We report measurements of the flow above a planar array of synthetic jets, firing upwards in a spatiotemporally random pattern to create turbulence at an air–water interface. The flow generated by this randomly actuated synthetic jet array (RASJA) is turbulent, with a large Reynolds number and a weak secondary (mean) flow. The turbulence is homogeneous over a large region and has similar isotropy characteristics to those of grid turbulence. These properties make the RASJA an ideal facility for studying the behaviour of turbulence at boundaries, which we do by measuring one-point statistics approaching the air–water interface (via particle image velocimetry). We explore the effects of different spatiotemporally random driving patterns, highlighting design conditions relevant to all randomly forced facilities. We find that the number of jets firing at a given instant, and the distribution of the duration for which each jet fires, greatly affect the resulting flow. We identify and study the driving pattern that is optimal given our tank geometry. In this optimal configuration, the flow is statistically highly repeatable and rapidly reaches steady state. With increasing distance from the jets, there is a jet merging region followed by a planar homogeneous region with a power-law decay of turbulent kinetic energy. In this homogeneous region, we find a Reynolds number of 314 based on the Taylor microscale. We measure all components of mean flow velocity to be less than 10% of the turbulent velocity fluctuation magnitude. The tank width includes roughly 10 integral length scales, and because wall effects persist for one to two integral length scales, there is sizable core region in which turbulent flow is unaffected by the walls. We determine the dissipation rate of turbulent kinetic energy via three methods, the most robust using the velocity structure function. Having a precise value of dissipation and low mean flow allows us to measure the empirical constant in an existing model of the Eulerian velocity power spectrum. This model provides a method for determining the dissipation rate from velocity time series recorded at a single point, even when Taylor's frozen turbulence hypothesis does not hold. Because the jet array offers a high degree of flow control, we can quantify the effects of the mean flow in stirred tanks by intentionally forcing a mean flow and varying its strength. We demonstrate this technique with measurements of gas transfer across the free surface, and find a threshold below which mean flow no longer contributes significantly to the gas transfer velocity.

The two-dimensional motion of a circular cylinder freely falling or rising in an infinite fluid is investigated numerically for the range of Reynolds number Re, < 188 (Galileo number G < 163), where the wake behind the cylinder remains two-dimensional, using a combined formulation of the governing equations for the fluid and the dynamic equations for the cylinder. The effect of vortex shedding on the motion of the freely falling or rising cylinder is clearly shown. As the streamwise velocity of the cylinder increases due to gravity, the periodic vortex shedding induces a periodic motion of the cylinder, which is manifested by the generation of the angular velocity vector of the cylinder parallel to the cross-product of the gravitational acceleration vector and the transverse velocity vector of the cylinder. Correlations of the Strouhal–Reynolds-number and Strouhal–Galileo-number relationship are deduced from the results. The Strouhal number is found to be smaller than that for the corresponding fixed circular cylinder when the two Reynolds numbers based on the streamwise terminal velocity of the freely falling or rising circular cylinder and the free-stream velocity of the fixed one are the same. From numerical experiments, it is shown that the transverse motion of the cylinder plays a crucial role in reducing the Strouhal number. The effect of the transverse motion is similar to that of suction flow on the low-pressure side, where a vortex is generated and then separates, so that the pressure on this side recovers with the vortex separation retarded. The effects of the transverse motion on the lift, drag and moment coefficients are also discussed. Finally, the effect of the solid/fluid density ratio on Strouhal–Reynolds-number relationship is investigated and a plausible correlation is proposed.

The dynamics of receding contact lines is investigated experimentally through controlled perturbations of a meniscus in a dip-coating experiment. We first describe stationary menisci and their breakdown at the coating transition. Above this transition where liquid is deposited, it is found that the dynamics of the interface can be interpreted as a quasi-steady succession of stationary states. This provides the first experimental access to the entire bifurcation diagram of dynamical wetting, confirming the hydrodynamic theory developed in Part 1. In contrast to quasi-static theories based on a dynamic contact angle, we demonstrate that the transition strongly depends on the large-scale flow geometry. We then establish the dispersion relation for large wavenumbers, for which we find a decay rate σ proportional to wavenumber |q|. The speed dependence of σ is described well by hydrodynamic theory, in particular the absence of diverging time scales at the critical point. Finally, we highlight some open problems related to contact angle hysteresis that lead beyond the current description.

A modified Lundgren model is applied for the description of stationary bathtub vortices in a viscous liquid with a free surface. Laminar liquid flow through the circular bottom orifice is considered in the horizontally unbounded domain. The liquid is assumed to be undisturbed at infinity and its depth is taken to be constant. Three different drainage regimes are studied: (i) subcritical, where whirlpool dents are less than the fluid depth; (ii) critical, where the whirlpool tips touch the outlet orifice; and (iii) supercritical, where surface vortices entrain air into the intake pipe. Particular attention is paid to critical vortices; the condition for their existence is determined and analysed. The influence of surface tension on subcritical whirlpools is investigated. Comparison of results with known experimental data is discussed.

We examine the development of a thermal plume originating from a localized heat source using direct numerical simulation. The Reynolds number of the plume, based on source diameter and the characteristic buoyancy velocity, is chosen to be 7700, which is sufficiently large so that the flow turns to a fully turbulent state. A highly resolved grid of 622 million points is used to capture the entire range of turbulent scales in the plume. Here at the source, only heat has been added with no mass or momentum addition and accordingly the vertical evolution of the mass, momentum and buoyancy fluxes computed from the simulation have been verified to follow those of a pure thermal plume. The computed vertical evolution of the time-averaged centreline velocity and temperature are in good agreement with available experimental measurements. Investigation of the time evolution of the plume shows periodic formation of vortex ring structure surrounding the main ascending column of hot fluid. The vortex ring forms very close to the heat source and even at formation it is three-dimensional. The vortex ring ascends with the plume and at an elevation of about two diameters it strongly interacts with and destabilizes the central column and subsequently a complex turbulent flow arises. Thus, relatively laminar, transitional and fully turbulent regimes of the plume evolution can be identified. In the fully turbulent regime, complex three-dimensional hairpin-like vortex structures are observed; but vestiges of the coherent vortex rolls that form close to the source can be observed in the turbulent statistics. It is shown that local entrainment consists of contraction and expulsion phases. Such instantaneous mechanisms drive the entrainment process, and the instantaneous entrainment coefficient shows large variation in both time and space with local values up to three times higher than the average entrainment level. Such findings support the view that entrainment mechanisms in plumes should be considered from an unsteady point of view. Movies are available with the online version of the paper.

Three new dynamic tensor thermal diffusivity subgrid-scale (SGS) heat flux (HF) models are proposed for large-eddy simulation of thermal convection. The constitutive relations for the proposed modelling approaches represent the most general explicit algebraic formulations possible for the family of SGS HF models constructed using the resolved temperature gradient and SGS stress tensor. As a result, these three new models include a number of previously proposed dynamic SGS HF models as special cases. In contrast to the classical dynamic eddy thermal diffusivity SGS HF model, which strictly requires the SGS heat flux be aligned with the negative of the resolved temperature gradient, the three new models proposed here admit more degrees of freedom, and consequently provide a more realistic geometrical and physical representation of the SGS HF vector. To validate the proposed models, numerical simulations have been performed based on two benchmark test cases of neutrally and unstably stratified horizontal channel flows.

The relevance of surface quasi-geostrophic dynamics (SQG) to the upper ocean and the atmospheric tropopause has been recently demonstrated in a wide range of conditions. Within this context, the properties of SQG in terms of kinetic energy (KE) transfers at the surface are revisited and further explored. Two well-known and important properties of SQG characterize the surface dynamics: (i) the identity between surface velocity and density spectra (when appropriately scaled) and (ii) the existence of a forward cascade for surface density variance. Here we show numerically and analytically that (i) and (ii) do not imply a forward cascade of surface KE (through the advection term in the KE budget). On the contrary, advection by the geostrophic flow primarily induces an inverse cascade of surface KE on a large range of scales. This spectral flux is locally compensated by a KE source that is related to surface frontogenesis. The subsequent spectral budget resembles those exhibited by more complex systems (primitive equations or Boussinesq models) and observations, which strengthens the relevance of SQG for the description of ocean/atmosphere dynamics near vertical boundaries. The main weakness of SQG however is in the small-scale range (scales smaller than 20–30 km in the ocean) where it poorly represents the forward KE cascade observed in non-QG numerical simulations.

The magnetostrophic regime, in which Lorentz and Coriolis forces are in balance, has been investigated in a rapidly rotating spherical Couette flow experiment. The spherical shell is filled with liquid sodium and permeated by a strong imposed dipolar magnetic field. Azimuthally travelling hydromagnetic waves have been put in evidence through a detailed analysis of electric potential differences measured on the outer sphere, and their properties have been determined. Several types of wave have been identified depending on the relative rotation rates of the inner and outer spheres: they differ by their dispersion relation and by their selection of azimuthal wavenumbers. In addition, these waves constitute the largest contribution to the observed fluctuations, and all of them travel in the retrograde direction in the frame of reference bound to the fluid. We identify these waves as magneto-inertial waves by virtue of the close proximity of the magnetic and inertial characteristic time scales of relevance in our experiment.

The natural and bypass routes to boundary-layer turbulence have traditionally been studied independently. In certain flow regimes, both transition mechanisms might coexist, and, if so, can interact. A nonlinear interaction of discrete and continuous Orr-Sommerfeld modes, which are at the origin of orderly and bypass transition, respectively, is found. It causes breakdown to turbulence, even though neither mode alone is sufficient. Direct numerical simulations of the interaction shows that breakdown occurs through a pattern of Λ-structures, similar to the secondary instability of Tollmien–Schlichting waves. However, the streaks produced by the Orr-Sommerfeld continuous mode set the spanwise length scale, which is much smaller than that of the secondary instability of Tollmien–Schlichting waves. Floquet analysis explains some of the features seen in the simulations as a competition between destabilizing and stabilizing interactions between finite-amplitude distortions.

This paper makes two new propositions regarding the modelling of rarefied (non-equilibrium) isothermal gas flows at the microscale. The first is a new test case for benchmarking high-order, or extended, hydrodynamic models for these flows. This standing time-varying shear-wave problem does not require boundary conditions to be specified at a solid surface, so is useful for assessing whether fluid models can capture rarefaction effects in the bulk flow. We assess a number of different proposed extended hydrodynamic models, and we find the R13 equations perform the best in this case.

Our second proposition is a simple technique for introducing non-equilibrium effects caused by the presence of solid surfaces into the computational fluid dynamics framework. By combining a new model for slip boundary conditions with a near-wall scaling of the Navier--Stokes constitutive relations, we obtain a model that is much more accurate at higher Knudsen numbers than the conventional second-order slip model. We show that this provides good results for combined Couette/Poiseuille flow, and that the model can predict the stress/strain-rate inversion that is evident from molecular simulations. The model's generality to non-planar geometries is demonstrated by examining low-speed flow around a micro-sphere. It shows a marked improvement over conventional predictions of the drag on the sphere, although there are some questions regarding its stability at the highest Knudsen numbers.

The paper examines the effect of the bottom stress on the weakly nonlinear evolution of a narrow-band wave field, as a potential mechanism of suppression of ‘freak’ wave formation in water of moderate depth. Relying upon established experimental studies the bottom stress is modelled by the quadratic drag law with an amplitude/bottom roughness-dependent drag coefficient. The asymptotic analysis yields Davey–Stewartson-type equations with an added nonlinear complex friction term in the envelope equation. The friction leads to a power-law decay of the spatially uniform wave amplitude. It also affects the modulational (Benjamin–Feir) instability, e.g. alters the growth rates of sideband perturbations and the boundaries of the linearized stability domains in the modulation wavevector space. Moreover, the instability occurs only if the amplitude of the background wave exceeds a certain threshold. Since the friction is nonlinear and increases with wave amplitude, its effect on the formation of nonlinear patterns is more dramatic. Numerical experiments show that even when the friction is small compared to the nonlinear term, it hampers formation of the Akhmediev/Ma-type breathers (believed to be weakly nonlinear ‘prototypes’ of freak waves) at the nonlinear stage of instability. The specific predictions for a particular location depend on the bottom roughness ks in addition to the water depth and wave field characteristics.

We investigate the derivation of Euler's equation from Hamilton's variational principle for flows decomposed into their mean and fluctuating parts. Our particular concern is with the flow decomposition used in the derivation of the Navier–Stokes–α equation which expresses the fluctuating velocity in terms of the mean flow and a small fluctuating displacement. In the past the derivation has retained terms up to second order in the Lagrangian which is then averaged. The variation is effected by incrementing the mean velocity, while holding the moments of the products of the displacements fixed. The process leads to a mean Euler equation for the mean velocity. The Navier–Stokes–α equation is only obtained after making a further closure approximation, which is not the concern of this paper. Instead attention is restricted here to the exact analysis of Euler's equation. We show that a proper implementation of Hamilton's principle, which concerns the virtual variation of particle paths, can only be achieved when the fluctuating displacement and mean velocity are varied in concert. This leads to an exact form of Euler's equation. If, on the other hand, the displacement is held fixed under the variation, a term in Euler's equation is lost. Averaging that erroneous form provides the basis of the Navier–Stokes–α equation. We explore the implications of the correct mean equation, particularly with regard to Kelvin's circulation theorem, comparing it with the so called GLM and glm-equations.

Geometrical shock dynamics, also called CCW theory, yields approximate equations for shock propagation in which only the conditions at the shock appear explicitly; the post-shock flow is presumed approximately uniform and enters implicitly via a Riemann invariant. The non-relativistic theory, formulated by G. B. Whitham and others, matches many experimental results surprisingly well. Motivated by astrophysical applications, we adapt the theory to ultra-relativistic shocks advancing into an ideal fluid whose pressure is negligible ahead of the shock, but is one third of its proper energy density behind the shock. Exact results are recovered for some self-similar cylindrical and spherical shocks with power-law pre-shock density profiles. Comparison is made with numerical solutions of the full hydrodynamic equations. We review relativistic vorticity and circulation. In an ultra-relativistic ideal fluid, circulation can be defined so that it changes only at shocks, notwithstanding entropy gradients in smooth parts of the flow.

Electrospray atomization of liquids in the cone-jet mode generates narrow droplet distributions with average diameters as small as a few nanometres. This ability is important for technologies such as colloid thrusters, nanoparticle generation and ion beam processes, and the optimization of these applications requires an understanding of the physics and structure of the associated beams. This paper presents a detailed experimental characterization of electrosprays in vacuum and formulates an analytical model of the beam. A key feature of our model is the use of a simplified expression for the electric field induced by the space charge. This simplification leads to a time-independent Eulerian formulation compatible with an analytical solution, in contrast to the direct simulation of a multitude of droplets which must be simultaneously tracked to account for Coulombic interactions. We find that the beams open up in an initial region relatively insensitive to the external electrodes, a process dominated by the electric repulsion between droplets and the initial droplet inertia. Although the external electric field modifies the trajectories of the droplets downstream of this initial region, the effect is moderate in our typical electrospray source and the analytical solution in the space charge region explains well the far-field beam structure observed experimentally. We also describe a numerical scheme that implements the full effect of the external electric field and provides a more accurate solution.

We discuss laboratory experiments investigating mixing in a density-driven current flowing down a sloping bottom, in a rotating homogenous fluid. A systematic study spanning a wide range of Froude, 0.8 < Fr < 10, and Reynolds, 10 < Re < 1400, numbers was conducted by varying three parameters: the bottom slope; the flow rate; and the density of the dense fluid. Different flow regimes were observed, i.e. waves (non-breaking and breaking) and turbulent regimes, while changing the above parameters. Mixing in the density-driven current has been quantified within the observed regimes, and at different locations on the slope. The dependence of mixing on the relevant non-dimensional numbers, i.e. slope, Fr and Re, is discussed. The entrainment parameter, E, was found to be dependent not only on Fr, as assumed in previous studies, but also on Re. In particular, mixing increased with increasing Fr and Re. For low Fr and Re, the magnitude of the mixing was comparable to mixing in the ocean. For large Fr and Re, mixing was comparable to that observed in previous laboratory experiments that exhibited the classic turbulent entrainment behaviour.

Direct numerical simulation is used to study the effect of crossflow on the dynamics, entrainment and mixing characteristics of vortex rings issuing from a circular nozzle. Three distinct regimes exist, depending on the velocity ratio (ratio of the average nozzle exit velocity to free-stream crossflow velocity) and stroke ratio (ratio of stroke length to nozzle exit diameter). Coherent vortex rings are not obtained at velocity ratios below approximately 2. At these low velocity ratios, the vorticity in the crossflow boundary layer inhibits roll-up of the nozzle boundary layer at the leading edge. As a result, a hairpin vortex forms instead of a vortex ring. For large stroke ratios and velocity ratio below 2, a series of hairpin vortices is shed downstream. The shedding is quite periodic for very low Reynolds numbers. For velocity ratios above 2, two regimes are obtained depending upon the stroke ratio. Lower stroke ratios yield a coherent asymmetric vortex ring, while higher stroke ratios yield an asymmetric vortex ring accompanied by a trailing column of vorticity. These two regimes are separated by a transition stroke ratio whose value decreases with decreasing velocity ratio. For very high values of the velocity ratio, the transition stroke ratio approaches the ‘formation number’. In the absence of trailing vorticity, the vortex ring tilts towards the upstream direction, while the presence of a trailing column causes it to tilt downstream. This behaviour is explained. In the absence of crossflow, the trailing column is not very effective at entrainment, and is best avoided for optimal mixing and entrainment. However, in the presence of crossflow, the trailing column is found to contribute significantly to the overall mixing and entrainment. The trailing column interacts with the crossflow to generate a region of high pressure downstream of the nozzle that drives crossflow fluid towards the vortex ring. There is an optimal length of the trailing column for maximum downstream entrainment. A classification map which categorizes the different regimes is developed.

We present the three-dimensional linear stability analysis of a pressure-driven, incompressible, fully developed, laminar flow in a channel delimited by rigid, homogeneous, isotropic, porous layers. We consider porous materials of small permeability in which the maximum fluid velocity is small compared to the mean velocity in the channel region and for which inertial effects may be neglected. We analyse the linear stability of symmetric laminar velocity profiles in channels with two identical porous walls as well as skewed laminar velocity profiles in channels with only one porous wall. We solve the fully coupled linear stability problem, arising from the adjacent channel and porous flows, using a spectral collocation technique. We validate our results by recovering the linear stability results of a flow in a channel with impermeable walls as the permeabilities of the porous layers tend to zero. We also verify that our results are consistent with the assumption of negligible inertial effects in the porous regions. We characterize the stability of pressure-driven flows by performing a parametric study in which we vary the permeability, porosity, and height of the porous layers as well as an interface coefficient, τ, associated with the momentum transfer process at the interfaces between the channel and porous regions. We find that very small amounts of wall permeability significantly affect the Orr–Sommerfeld spectrum and can dramatically decrease the stability of the channel flow. Within our assumptions, in channels with two porous walls, permeability destabilizes up to two Orr–Sommerfeld wall modes and introduces two new damped wall modes on the left branch of the spectrum. In channels with only one porous wall, permeability destabilizes up to one wall mode and introduces one new damped wall mode on the left branch of the spectrum. In both cases, permeability also introduces a new class of damped modes associated with the porous regions. The size of the unstable region delimited by the neutral curve grows substantially, and the critical Reynolds number can decrease to only 10% of the corresponding value for a channel flow with impermeable walls. We conclude our study by considering two real materials: foametal and aloxite. We fit the porosity and interface coefficient τ to published data so that the porous materials we model behave like foametal and aloxite, and we compare our results with previously published numerical and experimental results.

In problems involving the dispersion of hazardous gases in the atmosphere, the distribution of high concentrations is often of particular interest. We address the modelling of the distribution of high concentrations of a dispersing passive scalar at large Péclet number, concentrating on the case of steady releases. We argue, from the physical character of the small-scale processes, and from the statistical theory of extreme values, that the high concentrations can be fitted well by a Generalized Pareto Distribution (GPD). This is supported by evidence from a range of experiments. We show, furthermore, that if this is the case then the ratios of successive high-order absolute moments of the scalar concentration are linearly related to the reciprocal of the order. The linear fit thus obtained allows the GPD parameters to be determined from the moments. In this way the moments can be used to deduce the properties of the high concentrations, in particular the maximum possible concentration θmax = θmax(x). We argue, on general physical grounds, that θmax/C0 (where C0 = C0(X) is the centreline mean concentration, and X is the downstream distance from the source) decreases to zero very far from the centreline, but that the decrease takes place on a length scale much larger than the mean plume width (because it is controlled by the relatively slowly acting molecular diffusion, rather than the fast turbulent advection). Thus, over the distances for which accurate measurements can be made, we expect θmax/C0 to be approximately constant throughout the plume cross-section. On the centreline, we argue that θmax/C0 increases downstream from the source, reaches a maximum and then decreases, ultimately tending to 1 far downstream. In support of these deductions we present results for some high-quality data for a steady line source in wind tunnel grid turbulence. Finally, we apply to this problem some existing models for the relationships between moments. By considering the behaviour far from the centreline in these models, and linking the moments to the high concentrations, we derive relationships between the model parameters. This allows us to derive an expression for θmax/C0 which depends on a total of 5 parameters, and (weakly) on C/C0 (where C = C(x) is the local mean concentration). Comparison with the data is encouraging. We also discuss possible methods for modelling the spatial variation of these 5 parameters.