A viscous incompressible stably stratified fluid with a buoyancy frequency which varies slowly with altitude is considered. A simple harmonic localized disturbance generates an internal wave in which the energy propagates along curved paths. Small amplitude similarity solutions are obtained for two-dimensional and axisymmetric waves. It is found that under certain conditions the wave amplitude can increase with height. The two-dimensional theory compares quite well with experimental measurements.

An isolated buoyant thermal in very viscous fluid has been shown to attain a self-similar form at large times which grows as $t^{1/2}$ (Whittaker & Lister J. Fluid Mech., vol. 606, 2008, pp. 295–324). For large values of the Rayleigh number $\mathit{Ra}$ (based on the conserved total buoyancy), the similarity solution is slender with a roughly spherical head at the top and a long tail that contains most of the buoyancy and extends down to the origin. We investigate the time-dependent behaviour of the thermal numerically; both the long-time behaviour in terms of perturbations to the similarity solution and the short-time evolution from a spherical initial condition. Using a spectral method, we find the growth rates of the linear perturbations and their spatial structure in similarity space. All eigenmodes decay monotonically for $\mathit{Ra}\lesssim 360$, while for larger $\mathit{Ra}$ the dominant (slowest decaying or fastest growing) eigenmodes are oscillatory with waves propagating up the tail. Above a critical value $\mathit{Ra}_c \approx 10\, 000$, the steady solution becomes unstable to a limit cycle. A one-dimensional reduction to horizontally integrated quantities hints at a theoretical explanation for the oscillatory behaviour, but does not reproduce the loss of stability. Investigation of the initial transient at large $\mathit{Ra}$ reveals that an initially spherical thermal can rise $O(100)$ times its initial diameter before approaching its final self-similar shape. The presence of a rigid horizontal floor below the thermal makes a quantitative difference of around 10 % to the rate of rise at large $\mathit{Ra}$.

In this paper, we present computational results of some two-fold azimuthally symmetric travelling waves and their stability. Calculations over a range of Reynolds numbers ($Re$) reveal connections between a class of solutions computed by Wedin & Kerswell (J. Fluid Mech., vol. 508, 2004, pp. 333–371) (henceforth called the WK solution) and the $Re\rightarrow \infty$ vortex–wave interaction theory of Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In particular, the continuation of the WK solutions to larger values of $Re$ shows that the WK solution bifurcates from a shift-and-rotate symmetric solution, which we call the WK2 state. The WK2 solution computed for $Re\leqslant 1.19\times 10^{6}$ shows excellent agreement with the theoretical $Re^{-5/6}$, $Re^{-1}$ and $O(1)$ scalings of the waves, rolls and streaks respectively. Furthermore, these states are found to have only two unstable modes in the large $Re$ regime, with growth rates estimated to be $O(Re^{-0.42})$ and $O(Re^{-0.92})$, close to the theoretical $O(Re^{-1/2})$ and $O(Re^{-1})$ asymptotic results for edge and sinuous instability modes of vortex–wave interaction states (Deguchi & Hall, J. Fluid Mech., vol. 802, 2016, pp. 634–666) in plane Couette flow. For the nonlinear viscous core states (Ozcakir et al., J. Fluid Mech., vol. 791, 2016, pp. 284–328), characterized by spatial a shrinking of the wave and roll structure towards the pipe centre with increasing $Re$, we continued the solution to $Re\leqslant 8\times 10^{6}$ and we find only one unstable mode in the large Reynolds number regime, with growth rate scaling as $Re^{-0.46}$ within the class of symmetry-preserving disturbances.

A stream with constant velocity U is impulsively started at time t = 0 past the trailing edge of a semi-infinite flat plate. According to boundary-layer theory, it is found that the flow at a distance x downstream from the trailing edge is unaware of the presence of the plate when x > Ut; at time t = x/U there is then a discontinuity in the velocity normal to the plate. It is the neglect of diffusion parallel to the axis of the plate that introduces the discontinuity, and when the complete Navier–Stokes equations are considered for t ≃ x/U, a solution is found that can be matched with that gained from boundary-layer arguments.

The theory of two-layer hydraulics is extended to topography with co-varying width and height. When these variations of the non-dimensional width and total depth have a power law relationship, the solutions can still be presented in the Froude-number plane for both unidirectional and exchange flows. These differ from previous solutions, which were limited to treating width and height variations separately.

The subgrid-scale (SGS) physics in the convective atmospheric surface layer is studied using the SGS stress and SGS scalar flux. We derive the budget equations for the conditional mean SGS stress and SGS temperature flux and show that, for transport-equation-based SGS models, the budget terms must be correctly predicted by the SGS model in order for large-eddy simulation (LES) to reproduce the resolvable-scale velocity and temperature probability density functions. Field data from the Advection Horizontal Array Turbulence Study, which notably includes measurements of the fluctuating pressure and the advection of the velocity and temperature fields, are then used to analyse the budget equations. The results reveal the complex behaviour of the dynamics of the convective atmospheric surface layer. The budgets of the conditional mean SGS shear stress and SGS temperature flux are an approximate balance between the conditional mean production and pressure destruction, with the latter causing return to isotropy. The budgets of the normal SGS stress components are more complex. For strongly convective surface layers, energy is redistributed from the (smaller) vertical to the (larger) horizontal stress components during downdrafts, resulting in generation of anisotropy by the conditional mean SGS pressure–strain-rate correlation; wall pressure reflections can also enhance the anisotropy. The conditional mean SGS pressure transport, meanwhile, is a significant source of energy during updrafts as a result of the near-wall pressure minima. The vertical advection also plays a significant role in the transfer of SGS energy. For weakly convective surface layers, pressure transport is small while the SGS pressure–strain-rate correlation reverts to its usual role of causing return to isotropy. The results of the present study, particularly for the conditional mean SGS pressure–strain-rate correlation, provide new insights into the SGS physics first educed in a recent analysis of the mean SGS budgets by Nguyen et al. (J. Fluid Mech., vol. 729, 2013, pp. 388–422) and have important implications for near-wall models utilizing SGS transport equations in the convective atmospheric surface layer.

A Hermite series expansion is used to describe the time evolution of the contaminant concentration at a fixed location in a flow. Exact expressions are derived for the dosage, centroid and temporal variance. A two-mode approximation is proposed which gives accurate results at moderate to large distances downstream of the discharge.

An important phase of turbulence production in the flow past a wall occurs with the intermittent inflexional instability of the streamwise current. But according to a linearized inviscid calculation this instability (Kelvin-Helmholtz) can be reduced or eliminated by the presence of a (slippery) wall. Attention is therefore directed to the temporal evolution of a finite-amplitude patch of inflected fluid, i.e. one that is localized in the downstream direction. The model has piecewise uniform vorticity, and the contour-dynamical method is used. Numerical integrations show that sufficiently wide initial patches will eject slow fluid near the wall until it comes into close contact with the free stream, whereupon the ejection is deflected around a large eddy which is surrounded by a stable shear flow. The parametric regime in which this kind of finite instability occurs is sketched, and the Reynolds stress is computed. The initial condition assumed in this calculation depends on the prior existence and intensification of a local spanwise circulation, and this process is briefly discussed using a separate two-dimensional calculation. This shows that widely separated vorticity isopleths tend to completely merge, implying that such fronts in a real fluid may only be viscously limited. The analogous process of potential vorticity frontogenesis may be important in oceanic coastal currents.

A formal derivation of evolution equations is given for viscous gravity waves and viscous capillary—gravity waves with surfactants in water of infinite depth. Multiple scales are used to describe the slow modulation of a wave packet, and matched asymptotic expansions are introduced to represent the viscous boundary layer at the free surface. The resulting dissipative nonlinear Schrödinger equations show that the largest terms in the damping coefficients are unaltered from previous linear results up to third order in the amplitude expansions. The modulational instability of infinite wavetrains of small but finite amplitude is studied numerically. The results show the effect of viscosity and surfactants on the Benjamin—Feir instability and subsequent nonlinear evolution. In an inviscid limit for capillary—gravity waves, a small-amplitude recurrence is observed that is not directly related to the Benjamin—Feir instability.

This paper presents a model for film-condensation heat transfer on a general cylindrical surface in the presence of a non-uniform electrostatic field. The steady-state liquid-film flow is assumed to be fully developed and smooth. The liquid is electrically non-conducting. The vapour in contact with the film is pure and saturated. We also describe two experiments in which R113 is condensed in a tube. In one case condensation occurs in an asymmetric electric field as described by the model. In the other case, done for comparison, it occurs in an axisymmetric one. We report average heat-transfer measurements and visual observations in both cases. Comparison of the theory with the first experiment is good, despite the model not accounting for the film's waviness.

We investigate the linear stability of inviscid flows which are subject to a conservative body force. This includes a broad range of familiar conservative systems, such as ideal MHD, natural convection, flows driven by electrostatic forces and axisymmetric, swirling, recirculating flow. We provide a simple, unified, linear stability criterion valid for any conservative system. In particular, we establish a principle of maximum action of the form

formula here

where η is the Lagrangian displacement,e is a measure of the disturbance energy, T and V are the kinetic and potential energies, and L is the Lagrangian. Here d represents a variation of the type normally associated with Hamilton's principle, in which the particle trajectories are perturbed in such a way that the time of flight for each particle remains the same. (In practice this may be achieved by advecting the streamlines of the base flow in a frozen-in manner.) A simple test for stability is that e is positive definite and this is achieved if L(η) is a maximum at equilibrium. This captures many familiar criteria, such as Rayleigh's circulation criterion, the Rayleigh–Taylor criterion for stratified fluids, Bernstein's principle for magnetostatics, Frieman & Rotenberg's stability test for ideal MHD equilibria, and Arnold's variational principle applied to Euler flows and to ideal MHD. There are three advantages to our test: (i) d2T(η) has a particularly simple quadratic form so the test is easy to apply; (ii) the test is universal and applies to any conservative system; and (iii) unlike other energy principles, such as the energy-Casimir method or the Kelvin–Arnold variational principle, there is no need to identify all of the integral invariants of the flow as a precursor to performing the stability analysis. We end by looking at the particular case of MHD equilibria. Here we note that when u and B are co-linear there exists a broad range of stable steady flows. Moreover, their stability may be assessed by examining the stability of an equivalent magnetostatic equilibrium. When u and B are non-parallel, however, the flow invariably violates the energy criterion and so could, but need not, be unstable. In such cases we identify one mode in which the Lagrangian displacement grows linearly in time. This is reminiscent of the short-wavelength instability of non-Beltrami Euler flows.

The problem is solved of the flow due to a rapidly oscillating source situated near a plane surface with a hemispherical indentation. The strength of the far-field flow is evaluated, and also the natural frequency of oscillation of a small bubble centred at the same position as the source.

The nonlinear stability of the channel flow of fluid with temperature-dependent viscosity is considered for the case of vanishing Péclet number for two viscosity models, μ(T), which vary monotonically with temperature, T. In each case the basic state is found to lose stability from the linear critical point in a subcritical Hopf bifurcation. We find two-dimensional nonlinear time-periodic flows that arise from these bifurcations. The disturbance to the basic flow has wavy streamlines meandering between a sequence of triangular-shaped vortices, with this pattern skewing towards the channel wall which the basic flow skews towards. For each of these secondary flows we identify a nonlinear critical Reynolds number (based on half-channel width and viscosity at one of the fixed wall temperatures) which represents the minimum Reynolds number at which a secondary flow may exist. In contrast to the results for the linear critical Reynolds number, the precise form of μ(T) is not found to be qualitatively important in determining the stability of the thermal flow relative to the isothermal flow. For the viscosity models considered here, we find that the secondary flow is destabilized relative to the corresponding isothermal flow when μ(T) decreases and vice versa. However, if we remove the bulk effect of the non-uniform change in viscosity by introducing a Reynolds number based on average viscosity, it is found that the form of μ(T) is important in determining whether the thermal secondary flow is stabilized or destabilized relative to the corresponding isothermal flow. We also consider the linear stability of the secondary flows and find that the most unstable modes are either superharmonic or subharmonic. All secondary disturbance modes are ultimately damped as the Floquet parameter in the spanwise direction increases, and the last mode to be damped is always a phase-locked subharmonic mode. None of the secondary flows is found to be stable to all secondary disturbance modes. Possible bifurcation points for tertiary flows are also identified.

An axisymmetric flow of a rotating fluid into a point sink was studied experimentally. The type of motion is mainly controlled by the value of the Rossby number R, a ratio of inertial and Coriolis forces. Experimental investigation shows that at a sufficiently large value of R the fluid motion resembles potential flow. However, as R falls below a critical number the withdrawal of the fluid starts to be selective. The flow field then divides into two regions; namely, a central flowing core and an almost stagnant region surrounding it.

It is observed that at a Rossby number below the critical value the flow field, induced by a sudden start of discharge at the sink, experiences several distinct stages during the course of each run. At the initial moment the flow exhibits a feature of potential flow. During the second stage, it develops into a state of selective withdrawal with an inviscid profile of a flowing core, which is the main interest of the present study. In the third stage, due to the unavoidable influence of the free or solid surface at the upstream, flow undergoes another change. The flowing core becomes a fast-spinning jet, in which the viscous force becomes important.

It is also found that once selective withdrawal begins, the angular velocity and flow rate of the flowing core differ substantially from the basic rotation and the actual discharge at the sink. During this second stage of flow development, the flowing core tends to adjust itself such that the intrinsic Rossby number R′(= Wmax/2Ωcδc) based on the properties of the flowing core, virtually remains constant for all values of R below the critical value. This constant value of R′ is found to be about 0·36. The critical value of R which marks the beginning of the selective withdrawal is found to be in the neighbourhood of 0·26.

Motivated by the relevance of edge state solutions as mediators of transition, we use direct numerical simulations to study the effect of spatially non-uniform viscosity on their energy and stability in minimal channel flows. What we seek is a theoretical support rooted in a fully nonlinear framework that explains the modified threshold for transition to turbulence in flows with temperature-dependent viscosity. Consistently over a range of subcritical Reynolds numbers, we find that decreasing viscosity away from the walls weakens the streamwise streaks and the vortical structures responsible for their regeneration. The entire self-sustained cycle of the edge state is maintained on a lower kinetic energy level with a smaller driving force, compared to a flow with constant viscosity. Increasing viscosity away from the walls has the opposite effect. In both cases, the effect is proportional to the strength of the viscosity gradient. The results presented highlight a local shift in the state space of the position of the edge state relative to the laminar attractor with the consequent modulation of its basin of attraction in the proximity of the edge state and of the surrounding manifold. The implication is that the threshold for transition is reduced for perturbations evolving in the neighbourhood of the edge state in the case that viscosity decreases away from the walls, and vice versa.

This paper is a continuation of our study on nonlinear processes in large-amplitude geostrophic (LAG) dynamics. Here, we examine the so-called weak-β models. These models arise when the intrinsic length scale is large enough so that the dynamics is geostrophic to leading order but not so large that the β-effect enters into the dynamics at leading order (but remains, nevertheless, dynamically non-negligible). In contrast to our previous analysis of strong-β LAG models in Part 1, we show that the weak-β models allow for vigorous linear baroclinic instability.

For two-layer weak-β LAG models in which the mean depths of both layers are approximately equal, the linear instability problem can exhibit an ultraviolet catastrophe. We argue that it is not possible to establish conditions for the nonlinear stability in the sense of Liapunov for a steady flow. We also show that the finite-amplitude evolution of a marginally unstable flow possesses explosively unstable modes, i.e. modes for which the amplitude becomes unbounded in finite time. Numerical simulations suggest that the development of large-amplitude meanders, squirts and eddies is correlated with the presence of these explosively unstable modes.

For two-layer weak-β LAG models in which one of the two layers is substantially thinner than the other, the linear stability problem does not exhibit an ultraviolet catastrophe and it is possible to establish conditions for the nonlinear stability in the sense of Liapunov for steady flows. A finite-amplitude analysis for a marginally unstable flow suggests that nonlinearities act to stabilize eastward and enhance the instability of westward flows. Numerical simulations are presented to illustrate these processes.

We explore the effect of an applied homogeneous magnetic field on the electrospraying characteristics of a ferrofluid in the cone-jet mode. A sulfolane-based ferrofluid mixed with the ionic liquid ethyl ammonium nitrate has been synthesized. These mixtures have negligible volatility under ambient conditions and remain stable under a very wide range of electrical conductivities $K$. Magnetized Taylor cones spray with the same current emission characteristics as their non-magnetized counterparts in the shared voltage and flow rate parameter space. However, the magnetized Taylor cones studied remained stable at voltages 23 % lower than the non-magnetized spray; they also access flow rates 30 % and 40 % lower in ferrofluids with $K=0.3$ and $0.01~\text{S}~\text{m}^{-1}$. In the lower voltage ranges available only to magnetized tips, unusually long stable cones are observed. The magnetic stabilization mechanism responsible for these two effects remains unclear. It is noteworthy that these strong effects arise even when the tip curvature of the strictly magnetized liquid is orders of magnitude smaller than that for the strictly electrified liquid.

Braginskii magnetohydrodynamics (MHD) is a single-fluid description of large-scale motions in strongly magnetised plasmas. The ion Larmor radius in these plasmas is much shorter than the mean free path between collisions, so momentum transport across magnetic field lines is strongly suppressed. The relation between the strain rate and the viscous stress becomes highly anisotropic, with the viscous stress being predominantly aligned parallel to the magnetic field. We present an analytical study of the steady planar flow across an imposed uniform magnetic field driven by a uniform pressure gradient along a straight channel, the configuration known as Hartmann flow, in Braginskii MHD. The global momentum balance cannot be satisfied by just the parallel viscous stress, so we include the viscous stress perpendicular to magnetic field lines as well. The ratio of perpendicular to parallel viscosities is the key small parameter in our analysis. When another parameter, the Hartmann number, is large the flow is uniform across most of the channel, with boundary layers on either wall that are modifications of the Hartmann layers in standard isotropic MHD. However, the Hartmann layer solution predicts an infinite current and infinite shear at the wall, consistent with a local series solution of the underlying differential equation that is valid for all Hartmann numbers. These singularities are resolved by inner boundary layers whose width scales as the three-quarters power of the viscosity ratio, while the maximum velocity scales as the inverse one-quarter power of the viscosity ratio. The inner wall layers fit between the Hartmann layers, if present, and the walls. The solution thus does not approach a limit as the viscosity ratio tends to zero. Essential features of the solution, such as the maximum current and maximum velocity, are determined by the size of the viscosity ratio, which is the regularising small parameter.

In view of several practical ramifications of this problem, computational-analytical techniques for calculating waves induced by heaving arbitrary bodies in narrow tanks have been developed, including nonlinear wave groups produced near tank resonance. These feature computational near-field solutions matched with appropriate far-field solutions. In the linear case, the far field is provided by linear mode superposition. In the nonlinear case, the far field is described by a suitable nonlinear evolution equation of the cubic Schrödinger type. Matching techniques were developed. Calculations were successfully carried out and the results confirm the important effect of tank walls on added mass and damping.

Results of computations have been compared with some data obtained with a conical wavemaker in a narrow tank. Pronounced nonlinear wave groups were obtained near resonance, and these are well reproduced in some detail by the nonlinear theory and computations, without considering any effects of dissipation.

The related problem of resonant wave groups produced by a segmented paddle wavemaker has also been treated by analysis and subject to computation, with good general agreement with past experiments. The technique features matching near- and far-field computations using energy considerations.