Riming, the process whereby ice crystals get coated by impacting supercooled liquid droplets, is one of the dominant processes leading to precipitation in mixed-phase clouds. How a settling crystal collides with very small water droplets has been mostly studied in laminar conditions. The present numerical study aims at providing further insight on how turbulent flow motion affects the riming of ice crystals. We model the crystals as narrow oblate ellipsoids, smaller than the Kolmogorov elementary scale. By neglecting the effect of fluid inertia on the motion of the crystals and droplets, and using direct numerical simulations of the Navier–Stokes equations in a moderately turbulent regime, over a range of kinetic energy dissipation $1~\text{cm}^{2}~\text{s}^{-3}\lesssim \unicode[STIX]{x1D700}\lesssim 256~\text{cm}^{2}~\text{s}^{-3}$ , we determine the collision rate between disk-shaped ice crystals and very small liquid water droplets. Whereas differential settling plays the dominant role in determining the collision rate at small turbulence intensity, the role of turbulence becomes more important at the large values of $\unicode[STIX]{x1D700}$ simulated, an effect that can be partly attributed to the increased role of inertia. We always find that collisions occur with a large probability on the rim of the ellipsoids, a phenomenon that can be explained to a large extent by kinematic considerations. The difference in the settling velocity of crystals and droplets induces a strong asymmetry in the probability of collision between the faces of the ellipsoids. Our results shed light on the physical mechanisms involved in the riming of ice crystals in clouds.

A steady, two-dimensional cellular convection modifies the morphological instability of a binary alloy that undergoes directional solidification. When the convection wavelength is far longer than that of the morphological cells, the behaviour of the moving front is described by a slow, spatial–temporal dynamics obtained through a multiple-scale analysis. The resulting system has a parametric-excitation structure in space, with complex parameters characterizing the interactions between flow, solute diffusion, and rejection. The convection in general stabilizes two-dimensional disturbances, but destabilizes three-dimensional disturbances. When the flow is weak, the morphological instability is incommensurate with the flow wavelength, but as the flow gets stronger, the instability becomes quantized and forced to fit into the flow box. At large flow strength the instability is localized, confined in narrow envelopes. In this case the solutions are discrete eigenstates in an unbounded space. Their stability boundaries and asymptotics are obtained by a WKB analysis. The weakly nonlinear interaction is delivered through the Lyapunov–Schmidt method.

There is a lack of rigour in the usual explanation for the scaling of the vertical velocity of shallow flows based on geometrical arguments and the continuity equation. In this paper we show, by studying shallow axisymmetric swirl flows, that the dynamics of the flow are crucial to determine the proper scaling. In addition, we present two characteristic scaling parameters for such flows: Reδ2 for the radial velocity and Reδ3 for the vertical velocity, where Re is the Reynolds number of the swirl flow and δ=H/L is the flow aspect ratio with H the fluid depth and L a typical horizontal length scale. This scaling contradicts the common assumption that the vertical velocity should scale with the primary motion proportional to the aspect ratio δ. Moreover, if this scaling applies, then the primary flow can be considered as quasi-two-dimensional. Numerical simulations of a decaying Lamb–Oseen vortex served to test the analytical results and to determine their range of validity. It was found that the primary flow can be considered as quasi-two-dimensional only if δRe1/2≲3 and δRe1/3≲1.

We study the response of a steady free-falling liquid curtain perturbed by focused air jets blowing perpendicularly against it. Asymmetric and symmetric perturbations are applied by using either a single pulsed jet or two identical steady jets facing each other. The response strongly depends on the curtain flow rate, and the location and strength of the perturbation. For pulsed asymmetric perturbations of increasing amplitude, sinuous wave, drop ejection, bubble ejection and hole opening are successively observed. For steady symmetric perturbations, a steady hole forms downstream in the wake. For this latter case, we present a model for the curtain thickness and the location of the hole in the wake in terms of the curtain flow rate and the size, flow rate and location of the jets. The adjustable-parameter-free expression we obtain compares favourably to the experiments provided that the perturbation is sufficiently small (jet stagnation pressure smaller than curtain stagnation pressure) and the liquid viscosity is negligible.

Solutions are obtained for the self-similar form of the incompressible boundary-layer equations for all four second-order contributors, i.e. vorticity interaction, displacement speed, longitudinal and transverse curvature. These results are found to contain all previous self-similar solutions as members of the much larger family of solutions presented here. Numerical solutions are presented for a large number of cases, and several closed form solutions, which may have special significance for the separation problem, are also discussed.

The flow of viscous incompressible fluid in a circular cone induced by a non-zero velocity prescribed at the boundary within a ring $0\,{<}\,a_1\,{<}\,r\,{<}\,a_2\,{<}\,\infty$, where $r$ is the distance from the vertex, is considered in the limits of the Stokes approximation. In the spherical coordinate system $(r,\theta,\phi)$ with the origin at the vertex and the axis $\theta\,{=}\,0$ coincident with the axis of the cone the velocity and pressure fields are represented in the form of a Fourier series on the trigonometric system $\cos m \phi$. The solution is constructed for each term by use of the Mellin transform. The contribution of each term of the Fourier expansion to the local velocity field near the vertex is studied. The kinematics of the local flows is illustrated by two examples. The flows are induced by the motion of two and three equally spaced segments, respectively.

This paper is concerned with the rates at which atoms and molecules react in the air that flows over a body flying through the atmosphere at hypersonic speeds. Using air as a working fluid, a series of shock tube experiments were carried out to provide information about these rates. Mach angle measurements were made to determine the state of the gas in three situations of interest.

1. Flow over flat plates was used to determine the state of the gas behind the incident normal shock; temperatures in the gas that passed through the shock varied between 2000 and 6000°:K and densities between standard and 1/80 of standard density.

2. Flow over wedges was employed to decelerate the flow behind the incident shock to a small supersonic Mach number; here temperatures downstream of the oblique shock increased, at most, 2000°:K above the free stream value.

3. A Prandtl-Meyer expansion was used to cool rapidly the dissociated gas, so that the recombination process could be investigated; temperatures dropped at most 2500°:K and the densities varied between standard and 1/200 of the standard value. In some cases, the initial degree of dissociation of air was over 45%.

The results (figure 11) indicate that the dissociation and recombination relaxation times of the chemical species found in air are very fast, when compared to the time it takes a particle of gas to flow either around a blunt body in hypersonic flight or past smtill models in shock tubes. Thus the shock tube is shown to be an instrument capable of supplying air at high temperatures in thermodynamic equilibrium (figure 5).

In the case of a non-melting blunt body of about 1 ft. diameter flying through the atmosphere at hypersonic speeds, the present results imply that, when the gas behind the detached shock is in thermodynamic equilibrium, the flow will also be in equilibrium as it expands around the body, provided its speed is greater than 10 000 ft./sec at altitudes below 180 000 ft. (figure 12).

Series of concentration and velocity patterns are found for the rotating suspension of non-Brownian settling particles in a completely filled horizontal cylinder. Individual flow states, or phases, are studied using both side and cross-sectional imaging to examine the detailed flow structures. The overall steady-state phase diagram of the system is mapped over a wide range of the rotation rate and fluid viscosity. Effects of the particle radius a, volume fraction φ, and cylinder radius R on the transition boundaries are examined. It is found that the phase diagram of the rotating suspensions can be divided into three regions, in which the transition boundaries obey different scaling laws. A theoretical attempt is made to understand the scaling behaviour of the transition boundaries. The theoretical understanding is achieved at three different levels: a general dimensional consideration, a scaling analysis on the continuum equations of motion, and a specific instability calculation for the transition boundary at the centrifugal limit.

When similarity solutions are used to describe convective plumes or thermals, there is always found to be a discrepancy between the work done by buoyancy forces and the kinetic energy of mean motion. It is the main purpose of this paper to set down the ratio of these quantities for a wide variety of forms of buoyant elements and environmental stabilities. For consistency, the remaining fraction of the energy must appear as turbulent kinetic energy and eventually be dissipated, but these processes are not investigated in detail. The results are shown to have some relevance to the problem of convectively driven mixing across a density interface, where the largest scales of motion are dominant, and to the understanding of the transition zone between two self-preserving states of turbulent convection.

The steady mixed convection non-axisymmetric (Homann, Z. Angew. Math. Mech., vol. 16, 1936, pp. 153–164) stagnation-point flow over a vertical flat wall placed in a viscous and incompressible fluid is considered. A similarity solution is derived which involves the dimensionless parameters $\unicode[STIX]{x1D6FE}$, representing the shear-to-strain-rate ratio, and $\unicode[STIX]{x1D706}$, a mixed convection parameter. Forced convection, $\unicode[STIX]{x1D706}=0$, is treated first where solutions additional to those given previously by Weidman (J. Fluid Mech., vol. 702, 2012, pp. 460–469) are found arising from singularities as $\unicode[STIX]{x1D6FE}\rightarrow \pm 1$. Numerical solutions are obtained for representative values of both $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D706}$. Critical values $\unicode[STIX]{x1D706}_{c}$ of $\unicode[STIX]{x1D706}$ are seen in opposing flow and these are treated in detail. Asymptotic results for large $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D6FE}$ are derived.

An Eulerian multifluid model is used to describe the evolution of an electrospray plume and the flow induced in the surrounding gas by the drag of the electrically charged spray droplets in the space between an injection electrode containing the electrospray source and a collector electrode. The spray is driven by the voltage applied between the two electrodes. Numerical computations and order-of-magnitude estimates for a quiescent gas show that the droplets begin to fly back toward the injection electrode at a certain critical value of the flux of droplets in the spray, which depends very much on the electrical conditions at the injection electrode. As the flux is increased toward its critical value, the electric field induced by the charge of the droplets partially balances the field due to the applied voltage in the vicinity of the injection electrode, leading to a spray that rapidly broadens at a distance from its origin of the order of the stopping distance at which the droplets lose their initial momentum and the effect of their inertia becomes negligible. The axial component of the electric field first changes sign in this region, causing the fly back. The flow induced in the gas significantly changes this picture in the conditions of typical experiments. A gas plume is induced by the drag of the droplets whose entrainment makes the radius of the spray away from the injection electrode smaller than in a quiescent gas, and convects the droplets across the region of negative axial electric field that appears around the origin of the spray when the flux of droplets is increased. This suppresses fly back and allows much higher fluxes to be reached than are possible in a quiescent gas. The limit of large droplet-to-gas mass ratio is discussed. Migration of satellite droplets to the shroud of the spray is reproduced by the Eulerian model, but this process is also affected by the motion of the gas. The gas flow preferentially pushes satellite droplets from the shroud to the core of the spray when the effect of the inertia of the droplets becomes negligible, and thus opposes the well-established electrostatic/inertial mechanism of segregation and may end up concentrating satellite droplets in an intermediate radial region of the spray.

The velocity distribution function for the steady shear flow of disks (in two dimensions) and spheres (in three dimensions) in a channel is determined in the limit where the frequency of particle–wall collisions is large compared to particle–particle collisions. An asymptotic analysis is used in the small parameter ε, which is naL in two dimensions and n2L in three dimensions, where n is the number density of particles (per unit area in two dimensions and per unit volume in three dimensions), L is the separation of the walls of the channel and a is the particle diameter. The particle–wall collisions are inelastic, and are described by simple relations which involve coefficients of restitution et and en in the tangential and normal directions, and both elastic and inelastic binary collisions between particles are considered. In the absence of binary collisions between particles, it is found that the particle velocities converge to two constant values (ux, uy) =(±V, 0) after repeated collisions with the wall, where ux and uy are the velocities tangential and normal to the wall, V=(1−et) Vw/(1+et), and Vw and −Vw are the tangential velocities of the walls of the channel. The effect of binary collisions is included using a self-consistent calculation, and the distribution function is determined using the condition that the net collisional flux of particles at any point in velocity space is zero at steady state. Certain approximations are made regarding the velocities of particles undergoing binary collisions in order to obtain analytical results for the distribution function, and these approximations are justified analytically by showing that the error incurred decreases proportional to ε1/2 in the limit ε→0. A numerical calculation of the mean square of the difference between the exact flux and the approximate flux confirms that the error decreases proportional to ε1/2 in the limit ε→0. The moments of the velocity distribution function are evaluated, and it is found that 〈u2x〉→V2, 〈u2y〉 ∼V2ε and − 〈uxuy〉 ∼ V2εlog(ε−1) in the limit ε→0. It is found that the distribution function and the scaling laws for the velocity moments are similar for both two- and three-dimensional systems.

The slope and curvature distributions of wind waves along two principal axes (upwind-downwind and cross-wind) have been measured in a laboratory tank under various wind velocities. In both directions, the slope distributions are very closely Gaussian, and the components of the mean-square water-surface slope vary loga,rithmically with the friction velocity of the wind. As the windvelocityincreases, the ratio of the upwind-downwind and cross-wind components increases and lies between 0.5 and 0.6 at high wind velocities in the gravity-governed regime of wind-wave interaction. The radius of .water-surface curvature, along either direction of measurement, is generally found to be greater at a steeper viewing angle from the normal to the mean water surface. The average radius of curvature of the disturbed surface varies inversely with the friction velocity of the wind. The ratio of the upwind-downwind and cross-wind components of the average radius of curvature is unity at all wind velocities, indicating that the wind-disturbed water surface is isotropic on the smallest scale. Other results show that both the slope and the curvature distributions are asymmetric along the upwind-downwind direction, either because of the presence of parasitic capillaries or because of the occurrence of wave breaking. The results also indicate that even the high frequency portion of the spectrum is saturated locally but the spectrum is not universal, and that the long waves suppress the growth of the nearly saturated ripples.

The effect of gravity modulation on the nonlinear evolution of long-wavelength disturbances at the free surface of a surfactant-covered thin liquid layer is studied. The surfactants, which are assumed to be insoluble, give rise to interfacial concentration gradients and associated Marangoni flow in the underlying liquid film. A coupled system of lubrication-theory-based evolution equations for the film height and surfactant concentration is solved numerically using spectral methods. Previous work using Floquet theory had determined that small-amplitude long-wavelength disturbances are destabilized by gravity modulation in the presence of surfactant; uncontaminated films were found to be linearly stable. Our numerical results indicate that uncontaminated free surfaces are destabilized by nonlinearities and exhibit a harmonic response. The interface exhibits complex dynamics during a forcing cycle, characterized by numerous coalescence events between thickened fluid ridges leading to coarsening. The presence of surfactant-induced Marangoni flow gives rise to a harmonic response, larger scale fluid structures of reduced amplitude, less frequent coalescence events, and less complicated film dynamics.

The measured pressure distributions over the front hemisphere of a hemispherecylinder combination placed in an aligned fields magneto-fluid dynamic flow are presented. Integration of the pressure profiles show that the form or pressure drag of the composite body first increases as the magnetic field strength increases, at a fixed flow velocity, but then decreases as the magnetic field becomes still stronger.

In rough-wall boundary layers, wall-parallel non-homogeneous mean-flow solutions exist that lead to so-called dispersive velocity components and dispersive stresses. They play a significant role in the mean-flow momentum balance near the wall, but typically disappear in the outer layer. A theoretical framework is presented to study the decay of dispersive motions in the outer layer. To this end, the problem is formulated in Fourier space, and a set of governing ordinary differential equations per mode in wavenumber space is derived by linearizing the Reynolds-averaged Navier–Stokes equations around a constant background velocity. With further simplifications, analytically tractable solutions are found consisting of linear combinations of $\exp (-kz)$ and $\exp (-Kz)$, with $z$ the wall distance, $k$ the magnitude of the horizontal wavevector $\boldsymbol{k}$, and where $K(\boldsymbol{k},Re)$ is a function of $\boldsymbol{k}$ and the Reynolds number $Re$. Moreover, for $k\rightarrow \infty$ or $k_{1}\rightarrow 0$ (with $k_{1}$ the stream-wise wavenumber), $K\rightarrow k$ is found, in which case solutions consist of a linear combination of $\exp (-kz)$ and $z\exp (-kz)$, and are independent of the Reynolds number. These analytical relations are compared in the limit of $k_{1}=0$ to the rough boundary layer experiments by Vanderwel & Ganapathisubramani (J. Fluid Mech., vol. 774, 2015, R2) and are in reasonable agreement for $\ell _{k}/\unicode[STIX]{x1D6FF}\leqslant 0.5$, with $\unicode[STIX]{x1D6FF}$ the boundary-layer thickness and $\ell _{k}=2\unicode[STIX]{x03C0}/k$.

Results of an experimental study on the solidification and convection of a ternary solution of H$_2$O–CuSO$_4$–Na$_2$SO$_4$ when cooled from a vertical side boundary are presented. Cooling and crystallization at a sidewall result in the formation of horizontal gradients of composition and temperature which are restricted to thin boundary layers adjacent to the solid. The dynamics of the buoyancy-induced flows in the compositional and thermal boundary layers is shown to depend critically on the morphology of the growing solid as well as on the density changes in the fluid released on solidification. In a ternary system, the possibility of a density reversal in the residual fluid as a second component solidifies leads to a wide range of possible behaviours. Three broad regimes are identified and the flows which arise in two of these are studied in detail; the behaviour in the third regime can be inferred from these results. When the thermal and compositional boundary layers are both heavy and flow to the base of the tank (regime I), the evolution of the interior fluid is shown to be initially quantitatively similar to that arising from thermal convection only. Slow compositional changes in the interior only later result in a number of double-diffusive effects. When the compositional and thermal buoyancies in the boundary layers are opposed (regimes II and III), a unidirectional downflow, counterflow and upflow are all observed at various stages of the experiment. The onset and duration of these different flows is shown to depend on the ratio of compositional to thermal buoyancies, the relative fluxes in the boundary layers, and the crystal morphology. The effect of surface roughness for the particular ternary system investigated is quantified by measuring its effect on the fluxes in the boundary layers, which allows the modification of existing theories to predict the conditions under which the different boundary layer dynamics arise.

A series of experiments has been undertaken in which three major properties of a surface tension-gravity wave system have been examined. The results of these experiments have been compared with existing theories. The three properties are: (i) viscous decay in the absence of mean flow, the relevant theory being given in Lamb (1932, § 348); (ii) propagation velocities in the presence and absence of flow (Lamb 1932, § 267), and (iii) the change of wave energy on crossing a stable Couette shear flow. The measurements in the last case were compared with two theories, one obtained from the Navier–Stokes equations including terms up to second order in wave slope, the other, following previous authors, obtained on the assumption that any direct interaction of the waves with the shear flow is negligible. According to the theory obtained from the Navier–Stokes equations, the divergence of surface wave energy is equal to the rate of change of wave energy due to the interaction between the mean flow and the wave system plus the rate of change of wave energy due to viscous decay.

An optical system was used to measure the maximum wave slopes, the wave-numbers and the shear velocities.

Results indicate that an anomalous region of wave properties exists for wave-numbers near 2.7 cm−1. For a set of data in which the wave-numbers were always less than 1.8 cm−1, it was found that the viscous decay rate and the propagation laws agree with theory to within the experimental error, and the interaction measurements fit the theory with the non-linear term included rather than the traditional theories.