Stokesian dynamics is used to simulate the dynamics of a monolayer of a suspension of bimodally distributed spherical particles subjected to simple shearing flow. Hydrodynamic forces only are considered. Many-body far-field effects are calculated using the inverse of the grand mobility matrix. Near-field effects are calculated from the exact equations for the interaction between two unequal-sized spheres. Both the detailed microstructure (e.g. pair-distribution function and cluster formation) and the relative viscosity are determined for bimodal suspensions having particle size ratios of 2 and 4. The flow of an ‘infinite’ suspension is simulated by considering 25, 49, 64, and 100 particles to be ‘one’ cell of an infinite periodic array. The effects of both the size ratio and the relative fractions of the different-sized particles are examined. When the area fraction, ϕa, is less than 0.4 the particle size distribution does not affect the calculated viscosity. For ϕa > 0.4, and for a fixed fraction of small spheres, the bimodal suspensions generally have lower viscosities than monodispersed suspensions, with the size of this effect increasing with ϕa. These results compare favourably with experiment when ϕa and the volume fraction, ϕv, are normalized by the maximum packing values in two and three dimensions, respectively. At the microstructural level, viscosity reduction is related to the influence of particle size distribution on the average number of particles in clusters. At a fixed area fraction, the presence of smaller particles tends to reduce average cluster size, particularly at larger ϕa, where significant viscosity reductions are observed. Since the presence of large clusters in monodispersed suspensions has been directly linked to high viscosities, this provides a dynamic mechanism for the viscosity reduction in bimodal suspensions.

The generation of a standing wave of vortices in thin channels has been experimentally observed and discussed in the literature for the last several years. The specific cause of the wave and its response to various conditions remains largely unexplored. In this paper we model pulsatile flow through thin channels with inserted deflectors to generate the vortex wave, and we examine various measures to quantify its effects. We focus on the numerical solution of the transient vortex wave phenomenon and its response to a superimposed bulk flow, variations of pulsation, deflector spacing and shape as well as transverse suction. The quantifying measures are mapped over a Reynolds’ number–Strouhal number domain.

A two-layer planetary geostrophic model is adopted to study the breaking of planetary waves in the presence of Ekman pumping and the associated-mean flow. The governing equation for the interface is a quasi-linear equation, which is solved analytically by the method of characteristics. The waves are forced by annual or interannual upwelling or downwelling along the eastern boundary of a subtropical gyre. It is found that the time and position at which breaking occurs is mainly determined by the speed and depth of the eastern boundary perturbation, while the intensity of a breaking front is mainly determined by the amplitude of the perturbation. The breaking of a planetary wave is affected significantly by Ekman pumping and the associated mean flow, particularly for annual and interannual forcing. Downward Ekman pumping, as in a subtropical gyre, suppresses breaking in downwelling waves caused by eastern boundary upwelling, but enhances breaking in upwelling waves caused by eastern boundary downwelling. In the presence of steady downward Ekman pumping, downwelling breaking will not occur except for interfaces near the surface. The structure and intensity of a breaking front is also discussed.

The leading-order fluid motions and frequencies in resonance tubes coupled to a combustion-driven flow source, such as occurs in various types of pulse combustors, are usually strongly related to those predicted by linear acoustics. However, in order to determine the amplitudes of the infinite number of classical acoustic modes predicted by linear theory alone, and hence the complete solution, a nonlinear analysis is required. In the present work, we adopt a formal perturbation approach based on the smallness of the mean-flow Mach number which, as a consequence of solvability conditions at higher orders in the analysis, results in an infinitely coupled system of nonlinear evolution equations for the amplitudes of the linear acoustic modes. An analysis of these amplitude equations then shows that the combination of driving processes, such as combustion, that supply energy to the acoustic oscillations and those, such as viscous effects, that dampen such motions, in conjunction with the manner in which the resonance tube is coupled to its flow source, provides an effective mode-selection mechanism that inhibits the (linear) growth of all but a few of the lower-frequency modes. For the common case of long resonance tubes, the lowest frequencies correspond to purely longitudinal modes, and we analyse in detail the solution behaviour for a typical situation in which only the first of these has a positive linear growth rate. Basic truncation strategies for the infinitely coupled amplitudes are discussed, and we demonstrate, based on analyses with both two and three modes, the stable bifurcation of an acoustic oscillation, or limit cycle, at a critical value of an appropriate bifurcation parameter. In addition, we show that the bifurcated solution branch has a turning point at a second critical value of the bifurcation parameter beyond which no stable bounded solutions exist.

An experimental investigation of the circumstances leading to the creation and subsequent evolution of the low-speed streaks and other quasi-coherent structures on a long cylindrical body immersed in a sinusoidally oscillating flow (Stokes flow) is described. The wall shear stress and the phase lead of the maximum wall shear over the maximum free-stream velocity have been measured to characterize the unsteady boundary layer. The evolution of a sinuous streak, from its inception to its ultimate demise, and the generation of multiple streaks, arches, hairpins, and other vortical structures have been traced through flow visualization. The results have shown that the strong pressure gradients, inflexion points in the velocity profile, and the reversal of the shear stress have profound effects on the stability of the flow. The Reynolds number (Reδ = Umaxδ/ν) delineates the boundaries of the laminar stable flow, transitional flow, and turbulent flow at the start of which the phase angle decreases sharply, the friction coefficient increases rapidly, and the turbulent motion prevails over larger fractions of the flow cycle. The transitional and turbulent states are rich with vortical motions which burst themselves into existence most intensely during the later stages of the deceleration phase. The effect of the manipulation of the viscosity of the wall-layer fluid on the creation and bifurcation of the low-speed streaks is discussed in some detail.

The characteristics of thermal convection in a fluid whose viscosity varies strongly with temperature are studied in the laboratory. At the start of an experiment, the upper boundary of an isothermal layer of Golden Syrup is cooled rapidly and maintained at a fixed temperature. The fluid layer is insulated at the bottom and cools continuously. Rayleigh numbers calculated with the viscosity of the well-mixed interior are between 106 and 108 and viscosity contrasts are up to 106. Thermal convection develops only in the lower part of the thermal boundary layer, and the upper part remains stagnant. Vertical profiles of temperature are measured with precision, allowing deduction of the thickness of the stagnant lid and the convective heat flux. At the onset of convection, the viscosity contrast across the unstable boundary layer has a value of about 3. In fully developed convection, this viscosity contrast is higher, with a typical value of 10. The heat flux through the top of the layer depends solely on local conditions in the unstable boundary layer and may be written \[Q_{\rm s} = - CK_{\rm m} (\alpha g/\kappa \nu_{\rm m})^{\frac{1}{3}} \Delta T^{\frac{4}{3}}_{\rm v}\], where km and νm are thermal conductivity and kinematic viscosity at the temperature of the well-mixed interior, κ thermal diffusivity, α the coefficient of thermal expansion, g the acceleration due to gravity. ΔTv, is the ‘viscous’ temperature scale defined by \[\Delta T_{\rm v} = - \frac{\mu (T_{\rm m})}{({\rm d}\mu /{\rm d}T)(T_{\rm m})}\] where μ(T) is the fluid viscosity and Tm the temperature of the well-mixed interior. Constant C takes a value of 0.47 ± 0.03. Using these relations, the magnitude of temperature fluctuations and the thickness of the stagnant lid are calculated to be in excellent agreement with the experimental data. One condition for the existence of a stagnant lid is that the applied temperature difference exceeds a threshold value equal to (2ΔTv).

An exact analytic solution of the shallow water equations for the motion of a bore over a uniformly sloping beach is derived. This solution is valid only when the initial bore is supercritical. The results agree very well with those of Keller, Levine & Whitham (1960).

The linear stability and subsequent nonlinear evolution and acoustic radiation of a planar inviscid compressible vortex is examined. Linear-stability analysis shows that vortices with smoother vorticity profiles than the Rankine vortex considered by Broadbent & Moore (1979) are also unstable. However, only neutrally stable waves are found for a Gaussian vorticity profile. The effects of entropy gradient are investigated and for the particular entropy profile chosen, positive average entropy gradient in the vortex core is destabilizing while the opposite is true for negative average entropy gradient.

The linear initial-value problem is studied by finite-difference methods. It is found that these methods are capable of accurately computing the frequencies and weak growth rates of the normal modes. When the initial condition consists of random perturbations, the long-time behaviour is found to correspond to the most unstable normal mode in all cases. In particular, the Gaussian vortex has no algebraically growing modes. This procedure also reveals the existence of weakly decaying and neutrally stable waves rotating in the direction opposite to the vortex core, which were not observed previously.

The nonlinear development of an elliptic-mode perturbation is studied by numerical solution of the Euler equations. The vortex elongates and forms shocklets; eventually, the core splits into two corotating vortices. The individual vortices then gradually move away from each other while their rate of rotation about their mid-point slowly decreases. The acoustic flux reaches a maximum at the time of fission and decreases as the vortices move apart.

In the present work, we apply new tools from the field of adiabatic dynamical systems theory to make quantitative predictions of various important mixing quantities in quasi-steady Stokes flows which possess slowly varying saddle stagnation points. Many of these quantities can be obtained before experiments or numerical simulations are performed using only knowledge of the streamlines in steady-state flows and the externally determined flow parameters. The location and size of the main region in which mixing occurs is determined to leading order by the slowly sweeping instantaneous stagnation streamlines. Tracer patches get stretched by an amount O(1/ε) during each period, and the average striation thickness of the patch decreases by a factor of ε in the same time. Also, the rate of stretching of material interfaces is bounded from below with an analytically obtained exponentially growing lower bound. Finally, the highly regular appearance of islands in quasi-steady Stokes’ flows is explained using an extension of the KAM theory. As an example to illustrate these results, we study the transport of passive scalars in a low Reynolds number flow in the two-dimensional eccentric journal bearing when the angular velocities of the cylinders are slowly modulated, continuously and periodically in time, with frequency ε. In contrast to the flows usually studied with dynamical systems, these slowly varying systems are singular perturbation (apparently far from integrable) problems exhibiting large-scale chaos, in which the non-integrability is due to the slow, continuous O(1) modulation of the position of the saddle stagnation point and the two streamlines stagnating on it.

A phenomenological study of the processes occurring when a shock wave interacts with porous polyester and polyether foams has been undertaken. Plane shock waves generated in a shock tube were reflected off a slab of foam mounted against the back wall of the tube. Tests were conducted with an initial shock wave Mach number of 1.4 and a 70 mm thick slab of foam. The reduction in reflected shock wave strength and substantial increase in the back wall pressure over that for rigid wall reflection, found by other workers, were confirmed.

Piezoelectric pressure transducers were used to record the pressure before, alongside and behind the foam specimen. Schlieren photographs of the flow were made and showed some features not previously reported. In particular it is shown that there is a flow of gas across the face of the foam at some point of the process. Previous investigations of this interaction process have assumed that the face of the foam is a contact surface.

Short duration photographs of the distortion of the foam were taken, enabling the wave propagation in the foam material itself to be studied. It is established that the front of this compaction wave in the foam material moves at considerably lower velocity (∼ 90 m/s) than the gas wave as detected by the pressure transducers (∼ 200 m/s). This result contrasts with the assumption made in previous work that the two-phase medium behaves essentially as a homogeneous substance.

A simple physical model based on a zone of compacted material in the foam acting as a high-resistance flow barrier, is proposed to explain the observed phenomena.

The two-dimensional wave instability of convection induced by a semi-infinite heated surface embedded in a fluid-saturated porous medium is studied. Owing to the inadequacy of parallel-flow theories and the inaccuracy of the leading-order boundary-layer approximation at the point of incipient instability given by these theories, the problem has been re-examined using numerical simulations of the full time-dependent nonlinear equations of motion. Small-amplitude localized disturbances placed in the steady boundary layer are shown to propagate upstream much faster than they advect downstream. There seems to be a preferred wavelength for the evolving disturbance while it is in the linear regime, but the local growth rate depends on the distance downstream and there is a smooth, rather than an abrupt, spatial transition to convection.

The starting problem, where the temperature of the horizontal surface is instantaneously raised from the ambient, is found to give rise to a particularly violent fluid motion near the leading edge. A strong thermal plume is generated which is eventually advected downstream. The long-term evolution of the instability is computed. The flow does not settle down to a steady or a time-periodic state, and evidence is presented which suggests that it is inherently chaotic. The evolving flow field exhibits a wide range of dynamical behaviour including cell merging, the ejection of hot fluid from the boundary layer, and short periods of relatively intense fluid motion accompanied by boundary-layer thinning and short-wavelength waves.

The process of flux rope formation in a convecting cell is studied. The magnetic field has both a meridional and an azimuthal component, and so corresponds to a twisted field. Convection occurs in this cylindrical cell because of heating from below, and is assumed to take an axisymmetric form. Only the Boussinesq problem is studied here, but both the kinematic and the dynamic regimes are considered.

The two cases where the twisted field is due to (a) an imposed flux of vertical current and (b) an imposed flux of vertical vorticity are considered. Strongly twisted ropes can be generated more easily in case (b) than in case (a).

We show that convection can produce ropes twisted in the opposite direction from that of the initial field. We also find that solutions can be oscillatory even when linear theory predicts steady solutions.

A time-continuous, constant-resistivity version of the fast dynamo model introduced by Bayly & Childress (1988) is studied numerically. The expected dynamo mechanism in this context is described and is shown to be operative in the simulations. Exponential growth of the fastest growing mode is observed, with the growth rate for the smallest resistivity attempted (1/Rm = 10-4) agreeing well with the Bayly–Childress model. It is argued, based on the long- and short-wavelength behaviour of the mode for different resistivities, that the growth rates obtained for the Rm = 104 case should persist as Rm → ∞.

It is commonly assumed that heat flux and temperature diffusivity coefficients obtained in steady-state measurements can be used in the derivation of the heat conduction equation for fluid flows. Meanwhile it is also known that the steady-state heat flux as a function of temperature gradient in stably stratified turbulent shear flow is not monotone: at small values of temperature gradient the flux is increasing, whereas it is decreasing after a certain critical value of the temperature gradient. Therefore the problem of heat conduction for large values of temperature gradient becomes mathematically ill-posed, so that its solution (if it exists) is unstable.

In the present paper it is shown that a well-posed mathematical model is obtained if the finiteness of the adjustment time of the turbulence field to the variations of temperature gradient is taken into account. An evolution-type equation is obtained for the temperature distribution (a similar equation can be derived for the concentration if the stratification is due to salinity or suspended particles). The characteristic property which is obtained from a rigorous mathematical investigation is the formation of stepwise distributions of temperature and/or concentration from continuous initial distributions.

This paper considers hydraulic control and upstream influence in systems where the only wave propagation mechanism arises from the variation of vorticity or potential vorticity. These systems include two-dimensional shear flows as well as many simple paradigms for large-scale geophysical flows. The simplest is a flow in which the vorticity or potential vorticity is piecewise constant. We consider such a flow confined to a rotating channel and disturbed by a topographic perturbation. We analyse the behaviour of the system using steady nonlinear long-wave theory and demonstrate that it exhibits behaviour analogous to open-channel hydraulics, with the possibility of different upstream and downstream states. The manner by which the system achieves such states is examined using time-dependent long-wave theory via integration along characteristics and using full numerical solution via the contour-dynamics technique.

The full integrations agree well with the hydraulic interpretation of the steady-state theory. One aspect of the behaviour of the system that is not seen in open-channel hydraulics is that for strong subcritical flows there is a critical topographic amplitude beyond which information from the control cannot propagate far upstream. Instead flow upstream of the topographic perturbation adjusts until the long-wave speed is zero, the control moves to the leading edge of the obstacle and flow downstream of the control is supercritical, with a transition from one supercritical branch to another on the downstream slope of the obstacle.

The physical mechanisms by which the Reynolds shear stress is produced from dynamically evolving vortical structures in the wall region of a direct numerical simulation of turbulent channel flow are explored. The complete set of quasistreamwise vortices are systematically located and tracked through the flow by the locus of the points of intersection of their centres of rotation with the (y, z) numerical grid planes. This approach assures positive identification of vortices of widely differing strengths, including those whose amplitude changes significantly in time. The process of vortex regeneration, and the means by which vortices grow, distort and interact over time are noted. Ensembles of particle paths arriving on fixed planes in the flow are used to represent the physical processes of displacement and acceleration transport (Bernard & Handler 1990a) from which the Reynolds stress is produced. By interweaving the most dynamically significant of the particle paths with the evolving vortical structures, the dynamical role of the vortices in producing Reynolds stress is exposed. This is found to include ejections of low-speed fluid particles by convecting structures and the acceleration and deceleration of fluid particles in the cores of vortices. Sweep dominated Reynolds stress close to the wall appears to be a manifestation of the regeneration process by which new vortices are created in the flow.

We investigate experimentally the up- and downstream flow around an insulating cylinder in a conducting fluid subjected to an aligned magnetic field for low values of the magnetic Reynolds number Rm. For high values of the Alfvén number α = B0/U0(ρμ0)½ the upstream flow is characterized by magnetic and kinematic wakes. In this configuration we have measured, for the first time, the local values of the induced magnetic field. The results were analysed in a confined situation and show that the Oseen number k = ½Rm(1-α2) is the main parameter that characterizes the perturbation. In the downstream flow the two fields of perturbations (magnetic and kinetic) are characterized by von Kármán eddies. Our experiments were focused on the evolution of these eddies and show in particular that the critical Reynolds number increases strongly with the intensity of the magnetic field.

The temporal development of two-dimensional viscous incompressible flow generated by a circular cylinder impulsively started into steady rotatory and rectilinear motion at Re = 200 (based on the cylinder diameter 2a and the magnitude U of the rectilinear velocity) is studied computationally. We use an explicit finite-difference/pseudospectral technique and a new implementation of the Biot–Savart law to integrate a velocity/vorticity formulation of the Navier–Stokes equations. Results are presented for the four angular: rectilinear speed ratios α = Ωa/U (where Ω is the angular speed) considered experimentally by Coutanceau & Ménard (1985). For α ≤ 1, extension of the computations to dimensionless times larger than achieved either in the experimental work or in the computations of Badr & Dennis (1985) allows for a more complete discussion of the temporal development of the wake. Using the frame-invariant vorticity distribution, we discuss several aspects of the vortex kinematics and dynamics not revealed by the earlier work, in which vortex cores were identified from frame-dependent streamline and streamfunction information. Consideration of the flow in the absence of sidewalls confirms the artifactual nature of the trajectory of the first vortex reported by Coutanceau & Ménard for α = 3.25. For α greater than unity (the largest value considered by Badr & Dennis), our results indicate that at Re = 200 shedding of more than one vortex does indeed occur for α = 3.25 (and possibly for larger α), in contrast to the conclusion of Coutanceau & Ménard. Moreover, the shedding process is very different from that associated with the usual Kármán vortex street for α = 0. Specifically, consecutive vortices can be shed from one side of the cylinder and be of the same sense, in contrast to the non-rotating case, in which mirror-image vortices of opposite sense are shed alternately from opposite sides of the cylinder. The results are discussed in relation to the possibility of suppressing vortex shedding by open- or closed-loop control of the rotation rate.

This study involves a numerical simulation of spatially evolving secondary instability in plane channel flow. The computational algorithm integrates the time-dependent, three-dimensional, incompressible Navier–Stokes equations by a mixed finite-difference/spectral technique. In particular, we are interested in the differences between instabilities instigated by Klebanoff (K-) type and Herbert (H-) type inflow conditions, and in comparing the present spatial results with previous temporal models. It is found that for the present inflow conditions, H-type instability is biased towards one of the channel walls, while K-type instability evolves on both walls. For low initial perturbation amplitudes, H-type instability exhibits higher growth rates than K-type instability while higher initial amplitudes lead to comparable growth rates of both H-and K-type instability. In H-type instability, spectral analysis reveals the presence of the subharmonic two-dimensional mode which promotes the growth of the three-dimensional spanwise and fundamental modes through nonlinear interactions. An intermodal energy transfer study demonstrates that there is a net energy transfer from the three-dimensional modes to the two-dimensional mode. This analysis also indicates that the mean mode transfers net energy to the two-dimensional subharmonic mode and to the three-dimensional modes.

Mathematical methods of control theory are applied to the problem of control of fluid flow with the long-range objective of developing effective methods for the control of turbulent flows. The procedure of how to cast the problem of controlling turbulence into a problem in optimal control theory is presented using model problems through the formalism and language of control theory. Then we present a suboptimal control and feedback procedure for general stationary and time-dependent problems using methods of calculus of variations through the adjoint state and gradient algorithms. This suboptimal feedback control procedure is applied to the stochastic Burgers equation. Two types of controls are investigated: distributed and boundary controls. The control inputs are the momentum forcing for the distributed control and the boundary velocity for the boundary control. Costs to be minimized are defined as the sum of the mean-square velocity gradient inside the domain for the distributed control or the square velocity gradient at the wall for the boundary control; and in both cases a term was added to account for the implementation cost. Several cases of both controls have been numerically simulated to investigate the performances of the control algorithm. Most cases considered show significant reductions of the costs. Another version of the feedback procedure more effective for practical implementation has been considered and implemented, and the application of this algorithm also shows significant reductions of the costs. Finally, dependence of the control algorithm on the time-discretization method is discussed.