A general theory of unsteady cavitating flow past hydrofoils and other obstacles is given for the case of cavities of finite length L. If the circulation Γ, the cavity volume V and L are known as functions of time, the theory yields explicit formulae for the velocity over the wetted surface and for the cavitation number σ. The theory is based on the approximation that the cavity is bounded by stream-lines, and so is valid only for slow rates of change of L, V and Γ. The possibility of allowing for the presence of a vortex sheet behind the cavity is discussed. The theory is extended to the case of a cascade of hydrofoils behind which extend growing cavities.

Two examples of the theory are discussed, namely the unsteady flow past a symmetrical wedge, and the unsteady flow past a flat plate hydrofoil cavitating from the leading edge.

A theoretical study of the interaction term in the two-fluid model equations is presented. The relevant Navier–Stokes equation is volume-integrated in a control volume fixed in a field of dispersed two-phase flow; then it is time-integrated. An expression for the interaction term is obtained in the limit of infinitesimal control volume, which rigorously fits to the two-fluid model equation based on time averaging. The interaction term is then analysed for dispersed two-phase flow with homogeneous particle size. The mathematical expression of the resulting interaction term clearly shows its property, which has been overlooked for more than 40 years. It can be decomposed into the conventional interaction term and an additional ‘virtual force’ term. The virtual force term is evaluated approximately for two types of dispersed multiphase flow in order to demonstrate its effectiveness. The first is solid–liquid dispersed two-phase flow with spherical solid particles undergoing creeping flow, and the second is gas–liquid dispersed two-phase flow with large bubbles in a highly turbulent flow field. For solid spherical particles in a homogeneous creeping flow, the term vanishes, as was found numerically by Ten Cate and Sundaresan (Intl J. Multiphase Flow, vol. 32, 2006, pp. 106–131), although the term could be significant for a flow field with velocity gradient. For large bubbles in bubble columns in a recirculating turbulent flow regime, the term is significant. The two-fluid model equations are corrected by the introduction of these virtual force terms, which in some circumstances are important in simulating the macroscopic properties of dispersed two-phase flow with spatial variations.

The multi-point Monin–Obukhov similarity (MMO) was recently proposed (Tong & Nguyen, J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348) to address the issue of incomplete similarity in the framework of the original Monin–Obukhov similarity theory (MOST). MMO hypothesizes the following: (1) The surface-layer turbulence, defined to consist of eddies that are entirely inside the surface layer, has complete similarity, which however can only be represented by multi-point statistics, requiring a horizontal characteristic length scale (absent in MOST). (2) The Obukhov length $L$ is also the characteristic horizontal length scale; therefore, all surface-layer multi-point statistics, non-dimensionalized using the surface-layer parameters, depend only on the height and separations between the points, non-dimensionalized using $L$. However, similar to MOST, MMO was also proposed as a hypothesis based on phenomenology. In this work we derive MMO analytically for the case of the horizontal Fourier transforms of the velocity and potential temperature fluctuations, which are equivalent to the two-point horizontal differences of these variables, using the spectral forms of the Navier–Stokes and the potential temperature equations. We show that, for the large-scale motions (wavenumber $k<1/z$) in a convective surface layer, the solution is uniformly valid with respect to $z$ (i.e. as $z$ decreases from $z>-L$ to $z<-L$), where $z$ is the height from the surface. However, for $z<-L$ the solution is not uniformly valid with respective to $k$ as it increases from $k<-1/L$ to $k>-1/L$, resulting in a singular perturbation problem, which we analyse using the method of matched asymptotic expansions. We show that (1) $-L$ is the characteristic horizontal length scale, and (2) the Fourier transforms satisfy MMO with the non-dimensional wavenumber $-kL$ as the independent similarity variable. Two scaling ranges, the convective range and the dynamic range, discovered for $z\ll -L$ in Tong & Nguyen (2015) are obtained. We derive the leading-order spectral scaling exponents for the two scaling ranges and the corrections to the scaling ranges for finite ratios of the length scales. The analysis also reveals the dominant dynamics in each scaling range. The analytical derivations of the characteristic horizontal length scale ($L$) and the validity of MMO for the case of two-point horizontal separations provide strong support to MMO for general multi-point velocity and temperature differences.

A localized, insoluble, surfactant monolayer, spreading under the action of surface-tension gradients over a thin liquid film, has at its leading edge an integrable stress singularity which renders conventional thin-film approximations locally non-uniform. Here high-Reynolds-number asymptotics are used to explore the quasi-steady two-dimensional developing flow near the monolayer tip, assuming that gravity keeps the free surface almost flat, that weak ‘contaminant’ surfactant regularizes the singularity and that the monolayer spreads fast enough for inertial effects to be important in a region which is long compared to the film depth but which is short compared to the length of the monolayer. It is shown how downward displacement of the inviscid core flow by the subsurface viscous boundary layer yields a non-uniform pressure distribution which, when the monolayer is spreading fast enough for cross-stream pressure gradients to be significant at its tip, creates a short free-surface hump which is the thin-film version of a Reynolds ridge. The ridge and other singular flow structures are smoothed as the monolayer slows and levels of contaminant are increased. The conditions under which lubrication theory provides a uniformly accurate approximation for this class of surfactant-spreading flows are established.

We investigate weakly two-dimensional weakly nonlinear weakly dispersive surface waves propagating in a turbulent flow over a gradually sloping bottom. The waves are shown to be governed by a turbulently damped variable-coefficient Kadomtsev–Petviashvili equation with periodic boundary conditions. Equations governing the lowest-order mean currents in both directions as well as the equation describing the lowest-order mean surface elevation are also derived. Solutions for the wave equation are found numerically using a Fourier pseudospectral technique in space and finite differencing in the time-like variable.

The laminar pulsatile flow over a semi-infinite flat plate, on which is located a small (steady) surface distortion is investigated; triple-deck theory provides the basis for the study. The problem is of direct relevance to the externally imposed acoustic excitation of boundary layers. The investigation is primarily numerical and involves the solution of the nonlinear, unsteady boundary-layer equations which arise from the lower deck. The numerical method involves the use of finite differencing in the transverse direction, Crank-Nicolson marching in time, and Fourier transforms in the streamwise direction, and as such is an extension of the spectral method of Burggraf & Duck (1982). Supersonic and incompressible flows are studied. A number of the computations presented suggest that the small surface distortion can excite a large-wavenumber, rapidly growing instability, leading to a breakdown of the solution, with the wall shear at a point seeming to increase without bound as a finite time is approached. Rayleigh modes for the basic (undisturbed) velocity profile are computed and there is some correlation between the existence and magnitude of the growth rate of these unstable modes, and the occurrence of the apparent singularity. Streamline plots indicate that this phenomenon is linked to the formation of closed (or ‘cats-eye’) eddies in the main body of the boundary layer, away from the wall. Tollmien-Schlichting instabilities are clearly seen in the case of incompressible flows.

The fluid is incompressible, inviscid and non-diffusive. It has a uniform Brunt–Väisälä frequency, N, and is of constant depth, D. A body or wing moves horizontally through the fluid at velocity U in a straight line, exerting a vertical force during a given time interval. The force is constant, or oscillatory with frequency σ. The vertical average of the strain rate in a thin surface layer is calculated for a network of points behind the body.

The linearized analysis is first applied with tank walls, then modified for remote walls and a vertical force of long duration.

For moderately high velocity and forcing frequency (U/ND = 5, σ/N≅ 4−16) the recurring internal wave pattern just behind the body is well established in one cycle of the oscillatory force. A tank width one or two times the depth gives good agreement between tank and no-wall calculations for the chosen examples.

For a stationary wing (U/ND = 0) in a cubic tank with forcing frequency one-half the natural frequency (σ/N = ½) the strain rates after one cycle are 103 times greater than for the moving wing case. After five cycles the magnitudes are twenty times larger than after one cycle. Presumably these large increases are due to the continuous and efficient feeding of energy into a small fluid volume which occurs for the stationary wing. No-wall calculations for many cycles give amplitudes roughly one-half those for five cycles in the tank, showing the effect of escaping energy.

The relation of these developments to stationary phase analysis and preferred directions is discussed.

This study investigates the behaviour of a turbulent thermal undergoing a second-order chemical reaction with the fluid entrained from the environment. Environments with uniform and stratified density are considered. We show that the dynamics of such a reactive thermal is fully determined by three dimensionless groups, $N/E$, $G/R$ and $R/E$, where $N$ is the buoyancy frequency of the environment, $G$ measures the ability of the reaction to change buoyancy, $R$ reflects the rate of consumption of the chemical species and $E$ is the rate of entrainment of reactive species from the environment. Exact analytical solutions are found for the limiting cases of slow and instantaneous chemical reaction. The effect of each governing group on the time for neutral buoyancy and depletion of the source chemical is assessed numerically. Our theoretical predictions compare well with new experimental results for the limits of a moderately slow chemical reaction and an instantaneous reaction. It is shown that fast reactions, with $R/E\gg 1$, occur only in a fraction of the total volume of the thermal due to incomplete mixing. Finally, our model is applied to study the dynamics of a radioactive cloud formed after a nuclear accident.

A straightforward method that yields explicit transmission amplitudes is presented for Kelvin wave scattering by topography whose isobaths are parallel sufficiently far from the vertical, but not necessarily planar, wall supporting the incident wave. These results are obtained by first restricting attention to the low-frequency limit in which the flow splits naturally into three regions: an outer-x region containing the incident and transmitted Kelvin waves, an outer-y region containing outwardly propagating long topographic waves and an inner quasi-steady geostrophic region whose structure follows from earlier time-dependent analyses. The present analysis is further simplified by approximating general smooth features by stepped profiles with no restriction on the size, number or order of steps. Various qualitative results on the transmission amplitudes and flow fields are deduced from the explicit solutions and results are given on orthogonality, completeness and direction of propagation of the scattered long waves.

The fluid motion, temperature distribution and the mass-transfer problem of a binary gas mixture in a rapidly rotating centrifuge are investigated. The model centrifuge considered consists of a pair of concentric circular cylinders bounded on the top and bottom by horizontal end plates; the apparatus rotates rapidly about the axis of the cylinders. During steady operation a binary gas mixture containing species A and B is injected into and withdrawn from the centrifuge through axisymmetric slots located on the sidewalls. Solutions for the velocity, temperature and mass-fraction fields within the centrifuge are obtained for mechanically or thermally driven centrifuges. For the mass-transfer problem, a detailed analysis of the fluid-mechanical boundary layers is required, and, in particular, mass fluxes within the boundary layers are obtained for a wide range of source-sink geometries. Solutions to the mass-transfer problem are obtained for moderately and strongly forced flows in the container; the dependence of the separation (or enrichment) factor on centrifuge configuration, rotational speed and fraction of the volumetric flow rate extracted at the product port (the cut) are predicted.

We consider the thermocapillary motion of a liquid layer which is bounded between two superhydrophobic surfaces, each made up of a periodic array of highly conducting solid slats, with flat bubbles trapped in the grooves between them. Following the recent analysis of the longitudinal problem (Yariv, J. Fluid Mech., vol. 855, 2018, pp. 574–594), we address here the transverse problem, where the macroscopic temperature gradient that drives the flow is applied perpendicular to the grooves, with the goal of calculating the volumetric flux between the two surfaces. We focus upon the situation where the slats separating the grooves are long relative to the groove-array period, for which case the temperature in the solid portions of the superhydrophobic plane is piecewise uniform. This scenario, which was investigated numerically by Baier et al. (Phys. Rev. E, vol. 82 (3), 2010, 037301), allows for a surprising analogy between the harmonic conjugate of the temperature field in the present problem and the unidirectional velocity in a comparable longitudinal pressure-driven flow problem over an interchanged boundary. The main body of the paper is concerned with the limit of deep channels, where the problem reduces to the calculation of the heat transport and flow about a single surface and the associated ‘slip’ velocity at large distance from that surface. Making use of Lorentz’s reciprocity, we obtain that velocity as a simple quadrature, providing the analogue to the expression obtained by Baier et al. (2010) in the comparable longitudinal problem. The rest of the paper is devoted to the diametric limit of shallow channels, which is analysed using a Hele-Shaw approximation, and the singular limit of small solid fractions, where we find a logarithmic scaling of the flux with the solid fraction. The latter two limits do not commute.

Flow complexity is related to a number of phenomena in science and engineering and has been approached from the perspective of chaotic dynamical systems, ergodic processes or mixing of fluids, just to name a few. To the best of our knowledge, all existing methods to quantify flow complexity are only valid for infinite time evolution, for closed systems or for mixing of two substances. We introduce an index of flow complexity coined interlacing complexity index (ICI), valid for a single-phase flow in an open system with inlet and outlet regions, involving finite times. ICI is based on Shannon’s mutual information (MI), and inspired by an analogy between inlet–outlet open flow systems and communication systems in communication theory. The roles of transmitter, receiver and communication channel are played, respectively, by the inlet, the outlet and the flow transport between them. A perfectly laminar flow in a straight tube can be compared to an ideal communication channel where the transmitted and received messages are identical and hence the MI between input and output is maximal. For more complex flows, generated by more intricate conditions or geometries, the ability to discriminate the outlet position by knowing the inlet position is decreased, reducing the corresponding MI. The behaviour of the ICI has been tested with numerical experiments on diverse flows cases. The results indicate that the ICI provides a sensitive complexity measure with intuitive interpretation in a diversity of conditions and in agreement with other observations, such as Dean vortices and subjective visual assessments. As a crucial component of the ICI formulation, we also introduce the natural distribution of streamlines and the natural distribution of world-lines, with invariance properties with respect to the cross-section used to parameterize them, valid for any type of mass-preserving flow.

The astrophysical importance of the hypothesis that vorticity is expelled from regions of strong turbulence is briefly reviewed. An experiment from which Gough & Lynden-Bell claimed to provide positive evidence of vorticity expulsion has been repeated. The results are reproduced here and are shown to be due to thermally driven circulation currents. No unambiguous laboratory evidence for (or against) the vorticity expulsion hypothesis therefore exists at present.

We study buoyancy-driven exchange flow of two mixtures in a vertical narrow duct (two-dimensional channel as well as pipe) theoretically. While the light mixture is assumed always to be a pure fluid, the heavy mixture can be selected as either a pure or a particle-laden fluid. A small width-to-length ratio considered for the duct ($\unicode[STIX]{x1D6FF}\ll 1$) has been used as the perturbation parameter in developing a lubrication model (negligible inertia). In particular, we have adopted the methodology of Zhou et al. (Phys. Rev. Lett., vol. 94, 2005, 117803) for free-surface particle-laden film flows and extended it to a lock exchange system in confined geometry under the Boussinesq approximation. The resulting model is in the form of the classical Riemann problem and has been solved numerically using a robust total variation diminishing finite difference scheme. Both pure and particle-laden cases are investigated in detail. It is observed that the interface between the two fluids takes a self-similar shape at long times. In the case that both heavy and light fluids are pure, the dynamics of the flow is governed by two dimensionless quantities, namely the Reynolds number, $Re$, and the viscosity ratio, $\unicode[STIX]{x1D705}$, of the light and heavy fluids. The interpenetration of the heavy and light layers increases with $Re$ but decreases with $\unicode[STIX]{x1D705}$. Also, the heights of the heavy and light fronts change with $\unicode[STIX]{x1D705}$ but remain unchanged with $Re$. In the case of the particle-laden flow, however, four additional dimensionless parameters emerge, namely the initial volume fraction of particles, $\unicode[STIX]{x1D719}_{0}$, the ratio of particle diameter to duct width, $r_{p}$, and the density ratios of particles to carrying fluid, $\unicode[STIX]{x1D709}$, and of light fluid to carrying fluid, $\unicode[STIX]{x1D702}$. The effect of these parameters on the dynamics of the flow has been quantified through a systematic approach. In the presence of solid particles, the interface between the heavy and light layers becomes more curved compared to the case of pure fluids. This modification occurs due to the change of heavy mixture viscosity alongside the duct. Novel particle-rich zones are further discovered in the vicinity of the advancing heavy and light fronts. These zones are associated with different transport rates of the fluid and solid particles. The degree of particle enrichment remains the same with $Re$, is enhanced by $\unicode[STIX]{x1D705}$, $r_{p}$ and $\unicode[STIX]{x1D702}$, and is slightly diminished with $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x1D709}$. On the other hand, the stretched exchange zone between the heavy and light fronts grows with $r_{p}$, $\unicode[STIX]{x1D702}$ and $Re$, but decays with $\unicode[STIX]{x1D719}_{0}$, $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$.

Experimental results for instabilities present in a rotating Hagen–Poiseuille flow are reported in this study through fluid flow visualization. First, we found a very good agreement between the experimental and the theoretical predictions for the onset of convective hydrodynamic instabilities. Our analysis in a space–time domain is able to obtain quantitative data, so the wavelengths and the frequencies are also estimated. The comparison of the predicted theoretical frequencies with the experimental ones shows the suitability of the parallel, spatial and linear stability analysis, even though the problem is spatially developing. Special attention is focused on the transition from convective to absolute instabilities, where we observe that the entire pipe presents wavy patterns, and the experimental frequencies collapse with the theoretical results for the absolute frequencies. Thus, we provide experimental evidence of absolute instabilities in a pipe flow, confirming that the rotating pipe flow may be absolutely unstable for moderate values of Reynolds numbers and low values of the swirl parameter.