The effects of chemical heat release on the large-scale structure in a chemically reacting, turbulent mixing layer are investigated using direct numerical simulations. Three-dimensional, time-dependent simulations are performed for a binary, single-step chemical reaction occurring across a temporally developing turbulent mixing layer. It is found that moderate heat release slows the development of the large-scale structures and shifts their wavelengths to larger scales. The resulting entrainment of reactants is reduced, decreasing the overall chemical product formation rate. The simulation results are interpreted in terms of turbulence energetics, vorticity dynamics, and stability theory. The baroclinic torque and thermal expansion in the mixing layer produce changes in the flame vortex structure that result in more diffuse vortices than in the constant-density case, resulting in lower rotation rates of the large-scale structures. Previously unexplained anomalies observed in the mean velocity profiles of reacting jets and mixing layers are shown to result from vorticity generation by baroclinic torques.

The previous work of Thomas (1956) on the turbulent convection over a single, heated, horizontal surface has been extended using improved methods of analysis of the temperature fluctuations, and it has been possible to measure the distributions of mean temperature, the mean squares of the temperature fluctuation, the temperature gradient and the rate-of-change of temperature, and the statistical distributions of these quantities. These measurements were made for three different values of the convective heat flux, and the results are consistent with the dimensional consequences of assuming that the convection near the surface is independent of the distant boundaries and determined by the heat flux and the viscosity and conductivity of the fluid.

The most striking feature of the observations is that the fluctuations of temperature, temperature gradient, and rate-of-change of temperature, all show periods of activity, characterized by large fluctuations, alternating with periods of quiescence with comparatively small ones. Both the proportion and frequency of occurrence of the active periods decrease with increasing distance from the surface and they probably occur when rising columns of hot air pass through the point of observation. The quiescent periods occur when the point of observation lies outside the columns, and analysis of the statistical distributions of the various fluctuations shows that, during these periods, they are nearly independent of height. It is concluded that the quiescent fluctuations are typical of the turbulent convection far from the surface while the active fluctuations are the manifesta- tion of the convective processes arising near the rigid boundary. These processes may be described as the detachment of columns of hot air from the edge of the conduction layer and the erosion of these rising columns by contact with the surrounding air which is in vigorous turbulent motion. Since the variations of the intensities with height is dominated by the contributions of the active periods, it is not surprising that no agreement is found with the predictions of the similarity theory which assumes the convection to be independent of the conduction layer at a sufficient distance from it. The Malkus theory of turbulence, which empha- sizes dependence on the conduction layer, is in qualitative agreement with this inferred mechanism of the convection and is in quantitative agreement with the observed distribution of mean temperature. A brief discussion is given of the effect of a horizontal shearing motion on the convection and of the relation of these measurements to measurements of the temperature distribution in the earth's boundary layer with upward flux of total heat.

In this paper, we continue to study the mechanisms of the receptivity of the supersonic boundary layer to free-stream disturbances by using both direct numerical simulation and linear stability theory. Specifically, the receptivity of a Mach 4.5 flow over a flat plate to free-stream fast acoustic waves is studied. The receptivity to free-stream slow acoustic waves, entropy waves and vorticity waves will be studied in the future. The oblique shock wave induced by the boundary-layer displacement plays an important role in the receptivity because the free-stream disturbance waves first pass through the shock before entering the boundary layer. A high-order shock-fitting scheme is used in the numerical simulations in order to account for the effects of interactions between free-stream disturbance waves and the oblique shock wave. The results show that the receptivity of the flat-plate boundary layer to free-stream fast acoustic waves leads to the excitation of both Mack modes and a family of stable modes, i.e. mode I, mode II, etc. It is found that the forcing fast acoustic waves do not interact directly with the unstable Mack modes. Instead, the stable mode I waves play an important role in the receptivity process because they interact with both the forcing acoustic waves and the unstable Mack-mode waves. Through the interactions, the stable mode I waves transfer wave energy from the forcing fast acoustic waves to the second Mack-mode waves. The effects of incident wave angles, forcing wave frequencies, and wall temperature perturbation conditions on the receptivity are studied. The results show that the receptivity mechanisms of the second mode are very different from those of modes I and II, which leads to very different receptivity properties of these discrete wave modes to free-stream fast acoustic waves with different incident wave angles, frequencies, and different wall boundary conditions. The maximum receptivities of the second mode, mode I and mode II to planar free-stream fast acoustic waves are obtained when incident wave angles approximately equal 26°, 45°, and 18°, respectively. The results of receptivity to a beam of free-stream fast acoustic waves show that the leading edge is one of the most efficient regions for receptivity.

New measurements of the Nusselt number have been made in turbulent thermal convection confined in a cylindrical container of aspect ratio unity. The apparatus is essentially the same as that used by Niemela et al. (2000), except that the height was halved. The measurement techniques were also identical but the mean temperature of the flow was held fixed for all Rayleigh numbers. The highest Rayleigh number was $2 \times 10^{15}$. Together with existing data, the new measurements are analysed with the purpose of understanding the relation between the Nusselt number and the Rayleigh number, when the latter is large. In particular, the roles played by Prandtl number, aspect ratio, mean wind, boundary layers, sidewalls, and non-Boussinesq effects are discussed. Nusselt numbers, measured at the highest Rayleigh numbers for which Boussinesq conditions hold and sidewall forcing is negligible, are shown to vary approximately as a 1/3-power of the Rayleigh number. Much of the complexity in interpreting experimental data appears to arise from aspects of the mean flow, including complex coupling of its dynamics to sidewall boundary conditions of the container. Despite the obvious practical difficulties, we conclude that the next generation of experiments will be considerably more useful if they focus on large aspect ratios.

The interaction between a fixed particle and decaying homogeneous isotropic turbulence is studied numerically using an overset grid that provides resolution of all scales of fluid motion. A description of the numerical technique and validation of the solution procedure are presented. An ensemble of 64 simulations with the particle in different regions of the flow is computed. The particle diameter in the simulations is approximately twice the size of the unladen Kolmogorov length scale, and the maximum value of the particle Reynolds numbers due to the turbulent fluctuations is close to 20. Ensemble averages of quantities from the numerical solutions are used to investigate the turbulence modification and the fluid forces on the particle. Volume-averaged profiles of the turbulent kinetic energy and dissipation rate from the overset grid simulations reveal that the displacement of fluid by the particle and the formation of the boundary layer at the particle surface lead to turbulence modification in a local region. Time histories of the force applied to the particle from each overset grid simulation are compared to those predicted by a particle equation of motion. The particle equation of motion is shown to underpredict the root mean square (RMS) force applied to the particle by the turbulence. RMS errors between the forces from the overset grid simulation and those predicted by the particle equation of motion are shown to be between 15% and 30% of the RMS force on the particle. The steady viscous drag force is shown to be the dominant term in the particle equation of motion while the history integral term is negligible.

Substantial discrepancy in the conditions for attainment of different closure modes of a ventilated supercavity has existed widely in the published literature. In this study, supercavity closure is investigated with an objective to understand the physical mechanisms determining closure formation and transition between different closure modes and to reconcile the observations from prior studies under various flow settings. The experiments are conducted in a closed-wall recirculating water tunnel to image ventilated supercavity closure using high speed and high-resolution photography and simultaneously measure pressure inside the cavity. The flow conditions are varied systematically to cover a broad range of water velocity, ventilation flow rate and cavitator size, which correspond to different Froude numbers, air entrainment coefficients and blockage ratios, respectively. In addition to the classical closure modes reported in the literature (e.g. re-entrant jet, twin vortex, quad vortex, etc.), the study has revealed a number of new closure modes that occur during the transition between classical modes, or under very specific flow conditions. Closure maps are constructed to depict the flow regimes, i.e. the range of Froude number and air entrainment coefficient, for various closure modes at different blockage ratios. From the closure map at each blockage ratio, a critical ventilation flow rate, below which the supercavity collapses into foamy cavity upon reduction of Froude number, is identified. The air entrainment coefficients corresponding to such critical ventilation rate are found to be independent of blockage ratio. It has been observed that in the process of generating a supercavity by increasing ventilation flow rate, the cavitation number gradually reduces to a minimum value and stays fixed upon further increments in the ventilation rate. Once a supercavity is formed, the ventilation rate can be decreased to a much lower value with no change in cavitation number while still maintaining a supercavity. This process is accompanied by a change in closure modes, which generally goes from twin vortex, to quad vortex, and then to re-entrant jet. In addition, the blockage effect is shown to play an important role in promoting the occurrence of twin-vortex closure modes. Subsequently, a physical framework governing the variation of different closure modes is proposed, and is used to explain mode transition upon the change of flow conditions, including the blockage effect. This framework is further extended to shed light on the occurrence of closure modes for ventilated supercavitation experiments across different types of flow facilities, the natural supercavity closure and the pulsating supercavity reported in the literature. Finally, in combination with a recent numerical study, our research discusses the role of the internal flow physics on the observed features during supercavity formation and closure-mode transition, paving the way for future investigations in this direction.

We numerically characterize the temporal regimes for solutal convection from almost first contact to high dissolved solute concentration in a two-dimensional ideal porous layer for Rayleigh numbers $\mathcal{R}$ between $100$ and $5\times 10^4$. The lower boundary is impenetrable. The upper boundary is saturated with dissolved solute and either impermeable or partially permeable to fluid flow. In the impermeable case, initially there is pure diffusion of solute away from the upper boundary, followed by the birth and growth of convective fingers. Eventually fingers interact and merge, generating complex downwelling plumes. Once the inter-plume spacing is sufficient, small protoplumes reinitiate on the boundary layer and are swept into the primary plumes. The flow is now in a universal regime characterized by a constant (dimensionless) dissolution flux $F=0.017$ (the rate at which solute dissolves from the upper boundary). The horizontally averaged concentration profile stretches as a simple self-similar wedge beneath a diffusive horizontal boundary layer. Throughout, the plume width broadens proportionally to $\sqrt{t}$, where $t$ is (dimensionless) time. The above behaviour is parameter independent; the Rayleigh number only controls when transition occurs to a final $\mathcal{R}$-dependent shut-down regime. For the constant-flux and shut-down regimes, we rigourously derive upscaled equations connecting the horizontally averaged concentration, vertical advective flux and plume widths. These are partially complete; a universal expression for the plume width remains elusive. We complement these governing equations with phenomenological boundary conditions based on a marginally stable diffusive boundary layer at the top and zero advective flux at the bottom. Making appropriate approximations in each regime, we find good agreement between predictions from this model and simulated results for both solutal and thermal convection. In the partially permeable upper boundary case, fluid from the convecting layer can penetrate an overlying separate-phase-solute bearing layer where it immediately saturates. The regime diagram remains almost the same as for the impermeable case, but the dissolution flux is significantly augmented. Our work is motivated by dissolution of carbon dioxide relevant to geological storage, and we conclude with a simple flux parameterization for inclusion in gravity current models and suggest that the upscaled equations could lay the foundation for accurate inclusion of dissolution in reservoir simulators.

An expansion scheme is developed to describe the steady axisymmetric flow of a thin tubular liquid film of varying radius; the necessary small parameter is provided by the ratio between the characteristic film thickness and the characteristic tube radius. The co-ordinate system used is an orthogonal one based on the fluid interface and the fluid streamlines. The differential equations that arise thus treat the metric as an unknown set of variables. The method is restricted to situations dominated by viscous forces. Reference is made to numerical solutions that have been obtained in connexion with an industrial polymer-film-blowing process.

A calculation is given of the velocity which a cloud of identical gas bubbles acquires when the liquid in which the cloud is immersed is impulsively accelerated. From the results an expression follows for the effective virtual mass of a bubble in a gas-bubble/liquid mixture. Further consideration is given to that part of the momentum flux in the mixture associated with relative motion between liquid and bubbles. An expression for this quantity is derived which appears to differ from the one used in practice. It is shown that qualitative support for the expression obtained here is provided by experimental observations reported in the literature.

The current study investigates turbulence modification by particles in a backward-facing step flow with a fully developed channel flow inlet. This flow provides a range of flow regimes in which to compare turbulence modification under the same experimental conditions. Gas-phase velocities in the presence of 3–40% mass loadings of three different particle classes (90 and 150 μm diameter glass and 70 μm diameter copper spheres) were measured. Attenuation of the streamwise fluid turbulence of up to 35% was observed in the channel-flow extension region of the flow for a 40% mass loading of the largest particles. The level of attenuation decreased with decreasing particle Stokes number, particle Reynolds number and mass loading. No modification of the turbulence was found in the separated shear layer or in the redevelopment region behind the step, although there were significant particle loadings in these regions.

Exact solutions are presented for the three-dimensional creeping motion of a sphere of arbitrary size and position between two plane parallel walls for the following conditions: (a) pure translation parallel to two stationary walls, (b) pure rotation about an axis parallel to the walls, (c) Couette flow past a rigidly held sphere induced by the motion of one of the boundaries and (d) two-dimensional Poiseuille flow past a rigidly held sphere in a channel. The combined analytic and numerical solution procedure is the first application for bounded flow of the three-dimensional boundary collocation theory developed in Ganatos, Pfeffer & Weinbaum (1978). The accuracy of the solution technique is tested by detailed comparison with the exact bipolar co-ordinate solutions of Goldman, Cox & Brenner (1967a, b) for the drag and torque on a sphere translating parallel to a single plane wall, rotating adjacent to the wall or in the presence of a shear field. In all cases, the converged collocation solutions are in perfect agreement with the exact solutions for all spacings tested. The new collocation solutions have also been used to test the accuracy of existing solutions for the motion of a sphere parallel to two walls using the method of reflexions technique. The first-order reflexion theory of Ho & Leal (1974) provides reasonable agreement with the present results for the drag when the sphere is five or more radii from both walls. At closer spacings first-order reflexion theory is highly inaccurate and predicts an erroneous direction for the torque on the sphere for a wide range of sphere positions. Comparison with the classical higher-order method of reflexions solutions of Faxen (1923) reveals that the convergence of the multiple reflexion series solution is poor when the sphere centre is less than two radii from either boundary.

Solutions have also been obtained for the fluid velocity field. These solutions show that, for certain wall spacings and particle positions, a separated region of closed streamlines forms adjacent to the sphere which reverses the direction of the torque acting on a translating sphere.

A largely qualitative experimental investigation has been made of cellular convection patterns produced by the instability of a horizontal layer of fluid heated uniformly throughout its body and cooled from above. The fluid was aqueous zinc sulphate solution, and the heating was produced by an electrolytic current. The flow was visualized and photographed using polystyrene beads which, because of a differential expansion, came out of suspension in regions where the fluid was hotter or colder than average.

The development of the cellular patterns as the Rayleigh number (a modified form) was varied was in many respects similar to that for Bénard convection. There were, however, two striking differences. First, the fluid descended in the centres of the cells and ascended at the peripheries; this was probably associated with the asymmetry between the heating and cooling. Secondly, and more surprisingly, the horizontal scale of the convection patterns was unusually large (except at the lowest Rayleigh numbers); the distance between rising and falling currents was observed to reach typically five times the depth of the layer.

This last observation may be relevant to theories of convection within the Earth's mantle, to mesoscale convection in the atmosphere, and to patterns formed in ice.

The classical problem of water waves on an incompressible irrotational flow is considered. By introducing an appropriate non-dimensionalization, we derive four Korteweg–de Vries equations: two expressed in Cartesian co-ordinates and two in plane polars. The equations are: the classical (plane) KdV equation, the two-dimensional ‘nearly-plane’ equation, the concentric equation and a new ‘nearly-concentric’ equation. On the basis of the underlying water-wave equations, it is seen that two simple transformations exist between these KdV equations.

By constructing appropriate asymptotic regions defined in terms of the relevant small parameters, we show how various initial value problems give rise to certain solutions of the KdV equations. In particular, the generation of the similarity solutions is examined in detail and it is found that these solutions must eventually match to a solution of the full water-wave equations in a neighbourhood of the origin.

In this paper we study some instabilities and bifurcations of two-dimensional channel flows. We use analytical, numerical and experimental methods. We start by recapitulating some basic results in linear and nonlinear stability and drawing a connection with bifurcation theory. We then examine Jeffery–Hamel flows and discover new results about the stability of such flows. Next we consider two-dimensional indented channels and their symmetric and asymmetric flows. We demonstrate that the unique symmetric flow which exists at small Reynolds number is not stable at larger Reynolds number, there being a pitchfork bifurcation so that two stable asymmetric steady flows occur. At larger Reynolds number we find as many as eight asymmetric stable steady solutions, and infer the existence of another seven unstable solutions. When the Reynolds number is sufficiently large we find time-periodic solutions and deduce the existence of a Hopf bifurcation. These results show a rich and unexpected structure to solutions of the Navier–Stokes equations at Reynolds numbers of less than a few hundred.

A systematic dimensional rationale is proposed here to analyse the electrohydro-dynamic equations governing liquid electrospraying phenomena in the well-known steady cone-jet mode with no ambient discharges. As a result, a general, unified description of the complete parametrical space for the emitted current and droplet size is given. Four main distinct subspaces, their relevant boundaries and corresponding scaling laws are identified. Laws already proposed fit in their appropriate region, and previously unknown laws are found. A closed solution for the electric current I when inertia and polarization forces dominate is obtained, in agreement with published experimental results.

For fully-developed turbulent flow in straight channels of non-circular cross-section, there exists a transverse mean flow superimposed upon the axial mean flow. This transverse flow, commonly known as secondary flow, interacts with the axial mean flow and turbulence structure in a complex manner. In this paper several heretofore unexplored aspects of this type of secondary flow are discussed on the basis of results of an extensive experimental programme which was conducted for steady, incompressible, fully-developed turbulent air flow in both square and rectangular channels. Specifically, the following aspects are examined: (a) the Reynolds-number effect on secondary flow, (b) the directional characteristics of local wall shear stress, (c) the orientation of Reynolds-stress principal planes in a plane normal to the axial flow direction, and (d) the Reynolds equation along a secondary-flow streamline.

Within the Reynolds-number range of the investigation, the results indicate that secondary-flow velocities, when non-dimensionalized with either the bulk velocity or the axial mean-flow velocity at the channel centreline, decrease for an increase in Reynolds number. Also, the greatest skewness of local wall shear-stress vectors is shown to occur in the near vicinity of corners where secondary flow is maximum. In addition, it is shown that in planes normal to the axial flow direction, traces of Reynolds stress principal planes are not tangent and normal to lines of constant axial mean-flow velocity. This behaviour is in contrast to that for less complicated turbulent flows, for example, two-dimensional channel flow or circular-pipe flow where such traces are always tangent and normal to lines of constant axial mean-flow velocity in accordance with symmetry considerations. Finally, through experimental evaluation of terms in a momentum balance along a typical secondary-flow streamline, it is shown that secondary flow is the result of small differences in magnitude of opposing forces exerted by the Reynolds stresses and static pressure gradients in planes normal to the axial flow direction.

A study has been made of the factors influencing the break-down of sheets formed by the impingement of two liquid jets. It is shown that disintegration generally results from the formation of unstable waves of aerodynamic or hydrodynamic origin. While the characteristics of the former waves are fairly well understood, little is known about the latter. The results of this study indicate that hydrodynamic (or ‘impact’) waves are generated when the Weber number of each jet (ρlDV2 sin 2θ/γ, where ρl is the liquid density, D the jet diameter, V the mean jet velocity, θ the half-angle of impingement, and γ is the surface tension) is above a critical value, and that their formation is independent of the Reynolds number. Drop sizes have been measured and are shown to be critically dependent upon the mechanism of disintegration.

The suppression of initially isotropic turbulence by the sudden application of a uniform magnetic field is considered. The problem is characterized by three dimensionless numbers, a Reynolds number R, a magnetic Reynolds number Rm and a magnetic interaction parameter N (for definitions, see equations (1.2) and (1.4)). It is supposed that R [Gt ] 1, Rm [Lt ] 1 and N [Gt ] 1. There are two important time scales, ta a time characteristic of magnetic suppression, and t0 = Ntd, the ‘turn-over’ time of the turbulent energy-containing eddies. For 0 < t [Lt ] t0 the response of the energy-containing components of the turbulence to the applied field is linear and the time dependence of the kinetic energy density K(t) and magnetic energy density M(t) are analysed. There are essentially two distinct contributions to each from two domains of wave-number space D1 and D2 (defined in figure 2). In D1 the response is severely anisotropic, while in D2 it is nearly isotropic. The relative importance of the contributions K1(t) (from D1) and K2(t) (from D2) to K(t) depends on the value of the Lundquist number S = (NRm)½. If S [Lt ] 1, then K1(t) dominates for all t [lsim ] t0 and K(t) ∝ t−½ for td [Lt ] t [Lt ] t0. If 1 [Lt ] S [Lt ] R−2m, then a changeover in the dominant contribution occurs when t = O(S½Rm) t0. Analogous results are obtained for the magnetic energy density.

In order to model the evolution of a solitary wave near an obstacle or over an uneven bottom, the long-wave equations including curvature effects are introduced to describe the deformation and fission of a barotropic solitary wave passing over a shelf or an obstacle. The numerical results obtained from these equations are shown to be in good agreement with an analytical model derived by Germain (1984) in the framework of a generalized shallow-water theory, and with experimental results collected in a large channel equipped with a wave generator. Given the initial conditions, i.e. amplitude of the incident solitary wave, water depth in the deep region, and height of the shelf or the barrier, it is possible to predict the amplitude and number of the transmitted solitary waves as well as the amplitude of the reflected wave, and to describe the shape of the free surface at any time.

This article provides theoretical conditions for the use and meaning of a stability analysis around a mean flow. As such, it may be considered as an extension of the works by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to non-parallel flows and by Turton et al. (Phys. Rev. E, vol. 91 (4), 2015, 043009) to broadband flows. Considering a Reynolds decomposition of the flow field, the spectral (or temporal Fourier) mode of the fluctuation field is found to be equal to the action on a turbulent forcing term by the resolvent operator arising from linearisation about the mean flow. The main result of the article states that if, at a particular frequency, the dominant singular value of the resolvent is much larger than all others and if the turbulent forcing at this frequency does not display any preferential direction toward one of the suboptimal forcings, then the spectral mode is directly proportional to the dominant optimal response mode of the resolvent at this frequency. Such conditions are generally met in the case of weakly non-parallel open flows exhibiting a convectively unstable mean flow. The spatial structure of the singular mode may in these cases be approximated by a local spatial stability analysis based on parabolised stability equations (PSE). We have also shown that the frequency spectrum of the flow field at any arbitrary location of the domain may be predicted from the frequency evolution of the dominant optimal response mode and the knowledge of the frequency spectrum at one or more points. Results are illustrated in the case of a high Reynolds number turbulent backward facing step flow.