When a charged particle moves along a charged wall in a polar fluid, it experiences an electroviscous lift force normal to the surface and an electroviscous drag, superimposed on the viscous drag, parallel to the surface. Here a theoretical analysis is presented to determine the electroviscous drag on a charged spherical particle surrounded by a thin electrical double layer near a charged plane wall, when the particle translates parallel to the wall without rotation, in a symmetric electrolyte solution at rest. The electroviscous (electro-hydrodynamic) forces, arising from the coupling between the electrical and hydrodynamic equations, are determined as a solution of three partial differential equations, for electroviscous ion concentration (perturbed ion clouds), electroviscous potential (perturbed electric potential) and electroviscous or electro-hydrodynamic flow field (perturbed flow field). The problem was previously solved for small gap widths and low Peclet numbers in the inner region around the gap between the sphere and the wall, using lubrication theory. Here the restriction on the particle–wall distances is removed, and an analytical and numerical solution is obtained valid for the whole domain of interest. For large sphere–wall separations the solution approaches that for the electroviscous drag on an isolated sphere in an unbounded fluid. For small particle–wall distances it differs from that obtained by the use of lubrication theory, showing that lubrication theory is inadequate for electroviscous problems. The analytical results are in complete agreement with the full numerical calculations. For small particle–wall distances a model is given which provides both physical insight and an easy way to calculate the force with high precision.

Unstable equilibrium solutions in a homogeneous shear flow with sinuous (streamwise-shift-reflection and spanwise-shift-rotation) symmetry are numerically found in large-eddy simulations (LES) with no kinetic viscosity. The small-scale properties are determined by the mixing length scale $l_{S}$ used to define eddy viscosity, and the large-scale motion is induced by the mean shear at the integral scale, which is limited by the spanwise box dimension $L_{z}$. The fraction $R_{S}=L_{z}/l_{S}$, which plays the role of a Reynolds number, is used as a numerical continuation parameter. It is shown that equilibrium solutions appear by a saddle-node bifurcation as $R_{S}$ increases, and that the flow structures resemble those in plane Couette flow with the same sinuous symmetry. The vortical structures of both lower- and upper-branch solutions become spontaneously localised in the vertical direction. The lower-branch solution is an edge state at low $R_{S}$, and takes the form of a thin critical layer as $R_{S}$ increases, as in the asymptotic theory of generic shear flow at high Reynolds numbers. On the other hand, the upper-branch solutions are characterised by a tall velocity streak with multiscale multiple vortical structures. At the higher end of $R_{S}$, an incipient multiscale structure is found. The LES turbulence occasionally visits vertically localised states whose vortical structure resembles the present vertically localised LES equilibria.

The two-dimensional adiabatic transverse normal modes of an inviscid compressible fluid, having solid body rotation about the axis of its cylindrical container, are considered. Relative to the rotating fluid there are two trains of harmonic waves, propagating in opposite directions. The first five modes of the first-order harmonic wave and the first mode of the harmonic waves of order two, five, ten, fifteen and twenty have been considered. The period and amplitude of the waves is considerably modified by the rotation. Relative to a fixed coordinate system the angular velocity of the waves initially propagating in the same direction as the rotating fluid, is larger than that of the waves propagating in the opposite direction, as might be expected. However, in the case of the first mode, relative to the rotating fluid, the waves propagating in the opposite direction are the faster.

Traditional a posteriori tests of subgrid-scale (SGS) models often compare large eddy simulation (LES) profiles of various statistics with measurements. In this study we propose and employ a new a posteriori test to study SGS model performance. We compare the conditional means of the LES-generated SGS stress and stress production rate conditional on the resolvable-scale velocity with measurements. These statistics must be reproduced by the SGS model for LES to correctly predict the one-point resolvable-scale velocity joint probability density function. Our tests using data obtained in convective atmospheric boundary layers show that the a posteriori results are consistent with our a priori tests based on the same conditional statistics. The strengths and deficiencies of the models observed here were also identified in our a priori tests. The remarkable consistency between the two types of tests suggests that statistical model tests based on the conditional SGS stress and its production rate are a highly capable approach for identifying specific model deficiencies and for evaluating SGS model performance in simulations.

The aim of this paper is to show that the viscous shear instability identified in Part 1 is intrinsically coupled with internal gravity waves when a localized surface topography is present within a boundary layer. The coupling involves two aspects: receptivity and radiation. The former refers to excitation of shear instability modes by gravity waves, and the latter to emission of gravity waves by instability modes. Both physical processes are studied using triple-deck theory. In particular, the radiated gravity waves are found to produce a leading-order back action on the source, and this feedback effect, completely ignored in the acoustic analogy type of approach, is naturally taken into account by the triple-deck formalism. A by-product is that for certain incident angles, gravity waves are over-reflected by the boundary layer, i.e. the reflected waves are stronger than the incident waves.

We investigate the weakly nonlinear dynamics of transient gravity waves at infinite depth under the influence of a shear current varying linearly with depth. The shear field makes this problem three-dimensional and rotational in nature, but an analytical solution is permitted via integration of the Euler equations. Although similar problems were investigated in the 1960s and 70s for special cases of resonance, this is to our knowledge the first general wave interaction (mode coupling) solution derived to second order with a shear current present. Wave interactions are integrable in a spectral convolution to yield the second-order dynamics of initial value problems. To second order, irrotational wave dynamics interacts with the background vorticity field in a way that creates new vortex structures. A notable example is the large parallel vortices which drive Langmuir circulation as oblique plane waves interact with an ocean current. We also investigate the effect on wave pairs which are misaligned with the shear current to find that similar, but skewed, vortex structures are generated in every case except when the mean wave direction is precisely perpendicular to the direction of the current. This is in contrast to a conjecture by Leibovich (Annu. Rev. Fluid Mech., vol. 15, 1983, pp. 391–427). Similar nonlinear wave–shear interactions are found to also generate near-field vortex structures in the Cauchy–Poisson problem with an initial surface elevation. These interactions create further groups of dispersive ring waves in addition to those present in linear theory. The second-order solution is derived in a general manner which accommodates any initial condition through mode coupling over a continuous wave spectrum. It is therefore applicable to a range of problems including special cases of resonance. As a by-product of the general theory, a simple expression for the Stokes drift due to a monochromatic wave propagating at oblique angle with a current of uniform vorticity is derived, for the first time to our knowledge.

A weakly nonlinear analysis is presented of parametric instability in a rotating cylinder subject to periodic axial compression by small sinusoidal oscillations of one of its ends (‘the piston’). Amplitude equations are derived for the pair of parametrically resonant (primary) inertial modes which were found to arise from linear instability in Part 1. These equations introduce an infinity of geostrophic mode amplitudes, representing a nonlinear modification of the mean flow, for which evolution equations are also derived. Consequences of the total system of equations are investigated for axisymmetric modes. Different possible outcomes are found at large times: (a) a fixed point, representing a saturated state in which the oscillatory toroidal vortices of the primary mode are phase-locked to the piston motion with half its frequency; (b) a limit cycle or chaotic attractor, corresponding to slow-time oscillations of the primary mode; or (c) exponential divergence of the amplitudes to infinity. The latter outcome, a necessary condition for which is derived in the form of a threshold piston amplitude for divergence, invalidates the theory, inducing a gross change in the character of the flow and providing a route out of the weakly nonlinear regime. Non-zero fixed-point branches arise via bifurcations from both sides of the linear neutral curve, where the basic flow changes local stability. The lower-amplitude branch is shown to be unstable, while the upper one may lose local stability, resulting in a Hopf bifurcation to a limit cycle, which can subsequently become aperiodic via a series of further bifurcations. Typically, during the resulting oscillations, whether periodic or not, the perturbation first grows from small amplitude owing to basic-flow instability, then nonlinear detuning of the parametric resonance causes decay back to small amplitude in the second half of the cycle, which then restarts.

In the present study, the effects of an isolated stationary spherical particle on the transition process in a flat-plate boundary layer are examined by a spatial direct numerical simulation. The full three-dimensional time-dependent incompressible Navier–Stokes equations are integrated by a time-splitting method and discretized spatially by a high-order finite difference/spectral method. A virtual boundary technique defining the no-slip boundary of a sphere is implemented within the Cartesian geometry of the computational grid.

Two numerical simulations which consider the effects of the sphere on the boundary layer are presented. The subcritical Reynolds number case reveals the appearance of hairpin vortices shed into the sphere wake which decay as they are convected downstream. The initial interaction of the sphere and the boundary layer produces a three-dimensional isolated disturbance comprising a wave part and a transient part. The decaying transient part is convected downstream at the local mean velocity, while the wave part induces a decaying Tollmien–Schlichting wave in the flow field.

In the second case, an increase in the Reynolds number results in a wedge of incipient turbulent flow downstream of the sphere. The development of the wake of the sphere is dominated by the appearance of an isolated disturbance which rapidly breaks down forming a structure resembling a turbulent spot. It is demonstrated that the transition induced by a sphere in the boundary layer is due to a mechanism related to bypass transition.

A mathematical model is constructed for cavitating flow past a wedge with sides of equal length but with its axis of symmetry placed at an angle to the incident stream. The model involves a subsidiary cavity with a re-entrant jet at the vertex. Only the case of zero cavitation number is considered. The flow field is worked out in some detail for small angles of incidence, and the lift, drag and moment coefficients are calculated as far as first-order terms in the angle of incidence. It is shown that the effect of the rate of loss of momentum in the re-entrant jet on these force coefficients is negligible to this order.

Experimentally, it is shown that the secondary cavity does exist under suitable conditions, and the force coefficients obtained agree with the theory.

We consider the unsteady three-dimensional Kármán flow induced by the impulsive rotation of an infinite rotating plane immersed in an incompressible viscous fluid with a dilute suspension of small solid monodisperse spherical particles. The flow is described in terms of a ‘dusty gas’ model, which treats the discrete phase (particles) and the continuous phase (fluid) as two continua occupying the same space and interacting through a Stokes drag mechanism. The model is extended to allow for a local gravitational acceleration in a direction parallel to the axis of rotation, and is valid for cases in which gravity acts either in the same direction as or in the opposite direction to the Ekman axial flow induced by the rotation of the plane.

Analysis based on the theory of characteristics shows that the role of gravity is crucial to the treatment of the discrete-phase equations, particularly in regard to the appropriate boundary conditions to be applied at the solid surface. Other notable features include the presence of an essential singularity in the solution when gravity is absent; indeed this phenomenon may help to explain some of the difficulties encountered in previous studies of this type. If the gravitational force is directed away from the rotating surface, a number of other interesting features arise, including the development of discontinuities in the particle distribution profiles, with corresponding particle-free regions contained between the interface and the rotating boundary. These ‘shock’ features can be associated with a critical axial location in the boundary layer at which a balance is achieved between Ekman suction induced by the rotating boundary and the influence of gravitational effects acting to move particles away from the boundary.

Internal hydraulic theory is often used to describe idealized bi-directional exchange flow through a constricted channel. This approach is formally applicable to layered flows in which velocity and density are represented by discontinuous functions that are constant within discrete layers. The theory relies on the determination of flow conditions at points of hydraulic control, where long interfacial waves have zero phase speed. In this paper, we consider hydraulic control in continuously stratified exchange flows. Such flows occur, for example, in channels connecting stratified reservoirs and between homogeneous basins when interfacial mixing is significant. Our focus here is on the propagation characteristics of the gravest vertical-mode internal waves within a laterally contracting channel.

Two approaches are used to determine the behaviour of waves propagating through a steady, continuously sheared and stratified exchange flow. In the first, waves are mechanically excited at discrete locations within a numerically simulated bi-directional exchange flow and allowed to evolve under linear dynamics. These waves are then tracked in space and time to determine propagation speeds. A second approach, based on the stability theory of parallel shear flows and examination of solutions to a sixth-order eigenvalue problem, is used to interpret the direct excitation experiments. Two types of gravest mode eigensolutions are identified: vorticity modes, with eigenfunction maxima centred above and below the region of maximum density gradient, and density modes with maxima centred on the strongly stratified layer. Density modes have phase speeds that change sign within the channel and are analogous to the interfacial waves in hydraulic theory. Vorticity modes have finite propagation speed throughout the channel but undergo a transition in form: upwind of the transition point the vorticity mode is trapped in one layer. It is argued that modes trapped in one layer are not capable of communicating interfacial information, and therefore that the transition points are analogous to control points. The location of transition points are identified and used to generalize the notion of hydraulic control in continuously stratified flows.

Nonlinear interactions between the two wakes behind a pair of square cylinders, which are placed side by side in a uniform flow, are investigated by the linear and weakly nonlinear stability analyses and numerical simulations. It is known from the linear stability analysis that the flow past a pair of cylinders becomes unstable to a symmetric or an antisymmetric mode of disturbance, depending on the gap ratio, the ratio of the gap distance between the two cylinders to the cylinder diameter. The antisymmetric mode gives the critical condition for smaller gap ratios than a threshold value, and for larger gap ratios the symmetric mode becomes the most unstable. We focus on the flow pattern arising through the nonlinear interactions of the two modes of disturbance for gap ratios around the threshold value when both modes are growing. We derive a couple of amplitude equations for the two modes to properly describe the nonlinear interaction between them by applying the weakly nonlinear stability theory. The amplitude equations are shown to have three equilibrium solutions except the null solution such as a mixed-mode solution, symmetric and antisymmetric single-mode solutions. Examination of the stability of each equilibrium solution leads to a conclusion that the mixed-mode solution exchange its stability with both the symmetric and the antisymmetric single-mode solutions simultaneously. In the case where the mixed-mode solution is stable, both the symmetric and antisymmetric modes have finite amplitudes, and the resultant flow has an asymmetric flow pattern comprising of finite amplitudes of the two modes of disturbance superposed on the steady symmetric flow. While in the case where both the single-mode solutions are stable, either of the symmetric- and antisymmetric-mode solutions survives, overwhelming the other. Then, if the symmetric mode attains at an equilibrium finite amplitude and the antisymmetric mode vanishes, the resultant flow is symmetric, and if the antisymmetric mode survives and the symmetric mode decays out, the flow becomes asymmetric with the antisymmetric mode of disturbance superposed on the steady symmetric flow. Thus, the flow appearing due to instability differs depending on the initial condition, not uniquely determined, when both single-mode solutions are stable. We numerically delineated the region in the parameter space of the gap ratio and the Reynolds number where the mixed-mode solution is stable. The theoretical results obtained from the weakly nonlinear stability analyses are confirmed by numerical simulations. The conclusion derived from the stability analysis of the equilibrium solutions of the amplitude equations is widely applicable also to other double Hopf bifurcation problems.

We derive the local induction approximation (LIA) for a quantum vortex filament in the arclength coordinate frame where the tangent vector is the unknown function. The equation for the tangent vector to the filament is then converted to a potential form, which ends up being a type of nonlinear Schrödinger equation that governs the tangential LIA model (T-LIA). Such a formulation was previously derived by Umeki for the standard fluid model (in the absence of superfluid friction terms). While it is challenging to generalize many of the exact solutions found for the standard LIA to the quantum LIA model, we demonstrate that the T-LIA model facilitates this generalization nicely. Indeed, under the T-LIA model, we are able to construct a variety of solutions. The Hasimoto solution related to elastica is one of the fundamental solutions present for the standard fluid model; however, using the T-LIA model, we are able to demonstrate the existence of such a solution, thereby extending the Hasimoto solution to the superfluid case. In the zero-temperature limit, purely self-similar solutions are shown to exist for the T-LIA model. As the superfluid warms (so that the influence of the normal flow is no longer negligible), the analogue to the self-similar solution is a new class of solutions, which depend on the similarity variable as well as a time-dependent additive scaling. In other words, the self-similar structures gradually deform as the magnitude of the normal-fluid velocity increases, which makes complete physical sense. When dealing with small deviations from the central axis of alignment, we can describe such solutions analytically. There exists a family of helical vortex filaments in the presence of a normal fluid impinging on the vortex, in complete agreement with the previously studied results for the LIA model. Finally, a number of soliton solutions are shown to exist in different regimes of the T-LIA model.

An extension to the theory of detonation shock dynamics is made and new propagation laws are derived for steady, near-CJ (Chapman–Jouguet), weakly curved detonations for a chain-branching reaction model having two components. The first is a thermally neutral induction stage governed by an Arrhenius reaction with a large activation energy, which terminates at a location called the transition interface, where instantaneous conversion of fuel into an intermediate species (chain radical) occurs. The second is an exothermic main reaction layer (or chain-recombination zone) having a temperature-independent reaction rate. We make an ansatz that the shock curvature is sufficiently large to have a leading-order influence on the induction zone structure, whereupon it is shown that multi-dimensional effects must necessarily be accounted for in the main reaction layer. Only for exactly cylindrical or spherical waves can such multi-dimensional effects be omitted. A requirement that the main reaction layer structure pass smoothly through a sonic plane leads to a propagation law for the detonation front: a relationship between the detonation velocity, the shock curvature and various shock arclength derivatives of the position of the transition interface.

For exactly cylindrically or spherically expanding waves, a multi-valued detonation velocity–curvature relationship is found, similar to that found previously for a state-sensitive one-step reaction. The change in this relationship is investigated as the ratio of the length of the main reaction layer to the induction layer is changed. We also discuss the implications of chain-branching reaction kinetics for the prediction of critical detonation initiation energy based on detonation-velocity curvature laws. Finally several calculations that illustrate the important effect that arclength and transverse flow variations may have on the steady propagation of non-planar detonation fronts are presented. Such variations may be important for the propagation of cellular gaseous detonation fronts and for the axial propagation of detonations in a cylindrical stick of condensed-phase explosive. We also show that the arclength variations provide a formal mechanism for the existence of steady non-planar detonation fronts having converging sections, a possibility ruled out for simple irreversible one-step reaction mechanisms where only diverging steady waves are admissible.

The flow of a gas stream past a flat plate under the influence of rainfall is investigated. As raindrops sediment on the flat plate, they coalesce to form a water film that flows under the action of shear from the surrounding gas stream. In the limit of (a) large Reynolds number, Re, in the gas phase, (b) small rainfall rate, r˙, compared to the free-stream velocity, U∞, and (c) small film thickness compared to the thickness of the boundary layer that surrounds it, a similarity solution is obtained that predicts growth of the liquid film like x3/4; x denotes dimensionless distance from the leading edge. The flow in the gas stream closely resembles the Blasius solution, whereas viscous dissipation dominates inside the film. Local linear stability analysis is performed, assuming nearly parallel base flow in the two streams, and operating in the triple-deck regime. Two distinct families of eigenvalues are identified, one corresponding to the well-known Tollmien–Schlichting (TS) waves that originate in the gas stream, and the other corresponding to an interfacial instability. It is shown that, for the air–water system, the TS waves are convectively unstable whereas the interfacial waves exhibit a pocket of absolute instability, at the streamwise location of the applied disturbance. Moreover, it is found that as the inverse Weber number (We−1) increases, indicating the increasing effect of surface tension compared to inertia, the pocket of absolute instability is translated towards larger distances from the leading edge and the growth rate of unstable waves decreases, until a critical value is reached, We−1 ≈ We−1c, beyond which the family of interfacial waves becomes convectively unstable. Increasing the inverse Froude number (Fr−1), indicating the increasing effect of gravity compared to inertia, results in the pocket of absolute instability shrinking until a critical value is reached, Fr−1 ≈ Fr−1c, beyond which the family of interfacial waves becomes convectively unstable. As We−1 and Fr−1 are further increased, interfacial waves are eventually stabilized, as expected. In this context, increasing the rainfall rate or the free-stream velocity results in extending the region of absolute instability over most of the airfoil surface. Owing to this behaviour it is conjectured that a global mode that interacts with the boundary layer may arise at the interface and, eventually, lead to three-dimensional waves (rivulets), or, under extreme conditions, even premature separation.

A new two-parameter integral method is presented which is applicable to unsteady two-dimensional laminar boundary layers whether they are separated or not. The governing equations consist of three moments of the boundary-layer equation, and the assumed velocity profiles are those of unsteady trailing-edge flow and Falkner-Skan flow with slip. The governing equation system being hyperbolic, the spontaneous generation of a singularity associated with unsteady separation is confirmed as the focusing of characteristics. The obtained results of the boundary-layer quantities as well as the generation of separation singularity are in good agreement with those of exact methods (e.g. van Dommelen & Shen 1980) for starting flows of cylinders.

In this paper, the geometrical properties of the subgrid-scale (SGS) stress tensor are investigated through its eigenvalues and eigenvectors. The concepts of Euler rotation angle and axis are utilized to investigate the relative rotation of the eigenframe of the SGS stress tensor with respect to that of the resolved strain rate tensor. Both Euler rotation angle and axis are natural invariants of the rotation matrix, which uniquely describe the topological relation between the eigenframes of these two tensors. Different from the reference frame fixed to a rigid body, the eigenframe of a tensor consists of three orthonormal eigenvectors, which by their nature are subjected to directional aliasing. In order to describe the geometric relationship between the SGS stress and resolved strain rate tensors, an effective method is proposed to uniquely determine the topology of the eigenframes. The proposed method has been used for testing three SGS stress models in the context of homogeneous isotropic turbulence at three Reynolds numbers, using both a priori and a posteriori approaches.

In the present paper turbulent mixing in the Stokes–Ekman bottom boundary layer is investigated analytically and by wall-resolving large-eddy simulations (LES). The analytical solution shows that when the Rossby number oscillation and rotation interact with each other, this gives rise to a thickening of the boundary layer compared with the purely oscillating or the purely rotating case. The solution also shows the presence of elliptical patterns developing on the horizontal planes that shrink when approaching the low latitudes. In the turbulent regime, a LES was applied for an east–west tidal current, considered at three different latitudes in the Northern Hemisphere, namely, the polar case, the mid-latitude case () and the quasi-equatorial case (). The Reynolds number of the simulation, based on the viscous penetration depth and the frequency of the purely oscillatory flow, was set equal to . The analysis suggests that rotation has two main effects on the flow field: in the polar case, rotation tends to delay the cyclic re-transition to turbulence and to narrow the turbulent phases of the cycle. Also, rotation suppresses vertical fluctuations of velocity and redistributes energy from the streamwise direction to the spanwise direction. It is noteworthy that the high-latitude effect makes the turbulent field substantially different from the reference Stokes boundary layer case, whereas the low-latitude effects appear to be of secondary importance, owing to the weakness of the rotation rate. Consequently, the study shows that the Stokes boundary layer may be representative of the oceanic bottom boundary layer in the low-latitude cases ( cases in our simulations). Conversely, it cannot be considered as archetypal of the oceanic boundary layer at high latitudes ( case of our study), where the vertical background vorticity profoundly modifies the turbulent field.

A NATO Advanced Study Institute on the topic of transition from laminar flow to turbulence was held at Imperial College, London, from 1 to 6 July 1968. Each morning's session was started with a one-hour general lecture, and was followed by five or six half-hour lectures interspaced with discussion periods. The main lecturers were C. C. Lin (general survey), S. Rosenblat (stability of time-dependent flows), L. S. G. Kovasznay (turbulent, non-turbulent interfaces), L. E. Scriven (free surface effects) and A. A. Townsend (shear turbulence). The idea of the meeting was to bring forth and to discuss current ideas in the subject, both from the point of view of developments out of laminar flow and from that of developments into real turbulence. To this end speakers were chosen to introduce a variety of topics ranging from laminar-flow instabilities (with emphasis on aspects at present imperfectly understood), through non-linear effects to the processes affecting turbulence itself.

Many ideas recurred throughout the meeting, both at lectures and in discussion periods. This is true, for example, of several relevant points forcefully made by C. C. Lin. For this reason the present account does not attempt to describe the proceedings of the meeting in chronological order, but rather takes an overall view of the subject matter and points to the areas of agreement and of controversy in relation to various problems.