The analysis of density jumps is reconsidered and a new approximate model, based on an estimate that most of the turbulent energy produced by the jump is dissipated, is presented. The model suggests new upper bounds for the upstream Froude number, the entrainment and the downstream height. These bounds are shown to be consistent with earlier measurements.

Derivative-free optimization techniques are applied in conjunction with large-eddy simulation (LES) to reduce the noise generated by turbulent flow over a hydrofoil trailing edge. A cost function proportional to the radiated acoustic power is derived based on the Ffowcs Williams and Hall solution to Lighthill's equation. Optimization is performed using the surrogate-management framework with filter-based constraints for lift and drag. To make the optimization more efficient, a novel method has been developed to incorporate Reynolds-averaged Navier–Stokes (RANS) calculations for constraint evaluation. Separation of the constraint and cost-function computations using this method results in fewer expensive LES computations. This work demonstrates the ability to fully couple optimization to large-eddy simulation for time-accurate turbulent flow. The results demonstrate an 89% reduction in noise power, which comes about primarily by the elimination of low-frequency vortex shedding. The higher-frequency broadband noise is reduced as well, by a subtle change in the lower surface near the trailing edge.

Experimentation and theory are used to study the long-term dynamics of a two-dimensional density current flowing into a two-layer stratified basin. When the initial Richardson number, , characterizing the ratio of the background stratification to the buoyancy flux of the density current, is less than the critical value of , it is found that the density current penetrates the stratified interface. This result is ostensibly independent of slope for angles between 30° and 90°. If the current does not initially penetrate the interface, then it slowly increases the density of the top layer until the interfacial density difference is reduced sufficiently to drive penetration. The time scale for this to occur, , is explicitly a function of the buoyancy flux B and the length of the basin L. The initial Richardson number, , is a function of depth, the initial reduced gravity of the interface and a weak function of slope angle. In the absence of initial penetration for very steep slopes of 75° and 90°, we observe that penetrative convection at the interface leads to significant local entrainment. In consequence, the top layer thickens and the interfacial entrainment rate increases as the fifth power of the interfacial Froude number. In contrast, such a process is not observed at comparable interfacial Froude numbers on lower slopes of 30°, 45° and 60°, thereby demonstrating the important role of impact angle on penetrative convection. We attribute the increased interfacial entrainment by the steep density currents as the result of the transition from an undular bore to a turbulent hydraulic jump at the point where the density current intrudes. We discuss the applicability of the observed circulation to the stability of the Arctic halocline where we find years for a range of contemporary oceanographic conditions.

The theory of Wagner from 1932 for the normal symmetric impact of a two-dimensional body of small deadrise angle on a half-space of ideal and incompressible liquid is extended to derive the second-order corrections for the locations of the higher-pressure jet-root regions and for the upward force on the impactor using a systematic matched-asymptotic analysis. The second-order predictions for the upward force on an entering wedge and parabola are compared with numerical and experimental data, respectively, and it is concluded that a significant improvement in the predictive capability of Wagner's theory is afforded by proceeding to second order.

We develop a new methodology to examine the conditional and unconditional vertical velocity induced by high-Reynolds-number bubbles rising in a uniform flow, at low to moderate void fraction α (up to 15%). These statistics provide a local description of the perturbation of the liquid velocity around a test bubble in the swarm. In particular, the attenuation of the length of the wakes with increasing void fraction is measured for a large range of void fraction. The strong attenuation of the wakes is related to wake intermingling mechanisms. The methodology also enables a definition of the interstitial liquid flow. The velocity of the fluid averaged over all the interstitial volume far away from the bubbles is introduced. It is a useful concept, in particular to define the relative velocity, or for drift models. Our experimental results allow a discussion of the predictions of irrotational drift models. For low void fraction (α≤2%), potential flow models provide practical estimates of the interstitial velocity field. At higher void fractions, the effect of vorticity is important. A simple phenomenological model is proposed to include the effect of the flow generated by the bubble wakes.

We use direct Lyapunov exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find that despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold. As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow. We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures.

This paper examines a flashing liquid regime that takes place at very high ratios of injection to discharge pressures in flow restrictions. Typically, the flashing phenomenon has been observed in laboratory experiments where a liquid flows through a short nozzle into a low-pressure chamber at a pressure value considerably lower than the liquid saturation pressure at the injection temperature. By using two visualization techniques, the schlieren and the back-lighting methods, it was possible to identify some compressible phenomena associated with the liquid flashing process from the nozzle exit section. The schlieren method was used to capture the image of a shock-wave structure surrounding a liquid core from which the phase change takes place, and the optical technique allowed us to observe the central liquid core itself. The work corroborates previous physical descriptions of flashing liquid jets to explain an observed choking behaviour as well as the presence of shock waves. According to the present analysis, flashing takes place on the surface of the liquid core through an evaporation wave process, which results from a sudden liquid evaporation in a discontinuous process. Downstream of the evaporation discontinuity, the two-phase flow reaches very high velocities, up to the local sonic speed that typically occurs at high expansion conditions, as inferred from experiments and the physical model. That sonic state is also a point of maximum mass flow rate and it is known as the Chapman–Jouguet condition. The freshly sonic two-phase flow expands freely to increasing supersonic velocities and eventually terminates the expansion process through a shock-wave structure. This paper presents experimental results at several test conditions with iso-octane.

The paper describes a numerical study of the effects of microbubbles on the vorticity dynamics in a Taylor–Green vortex flow (TGV) using the two-fluid approach. The results show that bubbles with a volume fraction ∼10−2 enhance the decay rate of the vorticity at the centre of the vortex. Analysis of the vorticity equation of the bubble-laden flow shows that the local positive velocity divergence of the fluid velocity, ∇·U, created in the vortex core by bubble clustering, is responsible for the vorticity decay. At the centre of the vortex, the vorticity ωc(t) decreases nearly linearly with the bubble concentration Cm(t). Similarly, the enstrophy in the core of the vortex, ω2(t), decays nearly linearly with C2(t). The approximate mean-enstrophy equation shows that bubble accumulation in the high-enstrophy core regions produces a positive correlation between ω2 and ∇·U, which enhances the decay rate of the mean enstrophy.

Mean flow properties of turbulent magnetohydrodynamic channel flow with electrically insulating channel walls are studied using high-resolution direct numerical simulations. The Lorentz force due to the homogeneous wall-normal magnetic field is computed in the quasi-static approximation. For strong magnetic fields, the mean velocity profile shows a clear three-layer structure consisting of a viscous region near each wall and a plateau in the middle connected by logarithmic layers. This structure reflects the significance of viscous, turbulent, and electromagnetic stresses in the streamwise momentum balance dominating the viscous, logarithmic, and plateau regions, respectively. The width of the logarithmic layers changes with the ratio of Reynolds- and Hartmann numbers. Turbulent stresses typically decay more rapidly away from the walls than predicted by mixing-length models.

Interaction of a deep-water wave with a cylinder gives rise to ordered patterns of the flow structure, which are quantitatively characterized using a technique of high-image-density particle image velocimetry. When the cylinder is stationary, the patterns of instantaneous flow structure take on increasingly complex forms for increasing Keulegan--Carpenter number KC. These patterns involve stacking of small-scale vorticity concentrations, as well as large-scale vortex shedding. The time-averaged consequence of these patterns involves, at sufficiently high KC, an array of vorticity concentrations about the cylinder.

When the lightly damped cylinder is allowed to undergo bidirectional oscillations, the trajectories can be classified according to ranges of KC. At low values of KC, the trajectory is elliptical, and further increases of KC allow, first of all, both elliptical and in-line trajectories as possibilities, followed by predominantly in-line and figure-of-eight oscillations at the largest value of KC.

Representations of the quantitative flow structure, in relation to the instantaneous cylinder position on its oscillation trajectory, show basic classes of patterns. When the trajectory is elliptical, layers of vorticity rotate about the cylinder surface, in accordance with rotation of the relative velocity vector of the wave motion with respect to the oscillating cylinder. Simultaneously, the patterns of streamline topology take the form of large-scale bubbles, which also rotate about the cylinder. When the cylinder trajectory is predominantly in-line with the wave motion, generic classes of vortex formation and shedding can be identified; they include sweeping of previously shed vorticity concentrations past the cylinder to the opposite side. Certain of these patterns are directly analogous to those from the stationary cylinder.

The decay of a passive scalar in a sinusoidal shear flow translating in the cross-stream direction at a constant speed u is studied in the limit of small diffusivity κ. The decay rate, obtained by solving an eigenvalue problem, is found to tend to a non-zero constant as κ→0 when u is of order κ1/2. This result, establishing that fast decay is possible in shear flows, is fragile however: because of the existence of pseudomodes, the addition of a small noise leads to decay rates that decrease to zero with κ as κ2/5.

We report high-resolution local-temperature measurements in the upper boundary layer of turbulent Rayleigh–Bénard (RB) convection with variable Rayleigh number Ra and aspect ratio Γ. The primary purpose of the work is to create a comprehensive data set of temperature profiles against which various phenomenological theories and numerical simulations can be tested. We performed two series of measurements for air (Pr = 0.7) in a cylindrical container, which cover a range from Ra≈109 to Ra≈1012 and from Γ≈1 to Γ≈10. In the first series Γ was varied while the temperature difference was kept constant, whereas in the second series the aspect ratio was set to its lowest possible value, Γ=1.13, and Ra was varied by changing the temperature difference. We present the profiles of the mean temperature, root-mean-square (r.m.s.) temperature fluctuation, skewness and kurtosis as functions of the vertical distance z from the cooling plate. Outside the (very short) linear part of the thermal boundary layer the non-dimensional mean temperature Θ is found to scale as Θ(z)∼zα, the exponent α≈0.5 depending only weakly on Ra and Γ. This result supports neither Prandtl's one-third law nor a logarithmic scaling law for the mean temperature. The r.m.s. temperature fluctuation σ is found to decay with increasing distance from the cooling plate according to σ(z)∼zβ, where the value of β is in the range -0.30>β>-0.42 and depends on both Ra and Γ. Priestley's β=−1/3 law is consistent with this finding but cannot explain the variation in the scaling exponent. In addition to profiles we also present and discuss boundary-layer thicknesses, Nusselt numbers and their scaling with Ra and Γ.

We derive two third-order structure function relations for quasi-geostrophic turbulence, one for the forward cascade of potential enstrophy and one for the inverse cascade of energy. These relations are the counterparts of Kolmovorov's (1941) four-fifths law for the third-order longitudinal structure functions of three-dimensional turbulence.

An investigation of the linear temporal stability of a uniformly rotating viscous liquid column in the absence of gravity is presented. The governing parameters are the rotational Reynolds number Re and the Hocking parameter L, defined as the ratio of surface tension to centrifugal forces. Though the viscosity-independent condition L≥(k2 + n2-1)−1 for stability to disturbances of axial wavenumber k and azimuthal mode number n has been known for some time, the preferred modes, growth rates and frequencies at onset of instability have not been reported. We compute these results over a wide range of L–Re space and determine certain asymptotic behaviours in the limits of L→0, L→∞ and Re→∞. The computations exhibit a continuous evolution toward known inviscid stability results in the large-Re limit and their ultimate transition to an n = 1 spiral mode at small Re. While viscosity is shown to reduce growth rates for axisymmetric disturbances, it also produces a destabilizing effect for n = 2 planar and n = 1 spiral disturbances in certain regions of parameter space. A special feature is the appearance of a tricritical point in L–Re space at which maximum growth rates of the axisymmetric, n = 1 spiral, and n = 2 planar disturbances are equal.

We present an experimental study of the reopening mechanics of a collapsed liquid-filled elastic tube. The experiment is a simple mechanical model of pulmonary airway reopening and aims to assess the robustness of existing theoretical models. A metre-long horizontal elastic tube of inner radius Ri=4.88 ± 0.14mm is filled with silicone oil and is carefully collapsed mechanically. The injection of nitrogen at a constant flow rate results in the steady propagation of an air finger, after the decay of initial transients. This behaviour is observed over the realizable range of the capillary numbers Ca, which measures the ratio of viscous and capillary forces. With increasing Ca, the transition region between the collapsed and reopened sections of the tube shortens, and the height of the tube behind the bubble tip increases. We also find that air fingers can propagate in partially reopened tubes, in which the transmural pressure is negative far behind the finger tip.

The effect of viscosity on the reopening dynamics was explored by performing experiments using three different grades of silicone oil, with kinematic viscosities of 1000cS, 200cS and 100cS. A direct comparison between the experimental pressure dependence on Ca and numerical simulations of the zero-gravity three-dimensional airway-reopening model of Hazel & Heil (Trans. ASME: J. Biomech. Engng, vol. 128, 2006, p. 473) highlights some significant differences. Within the experimental parameter range, gravity profoundly influences the reopening mechanics in several ways. The reopening tube is supported by a rigid base, which induces an asymmetry about the horizontal mid-plane of the collapsed tube, resulting in distinct phases of reopening as Ca increases. In addition, buoyancy forces act on the air finger, which is observed to propagate near the top of the cross-section of the tube, leaving a thicker fluid-lining below. In the limit of small Ca, the height of the reopened tube increases significantly with viscosity. Experimental evidence suggests that this increase in viscosity leads to significant changes in the film configuration behind the propagating finger, caused by the increased contribution of buoyancy forces. The altered film configuration changes the mechanical load on the tube walls and, hence, the shape of the reopened tube.

Direct numerical simulations of the flow past a fixed oblate spheroidal bubble are carried out to determine the range of parameters within which the flow may be unstable, and to gain some insight into the instability mechanism. The bubble aspect ratio χ (i.e. the ratio of the major axis length over the minor axis length) is varied from 2.0 to 2.5 while the Reynolds number (based on the upstream velocity and equivalent bubble diameter) is varied in the range 102 ≤ Re ≤ 3 × 103. As vorticity generation at the bubble surface is at the root of the instability, theoretical estimates for the maximum of the surface vorticity and the surface vorticity flux are first derived. It is shown that, for large aspect ratios and high Reynolds numbers, the former evolves as χ8/3 while the latter is proportional to χ7/2Re−1/2. Then it is found numerically that the flow first becomes unstable for χ = χc ≈ 2.21. As the surface vorticity becomes independent of Re for large enough Reynolds number, the flow is unstable only within a finite range of Re, this range being an increasing function of χ − χc. An empirical criterion based on the maximum of the vorticity generated at the body surface is built to determine whether the flow is stable or not. It is shown that this criterion also predicts the correct threshold for the wake instability past a rigid sphere, suggesting that the nature of the body surface does not really matter in the instability mechanism. Also the first two bifurcations of the flow are similar in nature to those found in flows past rigid axisymmetric bluff bodies, such as a sphere or a disk. Wake dynamics become more complex at higher Reynolds number, until the Re−1/2-dependency of the surface vorticity flux makes the flow recover its steadiness and eventually its axisymmetry. A qualitative analysis of the azimuthal vorticity field in the base flow at the rear of the bubble is finally carried out to make some progress in the understanding of the primary instability. It is suggested that the instability originates in a thin region of the flow where the vorticity gradients have to turn almost at right angle to satisfy two different constraints, one at the bubble surface, the other within the standing eddy.

Effects of homogeneous nucleation and subsequent droplet growth in compressible flows in humid nitrogen are investigated numerically and experimentally. A Ludwieg tube is employed to produce expansion flows. Corresponding to different configurations, three types of experiment are carried out in such a tube. First, the phase transition in a strong unsteady expansion wave is investigated to demonstrate the mutual interaction between the unsteady flow and the condensation process and also the formation of condensation-induced shock waves. The role of condensation-induced shocks in the gradual transition from a frozen initial structure to an equilibrium structure is explained. Second, the condensing flow in a slender supersonic nozzle G2 is considered. Particular attention is given to condensation-induced oscillations and to the transition from symmetrical mode-1 oscillations to asymmetrical mode-2 oscillations in a starting nozzle flow, as first observed by Adam & Schnerr. The transition is also found numerically, but the amplitude, frequency and transition time are not yet well predicted. Third, a sharp-edged obstacle is placed in the tube to generate a starting vortex. Condensation in the vortex is found. Owing to the release of latent heat of condensation, an increase in the pressure and temperature in the vortex core is observed. Condensation-induced shock waves are found, for a sufficiently high initial saturation ratio, which interact with the starting vortex, resulting in a very complex flow. As time proceeds, a subsonic or transonic free jet is formed downstream of the sharp-edged obstacle, which becomes oscillatory for a relatively high main-flow velocity and for a sufficiently high humidity.

The spreading of a liquid droplet on a smooth solid surface in the partially wetting regime is studied using a diffuse-interface model based on the Cahn--Hilliard theory. The model is extended to include non-90 contact angles. The diffuse-interface model considers the ambient fluid displaced by the droplet while spreading as a liquid. The governing equations of the model for the axisymmetric case are solved numerically using a finite-spectral-element method. The viscosity of the ambient fluid is found to affect the time scale of spreading, but the general spreading behaviour remains unchanged. The wettability expressed in terms of the equilibrium contact angle is seen to influence the spreading kinetics from the early stages of spreading. The results show agreement with the experimental data reported in the literature.

The entire free-surface elevation field of a rotating fluid in the laboratory can be imaged and analysed, by using it as a parabolic Newtonian telescope mirror. This ‘optical altimetry’ readily achieves a precision of better than 1 μm of surface elevation. The surface topography corresponds to the pressure field just beneath the surface. It is the streamfunction for the geostrophic hydrostatic circulation, which can be resolved to better than 0.1 mm s−1. Still and animated images thus produced, of the entire surface elevation field, are of value in themselves, and using a projected image (a speckle pattern), have the promise of providing quantitative slope and height field data recovered by PIV (particle imaging velocimetry) techniques. With homogeneous fluid, geostrophic flow is the same at all depths. Yet of equal interest are sheared stratified rotating flows where the surface pressure is associated with inertial waves, convection, and other motions, geostrophic or ageostrophic.

Although the technique is designed for experiments in which Coriolis effects are strong, it is possible to use reflective imaging for flows at such high Rossby number that Coriolis effects are negligible, and hence this becomes a tool of more general interest in non-rotating fluid dynamics (for example, illuminating surface gravity waves).

Examples are given, involving (i) the Taylor–Proudman effect with very slow flows over topography; (ii) quasi-geostrophic and inertial-wave flows over a mountain (f-plane); (iii) inertial waves generated by oscillatory forcing; (iv) Kelvin waves (v) free oscillatory Rossby waves on a polar β-plane; and (vi) stationary waves, blocking, jets and wakes with β-plane zonal flow past a mountain. Movies are available with the online version of the paper.

When a fluid jet hits a solid surface, a hydraulic jumps occurs. This jump sharply delimits a thin film of liquid from a thicker film. We show here that a granular jet impinging on a solid surface also gives rise to several features reminiscent of the hydraulic jump and we refer to this situation as the granular jump. We describe, in detail, this phenomenon and show that if many of its features can be understood in analogy with the hydraulic jump, others are directly related to the granular nature of the medium and, in particular, the small-scale dynamics of the jump.