A quantitative definition of the receptivity of the Görtler instability is given by the Green's functions that external disturbances must be scalarly multiplied by in order to yield the amplitude of the most amplified instability mode, defined sufficiently far downstream of the plate's leading edge. These Green's functions (one for each kind of external disturbance, either coming from the free stream or from the wall) are here displayed for the first time. Calculating such functions from a numerical solution of the instability equations would require repeating the calculation for each of a complete set of different initial and boundary conditions; although numerical simulations of the Görtler instability abound in the literature, such a systematic screening has never been attempted. Here, instead, we calculate the Green's functions directly from a numerical solution of the adjoint of the linearized boundary-layer equations, which exploits the fact that the direct and adjoint parabolic problems have opposite directions of stable time-like evolution. The Green's functions can thus be obtained by marching backward in time at the same computational cost as a single forward-in-time integration of the direct problem. The backward-in-time technique is not limited to the Görtler problem; quantitative receptivity calculations for other types of instability can easily be envisioned.

A modified laser-Doppler velocimetry method is utilized to measure fully developed particle velocity and concentration profiles, as well as the mean-square amplitudes of velocity fluctuations (i.e. one component of the so-called particle temperature), for concentrated monodisperse suspensions across the narrow gap of a rectangular channel. A stable index-of-refraction match of the suspending and particulate phases in conjunction with short-focal-length focusing optics has enabled data acquisition up to particle volume fractions of 0.50. In general, the particle concentration distributions possess a maximum near the channel centreline and a minimum at the channel walls. Coupled to these concentration distributions were blunted velocity profiles, and particle velocity fluctuation distributions that had a sharp maximum at gap positions approximately 80% of the way from the channel axis towards the walls. The particle velocity distributions were consistent with the absence of slip between particles and the suspending fluid.

The experimental data were compared with theoretical predictions from the diffusive flux model (Leighton & Acrivos 1987; Phillips et al. 1992), a model due to Mills & Snabre (1995), and the suspensions balance model (McTigue & Jenkins 1992; Nott & Brady 1994). The influence of bulk particle concentration, suspension volumetric flow rate, and ratio of channel gap width to particle diameter on the fully developed profiles was qualitatively consistent with the theoretical predictions from all three models. For the diffusive flux and suspension balance models, we used both literature values for model parameters, and values obtained from a best fit to our entire set of experimental data. Overall, the Mills & Snabre and suspension balance models were found to provide a better quantitative fit to the experimental data than the diffusive flux model.

In this paper we report experimental velocity and concentration profiles for suspensions possessing a bidisperse distribution of particle size undergoing pressure-driven flow through a parallel-wall channel. In addition to the overall concentration distributions determined by implementing the modified laser Doppler velocimetry method described in Part 1 (Lyon & Leal 1998), concentration profiles for the particles of each size were measured by sampling the position of marked tracer particles across 60% of the channel gap. Non-uniform overall particle concentration distributions and blunted velocity profiles were found at bulk particle volume fractions of 0.30 and 0.40, which were equal to the monodisperse data of Part 1, within experimental uncertainty. The large-particle concentration profiles were non-uniform down to a large-particle bulk volume fraction of 0.075, while non-uniform distributions of the small particles were only found when the volume fraction of small particles in the bulk was greater than or equal to 0.20. Experiments in which at least half the suspended particulate volume was occupied by large particles revealed enrichment of the large particles in the centreline region of the channel. This size segregation was found to increase as the total number of suspended particles decreased. Finally, the data from experiments in which a uniform small-particle concentration profile was measured were compared with suspension balance model (McTigue & Jenkins 1992; Nott & Brady 1994) predictions for parameter values that corresponded only to the large particles. While close agreement with the large-particle concentration profiles was found, this comparison also reflected the fact that the small particles bring the suspension viscosity to a regime that is more sensitive to the particle concentration, rather than simply providing an increment in background viscosity to the suspending liquid.

Visualization studies of the flow behind an oscillating tapered cylinder are performed at Reynolds numbers from 400 to 1500. The cylinder has taper ratio 40[ratio ]1 and is moving at constant forward speed U while being forced to oscillate harmonically in the transverse direction. It is shown that within the lock-in region and above a threshold amplitude, no cells form and, instead, a single frequency of response dominates the entire span. Within certain frequency ranges a single mode dominates in the wake, consisting of shedding along the entire span of either two vortices per cycle (‘2S’ mode), or four vortices per cycle (‘2P’ mode); but within specific parametric ranges a hybrid mode is observed, consisting of a ‘2S’ pattern along the part of the span with the larger diameter and a ‘2P’ pattern along the part of the span with the smaller diameter. A distinct vortex split connects the two patterns which are phase-locked and have the same frequency. The hybrid mode is periodic, unlike vortex dislocations, and the location of the vortex split remains stable and repeatable, within one to two diameters, depending on the amplitude and frequency of oscillation and the Reynolds number.

Forces are measured at both ends of rigid cylinders with span 60 cm, performing transverse oscillations within an oncoming stream of water, at Reynolds number Re≈3800. Forced harmonic motions and free vibrations of uniform and tapered cylinders are studied. To study free motions, a novel force-feedback control system has been developed, consisting of: (a) a force transducer, which measures forces on a section of a cylinder moving forward at constant speed; (b) a computer using the measured force signal to drive in real time a numerical simulation of an equivalent mass-dashpot-spring system; (c) a servomotor and linear table which impose, also in real time, the numerically calculated motion on the cylinder section. The apparatus allows very low equivalent system damping and strict control of the parametric values and structure of the equivalent system.

Calculation of the cross-correlation coefficient between forces at the two ends of the uniform cylinder reveals five distinct regimes as a function of the nominal reduced velocity Vrn: two regimes, for low and high values of Vrn, and far away from the value of VrS corresponding to the Strouhal frequency, show small correlation; two regimes immediately adjacent to, but excluding, VrS show strong correlation, close to 1; surprisingly, there is a regime containing the Strouhal frequency, within which correlation is low. Free vibrations with a 40[ratio ]1 tapered cylinder show that the regime of low correlation, containing the Strouhal frequency, stretches to higher reduced velocities, while lock-in starts at lower reduced velocities.

When comparing the amplitude and phase of the lift coefficient measured for free and then for forced vibrations, we obtain close agreement, both for tapered and uniform cylinders. When comparing the cross-correlation coefficient, however, we find that it is much higher in the forced oscillations, especially for the uniform cylinder. Hence, although the force magnitude and phase may be replicated well in forced vibrations, the correlation data suggest that differences exist between free and forced vibration cases.

The effects of riblets on one wall of a channel bounding fully developed turbulent flow are investigated. Various perturbation elements including wires, fins and slots are modelled in order to understand the effects of riblets. It is found that widely spaced riblets, fins and wires create a substantial increase in turbulent activity just above the element. These elements are also found to produce a remarkable pattern of secondary mean flows consisting of matched pairs of streamwise vortices. The secondary flows occur only if the bulk flow is turbulent and their characteristics depend on element geometry. It is suggested that these secondary flows are strongly linked with the increase in drag experienced by widely spaced riblets in experimental studies. The secondary flows are probably caused by two-dimensional spanwise sloshing of the flow, inherent in a turbulent boundary layer, interacting with the stream-aligned element. This two-dimensional mechanism is investigated with a series of two-dimensional simulations of sloshing flow over isolated elements. Grid resolution and domain size checks are made throughout the investigation.

It is demonstrated experimentally that through the use of feedback control, it is possible to stabilize the no-motion (conductive) state of a fluid layer confined in a circular cylinder heated from below and cooled from above (the Rayleigh–Bénard problem), thereby postponing the transition from a no-motion state to cellular convection. The control system utilizes multiple sensors and actuators. The actuators consist of individually controlled heaters microfabricated on a silicon wafer which forms the bottom of the test cell. The sensors are diodes installed at the fluid's midheight. The sensors monitor the deviation of the fluid temperatures from preset, desired values and direct the actuators to act in such a way as to eliminate these deviations.

The periodic oscillation of the shock waves in screeching, underexpanded, supersonic jets, issuing from a choked, axisymmetric, nozzle at fully expanded Mach numbers (Mj) of 1.19 and 1.42, is studied experimentally and analytically. The experimental part uses schlieren photography and a new shock detection technique which depends on a recently observed phenomenon of laser light scattering by shock waves. A narrow laser beam is traversed from point to point in the flow field and the appearance of the scattered light is sensed by a photomultiplier tube (PMT). The time-averaged and phase-averaged statistics of the PMT data provide significant insight into the shock motion. It is found that the shocks move the most in the jet core and the least in the shear layer. This is opposite to the intuitive expectation of a larger-amplitude shock motion in the shear layer where organized vortices interact with the shock. The mode of shock motion is the same as that of the emitted screech tone. The instantaneous profiles of the first four shocks over an oscillation cycle were constructed through a detailed phase averaged measurement. Such data show a splitting of each shock (except for the first one) into two weaker ones through a ‘moving staircase-like’ motion. During a cycle of motion the downstream shock progressively fades away while a new shock appears upstream. Spark schlieren photographs demonstrate that a periodic convection of large organized vortices over the shock train results in the above described behaviour. An analytical formulation is constructed to determine the self-excitation of the jet column by the screech sound. The screech waves, while propagating over the jet column, add a periodic pressure fluctuation to the ambient level, which in turn perturbs the pressure distribution inside the jet. The oscillation amplitude of the first shock predicted from this linear analysis shows reasonable agreement with the measured data. Additional reasons for shock oscillation, such as a periodic perturbation of the shock formation mechanism owing to the passage of the organized structures, are also discussed.

We investigate, numerically and analytically, the structure and stability of steady and quasi-steady solutions of the Navier–Stokes equations corresponding to stretched vortices embedded in a uniform non-symmetric straining field, (αx, βy, γz), α+β+γ=0, one principal axis of extensional strain of which is aligned with the vorticity. These are known as non-symmetric Burgers vortices (Robinson & Saffman 1984). We consider vortex Reynolds numbers R=Γ/(2πv) where Γ is the vortex circulation and v the kinematic viscosity, in the range R=1−104, and a broad range of strain ratios λ=(β−α)/(β+α) including λ>1, and in some cases λ[Gt ]1. A pseudo-spectral method is used to obtain numerical solutions corresponding to steady and quasi-steady vortex states over our whole (R, λ) parameter space including λ where arguments proposed by Moffatt, Kida & Ohkitani (1994) demonstrate the non-existence of strictly steady solutions. When λ[Gt ]1, R[Gt ]1 and ε≡λ/R[Lt ]1, we find an accurate asymptotic form for the vorticity in a region 1<r/(2v/γ)1/2[les ]ε1/2, giving very good agreement with our numerical solutions. This suggests the existence of an extended region where the exponentially small vorticity is confined to a nearly cat's-eye-shaped region of the almost two-dimensional flow, and takes a constant value nearly equal to Γγ/(4πv)exp[−1/(2eε)] on bounding streamlines. This allows an estimate of the leakage rate of circulation to infinity as ∂Γ/∂t =(0.48475/4π)γε−1Γ exp (−1/2eε) with corresponding exponentially slow decay of the vortex when λ>1. An iterative technique based on the power method is used to estimate the largest eigenvalues for the non-symmetric case λ>0. Stability is found for 0[les ]λ[les ]1, and a neutrally convective mode of instability is found and analysed for λ>1. Our general conclusion is that the generalized non-symmetric Burgers vortex is unconditionally stable to two-dimensional disturbances for all R, 0[les ]λ[les ]1, and that when λ>1, the vortex will decay only through exponentially slow leakage of vorticity, indicating extreme robustness in this case.

Properties of the flow generated by a continuous source of dense fluid on a slope in a rotating system are investigated with a variety of laboratory experiments. The dense fluid may initially flow down the slope but it turns (under the influence of rotation) to flow along the slope, and initial geostrophic adjustment gives it an anticyclonic velocity profile. Some of the dense fluid drains downslope in a viscous Ekman layer, which may become unstable to growing waves. Provided that the viscous draining is not too strong, cyclonic vortices form periodically in the upper layer and the dense flow breaks up into a series of domes. Three processes may contribute to the formation of these eddies. First, initial downslope flow of the dense current may stretch columns of ambient fluid by the ‘Taylor column’ process (which we term ‘capture’). Secondly, the initial geostrophic adjustment implies lower-layer collapse which may stretch the fluid column, and thirdly, viscous drainage will progressively stretch and spin up a captured water column. Overall this last process may be the most significant, but viscous drainage has contradictory effects, in that it progressively removes dense lower-layer fluid which terminates the process when the layer thickness approaches that of the Ekman layer. The eddies produced propagate along the slope owing to the combined effects of buoyancy–Coriolis balance and ‘beta-gyres’. This removes fluid from the vicinity of the source and causes the cycle to repeat. The vorticity of the upper-layer cyclones increases linearly with Γ=Lα/D (where L is the Rossby deformation radius, α the bottom slope and D the total depth), reaching approximately 2f in the experiments presented here. The frequency at which the eddy/dome structures are produced also increases with Γ, while the speed at which the structures propagate along the slope is reduced by viscous effects. The flow of dense fluid on slopes is a very important part of the global ocean circulation system and the implications of the laboratory experiments for oceanographic flows are discussed.

The effect of wall inertia on the self-excited oscillations in a collapsible channel flow is investigated by solving the full coupled two-dimensional membrane–flow equations. This is the continuation of a previous study in which self-excited oscillations were predicted in an asymmetric channel with a tensioned massless elastic membrane (Luo & Pedley 1996). It is found that a different type of self-excited oscillation, a form of flutter, is superposed on the original large-amplitude, low-frequency oscillations. Unlike the tension-induced oscillations, the flutter has high frequency, and grows with time from a small amplitude until it dominates the original slower mode. The critical value of tension below which oscillations arise (at fixed Reynolds number) is found to increase as the wall inertia is increased. The rate at which energy is (a) dissipated in the flow field and (b) transferred to the wall during the flutter is discussed, and results at different parameter values are compared with those of a massless membrane. There is also a discussion of whether the onset of flutter, or that of the slower oscillations, is correlated with the appearance of flow limitation, as is thought to be the case in the context of wheezing during forced expiration of air from the lungs.

We present a numerical strategy that allows us to explore the full scope of the Doering–Constantin variational principle for computing rigorous upper bounds on energy dissipation in turbulent shear flow. The key is the reformulation of this principle's spectral constraint as a boundary value problem that can be solved efficiently for all Reynolds numbers of practical interest. We state results obtained for the plane Couette flow, and investigate in detail a simplified model problem that can serve as a definite guide for the application of the variational principle to other flows. The most notable findings are a bifurcation of the minimizing wavenumber and a pronounced minimum of the bound at intermediate Reynolds numbers, and a distinct asymptotic scaling of the optimized variational parameters.

We study analytically the asymptotics of the upper bound on energy dissipation for the two-dimensional plane Couette flow considered numerically in Part 1 of this work, in order to identify the mechanisms underlying the variational approach. With the help of shape functions that specify the variational profiles either in the interior or in the boundary layers, it becomes possible to quantitatively explain all numerically observed features, from the occurrence of two branches of minimizing wavenumbers to the asymptotic parameter scaling with the Reynolds number. In addition, we derive a new variational principle for the asymptotic bound on the dissipation rate. The analysis of this principle reveals that the best possible bound can only be attained if the variational profiles allow the shape of the boundary layers to change with increasing Reynolds number.

A theorem on helicity conservation proved by Moffatt (1969) for the flows of inviscid barotropic fluids is generalized, for steady flows, to any fluid in which vorticity field lines are material. To make this generalization, the helicity within a volume V enclosed by a material surface S must be defined by the volume integral,

formula here

where v is the fluid velocity, m is a unit vector tangent to a vorticity line, λ is the vorticity line stretch (Casey & Naghdi 1991), and J is the determinant of the deformation gradient tensor. For the case of an inviscid barotropic fluid, ([Hscr ]′S differs only by a constant factor from the helicity integral defined originally by Moffatt (1969). The condition under which ([Hscr ]′S is invariant under steady fluid motion is also the condition necessary and sufficient for the existence of a permanent system of surfaces on which both the stream lines and the vorticity lines lie (Sposito 1997). These surfaces and the helicity invariant ([Hscr ]′S figure importantly in the topological classification of integrable steady fluid flows, including flows with dissipation, in which vorticity lines are material.

The Hamiltonian formalism for surface waves associated with the method of Watson & West (1975) is extended to handle the case of spatially varying bottom depth. This description models moderately nonlinear waves over a wider range of scales than Boussinesq-type approximations. A pseudospectral simulation code has been developed using this formalism in two horizontal dimensions. Computations using the model compare well with measurements of waves over a bar, diffractive focusing by topography, and shoaling of solitary waves. Waveforms are computed accurately until near the point of breaking.