Classical linear stability theory is extended to include the effects of temperature- and pressure-dependent fluid properties. These effects are studied asymptotically by using Taylor series expansions for all the properties with respect to temperature and pressure. In this asymptotic approach all effects are well separated from each other, and only the Prandtl number remains as a parameter. In their general form the asymptotic solutions hold for all Newtonian fluids. A shooting technique with Gram-Schmidt orthonormalization for the zero-order equation (classical Orr-Sommerfeld problem) and a multiple shooting method for all other equations is applied to solve the stiff differential equations. In particular the zero- and first-order equations are solved for a flat-plate boundary-layer flow with temperature-dependent viscosity. PhysicalhT. this corresponds to a fluid with a linear viscosity/temperature relation. The results show that decreasing the viscosity in the near-wall region of the boundary layer stabilizes the flow, whereas it would be destabilized for a uniformly decreased viscosity.

This paper describes the results of numerical simulations for polydisperse sedimentation of equal-sized spheres, e.g. particles of different density. Using the Stokesian dynamics algorithm, mobility matrices are computed for random particle configurations and ensemble averages taken to calculate the mean mobility matrices. It is shown that the settling velocities of individual particles species may be expressed in terms of two scalar functions of total volume fraction. These are the selfmobility Mo, (∼ short-time self-diffusion coefficient) and the interaction mobility MI. This latter quality is related to the velocity of a force-free tracer particle in a suspension of identical particles subjected to a unit force. Numerical values for Mo and MI are calculated for a range of volume fractions from ϕ = 0.025 to 0.50. All results show excellent agreement with the dilute theory of Batchelor. Simple algebraic expressions are given which well correlate the numerical results.

The transient response of a contained stratified rapidly rotating fluid to an impulsive surface stress has been studied theoretically and experimentally. The analysis predicts, and the experiments confirm, that for low values of the Burger number S the initial fluid adjustment within the E−½Ω−1 timescale is characterized by a barotropic response in which the magnitude of the interior velocity is independent of depth. (Here E and Ω are the Ekman number and rotation rate respectively.) The period of the barotropic response decreases as S increases. For large S, the barotropic flow adjusts subsequently to a baroclinic flow within the E−½Ω−1 scale, and during this later stage the excess and deficit in velocity in the lower and upper parts respectively of the fluid are removed. The baroclinic flow forced by the surface stress in these cases is thereby established in a timescale which is typically less than the spin-up time for a homogeneous fluid. The agreement between theory and experiment is shown to be qualitatively good, and the quantitative discrepancies observed between the predicted and measured interior velocities are discussed.

The modulated nonlinear Schrödinger equation (Zhang & Melville 1990), describing the evolution of a weakly nonlinear short-gravity-wave train riding on a longer finite-amplitude gravity-wave train is used to study the stability of steady envelope solutions of the short-wave train. The formulation of the stability problem reduces to the solution of a pair of coupled equations for the disturbance amplitude and (relative) phase. Approximate analytical solutions and numerical solutions show that the conventional sideband (Benjamin-Feir) instability is just the first in a series of resonantly unstable regions which increase in number with increasing perturbation wavenumber. The first of these new instabilities is the result of a quintet resonance between four short waves and one long wave. Subsequent unstable regions correspond to sextet or higher-order resonances. The results presented here suggest that steady envelope solutions for unforced irrotational short waves on longer irrotational gravity waves may be unstable for a wide range of conditions.

Multiple steady flow patterns occur in surface-tension/buoyancy-driven convection in a liquid layer heated from below (Rayleigh–Bénard–Marangoni flows). Techniques of numerical bifurcation theory are used to study the multiplicity and stability of two-dimensional steady flow patterns (rolls) in rectangular small-aspect-ratio containers as the aspect ratio is varied. For pure Marangoni flows at moderate Biot and Prandtl number, the transitions occurring when paths of codimension 1 singularities intersect determine to a large extent the multiplicity of stable patterns. These transitions also lead, for example, to Hopf bifurcations and stable periodic flows for a small range in aspect ratio. The influence of the type of lateral walls on the multiplicity of steady states is considered. ‘No-slip’ lateral walls lead to hysteresis effects and typically restrict the number of stable flow patterns (with respect to ‘slippery’ sidewalls) through the occurrence of saddle node bifurcations. In this way ‘no-slip’ sidewalls induce a selection of certain patterns, which typically have the largest Nusselt number, through secondary bifurcation.

An asymptotic theory of marginal thermal convection in rotating systems is constructed for the limit of rapid rotation. Many self-gravitating astronomical bodies, including the major planets, the Sun, and the Earth's liquid core, correspond to this limit. In the laboratory, an analogous system can be constructed with a very rapidly rotating apparatus, in which the centrifugal force plays the role of self-gravitation. The formulation is offered in such a way that both these geophysical systems and laboratory analogues are included as special cases. When the inclination of the outer boundaries relative to the equatorial plane is considered weak, the two types of system are identical at leading order. In this limit, the asymptotic analysis is profoundly simplified, because the system satisfies the Taylor-Proudman theorem to leading order. Nevertheless the system contains a very peculiar property: the mode defined by a conventional WKBJ theory implicitly assuming a locality of convection in the radial direction perpendicular to the axis of rotation cannot be accepted as a correct marginal mode, because a modulation equation gives an exponential growth in the radial direction, which contradicts an implicit initial assumption. The erroneous behaviour is traced to a spatial dispersion of thermal Rossby waves, which governs the marginal mode. The difficulty is resolved by extending the analysis to a complex plane of the radial coordinate of the point where convection amplitude attains its maximum. Such a complex radial distance is defined as the point where the wave dispersion disappears locally. The projection of the solution onto the real axis results in an inclination of the Taylor columns with respect to the radial direction. This is in good agreement with the most recent numerical studies. The isolation of convective Taylor columns in the radial direction weakens and the spiralling gets stronger as the Prandtl number decreases, as a result of the need to displace the critical radial distance further from the real axis.

Laboratory experiments have been conducted to study the shoaling of internal solitary waves of depression in a two-layer system on a uniform slope. The shoaling of a single solitary wave results in wave breaking and the production of multiple turbulent surges, or boluses, which propagate up the slope. Significant vertical mixing occurs everywhere inshore of the breaking location. The kinematics of the breaking and bolus runup are described and a breaking criterion is found. The energetics of the breaking are investigated. Over the range of parameters examined, 15 (±5) % of the energy lost from first-mode wave motion inshore of the break point goes into vertical mixing.

Experimental investigations on the origin and evolution of structure in circular shear flows, made under different conditions by different groups of authors. reveal a number of common regularities. (i) When the difference ΔΩ between the angular velocities of the centre and the periphery is smaller than a certain critical value (ΔΩ)c, the flow is axisymmetric. (ii) When ΔΩ = (ΔΩ)c, a pattern appears consisting of mc vortices. (iii) With a subsequent adiabatic growth of ΔΩ (at a certain (ΔΩ)mc+ > (ΔΩ)c), transition to a pattern with mc − 1 vortices occurs, but a pattern with mc + 1 vortices never arises (although in terms of linear theory the modes mc − 1 and mc + 1 are equivalent). Subsequent growth of ΔΩ leads to the transition (mc − 1) → (mc − 2), etc. (iv) As ΔΩ decreases, a cascade of inverse transitions of the form m − 1 → m up to m = mc occurs, and the transition m − 1 → m proceeds at a smaller value of ΔΩ compared with the transition m → m − 1, i.e. hysteresis occurs.

This paper offers a weakly nonlinear theory which makes it possible to describe the change of the order of symmetry of the wave pattern (number of vortices) with a change of ΔΩ and to ascertain conditions under which the above regularities occur. Some particular examples of the calculation of several models of shear flows are given, and it is shown that direct transitions (m → m −1) can be described in terms of a weakly nonlinear theory only for flows with a sufficiently large curvature of the shear layer, i.e. when D ≡ L/R = O(1), where L is the width of the shear layer, and R is its radius, and at not too large m (mc = 4,5). If D [Lt ] 1, a description of direct transitions requires a strongly nonlinear theory and is beyond the scope of this paper. Inverse transitions (m − 1 → m,m [les ] mc) admit a weakly nonlinear treatment at any D.

The coalescence of a cylinder with half-space by creeping viscous flow driven solely by surface tension is analysed using methods developed previously. The evolution of the shape with time is described, exactly, in terms of a time-dependent mapping function z = ω(ζ,t) of the upper half-plane, conformal on Im ζ [Gt ] 0. The results are in closed analytic form except for the time, which requires a quadrature. The height of the figure decays as t−1 as t → ∞, which is consistent with Kuiken's analysis of an isolated disturbance. (Previously, the author reported an erroneous solution which behaved otherwise.) The results are compared with the coalescence of equal cylinders obtained previously. For a modest degree of coalescence, the shapes are rather alike. In the limit as t → 0. the time dependence of the minimum widths (necks) are the same. At the times when the minimum widths disappear, the heights of the two shapes are equal.

Appended is a note providing a counter-example to earlier conjecture. A simply connected region undergoing this type of flow need not remain so.

The Kelvin–Helmholtz rollup of three-dimensional temporally evolving plane mixing layers with an initial Reynolds number of 500 based on vorticity thickness and half the velocity difference have been simulated numerically. All simulations were begun from a few low-wavenumber disturbances, usually derived from linear stability theory, in addition to the mean velocity profile. A standard set of ‘clean’ structures develops in the majority of the simulations. The spanwise vorticity rolls up into a corrugated spanwise roller with vortex stretching creating strong spanwise vorticity in a cup-shaped region at the bends of the roller. Predominantly streamwise rib vortices develop in the braid region between the rollers. For sufficiently strong initial three-dimensional disturbances these ribs ‘collapse’ into compact axi-symmetric vortices. The rib vortex lines connect to neighbouring ribs and are kinked in the direction opposite to that of the roller vortex lines. Because of this, these two sets of vortex lines remain distinct. For certain initial conditions, persistent ribs do not develop. In such cases, the development of significant three-dimensionality is delayed.

In addition, simulations of infinitesimal three-dimensional disturbances evolving in a two-dimensional mixing layer were performed. Many features of the fully nonlinear flows are remarkably well predicted by the linear computations. Such computations can thus be used to predict the degree of three-dimensionality in the mixing layer even after the onset of nonlinearity. Several nonlinear effects can also be identified by comparing linear and nonlinear computations. These include the collapse of rib vortices, the formation of cups of spanwise vorticity, and the appearance of spanwise vorticity with sign opposite that of the mean vorticity. These nonlinear effects have been identified as precursors of the transition to turbulence (Moser & Rogers 1991).

When a well-mixed suspension of small particles is emplaced below a clear fluid whose density is greater than that of the interstitial fluid, but less than that of the bulk suspension, the subsequent settling of the dense particles releases buoyant interstitial fluid and drives convection in the overlying layer. Mixing of interstitial fluid and some entrained particles into the overlying fluid causes the density of the overlying fluid to evolve with time and changes the rate of descent of the interface between the sedimenting and convecting regions. These effects are investigated experimentally in a simple rectangular geometry using suspensions of spherical glass particles. It is found that the convecting region is well mixed in both composition and particle concentration and that the interfacial velocity may be predicted from the instantaneous (uniform) bulk density of the upper layer and the distribution of the particle settling velocities. In the case of an overlying density gradient, the convection does not extend through the depth of the overlying fluid but erodes the base of the gradient to form a well-mixed layer between the gradient and the sedimenting fluid. On completion of the first cycle of sedimentation-driven convection, sedimentation from this well-mixed layer produces further cycles of sedimentation-driven convection, which are of successively decreasing intensity and increasing duration. Whether the overlying fluid is uniform or stratified, both theory and experiment show that the particles that are lifted into the convection are smaller on average than those which settle at the base of the lower layer. Thus, when the lifted particles are eventually allowed to settle there is a discontinuity generated in the variation of the size distribution of particles with height in the final sedimented pile. This phenomenon may be an important mechanism for secondary layering in the deposits from turbidity currents and pyroclastic flows.

Systematic measurements of the surface drift current, the wind profile over the water surface and the wave spectra have been made for (i) pure wind-waves, (ii) a coexisting system of wind-waves and swell propagating against the wind, and (iii) a coexisting system of wind-waves and swell propagating in the direction of the wind. The surface drift current is gradually intensified by the swell propagating against the wind when the swell steepness increases. The maximum increase of the surface drift velocity caused by the opposing swell is about 46 % of the surface drift velocity for pure wind-waves at the same wind speed. Such a phenomenon was not observed when the swell was propagating in the direction of the wind.

A series of experimental studies have been made of the fluid behaviour in a completely filled, precessing, right circular cylinder. The tank was spun about its axis of symmetry and subjected to a forced precession at various excitation frequencies ω, nutation angles θ and at various Ekman numbers. This forcing excites a subset of the modes, called inertia waves, that are made possible by the Coriolis force that arises in a spinning environment. In these experiments, the fluid flow breakdown phenomena are investigated. Here the fluid, when forced near a resonant frequency, exhibits a transition to disordered or turbulent flow. This paper presents a categorization of some of the breakdown regimes, of which the ‘resonant collapses’ (McEwan 1970) are the most catastrophic members.

The studies reported in this paper used entirely visual observations and measurements. The experimental observations employed a visualization technique that gave no information on fluid velocities, but provided an excellent picture of the flow structure. Quantitative data were extracted in the form of the time for the breakdown to occur. The breakdown phenomena, while readily produced over a large region of parameter space, are complex and varied. The observations show that our system is extraordinarily rich, exhibiting, for example, recurrent breakdowns which may be explained in terms of chaotic intermittency. A detailed description of some of the different breakdown regimes indicates that no single model will explain the behaviour throughout parameter space. This research is motivated by the instability problems of spinning spacecraft containing liquid fuels.

An investigation of the interfacial-tension-driven fragmentation of a very long fluid filament in a quiescent viscous fluid is presented. Experiments covering almost three orders of magnitude in viscosity ratio reveal as many as 19 satellite droplets in between the largest droplets; complementary boundary-integral calculations are used to study numerically the evolution of the filament as a function of the viscosity ratio of the fluids and the initial wavenumber of the interface perturbation. Satellite drops are generated owing to multiple breakup sequences around the neck region of a highly deformed filament. In low-viscosity ratio systems, p < O(0.1), the breakup mechanism is self-repeating in the sense that every pinch-off is always associated with the formation of a neck, the neck undergoes pinch-off, and the process repeats. In general the agreement between computations and experiments is excellent; both indicate that the initial wavenumber of the disturbance is important in the quantitative details of the generated drop size distributions. However, these details are insignificant when compared with the large variations produced in the drop size distributions owing to variation in the viscosity ratio.

Using the parallel-flow approximation the prechaotic bifurcation behaviour of Blasius boundary-layer flow is studied at finite Reynolds numbers. The objective of this investigation is to search for qualitative solution changes which might be linked to the rapid breakdown at transition. As a first step this requires the computation of the two-dimensional primary equilibrium surface of nonlinear Tollmien-Schlichting waves. This two-dimensional neutral surface exhibits a period-halving bifurcation which is qualitatively different from the situation for plane Poiseuille flow. At the same time the numerically computed equilibrium solution offers the possibility of assessing the range of convergence of weakly nonlinear results.

In a second step the stability of this nonlinear equilibrium solution is investigated with respect to three-dimensional disturbances. Of particular importance is the existence of a modal degeneracy between amplified secondary instability modes, implying locally algebraic growth. On decreasing the Reynolds number, the amplification rate of this direct resonance point switches from being amplified to being damped. Interestingly, the Reynolds number corresponding to this zero-amplification point seems to be in the vicinity of the experimentally observed transition Reynolds number for Blasius flow.

Numerical simulations of a three-dimensional temporally growing shear layer are obtained at high Reynolds number and zero Froude number using a vortex scheme modified for a variable-density flow. Attention is focused on the effect of initial vorticity and density distributions on the interaction between instability modes which lead to the generation and intensification of streamwise vorticity. Results show that the three-dimensional instabilities evolve following the formation of concentrated span wise vorticity cores. The deformation of each core along its span resembles the amplification of the translative instability. The generation of vortex rods, which wrap around individual cores while stretching between neighbouring cores, suggest a mode similar to the Corcos instability. The instability modes leading to the formation of both structures, energized by the extensional strain generated by the cores, grow simultaneously. A similar series of events occurs in variable-density shear layers and in shear layers which start with an asymmetric vorticity distribution. Baroclinic vorticity generation in the variable-density layer leads to the formation of asymmetric cores whose volumetric composition is biased towards the lighter fluid. The structures are propelled, by their asymmetric vorticity distribution, in the direction of the heavier stream while their eccentric spinning forces an uneven stretching of the vortex rods. The origin of the asymmetry is established by comparing these with the results of a shear layer with an initially asymmetric vorticity distribution in a uniform-density flow. The strong late-stage asymmetry exhibited by the former is not observed in the latter. Thus, baroclinic vorticity generation is responsible for the observed symmetry. We also find that initially asymmetric vorticity distribution does not, as suggested before, lead to asymmetric spacing between the streamwise rods, it is concluded that the experimentally observed asymmetric spacing must arise after pairing.

The three-dimensional transition of the flow behind a bluff body is studied, with an emphasis placed on the evolution of large-scale structures in the wake. It has previously been found that there are two fundamental modes of three-dimensional vortex shedding in the wake of a circular cylinder (each mode being dependent on the range of Reynolds number), with a spanwise lengthscale of the same order as the primary streamwise wavelength of the vortex street. However. it is shown in the present study that the wake transition also involves the appearance of large-scale spot-like ‘vortex dislocations’, that grow downstream to a size of the order of 10–20 primary wavelengths. Vortex dislocations are generated between spanwise vortex-shedding cells of different frequency. The presence of these dislocations explains the large intermittent velocity irregularities that were originally found by Roshko (1954) and later by Bloor (1964) to characterize transition. The presence of these vortex dislocations in wake transition is largely responsible for the break-up to turbulence of the wake as it travels downstream.

In order to study their evolution in detail, dislocations have been (passively) forced to occur at a local spanwise position with the use of a small ring disturbance. It is found that ‘two-sided’ dislocations are stable in a symmetric in-phase configuration, and that they induce quasi-periodic velocity spectra and (beat) dislocation-frequency oscillations in the near wake. Intrinsic to these dislocations is a mechanism by which they spread rapidly in the spanwise direction, involving helical twisting of the vortices and axial core flows. This is felt to be a fundamental mechanism by which vortices develop large-scale distortions in natural transition. As the wake travels downstream, the energy at the low dislocation frequency decays slowly (in contrast to the rapid decay of other frequencies), leaving the downstream wake dominated by the large dislocation structures. Distinct similarities are found between the periodic forced dislocations and the intermittent dislocations that occur in natural transition. Further similarities of dislocations in different types of flow suggest that vortex or phase dislocations could conceivably be a generic feature of transition in all shear flows.

A linear stability analysis has been implemented for Taylor–Dean flow. a viscous flow between rotating concentric cylinders with a pressure gradient acting in the azimuthal direction. The analysis is made under the assumption that the gap spacing between the cylinders is small compared to the mean radius (small-gap approximation). A parametric study covering wide ranges of μ. the ratio of angular velocity of the outer cylinder to that of inner cylinder. and β. a parameter characterizing the ratio of representative pumping and rotation velocities is conducted. For − 1 [les ] μ < 1, results show that non-axisymmetric instability modes prevail in a wide range of β. The most stable state is found to occur within − 3.9 < β < − 3.6 for μ < 0.3 and at − β ≈ 1.59μ + 3.5 for μ > 0.3. The most stable state is always accompanied by a shortest critical axial wavelength. Instability modes with different azimuthal wavenumber have similar stability characteristics because the basic state is either close to or at the most stable situation. This similarity is absent from either Taylor or Dean flow.

We study the three-dimensional evolution of a nominally axisymmetric jet subject to helical perturbations. Our approach is a computational one, employing an inviscid vortex filament technique to gain insight into the vorticity dynamics of jets dominated by helical vortices. For the case of a helical perturbation only, the streamwise vorticity forming in the braid is of the same sign everywhere, with the vortex helix representing streamwise vorticity of opposite sign. Owing to the helical symmetry, concentrated structures do not form in the braid. By introducing an additional periodic perturbation in the azimuthal direction, the helical symmetry is broken and we observe the emergence of concentrated streamwise braid vortices all of the same sign, in contrast to the counter-rotating braid vortices of ring-dominated jets. A Kelvin—Helmholtz-like instability of the braid vorticity layer plays a significant role in their generation. We furthermore find that the initial evolution of the braid vorticity is strongly dependent upon the ratio between the helical and azimuthal perturbation amplitudes. Smaller azimuthal perturbation amplitudes slow down the concentration process of the braid vorticity. However, we find that the long-time strength of the streamwise braid vortices should not depend on the amplitudes of the streamwise and azimuthal perturbation waves, but rather on their wavenumbers. The evolution of the helical vortex varies with the ratio between jet radius R and shear-layer momentum thickness θ. While for a jet with R/θ = 22.6 and azimuthal wavenumber five, the emerging helix continuously rotates and thereby avoids instability, we observe in a jet with R/θ = 11.3 the reduction of this rotation and the near exponential growth of waves on the helical vortex, characteristic of vortex helix instability.

In this paper, an extension of Miyazaki & Hasimoto's (1984) Green function for the slow flow created by a point force of arbitrary direction above an infinite plane wall with a circular hole was used to formulate a set of boundary-integral equations for the motion, at low Reynolds and Stokes numbers, of a finite rigid sphere in a simple shear flow with suction past an infinite wall containing a circular side hole. The equations were solved numerically by discretizing the surface of the sphere into a finite number of elements and then using a constant-density approximation for the unknown surface force distribution and a boundary collocation technique to satisfy the no-slip boundary condition at the centre of each element. Numerical tests and comparisons with available exact and numerical results show that convergence to three or four significant figures can be achieved for all the 21 independent unknown force and torque coefficients. Numerical values for these coefficients were obtained throughout the flow field for sphere–hole radii ratios of $\frac{1}{10}, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}$ and 1, and the neutrally buoyant velocities and trajectories of individuals spheres were then computed for a range of initial upstream positions, and for various values of a suction parameter defined in Yan et al. (1991 a) which refers to the relative strengths of the suction and shear flows. In turn, these trajectories were used to map out the particle capture tube and its upstream cross-section and thereby determine the particle screening effect, one of the underlying mechanisms responsible for the well-known exit concentration defect observed when particles enter a side pore. The other mechanism, the fluid skimming effect due to the presence of a particle-free layer on the upstream wall, was considered recently in a companion paper (Yan et al. 1991 a). It is shown here that the fluid skimming effect provides a lower bound for this concentration defect under the conditions of this analysis. The theoretical predictions exhibit features that are qualitatively similar to the experimental observations of the hematocrit (red cell) defect in the microcirculation, although the dilute suspension limit considered herein is well below the observed hematocrit in the microcirculation and the particles are modelled as rigid spheres.