Results of linear stability calculations and post-stability experimental observations are reported for horizontal fluid layers with upward heat flux bounded below by a stably stratified fluid. Stability calculations were done for several families of continuous and discontinuous temperature distributions, and it was found that as a rule the flow originating in the unstable layer penetrates into the stably stratified region, resulting in increased critical cell size and correspondingly decreased critical Rayleigh number. A notable exception to this occurs for an unstable layer with a linear temperature distribution adjacent to a stable layer of very high stable density gradient. In this case energy pumped from the unstable to the stable region is sufficient to raise the critical Rayleigh number above that of a solid boundary. It is also found that, for density distributions with a more gradual transition between the stable and the unstable regions, the effect of increased cell size upon the critical Rayleigh number is sometimes masked by effects of curvature in the density profile of the unstable region, which tends to increase the critical Rayleigh number. The inadequacy of the usual definition of Rayleigh number to characterize the stability of such complex systems is discussed. Experimentally, such a temperature distribution was produced by radiant energy from above as it was absorbed by the top few centimetres of the fluid. Within an uncertainty of ± 20%, it was found that the critical experimental Rayleigh number agreed with neutral stability calculations. The supercritical convective motion consisted of vertical jets of cool surface fluid which plunged downward into the interior of the fluid. The jets were not arranged in an orderly lattice but were in a constant state of change, each jet having a tendency to merge with a close neighbour. The net loss of jets due to merging was balanced by new jets spontaneously appearing. As Rayleigh number was increased, the mean number of jets and the intermittancy increased proportionally. Temperature scans taken with a movable probe showed that cool surface fluid plunging downward in the jets was confined to a fairly restricted region, the surrounding fluid being quite isothermal.

Shock wave propagation arising from steady one-dimensional motion of a piston in a granular gas composed of inelastically colliding particles is treated theoretically. A self-similar long-time solution is obtained in the strong shock wave approximation for all values of the upstream gas volumetric concentration v0. Closed form expressions for the long-time shock wave speed and the granular pressure on the piston are obtained. These quantities are shown to be independent of the particle collisional properties, provided their impacts are accompanied by kinetic energy losses. The shock wave speed of such non-conservative gases is shown to be less than that for molecular gases by a factor of about 2.

The effect of particle kinetic energy dissipation is to form a stagnant layer (solid block), on the surface of the moving piston, with density equal to the maximal packing density, vM. The thickness of this densely packed layer increases indefinitely with time. The layer is separated from the shock front by a fluidized region of agitated (chaotically moving) particles. The (long-time, constant) thickness of this layer, as well as the kinetic energy (granular temperature) distribution within it are calculated for various values of particle restitution and surface roughness coefficients. The asymptotic cases of dilute (v0 [Lt ] 1) and dense (v0 ∼ vM) granular gases are treated analytically, using the corresponding expressions for the equilibrium radial distribution functions and the pertinent equations of state. The thickness of the fluidized region is shown to be independent of the piston velocity.

The calculated results are discussed in relation to the problem of vibrofluidized granular layers, wherein shock and expansion waves were registered. The average granular kinetic energy in the fluidized region behind the shock front calculated here compared favourably with that measured and calculated (Goldshtein et al. 1995) for vibrofluidized layers of spherical granules.

This paper investigates the non-uniform motion of a thin plate of finite aspect ratio, with a rounded leading edge and sharp trailing edge, executing heaving and pitching oscillations at zero mean lift. Such vertical motions characterize the horizontal lunate tails with which cetacean mammals propel themselves, and the same motions, turned through 90° to become horizontal motions of sideslip and yaw, characterize the vertical lunate tails of certain fast-swimming fishes. An oscillating vortex sheet consisting of streamwise and spanwise components is shed to trail behind the body and it is this additional feature of the streamwise component resulting from the finiteness of the plate that makes this study a generalization of the two-dimensional treatment of lunate-tail propulsion by Lighthill (1970). The forward thrust, the power required, the energy imparted to the wake and the hydromechanical propulsive efficiency are determined for this general motion as functions of the physical parameters defining the problem: namely the aspect ratio, the reduced frequency, the feathering parameter and the position of the pitching axis. The dependence of the thrust coefficient and propulsive efficiency on these physical parameters, for the complete range of variation consistent with the assumptions of the problem, has been depicted graphically.

We report direct numerical simulations of incompressible unsteady open-channel flow. Two mechanisms of turbulence production are considered: shear at the bottom and externally imposed stress at the free surface. We concentrate upon the effects of mutual interaction of small-amplitude gravity waves with in-depth turbulence and statistical properties of the near-free-surface region. Extensions of our approach can be used to study turbulent mixing in the upper ocean and wind–sea interaction, and to provide diagnostics of bulk turbulence.

An axisymmetric capsule, consisting of an incompressible liquid droplet, surrounded by an infinitely thin elastic membrane having a Mooney constitutive behaviour, is suspended into another incompressible Newtonian liquid subjected to an elongational shear flow. The motion and the deformation of the capsule are determined numerically by means of a boundary-integral technique. It is thus possible to reach large deformations, and to study the influence of the initial geometry of the particle, as well as that of the constitutive behaviour of the membrane. In all cases considered here, it appears that there exists a critical value of the non-dimensional shear rate (the capillary number) above which no steady solution can be obtained, and where the capsule continuously deforms. This phenomenon is interpreted as the outset of burst. The model shows also the importance of the sphericity index for the determination of the overall capsule deformability.

A laser light-scattering technique has been used to study the relative particle concentration field in a round turbulent air jet. Measurements were made in the far field of a smoke-marked turbulent jet exhausting into a secondary air stream. Radial distributions of mean particle concentration, concentration fluctuation intensity and intermittency were measured at several streamwise locations. Concentration fluctuation power spectra and the micro- and integral scales of the concentration fluctuations were measured on the jet axis. The effects of ambiguity noise and noise due to optical path attenuation on the performance of the laser light-scattering system are discussed.

We report the results of a complete modal and non-modal linear stability analysis of the electrohydrodynamic flow for the problem of electroconvection in the strong-injection region. Convective cells are formed by the Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in electrohydrodynamics, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable flow. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$\mathit{Re}$ Poiseuille flow yields a more unstable flow in both modal and non-modal stability analyses. Even though the energy analysis and the input–output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centreline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centres of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field.

The destabilization of the stationary basic flow occurring between two disks enclosed by a cylinder is studied experimentally when the radius of the disks is large compared to the spacing. In the explored range of the cell aspect ratio, when one disk only is rotating, circular vortices propagating to the centre are observed above a critical angular velocity. These structures occur naturally but can also be forced by small modulations of the angular velocity of the disk. For each rotation rate the dispersion relation of the instability is experimentally reconstructed from visualizations and it is shown that this dispersion relation can be scaled by the boundary layer thickness measured over the disk at rest. The bifurcation is found to be of supercritical nature. The effect of the forcing amplitude is in favour of a linear convective nature of this instability of the non-parallel inward flow existing above the stationary disk. The most unstable temporal frequency is found to be about four times the frequency of the rotating disk. The evolution of the threshold of this primary instability is described for different aspect ratios of the cell. Finally, two sets of experiments made under transient conditions are presented: one in order to investigate further a possible convective/absolute transition for the instability, and the other to compare with the impulsive spin-down-to-rest experiments of Savas (1983).

We report theoretical and numerical studies of the flow behaviour when a fluid is injected into a confined porous medium saturated with another fluid of different density and viscosity. For a two-dimensional configuration with point source injection, a nonlinear convection–diffusion equation is derived to describe the time evolution of the fluid–fluid interface. In the early time period, the fluid motion is mainly driven by the buoyancy force and the governing equation is reduced to a nonlinear diffusion equation with a well-known self-similar solution. In the late time period, the fluid flow is mainly driven by the injection, and the governing equation is approximated by a nonlinear hyperbolic equation that determines the global spreading rate; a shock solution is obtained when the injected fluid is more viscous than the displaced fluid, whereas a rarefaction wave solution is found when the injected fluid is less viscous. In the late time period, we also obtain analytical solutions including the diffusive term associated with the buoyancy effects (for an injected fluid with a viscosity higher than or equal to that of the displaced fluid), which provide the structure of the moving front. Numerical simulations of the convection–diffusion equation are performed; the various analytical solutions are verified as appropriate asymptotic limits, and the transition processes between the individual limits are demonstrated. The flow behaviour is summarized in a diagram with five distinct dynamical regimes: a nonlinear diffusion regime, a transition regime, a travelling wave regime, an equal-viscosity regime, and a rarefaction regime.

An experimental and theoretical investigation has been carried out to determine the effect of a stable ambient thermal stratification on the developing buoyancyinduced flow adjacent to a flat vertical surface dissipating a uniform heat flux. The nature of the resulting base flow and its instability characteristics, linearized for two-dimensional disturbances, were analysed for Prandtl numbers Pr of 6·7 and 0·733, for several levels of ambient stratification. Stratification was found to cause initial stabilization of the flow but later destabilization downstream. Disturbance growth rates, frequency filtering and amplitude distributions across the boundary region were calculated. These aspects of the disturbance field were measured in a flow generated by an electrically heated metal foil, with artificially introduced two-dimensional disturbances, in water (Pr = 6–7). The experimental results are in very good agreement with the calculations. Measurements of natural transition indicate that a stable ambient stratification delays the onset of transition. A tentative transition-correlating parameter is generalized to include the effect of stratification.

Compound threads and jets consist of a core liquid surrounded by an annulus of a second immiscible liquid. Capillary forces derived from axisymmetric disturbances in the circumferential curvatures of the two interfaces destabilize cylindrical base states of compound threads and jets (with inner and outer radii R1 and aR1 respectively). The capillary instability causes breakup into drops; the presence of the annular phase allows both the annular- and core-phase properties to influence the drop size. Of technological interest is breakup where the core snaps first, and then the annulus. This results in compound drops. With jets, this pattern can form composite particles, or if the annular fluid is evaporatively removed, single drops whose size is modulated by both fluids.

This paper is a study of the linear temporal instability of compound threads and jets to understand how annular fluid properties control drop size in jet breakup, and to determine conditions which favour compound drop formation. The temporal dispersion equation is solved numerically for non-dimensional annular thicknesses a of order one, and analytically for thin annuli (a – 1 = ε [Lt ] 1) by asymptotic expansion in ε. There are two temporally growing modes: a stretching mode, unstable for wavelengths greater than the undisturbed inner circumference 2πR1, in which the two interfaces grow in phase; and a squeezing mode, unstable for wavelengths greater than 2πaR1, which grows exactly out of phase. Growth rates are always real, indicating that in jetting configurations disturbances convect downstream with the base velocity. For order-one thicknesses, the growth rate of the stretching mode is higher for the entire range of system parameters examined. The drop size scales with the wavenumber of the maximally growing wave (kmax). We find that for the dominant stretching mode and a = 2, variations from 0.1 to 10 in the ratios of the annulus to core viscosity, or the tension of the outer surface to that of the inner interface, can result in changes in kmax by a factor of approximately 2. However, for these changes in the system ratios, the growth rate (smax) and the ratio of the amplitude of the outer to the inner interface (Amax) for the fastest growing wave only change marginally, with Amax near one. The system appears most sensitive to the ratio of the density of the annulus to the core fluid. For a variation between 0.1 and 10, kmax again changes by a factor of 2, but Amax and smax vary more significantly with large amplitude ratios for low density ratios. The amplitude ratio of the stretching mode at the maximally growing wave (Amax) indicates whether the film or core will break first. When this ratio is near one, linear theory predicts that the core breaks with the annulus intact, forming compound drops. Except for low values of the density ratio, our results indicate that most system conditions promote compound drop formation.

For thin annuli, the growth rate disparity between modes becomes even greater. In the limit ε → 0, the squeezing growth rate is roughly proportional to ε2 while the stretching mode growth rate is roughly proportional to ε0 and asymptotes to a single jet with radius R1 and tension equal to the sum of the two tensions. Thus, in this limit the growth rate and kmax are independent of the film density and viscosity. The amplitude ratio of the stretching mode becomes equal to one for all wavenumbers; so thin films break as compound drops. Our results compare favourably with previously published measurements on unstable waves in compound jets.

The calculus of variations is employed to find steady divergence-free velocity fields that maximize transport of a tracer between two parallel walls held at fixed concentration for one of two constraints on flow strength: a fixed value of the kinetic energy (mean square velocity) or a fixed value of the enstrophy (mean square vorticity). The optimizing flows consist of an array of (convection) cells of a particular aspect ratio $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\varGamma $. We solve the nonlinear Euler–Lagrange equations analytically for weak flows and numerically – as well as via matched asymptotic analysis in the fixed energy case – for strong flows. We report the results in terms of the Nusselt number ${\mathit{Nu}}$, a dimensionless measure of the tracer transport, as a function of the Péclet number ${\mathit{Pe}}$, a dimensionless measure of the strength of the flow. For both constraints, the maximum transport ${\mathit{Nu}}_{\mathit{MAX}}({\mathit{Pe}})$ is realized in cells of decreasing aspect ratio $\varGamma _{\mathit{opt}}({\mathit{Pe}})$ as ${\mathit{Pe}}$ increases. For the fixed energy problem, ${\mathit{Nu}}_{\mathit{MAX}} \sim {\mathit{Pe}}$ and $\varGamma _{\mathit{opt}} \sim {\mathit{Pe}}^{-1/2}$, while for the fixed enstrophy scenario, ${\mathit{Nu}}_{\mathit{MAX}} \sim {\mathit{Pe}}^{10/17}$ and $\varGamma _{\mathit{opt}} \sim {\mathit{Pe}}^{-0.36}$. We interpret our results in the context of buoyancy-driven Rayleigh–Bénard convection problems that satisfy the flow intensity constraints, enabling us to investigate how the transport scalings compare with upper bounds on ${\mathit{Nu}}$ expressed as a function of the Rayleigh number ${\mathit{Ra}}$. For steady convection in porous media, corresponding to the fixed energy problem, we find ${\mathit{Nu}}_{\mathit{MAX}} \sim {\mathit{Ra}}$ and $\varGamma _{\mathit{opt}} \sim {\mathit{Ra}}^{-1/2}$, while for steady convection in a pure fluid layer between stress-free isothermal walls, corresponding to fixed enstrophy transport, ${\mathit{Nu}}_{\mathit{MAX}} \sim {\mathit{Ra}}^{5/12}$ and $\varGamma _{\mathit{opt}} \sim {\mathit{Ra}}^{-1/4}$.

We investigate experimentally the spontaneous motion of drops and bubbles confined between two plates forming a narrow wedge. Such discoidal objects migrate under the gradient in interfacial energy induced by the non-homogeneous confinement. The resulting capillary driving force is balanced by viscous resistance. The viscous friction on a drop bridging parallel plates is estimated by measuring its sliding velocity under gravity. The viscous forces are the sum of two contributions, from the bulk of the liquid and from contact lines, the relative strength of which depends on the drop size and velocity and the physical properties of the liquid. The balance of capillarity and viscosity quantitatively explains the dynamics of spontaneous migration of a drop in a wedge. Close the tip of the wedge, bulk dissipation dominates and the migrating velocity of drops is constant and independent of drop volume. The distance between the drop and the tip of the wedge is thus linear with time $t$, $x(t) \sim t_0-t$, where $t_0$ is the time at which the drop reaches the tip of the wedge. Far away from the apex, contact lines dominate the friction, the motion is accelerated toward the tip of the wedge and velocities are higher for larger drops. In this regime, it is shown that $x(t) \sim (t_0-t)^{4/13}$. The position and time of the crossover between the two dissipation regimes are used to write a dimensionless equation of motion. Plotted in rescaled variables, all experimental trajectories collapse to the prediction of our model. In contrast to drops, gas bubbles in a liquid-filled wedge behave as non-wetting objects. They thus escape the confinement of the wedge to reduce their surface area. The physical mechanisms involved are similar for drops and bubbles, so that the forces acting have the same mathematical structures in both cases, except for the sign of the capillary driving force and a numerical factor. We thus predict and show experimentally that the trajectories of drops and bubbles obey the same equation of motion, except for a change in the sign of $t_0-t$.

The effects of a favourable pressure gradient (K[les ]4×10−6) and of the Reynolds number (862[les ]Reδ2[les ]5800) on the mean and fluctuating quantities of four turbulent boundary layers were studied experimentally and are presented in this paper and a companion paper (Part 2). The measurements consist of extensive hot-wire and skin-friction data. The former comprise mean and fluctuating velocities, their correlations and spectra, the latter wall-shear stress measurements obtained by four different techniques which allow testing of calibrations in both laminar-like and turbulent flows for the first time. The measurements provide complete data sets, obtained in an axisymmetric test section, which can serve as test cases as specified by the 1981 Stanford conference.

Two different types of accelerated boundary layers were investigated and are described: in this paper (Part 1) the fully turbulent, accelerated boundary layer (sometimes denoted laminarescent) with approximately local equilibrium between the production and dissipation of the turbulent energy and with relaxation to a zero pressure gradient flow (cases 1 and 3); and in Part 2 the strongly accelerated boundary layer with ‘inactive’ turbulence, laminar-like mean flow behaviour (relaminarized), and reversion to the turbulent state (cases 2 and 4). In all four cases the standard logarithmic law fails but there is no single parametric criterion which denotes the beginning or the end of this breakdown. However, it can be demonstrated that the departure of the mean-velocity profile is accompanied by characteristic changes of turbulent quantities, such as the maxima of the Reynolds stresses or the fluctuating value of the skin friction.

The boundary layers described here are maintained in the laminarescent state just up to the beginning of relaminarization and then relaxed to the turbulent state in a zero pressure gradient. The relaxation of the turbulence structure occurs much faster than in an adverse pressure gradient. In the accelerating boundary layer absolute values of the Reynolds stresses remain more or less constant in the outer region of the boundary layer in accordance with the results of Blackwelder & Kovasznay (1972), and rise both in the vincinity of the wall in conjunction with the rising wall shear stress and in the centre region of the boundary layer with the increase of production.

A computationally efficient numerical method is developed to investigate nonlinear interactions between steep surface gravity waves and depth-varying ocean currents. The free-surface boundary conditions are used to derive a coupled set of equations that are integrated in time for the evolution of the free-surface elevation and tangential component of the fluid velocity at the free surface. The vector form of Green's second identity is used to close the system of equations. The closure relationship is consistent with Helmholtz's decomposition of the velocity field into rotational and irrotational components. The rotational component of the flow field is given by the Biot–Savart integral, while the irrotational component is obtained from an integral of a mixed distribution of sources and vortices over the free surface. Wave-induced changes to the vorticity field are modelled using the vorticity transport equation. For weak currents, an explicit expression is derived for the wave-induced vorticity field in Fourier space that negates the need to numerically solve the vorticity transport equation. The computational efficiency of the numerical scheme is further improved by expanding the kernels of the boundary and volume integrals in the closure relationship as a power series in a wave steepness parameter and using the fast Fourier transform method to evaluate the leading-order contribution to the convolution integrals. This reduces the number of operations at each time step from O(N2) to O(NlogN) for the boundary integrals and O[(NM)2] to O(NlogN) for the volume integrals, where N is the number of horizontal grid points and M is the number of vertical layers, making the model an order of magnitude faster than traditional boundary/volume integral methods. The numerical model is used to investigate nonlinear wave–current interaction in depth-uniform current fields and the modulational instability of gravity waves in an exponentially sheared current in deep water. The numerical results demonstrate that the mean flow vorticity can significantly affect the growth rate of extreme waves in narrowband sea states.

The flow around different prolate (needle-like) and oblate (disc-like) spheroids is studied using a multi-relaxation-time lattice Boltzmann method. We compute the mean drag coefficient $C_{D,\unicode[STIX]{x1D719}}$ at different incident angles $\unicode[STIX]{x1D719}$ for a wide range of Reynolds numbers ($\mathit{Re}$). We show that the sine-squared drag law $C_{D,\unicode[STIX]{x1D719}}=C_{D,\unicode[STIX]{x1D719}=0^{\circ }}+(C_{D,\unicode[STIX]{x1D719}=90^{\circ }}-C_{D,\unicode[STIX]{x1D719}=0^{\circ }})\sin ^{2}\unicode[STIX]{x1D719}$ holds up to large Reynolds numbers, $\mathit{Re}=2000$. Further, we explore the physical origin behind the sine-squared law, and reveal that, surprisingly, this does not occur due to linearity of flow fields. Instead, it occurs due to an interesting pattern of pressure distribution contributing to the drag at higher $\mathit{Re}$ for different incident angles. The present results demonstrate that it is possible to perform just two simulations at $\unicode[STIX]{x1D719}=0^{\circ }$ and $\unicode[STIX]{x1D719}=90^{\circ }$ for a given $\mathit{Re}$ and obtain particle-shape-specific $C_{D}$ at arbitrary incident angles. However, the model has limited applicability to flatter oblate spheroids, which do not exhibit the sine-squared interpolation, even for $\mathit{Re}=100$, due to stronger wake-induced drag. Regarding lift coefficients, we find that the equivalent theoretical equation can provide a reasonable approximation, even at high $\mathit{Re}$, for prolate spheroids.

We have obtained a closed-form solution for the sound radiation from multipole sources imbedded in an infinite cylindrical jet with an arbitrary velocity profile. It is valid in the limit where the wavelength is large compared with the jet radius. Simple formulae for the acoustic pressure field due to convected point sources are also obtained. The results show (in a simple way) how the mean flow affects the radiation pattern from the sources. For convected lateral quadrupoles it causes the exponent n of the Doppler factor (1 - M cosθ)−n multiplying the far-field pressure signal to be increased from the value of 3 used by Lighthill to 5.

Kinematics of nominally periodic surf zone waves have been measured in the laboratory using LDA (laser Doppler anemometry), above trough level as well as below, for weakly plunging breakers transforming into bores in shallower water. The aim was to determine, through phase- or ensemble-averaging, periodic flow structures in a two-dimensional vertical plane, from large-scale down to small-scale vortical structures. Coherent multiple vortical structures were evident at the initiation of breaking, becoming elongated along the surface during bore propagation. The initial region is likely to become more extensive as waves become more strongly plunging and could explain the difference in turbulence characteristics between plunging and spilling breakers observed elsewhere. Comparison of vorticity magnitudes with hydraulic-jump measurements showed some similarities during the initial stages of breaking, but these quickly grew less as breaking progressed into shallower water. Period-averaged kinematics and vorticity were also obtained showing shoreward mass transport above trough level and undertow below, with a thick layer of vorticity at trough level and a thin layer of vorticity of opposite rotation at the bed. There were also concentrated regions of mean vorticity near the end of the plunging region. Residual turbulence of relatively high frequency was presented as Reynolds stresses, showing marked anisotrophy. Dynamic pressure (pressure minus its hydrostatic component) was determined from the kinematics. The magnitudes of different effects were evaluated through the depth-integrated Reynolds-averaged Navier–Stokes (RANS) equations, which may be reduced to nine terms (the standard inviscid terms of the shallow-water equations conserving mass and momentum with hydrostatic pressure, and six additional terms), assuming that the complex, often aerated, free surface is treated as a simple interface. All terms were evaluated, assuming that a space/time transformation was justified with a slowly varying phase speed, and the net balance was always small in relation to the maxima of the larger terms. Terms due to dynamic pressure and vertical dispersion (due to the vertical variation of velocity) were as significant as the three terms in the inviscid shallow-water equations; terms involving residual turbulence were insignificant. The r.m.s. (root mean square) variation of each along the slope is highly irregular, with the inertia term due to (Eulerian) acceleration always greatest. This is consistent with complex, though repetitive, coherent structures. Modelling the flow with the shallow-water equations, using the surface elevation variation at the break point as input, nevertheless gave a good prediction of the wave height variation up the slope.

We consider a disturbance that evolves from a strictly linear finite-growth-rate instability wave, with nonlinear effects first becoming important in the critical layer. The local Reynolds number is assumed to be just small enough so that the spatial-evolution, nonlinear-convection, and viscous-diffusion terms are of the same order of magnitude in the interactive critical-layer vorticity equation. The numerical results show that viscous effects eventually become important even when the viscosity is very small due to continually decreasing scales generated by the nonlinear effects. The vorticity distribution diffuses into a more regular pattern vis-a-vis the inviscid case, and the instability-wave growth ultimately becomes algebraic. This leads to a new dominant balance between linear- and nonlinear-convection terms and an equilibrium critical layer of the Benney & Bergeron (1969) type begins to emerge, but the detailed flow field, which has variable vorticity within the cat's-eye boundary, turns out to be somewhat different from theirs. The solution to this rescaled problem is compared with the numerical results and is then used to infer the scaling for the next stage of evolution of the flow. The instability-wave growth is simultaneously affected by mean-flow divergence and nonlinear critical-layer effects in this latter stage of development and is eventually converted to decay. The neutral stability point is the same as in the corresponding linear case, however.

The propagation of gravity fronts of high density ratios has been studied experimentally (exchange flow) and by computer simulation. Non-Boussinesq fronts are known to occur in certain safety problems (chemical spills and fires), and we have investigated seven gas combinations giving density ratios from near unity to well over 20. The results are presented in terms of a density parameter ρ* which remains finite both in the weak (ρ* = 0) and the strong (ρ* = 1) limit. The front velocities, measured by means of hot wires, are found to fall on two distinct curves, one for the slower lightgas fronts and one for the faster heavy-gas fronts. Two fractional depths, Φ = ½ (lock exchange) and Φ = ⅙, have been investigated in detail and results for the interesting case Φ → 0 have been obtained by extrapolation. To aid in the extrapolation and for comparison, all experimental (and some intermediate) cases have been simulated by means of a general purpose CFD-code (PHOENICS). Good agreement is found for cases without convergence problems, i.e. for heavy-gas fronts of density ratio less than 5. Further information on frontal shape etc. has been obtained from visualization. The extrapolations to infinite depth indicate a limiting speed for both the heavy- and light-gas fronts close to the values predicted from shallow-layer theory for the analogous dam-break problem.