An investigation is made of the stability of time-periodic azimuthal flows between coaxial, circular cylinders. The disturbance equations are linearized and consideration is limited to the effects of axisymmetric disturbances in a fluid with zero viscosity. It is found convenient to examine separately the cases of flows with zero time-mean and non-zero time-mean respectively. Some remarks are made concerning the definition of stability in relation to such flows and their general stability characteristics are evaluated and discussed.

Power spectral density measurements were made of the water surface displacement in a wind-generated wave train in the Stanford wind, water-wave research facility. Growth rates of different components were determined from the measured spectra. The resulting values were compared with those predicted by the viscous Reynolds stress mechanism of energy transfer from wind to wave which was proposed by Miles. Where the growth was exponential the theory could be made to predict growth rates successfully at wave frequencies less than 3·5 c/s. At higher frequencies the theory predicted values an order of magnitude larger than those measured. Limited regions of linear growth were found at the lowest wind speed for components with frequencies less than 2·5 c/s. The scatter in the data did not permit a quantitative comparison with theory to be made for this range.

The stability of liquid layers in a porous medium under the action of viscous and surface forces is described. An extension of previous studies on the stability of a single interface in a porous medium is presented as the basis for solutions to many problems of practical interest where flow in porous media are involved.

The structure of plane steady shock waves in a gas with several internal energy modes which relax in parallel is investigated. Transport effects are neglected. Conditions for continuity and monotonicity of the velocity profile are discussed; when all modes have constant specific heats and relaxation times it is established that velocity must decrease monotonically. Internal mode energy contents may overshoot their local equilibrium values.

Numerical results for waves in a hypothetical gas with two relaxing modes are presented for purposes of illustration.

Liu & Chan (J. Fluid Mech. vol. 579, 2007, p. 467, hereinafter referred to as LC) derived analytical solutions for the interactions between shallow water waves and a viscous fluid seabed. In this paper we present a set of new experimental data on the solitary-wave-induced flows in a viscous muddy seabed so as to validate LC's theory and the approximations employed. In the experiments a clear silicone fluid was used as the viscous mud and particle image velocimetry was employed to measure the velocity field inside the viscous mud. The shear stress along the bottom of the mud bed and the displacement of the water--mud interface were also deduced from data. Experimental results showed excellent agreement with the theoretical solutions. Additional analyses were performed to show that the ratio of the muddy seabed thickness to the corresponding bottom boundary-layer thickness, , plays an important role in characterizing mud flow regimes. When , the vertical profile of the horizontal velocity in the mud bed can be parameterized as a parabola. On the other hand, when , the velocity profile appears as a plug flow above a thin viscous layer. When , the flow patterns are more complex than the other two regimes and flow reversal can occur inside the viscous mud bed.

A multi-layer model is used to describe a ‘two-dimensional’ continuously stratified fluid. We use a momentum theorem in each layer to derive an ordinary differential equation describing the vertical structure behind a jump. This equation is compared with the corresponding equation for continuous flow. As one would expect from the classical one-layer theory, they are identical up to second order for weak disturbances. The energy change across a jump is also derived. By requiring that energy be lost through a jump, we calculate when a weak jump is possible in general. Algorithms for computing jumps of arbitrary strength are given. To ensure that the flow after the jump is stable and also for a numerical reason to be stated in § 8, we require that the Richardson number after the jump be equal to or greater than S¼S. Numerical examples are given to show the range of parameters within which jumps are possible; the velocity profiles related to different kinds of jumps also appear. Since hydraulic jumps in a continuously stratified fluid have not yet been observed in any laboratory, it should be of interest to verify these calculations experimentally.

Planar velocity fields of mixing-enhanced compressible planar shear layers are measured via particle image velocimetry (PIV) in order to investigate the mechanism of mixing enhancement by sub-boundary-layer triangular disturbances. The measurements are conducted at convective Mach numbers, $M_{{c}}$, of 0.62 and 0.24 to examine compressibility effects on effectiveness of the mixing enhancement technique. Instantaneous side- and plan-view vector maps of the shear layers are obtained, and turbulence statistical quantities are derived from the instantaneous velocity data. Schlieren and planar laser Mie scattering (PLMS) techniques are also used to measure the shear-layer thickness and growth rate as well as surveying the qualitative flow fields. The velocity fields for several disturbance configurations with different shape, size, or thickness are compared in terms of the shear-layer thickness and growth rate in order to investigate the effects of the configuration variation on the mixing enhancement strategy. Configuration parameters include thickness, the semi-vertex angle of the triangular disturbance, and the streamwise offset of the disturbance from the splitter tip. The measured transverse profile of the mean streamwise velocity shows a characteristic shape with triple inflection points for the effective mixing-enhanced cases at the two different compressibility conditions, while periodic inflection points are observed in the spanwise direction. A pair of stationary counter-rotating streamwise vortices introduced by the subboundary-layer disturbances are also observed, even in the fully developed region of the shear layers. At $M_{{c}}\,{=}\,0.62$, it is found that in successfully enhanced cases, regardless of the disturbance configurations, the present mixing-enhancement strategy has the effect of increasing the turbulence intensity and Reynolds stress, and suppressing the turbulence anisotropy increase with increasing compressibility, i.e. alleviating the compressibility effect which intrinsically reduces pressure–strain-rate redistribution, leading to effective mixing enhancement. Comparison of the results at the two compressibility conditions reveals that the growth rate of the layer is almost constant in the streamwise direction for all cases at $M_{{c}}\,{=}\,0.62$, while for all disturbed cases at $M_{{c}}\,{=}\,0.24$, after an initial layer thickening, growth rate decreases with downstream distance to the value for the undisturbed case, indicating that the present mixing enhancement is less effective at nearly incompressible conditions.

We consider the effect of a wetting condition at the moving contact line on the frequency and damping of surface waves on an inviscid liquid in a circular cylinder. The velocity potential φ and the free surface elevation η are sought as complex eigenfunction expansions. The φ eigenvalues are the classical ones whereas the η eigenvalues are unknown and have to be computed so as to satisfy the wetting condition on the contact line and the other free surface conditions – these turn out to be complex in general. A projection of the latter conditions on to an appropriate basis leads to an eigenvalue problem, for the complex frequency Ω, which has to be solved iteratively with the wetting condition. The variation of Ω with liquid depth h, Bond number Bo, capillary coefficient λ and static contact angle θc0 is explored for the (1, 0),(2, 0),(0, 1),(3, 0) and (4, 0) modes. The damping vanishes for λ = 0 (pinned-end edge condition) and λ = ∞ (free-end edge condition) with a maximum in the interior while the frequency decreases with increasing λ, approaching limiting values at the endpoints. A comparison with the analytic results of Miles (J. Fluid Mech., vol. 222, 1991, p. 197) for the no-meniscus case and the experimental results of Cocciaro, Faetti, & Festa (J. Fluid Mech., vol. 246, 1993, p. 43), where a meniscus is present, is good. The study provides a simple procedure for calculating the inviscid capillary damping associated with the moving contact line in a circular cylinder of finite depth with meniscus effects also being considered.

Detonations usually form through either direct initiation or deflagration-to-detonation transition (DDT). In this work, a detonation initiation process is introduced that shows attributes from each of these two processes. Energy is deposited into a finite volume of fluid in an amount of time that is similar to the acoustic time scale of the heated fluid volume. Two-dimensional simulations of the reactive Euler equations are used to solve for the evolving detonation initiation process. The results show behaviour similar to both direct initiation and DDT. Localized reaction transients are shown to be intimately related to the appearance of a detonation. Thermomechanical concepts are used to provide physical interpretations of the computational results in terms of the interaction between compressibility phenomena on the acoustic time scale and localized, spatially resolved, chemical energy addition on a heat-addition time scale.

The interaction of the barotropic tide with a tall, two-dimensional ridge is examined analytically and numerically at latitudes where the tide is subinertial, and contrasted to when the tide is superinertial. When the tide is subinertial, the energy density associated with the response grows with latitude as both the oscillatory along-ridge flow and near-ridge isopycnal displacement become large. Where $f\neq 0$, nonlinear processes lead to the formation of along-ridge jets, which become faster at high latitudes. Dissipation and mixing is larger, and peaks later in the tidal cycle when the tide is subinertial compared with when the tide is superinertial. Mixing occurs mainly on the flanks of the topography in both cases, though a superinertial tide may additionally generate mixing above topography arising from convective breaking of radiating waves.

Shoreline boundary conditions for nearshore hydrodynamic models are discussed on the basis of the swash zone equations of Brocchini & Peregrine (1996). Swash zone flows are investigated further using the shallow water equations. Results from numerical computations are used to guide approximation to provide more practical boundary conditions for wave-averaged flows. Approximate boundary conditions, valid for small values of the rate of change of the mean water volume in the swash zone, are found which allow explicit computation of a non-zero mean water depth at the mean shoreline. This is computed in terms of the local height of the short waves. Implementation issues are also discussed.

We examine the rotor–oscillator flow, a slow viscous flow between long parallel plates driven by the rotation of a slender cylinder (the rotor) and the longitudinal oscillation of one of the plates (the oscillator). For rotor locations of interest to us, this flow exhibits a hyperbolic mixing region, characterized by homoclinic tangling associated with a hyperbolic fixed point, and a degenerate mixing region, characterized by heteroclinic tangling associated with two degenerate fixed points on one of the boundary plates (normally the oscillator). These mixing regions are investigated both theoretically, by applying various dynamical tools to a mathematical model of the flow, and experimentally, by observing the advection of a passive tracer in a specially constructed apparatus. Although degenerate mixing regions have been largely ignored or undervalued in previous research on chaotic mixing, our results demonstrate that more mixing is associated with the degenerate mixing region than the hyperbolic one in many cases. We have also discovered a peculiar phenomenon, which we call Melnikov resonance, involving a rapid fluctuation in the size of the hyperbolic mixing region as the frequency of the oscillator is varied.

Axisymmetric Rayleigh–Bénard convection in a cylinder with vertical axis is studied. The nonlinear behaviour is investigated near the onset of convection using an eigenfunction expansion. It is found that initially a steady target pattern develops; the number of rolls depends only on the aspect ratio of the box. At about 5% beyond onset, an oscillatory pattern develops, in which the number of rolls oscillates between two adjacent values. The transitions between the initial steady state and this oscillatory pattern are also investigated, and fall into two main categories. As the Rayleigh number is reduced to the transition point, either the period of the travelling wave tends to infinity, whilst its amplitude stays finite, or there is a sudden transition to a vascillating pattern, the amplitude of which becomes smaller and finally vanishes, whilst the period remains finite. The results are compared with experimental work.

We study numerically the scale-to-scale transfers of enstrophy and passive-tracer variance in two-dimensional turbulence, and show that these transfers display significant differences in the inertial range of the enstrophy cascade. While passive-tracer variance always cascades towards small scales, enstrophy is characterized by the simultaneous presence of a direct cascade in hyperbolic regions and of an inverse cascade in elliptic regions. The inverse enstrophy cascade is particularly intense in clusters of small-scales elliptic patches and vorticity filaments in the turbulent background, and it is associated with gradient-decreasing processes. The inversion of the enstrophy cascade, already noticed by Ohkitani (Phys. Fluids A, vol. 3, 1991, p. 1598), appears to be the main difference between vorticity and passive-tracer dynamics in incompressible two-dimensional turbulence.

This experimental study investigates the effect of imposed rotary oscillation on the flow-induced vibration of a sphere that is elastically mounted in the cross-flow direction, employing simultaneous displacement, force and vorticity measurements. The response is studied over a wide range of forcing parameters, including the frequency ratio $f_{R}$ and velocity ratio $\unicode[STIX]{x1D6FC}_{R}$ of the oscillatory forcing, which vary between $0\leqslant f_{R}\leqslant 5$ and $0\leqslant \unicode[STIX]{x1D6FC}_{R}\leqslant 2$. The effect of another important flow parameter, the reduced velocity, $U^{\ast }$, is also investigated by varying it in small increments between $0\leqslant U^{\ast }\leqslant 20$, corresponding to the Reynolds number range of $5000\lesssim Re\lesssim 30\,000$. It has been found that when the forcing frequency of the imposed rotary oscillations, $f_{r}$, is close to the natural frequency of the system, $f_{nw}$, (so that $f_{R}=f_{r}/f_{nw}\sim 1$), the sphere vibrations lock on to $f_{r}$ instead of $f_{nw}$. This inhibits the normal resonance or lock-in leading to a highly reduced vibration response amplitude. This phenomenon has been termed ‘rotary lock-on’, and occurs for only a narrow range of $f_{R}$ in the vicinity of $f_{R}=1$. When rotary lock-on occurs, the phase difference between the total transverse force coefficient and the sphere displacement, $\unicode[STIX]{x1D719}_{total}$, jumps from $0^{\circ }$ (in phase) to $180^{\circ }$ (out of phase). A corresponding dip in the total transverse force coefficient $C_{y\,(rms)}$ is also observed. Outside the lock-on boundaries, a highly modulated amplitude response is observed. Higher velocity ratios ($\unicode[STIX]{x1D6FC}_{R}\geqslant 0.5$) are more effective in reducing the vibration response of a sphere to much lower values. The mode I sphere vortex-induced vibration (VIV) response is found to resist suppression, requiring very high velocity ratios ($\unicode[STIX]{x1D6FC}_{R}>1.5$) to significantly suppress vibrations for the entire range of $f_{R}$ tested. On the other hand, mode II and mode III are suppressed for $\unicode[STIX]{x1D6FC}_{R}\geqslant 1$. The width of the lock-on region increases with an increase in $\unicode[STIX]{x1D6FC}_{R}$. Interestingly, a reduction of VIV is also observed in non-lock-on regions for high $f_{R}$ and $\unicode[STIX]{x1D6FC}_{R}$ values. For a fixed $\unicode[STIX]{x1D6FC}_{R}$, when $U^{\ast }$ is progressively increased, the response of the sphere is very rich, exhibiting characteristically different vibration responses for different $f_{R}$ values. The phase difference between the imposed rotary oscillation and the sphere displacement $\unicode[STIX]{x1D719}_{rot}$ is found to be crucial in determining the response. For selected $f_{R}$ values, the vibration amplitude increases monotonically with an increase in flow velocity, reaching magnitudes much higher than the peak VIV response for a non-rotating sphere. For these cases, the vibrations are always locked to the forcing frequency, and there is a linear decrease in $\unicode[STIX]{x1D719}_{rot}$. Such vibrations have been termed ‘rotary-induced vibrations’. The wake measurements in the cross-plane $1.5D$ downstream of the sphere position reveal that the sphere wake consists of vortex loops, similar to the wake of a sphere without any imposed rotation; however, there is a change in the timing of vortex formation. On the other hand, for high $f_{R}$ values, there is a reduction in the streamwise vorticity, presumably leading to a decreased total transverse force acting on the sphere and resulting in a reduced response.

In this paper, we consider shock reflection problems, occurring in supersonic and hypersonic intake flow under off-design conditions, in which the incident shock wave is disturbed by the lip-generated upstream expansion wave and the reflected shock wave intersects with a downstream cowl-turning deflected shock wave. The expansion wave and deflected shock wave are here generated with the same magnitude of flow deflection angle or turning angle. With the help of shock interaction theory and numerical simulation, the influence of the turning angle of the lip and cowl on the flow structure and the critical conditions for transition between regular reflection and Mach reflection are analysed. It is found that the dual-solution domain is significantly altered by the interference between the expansion wave and shock waves. The flow structure in the condition of Mach reflection is then analysed with a model updated from a previous study. It is shown that the Mach stem height is an increasing function of the turning angle, while the horizontal position of the Mach stem is shifted in the downstream direction for small turning angle and in the upstream direction for large turning angle.

Hydrodynamic diffusion in the absence of Brownian motion is studied via active microrheology in the ‘pure-hydrodynamic’ limit, with a view towards elucidating the transition from colloidal microrheology to the non-colloidal limit, falling-ball rheometry. The phenomenon of non-Brownian force-induced diffusion in falling-ball rheometry is strictly hydrodynamic in nature; in contrast, analogous force-induced diffusion in colloids is deeply connected to the presence of a diffusive boundary layer even when Brownian motion is very weak compared with the external force driving the ‘probe’ particle. To connect these two limits, we derive an expression for the force-induced diffusion in active microrheology of hydrodynamically interacting particles via the Smoluchowski equation, where thermal fluctuations play no role. While it is well known that the microstructure is spherically symmetric about the probe in this limit, fluctuations in the microstructure need not be – and indeed lead to a diffusive spread of the probe trajectory. The force-induced diffusion is anisotropic, with components along and transverse to the line of external force. The latter is identically zero owing to the fore–aft symmetry of pair trajectories in Stokes flow. In a naïve first approach, the vanishing relative hydrodynamic mobility at contact between the probe and an interacting bath particle was assumed to eliminate all physical contribution from interparticle forces, whereby advection alone drove structural evolution in pair density and microstructural fluctuations. With such an approach, longitudinal force-induced diffusion vanishes in the absence of Brownian motion, a result that contradicts well-known experimental measurements of such diffusion in falling-ball rheometry. To resolve this contradiction, the probe–bath-particle interaction at contact was carefully modelled via an excluded annulus. We find that interparticle forces play a crucial role in encounters between particles in the hydrodynamic limit – as they must, to balance the advective flux. Accounting for this force results in a longitudinal force-induced diffusion $D_{\Vert }=1.26aU_{S}{\it\phi}$, where $a$ is the probe size, $U_{S}$ is the Stokes velocity and ${\it\phi}$ is the volume fraction of bath particles, in excellent qualitative and quantitative agreement with experimental measurements in, and theoretical predictions for, macroscopic falling-ball rheometry. This new model thus provides a continuous connection between micro- and macroscale rheology, as well as providing important insight into the role of interparticle forces for diffusion and rheology even in the limit of pure hydrodynamics: interparticle forces give rise to non-Newtonian rheology in strongly forced suspensions. A connection is made between the flow-induced diffusivity and the intrinsic hydrodynamic microviscosity which recovers a precise balance between fluctuation and dissipation in far from equilibrium suspensions; that is, diffusion and drag arise from a common microstructural origin even far from equilibrium.

Recent experimental observations (Kühnen et al., Nat. Phys., vol. 14, 2018b, pp. 386–390) have shown that flattening a turbulent streamwise velocity profile in pipe flow destabilises the turbulence so that the flow relaminarises. We show that a similar phenomenon exists for laminar pipe flow profiles in the sense that the nonlinear stability of the laminar state is enhanced as the profile becomes more flattened. The flattening of the laminar base profile is produced by an artificial localised body force designed to mimic an obstacle used in the experiments of Kühnen et al. (Flow Turbul. Combust., vol. 100, 2018a, pp. 919–943) and the nonlinear stability measured by the size of the energy of the initial perturbations needed to trigger transition. Significant drag reduction is also observed for the turbulent flow when triggered by sufficiently large disturbances. In order to make the nonlinear stability computations more efficient, we examine how indicative the minimal seed – the disturbance of smallest energy for transition – is in measuring transition thresholds. We first show that the minimal seed is relatively robust to base profile changes and spectral filtering. We then compare the (unforced) transition behaviour of the minimal seed with several forms of randomised initial conditions in the range of Reynolds numbers $Re=2400$–$10\,000$ and find that the energy of the minimal seed after the Orr and oblique phases of its evolution is close to that of a critical localised random disturbance. In this sense, the minimal seed at the end of the oblique phase can be regarded as a good proxy for typical disturbances (here taken to be the localised random ones) and is thus used as initial condition in the simulations with the body force. The enhanced nonlinear stability and drag reduction predicted in the present study are an encouraging first step in modelling the experiments of Kühnen et al. and should motivate future developments to fully exploit the benefits of this promising direction for flow control.

A numerical study is conducted to investigate the influence of a uniform axial magnetic field on the global linear stability of the rotating-disc boundary layer. Simulation results obtained using a radially homogenized base flow were found to be in excellent agreement with an earlier linear stability analysis, which indicated that an axial magnetic field can locally suppress both convective and absolute instabilities. However, the numerical results obtained for the genuine, radially inhomogeneous, flow indicate that a global form of instability develops for sufficiently large magnetic fields. The qualitative nature of the global instability is similar to that which was observed in a previous study, where mass suction was applied at the rotating disc surface. It is shown that, just as for the case with mass suction, it is possible to explain the promotion of global instability by considering a model that includes detuning effects, which are associated with the radial variation of locally defined absolute temporal frequencies. The recurrence of the same type of instability behaviour when two distinct flow control strategies are implemented, one using suction and the other an axial magnetic field, indicates that the phenomena described by the model may be considered generic.