The spreading, imbibition and solidification of a hot two-dimensional droplet on a cooler porous substrate is studied. Lubrication theory and asymptotic reduction are used to derive a coupled system of evolution equations for the droplet thickness and vertical extents of the solid/melt boundary and the saturation front within the porous medium. This medium is assumed to be a membrane composed of an array of pores having fixed width. A precursor layer model and a disjoining pressure are used to relieve the singularity at the contact line. When the solidification and imbibition time scales are similar to those associated with spreading, the dynamics follow two stages: spreading and imbibition accompanied by solidification within the pores leading to their blockage, followed by contact line arrest and basal solidification of the droplet; the possibility of crust formation at the gas–melt interface is excluded from the present study. The dependence of the dynamics on the relevant system parameters is elucidated. Furthermore, our modelling predictions compare favourably with experimental data also obtained as part of the present work.

Direct numerical simulations (DNS) have been performed of turbulent flow in a plane channel with a solid top wall and a permeable bottom wall. The permeable wall is a packed bed, which is characterized by the mean particle diameter and the porosity. The main objective is to study the influence of wall permeability on the structure and dynamics of turbulence. The flow inside the permeable wall is described by means of volume-averaged Navier–Stokes equations. Results from four simulations are shown, for which only the wall porosity ($\epsilon_c$) is changed. The Reynolds number based on the thickness of the boundary layer over the permeable wall and the friction velocity varies from $\hbox{\it Re}_{\tau}^p\,{=}\,176$ for $\epsilon_c\,{=}\,0$ to $\hbox{\it Re}_{\tau}^p\,{=}\,498$ for $\epsilon_c\,{=}\,0.95$. The influence of wall permeability can be characterized by the permeability Reynolds number, $\hbox{\it Re}_K$, which represents the ratio of the effective pore diameter to the typical thickness of the viscous sublayers over the individual wall elements. For small $\hbox{\it Re}_K$, the wall behaves like a solid wall. For large $\hbox{\it Re}_K$, the wall is classified as a highly permeable wall near which viscous effects are of minor importance. It is observed that streaks and the associated quasi-streamwise vortices are absent near a highly permeable wall. This is attributed to turbulent transport across the wall interface and the reduction in mean shear due to a weakening of, respectively, the wall-blocking and the wall-induced viscous effect. The absence of streaks is consistent with a decrease in the peak value of the streamwise root mean square (r.m.s.) velocity normalized by the friction velocity at the permeable wall. Despite the increase in the peak values of the spanwise and wall-normal r.m.s. velocities, the peak value of the turbulent kinetic energy is therefore smaller. Turbulence near a highly permeable wall is dominated by relatively large vortical structures, which originate from a Kelvin–Helmholtz type of instability. These structures are responsible for an exchange of momentum between the channel and the permeable wall. This process contributes strongly to the Reynolds-shear stress and thus to a large increase in the skin friction.

High-resolution three-dimensional numerical experiments show that initially balanced (void of waves) geophysical flows, static and inertially stable, generate spiral patterns of small-amplitude inertia–gravity waves (IGWs). The spiral wave patterns are due to the spontaneous generation of IGW packets emitted from fluid volumes (the IGW sources) experiencing large local changes of potential vorticity. The IGW packets spread away from the vortical flow and cause spiral wave patterns of the same sense of spiralling, cyclonic or anticyclonic, as the moving IGW sources. The spiral patterns are noticeable in the vertical velocity in deep layers, away from the large-amplitude balanced vertical velocity. The generation of the spiral wave patterns is illustrated through several examples, namely, the single ellipsoidal vortex (cyclone and anticyclone), the merging of two spherical vortices, the dipole, and the anticyclonic shear instability.

An asymptotic quasi-normal Markovian (AQNM) model is developed in the limit of small Rossby number $Ro$ and high Reynolds number, i.e. for rapidly rotating turbulent flow. Based on the ‘slow’ amplitudes of inertial waves, the kinetic equations are close to those that would be derived from Eulerian wave-turbulence theory. However, for their derivation we start from an EDQNM statistical closure model in which the velocity field is expanded in terms of the eigenmodes of the linear wave regime. Unlike most wave-turbulence studies, our model accounts for the detailed anisotropy as the angular dependence in Fourier space. Nonlinear equations at small Rossby number are derived for the set $e$, $Z$, $h$ – energy, polarization anisotropy, helicity – of spectral quantities which characterize second-order two-point statistics in anisotropic turbulence, and which generate every quadratic moment of inertial wave amplitudes. In the simplest symmetry consistent with the background equations, i.e. axisymmetry without mirror symmetry, $e$, $Z$ and $h$ depend on both the wavevector modulus $k$ and its orientation $\theta$ to the rotation axis. We put the emphasis on obtaining accurate numerical simulations of a generalized Lin equation for the angular-dependent energy spectrum $e(k, \theta, t)$, in which the energy transfer reduces to integrals over surfaces given by the triadic resonant conditions of inertial waves. Starting from a pure three-dimensional isotropic state in which $e$ depends only on $k$ and $Z\,{=}\,h\,{=}\,0$, the spectrum develops an inertial range in the usual fashion as well as angular anisotropy. After the development phase, we observe the following features:

1. A $k^{-3}$ power law for the spherically averaged energy spectrum. However, this is the average of power laws whose exponents vary with the direction of the wavevector from $k^{-2}$ for wavevectors near the plane perpendicular to the rotation axis, to $k^{-4}$ for parallel wavevectors.

2. The spectral evolution is self-similar. This excludes the possibility of a purely two-dimensional large-time limit.

3. The energy density is very large near the perpendicular wavevector plane, but this singularity is integrable. As a result, the total energy has contributions from all directions and is not dominated by this singular contribution.

4. The kinetic energy decays as $t^{-0.8}$, an exponent which is about half that one without rotation.

The response of turbulence subjected to planar straining and de-straining is studied experimentally, and the impact of the applied distortions on the energy transfer across different length scales is quantified. The data are obtained using planar particle image velocimetry (PIV) in a water tank, in which high-Reynolds-number turbulence with very low mean velocity is generated by an array of spinning grids. Planar straining and de-straining mean flows are produced by pushing and pulling a rectangular piston towards, and away from, the bottom wall of the tank. The data are processed to yield the time evolution of Reynolds stresses, anisotropy tensors, turbulence kinetic energy production, and mean subgrid-scale (SGS) dissipation rate at various scales. During straining, the production rises rapidly. After the relaxation period the small-scale SGS stresses recover isotropy, but the Reynolds stresses still display significant anisotropy. Thus when destraining is applied, a strong negative production (mean backscatter) occurs, i.e. the turbulence returns kinetic energy to the mean flow. The SGS dissipation displays similar behaviour at large filter scales, but the mean backscatter gradually disappears with decreasing filter scales. Energy spectra are compared to predictions of rapid distortion theory (RDT). Good agreement is found for the initial response but, as expected for the time-scale ratios of the experiment, turbulence relaxation causes discrepancies between measurements and RDT at later times.

The equilibrium conditions of a point vortex in the separated flow past a locally deformed wall is studied in the framework of the two-dimensional potential flow. Equilibrium locations are represented as fixed points of the vortex Hamiltonian contour line map. Their pattern is ascribable to the Poincaré–Birkhoff fixed-point theorem. An ‘equilibrium manifold’, representing the generalization of the Föppl curve for circular cylinders, is defined for arbitrary bodies. The property $\partial\omega/\partial\skew3\tilde\psi\,{=}\,0$ holds on it, with $\skew3\tilde\psi$ being the stream function and $\omega$ the streamline slope of the pure potential flow.

A ‘Kutta manifold’ is defined as the locus of vortices in flows that separate at a prescribed point (Kutta condition). The existence of standing vortices that satisfy the Kutta condition is discussed for symmetric bodies. On the basis of an asymptotic expansion of the equilibrium manifold, Kutta manifold and body geometry, it is shown that different classes of symmetric bodies exist which are ranked by the number of allowable standing vortices that satisfy the Kutta condition.

We investigate flow in a tube driven by axial and radial wall oscillations that are periodic in time. Furthermore, the walls undergo periodic circumferential oscillations, the amplitude of which increases with axial distance along the tube; this wall motion results in a new mechanism for the generation of axial flows within the tube. The axial steady-streaming flows that arise as a result of circumferential oscillations are opposed to those due to the axial and radial pulsations, and are potentially as significant. We then consider the uptake of a passive solute through the walls of such a tube into an adjacent medium in which the solute diffuses and is consumed at a constant rate. The tube ends are open to well-mixed fluid containing the solute. The effect of the streaming flows on the solute flux into the tube is determined. The solute disperses along the tube due to the interaction between advection and transverse diffusion, and the time-mean solute distribution throughout the tube is determined for a wide range of parameters.

In a previous work, two-dimensional film flows were modelled using a weighted-residual approach that led to a four-equation model consistent at order $\epsilon^2$. A two-equation model resulted from a subsequent simplification but at the cost of lowering the degree of the approximation to order $\epsilon$ only. A Padé approximant technique is applied here to derive a refined two-equation model consistent at order $\epsilon^2$. This model, formulated in terms of coupled evolution equations for the film thickness $h$ and the flow rate $q$, accounts for inertia effects due to the deviations of the velocity profile from the parabolic shape, and closely follows the asymptotic long-wave expansion in the appropriate limit. Comparisons of two-dimensional wave properties with experiments and direct numerical simulations show good agreement for the range of parameters in which a two-dimensional wavy motion is reported in experiments.

The stability of two-dimensional travelling waves to three-dimensional pertur-bations is investigated based on the extension of the models to include spanwise dependence. The secondary instability is found to be not very selective, which explains the widespread presence of the synchronous instability observed in the experiments by Liu et al.(1995) whereas Floquet analysis predicts a subharmonic scenario in most cases. Three-dimensional wave patterns are computed next assuming periodic boundary conditions. Transition from two- to three-dimensional flows is shown to be strongly dependent on initial conditions. The herringbone patterns, the synchronously deformed fronts and the three-dimensional solitary waves observed in experiments are recovered using our regularized model, which is found to be an excellent compromise between the complete model, which has seven equations, and the simplified model, which does not include the second-order inertia corrections. Those corrections are found to play a role in the selection of the type of secondary instability as well as of the spanwise wavelength of the emerging pattern.

We discuss the procedure for the exact solution of the Riemann problem in special relativistic magnetohydrodynamics (MHD). We consider both initial states leading to a set of only three waves analogous to the ones in relativistic hydrodynamics, as well as generic initial states leading to the full set of seven MHD waves. Because of its generality, the solution presented here could serve as an important test for those numerical codes solving the MHD equations in relativistic regimes.

We generalize the linear stability analysis of the axisymmetric self-similar solution of gravity currents from finite-volume releases to include perturbations that depend on both radial and azimuthal coordinates. We show that the similarity solution is stable to sufficiently small perturbations by proving that all perturbation eigenfunctions decay in time. Moreover, asymmetric perturbations are shown to decay more rapidly than axisymmetric perturbations in general. An asymptotic formula for the eigenvalues is derived, which indicates that asymptotic rates of decay of perturbations are given by $t^{-\sigma}$ where $0\,{<}\,\sigma\,{<}\,\frac{1}{4}$ as the Froude number decreases from $\sqrt2$ to $0$. We demonstrate that this formula agrees closely with numerically calculated eigenvalues and, in the absence of azimuthal dependence, it reduces to an expression that improves on the asymptotic formula obtained by Grundy & Rottman (1985). For two-dimensional (planar) currents, we further prove analytically that all perturbation eigenfunctions decay like $t^{-{1/2}}.$

A new method is developed for approximating the scattering of linear surface gravity waves on water of varying quiescent depth in two dimensions. A conformal mapping of the fluid domain onto a uniform rectangular strip transforms steep and discontinuous bed profiles into relatively slowly varying, smooth functions in the transformed free-surface condition. By analogy with the mild-slope approach used extensively in unmapped domains, an approximate solution of the transformed problem is sought in the form of a modulated propagating wave which is determined by solving a second-order ordinary differential equation. This can be achieved numerically, but an analytic solution in the form of a rapidly convergent infinite series is also derived and provides simple explicit formulae for the scattered wave amplitudes. Small-amplitude and slow variations in the bedform that are excluded from the mapping procedure are incorporated in the approximation by a straightforward extension of the theory. The error incurred in using the method is established by means of a rigorous numerical investigation and it is found that remarkably accurate estimates of the scattered wave amplitudes are given for a wide range of bedforms and frequencies.

We have analysed the structure of the irrotational flow near the minimum radius of an axisymmetric bubble at the final instants before pinch-off. The neglect of gas inertia leads to the geometry of the liquid–gas interface near the point of minimum radius being slender and symmetric with respect to the plane $z\,{=}\,0$. The results reproduce our previous finding that the asymptotic time evolution for the minimum radius, $R_o(t)$, is $\tau\propto R^2_o\sqrt{-\,{\rm ln}\,R^2_o}$, $\tau$ being the time to breakup, and that the interface is locally described, for times sufficiently close to pinch-off, by $f(z,t)/R_o(t)\,{=}\,1\,{-}\,(6\,{\rm ln}\,R_o)^{-1}(z/R_o)^2$. These asymptotic solutions correspond to the attractor of a system of ordinary differential equations governing the flow during the final stages before pinch-off. However, we find that, depending on initial conditions, the solution converges to the attractor so slowly (with a logarithmic behaviour) that the universal laws given above may hold only for times so close to the singularity that they might not be experimentally observed.

The classical equations of water waves are reformulated as a system of two equations, one of which is an explicit non-local equation, for the wave height and for the velocity potential evaluated on the free surface. Evaluation of the velocity potential as a function of the depth is not required in order to calculate the wave height and the velocity potential on the free surface. The non-local system yields integral relations related to mass and centre of mass, and is shown to reduce to known asymptotic limits in shallow and deep water. Included in these asymptotic reductions are the Boussinesq, Benney–Luke and nonlinear Schrödinger equations. Two-dimensional lumps with sufficient surface tension are obtained numerically. The extension of this non-local formulation to the case of a variable bottom is also presented.

Effects of insoluble surfactants on the stability of film flow driven by an oscillatory plate are investigated in the limit of long-wavelength perturbations. Two particular Floquet modes are identified and the corresponding growth rates are obtained by solving a quadratic equation. Results show that the oscillatory film flow can be stabilized by surface surfactant in the sense of raising the critical Froude number and narrowing the bandwidths of the unstable frequencies.

We consider the interaction of free-stream disturbances with the leading edge of a body and its effect on the transition point. We present a method which combines an asymptotic receptivity approach, and a numerical method which marches through the Orr–Sommerfeld region. The asymptotic receptivity analysis produces a three-deck eigensolution which in its far downstream limiting form produces an upstream boundary condition for our numerical parabolized stability equation (PSE). We discuss the advantages of this method compared to existing numerical and asymptotic analysis and present results which justify this method for the case of a semi-infinite flat plate, where asymptotic results exist in the Orr–Sommerfeld region. We also discuss the limitations of the PSE and comment on the validity of the upstream boundary conditions. Good agreement is found between the present results and the numerical results of Haddad & Corke (1998).

The turbulent channel flow with streamwise rotation has been investigated by means of several different analytical, numerical, and modelling approaches. Lie group analysis of the two-point correlation equations led to linear scaling laws for the streamwise mean velocity. In addition it was found that a cross-flow in the spanwise direction is induced, which may also exhibit a linear region. By further analysis of the two-point correlation equation, it is shown that all six components of the Reynolds stress tensor are non-zero. In addition certain symmetries and skew-symmetries about the centreline have been established for all flow quantities. All these findings of the analysis have been verified very well by means of direct numerical simulations (DNS). The flow has also been calculated with large-eddy simulations (LES) and second-moment closure models. The dynamic LES captured most of the theoretical and DNS findings quantitatively. Except for one stress component the second-moment closure model was able to capture most of the basic trends, but no quantitative agreement could be achieved.

Two parameters are introduced that uniquely characterize the state of a third-order symmetric tensor. We show that the proposed parameters arise from the uniform metric in the matrix space; thus the joint PDF of these parameters can be used to determine the geometrical statistics of any third-order symmetric tensor. We use this joint PDF to describe the states of the subgrid-scale stress, which is of central interest in large-eddy simulation. Direct numerical simulation of forced isotropic turbulence is used in our a priori tests. With the proposed parameterization we can also assess the most probable flow configuration at the scales of motion just above the Kolmogorov scale. We test four different subgrid-scale models in terms of how well they predict the structure, or state, of the subgrid-scale stress. It is found that models based on truncated Taylor series do not produce an adequate distribution of states, even if augmented by a turbulent viscosity term. On the other hand, models based on the scale-similarity assumption predict a distribution of states that is close to actual.

Experiments are reported on the hydrodynamic performance of a flexible fin. The fin replicates some features of the pectoral fin of a batoid fish (such as a ray or skate) in that it is actuated in a travelling wave motion, with the amplitude of the motion increasing linearly along the span from root to tip. Thrust is found to increase with non-dimensional frequency, and an optimal oscillatory gait is identified. Power consumption measurements lead to the computation of propulsive efficiency, and an optimal efficiency condition is evaluated. Wake visualizations are presented, and a vortex model of the wake near zero net thrust is suggested. Strouhal number effects on the wake topology are also illustrated.

A new description of two-dimensional continuous free-surface flows in Lagrangian coordinates is proposed. It is shown that the position of a fluid particle in such flows can be represented as a fixed point of a transformation in $\mathbb{R}^2$. Components of the transformation function satisfy the linear Euler-type continuity equation and can be expressed via a single function analogous to an Eulerian stream function. Fixed-point iterations lead to a simple recursive representation of a solution satisfying the Lagrangian continuity equation. Expanding the unknown function in a small-perturbation asymptotic expansion we obtain the complete asymptotic formulation of the problem in a fixed domain of Lagrangian labels. The method is then applied to the classical problem of a regular wave travelling in deep water, and the fifth-order Lagrangian asymptotic solution is constructed, which provides a much better approximation of steep waves than the corresponding Eulerian Stokes expansion. In contrast with early attempts at Lagrangian regular-wave expansions, the asymptotic solution presented is uniformly valid at large times.

We study the effects of gravity modulation on the mixing characteristics of two interdiffusing miscible fluids initially in two vertical regions separated by a thin diffusion layer. We formulate the case of general gravity modulation of arbitrary orientation, amplitude $g$ and characteristic frequency $\omega$. For harmonic vertical modulation in two dimensions, the time-dependent Boussinesq equations are solved numerically and the evolution of the interface between the fluids is observed. The problem is governed by six parameters: the Grashof number, $\hbox{\it Gr}\,{=}\,({\Delta\rho}/{\bar{\rho}})g({l^{3}_{\nu}}/{\nu^{2}})$, based on the viscous length scale, $l_{\nu}\,{=}\,\sqrt{{\nu}/{\omega}}$; the Schmidt number, $\hbox{\it Sc}\,{=}\,{\nu}/{D}$; the aspect ratio, $A$; the non-dimensional length of the domain, $l$; the steepness of the initial concentration profile, $\delta$; and the phase angle of the harmonic modulation, $\phi$. When $\phi\,{=}\,0,\;\pi$, we observe four different flow regimes with increasing $\hbox{\it Gr}$: neutral oscillations at the forcing frequency; successive folds which propagate diffusively; localized shear instabilities; and both shear and convective instabilities leading to rapid mixing. In the last regime, the flow is disordered but not chaotic. By varying $\hbox{\it Sc}$, it was determined that the mechanism for the formation of these shear and convective instabilities is inertial. When $\phi \neq 0$ or $\pi$, the flow is similar to a modulated lock exchange flow.