A vertically oscillating grid is used to simulate boundary mixing in the laboratory. The oscillation of the grid creates a turbulent mixing region in its vicinity, and mixing within this region creates a step-like structure in an initial density distribution which varies linearly with depth. If the initial density varies only at the boundary between two homogeneous layers, the same grid turbulence generates additional steps above and below the initial one. The steps, in turn, drive multiple intrusions of mixed fluid away from the boundary and into the non-turbulent interior of the fluid. A compensat- ing return flow carries fluid from the interior into the turbulent mixing region. From the data, the inference is made that the intrusions make a negligible direct contribution to the vertical mass transport. An analytical model of the intrusions, which employs only molecular values of the transport coefficients and also demonstrates negligible vertical mass transport, is consissent with the laboratory observations.

Nevertheless, the data indicate that the fluid eventually reaches a homogeneous density by means of the gradual change of the gradient at a rate which is essentially the same both near the grid and far from it. For an initially uniform density profile this change occurs at all heights simultaneously, and for an initial density step it occurs preferentially near the step. Thus in both cases the interior flow must include slow vertical advection away from the horizontal centre plane. These advective currents can be made part of a consistent dynamic model, the buoyant equivalent of the spin- down in a rotating flow, provided that the net effect of the grid mixing includes a decrease in the local slope of the density gradient. This mode of adjustment explains satisfactorily the experimentally observed negligible horizontal density gradients. For the cme of an initially uniform density stratification, the shape of the evolving density gradient is not accounted for. In particular, it is not clear why the gradient changes at mid-depth almost simultaneously with variations at top and bottom boundaries.

The vertical mass flux is found to be independent of container length, and it increases with grid frequency of oscillation, amplitude of oscillation and with the mean density gradient.

Measurements of the spreading rates of gravity-driven currents at both the surface and the bottom of a fluid layer of different density are reported. For the case of a constant inflow rate the spreading relations are derived by estimating the order of magnitude of the forces involved. After an initial balance between gravity and inertia forces the final spreading phase is governed by the balance between gravity and viscous forces. For the latter flow regime, measurements in plane and axisymmetric flow geometries agree well with the spreading relations for gravity currents with a no-slip boundary. The proportionality factor, which is not predicted from this model, is then determined from the measurements and a good agreement is found with the theoretical value derived in the accompanying paper by Huppert (1982).

The viscous gravity current that results when fluid flows along a rigid horizontal surface below fluid of lesser density is analysed using a lubrication-theory approximation. It is shown that the effect on the gravity current of the motion in the upper fluid can be expressed as a condition of zero shear on the unknown upper surface of the gravity current. With the supposition that the volume of heavy fluid increases with time like tα, where α is a constant, a similarity solution to the governing nonlinear partial differential equations is obtained, which describes the shape and rate of propagation of the current. The viscous theory is shown to be valid for t [Gt ] t1, when α < αc and for t [Lt ] t1 when α > αc, where t1, is the transition time at which the inertial and viscous forces are equal, with $\alpha_{\rm c} = \frac{7}{4}$ for a two-dimensional current and αc = 3 for an axisymmetric current. The solutions confirm the functional forms for the spreading relationships determined for α = 1 in the preceding paper by Didden & Maxworthy (1982), as well as evaluating the multiplicative factors appearing in the relationships. The relationships compare very well with experimental measurements of the axisymmetric spreading of silicone oils into air for α = 0 and 1. There is also very good agreement between the theoretical predictions and the measurements of the axisymmetric spreading of salt water into fresh water reported by Didden & Maxworthy and by Britter (1979). The predicted multiplicative constant is within 10% of that measured by Didden & Maxworthy for the spreading of salt water into fresh water in a channel.

In this paper we examine the low-frequency sound radiated when various types of unsteady flow interact with a jet pipe. In each case we solve the problem exactly by the Wiener-Hopf technique, producing results valid for arbitrary internal and ex- ternal Mach numbers and temperatures, discuss the importance of a Kutta condition at the duct exit, and provide an interpretation, in elementary terms, of the radiated sound field using the Lighthill acoustic analogy. A central feature is that the solutions are always obtained subject to a causality requirement, regardless of whether or not a Kutta condition is imposed at the pipe lip.

When low-frequency sound propagates down the jet pipe, little of it reaches the far field, and the major disturbance outside the pipe is that associated with the jet instability waves. At subsonic jet speeds and low-enough Strouhal number these waves transport kinetic energy at a rate precisely balancing the loss of acoustic energy from the pipe, resulting in a net attenuation of the sound power. For supersonic jet condi- tions a further wave motion, the unsteady-flow counterpart of the steady wave struc- ture of an imperfectly expanded jet, is present in addition to the instability wave. We use the Lighthill acoustic analogy to show that, for high-enough jet Mach number and temperature, the sound radiation is caused largely by quadrupole sources arising from the jet instability waves. An alternative interpretation uses the acoustic analogy incorporating a mean flow due to Dowling, Ffowcs Williams and Goldstein, and expresses the far-field sound as the sum of contributions from monopoles and dipoles distributed over the duct exit. The directivity and power of the calculated far-field sound are in good agreement with experiments.

We also calculate the sound scattered by the jet pipe when there is an incident external sound field, and show a previously published result to be in error. In general, the flbw phenomena produced by internal and external incident sound fields are similar. Finally, we discuss the effects of nozzle contraction. We find that the radiated sound field is little changed in character, but that the reflection properties of the nozzle may be drastically altered.

Pressure-driven flow along a channel in the presence of an applied magnetic field which is periodic in the streamwise direction is considered. The configuration is such that the transverse component of field By, is non-zero on the centreline y = 0, but its streamwise average 〈By〉 is zero. In this situation, flux expulsion due to reconnection of field lines occurs when the pressure gradient is sufficiently large. This leads to a decrease in the Lorentz forces, hence to an acceleration of the flow, and hence to stronger flux expulsion. When viscous effects are weak (i.e. at high Hartmann number) this creates a runaway effect, which appears at a critical value of the pressure gradient. This critical value is determined in the inviscid limit, and numerical and analytical methods are used to explore the associated ‘cusp-catastrophe’ behaviour when effects of weak viscosity are taken into account.

Conditional averages of longitudinal, normal velocity and temperature fluctuations and of their products have been obtained in a slightly heated boundary layer with zero pressure gradient over a momentum-thickness Reynolds-number range 990 ≤ Rm ≤ 7100. These averages are based on the identification of coherent temperature fronts that extend across most of the layer. The average period between fronts is approximately independent of Rm when Rm is greater than about 1500. The streamwise length scale of the fronts and the magnitude of velocity and temperature derivatives associated with the fronts scale on the thickness of the layer except for Rm, less than about 3000. This scaling is consistent with the Reynolds-number independence, for Rm greater than about 3000, of both mean and turbulent velocity and temperature fields. Conditional averages are discussed in the context of Head & Bandyopadhyay's (1978) suggestion, based on smoke-flow visualization, that the boundary layer consists almost exclusively of hairpin eddies.

Characteristics of the premixed flame in uniform straining flow are investigated by the technique of activation-energy asymptotics. An inverse method is used, which avoids some of the restrictions of previous analyses. It is shown that this method recovers known results for adiabatic flames. New results for flames with heat loss are obtained, and it is shown that, in the presence of finite heat loss, straining can extinguish flames. A stability analysis shows that straining can suppress the cellular instability of flames with Lewis number less than unity. Strain can produce instability of flames with Lewis number greater than unity. A comparison shows quite good agreement between theoretical deductions and experimental observations of Ishizuka, Miyasaka & Law (1981).

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.

A prescribed shear stress applied to the free surface of a thin liquid layer sets up a steady shear flow. When the shear flow has a linear velocity, profile, Miles, using asymptotic analysis, finds critical values Rc of the Reynolds number above which unstable travelling waves exist. However, Miles omits a term in the normal-stress boundary condition. We correct this omission and solve the appropriate Orr-Sommer-feld system numerically to obtain the critical conditions. For the case of a zero-surface-tension interface, we find that Rc = 34.2, as compared with Miles’ value of Rc = 203. As surface tension increases, Rc asymptotes to the inviscid limit developed by Miles. The critical Reynolds number, critical wavenumber and critical phase speed are presented as functions of a non-dimensional surface tension. We investigate the mechanism of the instability through an examination of the disturbance-energy equation. When the shear flow has a parabolic velocity profile, we find a long-wave instability at small values of the Reynolds number. Numerical methods are used to extend these results to larger values of the wavenumber. Examination is made of the relation between this long-wave instability and profile curvature.

Aspects of the various steady states of Taylor-vortex flow between concentric cylinders have been investigated by means of flow visualization. The experiments have focused principally on the evolution of the primary flow, that is, on the continuum of steady states parametrized by the Reynolds number R, beginning at small R where the primary flow is the only one possible. For any particular aspect ratio Γ, the primary flow develops a well-defined pattern of cells at higher R, but then other steady cellular flows (secondary modes) are also possible. The observations presented demonstrate mutations of the primary flow as Γ is varied through critical values: its R-dependent evolution is thereby switched from one to another array of cells realized at higher R. In each of four cases (4–6, 6–8, 8–10 and 10–12 cells), the mutation is shown to involve hysteresis of the primary-flow locus and complicated interactions with secondary modes.

Following a description of the apparatus in §2, a discussion of the experimental method used to observe the often delicate hysteresis effects is given in §3. The experimental results in §4 are in broad agreement with abstract mathematical ideas that have been previously shown to bear on the Taylor experiments, but several new and surprising features, such as the coupling between pairs of cells, have been uncovered.

A representative set of experimental observations is presented demonstrating a remarkably large number of distinct steady flows that all subsist on the same boundary conditions. Commentary on the significance of these and related findings re-emphasizes previous proposals about the interpretation of the Taylor experiment.

Taking account of slenderness of many lakes, a hydrodynamic model is developed. The three-dimensional differential equations are formulated in a curvilinear Co-ordinate system along the ‘long’ axis of the lake. Applying the method of weighted residuals and expanding the field variables with shape functions over the cross-sections, approximate equations for the fluid motion are derived. The emerging equations form a cross-sectionally discretized set of spatially one-dimensional partial differential equations in the longitudinal lake direction. At first, these channel equations are presented for unspecified fluid properties and arbitrary shape functions, leaving applications possible for inviscid or viscous fluids with arbitrary closure conditions. The channel equations are subsequently specialized for Cauchy series as shape functions. For the free oscillation the simplest channel model is shown to reduce to the classical Chrystal equation. A first-order linear channel model is deduced. It exhibits the essential features of gravitational oscillations in rotating basins, in that it provides wave-type solutions with the characteristics of Kelvin and Poincaré waves. This paper presents the derivation of the equations. Their application to ideal and real basins is deferred to several further papers.

A first-order channel model for fluid motion in long homogeneous lakes, as derived in detail by Raggio & Hutter (1982a), is presented. This model describes the motion through spatially one-dimensional boundary-value problems and is deduced by representing each field variable by cross-sectional expansions with a constant and a linear term. Various wave solutions of the governing equations applied to rectangular basins with flat bottom are presented. It is demonstrated that for moderate rotation speeds of the Earth and for elongated basins of a homogeneous fluid the main features of gravitational oscillations are predicted by the model. In particular Kelvin- and Poincaré-type waves are shown to exist. Furthermore, conditions of complete and incomplete reflections of Kelvin waves and free oscillations are discussed. The results corroborate the suitability of the model as far as wave motion in rectangular basins is concerned, but equally elucidate the physics behind them, which is less transparent when attacked with the full theory. The application of the model to basins of different shapes and to a real lake is reserved to a companion paper.

The extended channel model derived and analysed in two previous articles is further developed by investigating free oscillations and forced motion in natural enclosed basins. Firstly, a zeroth-order model is analysed. In this model, field variables are expressed as a product of a single known cross-sectional shape function and an unknown function of time and the co-ordinate along the lake axis. Conditions are discussed under which this zeroth-order model is meaningful, and it is shown that under normal circumstances Coriolis effects must be ignored. Subsequently the general Nth-order channel model is applied to the Lake of Lugano. It is shown that eigedrequencies and amphidromic systems are well predicted in such channel-like lakes. The paper ends with a discussion on the selection of shape functions and with further applications and limitations of the channel model.

Numerical simulations of single-cell, two-dimensional, time-dependent thermal convection in a square cross-section of fluid-saturated porous material heated uniformly from below reveal a series of transitions between distinct oscillatory dynamical regimes. With increasing Rayleigh number R, the flow first evolves from steady-state behaviour into periodic motion with a single frequency f which depends on R approximately according to $f\propto R^{\frac{7}{8}}$ the transition Rayleigh number lies between about 380 and 400. At a value of R between about 480 and 500 the flow transforms into a fluctuating state characterized by two frequencies. Soon thereafter, for R between about 500 and 520, it reverts back to single-frequency periodic behaviour with f approximately proportional to $R^{\frac{3}{2}}$. The two frequencies in the narrow transition regime may be locked to a rational ratio, in which case the flow is periodic, or they may be commensurate, in which case the flow is quasi-periodic. The spectral characteristics of numerical realizations of unsteady convection and the occurrences of transitions therein are highly dependent on truncation level in Galerkin schemes or resolution in finite-difference approaches.

A quantitative theoretical model is developed to describe the time-dependent draining of an initially uniform-thickness fluid squeeze film between an infinitely flexible membrane-bound fluid cell and a planar rigid surface or between two symmetrically loaded cells subject to impulsive loading. The solution of the coupled nonlinear membrane-fluid-film equations shows that two characteristic times and lengthscales are required to describe the membrane deformation and draining behaviour of the fluid film. The early-time behaviour is strikingly different from that predicted by elastohydrodynamic squeeze-film theory (Christensen 1962), where the local elastic deformation of the boundary is not controlled by membrane tension but is pro-portional to the local film pressure. While fluid trapping occurs in both cases, a bidirectional flow is set up during the early-time period in the membrane squeeze film owing to the establishment of an off-axis pressure maximum near the edge of the near-contact area. Fluid is driven radially inward, causing upwelling of the membrane in the central region, and driven radially outward near the edge of the contact area., causing this region to form a narrow fluid gap. After the narrow-edge region hasformed, the off-axis pressure maximum gradually disappears and is replaced by a pressure plateau in the interior and a radial outflow at all locations that is similar to the elasto-hydrodynamic squeeze film. The present problem is closely related to the fluid films studied by Hartland (1967, 1968, 1969), Jones & Wilson (1978) and others when a, small spherical particle or fluid droplet rises or settles under gravity towards a uniform-tension fluid-fluid interface. These studies have theoretically and experimentally examined the long-time drainage of the film after the narrow edge region has formed and the fluid-trapping phenomenon is established. The solutions to the initial-value problem described herein show how this asymptotic quasi-steady drainage state is reached.

A simple experiment has been constructed to confirm qualitatively the theoretically predicted short-time behaviour. Experimental photographs graphically illustrate the gradual thickening of the lubricating layer near the origin and the formation and draining of the edge region as predicted by the membrane squeeze-film theory.

Jeffery's solution in bi-polar co-ordinates of the two-dimensional Stokes equations cannot be applied to the low-Reynolds-number flow past two parallel circular cylinders because of severe mathematical difficulties. These difficulties can be overcome by considering the flow field far from the cylinders and then modifying the solution near the cylinders so that it becomes the inner expansion for an application of the method of matched asymptotic expansions. After the calculation of the drag, lift and moment coefficients of two adjacent equal circular cylinders to O(1) in the Reynolds number R, the analysis is extended to incorporate partially the effects of fluid inertia of order R. The results show fairly good agreement with Taneda's experimental data.

An earlier paper (Gaster 1981) discussed different asymptotic representations of the isolated wave packet that evolved from an impulsive point excitation of a laminar boundary layer. Comparisons of the various asymptotic representations of the integral describing these packets were made on the basis of numerical evbluations of the various approximations together with the direct numerical solutions of the integral. Here the problem is pursued by analytical means, and error estimates are obtained for the different methods used.

In an experimental investigation of the transient processes that occur when a single droplet of butane at the superheat limit vaporizes explosively, short-exposure photographs and fast-response pressure measurements have been used to construct a description of the complete explosion process. It is observed that only a single bubble forms within the drop during each explosion, and that the growth proceeds on a microsecond time scale. An interfacial instability driven by rapid evaporation has been observed on the surface of the bubbles. It is suggested that the Landau mechanism of instability, originally described in connection with the instability of laminar flames, also applies to rapid evaporation at the superheat limit.

The photographic evidence and the pressure data are used to estimate the evapora- tive mass flux across the liquid-vapour interface after the onset of instability. The rate of evaporation is shown to be two orders of magnitude greater than would be predicted by conventional bubble-growth theories that do not account for the effects of instability. An estimate of the mean density within the bubbles during the evaporative stage indicates that it is more than one half of the critical density of butane.

Additional interesting dynamical effects that are observed include a series of toroidal waves that form on the interface between the butane vapour and the external host liquid in the bubble column apparatus after the bubble has grown large enough to contact the outer edge of the drop, and violent oscillations of the bubble that occur on a millisecond time scale, after evaporation of the liquid butane is complete, that cause the disintegration of the bubble into a cloud of tiny bubbles by Rayleigh–Taylor instability.

Time-dependent flows such as occur in breaking surface waves are often most con- veniently described in parametric form, with the coordinate z and velocity potential χ each expressed in terms of a third complex variable ω and the time t.

In this paper we discuss some interesting flows given in terms of elementary functions of ω and t. Included are the Stokes 120° corner flow, the 45° rotor or rotating wedge, and a decelerated upwelling flow, with an exactly plane surface.

Lastly it is shown that a class of cubic flows, which are related to the plane upwelling flow just mentioned, has a free surface that corresponds with remarkable accuracy to the forward face of an overturning, or plunging, breaker.