This work determines analytically the drag on, and heat flux out of, a hot sphere that translates steadily in a fluid of strongly temperature-dependent viscosity. There is no dissipative heating. The essentials are illustrated by an exact solution for the flow induced by slowly squeezing two parallel planes together. The lower plane is hot and stationary; the upper is cold and advances in a direction normal to itself at uniform speed U. The gap is completely filled by a fluid of strongly temperature-dependent viscosity. We find the temperature and velocity profiles, and determine the Nusselt number N and Péklet number P as functions of the normal force D on the lower plane. The large viscosity variation tries to concentrate the flow into a relatively thin softened layer in which the viscosity is of order its value μ0 at the hot plane. In the limit of infinite viscosity ratio (fixed P), it succeeds (lubrication limit): if P [Lt ] 1, the width of the softened layer is determined by conduction and D ∝ μ0U; but D ∝ μ0U4 when forced convection is important. If P → ∞ (fixed viscosity ratio), the softened layer is so thin that it chokes, and all the deformation occurs outside the thermal layer in the fluid of uniform viscosity μ∞ (Stokes limit); then D ∝ μ∞U. These mechanisms appear as three distinct legs in our plot of log P against log D. There are similar transitions in the plot of log N against log D. The solution gives an estimate of the drag on a sphere. We test this estimate against an analytical solution for the sphere in the lubrication limit. Then we extend the solution to cover power-law fluids, and apply it to a model (by Marsh) of magma transport beneath island-arc volcanoes. The results suggest that the magma covers the first 50 km of its ascent by an isoviscous mechanism, with the lubrication mechanism operating in the remaining 50 km. To open a fresh pathway from the source to the surface takes about 106 years and uses about 1027 erg.

The strong-interaction theory developed in Ganatos, Weinbaum & Pfeffer (1 9804 and Ganatos, Pfeffer & Weinbaum (1980b) for the normal and parallel creeping motion of a sphere of arbitrary size between two infinite plane-parallel walls is applied to several particle-boundary interaction problems of long-standing interest. The first highly accurate solutions are presented for the slip and angular velocity of a neutrally buoyant sphere carried by the fluid in a Couette or Poiseuille channel flow and the gravitational settling of a non-neutrally buoyant sphere in an inclined channel. The latter problem clearly illustrates the non-isotropy of the frictional resistance tensor on the particle motion. The solutions for the fluid velocity field exhibit an induced circulation extending to infinity fore and aft of the sphere for a neutrally buoyant sphere in Couette flow and an induced back-flow ahead of the sphere for the Poiseuille flow geometry. Approximate but highly accurate solutions are presented for small gap widths between a sphere and the neighbouring boundary, which take account of the influence of the second wall.

A series of experiments on strong nonlinear instabilities of gravity-wave trains of large steepness 0·25 ≤ ak ≤ 0·34 in deep water in a long tow tank and a wide basin were performed. These experiments clarified several phenomena such as subharmonic instabilities, wave breaking, evolution of power spectra and directional energy spreading. It was found that an initial two-dimensional wavetrain of large steepness evolved into a series of three-dimensional spilling breakers, and was followed by a transition to a two-dimensional modulated wave train during which a series of oblique wave groups were radiated. The final stage of evolution was a series of modulated, two-dimensional wave groups with lower steepness and frequency. The three most frequent normal modes of two-dimensional subharmonic instabilities underlying the frequency downshifting, in the order of their relative occurrence, are found to be about 4/3, 5/4 and 3/2 in terms of the ratio of the fundamental to perturbed wave frequencies. The Bimescale for the phenomenon of the frequency downshifting is from 40 to 60 fundamental wave periods. Comparisons of the experimental results with existing theories showed that observed processes have several qualitative similarities with theoretical computations.

The three-dimensional structures of symmetric and skew wave patterns in deep water observed in a tow tank and a wide basin are described. These symmetric waves are the result of three-dimensional subharmonic bifurcation of two-dimensional wave-trains with steepness a0k0 ≥ 0·25, where a0, and k0, are the wave amplitude and wave-number respectively. The wave profiles, local surface slopes and amplitude spectra at various cross-sections of the crescent-shaped symmetric waves are presented and discussed.

The skew waves are another type of three-dimensional bifurcation from a uniform wavetrain, and occur most clearly when 0·16 [siml ] a0k0 [siml ] 0·18. These skew wave patterns are found to interact between two sets of themselves propagating from different directions, and to be subject to the Benjamin–Feir type modulations. The interactions cause compact three-dimensional wave packets. Agreement between experiment and theory is found for the case of symmetric wave patterns. The bifurcation of uniform Stokes waves into symmetric wave patterns provide a new physical process for directional energy spreading.

Steady three-dimensional symmetric wave patterns for finite-amplitude gravity waves on deep water are calculated from the full unapproximated water-wave equations as well as from an approximate equation due to Zakharov. These solutions are obtained as bifurcations from plane Stokes waves. The results are in good agreement with the experimental observations of Su.

Two types of singularity that occur at the upper pole of a heated sphere in a fluid at rest when the Grashof number is large are discussed. The first is a property of a limit solution of the unsteady boundary-layer equations and is indicative of the fact that the boundary layer growing from the lower pole does not remain empty for all time but erupts into a plume above the sphere. The second arises from a solution of the steady boundary-layer equations and illustrates the phenomenon of an axisymmetric boundary layer converging at a point, with a velocity component parallel to the sphere that is non-zero over the major part of the boundary layer. An analysis is prcscnted for each situation and comparison made with a numerical integration of the appropriate equations.

The Ranque–Hilsch effect, observed in swirling flow within a single tube, is a spontaneous separation of total temperature, with the colder stream near the tube centreline and the hotter air near its periphery. Despite its simplicity, the mechanism of the Ranque–Hilsch effect has been a matter of long-standing dispute. Here we demonstrate, through analysis and experiment, that the acoustic streaming, induced by orderly disturbances within the swirling flow is, to a substantial degree, a cause of the Ranque–Hilsch effect. The analysis predicts that the streaming induced by the pure tone, a spinning wave corresponding to the first tangential mode, deforms the base Rankine vortex into a forced vortex, resulting in total temperature separation in the radial direction. This is confirmed by experiments, where, in the Ranque–Hilsch tube of uniflow arrangement, we install acoustic suppressors of organ-pipe type, tuned to the discrete frequency of the first tangential mode, attenuate its amplitude, and show that this does indeed reduce the total temperature separation.

An experimental and theoretical study is presented of the inlet-vortex (or ground-vortex) phenomenon. The experiments were carried out in a water tunnel using hydrogen-bubble flow visualization. The theoretical study is based on a secondary-flow approach in which vortex filaments in a (weak) shear flow are viewed as convected (and deformed) by a three-dimensional irrotational primary flow; the latter being calculated numerically using a three-dimensional panel method. Two basic mechanisms of inlet-vortex generation are identified. The first of these, which has been alluded to qualitatively by other investigators, is the amplification of ambient (i.e. far-upstream) vorticity as the vortex lines are stretched and drawn into the inlet. Quantitative calculations have been carried out to illustrate the central features connected with this amplification. I n contrast with what has been supposed, however, there is another mechanism of inlet-vortex formation, which does not appear to have been recognized previously and which does not require the presence of ambient vorticity. It is thus shown that an inlet vortex can arise in an (upstream) irrotational flow, for an inlet in cross wind. In this situation, the vortex is accompanied by a variation in circulation along the length of the inlet. The ratio of inlet velocity to upstream veIocity is an important parameter for both mechanisms.

The forces on two spherical particles moving in a fluid are investigated by the method of matched asymptotic expansions in the small Reynolds number, for the case when the particles are within each other's inner region of expansion. The particular case in which the distance l between the sphere centres is very much larger than the sphere radii a and b is studied in detail. The asymptotic expansion of the force on one of the spheres for small a/l and b/l is obtained. Some properties of the force, not to be expected from the Stokes equation, are revealed.

An analytical theory is developed for the stability properties of planar fronts of premixed laminar flames freely propagating downwards in a uniform reacting mixture. The coupling between the hydrodynamics and the diffusion process is described for an arbitrary expansion of the gas across the flame. Viscous effects are included with an arbitrary Prandtl number. The flame structure is described for a large value of the reduced activation energy and for a Lewis number close to unity. The flame thickness is assumed to be small compared with the wavelength of the wrinkles of the front, this wavelength being also the characteristic lengthscale of the perturbations of the flow field outside the flame. A two-scale method is then used to solve the problem. The results show that the acceleration of gravity associated with the diffusion mechanisms inside the front can counterbalance the hydrodynamical instability when the laminar-flame velocity is low enough. The theory provides predictions concerning the instability threshold. In particular, the dimensions of the cells are predicted to be large compared with the flame thickness, and thus the basic assumption of the theory is verified. Furthermore, the quantitative predictions are in good agreement with the existing experimental data.

The bifurcation is shown to be of a different nature than predicted by the purely diffusive–thermal model.

The viscous diffusivities are supposed to be independent of the temperature, and then the viscosity is proved to have no effect at all on the dynamical properties of the flame front.

Early treatments of flames as gasdynamic discontinuities in a fluid flow are based on several hypotheses and/or on phenomenological assumptions. The simplest and earliest of such analyses, by Landau and by Darrieus prescribed the flame speed to be constant. Thus, in their analysis they ignored the structure of the flame, i.e. the details of chemical reactions, and transport processes. Employing this model to study the stability of a plane flame, they concluded that plane flames are unconditionally unstable. Yet plane flames are observed in the laboratory. To overcome this difficulty, others have attempted to improve on this model, generally through phenomenological assumptions to replace the assumption of constant velocity. In the present work we take flame structure into account and derive an equation for the propagation of the discontinuity surface for arbitrary flame shapes in general fluid flows. The structure of the flame is considered to consist of a boundary layer in which the chemical reactions occur, located inside another boundary layer in which transport processes dominate. We employ the method of matched asymptotic expansions to obtain an equation for the evolution of the shape and location of the flame front. Matching the boundary-layer solutions to the outer gasdynamic flow, we derive the appropriate jump conditions across the front. We also derive an equation for the vorticity produced in the flame, and briefly discuss the stability of a plane flame, obtaining corrections to the formula of Landau and Darrieus.

Velocity moments have been calculated for the rapid distortion of axisymmetric turbulence in a uniform mean shear. The moments are compared with data on steady pipe flows and two-dimensional channel flows, and the turbulence structure of these flows is summarized in terms of effective distortion strain. The centre-line structure is taken to be characteristic of the undistorted state which from the data is more nearly axisymmetric rather than isotropic. A closer comparison is found than that by Townsend (1970), and in particular the differences in stress ratios $\tau/\rho\overline{u^2_1},\tau/\rho\overline{q^2} $ found between different experiments can be accounted for with the hypothesis of initially axisymmetric turbulence. Profiles for the effective strain are derived from the experiments and are shown to have the same form and to indicate the existence of a relaxation timescale for the large eddies, comparable to the energy decay timescale. An equation for the effective distortion strain is formulated that can be incorporated into a turbulence model.

An algorithm is formulated for computing perturbation-series solutions for standing waves on the interface between two semi-infinite fluids of different but uniform densities. Using a comppter, the series solutions are computed to fifth order for a general value of r, the ratio of the density of the upper fluid to that of the lower fluid (0 ≤ r ≤ l), and to 21st order for five specific values of this ratio: r = 0, 10−3, 0·1, 5·0, 1·0. The series for the period, the energy, and the interface profile of the waves are summed using Padé approximants. The maximum wave height for each of the above five density ratios is estimated from the locations of the poles of the Padé approximants for the wave period and the wave energy. At maximum height the interface appears to be vertical at a point on the interface that is very near the crest for r = 10−3 and approaches the midpoint between the crest and the trough as r approaches 1·0.

The central features of lihear and nonlinear disturbance growth in the unstable shear layer, mechanisms of impingement of the resultant vortices on the edge, induced force on the wedge, and upstream influence in the form of induced velocity fluctuations at separation are examined by simultaneous visualization, velocity, and force-measurement techniques.

The nature of the vortex–wedge interaction, and the associated force on the wedge, are directly related to the induced velocity at the upstream separation edge, thereby providing the essential ‘feedback’ for the self-sustained oscillation. Velocity fluctuations at the upper and lower sides of the separation edge tend to be π out of phase, a condition that is maintained along the outer boundaries of the downstream shear layer. Moreover, the phase between velocity fluctuations at separation and impingement satisfies the relation 2nπ, where n is an integer.

The shear layer downstream of the separation edge initially forms an asymmetric wake, which evolves into large-scale vortices, all of which have a circulation appropriate to the high-speed side. The disturbance amplification associated with the high-speed side dominates from the separation edge onwards, precluding development of instabilities associated with the low-speed side.

Regardless of the initial amplitude of the disturbance induced at the separation edge, the same saturation amplitude is attained in the downstream (nonlinear) region of the shear layer, underscoring the fact that variations in force amplitude at the wedge are dominated by the type of vortex–edge interaction mechanism. The sensitivity of this interaction to small offsets between the vortex centre and the leading edge entails that jumps in frequency of oscillation are also associated with jumps in the force amplitude.

Steady two-dimensional jets of inviscid incompressible fluid, rising and falling under gravity, are calculated numkrically. The shape of each jet depends upon a single parameter, the Froude number $\lambda = q_{r\m c}(Qg)^{-\frac{1}{3}}$, which ranges from zero to infinity. Here qc is the velocity at the crest of the jet, i.e. the highest point of the upper surface, Q is the flux in the jet. and g is the acceleration of gravity. For λ = ∞ the jet is slender and parabolic. It becomes thicker as λ decreases, and reaches a limiting form at λ = 0. Then there is a stagnation point at the crest, where the surface makes a 120° angle with itself. This angle is predicted by the same argument Stokes used in his study of water waves.

The problem is formulated as an integro-differential equation for the two free surfaces of the jet, This equation is dlscretized to yield a set of nonlinear equations, which are solved numerically by Newton's method. In addition, asymptotic results for large λ are obtained analytically. Graphs of the results are presented.

The steady, laminar, incompressible flow over a periodic wavy surface with a prescribed surface-velocity distribution is found from the solution (via Newton's method) of the two-dimensional Navier–Stokes equations. Validation runs have shown excellent agreement with known analytical (Benjamin 1959) and analytico-numerical (Bordner 1978) solutions for small-amplitude wavy surfaces: For steeper waves, significant changes are observed in the computed surface-pressure distribution (and consequently in the nature of the momentum flux across the interface) when a surface orbital velocity distribution, of the type found in water waves, is included,

Multiprobe techniques were used to study the coherent flow-oriented eddy pattern that exists close to a wall in a turbulent flow. Simultaneous measurements of the spanwise component sz of the fluctuating wall velocity gradient at a number of locations in the spanwise direction were used to detect different aspects of the secondary flows that are superimposed on the mean flow. Conditionally averaged measurements of the streamwise component of the wall velocity gradient and of the streamwise component of the velocity at a number of locations on a line perpendicular to the wall show how the streamwise velocity fluctuations are related to the eddy structure, as detected by the sz(z) variation.

The results are consistent with the suggestion by Sirkar & Hanratty (1970a) that the flow-oriented eddies are approximately homogeneous in the flow direction and, on average, have inflows and outflows of equal magnitude coupled in the spanwise direction; i.e. they may be modelled, on average, by a sinusoidal sz(z) variation with dimensionless wavelength λ+ = 100. The results are also consistent with explanations by Fortuna and Hanratty (see Fortuna 1971; Hanratty, Chorn & Hatziavramidis 1977) and by Hatziavramidis & Hanratty (1979) of how these secondary flows are affecting momentum transport to the wall.

Experiments have been made to examine the fluid motion and density perturbations caused by oscillating a grid of vertical bars horizontally in a uniformly stratified fluid at frequency ω greatly exceeding the buoyancy frequency N. A highly turbulent region is produced near the grid. Beyond this lies a region of intrusive layers as described by Ivey & Corcos (1982), while further from the grid the motion is dominated by internal waves. The boundary between the turbulent region and the intrusive region is clearly defined, and the width of the former region appears to be proportional to $y = a^{\frac{3}{4}}M^{\frac{1}{4}}(\omega/N)^{\frac{1}{2}}$, where a is the amplitude and M the mesh length of the grid. The vertical scale of the layers, which sometimes form to give a regular sequence of high and low density gradients, is also proportional to y. This scaling is shown to be consistent with that found by Hopfinger & Toly (1976) in their study of motion produced by grids oscillating in homogeneous fluids, while the abrupt change in character of the motion at the edge of the turbulent region is in accord with measurements made by Dickey & Mellor (1980) in the wake of a grid drawn steadily through a stratified fluid. The scaling is also in accord with Ivey & Corcos’ observations of the width of the turbulent region, the thickness of the layers, and the vertical flux of density.

The observation of the layers is considered in the light of conjectures about the instability of turbulent motion in stratified fluids. The character and conditions of generation of the layers are consistent with their being a manifestation of the instability, but the identification is not conclusive.

We compare - both analytically and numerically – two related spectral (≡ two-point) closures for the problem of the decay of temperature fluctuations convected by isotropic turbulence. The methods are the test-field model (TFM) (Kraichnan 1971; Newman & Herring 1979) and the eddy-damped quasinormal Markovian (ENQNM) approximation (Orszag 1974; Lesieur & Schertzer 1978). We show that EDQNM may be regarded as a rational approximation to, and simplification of, the TFM, except at small wavenumbers, where an additional eddy-dissipative term is needed to produce satisfactory results for the former. We consider three available methods for determining the relaxation timescales: (i) comparison with experiments, (ii) comparison with the direct-interaction approximation (DIA) in thermal equilibrium, and (iii) comparison with DIA at very small wavenumber, where it is believed to represent the dynamics properly. Comparison with both large Reynolds number and wind-tunnel Reynolds numbers is presented. For the latter, we discuss the relationship of the present theoretical results to the experiments of Warhaft & Lumley (1978) and Sreenivasan et al. (1980), and to the theoretical analysis of Corrsin (1964), Kerr & Nelkin (1980) and Antonopolos-Domis (1981).

The eigenvalue problem of Kuo governing the linear stability of a parallel zonal flow of an inviscid incompressible fluid on a β-plane is treated in this paper. First, a synthesis of the known properties of the normal modes is presented, as a short summary. The rest of the paper is a study of the properties of one of the classes of stable modes, namely Rossby waves modified by the basic shear. These modes are found as the solution of the inverse of a regular Sturm–Liouville problem. Several asymptotic results for small and for increasing values of the basic shear (i.e. equivalently for large and decreasing values of β) are found for quite general velocity profiles. These are illustrated by some numerical calculations of the wave characteristics for a few particular basic velocity profiles.