The process of mixing at the top of a turbulent cloud layer contains a new feature which has not been considered in previous studies of mixing; evaporation of liquid water can cause density changes which may affect the dynamics. A model experiment has been devised to study this problem, using liquids whose density behaviour is non-linear to simulate evaporation.

The existence of a moist, stable, turbulent region above cloudtop can be explained using the laboratory results, which suggest that this region can be regarded dynamically as part of the cloud. Comparison of the rates of mixing in the model experiments with and without ‘evaporation’ suggests that evaporation could cause a small decrease in the mixing rate for a given density difference, but the change would be negligible in practice. This result also sheds some light on the mechanism of mixing, in both the linear and non-linear cases.

We consider the dynamics of an axisymmetric, partially wetting droplet of a one-component volatile liquid. The droplet is supported on a smooth superheated substrate and evaporates into a pure vapour atmosphere. In this process, we take the liquid properties to be constant and assume that the vapour phase has poor thermal conductivity and small dynamic viscosity so that we may decouple its dynamics from the dynamics of the liquid phase. This leads to a so-called ‘one-sided’ lubrication-type model for the evolution of the droplet thickness, which accounts for the effects of evaporation, capillarity, gravity, slip and kinetic resistance to evaporation. By asymptotically matching the flow near the contact line region and the bulk of the droplet in the limit of a small slip length and commensurably small evaporation and kinetic resistance effects, we obtain coupled evolution equations for the droplet radius and volume. The predictions of our asymptotic analysis, which also include an estimate of the evaporation time, are found to be in excellent agreement with numerical simulations of the governing lubrication model for a broad range of parameter regimes.

We investigate the generation conditions of a high-Reynolds-number swirling jet experiment, based on a rotating honeycomb device and using a final contraction. Using hot-wire measurements, we first show that for high swirl levels, the flow at the jet exhaust may exhibit fully developed turbulence in the whole plane. By analysing the fluctuation levels obtained for several values of the contraction ratio, ranging from 4 to 18.4, we prove that this turbulence does not result from upstream-propagating disturbances initiated in the jet, but originates in the pipe flow upstream of the exit plane. Using stereo particle image velocimetry, we then measure the flow in the constant-cross-section pipe located between the rotating honeycomb outlet and the contraction. This investigation is supplemented with simplified numerical simulations of the mean flow. The pipe flow dynamics is found to result from the interplay of a rich variety of complex phenomena, which are independent of the contraction ratio in the range considered here. In the near-wall region, centrifugal instability occurs in the form of intermittent azimuthal vortices, starting from moderate swirl levels and persisting for all higher levels. As the flow exiting from the honeycomb has a swirl level high enough to reach the subcritical regime, a complex mean flow organization is observed, dominated by the presence of large-amplitude axisymmetric Kelvin wave trains. Gradients in the resulting flow lead to the appearance of generalized centrifugal instabilities in an annular region in the rotational core, starting in the early subcritical regime. As the swirl level is further increased, large-scale, high-amplitude axisymmetric and simple spiral perturbations add to the global dynamics, leading to an overall very high fluctuation level. Consideration of the turbulent spectra in the jet exit planes suggests that the simple spiral coherent structure could be the resonant response of the flow to the periodic excitation by the rotating honeycomb. Overall, the study illustrates why a swirling jet experiment should exclude the use of a final contraction in order to guarantee smooth flow conditions in the exit at high swirl.

The unsteady response of a water free surface to a localized pressure source moving at constant speed $U$ in the range $0.95c_{min}\lesssim U\leqslant 1.02c_{min}$, where $c_{min}$ is the minimum phase speed of linear gravity–capillary waves in deep water, is investigated through experiments and numerical simulations. This unsteady response state, which consists of a V-shaped pattern behind the source, and features periodic shedding of pairs of depressions from the tips of the V, was first observed qualitatively by Diorio et al. (Phys. Rev. Lett., vol. 103, 2009, 214502) and called state III. In the present investigation, cinematic shadowgraph and refraction-based techniques are utilized to measure the temporal evolution of the free-surface deformation pattern downstream of the source as it moves along a towing tank, while numerical simulations of the model equation described by Cho et al. (J. Fluid Mech., vol. 672, 2011, pp. 288–306) are used to extend the experimental results over longer times than are possible in the experiments. From the experiments, it is found that the speed–amplitude characteristics and the shape of the depressions are nearly the same as those of the freely propagating gravity–capillary lumps of inviscid potential theory. The decay rate of the depressions is measured from their height–time characteristics, which are well fitted by an exponential decay law with an order one decay constant. It is found that the shedding period of the depression pairs decreases with increasing source strength and speed. As the source speed approaches $c_{min}$, this period tends to approximately 1 s for all source magnitudes. At the low-speed boundary of state III, a new response with unsteady asymmetric shedding of depressions is found. This response is also predicted by the model equation.

An experimental study has been made of the flow produced by an internal heat source as it moves around a fixed annulus of fluid. The mean surface flow relative to the annulus is found to be in the opposite direction to the movement of the heat source. The variation in its magnitude with changes in the external parameters is investigated. Thermocouple measurements and streak photography are used to examine the interior temperature and flow structure.

We conduct laboratory experiments on the time evolution of an ice layer cooled from below and subjected to a turbulent shear flow of warm water from above. Our study is motivated by observations of warm water intrusion into the ocean cavity under Antarctic ice shelves, accelerating the melting of their basal surfaces. The strength of the applied turbulent shear flow in our experiments is represented in terms of its Reynolds number $Re$, which is varied over the range $2.0\times 10^{3}\leqslant Re\leqslant 1.0\times 10^{4}$. Depending on the water temperature, partial transient melting of the ice occurs at the lower end of this range of $Re$ and complete transient melting of the ice occurs at the higher end. Following these episodes of transient melting, the ice reforms at a rate that is independent of $Re$. We fit our experimental measurements of ice thickness and temperature to a one-dimensional model for the evolution of the ice thickness in which the turbulent heat transfer is parameterized in terms of the friction velocity of the shear flow. Applying our model to field measurements at a site under the Antarctic Pine Island Glacier ice shelf yields a predicted melt rate that exceeds present-day observations.

We derive and test a new heuristic theory for third-order structure functions that resolves the forcing scale in the scenario of simultaneous spectral energy transfer to both small and large scales, which can occur naturally, for example, in rotating stratified turbulence or magnetohydrodynamical (MHD) turbulence. The theory has three parameters – namely the upscale/downscale energy transfer rates and the forcing scale – and it includes the classic inertial-range theories as local limits. When applied to measured data, our global-in-scale theory can deduce the energy transfer rates using the full range of data, therefore it has broader applications compared with the local theories, especially in situations where the data is imperfect. In addition, because of the resolution of forcing scales, the new theory can detect the scales of energy input, which was impossible before. We test our new theory with a two-dimensional simulation of MHD turbulence.

Motivated by the recent paper of Hills & Moffatt (2000), we investigate the Stokes flow in a trihedral corner formed by three mutually orthogonal planes, induced by a non-zero velocity distribution over one of the walls of the corner. It is shown that the local behaviour of the velocity field near the edges of the corner, where a discontinuity of the boundary velocity is assumed, coincides with the Goodier–Taylor solution for a two-dimensional wedge. Analysis of the streamline patterns confirms the existence of eddies near the stationary edge in the flow, induced either by uniform translation of one of the walls of the corner in the direction perpendicular to its bisectrix or by uniform rotation of a side about the vertex of the corner. These flows are shown to be quasi-two-dimensional. If the wall rotates about a centre displaced from the vertex, the induced flow is essentially three-dimensional. In the antisymmetric velocity field, a stagnation line appears composed of stagnation points of different types. Otherwise the three-dimensionality manifests itself in a non-closed spiral shape of the streamlines.

Using an observation by Lamb, namely that continuum sound wave propagation in sufficiently narrow channels is quasi-steady and isothermal, we obtain analytical predictions for the propagation of sound waves at small scales under non-continuum transport. We also extend Lamb's approach to include the effects of inertia and heat conduction for wave propagation at larger characterisitc scales descibed by continuum transport (no-slip and slip-flow regimes). Our theoretical predictions are compared to molecular-based direct Monte Carlo solutions of the Boltzmann equation. Very good agreement is found between theory and numerical solutions.

Observations show that the near-sill dynamics of dense abyssal overflows is variable and is governed, to a significant extent, by a balance between rotation, bottom friction and downslope acceleration due to gravity. Numerical simulations indicate that the near-sill downslope velocities are comparable to the phase/group velocities of long internal gravity waves. This suggests the possibility that overflows can become supercritical and destabilized by bottom friction. A theory is presented for the frictional destabilization of rotating abyssal overflows and the accompanying baroclinic coupling with the overlying ocean. This mode of transition allows for the formation of downslope and alongslope propagating periodic bores or pulses in the overflow and the generation of amplifying long internal gravity waves in the overlying ocean, and may help to explain aspects of the observed variability which seem unrelated to purely inertial baroclinic instability.

We consider the non-stationary dynamics of an intense localized vortex on a β-plane using a shallow-water model. An asymptotic theory for a vortex with piecewise-continuous potential vorticity is developed assuming the Rossby number to be small and the free surface elevation to be small but finite. Analogously to the well-known quasi-geostrophic model, the vortex translation is produced by a secondary dipole circulation (β-gyres) developed in the vortex vicinity and consisting of two parts. The first part (geostrophic β-gyres) coincides with the β-gyres in the geostrophic model, and the second (ageostrophic β-gyres) is due to ageostrophic terms in the governing equations. The time evolution of the ageostrophic β-gyres consists of fast and slow stages. During the fast stage the radiation of inertia–gravity waves results in the rapid development of the β-gyres from zero to a dipole field independent of the fast time variable. Correspondingly, the vortex accelerates practically instantaneously (compared to the typical swirling time) to some finite value of the translation speed. At the next slow stage the inertia–gravity wave radiation is insignificant and the β-gyres evolve with the typical swirling time. The total zonal translation speed induced by the geostrophic and ageostrophic β-gyres tends with increasing time to the speed of a steadily translating monopole exceeding (not exceeding) the drift velocity of Rossby waves for anticyclones (cyclones). This cyclone/anticyclone asymmetry generalizes the well-known finding about the greater longevity of anticyclones compared to cyclones to the case of non-stationary evolving monopoles. The influence of inertia–gravity waves upon the vortex evolution is analysed. The main role of these waves is to provide a ‘fast’ adjustment to the ‘slow’ vortex evolution. The energy of inertia–gravity waves is negligible compare to the energy of the geostrophic β-gyres. Yet another feature of the ageostrophic vortex evolution is that the area of the potential vorticity patch changes in the course of time, the cyclonic patch contracting and the anticyclonic one expanding.

The dispersion relations for infinitesimal internal gravity waves (A) and axisymmetric waves in swirling streams (B) are considered. In both cases the mainstream may be sheared and density stratified in the transverse (vertical in case A, radial in case B) direction. The following results are proved for either case: If the maximum speed Wmax (or minimum speed Wmin) (in a meridian plane in case B) of the mainstream occurs at an interior point in the fluid, then the phase speed of any mode takes all values from the Wmax (or Wmin) to +∞ ( —∞) as the overall Richardson number λ2 varies from 0 to ∞. If Wmax(Wmin) is attained at a boundary point with finite rate of strain, there is a positive non-zero critical Richardson number below which one or both branches of the dispersion relation terminate. These results employ variational methods and correct erroneous results concerning problem B stated in Chandrasekhar's treatise on hydrodynamic stability. Furthermore, bounds are given on the group velocity for both branches of the dispersion relation. From these bounds it is shown that in the absence of reversals of the mainstream (Wmin > 0) upstream propagation of wave energy is impossible whenever upstream propagation of constant phase surfaces is impossible.

An experimental study was conducted in a turbulent spray flame in which droplets were produced ultrasonically at low velocity relative to the host gas. In this fashion, injector-specific effects on the two-phase flow were minimized and a scenario generally characteristic of the far field of practical spray systems could be simulated. Close to the burner exit, the spray flame appeared as a dense column of drops burning with an envelope flame. Further downstream, it opened up slowly in the radial direction and developed a turbulent ‘brush’ appearance. Measurements of the size, velocity and concentration of the droplets, and of gas-phase velocity and temperature were made by combining a Phase-Doppler interferometric technique with Stokes/anti-Stokes Raman thermometry. The experimental data were used to derive scaling and self-similarity for the Reynolds-averaged continuity and momentum equations using suitable transformations.

Results showed three distinct regions, on the basis of the behaviour of the gas axial velocity in the spray flame. In the lower part of the flame, the gas momentum increased because of vaporization. In the intermediate region of the spray flame, the axial velocity decayed along the centreline as an inverse power of the distance from the virtual origin, with exponents smaller than unity. In the upper part of the spray flame, the flow field recovered the axial velocity decay that is typical of incompressible jets, namely as an inverse of the axial distance. Self-similar behaviour held for the axial velocity throughout the intermediate region. The vapour source term in the gas continuity equation scaled approximately as an inverse power of axial distance, and exhibited self-similarity throughout the spray flame. As a result, a simple model of the Reynolds stress term could be formulated, in which two competing contributions appear: one, that is due to turbulent transport, tends to increase the value of the velocity correlation; another, that is due to the vaporization term, tends to reduce the value of the velocity correlation and can be construed as a vaporization-induced tendency towards relaminarization. The first term is modelled by a classic gradient-transport approach yielding an empirical mixing length relating the velocity correlation to the average velocity gradient. Model and experiments are found to be in good agreement, especially sufficiently far from the injector, where one-way coupling between the two phases holds.

A fluid layer that lines the inner surface of a circular tube has motion induced by axial surface-tension gradients. The lubrication equations for the system are analysed and it is found that even for thin layers the motions differ markedly from those in planar layers. The planar case serves as a class of outer solutions. These approximate solutions are modified by a boundary-layer correction where the mean surface tension is important.

To investigate turbulent properties of the tidally induced bottom boundary layer (TBBL) in a rotating frame, we performed three-dimensional numerical experiments under unstratified conditions, varying the temporal Rossby number Rot = |σ*/f*|, where σ* and f* are the tidal frequency and the Coriolis parameter, respectively. The vertical profiles of the time-averaged currents and stresses showed good similarity and coincided well with the turbulent Ekman layer, when they were normalized by the modified ‘outer’ scales, the frictional velocity u*τ, T* = 1/|f* + σ*| and δ* = u*τ/|f* + σ*| for the velocity, time and length scales (σ* is positive when the tidal ellipse rotates anticlockwise). This means that the similarity in turbulent statistics is universally applicable to the TBBL in the world's ocean except near the equator. Although strong inertial waves contaminated the perturbation field when Rot ~ 1 and masked the similarity, the apparent diffusivity κ*ap estimated by tracer experiments again showed similarity, since the inertial waves did not affect the mixing process in the present experiments. Thus, κ*ap can be represented in terms of the three external parameters: the latitude (f*), the tidal frequency (σ*) and the tidal amplitude (u*τ). The obtained scaling of u*τ δ* = u*τ2/|f*+σ*| for κ*ap suggests that effective mixing may occur when Rot ~ 1, i.e. near the critical latitude.

In the low-Froude-number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Steady linearized flow past a submerged source is considered, and exponential asymptotic methods are applied to determine the behaviour of the free-surface gravity waves. The free surface is found to contain longitudinal and transverse waves that switch on rapidly across curves known as Stokes lines on the free surface. The longitudinal waves are present everywhere downstream of the singularity, while the transverse waves are restricted to two downstream wedges. As the depth of the source approaches the surface, the familiar Kelvin-wedge wave behaviour is recovered.

We present experimental results on the development of gravity currents moving onto sloping boundaries with slope angles $\unicode[STIX]{x1D703}=7^{\circ }$, $10^{\circ }$ and $15^{\circ }$. Different regimes of flow development are observed depending on the slope angle and on the initial velocity and density profiles, characterized by the Richardson number $J_{i}=\unicode[STIX]{x1D6FF}_{i}{g_{0}}^{\prime }/\unicode[STIX]{x0394}u_{i}^{2}$, where $\unicode[STIX]{x1D6FF}_{i}$, $\unicode[STIX]{x0394}u_{i}$ and $g_{0}^{\prime }$ are, respectively, the velocity interface thickness, the maximum velocity difference and reduced gravity at the beginning of the slope. For $J_{i}>0.7$ and the larger slope angle, the flow strongly accelerates, reaches a maximum at the beginning of the Kelvin–Helmholtz instability, then decelerates and re-accelerates again. For $0.3<J_{i}<0.6$, instability occurs earlier and velocity oscillations are less. When $J_{i}\leqslant 0.3$ the increase in velocity is smooth. The magnitude of velocity oscillation depends on the combined effect of $J_{i}$ and slope angle, expressed by an overall acceleration parameter $\overline{T_{a}}=(\unicode[STIX]{x1D6FF}_{i}/U_{i})((U_{c}-U_{i})/x_{c})$, which, to first order, is given by $J_{i}\sin \unicode[STIX]{x1D703}$, where $U_{c}$ and $x_{c}$ are, respectively, the velocity and position at instability onset. The velocity increases smoothly up to an equilibrium state when $\overline{T_{a}}\leqslant 0.06$ and exhibits an irregular behaviour at larger values of $\overline{T_{a}}$. The critical Richardson number $J_{c}$ decreases with increasing $J_{i}$ (increasing $\unicode[STIX]{x1D6FF}_{i}/h_{i}$) which is due to wall effects and $\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}\neq 1$. After the beginning of Kelvin–Helmholtz instability, entrainment rates are close to those of a mixing layer, decreasing to values of a gravity current after the mixing layer reaches the boundary. It is shown here that the interfacial instability during current development affects the bottom shear stress which can reach values of $c_{D}\approx 0.03$ regardless of initial conditions. By solving numerically the depth integrated governing equations, the gravity flow velocity, depth and buoyant acceleration in the flow direction can be well predicted for all the performed experiments over the full measurement domain. The numerical results for the experiments with $J_{i}>0.3$ predict that the current requires a distance of at least $x_{n}\approx 40h_{i}$ to reach a normal state of constant velocity, which is much larger than the distance $x_{n}\approx 10h_{i}$ required in the case of a current with $J_{i}\leqslant 0.3$ that is commonly assumed for downslope currents.

Large internal solitary waves with subsurface cores have recently been observed in the South China Sea. Here fully nonlinear solutions of the Dubreil–Jacotin–Long equation are used to study the conditions under which such cores exist. We find that the location of the cores, either at the surface or below the surface, is largely determined by the sign of the vorticity of the near-surface background current. The results of a numerical simulation of a two-dimensional shoaling internal solitary wave are presented which illustrate the formation of a subsurface core.