We report here visualizations and quantitative measurements of scalar transport, under the influence of rotation, through permeable sediments with an overlying fluid layer. The experimental set-up considered here is a stationary cylinder containing a fluid-saturated porous medium up to its midheight, with supernatant water on top. A rotating lid generates, in the upper fluid region, a flow that partially percolates into the porous layer below. The velocity field in the fluid layer is obtained using particle image velocimetry (PIV). Further, dye transport from the sediment is studied using two different techniques. The first one is positron emission tomography (PET), a non-invasive method which allowed us to ‘see’ through the opaque solid matrix, and to obtain full three-dimensional pictures of dye transport through the sediment. The second one is digital photographic visualization from outside, and subsequent image processing in order to obtain the near-wall dye-washout depth. The experimental data suggest that the temporal evolution of washout depth for different sediments follows near-logarithmic behaviour. This finding is of importance for the a priori estimation of the transport of fluid and other solute substances in sandy aquatic sediments.

Buoyancy-driven convection is modelled using the Navier–Stokes and entropy equations. It is first shown that the coefficient of heat capacity at constant pressure, ${c}_{p} $, must in general depend explicitly on pressure (i.e. is not a function of temperature alone) in order to resolve a dissipation inconsistency. It is shown that energy dissipation in a statistically steady state is the time-averaged volume integral of $- \mathrm{D} P/ \mathrm{D} t$ and not that of $- \alpha T(\mathrm{D} P/ \mathrm{D} t)$. Secondly, in the framework of the anelastic equations derived with respect to the adiabatic reference state, we obtain a condition when the anelastic liquid approximation can be made, $\gamma - 1\ll 1$, independent of the dissipation number.

Flow characteristics and fluxes in thermohaline staircases are measured in two tanks differing in aspect ratio A, where A is the ratio of tank width to fluid depth. In one tank (the ‘1 × 1’ tank) which is 30 cm deep and 30 cm wide, a staircase of one salt-finger layer and one convecting layer develops for a certain setting of the control parameters. The convecting layer has A ≃ 2. Shadowgraphs show convecting plumes that appear disorganized, and a large-scale flow never develops. Instead, the finger layer grows in height, overtakes the convecting layer and within a few days becomes one finger layer. The second tank (the ‘1 × 5’ tank) is also 30 cm deep but is 150 cm wide. For the same control parameter setting a similar staircase with a finger layer 20 cm deep and a convecting layer 10 cm deep develop. The convecting layer, with A = 15, has quite a different character. A large-scale flow develops so that the convecting layer has one cell, 10 cm deep and 150 cm wide. In this flow are large plumes which are transient and tilted; particle image velocimetry measurements of Reynolds stresses show they help to maintain the large-scale flow against viscous dissipation. Shadowgraphs show all the finger tips swept in the direction of the large-scale flow adjacent to the finger layer. Measurements show that the large-scale flow ‘collects’ the salt delivered by the many fingers so that the accumulated negative buoyancy leads to deep convection. This is a more stable arrangement, with the configuration lasting to the order of 102 days.

We present an investigation of the root-mean-square (r.m.s.) temperature $\unicode[STIX]{x1D70E}_{T}$ and the r.m.s. velocity $\unicode[STIX]{x1D70E}_{w}$ in the bulk of Rayleigh–Bénard turbulence, using new experimental data from the current study and experimental and numerical data from previous studies. We find that, once scaled by the convective temperature $\unicode[STIX]{x1D703}_{\ast }$, the value of $\unicode[STIX]{x1D70E}_{T}$ at the cell centre is a constant ($\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }\approx 0.85$) over a wide range of the Rayleigh number ($10^{8}\leqslant Ra\leqslant 10^{15}$) and the Prandtl number ($0.7\leqslant Pr\leqslant 23.34$), and is independent of the surface topographies of the top and bottom plates of the convection cell. A constant close to unity suggests that $\unicode[STIX]{x1D703}_{\ast }$ is a proper measure of the temperature fluctuation in the core region. On the other hand, $\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }$, the vertical r.m.s. velocity at the cell centre scaled by the convective velocity $w_{\ast }$, shows a weak $Ra$-dependence (${\sim}Ra^{0.07\pm 0.02}$) over $10^{8}\leqslant Ra\leqslant 10^{10}$ at $Pr\sim 4.3$ and is independent of plate topography. Similar to a previous finding by He & Xia (Phys. Rev. Lett., vol. 122, 2019, 014503), we find that the r.m.s. temperature profile $\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }$ in the region of the mixing zone with a mean horizontal shear exhibits a power-law dependence on the distance $z$ from the plate, but now the universal profile applies to both smooth and rough surface topographies and over a wider range of $Ra$. The vertical r.m.s. velocity profile $\unicode[STIX]{x1D70E}_{w}(z)/w_{\ast }$ obeys a logarithmic dependence on $z$. The study thus demonstrates that the typical scales for the temperature and the velocity are the convective temperature $\unicode[STIX]{x1D703}_{\ast }$ and the convective velocity $w_{\ast }$, respectively. Finally, we note that $\unicode[STIX]{x1D703}_{\ast }$ may be utilised to study the flow regime transitions in ultrahigh-$Ra$-number turbulent convection.

A velocity-defect law for channel flow is shown to result from the single requirement of Reynolds-stress spectral ‘smoothness’ for any mean profile maintained free of inflexions by transient instabilities. The deduced velocity is a logarithmic function of position near boundaries and parabolic in mid-regions of the flow, independent of the detailed mechanisms of momentum transport by the fluid. However, if the ‘smoothness’ of the spectral tail is decreased, a second logarithmic layer of steeper slope emerges inside the first layer. Profile data from drag-reduction experiments gathered by Virk (1975) exhibit the deduced inner log layer and its transition region to the usual outer flow.

This paper is a theoretical and experimental investigation of the onset of the buoyancydriven longitudinal roll cells that occur when a liquid layer flows over a heated horizontal plate. Linear stability theory is applied under the assumption that the spatially developing temperature profile can be treated locally, that is at each axial position, as being ‘frozen’. Using the film thickness as the length scaling factor, the critical Rayleigh numbers associated with the onset of longitudinal rolls are found to be considerably lower than measured values. A modified local stability analysis using the thermal boundary-layer thickness as the scaling factor is shown to agree with experiments. Predicted wavenumbers and the position of the onset of cellular convection are in agreement with wavenumbers measured by flow-visualization techniques. The position of the onset of cellular convection is also obtained from heat-transfer measurements at the heated surface. In the asymptotic limit of a linear undisturbed temperature profile the classical solutions of the Rayleigh-Bénard problem for the critical Rayleigh number and wavenumber are recovered, the only effect of the flow being the structure of the secondary flow that occurs when the system is unstable. Amplification theory is also compared with experimental data for the position at which the thermal effects of the convection are detectable and for the wavenumbers measured by flow visualization. The thermal amplification ratio $\overline{Nu}$ and a velocity-disturbance amplification ratio $\overline{w}$ are used to interpret the onset of discernible cellular convection. The data are not consistent with any single amplification ratio over the range of Rayleigh numbers studied (103 < Ra < 3 × 105), and the theoretical and experimental results suggest that a band of wavenumbers rather than a single wavenumber is encountered when cellular convection occurs.

This paper attempts to determine the optimum profile of a two-dimensional plate that produces the maximum hydrodynamic lift while planing on a water surface, under the condition of no spray formation and no gravitational effect, the latter assumption serving as a good approximation for operations at large Froude numbers. The lift of the sprayless planing surface is maximized under the isoperimetric constraints of fixed chord length and fixed wetted arc-length of the plate. Consideration of the extremization yields, as the Euler equation, a pair of coupled nonlinear singular integral equations of the Cauchy type. These equations are subsequently linearized to facilitate further analysis. The analytical solution of the linearized problem has a branch-type singularity, in both pressure and flow angle, at the two ends of plate. In a special limit, this singularity changes its type, emerging into a logarithmic one, which is the weakest type possible. Guided by this analytic solution of the linearized problem, approximate solutions have been calculated for the nonlinear problem using the Rayleigh-Ritz method and the numerical results compared with the linearized theory.

The internal structure of vortex rings and helical vortices is studied using asymptotic analysis and numerical simulations in cases where the core size of the vortex is small compared to its radius of curvature, or to the distance to other vortices. Several configurations are considered: a single vortex ring, an array of equally-spaced rings, a single helix and a regular array of helices. For such cases, the internal structure is assumed to be at leading order an axisymmetric concentrated vortex with an internal jet. A dipolar correction arises at first order and is shown to be the same for all cases, depending only on the local vortex curvature. A quadrupolar correction arises at second order. It is composed of two contributions, one associated with local curvature and another one arising from a non-local external 2-D strain field. This strain field itself is obtained by performing an asymptotic matching of the local internal solution with the external solution obtained from the Biot–Savart law. Only the amplitude of this strain field varies from one case to another. These asymptotic results are thereafter confronted with flow solutions obtained by direct numerical simulation (DNS) of the Navier–Stokes equations. Two different codes are used: for vortex rings, the simulations are performed in the axisymmetric framework; for helices, simulations are run using a dedicated code with built-in helical symmetry. Quantitative agreement is obtained. How these results can be used to theoretically predict the occurrence of both the elliptic instability and the curvature instability is finally addressed.

We investigate how a filter-medium microstructure influences filtration performance. We derive a theory that generalizes classical multiscale models for regular structures to account for filter media with more realistic microstructures, comprising random microstructures with polydisperse unidirectional fibres. Our multiscale model accounts for the fluid flow and contaminant transport at the microscale (over which the medium structure is fully resolved) and allows us to obtain macroscopic properties such as the effective permeability, diffusivity and fibre surface area. As the fibres grow due to contaminant adsorption, this leads to contact of neighbouring fibres. We propose an agglomeration algorithm that describes the resulting behaviour of the fibres upon contact, allowing us to explore the subsequent time evolution of the filter medium in a simple and robust way. We perform a comprehensive investigation of the influence of the filter-medium microstructure on filter performance in a spectrum of possible filtration scenarios.

The oscillatory motion of an electrically charged non-spherical colloidal particle in an oscillating electric field is investigated. The particle is immersed in an incompressible viscous fluid and assumed to have a thin electric double layer. For moderate-aspect-ration spheroids and cylinders, a simple algebraic expression is derived that accurately describes oscillatory electrophoretic particle motion in terms of the steady Stokes resistance, added mass, and Basset force. The effects of double-layer conduction and displacement currents within dielectric particles are included. The results indicate that electroacoustic measurements may be able to determine the ζ-potential, dielectric constant, surface conductivity (and microstructural information contained therein), size, density, volume fraction, and possibly shape of non-spherical particles in a dilute suspension. A simple formula is obtained for the high-frequency electrical conductivity of a dilute suspension of colloidal spheroids with arbitrary charge and dielectric constant; only the added mass and Basset force are required and the requisite parameters are given. The result is needed for electroacoustic measurements but it may also be independently useful for determining the dielectric constant, surface conductivity, volume fraction, and possibly the shape of non-spherical particles in a dilute suspension. Electroacoustic energy dissipation is described for a dilute colloidal suspension. It is shown that resistive electrical heating and viscous dissipation occur independently. Electrical and viscous dissipation coefficients that characterize the order volume fraction contributions of the suspended particles are calculated; the electrical dissipation coefficient is O(1) for all oscillation frequencies, whereas the latter vanishes at low- and high-frequencies. The fluid motion is shown to be a superposition of unsteady, viscous and potential flows past an oscillating particle with no applied electric field. The electro-osmotic flow field is insensitive to particle geometry and qualitatively different from the flow past an oscillating particle with no applied field.

We study convective motion in vertically vibrated three-dimensional granular beds by comparing the predictions of a model based on a hydrodynamic description to Navier–Stokes order with experimental results obtained using positron emission particle tracking (PEPT). The three-dimensional conservation equations relating mass, momentum and energy are solved using the finite element (FE) method for a viscous vibrofluidized bed by using only observable system parameters such as particle number, size, mass and coefficients of restitution. The mean velocity profiles from the viscous model show reasonable agreement with the experimental results at relatively low altitudes for the range of experimental values studied, though the velocity fields at higher altitudes were systematically underestimated by the model. We confirm that the convection rolls are influenced by the sidewall coefficient of restitution and demonstrate the scaling relationships that operate, where increasing amplitude of vibration leads to a reduction in the angular velocity of the rolls.

Results are presented from a numerical and analytical study of negatively buoyant thermals. The numerical study consists of large-eddy simulations of thermal descent and spread. The thermals are initiated by a spherical perturbation in the homogeneous background potential temperature. Simulations covering various release heights, thermal radii and thermal buoyancies are carried out. The analysis involves matching similarity models of a thermal and an axisymmetric gravity current, hence describing the flow evolution in terms of the initial conditions and flow coefficients only. The simulations demonstrate that the flow transition through the impingement region is relatively smooth, the main flow adjustment being in the initial post-release phase of the thermal. Comparison of the simulations and the model enables determination of the coefficients, and validation of the similarity approach to predict the radial speed, reduced gravity and depth of the spreading flow on the ground. The predictions of reduced gravity and depth also depend on quantification of the increase in gravity-current volume due to entrainment, which is obtained from the simulations.

A quasi-linear theory is presented for how randomly forced, barotropic velocity fluctuations cause an exponentially growing, large-scale (mean) magnetic dynamo in the presence of a uniform parallel shear flow. It is a ‘kinematic’ theory for the growth of the mean magnetic energy from a small initial seed, neglecting the saturation effects of the Lorentz force. The quasi-linear approximation is most broadly justifiable by its correspondence with computational solutions of nonlinear magnetohydrodynamics, and it is rigorously derived in the limit of small magnetic Reynolds number, . Dynamo action occurs even without mean helicity in the forcing or flow, but random helicity variance is then essential. In a sufficiently large domain and with a small seed wavenumber in the direction perpendicular to the mean shearing plane, a positive exponential growth rate can occur for arbitrary values of , viscous Reynolds number , and random-force correlation time and orientation angle in the shearing plane. The value of is independent of the domain size. The shear dynamo is ‘fast’, with finite in the limit of . Averaged over random realizations of the forcing history, the ensemble-mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (magnetic energy). In the limit of small and , the dynamo behaviour is related to the well-known alpha–omega ansatz when the force is slowly varying () and to the ‘incoherent’ alpha–omega ansatz when the force is more rapidly fluctuating.

We investigate the deformation of a linear viscoelastic compliant coating in a turbulent flow for a wide range of coating parameters. A one-way coupling model is proposed in which the turbulent surface stresses are expressed as a sum of streamwise-travelling waves with amplitudes determined from the stress spectra of the corresponding flow over a rigid wall. The analytically calculated coating deformation is analysed in terms of the root-mean-square (r.m.s.) surface displacement and the corresponding point frequency spectra. The present study systematically investigates the influence of five coating properties namely density, stiffness, thickness, viscoelasticity and compressibility. The surface displacements increase linearly with the fluid/solid density ratio. They are linearly proportional to the coating thickness for thin coatings, while they become independent of the thickness for thick coatings. Very soft coatings show resonant behaviour, but the displacement for stiffer coatings is proportional to the inverse of the shear modulus. The viscoelastic loss angle has only a significant influence when resonances occur in the coating response, while Poisson’s ratio has a minor effect for most cases. The modelled surface displacement is qualitatively compared with recent measurements on the deformation of three different coatings in a turbulent boundary-layer flow. The model predicts the order of magnitude of the surface displacement, and it captures the increase of the coating displacement with the Reynolds number and the coating softness. Finally, we propose a scaling that collapses all the experimental data for the r.m.s. of the vertical surface displacement onto a single curve.

The problem of a turbulent boundary layer that evolves into a wake flow at the sharp trailing edge of a thin flat plate is considered; the formal structure of the near-wake flow is investigated using matched expansions in the limit of infinite Reynolds number. The symmetric turbulent near wake is shown to develop a two-layer structure which is independent of turbulence model. The general asymptotic analysis shows that a thin layer at the wake centreline grows linearly with distance from the trailing edge while the centreline velocity varies logarithmically in a manner that is supported strongly by experimental measurements. The relatively thick outer layer of the near-wake flow is undisturbed by the evolution of the inner layer to leading order. An additional region near the trailing edge is required to resolve a non-uniformity in transverse velocity. The general asymptotic results are used to guide the development of a zonal turbulence model for the near wake in the form of a simple eddy viscosity formula. Analytic profiles for velocity and Reynolds stress are obtained for the near-wake region; these profiles are shown to provide accurate representations of available near-wake experimental data.

An infra-red heat-transfer gauge was used in a shock tube for end-wall measurements of the convective heat transfer from argon behind the reflected shock. The thermal conductivity of neutral (un-ionized) argon was measured before the ionization-relaxation time, and was fitted with the power-law temperature dependence 4·2 × 10−5(T/300)0·76±0·03 cal/sec cm°K, where T is measured in °K, and ±0·03 refers to the probable error The free-stream temperature ranged from 20,000 to 75,000°K, corresponding to incident-shock velocities from 3 to 6mm/μsec. At later times, after the free stream established equilibrium ionization, the convective-heat-transfer rate remained the same as the initial rate with neutral argon. Theoretical predictions of Fay & Kemp (1965), assuming equilibrium-boundary-layer conditions, are 20–30% below the experimental values. Also reported in this paper are measurements of the ionization times behind the reflected shock, and these are in agreement with an extrapolation of the Petschek & Byron (1957) measurements behind the incident shock. There is a discussion of the large changes in the gas conditions behind the reflected shock due to the ionization process. The final equilibrium conditions are reached abruptly, as indicated by the continuum-radiation emission which becomes constant immediately after ionization relaxation.

We consider the two-phase flow of a suspension in a rotating cylinder with inclined endplates for which inertial and viscous effects are small. It is shown that, when the Coriolis force is dominant, flow in the core is essentially unaffected by geometry. If a fluid particle can make a complete circuit about the rotation axis, the sedimentation velocity cannot be augmented by geometrical effects as it can in gravitational settling. However, with the insertion of a complete meridional barrier to block movement around the centre, separation becomes more sensitive to the shape of the container walls. In this case, behaviour similar to that in a gravitational field is possible once again.

The stability of a baroclinic zonal current to symmetric perturbations on a meridionally unbounded f-plane is considered. The lower boundary is at rest but the upper one moves with a constant velocity in keeping with the velocity of the zonal current. Following Stone (1966) a horizontal length scale O(Ro) is taken, where Ro is the Rossby number, with the Richardson number Ri = O(1). Instability sets in when the wavelength is O(E1/3), where E is the Ekman number based on the distance between the rigid horizontal boundaries, which corresponds to Stone's inviscid value zero, and to McIntyre's (1970) value infinity on a length scale O(E½).

A nonlinear analysis about the point of onset of instability yields the result that for the monotonic mode zonal momentum is convected polewards. The possible implications of this result for the dynamics of Jupiter's atmosphere are discussed.

We derive the John–Sclavounos equations, describing the motion of a fluid particle on the sea surface, from first principles using Lagrangian and Hamiltonian formalisms applied to the motion of a frictionless particle constrained on an unsteady surface. This framework leads to a number of new insights into the particle kinematics. The main result is that vorticity generated on a stress-free surface vanishes at a wave crest when the horizontal particle velocity equals the crest propagation speed, which is the kinematic criterion for wave breaking. If this holds for the largest crest, then the symplectic two-form associated with the Hamiltonian dynamics reduces instantaneously to that associated with the motion of a particle in free flight, as if the surface did not exist. Further, exploiting the conservation of the Hamiltonian function for steady surfaces and travelling waves, we show that particle velocities remain bounded at all times, ruling out the possibility of the finite-time blowup of solutions.

The stability properties of a natural convection boundary layer adjacent to an isothermally heated vertical wall, with Prandtl number 0.71, are numerically investigated in the configuration of a temporally evolving parallel flow. The instantaneous linear stability of the flow is first investigated by solving the eigenvalue problem with a quasi-steady assumption, whereby the unsteady base flow is frozen in time. Temporal responses of the discrete perturbation modes are numerically obtained by solving the two-dimensional linearized disturbance equations using a ‘frozen’ base flow as an initial-value problem at various $Gr_{\unicode[STIX]{x1D6FF}}$, where $Gr_{\unicode[STIX]{x1D6FF}}$ is the Grashof number based on the velocity integral boundary layer thickness $\unicode[STIX]{x1D6FF}$. The resultant amplification rates of the discrete modes are compared with the quasi-steady eigenvalue analysis, and both two-dimensional and three-dimensional direct numerical simulations (DNS) of the temporally evolving flow. The amplification rate predicted by the linear theory compares well with the result of direct numerical simulation up to a transition point. The extent of the linear regime where the perturbations linearly interact with the base flow is thus identified. The value of the transition $Gr_{\unicode[STIX]{x1D6FF}}$, according to the three-dimensional DNS results, is dependent on the initial perturbation amplitude. Beyond the transition point, the DNS results diverge from the linear stability predictions as nonlinear mechanisms become important.