Hamiltonian approximation methods yields approximate dynamical equations that apply to nearly geostrophic flow at scales larger than the internal Rossby deformation radius. These equations incorporate fluid inertia with the same order of accuracy as the semi-geostrophic equations, but are nearly as simple (in appropriate coordinates) as the equations obtained by completely omitting the inertia.

Employing matched asymptotics, we extend the recent theory of Mei (1986) to study the phenomenon of upstream influence by a slender ship moving near the critical speed. For a special class of channel width and ship slenderness, it is shown that the response on the sea surface is essentially one-dimensional with the wave crests perpendicular to the ship's axis. In particular, solitons are radiated upstream. The hydrodynamic pressure on the ship, as well as the total sinkage force, wave resistance and trimming moment are calculated. These forces are functions of time despite the constant speed of the ship. The sinkage and trim for a ship model fixed on an advancing carriage are computed and show rapid variations across the critical speed as in the reported experiments of Graff, Kracht & Weinblum (1964). Because of the assumed slenderness of the ship, this theory does not predict two-dimensional waves in the wake. Nevertheless, there is crude agreement in the time-averaged hydrodynamic forces between theory and experiment.

Of the canonical flow instabilities (Kelvin–Helmholtz, Holmboe-wave and Taylor–Caulfield) of stratified shear flow, the Taylor–Caulfield instability (TCI) has received relatively little attention, and forms the focus of the current study. First, a diagnostic of the linear instability dynamics is developed that exploits the net pseudomomentum to distinguish TCI from the other two instabilities for any given flow profile. Second, the nonlinear dynamics of TCI is studied across its range of unstable horizontal wavenumbers and bulk Richardson numbers using numerical simulation. At small bulk Richardson numbers, a cascade of billow structures of sequentially smaller size may form. For large bulk Richardson numbers, the primary nonlinear travelling waves formed by the linear instability break down via a small-scale, Kelvin–Helmholtz-like roll-up mechanism with an associated large amount of mixing. In all cases, secondary parasitic nonlinear Holmboe waves appear at late times for high Prandtl number. Third, a nonlinear diagnostic is proposed to distinguish between the saturated states of the three canonical instabilities based on their distinctive density–streamfunction and generalised vorticity–streamfunction relations.

Flow visualization has been used quantitatively to determine the flow relative to a piston and a free surface started from rest. The discharge of water from a cylindrical reservoir was investigated. Flow with a free surface started from rest was found to have a critical Reynolds number (based on tube diameter and surface speed) of about 450 above which a ring vortex was produced just below the surface.

Measurements at Reynolds numbers of 525 and 1200 were compared with computations made by the methods described in Part 1. The computed drift of tracer particles agreed well with observed values. The largest discrepancies occurred in the radial component of the drift in the early stages of the motion and amounted to 2½% of the tube diameter.

We investigate the influence of a soluble surfactant on the steady-state motion of a finger of air through a compliant channel. This study provides a basic model from which to understand the fluid–structure interactions and physicochemical hydrodynamics of pulmonary airway reopening. Airway closure occurs in lung diseases such as respiratory distress syndrome and acute respiratory distress syndrome as a result of fluid accumulation and surfactant insufficiency. This results in ‘compliant collapse’ with the airway walls buckled and held in apposition by a liquid occlusion that blocks the passage of air. Airway reopening is essential to the recovery of adequate ventilation, but has been associated with ventilator-induced lung injury because of the exposure of airway epithelial cells to large interfacial flow-induced pressure gradients. Surfactant replacement is helpful in modulating this deleterious mechanical stimulus, but is limited in its effectiveness owing to slow surfactant adsorption. We investigate the effect of surfactant on micro-scale models of reopening by computationally modelling the steady two-dimensional motion of a semi-infinite bubble propagating through a liquid-filled compliant channel doped with soluble surfactant. Many dimensionless parameters affect reopening, but we primarily investigate how the reopening pressure depends upon the capillary number (the ratio of viscous to surface tension forces), the adsorption depth parameter (a bulk concentration parameter) and the bulk Péclet number (the ratio of bulk convection to diffusion). These studies demonstrate a dependence of on , and suggest that a critical bulk concentration must be exceeded to operate as a low-surface-tension system. Normal and tangential stress gradients remain largely unaffected by physicochemical interactions – for this reason, further biological studies are suggested that will clarify the role of wall flexibility and surfactant on the protection of the lung from atelectrauma.

We consider the spatio-temporal evolution of patterns in the marginally unstable Ekman layer driven by an applied shear stress. Both the normal and tangential components of the Earth's angular velocity are included in a tangent plane approximation of the oceanic boundary layer at latitude $\lambda$. The fluid motion in a layer of finite depth as well as one of infinite depth is considered. The linear instability in the infinite depth case is known to depend on the direction of the applied stress for $\lambda \ne 90\dg$, but this dependence is weak for the stress-driven Ekman layer. By contrast, the weakly nonlinear motion exhibits for finite and infinite depths qualitatively different dynamics for different stress directions.

The problem is treated by the method of multiple scales. In the case of finite depth, this leads to the Davey–Hocking–Stewartson equation, an amplitude equation of complex Ginzburg–Landau type coupled to a Poisson equation. In the case of infinite depth, it leads to the anisotropic complex Ginzburg–Landau equation for the amplitude of the roll motion. Motions in both finite and infinite depth basins are explored by numerical simulation, and are shown to lead to chaotic dynamics for the modulation envelope in most cases. The statistics and the nature of the patterns produced in this motion are discussed.

Because the three-dimensional large-scale vortex structures dominate the dispersion of particles at intermediate Stokes numbers in shear layers, there is interest in understanding the two-way thermal coupling effect in droplet–gas flows for practical combustion applications. Using pseudo-spectral and Lagrangian approaches, three-dimensional two-way thermally coupled droplet-laden mixing layers are studied with hot air and cool water droplets. Higher air density is observed around the region of the interface of the two streams and, thus, thermal contraction occurs in this region. This thermal contraction results in an increase of the magnitude of the vorticity field, a more unstable flow, a higher droplet concentration and a lower droplet dispersion across the mixing layer.

The first generation of stars had very different properties than later stellar generations, as they formed from a ‘pristine’ gas that was completely free of heavy elements. Normal star formation took place only after the first stars had polluted the surrounding turbulent interstellar gas, increasing its local heavy-element mass concentration, , beyond a ‘critical’ threshold value, (). Motivated by this astrophysical problem, we investigate the fundamental physics of the pollution of pristine fluid elements in statistically homogeneous and isotropic compressible turbulence. Turbulence stretches the pollutants, produces concentration structures at small scales, and brings the pollutants and the unpolluted flow in closer contact. The pristine material is polluted when exposed to the pollutant sources or the fluid elements polluted by previous mixing events. Our theoretical approach employs the probability distribution function (p.d.f.) method for turbulent mixing, as the fraction of pristine mass corresponds to the low tail of the density-weighted concentration p.d.f. We adopt a number of p.d.f. closure models and derive evolution equations for the pristine fraction from the models. To test and constrain the prediction of theoretical models, we conduct numerical simulations for decaying passive scalars in isothermal turbulent flows with Mach numbers of 0.9 and 6.2, and compute the mass fraction, , of the flow with . In the Mach 0.9 flow, the evolution of is well-described by a continuous convolution model and goes as , if the mass fraction of the polluted flow is larger than If the initial pollutant fraction is smaller than an early phase exists during which the pristine fraction follows an equation derived from a nonlinear integral model: . The time scales and are measured from our simulations. When normalized to the flow dynamical time, the decay of in the Mach 6.2 flow is slower than at Mach 0.9 because the time scale for scalar variance decay is slightly larger and the low tail of the concentration p.d.f. broadens with increasing Mach number. We show that in the Mach 6.2 flow can be well fitted using a formula from a generalized version of the self-convolution model.

A semi-empirical model is presented that describes the development of a fully developed turbulent boundary layer in the presence of surface roughness with length scale $k_{s}$ that varies with streamwise distance $x$. Interest is centred on flows for which all terms of the von Kármán integral relation, including the ratio of outer velocity to friction velocity $U_{\infty }^{+}\equiv U_{\infty }/u_{\unicode[STIX]{x1D70F}}$, are streamwise constant. For $Re_{x}$ assumed large, use is made of a simple log-wake model of the local turbulent mean-velocity profile that contains a standard mean-velocity correction for the asymptotic fully rough regime and with assumed constant parameter values. It is then shown that, for a general power-law external velocity variation $U_{\infty }\sim x^{m}$, all measures of the boundary-layer thickness must be proportional to $x$ and that the surface sand-grain roughness scale variation must be the linear form $k_{s}(x)=\unicode[STIX]{x1D6FC}x$, where $x$ is the distance from the boundary layer of zero thickness and $\unicode[STIX]{x1D6FC}$ is a dimensionless constant. This is shown to give a two-parameter $(m,\unicode[STIX]{x1D6FC})$ family of solutions, for which $U_{\infty }^{+}$ (or equivalently $C_{f}$) and boundary-layer thicknesses can be simply calculated. These correspond to perfectly self-similar boundary-layer growth in the streamwise direction with similarity variable $z/(\unicode[STIX]{x1D6FC}x)$, where $z$ is the wall-normal coordinate. Results from this model over a range of $\unicode[STIX]{x1D6FC}$ are discussed for several cases, including the zero-pressure-gradient ($m=0$) and sink-flow ($m=-1$) boundary layers. Trends observed in the model are supported by wall-modelled large-eddy simulation of the zero-pressure-gradient case for $Re_{x}$ in the range $10^{8}{-}10^{10}$ and for four values of $\unicode[STIX]{x1D6FC}$. Linear streamwise growth of the displacement, momentum and nominal boundary-layer thicknesses is confirmed, while, for each $\unicode[STIX]{x1D6FC}$, the mean-velocity profiles and streamwise turbulent variances are found to collapse reasonably well onto $z/(\unicode[STIX]{x1D6FC}x)$. For given $\unicode[STIX]{x1D6FC}$, calculations of $U_{\infty }^{+}$ obtained from large-eddy simulations are streamwise constant and independent of $Re_{x}$ when this is large. The present results suggest that, in the sense that $U_{\infty }^{+}(\unicode[STIX]{x1D6FC},m)$ is constant, these flows can be interpreted as the fully rough limit for boundary layers in the presence of small-scale linear roughness.

The relative dispersion of a scalar plume is examined experimentally. A passive fluorescent tracer is continuously released from a flush-bed mounted source into the turbulent boundary layer of a laboratory-generated open channel flow. A two-dimensional particle image velocimetry–laser-induced florescence (PIV–LIF) technique is applied to measure the instantaneous horizontal velocity and concentration fields. Measured results are used to investigate the relationship between the boundary-layer turbulence and the evolution of the distance-neighbour function, namely the probability density distribution of the separation distance between two marked fluid particles within a cloud of particles. Special attention is paid to the hypothesis that a diffusion equation can describe the evolution of the distance-neighbour function. The diffusion coefficient in such an equation, termed the ‘relative diffusivity’, is directly calculated based on the concentration distribution. The results indicate that the relative diffusivity statistically depends on particle separation lengths instead of the overall size of the plume. Measurements at all stages of the dispersing plume collapse onto a single curve and follow a 4/3 power law in the inertial subrange. The Richardson–Obukhov constant is estimated from the presented dataset. The relationship between the one-dimensional (1D) representation of the distance-neighbour function and its three-dimensional (3D) representation is discussed. An extended model for relative diffusivity beyond the inertial subrange is proposed based on the structure of the turbulent velocity field, and it agrees well with measurements. The experimental evidence implies that, while the diffusion of the distance-neighbour function is completely determined by the underlying turbulence, the overall growth rate of the plume is affected by both the turbulent flow and its actual concentration distribution.

A theory is developed that describes microstreaming generated by two interacting gas bubbles in an acoustic field. The theory is used in numerical simulations to compare the characteristics of acoustic microstreaming at different frequencies, separation distances between the bubbles and bubble sizes. It is shown that the interaction of the bubbles leads to a considerable increase in the intensity of the velocity and stress fields of acoustic microstreaming if the bubbles are driven near the resonance frequencies that they have in the presence of each other. Patterns of streamlines for different situations are presented.

Whitham (1957, 1958) has developed a method for treating the propagation of shock waves into non-uniform media, which has been widely and successfully used. In shock diffraction problems, he considers the medium ahead of the shock to be at rest. The purpose of this note is to generalize Whitham's technique to problems having a uniform flow ahead of the shock, in which the inclination of the shock normal to the direction of flow ahead of the shock remains small. This extension was suggested to the author by Miles (1964) who considered the problem of diffraction of a strong shock by a thin supersonic wedge.

The mean electromotive force (EMF) associated with exponentially growing perturbations of an Euler flow with elliptic streamlines in a rotating frame of reference is studied. We are motivated by the possibility of dynamo action triggered by tidal deformation of astrophysical objects such as accretion discs, stars or planets. Ellipticity of the flow models such tidal deformations in the simplest way. Using analytical techniques developed by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312) in the limit of small elliptic (tidal) deformations, we find the EMF associated with each resonant instability described by Mizerski & Bajer (J. Fluid Mech., vol. 632, 2009, pp. 401–430), and for arbitrary ellipticity the EMF associated with unstable horizontal modes. Mixed resonance between unstable hydrodynamic and magnetic modes and resonance between unstable and oscillatory horizontal modes both lead to a non-vanishing mean EMF which grows exponentially in time. The essential conclusion is that interactions between unstable eigenmodes with the same wave-vector can lead to a non-vanishing mean EMF, without any need for viscous or magnetic dissipation. This applies generally (and not only to the elliptic instabilities considered here).