The character of transition from laminar to chaotic Rayleigh–Bénard convection in a fluid layer bounded by free-slip walls is studied numerically in two and three space dimensions. While the behaviour of finite-mode, limited-spatial-resolution dynamical systems may indicate the existence of two-dimensional chaotic solutions, we find that, this chaos is a product of inadequate spatial resolution. It is shown that as the order of a finite-mode model increases from three (the Lorenz model) to the full Boussinesq system, the degree of chaos increases irregularly at first and then abruptly decreases; no strong chaos is observed with sufficiently high resolution.

In high-Prandtl-number σ two-dimensional Boussinesq convection, it is found that there are finite critical Rayleigh numbers Ra for the onset of single- and two-frequency oscillatory motion, Ra [gsim ] 60 Rac and Ra [gsim ] 290 Rac respectively, for σ = 6.8. These critical Rayleigh numbers are much higher than those at which three-dimensional convection achieves multifrequency oscillatory states. However, in two dimensions no additional complicating fluctuations are found, and the system seems to revert to periodic, single-frequency convection at high Rayleigh number, e.g. when Ra [gsim ] 800Rac at σ = 6.8.

In three dimensions with σ = 10 and aspect ratio 1/√2, single-frequency convection begins at Ra ≈ 40Rac and two-frequency convection starts at Ra ≈ 50Rac. The onset of chaos seems coincident with the appearance of a third discrete frequency when Ra [gsim ] 65Rac. This three-dimensional transition process may be consistent with the scenario of Ruelle, Takens & Newhouse (1978).

As Ra increases through the chaotic regime, various measures of chaos show an increasing degree of small-scale structure, horizontal mixing and other characteristics of thermal turbulence. While the three-dimensional energy in these flows is still quite small, it is evidently sufficient to overcome the strong dynamical constraints imposed by two dimensions.

Gollub & Benson (1980) found experimentally that frequency modulation of lower boundary temperature Ra(t) = Ra(0) [1 + ε sin ωt] induces chaotic behaviour in a quasi-periodic flow close to transition. We investigate numerically the effects of finite modulation of Ra on the flow far below natural transition (R = 50Rac). By choosing ε = 0.1 and the Rayleigh-number oscillation frequency ω incommensurate with the frequencies of the quasi-periodic motion, transition to chaos is induced early. This result also seems consistent with the Ruelle et al. scenario and leads to the conjecture that periodic modulation of the Rayleigh number of the above form in a two-frequency flow may provide the third frequency necessary for chaotic flow.

For moderate Prandtl number, σ = 1, our results show that two-dimensional flow seems free of oscillation, while three-dimensional flow is vigorously turbulent for Ra [gsim ] 70Rac.

For a number of applications it is important to know the location of the boundary-layer transition from laminar to turbulent. At present it is generally recognized that the onset of turbulence is directly connected with the loss of stability of the initial laminar flow. In the overwhelming majority of cases experimental data on the influence of various factors upon the transition location agree well with the calculated data concerning the influence of these factors on the boundary-layer stability, i.e. the theory of stability may be used successfully to predict various experimental dependencies.

The boundary-layer stability and the transition are considerably affected by heat transfer from the surface of the streamlined body. But, in this case, experimental data on the transition do not always correspond to the results of the stability theory. In particular, experimental works concerning the effect of cooling of the model surface on the supersonic boundary layer transition yield contradictory results (see e.g. Gaponov & Maslov 1980; Morkovin 1969). Some of the contradictions were removed by Demetriades (1978) and Lysenko & Maslov (1981), but on the whole the problem cannot be considered solved, primarily owing to the fact that many theoretical results have not yet been experimentally confirmed.

In the present paper the experimental study of development of small natural disturbances in the boundary layer of a cooled flat plate for Mach numbers M = 2, 3 and 4 is described. It confirms the main conclusions of the linear theory of hydrodynamic stability concerning the fact that surface cooling: (i) stabilizes the first-mode disturbances; (ii) destabilizes the second-mode disturbances; (iii) may lead to the region of unstable frequencies of the first mode being divided into two; (iv) does not affect the interaction of acoustic waves and the supersonic boundary layer.

A mathematical formulation is presented for the transient development of the fluid-flow field and the temperature field in a liquid pool, generated by a spatially variable heat flux falling on an initially solid metal block. This physical situation is an idealized representation of a TIG (tungsten-inert-gas) welding process. In the formulation allowance is made for electromagnetic, buoyancy and surface forces and the resultant equations are solved numerically.

It is found that both the convective flow field and the temperature field are markedly affected by the nature of the heat flux and the flux of electric current falling on the free surface.

In the absence of surface-tension effects a broadly distributed heat flux and corresponding current distribution cause a situation where both electromagnetic and buoyancy forces are important in determining the fluid-flow field; however, in these systems the fluid-flow field does not play a significant role in defining the heat-transfer process. In contrast, a sharply focused heat flux and current density on the free surface give rise to strong electromagnetically driven flows, which play an important role in determining the shape of the weld pool.

Calculations are also done exploring the effect of surface-tension-driven flows. It is found that surface-tension gradients may produce quite high surface velocities and can have a profound effect on determining the weld-pool shape.

Local measurements have been carried out in a mercury-nitrogen two-phase flow using the hot-film anemometric technique, in a cylindrical pipe with electrically insulated walls. The flow was turbulent (Re = 63000) and in the bubbly regime (〈α〉 ≈ 0.03). The maximum intensity of the magnetic field was 0.94 T. Processing of the hot-film signal provided in this medium, for the first time, simultaneous quantitative data for the local time-averaged velocity of the liquid-phase turbulent fluctuations, as well as local void fractions. These data were checked against those obtained, for the same conditions, using a double conductivity probe. The latter also provided information on the bubble velocity and average bubble size. Turbulence spectra were also measured.

We consider the time evolution of a layer of fresh water placed on top of a layer of salt water in a laboratory tank when denser salt water is supplied to a nozzle at the free surface. The inflow is carried out slowly so as to form a pure plume, which at first does not penetrate through the interface. The two basic processes that govern the time evolution of the initially fresh-water layer are the filling-box process (Baines & Turner 1969) and the entrainment through the end of the plume which impinges upon the density interface. A theoretical model that takes these two processes into account is presented, with the numerical solution of the asymptotic state, valid at large times. The asymptotic solution and experiment are in good agreement; the theory describes well the vertical buoyancy profile, the change in buoyancy difference across the interface with time, and the time when the plume begins to penetrate through the interface. The entrainment rate obtained from changes in thickness of the upper layer with time can be expressed as a function of the Froude number. The functional dependence is close to Fr3 at small values of Fr, and it approaches a finite limit as Fr increases. The buoyancy flux across the interface, which is non-dimensionalized by the rate of buoyancy input, also changes as a function of Fr, taking a maximum value of 0.168 at Fr = 0.46 and decreasing sharply at larger and smaller Froude numbers. These values agree well with those found from field observations and experiments on the entrainment at the boundary of convectively mixed layers. It is pointed out that some earlier results of Baines (1975) are not consistent with the model presented here.

The relation of magnetic helicity to the topological structure of field lines is discussed. If space is divided into a collection of flux tubes, magnetic helicity arises from internal structure within a flux tube, such as twist and kinking, and external relations between flux tubes, i.e. linking and knotting. The concepts of twist number and writhing number are introduced from the mathematical-biology literature to describe the contributions to helicity from twist about the axis of a flux tube, and from the structure of the axes themselves.

There exists no absolute measure of the helicity within a subvolume of space if that subvolume is not bounded by a magnetic surface. However, a topologically meaningful and gauge-invariant relative measure of helicity for such volumes is presented here. The time derivative of this relative measure is calculated, which leads to an expression for the flow of topological structure across boundaries.

The process of three-dimensional distortion of previously two-dimensional disturbances was investigated in a rectangular channel. For the first time the three-dimensional structures at the breakdown station of the two-dimensional wave were studied by flow visualization. It was shown that the structures have forms identical with Λ-shaped vortices in a boundary layer of the flat plate. The spanwise spacing of the Λ-shaped vortices is quite independent of the mean-flow velocity and of the frequency of artificial disturbances.

The linear stability properties of the Ekman layer in a rapidly rotating gas have been computed numerically. The two types of instability present in an Ekman layer of a homogeneous fluid, which are usually called classes A and B, respectively, are significantly modified by the compressibility. The critical Reynolds number for the class A instability is found to first increase and then decrease for increasing values of the Mach number. The instability waves of class B are monotonically destabilized as the value of the Mach number increases. In addition, a new class of unstable waves appears for a finite value of the Mach number.

In this paper we study the stability of long, steady, two-dimensional salt fingers. It is already known that salt fingers carrying a large enough density flux are unstable to long-wavelength internal-wave perturbations. Stern (1969) studied the mechanism of this, the collective instability, and it was studied in more detail by Holyer (1981). We extend the earlier work to include perturbations of all wavelengths, as well as long-wavelength perturbations. By applying the methods of Floquet theory to the periodic salt fingers, the growth rates of perturbations are found. For both heat-salt and salt-sugar systems the collective instability, which can be recognized by its frequency of oscillation, does not have the largest growth rate. There is a new, non-oscillatory instability, which, according to linear theory, grows faster than the collective instability. We study the instabilities that arise by using a combination of analytical and numerical methods. Further work will be necessary in order to assess the importance of these instabilities in different physical situations and to examine their development as their amplitude increases.

The stability of two-dimensional infinitesimal disturbances of the inviscid Kármán vortex street of finite-area vortices is reexamined. Numerical results are obtained for the growth rate and oscillation frequencies of disturbances of arbitrary subharmonic wavenumber and the stability boundaries are calculated. The stabilization of the pairing instability by finite area demonstrated by Saffman & Schatzman (1982) is confirmed, and also Kida's (1982) result that this is not the most unstable disturbance when the area is finite. But, contrary to Kida's quantitative predictions, it is now found that finite area does not stabilize the street to infinitesimal two-dimensional disturbances of arbitrary wavelength and that it is always unstable except for one isolated value of the aspect ratio which depends upon the size of the vortices. This result does agree, however, with those of a modified version of Kida's analysis.

Head-on collision of two (KdV) solitary waves at the interface of an inviscid two-fluid system of rigid upper and lower boundaries is investigated by a perturbation method. We obtain the third-order solution and find a dispersive wavetrain trailing behind each emerging solitary wave. The wavetrain is of the same polarity (depression/elevation) as the main wave. Furthermore, the energy and amplitude of the wave-train are decreasing in time as long as it is still attached to the main wave. This implies an increase in energy of the main wave. Up to the third order of accuracy the solitary wave emerging from a head-on collision retains its initial profile save for a phase shift. This phase shift is found to be amplitude dependent to the second order. The transfer of energy from the wavetrain to the main wave explains the slow recovery of the incident profiles in existing numerical results on the head-on collision of two solitary waves at the surface of an infinite channel.

The present work analyses the effects of variable porosity and inertial forces on convective flow and heat transfer in porous media. Specific attention is given to forced convection in packed beds in the vicinity of an impermeable boundary. After establishing the governing equations, a thorough investigation of the channelling effect and its influence on flow and heat transfer through variable-porosity media is presented. Based on some analytical considerations, a numerical scheme for the solution of the governing equations is proposed to investigate the variable-porosity effects on the velocity and temperature fields inside the porous medium. The method of matched asymptotic expansions is used to show the qualitative aspects of variable porosity in producing the channelling effect. These qualitative features are also confirmed by the numerical solution. The qualitative effects of the controlling parameters on flow and heat transfer in variable-porosity media are discussed at length. The variable-porosity effects are shown to be significant for most cases. For the same conditions as the perturbation solution, the numerical results are in excellent agreement with the perturbation analysis. The numerical results are also in very good agreement with the available experimental data of previous studies.

Two- and three-dimensional non-stationary viscous-fluid flows in a plane channel are considered. By means of efficient computational algorithms for direct integration of the incompressible Navier-Stokes equations the evolution of these flows over large time intervals is simulated. Classes of two- and three-dimensional non-stationary flows with stationary integral characteristics (the flow rate, mean pressure gradient, total energy of pulsations etc.) were found. Such flows are called secondary flows. Two-dimensional secondary flows have only qualitative similarity to turbulent flows observed in experiments. Three-dimensional secondary flows agree very well, even quantitatively, with turbulent flows. The principal characteristics of turbulent flows such as drag coefficient, mean-velocity profile, the distributions of the pulsation velocity components and some others are reproduced in three-dimensional secondary flows with good accuracy.

This is a study of how certain geometrical and flow parameters affect the tendency of a fluid to flow around rather than over a single obstacle of simple shape in a homogeneous non-rotating fluid. A series of numerical experiments was conducted with a finite-difference model of such a shallow flow, assuming a hydrostatic pressure distribution. The results demonstrate how the flow over a three-dimensional obstacle deviates from the patterns established for a two-dimensional ridge. Measures are suggested for quantitative assessment of the tendency to flow around as a function of relative hill height and Froude number.

A series of laboratory experiments was also performed, examining the motions of two superposed homogeneous layers of fluid past an isolated obstacle in a towing tank. The resulting motion of the interface was found to agree with the results of the numerical experiments. The laboratory experiments also extended the understanding gained from the numerical experiments. Flow-visualization techniques were employed to aid in the qualitative assessment of the flow around the obstacles and its dependence on hill and flow parameters. In particular, these techniques demonstrated the impingement of the interface on the obstacle, and its dependence on flow speed and hill height.

The evolution of unsteady boundary layers on the plane of symmetry of a slender prolate spheroid in uniform motion at constant angle of attack after an impulsive start has been studied for the case of a prescribed pressure distribution. Calculated results have been obtained for angles of attack ranging from 30° to 50° and show, for example, that the unsteady-state solutions approach the steady-state solutions rapidly on the windward and leeward sides for α < αc (≈ 41°). This is also so on the windward side for α > αc. On the leeward side for α > αc, however, the unsteady boundary layer is initially unseparated, but develops a region of reversed flow with increasing time. Subsequently, the streamwise displacement thickness develops a pronounced peak, which leads to a singularity of the type observed by van Dommelen & Shen on a circular cylinder started impulsively from rest.

The self-excited oscillation of a planar jet impinging upon a wedge can give rise to not simply a single, but as many as seven well-defined frequency components in the range of Reynolds-number (based on nozzle width and mean velocity) 250 ≤ Re ≤ 1150. All of these components are traceable to the nonlinear distortion/interaction (i.e. sum and difference) frequencies of two primary components: the most stable frequency of the jet shear layer (β); and a low-frequency modulating component ($\frac{1}{3}$β).

The modulating component $\frac{1}{3}$β arises from vortex-vortex interaction at the impingement edge. This interaction involves vortices of both like and opposite sense; the vortices of opposite sense to those of the incident shear layer arise from eruption of the viscous layer at the edge and tend to form counter-rotating vortex pairs with the incident vortices. The details of the vortex-vortex interaction pattern vary with Reynolds number; however, the pattern adjusts itself to maintain the modulating component $\frac{1}{3}$β. Its upstream influence strongly modulates the sensitive region of the shear layer at the jet nozzle lip. Consequently, the linear-growth region of the disturbance near the lip is dominated by the component $\frac{1}{3}$β, which eventually gives way to the most unstable component β further downstream.

Measurements of a turbulent trailing vortex in zero pressure gradient are described. These include mean velocities and all the components of the Reynolds-stress tensor. The measurements were made using linearized hot wires at stations 45, 78 and 109 chordlengths downstream of the wing. Axisymmetric jets or wakes were added coaxially to the vortex while the total circulation was held constant, and their effect studied. It was found, as Poppleton and Mason & Marchman have reported, that increasing the flow force hastens the radial dispersion of vorticity; this is seen to be concurrent with higher turbulence intensities and Reynolds shear stresses. With the flux of excess axial momentum effectively zero, thereby approximating a turbulent line vortex, no discernible downstream change was observed in the velocity field and very little in the turbulence field.

A balance of terms in the mean-momentum equations is presented and discussed. It is seen that, in spite of the fact that the radial velocity is numerically much smaller than the axial velocity, terms that contain it, both in the axial- and tangential-momentum equations, cannot be ignored, unless the magnitude of the flow force (divided by the fluid density) is much less than the square of the total circulation.

Moderate-amplitude axisymmetric oscillations of charged inviscid drops held together by surface tension are calculated by a multiple-timescale expansion. The corrections to the drop shape and velocity caused by mode coupling at second order in amplitude are predicted for two-, three- and four-lobed motions of drops with net charge up to the Rayleigh limit Qc ≡ 4π½. Resonant oscillations between four- and six-lobed motions occur for total charge values near $Q_{\rm r}\equiv (\frac{32}{3}\pi)^{\frac{1}{2}}$ and are analysed. Both frequency and amplitude modulation of the oscillation are predicted for drop motions starting from general initial deformations.

Numerical solutions are presented of the governing equations for three plane flows: the laminar free jet; the developing turbulent free jet; and the turbulent impinging jet for different ratios h/b of the nozzle height h above the plate to the nozzle width b.

The accuracy of the numerical procedure is demonstrated by comparing the solution of the Navier-Stokes equations for the laminar-flow case with their analytical boundary-layer solution. For turbulent flows these equations are solved after Reynolds averaging. Closure is achieved by a two-equation turbulence model in conjunction with three alternative algebraic expressions for the turbulent stresses. The capabilities of such an approach are illustrated by the extent and consistency of the predictions and the satisfactory agreement of the measurable quantities with the more reliable experimental data in the literature. The limitations of the models employed, evident from their lack of universality, are discussed in the light of their derivation from more complex ‘single-point’ closures.

Features of the flows studied of interest include: the near-nozzle behaviour of a ‘finite’ laminar free jet; the potential core and transition regions of a turbulent free jet, along with the fully developed similarity profiles; the enhanced heat-transfer characteristics of impinging jet flows; and the similarity of the developing wall jet after impingement to the standard wall-jet configuration.

The channel induction furnace is an electrically efficient device for the heating and stirring of liquid metals. In this paper an axisymmetric model for the channel flow is proposed, in which the fluid is confined to the inside of a torus. An exact solution for the magnetic field is found in terms of toroidal harmonic functions. Finite-difference methods are used to calculate the primary, cross-channel motions under the assumptions of a small skin depth, a constant eddy viscosity and no thermal dependence. Non-axisymmetric perturbations to the channel shape are considered and the perturbed field calculated. The secondary circulation along the channel is discussed.