Small amplitude waves on a shallow layer of water are studied from the point of view used in aerodynamic sound theory. It is shown that many aspects of the generation and propagation of water waves are similar to those of sound waves in air. Certain differences are also discussed. It is concluded that shallow water simulation can be employed in the study of some aspects of aerodynamically generated sound.

The effect of boundary conditions on the ‘critical dynamics’ at the onset of Taylor vortices is investigated in a combined numerical and experimental study. Numerical calculations of Navier–Stokes equations with ‘stress-free’ boundary conditions show that the Landau amplitude equation provides a good model of the transient dynamics. However, this rapidly breaks down when the ‘no-slip’ condition is approached. Apparent ‘critical’ behaviour observed in experiments is shown to have a surprising dependence on the length of the system.

We consider the two-dimensional flow produced by the slow horizontal motion of a vertical plate of height 2b through a vertically stratified (ρ = ρ0(1 - βz)) non-diffusive viscous fluid. Our results are valid when U2 [Lt ] Ub/ν [Lt ] 1, where U is the speed of the plate and ν the kinematic viscosity of the fluid. Upstream of the body we find a blocking column of length 10−2b4/(Uν/βg. This column is composed of cells of closed streamlines. The convergence of these cells near the tips of the plate leads to alternate jets. The plate itself is embedded in a vertical shear layer of thickness (Uν/βg)1/3. In the upstream portion of this layer the vertical velocities are of order U and in the downstream portion of order Ub/(Uν/βg)1/3 ([Gt ] U). The flow is uniform and undisturbed downstream of this layer.

A numerical model is developed to study the effects of friction on the steady exchange flow that evolves when a barrier is removed from a constriction separating two reservoirs of slightly different densities. The model has excellent agreement with an analytical solution and laboratory measurements of exchange flows through channels of constant width and depth. The model reveals three viscous flow regimes for a convergent–divergent contraction of constant depth, and three additional viscous flow regimes when an offset sill is introduced. Each regime is characterized by a different set of internal hydraulic control locations. Examination of the predicted interface profiles reveals that it is not possible to distinguish between different flow regimes on the basis of these profiles alone.

The formation of doubly-periodic patterns on the surface of a fluid layer with a uniform velocity field and constant depth is considered. The fluid is assumed to be inviscid and the flow irrotational. The problem of steady patterns is shown to have a novel variational formulation, which includes a new characterization of steady uniform mean flow, and steady uniform flow coupled with steady doubly periodic patterns. A central observation is that mean flow can be characterized geometrically by associating it with symmetries. The theory gives precise information about the role of the ten natural parameters in the problem which govern the wave–mean flow interaction for steady patterns in finite depth. The formulation is applied to the problem of interaction of capillary–gravity short-crested waves with oblique travelling waves, leading to several new observations for this class of waves. Moreover, by including oblique travelling waves and short-crested waves in the same analysis, new bifurcations of short-crested waves are found, which give rise to mixed waves which may have complicated spatial structure.

A lubrication theory has been developed for the electro-osmotic flow of non-uniform buffers in narrow rectilinear channels. The analysis applies to systems in which the transverse dimensions of the channel are large compared with the Debye screening length of the electrolyte. In contrast with related theories of electrokinetic lubrication, here the streamwise variations of the velocity field stem from, and are nonlinearly coupled to, spatiotemporal variations in the electrolyte composition. Spatially non-uniform buffers are commonly employed in electrophoretic separation and transport schemes, including iso-electric focusing (IEF), isotachophoresis (ITP), field-amplified sample stacking (FASS), and high-ionic-strength electro-osmotic pumping. The fluid dynamics of these systems is controlled by a complex nonlinear coupling to the ion transport, driven by an applied electric field. Electrical conductivity gradients, attendent to the buffer non-uniformities, result in a variable electro-osmotic slip velocity and, in electric fields approaching 1 kV cm−1, Maxwell stresses drive the electrohydrodynamic circulation. Explicit semi-analytic expressions are derived for the fluid velocity, stream function, and electric field. The resulting approximations are found to be in good agreement with full numerical solutions for a prototype buffer, over a range of conditions typical of microfluidic systems. The approximations greatly simplify the computational analysis, reduce computation times by a factor 4–5, and, for the first time, provide general insight on the dominant fluid physics of two-dimensional electrically driven transport.

The investigation of the effects which a changing mean flow has on a uniform train of internal gravity waves (Thorpe 1978a) is continued by considering waves in a uniformly accelerating stratified plane Couette flow with constant density gradient. Experiments reveal a change in the mode structure and phase distribution of the waves, and their eventual breaking near the boundary where the mean flow is greatest, the phase speed of the waves being positive. A linear numerical model is devised which accurately describes the waves up to the onset of their breaking, and this is used to investigate their energetics. The working of the Reynolds stress against the mean velocity gradient results in a very rapid transfer of energy from the waves to the mean flow, so that by the time breaking occurs only a small fraction of their initial energy remains for possible transfer into potential energy of the fluid.

The consequences have important applications in oceanography and meteorology, to flow stability and flow generation, and explain some earlier laboratory observations.

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.

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.

A layer of viscous incompressible fluid is confined between two horizontal plates which rotate rapidly in their own plane with a constant angular velocity. A hemisphere has its plane face joined to the lower plate and when a uniform flow is forced past such an obstacle, a Taylor column bounded by thin detached vertical shear layers forms. The linear theory for this problem, wherein the Rossby number ε is set equal to zero on the assumption that the flow is slow, is examined in detail. The nonlinear modifications of the shear layers are then investigated for the case when ε ∼ E½, where E is the Ekman number. In particular, it is shown that provided that the Rossby number is large enough separation occurs in the free shear layers. The extension of the theory to flow past arbitrary spheroids is indicated.

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.

An analytical dispersion relation describing the linear stability of a plane detonation wave to low-frequency two-dimensional disturbances with arbitrary wavenumbers is derived using a normal mode approach and a combination of high activation energy and Newtonian limit asymptotics, where the ratio of specific heats γ→1. The reaction chemistry is characterized by one-step Arrhenius kinetics. The analysis assumes a large activation energy in the plane steady-state detonation wave and a characteristic linear disturbance wavelength which is longer than the fire-zone thickness. Newtonian limit asymptotics are employed to obtain a complete analytical description of the disturbance behaviour in the induction zone of the detonation wave. The analytical dispersion relation that is derived depends on the activation energy and exhibits favourable agreement with numerical solutions of the full linear stability problem for low-frequency one- and two-dimensional disturbances, even when the activation energy is only moderate. Moreover, the dispersion relation retains vitally important characteristics of the full problem such as the one-dimensional stability of the detonation wave to low-frequency disturbances for decreasing activation energies or increasing overdrives. When two-dimensional oscillatory disturbances are considered, the analytical dispersion relation predicts a monotonic increase in the disturbance growth rate with increasing wavenumber, until a maximum growth rate is reached at a finite wavenumber. Subsequently the growth rate decays with further increases in wavenumber until the detonation becomes stable to the two-dimensional disturbance. In addition, through a new detailed analysis of the behaviour of the perturbations near the fire front, the present analysis is found to be equally valid for detonation waves travelling at the Chapman–Jouguet velocity and for detonation waves which are overdriven. It is found that in contrast to the standard imposition of a radiation or piston condition on acoustic disturbances in the equilibrium zone for overdriven waves, a compatibility condition on the perturbation jump conditions across the fire zone must be satisfied for detonation waves propagating at the Chapman–Jouguet detonation velocity. An insight into the physical mechanisms of the one- and two-dimensional linear instability is also gained, and is found to involve an intricate coupling of acoustic and entropy wave propagation within the detonation wave.

We have investigated the dynamics of a monotrichous bacteria cell near a wall boundary, taking elastic hook flexibility into consideration. Combining theoretical linear stability analysis and direct numerical computations via the boundary element method, we have found that the elastohydrodynamic coupling between the hook elasticity and cell rotational motion enables a stable vertical spinning behaviour like a low-Reynolds-number spinning top. The forwardly rotated flagellum, which generates the force exertion pushing towards the cell body, typically destabilizes the vertical upright position and leads to a boundary-following motion. In contrast, the backward rotation of the flagellum, generating a force pulling the cell body, contributes to stable upright behaviour in a large range of hook rigidity. Further numerical investigations have demonstrated that the non-spherical geometry of the cell body and boundary adhesive interactions affect the bacterial dynamics, leading to complex behaviours such as horizontal spinning and unstable vertical spinning motions, both of which are experimentally observed in Pseudomonas aeruginosa bacteria. These results highlight the rich diversity of bacterial surface motility emerging from mechanical boundary interactions coupled with the cell swimming and hook flexibility.