Double-diffusive flux-gradient laws are commonly used to describe the development of large-scale structures driven by salt fingers – thermohaline staircases, collective instability waves and intrusions. The flux-gradient model assumes that the vertical transport is uniquely determined by the local background temperature and salinity gradients. While flux-gradient laws adequately capture mixing characteristics on scales that greatly exceed those of primary double-diffusive instabilities, their accuracy rapidly deteriorates when the scale separation between primary and secondary instabilities is reduced. This study examines conditions for the breakdown of the flux-gradient laws using a combination of analytical arguments and direct numerical simulations. The applicability (failure) of the flux-gradient laws at large (small) scales is illustrated through the example of layering instability, which results in the spontaneous formation of thermohaline staircases from uniform temperature and salinity gradients. Our inquiry is focused on the properties of the ‘point-of-failure’ scale ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H_{pof}$) at which the vertical transport becomes significantly affected by the non-uniformity of the background stratification. It is hypothesized that $H_{pof} $ can control some key characteristics of secondary double-diffusive phenomena, such as the thickness of high-gradient interfaces in thermohaline staircases. A more general parametrization of the vertical transport – the flux-gradient-aberrancy law – is proposed, which includes the selective damping of relatively short wavelengths that are inadequately represented by the flux-gradient models. The new formulation is free from the unphysical behaviour of the flux-gradient laws at small scales (e.g. the ultraviolet catastrophe) and can be readily implemented in theoretical and large-scale numerical models of double-diffusive convection.

The onset of thermal convection in a rapidly rotating spherical shell is studied by linear stability analysis based on the fully compressible Navier–Stokes equations. Compressibility is quantified by the number of density scale heights $N_{\unicode[STIX]{x1D70C}}$, which measures the intensity of density stratification of the motionless, polytropic base state. The nearly adiabatic flow with polytropic index $n=1.499<n_{a}=1.5$ is considered, where $n_{a}$ is the adiabatic polytropic index. By investigating the stability of the base state with respect to the disturbance of specified wavenumber, the instability process is found to be sensitive to the Prandtl number $Pr$ and to $N_{\unicode[STIX]{x1D70C}}$. For large $Pr$ and small $N_{\unicode[STIX]{x1D70C}}$, the quasi-geostrophic columnar mode loses stability first; while for relatively small $Pr$ a new quasi-geostrophic compressible mode is identified, which becomes unstable first under strong density stratification. The inertial mode can also occur first for relatively small $Pr$ and a certain intensity of density stratification in the parameter range considered. Although the Rayleigh numbers $Ra$ for the onsets of the quasi-geostrophic compressible mode and columnar mode are different by several orders of magnitude, we find that they follow very similar scaling laws with the Taylor number. The critical $Ra$ for convection onset is found to be always positive, in contrast with previous results based on the widely used anelastic model that convection can occur at negative $Ra$. By evaluating the relative magnitude of the time derivative of density perturbation in the continuity equation, we show that the anelastic approximation in the present system cannot be applied in the small-$Ra$ and large-$N_{\unicode[STIX]{x1D70C}}$ regime.

How internal flow states can influence the global transport properties in a turbulent system has always been an intriguing question. Weiss & Ahlers (J. Fluid Mech., this issue, vol. 676, 2011, pp. 5–40) have provided an example by measuring the instantaneous Nusselt number in turbulent Rayleigh-Bénard convection and correlating it to the different modes of large-scale flow.

It has been observed experimentally that when a free shear layer is perturbed by a disturbance consisting of two waves with frequencies and , components with the combination frequencies ( and being integers) develop to a significant level thereby causing flow randomization. This spectral broadening process is investigated theoretically for the case where the frequency difference is small, so that the perturbation can be treated as a modulated wavetrain. A nonlinear evolution system governing the spectral dynamics is derived by using the non-equilibrium nonlinear critical layer approach. The formulation provides an appropriate mathematical description of the physical concepts of sideband instability and amplitude–phase modulation, which were suggested by experimentalists. Numerical solutions of the nonlinear evolution system indicate that the present theory captures measurements and observations rather well.

A novel concept for simulation of turbulent mixing, termed hierarchical parcel swapping (HiPS), was recently proposed. The method involves either a parameterized representation of the turbulent flow or a more self-contained flow simulation. As a step toward turbulent mixing applications, the latter formulation is used for the first numerical demonstration of model performance. Owing to its suitability for this purpose and its role as a canonical benchmark, channel flow is the target application. Despite its idealized representation of this flow, HiPS is shown to capture salient features of the flow with a notable degree of quantitative accuracy. The implications of this finding with regard to flow physics and with regard to the applicability of HiPS to other problems are discussed.

Gaseous flow in a pie-shaped cylinder of infinite length rotating about the apex is considered. The horizontal flow is induced either by the temperature distribution or by the source/sink distribution on the walls θ = constant. It is found that along the vertical walls θ = constant the E½ boundary layer is formed, where E is the Ekman number. Although the equation governing the above boundary layer is very similar to that of the Ekman layer, it is a new type of boundary layer which may be called the buoyancy layer. Along the wall on which r is constant thermal boundary layers very similar to the Stewartson layers are found to be formed. The role of these layers is to mediate the temperature jump. These layers disappear in the incompressible limit.

A simple model is proposed to describe the molecular mixing characteristics of a two-dimensional turbulent jet that is discharged into a confined volume. The model, which is based on similarity and physical considerations of only the large-scale motions, derives the characteristic time for the problem and identifies the regime for which the mixing will be most rapid. Results are reported for experiments where helium and helium/argon mixtures were injected into a cylindrical volume initially containing air. Using an aspirating probe that measured transient helium concentrations in the volume, the mixing time was determined as a function of the size of the confining volume and the injection parameters. The experimental results are in general accord with the model, and validate the use of the model for the determination of the minimum mixing time.

Numerical solutions have been obtained for three-dimensional buoyant flow of air under laminar conditions in a slender, square-section cavity lying parallel to the gradient vector of the temperature field in which it is embedded. Velocity and temperature profiles in the cavity are presented in support of a flow model in which primary and secondary circulation are reconciled by an advective mechanism in the central region. The effect of the temperature gradient, represented by a Rayleigh number, is explored when the cavity is horizontal. The effect of inclining the cavity above and below this horizontal position is also explored. Comparisons are made with related work on cylindrical and two-dimensional rectangular cavities.

In this paper the effect of a small-amplitude progressive wave on the laminar boundary layer on a semi-infinite flat plate, due to a uniform supersonic free-stream flow is considered. The perturbation to the flow caused by the wave divides into two streamwise zones. In the first, relatively close to the leading edge of the plate, on a transverse scale comparable with the boundary-layer thickness, the perturbation flow is described by a form of the unsteady linearized compressible boundary-layer equations. In the free stream, this component of flow is governed by the wave equation, the solution of which provides the outer velocity conditions for the boundary layer. This boundary-layer system is solved numerically, and also the asymptotic structure in the far downstream limit is studied. This reveals a breakdown and a subsequent second streamwise zone, where the flow disturbance is predominantly inviscid. The two zones are shown to match in a proper asymptotic sense.

We consider the asymptotic structure of a steady developed viscous thin film passing the sharp trailing edge of a horizontally aligned flat plate under the weak action of gravity acting vertically and surface tension. The surprisingly rich details of the flow in the immediate vicinity of the trailing edge are elucidated both analytically and numerically. As a central innovation, we demonstrate how streamline curvature serves to regularise the edge singularity apparent on larger scales via generic viscous–inviscid interaction. This is shown to be provoked by weak disturbances of accordingly strong exponential downstream growth, which we trace from the virtual origin of the flow towards the trailing edge. They represent a prototype of the precursor to free interaction in the most general sense, which, interestingly, has not attracted due attention previously. Moreover, we delineate how an increased effect of gravity involves marginally choked flow at the edge.

A new mechanism for the instability of steady mountain waves is found through analysis of the linear stability problem. Steady flow of a hydrostatic stratified fluid is known to be unstable when the streamlines are at, or very close to, overturning. When the topography has multiple peaks, it is shown that this criterion can be superseded by an instability owing to a resonant triad interaction. For flow over two peaks, the threshold heights for instability are roughly half those which produce overturning streamlines. The mechanism behind the instability is the parametric amplification of counter-propagating gravity waves. The resonant nature of the instability is further illustrated by the existence of discrete peak-to-peak separation distances where the growth rate is a maximum.

The flow of a suspension through a bifurcating channel is studied experimentally and by computational methods. The geometry considered is an ‘asymmetric T’, as flow in the entering branch divides to either continue straight or to make a right angle turn. All branches are of the same square cross-section of side length $D$, with inlet and outlet section lengths $L$ yielding $L/D=58$ in the experiments. The suspensions are composed of neutrally buoyant spherical particles in a Newtonian liquid, with mean particle diameters of $d=250~\unicode[STIX]{x03BC}\text{m}$ and $480~\unicode[STIX]{x03BC}\text{m}$ resulting in $d/D\approx 0.1$ to $d/D\approx 0.2$ for $D=2.4~\text{mm}$. The flow rate ratio $\unicode[STIX]{x1D6FD}=Q_{\Vert }/Q_{0}$, defined for the bulk, fluid and particles, is used to characterize the flow behaviour; here $Q_{\Vert }$ and $Q_{0}$ are volumetric flow rates in the straight outlet branch and inlet branch, respectively. The channel Reynolds number $Re=(\unicode[STIX]{x1D70C}DU)/\unicode[STIX]{x1D702}$ was varied over $0<Re<900$, with $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D702}$ the fluid density and viscosity, respectively, and $U$ the mean velocity in the inlet channel; the inlet particle volume fraction was $0.05\leqslant \unicode[STIX]{x1D719}_{0}\leqslant 0.30$. Experimental and numerical results for single-phase Newtonian fluid both show $\unicode[STIX]{x1D6FD}$ increasing with $Re$, implying more material tending toward the straight branch as the inertia of the flow increases. In suspension flow at small $\unicode[STIX]{x1D719}_{0}$, inertial migration of particles in the inlet branch affects the flow rate ratio for particles ($\unicode[STIX]{x1D6FD}_{\mathit{particle}}$) and suspension ($\unicode[STIX]{x1D6FD}_{\mathit{suspension}}$). The flow split for the bulk suspension satisfies $\unicode[STIX]{x1D6FD}>0.5$ for $\unicode[STIX]{x1D719}_{0}<0.16$ while $\unicode[STIX]{x1D719}_{0}=0.16$ crosses from $\unicode[STIX]{x1D6FD}\approx 0.5$ to $\unicode[STIX]{x1D6FD}>0.5$ at $Re\approx 100$. For $\unicode[STIX]{x1D719}_{0}\geqslant 0.2$, $\unicode[STIX]{x1D6FD}<0.5$ at all $Re$ studied. A complex dependence of the mean solid fraction in the downstream branches upon inlet fraction $\unicode[STIX]{x1D719}_{0}$ and $Re$ is observed: for $\unicode[STIX]{x1D719}_{0}<0.1$, the solid fraction in the straight downstream branch initially decreases with $Re$, before increasing to surpass the inlet fraction at large $Re$ ($Re\approx 500$ for $\unicode[STIX]{x1D719}_{0}=0.05$). At $\unicode[STIX]{x1D719}_{0}>0.1$, the solid fraction in the straight branch satisfies $\unicode[STIX]{x1D719}_{\Vert }/\unicode[STIX]{x1D719}_{0}>1$, and this ratio grows with $Re$. Discrete-particle simulations employing immersed boundary and lattice-Boltzmann techniques are used to analyse these phenomena, allowing rationalization of aspects of this complex behaviour as being due to particle migration in the inlet branch.

The dispersion of fluid particles in a turbulent flow is described by transition-probability density functions. A renormalized expansion is made to establish the evolution equations of these functions. The resulting equations are nonlinear integrodifferential equations written in terms of Eulerian velocity-correlation functions. For the dispersion of a single particle, the equation at the zeroth order is the same as the one obtained by Roberts (1961). For the relative dispersion of a pair of particles, the equation is more convenient for applications than those of other theories. With this equation, Richardson's $\frac{4}{3}$-power law for relative diffusion is recovered analytically based upon the Kolmogoroff spectrum and numerically based upon the von Kármán spectrum and a smoothed experimental spectrum.

A numerical study based on the nonlinear Schrödinger equation, as applied to nonlinear resonant standing waves excited directly by a wavemaker in a rectangular tank, is presented. The stationary solutions of the problem serve as a starting point of the investigation. Bifurcations from a single steady state to multiple stationary solutions are obtained for several values of damping coefficients along the tank and at the wavemaker. The stability of the latter solutions is tested. Limit-cycle or fixed-point solutions are obtained. The results of the numerical study are discussed in connection with experimental data. The necessity of incorporation of dissipation at the wavemaker in the theoretical model in order to obtain qualitative agreement with experiment is demonstrated.

The steady, two dimensional, incompressible flow of a viscoelastic fluid past a wedge of 90 degrees coated with the viscoelastic material is studied theoretically using constitutive equations proposed by Oldroyd in 1958. The effect of diffusion of the coating as well as its material properties (viscosity, relaxation time, retardation time, etc.) on the frictional force is investigated.

A boundary-layer analysis is performed on the constitutive equations as well as on the momentum equations. A similarity transformation is found for the set of boundary-layer equations. Series expansion and Laplace's method are employed to obtain the solution in the asymptotic series of gamma functions.

The results obtained show that:

1. The thinner displacement thickness does not necessarily imply a large frictional force for the viscoelastic flow.

2. For a homogeneous viscoelastic flow, the frictional force increases as the degree of dilatancy of the material increases, and decreases with increasing degree of pseudo-plasticity of the material.

3. For a non-homogeneous viscoelastic flow with given material constants, depending on whether the material is pseudoplastic or dilatant and on the ratio of the material concentration of outer flow and the concentration at the body, the frictional coefficient will decrease or increase from that of the homogeneous flow with the concentration at the body as the Schmidt number increases, and will approach a limit when the Schmidt number becomes very large. This limit is the frictional coefficient of the homogeneous flow with the concentration of the outer flow.

Fourier and wavelet transformation techniques are utilized in a complementary manner in order to characterize temporal aspects of the transition of a planar jet shear layer. The subharmonic is found to exhibit an interesting temporal amplitude and phase variation that has not been previously reported. This takes the form of intermittent π-shifts in subharmonic phase between two fixed phase values. These phase jumps are highly correlated with local minima of the subharmonic amplitude. In contrast, the fundamental amplitude and phase show no such behaviour. The temporal phase behaviour of the subharmonic has the effect of intermittently disrupting the phase lock with the fundamental. A dynamical systems model is developed which is based on a classic vortex representation of the shear layer. The Hamiltonian formulation of the problem is shown to provide remarkable agreement with the experimental results. All the essential aspects of the temporal amplitude and phase behaviour of the subharmonic are reproduced by the model including amplitude-dependent effects. The model is also shown to provide a dynamical systems based explanation for time-averaged amplitude and phase behaviour observed in these as well as earlier experiments. The results of experiments involving both bimodal forcing at fundamental and subharmonic frequencies with prescribed initial effective phase angle as well as experiments involving only fundamental excitation over an amplitude range extending two orders of magnitude are presented. The temporal subharmonic amplitude and phase behaviour is observed in bimodal forcing experiments in those regions of the flow characterized by subharmonic mode suppression and vortex tearing events (even if the forcing amplitudes are quite large). In addition, temporal subharmonic amplitude and phase behaviour is the rule in experiments involving low-amplitude forcing of the fundamental and the natural development of the subharmonic.

Turbulence intensity variations in the outer region of turbulent shear flows are considered, in the context of the diagnostic plot first introduced by Alfredsson et al. (Phys. Fluids, vol. 23, 2011, 041702) and for both (smooth and rough) wall-bounded flows and classical free shear flows. With $U$ defined as the mean velocity within the flow, $U_{e}$ as a suitable reference velocity and $u^{\prime }$ as the root mean square of the fluctuating velocity, it is demonstrated that, for wall flows, the attached eddy hypothesis yields a closely linear diagnostic plot ($u^{\prime }/U$ versus $U/U_{e}$) over a certain Reynolds number range, explaining why the relation seems to work well for both boundary layers and channels despite its lack of any physical basis (Castro et al., J. Fluid Mech., vol. 727, 2013, pp. 119–131). It is shown that mixing layers, jets and wakes also exhibit linear variations of $u^{\prime }/U$ versus $U/U_{e}$ over much of the flows (starting roughly from where the turbulence production is a maximum), with slopes of these variations determined by the total mean strain rate, characterised by Townsend’s flow constant $R_{s}$. The diagnostic plot thus has a wider range of applicability than might have been anticipated.

Drop impact on a water surface can be followed by underwater sounds originating not at the drop impact but when the entrained bubbles oscillate. Although the sound mechanism in the regular bubble entrainment region is well-known, there is less knowledge on the impact phenomena in the irregular bubble entrainment region where various situations can exist, such as many types of bubble formation or even no bubble generation under some conditions. In the present study, the aim is to clarify the dynamics of the water surface after the impact of a primary drop, mainly with diameter 5.2, 5.7 and 6.2mm, each of which is accompanied by a single satellite drop. Special attention was paid to the breakup behaviour of the water surface for Froude number Fr < 300. It was found that three underwater sounds were generated for a single drop impact, besides the sound due to impact itself. The first two were audible to the human ear, but the third one was almost inaudible. The first underwater sound resulted from the oscillation of a single air bubble formed as a result of the satellite drop impact on the bottom of the contracting cavity, and the second sound was due to the oscillation of air bubbles generated during the collapse of the water column. The formation of these air bubbles strongly depends on the Froude number, Weber number (or Bond number) and the aspect ratio of the drop at impact, although involving probability characteristics. Furthermore it is suggested that an air bubble entrapped in a water column plays an important role in increasing the probability of contact between the column surface and the curved free surface. A Japanese Suikinkutsu was introduced as an application of drop-impact-induced sounds. Using an open-type Suikinkutsu an additional experiment was carried out with larger drops with average diameters of 6.2, 7.2 and 7.8, mm.

Non-parallelism, i.e. the effect of the slow variation of the boundary-layer flow in the chordwise and spanwise directions, in general produces a higher-order correction to the growth rate of instability modes. Here we investigate stationary crossflow vortices, which arise due to the instability of the three-dimensional boundary layer over a swept wing, focusing on a region near the leading edge where non-parallelism plays a leading-order role in their development. In this regime, the vortices align themselves with the local wall shear at leading order, and so have a marginally separated triple-deck structure, consisting of the inviscid main boundary layer, an upper deck and a viscous sublayer. We find that the streamwise (and spanwise) variations of both the base flow and the modal shape must be accounted for. An explicit expression for the growth rate is derived that shows a neutral point occurs in this regime, downstream of which non-parallelism has a stabilising effect. Stationary crossflow vortices thus have a viscous and non-parallel genesis near the leading edge. If an ‘effective pressure minimum’ occurs within this region then the growth rate becomes unbounded, and so the previous analysis is regularised within a localised region around it. A new instability is identified. The mode maintains its three-tiered structure, but the pressure perturbation now plays a passive role. Downstream, the instability evolves into a Cowley, Hocking & Tutty (Phys. Fluids, vol. 28, 1985, pp. 441–443) instability associated with a critical layer located in the lower deck. Finally, we consider the receptivity of the flow in the non-parallel regime: generation of stationary crossflow modes by arrays of chordwise-localised, spanwise-periodic surface roughness elements. The flow responds differently to different Fourier spectral content of the roughness, giving the lower deck a two-part structure. We find that roughness elements with sharper edges generate stronger modes. For roughness elements of fairly moderate height, the resulting nonlinear forcing leads to the so-called super-linearity of receptivity, namely, the amplitude of the generated crossflow mode deviates from the linear dependence on the roughness height even though the perturbation in the boundary layer remains linear.

A finite-element method has been developed to model inertial waves in a frustum of a cone, since analytical methods have proved inadequate. The governing Poincaré problem is posed as a variational principle and approximate eigensolutions are computed. The numerical results are used to complete the interpretation of the experimental results of Beardsley (1970) for a frustum of a cone. The significant role played by characteristic surfaces partly explains the enigmatic nature of the ill-posed boundary-value problem that describes inertial wave resonances.