presented for transitional bubbles on an NACA 663-018 airfoil at two angles of attack for a chord Reynolds number of 2 × 106. Comparisons wth experiment show that the flow properties of the separation bubbles can be predicted very well with this procedure provided that an accurate estimate of transition location is made.

Measurements of a steady free-stream, nominally two-dimensional, separating turbulent boundary layer have been reported in earlier parts of this work. Here measurements are reported that show the effects of frequency on sinusoidal unsteadiness of the free-stream velocity on this separating turbulent boundary layer at reduced frequencies of 0.61 and 0.90. The ratio of oscillation amplitude to mean velocity is about 1/3 for each flow.

Upstream of flow detachment, hot-wire anemometer measurements were obtained. A surface hot-wire anemometer was used to measure the phase-averaged skin friction. Measurements in the detached-flow zone of phase-averaged velocities and turbulence quantities were obtained with a directionally sensitive laser anemometer. The fraction of time that the flow moves downstream was measured by the LDV and by a thermal flow-direction probe.

Upstream of any flow reversal or backflow, each flow behaves in a quasisteady manner, i.e. the phase-averaged flow is described by the steady free-stream flow structure. The semilogarithmic law-of-the-wall velocity profiles applies at each phase of the cycle. The Perry & Schofield (1973) velocity-profile correlations fit the mean and ensemble-averaged velocity profiles near detachment.

After the beginning of detachment, large amplitude and phase variations develop through each flow. Unsteady effects produce hysteresis in relationships between flow parameters. As the free-stream velocity during a cycle begins to increase, the detached shear layer decreases in thickness, and the fraction of time $\hat{\gamma}_{{\rm p}u} $ that the flow moves downstream increases as backflow fluid is washed downstream. As the free-stream velocity nears the maximum value in a cycle, the increasingly adverse pressure gradient causes progressively greater near-wall backflow at downstream locations while $\hat{\gamma}_{{\rm p}u}$ remains high at the upstream part of the detached flow. After the free-stream velocity begins to decelerate, the detached shear layer grows in thickness, and the location where flow reversal begins moves upstream. This cycle is repeated as the free-stream velocity again increases.

In both unsteady flows, the ensemble-averaged detached-flow velocity profiles agree with steady free-stream profiles for the same $\hat{\gamma}_{{\rm p}u\min} $ value near the wall when $\partial\hat{\gamma}_{{\rm p}u\min}/\partial\hat{t} < 0 $. However, the reduced-frequency k = 0.90 flow has much larger hysteresis in ensemble-averaged velocity profile shapes when $\partial\hat{\gamma}_{{\rm p}u\min}/\partial{t} \geqslant 0 $. Larger and negative values of the profile shape factor $\hat{H}$ occur for this flow during phases when the non-dimensional backflow is greater and $\hat{\gamma}_{{\rm p}u\min}\rightarrow 0.01$.

We examine the structure of nominally wedge-shaped or conical premixed flames, of the type that stand at the mouth of a slot or Bunsen burner. A hydrodynamic analysis, justified for slender flames, accounts for the broad features of the flow field, but within a thin zone whose location defines the shape of the flame, heat-conduction, species diffusion and chemical reaction must be accounted for. A simple mathematical description is possible there in the asymptotic limit of infinite activation energy. Near the very extremity of the tip, where the reaction zone is close to the flame axis, this elementary description is no longer valid, and the combustion field is characterized by a free-boundary problem of Stefan type with nonlinear field equations. The numerical treatment of this problem is based on weak solution techniques.

A reciprocating oscillatory turbulent flow in a rectangular duct is investigated experimentally by making use of a laser-Doppler velocimeter, hot-wire anemometers as well as electronic digital sampling and processing equipments.

The profiles of the mean velocity, the turbulence intensities, the Reynolds stress and the turbulent-energy production rate are compared for the accelerating and decelerating phases.

The characteristics of such a flow are quite different from wall turbulence which is steady in the mean. In the accelerating phase, turbulence is triggered by the shear instability at a slight distance from the wall but is suppressed and cannot develop. However, with the beginning of flow deceleration, turbulence grows explosively and violently and is maintained by the bursting type of motion.

The turbulent-energy production becomes exceedingly high in the decelerating phase, but the turbulence is reduced to a very low level at the end of the decelerating phase and in the accelerating stage of reversal flow. Spectra and spatial correlations for the various phases are compared. The spectral decay in the high-frequency range for the decelerating phase with high turbulence is far steeper than that of Kolmogorov's −5/3 power law, indicating remarkably high energy dissipation by high-frequency turbulence.

Notwithstanding the great difference between the ensemble-averaged characteristics of the oscillatory flow and those of steady wall turbulence, its basic processes such as ejection, sweep and interactions directed towards and away from the wall are the same as those of ‘steady’ wall turbulence.

The existence of modes of compressible fluid flow involving a separation of variables into a similarity solution in two dimensions and one-dimensional flow in the third is demonstrated. The numerical integration of such flows by a modified von Neumann–Richtmyer scheme is proposed, and the stability conditions investigated, showing that a generalized Courant–Friedrichs–Lewy condition is necessary. The inclusion of dissipation in the forms of artificial viscosity and thermal conduction into the model is discussed. The results of some test calculations are presented to demonstrate the behaviour of this model.

By considering the evolution of a localized voidage disturbance imposed on an otherwise uniformly fluidized bed we are able to determine the dominant effects of the many terms in the continuum equations of motion governing a fluidized bed. For small perturbations a linearized theory is developed, showing that the stability of the uniform state is critically dependent upon the particle-phase collisional pressure and the flow rate of the uniform state, while the effect of particle phase viscosity is shown to be purely dispersive. When the uniform state is stable, the disturbance is shown to develop into a decaying pulse followed by a decaying wavetrain.

For finite-amplitude disturbance, nonlinear effects are considered. These are shown to give rise to the propagation of high voidage gradients through the bed. Having established that such voidage fronts will develop, a detailed study of their structure is made. This gives strong indications that, for flow rates at which the uniform state is unstable, the bed will restabilize into a quasisteady periodic state.

A theoretical study is made of the onset of cellular convection in a shallow two-dimensional container of fluid when the temperature difference between the horizontal boundaries is a monotonic function of horizontal distance. The typical lengthscale of the horizontal variation in the temperature difference is taken to be of the same order of magnitude as the length of the container, and both are much longer than the depth of the container. The resulting flow may be regarded as consisting of two parts, a steady base flow and a disturbed flow. It is found that a weak disturbance taking the form of transverse rolls is first set up near the endwalls, but as the temperature difference between the horizontal boundaries is uniformly increased, an instability in the form of longitudinal rolls takes place near the hotter end of the container. This description is in good qualitative agreement with experiment.