Scale analysis of unstable density-driven miscible convection in porous media is performed. The main conclusions for instabilities in the developed (long time scales) regime are that (i) large-scale structures are responsible for the bulk of the production of concentration variance, (ii) variance dissipation is dominated by the small (diffusive) scales and that (iii) both the production and dissipation rates are independent of the Rayleigh number. These findings provide a strong basis for a new modelling approach, namely, large-mode simulation (LMS), for which closure is achieved by replacing the actual diffusivity with an effective one. For validation, LMS results for vertical flow in a homogeneous rectangular domain are compared with direct numerical simulations (DNS). Some of the analysis is based on the derivation and closure of the concentration mean and variance equations, whereby averaging over the ensemble of all possible initial perturbations is considered. While self-similar solutions are obtained for vertical, statistically one-dimensional fingering, triple correlation of concentration and scalar dissipation rate (rate at which the concentration variance decays due to diffusion) have to be modelled in the general case. For this purpose, an ensemble-averaged Darcy modelling (EADM) approach is proposed.

When a confined bubble translates steadily in a cylindrical capillary tube, without the consideration of gravity effects, a uniform thin film of liquid separates the bubble surface and the tube wall. In this work, we investigate how this steady state is established by considering the transitional motion of the bubble as it adjusts its film thickness profile between two steady states, characterized by two different bubble speeds. During the transition, two uniform film regions coexist, separated by a step-like transitional region. The transitional motion also requires modification of the film solution near the rear of the bubble, which depends on the ratio of the two capillary numbers. These theoretical results are verified by experiments and numerical simulations.

This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.

When a torus oscillates horizontally in a linearly stratified fluid, the wave rays form a double cone, one upward and one downward, with two focal points where the wave amplitude has a maximum due to wave focusing. Following a former study on linear aspects of wave focusing (Ermanyuk et al., J. Fluid Mech., vol. 813, 2017, pp. 695–715), we here consider experimental results on the nonlinear aspects that occur in the focal region below the torus for higher-amplitude forcing. A new non-dimensional number that is based on heuristic arguments for the wave amplitude in the focal area is presented. This focusing number is defined as $Fo=(A/a)\unicode[STIX]{x1D716}^{-1/2}f(\unicode[STIX]{x1D703})$, with oscillation amplitude $A$, $f(\unicode[STIX]{x1D703})$ a function for the variation of the wave amplitude with wave angle $\unicode[STIX]{x1D703}$, and $\unicode[STIX]{x1D716}^{1/2}=\sqrt{b/a}$ the increase in amplitude due to the focusing, with $a$ and $b$, respectively, the minor and major radius of the torus. Nonlinear effects occur for $Fo\geqslant 0.1$, with the shear stress giving rise to a mean flow which results in the focal region in a central upward motion partially surrounded by a downward motion. With increasing $Fo$, the Richardson number $Ri$ measured from the wave steepness monotonically decreases. Wave breaking occurs at $Fo\approx 0.23$, corresponding to $Ri=0.25$. In this regime, the focal region is unstable due to triadic wave resonance. For the different tori sizes under consideration, the triadic resonant instability in these three-dimensional flows resembles closely the resonance observed by Bourget et al. (J. Fluid Mech., vol. 723, 2013, pp. 1–20) for a two-dimensional flow, with only minor differences. Application to internal tidal waves in the ocean are discussed.

A comprehensive study of the fundamental characteristics of leading-edge separation in rarefied hypersonic flows is undertaken and its salient features are elucidated. Separation of a boundary layer undergoing strong expansion is typical in many practical hypersonic applications such as base flows of re-entry vehicles and flows over deflected control surfaces. Boundary layer growth under such conditions is influenced by effects of rarefaction and thermal non-equilibrium, thereby differing significantly from the conventional no-slip Blasius type. A leading-edge separation configuration presents a fundamental case for studying the characteristics of such a flow separation but with minimal influence from a pre-existing boundary layer. In this work, direct simulation Monte Carlo computations have been performed to investigate flow separation and reattachment in a low-density hypersonic flow over such a configuration. Distinct features of leading-edge flow, limited boundary layer growth, separation, shear layer, flow structure in the recirculation region and reattachment are all explained in detail. The fully numerical shear layer profile after separation is compared against a semi-theoretical profile, which is obtained using the numerical separation profile as the initial condition on existing theoretical concepts of shear layer analysis based on continuum flow separation. Experimental studies have been carried out to determine the surface heat flux using thin-film gauges and computations showed good agreement with the experimental data. Flow visualisation experiments using the non-intrusive planar laser-induced fluorescence technique have been performed to image the fluorescence of nitric oxide, from which velocity and rotational temperature distributions of the separated flow region are determined.

In a balanced oceanic front, the possible directions of the group velocity vector for internal waves depart from the classic Saint Andrew’s cross as a consequence of sloping isopycnals and the associated thermal wind shear. However, for waves oscillating at the Coriolis frequency $f$, one of these directions remains horizontal, while the other direction allows for vertical propagation of energy. This implies the existence of critical reflections from the ocean surface, after which wave energy, having propagated from below, cannot propagate back down. This is similar to the reflection of internal waves, propagating in a quiescent medium, from a bottom that runs parallel to the group velocity vector. We first illustrate this phenomenon with a series of linear Boussinesq numerical experiments on waves with various frequencies, ${\it\omega}$, exploring critical (${\it\omega}=f$), forward (${\it\omega}>f$), and backward (${\it\omega}<f$) reflections. We then conduct the nonlinear equivalents of these simulations. In agreement with the classical case, backward reflection inhibits triadic resonances and does not exhibit prominent nonlinear effects, while forward reflection shows strong generation of harmonics that radiate energy away from the surface. Surprisingly though, critical reflections are associated with oscillatory motions that extend down from the surface. These motions are not freely propagating waves but instead take the form of a cluster of non-resonant triads which decays with depth through friction.

The spin-down of a barotropic axisymmetric vortex, such as observed in laboratory models, is examined analytically. In addition to the classical, self-similar Ekman decay due to viscous effects (Greenspan & Howard 1963), which is characterized by an azimuthal velocity profile with the position of its maximum velocity fixed and a decay time equal to the Ekman timescale. the effects of nonlinearity and a free surface are considered separately.

The Ekman circulation in the radial and vertical planes whose strength is determined by the vorticity of the overlying fluid, leads to radial advection of the azimuthal velocity. This nonlinearity results in a nonlinear kinematic wave equation for the circulation and leads to the outward/inward propagation of the position of maximum azimuthal velocity for cyclonic/anticyclonic vortices. The associated steepening of the azimuthal velocity profile may lead to a shock formation when the absolute vorticity of the initial profile is negative at a certain radius. For anticyclonic vortices having a monotonically increasing angular velocity profile this shock formation occurs at the core. For such vortices (or arbitrary cyclonic vortices) this dynamical ‘breaking’ criterion is, despite significant differences in the physics concerned, identical to Rayleigh's kinematical criterion for the onset of centrifugal instability.

For a dynamically active free-surface fluid the spin-down of a decaying vortex is prolonged by a radially dependent factor proportional to the Froude number. This conclusion holds both in a cylinder with a parabolic bottom (mimicking the shape of the free surface of a fluid in solid-body rotation) and in a flat-bottomed cylinder. In view of the constancy of background vorticity the former geometry is relevant for a comparison to geophysical f-plane vortices. The latter geometry, however, is more easily established in a laboratory experiment, but the evolution of the azimuthal velocity profile is much more complicated and depends on the initial azimuthal velocity profile in a highly convoluted way.

Off-axis collisions of equal-strength vortex rings were experimentally examined. Two equal-strength vortices were generated which moved toward each other along parallel, but offset, trajectories. Two colour laser-induced fluorescence visualization techniques were used to observe these phenomena and gain insight into their importance in vortex interactions. The most prominent features of this interaction were rapid growth and rotation of the rings and formation of evenly spaced ringlets around the cores of the original rings. Large-scale motions are described using simple vortex induction arguments. The small scales are caused by nonlinear amplification of instabilities during the asymmetric interaction.

The general method described in Baines 1988 has been applied to stratified flows of finite depth over long obstacles where the flow initially has uniform horizontal velocity. The fluid consists of a finite number of homogeneous layers of equal thickness and with equal density increments. This represents the state of continuous stratification with constant density gradient as closely as possible, for a given number of layers. Two-, three-, four- and sixty-four-layered models are studied in detail. The results are expressed in terms of the initial Froude number F0 (F0 = U/ĉ1 where U is the fluid speed and ĉ1 is the speed of the fastest long internal wave mode in the fluid at rest) and the obstacle height. In general, introduction of an obstacle into the flow causes disturbances to propagate upstream (columnar disturbance modes) which alter the velocity and density profiles there. These may accumulate to cause upstream blocking of some of the fluid layers if F0 is sufficiently small. As the number of fluid layers increases, so does the range of F0 for which this upstream blocked flow occurs. There are no upstream disturbances for F0 > 1, and for F0 < 1 the upstream disturbances are of the rarefaction type if upstream blocking does not occur. The results for three and four layers show how several coexisting modes may interact to affect the upstream profiles. The results for sixty-four-layers provide theoretical support for the observational criterion (Baines 1979b) that blocking in initially uniformly stratified flow occurs when Nhm/U > 2 (N is the Brunt—Väisälä frequency and hm the obstacle height), provided that more than two modes are present. In some situations, layered models are found to be inadequate as a representation of continuous stratification when one or more layers thicken to the extent that their discreteness is significant.

The dynamics of a plane air sheet surrounded by a coflowing water film, discharging into stagnant air, is investigated by means of experiments and linear stability theory. For fixed values of the water-to-air thickness ratio, h = hw,0*/ha,0* ≃ 5.27, and of the air-to-water density ratio, S = ρa/ρw ≃ 0.0012, two different flow regimes are experimentally observed depending on the values of two control parameters, namely the Weber number, defined as We = ρwuw,0*2ha,0*/σ, and the velocity ratio, Λ = uw,0*/ua,0*, where uw,0* and ua,0* are the water velocity and the mean air velocity at the exit slit, respectively, and ha,0* and hw,0* are the half-thicknesses of the air and water sheets at the exit. The study focuses on the characterization of the transition between the two regimes found experimentally: a bubbling regime, leading to the periodic breakup of the air sheet, and a jetting regime, where both sheets evolve slowly downstream without breaking. With the aim of exploring whether the transition from the jetting to the bubbling regime is related to a convective/absolute instability transition, we perform a linear spatiotemporal stability analysis. The base flow is described by a simple model that incorporates the downstream evolution of the sheets, which shows excellent agreement with our experiments if the existence of a sufficiently long region of absolute instability, of the order of one absolute wavelength evaluated at the nozzle exit, is imposed as an additional requirement. Finally, we show that the transition is also properly captured by two-dimensional numerical simulations using the volume of fluid technique.

A methodology is developed for modelling entrainment in two-layer shallow water flows using non-standard conserved quantities, replacing layerwise mass conservation by global energy conservation. Thus, the energy that the standard model would regularly dissipate at internal shocks is instead available to exchange fluid between the layers. Two models are considered for the upper boundary of the flow: a rigid lid and a free surface. The latter provides a selection principle for choosing physically relevant conservation laws among the infinitely many that the former possesses, when the ratio between the baroclinic and barotropic speeds tends to zero. Solutions of the equations are studied analytically and numerically, applied to the lock-exchange problem, and compared with other closures.

When a rotating fluid over sloping topography is heated from below and/or cooled from above, horizontal temperature gradients develop which drive convection cells aligned with isobaths. We refer to these cells as topographic Hadley cells. Laboratory experiments reveal that sinking occurs in small cyclonic vortices situated in relatively shallow regions. This is balanced by slower upwelling in adjacent deeper regions. The cross-isobath motions which connect the upwelling and downwelling are accelerated by Coriolis forces, resulting in strong jets which follow isobathic contours. For anticlockwise rotation, the surface jets keep the shallows to their left when looking in the direction of flow, which is opposite to both Kelvin and Rossby wave propagation. The width of the jets scales with the Rossby deformation radius and if this is much less than the width of the slope region then a number of parallel jets form. Motions on the deeper side of the jets where the flow is accelerating are adequately described by linear inviscid theory. However, the strong shears generated by this acceleration lead to baroclinic instability. The resulting cross-stream momentum fluxes broaden and flatten the velocity profile, allowing the flow on the shallow side of the jet to decelerate smoothly before sinking. Topographic Hadley cells are dynamically similar to terrestrial atmospheric Hadley cells and may also be relevant to the zonal jet motions observed on Jupiter and Saturn. It is also suggested that in coastal seas they may represent an important mode of heat (or salt) transfer where surface cooling (or evaporation) drives convection.

Recent work (Hunter & Riahi 1975) on nonlinear convection in a rotating fluid is extended to a multi-modal regime. The schematic multi-boundary-layer method of Busse (1969) and the upper-bound technique of Howard (1963) are used to obtain upper bounds on the Nusselt number N. It is shown that there are infinitely many modes in the range $Ta \ll R^{\frac{3}{2}}$, where Ta is the Taylor number and R is the Rayleigh number, and different types of mode optimize N in different regions of the parameter space (R, Ta). While the optimal N is independent of Ta for Ta [Lt ] R, it is found that it increases with Ta in $R \ll Ta \ll (R \log R)^{\frac{4}{3}}$ and decreases as Ta increases in $(R \log R)^{\frac{4}{3}} \ll Ta \ll R^{\frac{3}{2}}$, and that the functional dependence of the optimal N on R and Ta is continuous (within a logarithmic term) throughout the region of R, Ta space.

It is shown that the solution of the semi-geostrophic equations for shallow-water flow can be found and analysed in spherical geometry by methods similar to those used in the existing $f$-plane solutions. Stable states in geostrophic balance are identified as energy minimizers and a procedure for finding the minimizers is constructed, which is a form of potential vorticity inversion. This defines a generalization of the geostrophic coordinate transformation used in the $f$-plane theory. The procedure is demonstrated in computations.

The evolution equations take a simple form in the transformed coordinates, though, as expected from previous work in the literature, they cannot be expressed exactly as geostrophic motion. The associated potential vorticity does not obey a Lagrangian conservation law, but it does obey a flux conservation law, with an associated circulation theorem.

The divergence of the flow in the transformed coordinates is primarily that naturally associated with geostrophic motion, with additional terms coming from the curvature of the sphere and extra ‘curvature’ resulting from the variable Coriolis parameter in the generalized coordinate transformation. These terms are estimated, and are found to be very small for normal data. The estimate is verified in computations, confirming the accuracy of the local $f$-plane approximation usually made with semi-geostrophic theory.

This paper deals with the propagation of a sound pulse into a gas which initially has solid-body rotation and constant temperature, the initial pressure and density increasing outwards like ex, where x is the square of a certain dimensionless radial co-ordinate. The perturbations are due to a source-like disturbance on the axis of symmetry, which begins to act at time t = 0: most attention is paid to source strengths which vary in time like a Dirac pulse or a step function, but the following remarks apply generally.

Immediately behind the wave front the perturbation velocity and temperature decay like e-½x, while the (absolute) perturbation pressure and density grow like e½x (the relative pressure and density increments, which are referred to local conditions in the undisturbed state, then also decay like e-½x). The rotation also introduces oscillations in flows which, with the same disturbance at the origin and no rotation, would vary monotonically with time at a given point.