We consider free convection driven by a heated vertical plate immersed in a nonlinearly stratified medium. The plate supplies a uniform horizontal heat flux to a fluid, the bulk of which has a stable stratification, characterized by a non-uniform vertical temperature gradient. This gradient is assumed to have a typical length scale of variation, denoted Z0, while Ψ0 is the order of magnitude of the related heat flux that crosses the medium vertically.

We derive an analytic solution to the Boussinesq equations that extends the classical solution of Prandtl to the case of nonlinearly stratified media. This novel solution is asymptotically valid in the regime RaS ≫ 1, where RaS denotes the Rayleigh number of nonlinear stratification, based on Z0, Ψ0, and the physical properties of the medium.

We then apply the new theory to the natural convection affecting the vapour phase in a liquefied pure gas tank (e.g. the cryogenic storage of hydrogen). It is assumed that the cylindrical storage tank is subject to a constant uniform heat flux on its lateral and top walls. We are interested in the vapour motion above a residual layer of liquid in equilibrium with the vapour. High-precision axisymmetric numerical computations show that the flow remains steady for a large range of parameters, and that a bulk stratification characterized by a quadratic temperature profile is undoubtedly present. The application of the theory permits a comparison of the numerical and analytic results, showing that the theory satisfactorily predicts the primary dynamical and thermal properties of the storage tank.

We describe a mathematical model of double-diffusive (thermosolutal) convection in a saturated porous layer, when the solubility of the solute depends on the temperature, and the porosity and permeability of the porous medium evolve through dissolution and precipitation. We present the results of linear and weakly nonlinear stability analyses and explore the longer-term development of the system numerically. When the solutal concentration gradient is destabilising, the dynamics are somewhat similar to those previously found for single-species convection (Ritchie & Pritchard, J. Fluid Mech., vol. 673, 2011, pp. 286–317), including the occurrence of subcritical instabilities driven by a reaction–diffusion mechanism. However, when the solutal concentration gradient is stabilising and the thermal gradient is destabilising, novel dynamics emerge. These include a vertical segregation of circulation cells and porosity perturbations near the onset of convection, and over longer time scales the formation of a low-permeability region in the middle of the layer, pierced by occasional high-permeability channels. Under these conditions, convection may die away to nearly zero for extended periods before resuming vigorously in localised regions at later times.

A quasi-cylindrical approximation is used to analyse the axisymmetric swirling flow of a liquid with a hollow air core in the chamber of a pressure swirl atomizer. The liquid is injected into the chamber with an azimuthal velocity component through a number of slots at the periphery of one end of the chamber, and flows out as an annular sheet through a central orifice at the other end, following a conical convergence of the chamber wall. An effective inlet condition is used to model the effects of the slots and the boundary layer that develops at the nearby endwall of the chamber. An analysis is presented of the structure of the liquid sheet at the end of the exit orifice, where the flow becomes critical in the sense that upstream propagation of long-wave perturbations ceases to be possible. This analysis leads to a boundary condition at the end of the orifice that is an extension of the condition of maximum flux used with irrotational models of the flow. As is well known, the radial pressure gradient induced by the swirling flow in the bulk of the chamber causes the overpressure that drives the liquid towards the exit orifice, and also leads to Ekman pumping in the boundary layers of reduced azimuthal velocity at the convergent wall of the chamber and at the wall opposite to the exit orifice. The numerical results confirm the important role played by the boundary layers. They make the thickness of the liquid sheet at the end of the orifice larger than predicted by irrotational models, and at the same time tend to decrease the overpressure required to pass a given flow rate through the chamber, because the large axial velocity in the boundary layers takes care of part of the flow rate. The thickness of the boundary layers increases when the atomizer constant (the inverse of a swirl number, proportional to the flow rate scaled with the radius of the exit orifice and the circulation around the air core) decreases. A minimum value of this parameter is found below which the layer of reduced azimuthal velocity around the air core prevents the pressure from increasing and steadily driving the flow through the exit orifice. The effects of other parameters not accounted for by irrotational models are also analysed in terms of their influence on the boundary layers.

The characteristics of unbounded flow past an impulsively started planar energy extracting device, such as a wind or tidal turbine, are studied theoretically, numerically and experimentally. The initial thrust on an impulsively started device, which can be more than double the steady thrust, is an important consideration for design and safe operation. The energy sink is modelled here as an ‘actuator surface’ which imposes a uniform pressure discontinuity in the fluid proportional to the square of the fluid speed normal to the surface, the fluid density, and a dimensionless resistance coefficient. The flow past the actuator is studied theoretically for the case of weak resistance using an unsteady model which recovers steady linear momentum theory in the limit of long time. For the case of strong resistance the flow is studied numerically using the point vortex method. Experimental measurements of thrust on a mesh towed through static water are compared to the numerical results and show good agreement. The thrust on an impulsively started device is estimated, for a typical installation, to fall to within 10 % of the steady value within ∼1 min. The numerical model is also used to simulate the gradual startup of a device, yielding estimates of the time constant necessary in a control system in order to reduce peak thrusts in practice.

Boundary-layer receptivity in the leading-edge region of a cambered thin airfoil is analysed for the case of a low-Mach-number flow. Acoustic free-stream disturbances are considered. Asymptotic results based on large Reynolds number ($U^2 / \omega \nu \,{\gg}\, 1$) are presented, supplemented by numerical solutions. The influence of mean aerodynamic loading enters the theory through a parameter $\mu$, which provides a measure of the flow speed variations in the leading-edge region, due to flow around the leading edge from the lower surface to the upper. A Strouhal number based on airfoil nose radius, $S\,{=}\,\omega r_n/U$, also enters the theory. The variation of the receptivity level as a function of $\mu $ and $S$ is analysed. Modest levels of aerodynamic loading are found to decrease the receptivity level for the upper surface of the airfoil, while the receptivity is increased for the lower surface. For larger angles of attack close to the critical angle for boundary layer separation, a local rise in the receptivity occurs for the upper surface, while on the lower surface the receptivity decreases. These effects are more pronounced at larger values of $S$. While the Tollmien–Schlichting wave does not emerge until a downstream distance of $O((U^2 / \omega \nu)^{1/3} U / \omega)$, the amplitude of the Tollmien–Schlichting wave is influenced by the acoustic free-stream disturbances only in a relatively small region near the leading edge, of length approximately $4 U/\omega$. The numerical receptivity coefficients calculated, together with the asymptotic eigenfunctions presented, provide all the necessary information for transition analysis from the interaction of acoustic disturbances with leading-edge geometry.

Vortex-current filaments have been used to study phenomena such as coronal loops and solar flares as well as tokamaks, and recent experimental work has demonstrated dynamics akin to vortex-current filaments on a table-top plasma focus device. While MHD vortex dynamics and related applications to turbulence have attracted consideration in the literature due to a wide variety of applications, not much analytical progress has been made in this area, and the analysis of such vortex-current filament solutions under various geometries may motivate further experimental efforts. To this end, we consider the motion of open, isolated vortex-current filaments in the presence of magnetohydrodynamic (MHD) as well as the standard hydrodynamic effects. We begin with the vortex-current model of Yatsuyanagi, Hatori & Kato (J. Phys. Soc. Japan, vol. 65, 1996, pp. 745–759) giving the self-induced motion of a vortex-current filament. We give the ‘cutoff’ formulation of the Biot–Savart integrals used in this model, to avoid the singularity at the vortex core. We then study the motion of a variety of vortex-current filaments, including helical, planar and self-similar filament structures. In the case where MHD effects are weak relative to hydrodynamic effects, the filaments behave as expected from the pure hydrodynamic theory. However, when MHD effects are strong enough to dominate, then we observe structural changes to the filaments in all cases considered. The most common finding is reversal of vortex-current filament orientation for strong enough MHD effects. Kelvin waves along a vortex filament (as seen for helical and self-similar structures) will reverse their translational and rotational motion under strong MHD effects. Our findings support the view that vortex-current filaments can be studied in a manner similar to classical hydrodynamic vortex filaments, with the primary role of MHD effects being to change the filament motion, while preserving the overall geometric structure of such filaments.

We explore the direct modification of the pseudo-spectral truncation of two-dimensional, incompressible fluid dynamics to maintain a prescribed kinetic energy spectrum. The method provides a means of simulating fluid states with defined spectral properties, for the purpose of matching simulation statistics to given information, arising from observations, theoretical prediction or high-fidelity simulation. In the scheme outlined here, Nosé–Hoover thermostats, commonly used in molecular dynamics, are introduced as feedback controls applied to energy shells of the Fourier-discretized Navier–Stokes equations. As we demonstrate in numerical experiments, the dynamical properties (quantified using autocorrelation functions) are only modestly perturbed by our device, while ensemble dispersion is significantly enhanced compared with simulations of a corresponding truncation incorporating hyperviscosity.

Since Trösch (Proceedings of the 4th International Conference on Applied Numerical Modeling, Tainan, Taiwan, 1984 (ed. H. M. Hsia, Y. L. Chou, S. Y. Wang & S. J. Hsieh) Science and Technology Series, vol. 63, 1986, pp. 307–311. American Astronautical Society) found trapped sub-inertial oscillations in computations of low-frequency variability in the Lake of Lugano, models of trapping have generally considered evenly spaced isobaths parallel to shorelines with approximate boundary conditions at any shelf–ocean boundary. Here an asymptotic analysis for slowly varying topography and accurate spectral computations demonstrate trapping on non-rectilinear shelves. It is shown that changes in any of three factors, isobath curvature, distance from the coast and the shelf-break, and the slope at the shelf-break, are sufficient on their own to give trapping. Continental shelves that abut smoothly onto the open ocean are considered thus avoiding the shelf–ocean boundary condition approximation and allowing the accuracy of previous approximations to be assessed.

The in uence of the thermal conductivity of rigid boundaries is analysed for the case when the onset of convection occurs in the form of a time-dependent mode. A new form of long-wavelength convection is found which is disjunct from the high-wavenumber thermal Rossby wave in the presence of infinitely conducting boundaries usually considered.