Fluid residence time is a key concept in the understanding and design of chemically reacting flows. In order to investigate how turbulent mixing affects the residence time distribution within a flow, this study examines statistics of fluid residence time from a direct numerical simulation (DNS) of a statistically stationary turbulent round jet with a jet Reynolds number of 7290. The residence time distribution in the flow is characterised by solving transport equations for the residence time of the jet fluid and for the jet fluid mass fraction. The product of the jet fluid residence time and the jet fluid mass fraction, referred to as the mass-weighted stream age, gives a quantity that has stationary statistics in the turbulent jet. Based on the observation that the statistics of the mass fraction and velocity are self-similar downstream of an initial development region, the transport equation for the jet fluid residence time is used to derive a model describing a self-similar profile for the mean of the mass-weighted stream age. The self-similar profile predicted is dependent on, but different from, the self-similar profiles for the mass fraction and the axial velocity. The DNS data confirm that the first four moments and the shape of the one-point probability density function of mass-weighted stream age are indeed self-similar, and that the model derived for the mean mass-weighted stream-age profile provides a useful approximation. Using the self-similar form of the moments and probability density functions presented it is therefore possible to estimate the local residence time distribution in a wide range of practical situations in which fluid is introduced by a high-Reynolds-number jet of fluid.

We present simultaneous two-dimensional velocity and scalar measurements on a central vertical plane in an axisymmetric pure turbulent plume. We use an edge-detection algorithm to determine the edge of the plume, and compare the data obtained in both a fixed Eulerian frame and a frame relative to local coordinates defined in terms of the instantaneous plume edge. In an Eulerian frame we observe that the time-averaged distributions of vertical and horizontal velocity are self-similar, the vertical velocity being well represented by a Gaussian distribution. We condition these measurements on whether fluid is inside or outside of the plume, and whether fluid inside is mixed plume fluid or engulfed ambient fluid. We find that, on average, 5 % of the total vertical volume transport occurs outside the plume and this figure rises to nearly 14 % at heights between large-scale coherent structures. We show that the fluxes of engulfed fluid within the plume envelope are slightly larger than the vertical transport outside the plume – indicating that ambient fluid is engulfed into the plume envelope before being nibbled across the turbulent/non-turbulent interface (TNTI) and then ultimately irreversibly mixed. Our new measurements in the plume coordinate (following the meandering fluctuating plume) show the flow within the plume and in the nearby ambient fluid is strongly influenced by whether an eddy is present locally within the plume, or absent. When an eddy is present and the plume is wide, the vertical velocities near the plume edge are small and hence all vertical transport is inside the plume. In regions where the plume is narrow and there is no eddy, large vertical velocities and hence transport are observed outside the plume suggesting that pressure forces associated with the eddies accelerate ambient fluid which is then engulfed into the plume. Finally, we show that observing significant vertical velocities beyond the scalar edge of the plume does not suggest that the characteristic width of the velocity distribution is greater than that of the scalar field; on the contrary, we show our observations to be consistent with a buoyancy distribution that is up to 20 % wider than that of the velocity. Measurements in the plume coordinates show that the mixing of momentum across the plume results in a distribution for which the differential entropy is close to maximal and the mixing of momentum is uninhibited (i.e. not bounded) by the TNTI of the plume. Furthermore, our measurements suggest that the scalar mixing across the plume may also result in a distribution for which the differential entropy is close to maximal but, in contrast to the momentum, the scalar mixing is strictly bounded by the plume edge.

Convection in a rotating rectangular basin with differential thermal forcing at one horizontal boundary is examined using laboratory experiments. The experiments have an imposed heat flux boundary condition, are at large values of the flux Rayleigh number ($Ra_{F}\sim O(10^{13}{-}10^{14})$ based on the box length $L$), use water with Prandtl number $Pr\approx 4$ and have a small depth to length aspect ratio. The results show the conditions for transition from non-rotating horizontal convection governed by an inertial–buoyancy balance in the thermal boundary layer, to circulation governed by geostrophic flow in the boundary layer. The geostrophic balance constrains mean flow and reduces the heat transport as Nusselt number $Nu\sim (Ra_{F}Ro)^{1/6}$, where $Ro=B^{1/2}/f^{3/2}L$ is the convective Rossby number, $B$ is the imposed buoyancy flux and $f$ is the Coriolis parameter. Thus flow in the geostrophic boundary layer regime is governed by the relative roles of horizontal convective accelerations and Coriolis accelerations, or buoyancy and rotation, in the boundary layer. Experimental evidence suggests that for more rapid rotation there is another transition to a regime in which the momentum budget is dominated by fluctuating vertical accelerations in a region of vortical plumes, which we refer to as a ‘chimney’ following related discussion of regions of deep convection in the ocean. Coupling of the chimney convection in the region of destabilising boundary flux to the diffusive boundary layer of horizontal convection in the region of stabilising boundary flux gives heat transport independent of rotation in this ‘inertial chimney’ regime, and the new scaling $Nu\sim Ra_{F}^{1/4}$. Scaling analysis predicts the transition conditions observed in the experiments, as well as a further ‘geostrophic chimney’ regime in which the vertical plumes are controlled by local geostrophy. When $Ro<10^{-1}$, the convection is also observed to produce a set of large basin-scale gyres at all depths in the time-averaged flow.

Turbulent flow separation induced by a protuberance on one of the walls of an otherwise planar channel is investigated using direct numerical simulations. Different bulge geometries and Reynolds numbers – with the highest friction Reynolds number simulation reaching a peak of $Re_{\unicode[STIX]{x1D70F}}=900$ – are addressed to understand the effect of the wall curvature and of the Reynolds number on the dynamics of the recirculating bubble behind the bump. Global quantities reveal that most of the drag is due to the form contribution, whilst the friction contribution does not change appreciably with respect to an equivalent planar channel flow. The size and position of the separation bubble strongly depends on the bump shape and the Reynolds number. The most bluff geometry has a larger recirculation region, whilst the Reynolds number increase results in a smaller recirculation bubble and a shear layer more attached to the bump. The position of the reattachment point only depends on the Reynolds number, in agreement with experimental data available in the literature. Both the mean and the turbulent kinetic energy equations are addressed in such non-homogeneous conditions revealing a non-trivial behaviour of the energy fluxes. The energy introduced by the pressure drop follows two routes: part of it is transferred towards the walls to be dissipated and part feeds the turbulent production hence the velocity fluctuations in the separating shear layer. Spatial energy fluxes transfer the kinetic energy into the recirculation bubble and downstream near the wall where it is ultimately dissipated. Consistently, anisotropy concentrates at small scales near the walls irrespective of the value of the Reynolds number. In the bulk flow and in the recirculation bubble, isotropy is restored at small scales and the isotropy recovery rate is controlled by the Reynolds number. Anisotropy invariant maps are presented, showing the difficulty in developing suitable turbulence models to predict separated turbulent flow dynamics. Results shed light on the processes of production, transfer and dissipation of energy in this relatively complex turbulent flow where non-homogeneous effects overwhelm the classical picture of wall-bounded turbulent flows which typically exploits streamwise homogeneity.

The increasing size of aircraft engines is leading to reconsideration of their conventional integration in the under-wing configuration due to the strong interaction between the jet and the airframe components. As a consequence, more insight is needed into the complex mechanisms underlying the interaction phenomenon between the jet flow and a surface. The objective of this paper is to carry out a series of experimental tests on a simplified laboratory-scale model to approach/deepen the problem. This analysis is the continuation of a previous study (Di Marco et al., J. Fluid Mech., vol. 770, 2015, pp. 247–272) on a rigid flat plate installed tangentially to the axis of an incompressible jet. In the present work, the velocity and wall pressure fields were simultaneously measured for different radial distances of the plate from the nozzle axis. Pointwise hot-wire anemometer measurements were carried out to characterize the effect of the plate on the velocity field statistics up to the fourth-order moment. The analysis revealed that the presence of the plate brings about a deflection of the mean aerodynamic field over the surface and a reduction of the turbulence intensity. Wall pressure fluctuations induced by the jet flow over the plate were measured by a linear array of cavity-mounted microphones placed along the streamwise direction. The velocity/pressure cross-statistics are achieved in the time and frequency domains using cross-correlations and Fourier analysis. A wavelet-based conditional sampling procedure is applied as well to characterize the flow signatures related to the velocity and wall pressure fluctuations, revealing that the surface induces the breakdown of the large-scale turbulent structures. The dependence of the multivariate and conditioned statistics on the plate distance from the jet as well as on the streamwise and crosswise probe positions is extensively discussed. Different organized flow motions over the surface are found based on the wall pressure events detected. A scaling criterion for velocity signatures is also presented addressing the governing parameters of the jet–plate interaction phenomenon.

This paper describes a series of experiments using particle image velocimetry to investigate the dynamic stall resulting due to a rapid pitching motion of a flat plate. There exist in such unsteady separated flows multiple time-dependent coherent structures, whose interaction and evolution are complex and nonlinear. The experiments presented here are aimed at determining the behaviour of a dynamic stall vortex system in the Reynolds number range $10^{3}<Re<10^{4}$. Evidence is presented for the development of the three-dimensional structure associated with the dynamic stall vortex and its interaction with the no-slip boundary condition at the surface of the pitching plate. The analysis presented suggests that a centrifugal instability exists, and that the form of the three-dimensional structure is consistent with that expected of a centrifugal instability. The structure and scale dependence of the flow are explored using wavelet and Fourier methods, with the dependence of the flow on Reynolds number examined, as well as the influence of spanwise end boundary conditions.

The mixing properties of statically stable density interfaces subject to imposed vertical shear are studied using direct numerical simulations of stratified plane Couette flow. The simulations are designed to investigate possible self-maintaining mechanisms of sharp density interfaces motivated by Phillips’ argument (Deep-Sea Res., vol. 19, 1972, pp. 79–81) by which layers and interfaces can spontaneously form due to vertical variations of diapycnal flux. At the start of each simulation, a sharp density interface with the same initial thickness is introduced at the midplane between two flat, horizontal walls counter-moving at velocities $\pm U_{w}$. Particular attention is paid to the effects of varying Prandtl number $\mathit{Pr}\equiv \unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}$, where $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D705}$ are the molecular kinematic viscosity and diffusivity respectively, over two orders of magnitude from 0.7, 7 and 70. Varying $\mathit{Pr}$ enables the system to access a considerable range of characteristic turbulent Péclet numbers $\mathit{Pe}_{\ast }\equiv {\mathcal{U}}_{\ast }{\mathcal{L}}_{\ast }/\unicode[STIX]{x1D705}$, where ${\mathcal{U}}_{\ast }$ and ${\mathcal{L}}_{\ast }$ are characteristic velocity and length scales, respectively, of the motion which acts to ‘scour’ the density interface. The dynamics of the interface varies with the stability of the interface which is characterised by a bulk Richardson number $\mathit{Ri}\,\equiv \,b_{0}h/U_{w}^{2}$, where $b_{0}$ is half the initial buoyancy difference across the interface and $h$ is the half-height of the channel. Shear-induced turbulence occurs at small $\mathit{Ri}$, whereas internal waves propagating on the interface dominate at large $\mathit{Ri}$. For a highly stable (i.e. large $\mathit{Ri}$) interface at sufficiently large $\mathit{Pe}_{\ast }$, the complex interfacial dynamics allows the interface to remain sharp. This ‘self-sharpening’ is due to the combined effects of the ‘scouring’ induced by the turbulence external to the interface and comparatively weak molecular diffusion across the core region of the interface. The effective diapycnal diffusivity and irreversible buoyancy flux are quantified in the tracer-based reference coordinate proposed by Winters & D’Asaro (J. Fluid Mech., vol. 317, 1996, pp. 179–193) and Nakamura (J. Atmos. Sci., vol. 53, 1996, pp. 1524–1537), which enables a detailed investigation of the self-sharpening process by analysing the local budget of buoyancy gradient in the reference coordinate. We further discuss the dependence of the effective diffusivity and overall mixing efficiency on the characteristic parameters of the flow, such as the buoyancy Reynolds number and the local gradient Richardson number, and highlight the possible role of the molecular properties of fluids on diapycnal mixing.

An optimal sensor placement procedure is proposed within the framework of variational data assimilation (DA) for unsteady flows, with the aim of maximizing the efficiency of the DA procedure. It is dedicated to the a priori design of a sensor network, and relies on a first-order adjoint approach. The proposed methodology first consists in identifying, via optimal control, the locations in the flow that have the greatest sensitivity with respect to a change in the initial condition, boundary conditions or model parameters. In a second step, sensors are placed at these locations for DA purposes. The use of this optimal sensor placement procedure does not require extra development in the case where a variational DA suite is available. The proposed methodology is applied to the reconstruction of unsteady bidimensional flows past a rotationally oscillating cylinder. More precisely, the possibilities of reconstructing the rotational speed of the cylinder and the initial flow, which here encompasses upstream conditions, from various types of observations are investigated via variational DA. Then, the observation optimization procedure is employed to identify optimal locations for placing velocity sensors downstream of the cylinder. Both reduction in the computational cost and improvement in the quality of the reconstructed flow are achieved through optimal sensor placement, encouraging the application of the proposed methodology to more complex and realistic flows.

Snow avalanches are typically initiated on marginally stable slopes with a surface layer of fresh snow that may easily be incorporated into them. The erosion of snow at the front is fundamental to the dynamics and growth of snow avalanches and they may rapidly bulk up, making them much more destructive than the initial release. Snow may also deposit at the rear, base and sides of the flow and the net balance of erosion and deposition determines whether an avalanche grows or decays. In this paper, small-scale analogue experiments are performed on a rough inclined plane with a static erodible layer of carborundum grains. The static layer is prepared by slowly closing down a flow from a hopper at the top of the slope. This leaves behind a uniform-depth layer of thickness $h_{stop}$ at a given slope inclination. Due to the hysteresis of the rough bed friction law, this layer can then be inclined to higher angles provided that the thickness does not exceed $h_{start}$, which is the maximum depth that can be held static on a rough bed. An avalanche is then initiated on top of the static layer by releasing a fixed volume of carborundum grains. Dependent on the slope inclination and the depth of the static layer three different behaviours are observed. For initial deposit depths above $h_{stop}$, the avalanche rapidly grows in size by progressively entraining more and more grains at the front and sides, and depositing relatively few particles at the base and tail. This leaves behind a trough eroded to a depth below the initial deposit surface and whose maximal areal extent has a triangular shape. Conversely, a release on a shallower slope, with a deposit of thickness $h_{stop}$, leads to net deposition. This time the avalanche leaves behind a levee-flanked channel, the floor of which lies above the level of the initial deposit and narrows downstream. It is also possible to generate avalanches that have a perfect balance between net erosion and deposition. These avalanches propagate perfectly steadily downslope, leaving a constant-width trail with levees flanking a shallow trough cut slightly lower than the initial deposit surface. The cross-section of the trail therefore represents an exact redistribution of the mass reworked from the initial static layer. Granular flow problems involving erosion and deposition are notoriously difficult, because there is no accepted method of modelling the phase transition between static and moving particles. Remarkably, it is shown in this paper that by combining Pouliquen & Forterre’s (J. Fluid Mech., vol. 453, 2002, pp. 133–151) extended friction law with the depth-averaged $\unicode[STIX]{x1D707}(I)$-rheology of Gray & Edwards (J. Fluid Mech., vol. 755, 2014, pp. 503–544) it is possible to develop a two-dimensional shallow-water-like avalanche model that qualitatively captures all of the experimentally observed behaviour. Furthermore, the computed wavespeed, wave peak height and stationary layer thickness, as well as the distance travelled by decaying avalanches, are all in good quantitative agreement with the experiments. This model is therefore likely to have important practical implications for modelling the initiation, growth and decay of snow avalanches for hazard assessment and risk mitigation.

A simple criterion for water particles to surf an underlying surface gravity wave is presented. It is found that particles travelling near the phase speed of the wave, in a geometrically confined region on the forward face of the crest, increase in speed. The criterion is derived using the equation of John (Commun. Pure Appl. Maths, vol. 6, 1953, pp. 497–503) for the motion of a zero-stress free surface under the action of gravity. As an example, a breaking water wave is theoretically and numerically examined. Implications for upper-ocean processes, for both shallow- and deep-water waves, are discussed.

The fluctuations in power output from wind farms display significantly reduced spectra compared to single wind turbines due to power smoothing and averaging. In order to better understand these spectral features and to relate them to properties of turbulent boundary layers, we perform a wind tunnel experiment in which we measure spatio-temporal characteristics of an experimental surrogate of the power output from a micro wind farm with 100 porous disk models. The experimental results show that the frequency spectrum of the total wind farm power follows a power law with a slope between $-5/3$ and $-2$, and up to lower frequencies than seen for any individual turbine model. In agreement with previous studies in the literature, peaks in the spectrum are observed at frequencies corresponding to the mean flow convection time between consecutive turbines. In the current work we interpret the sum of power extraction from an array of turbines as a discrete spatial filtering of a turbulent boundary layer and derive the associated transfer function. We apply it to an existing model for the wavenumber–frequency spectrum of turbulent boundary layers. This approach allows us to verify the individual roles of Doppler shift and broadening of frequencies on the resulting spatially sampled frequency spectrum. Comparison with the wind tunnel data confirms that the approach captures and explains the main features in the spectrum, indicating the crucial role of the interaction between the spatial sampling and the space–time correlations inherently present in the flow. The frequency spectrum of the aggregated power from a wind farm thus depends on both the spectrum of the incoming turbulence and its modulation by the spatial distribution of turbines in the boundary layer flow.

Interaction of a compliant wall with a turbulent channel flow is investigated experimentally by simultaneously measuring the time-resolved, three-dimensional (3D) flow field and the two-dimensional (2D) surface deformation. The optical set-up integrates tomographic particle image velocimetry to measure the flow with Mach–Zehnder interferometry to map the deformation. The Reynolds number is $Re_{\unicode[STIX]{x1D70F}}=2300$, and the Young’s modulus of the wall is 0.93 MPa, resulting in a ratio of shear speed to the centreline velocity ($U_{0}$) of 6.8. The wavenumber–frequency spectra of deformation show the surface motions consist of a non-advected low-frequency component and advected modes, some travelling downstream at approximately $U_{0}$ and others at ${\sim}0.72U_{0}$. The r.m.s. values of the advected and non-advected modes are $0.04~\unicode[STIX]{x03BC}\text{m}$$(0.004\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})$ and $0.2~\unicode[STIX]{x03BC}\text{m}$ ($0.02\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$), respectively, much smaller than the wall unit ($\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$), hence they do not affect the flow. Trends in the wall dynamics are elucidated by correlating the deformation with flow variables, including the 3D pressure distribution calculated by spatially integrating the material acceleration. Predictions by the Chase [J. Acoust. Soc. Am., vol. 89 (6), pp. 2589–2596] linear model are also calculated and compared to the measured trends. The spatial deformation–pressure correlations peak at $y/h\approx 0.12$ ($h$ is half channel height), the elevation of Reynolds shear stress maximum in the log-layer. Streamwise lagging of the deformation behind the pressure is caused in part by phase lag of the pressure with decreasing distance from the wall, and in part by material damping. Positive deformations (bumps) caused by negative pressure fluctuations are preferentially associated with ejections involving spanwise vortices located downstream and quasi-streamwise vortices with spanwise offset. Results of conditional correlations are consistent with the presence of hairpin-like structures. The negative deformations (dimples) are preferentially associated with positive pressure fluctuations at the transition between an upstream sweep to a downstream ejection.

Problems are posed and solved for upper-ocean submesoscale density fronts and filaments in the presence of surface wind stress and the associated boundary-layer turbulent mixing, their associated geostrophic and secondary circulations and their instantaneous buoyancy fluxes and frontogenetic evolutionary tendencies in both velocity and buoyancy gradients. The analysis is diagnostic rather than prognostic, and it is based on a momentum-balanced approximation that assumes the ageostrophic acceleration is negligible, although the Rossby number is finite and ageostrophic advection is included, justified by the quasi-steady, coherent-structure flow configurations of fronts and filaments. Across a wide range of wind and buoyancy-gradient parameters, the ageostrophic secondary circulation for a front is a single overturning cell with downwelling on the dense side, hence with a positive (restratifying) vertical buoyancy flux. For a dense filament the circulation is a double cell with central downwelling and again positive vertical buoyancy flux. The primary explanation for these secondary-circulation cells is a ‘turbulent thermal wind’ linear momentum balance. These circulation patterns, and their associated frontogenetic tendencies in both the velocity and buoyancy gradients, are qualitatively similar to those due to the ‘classical’ mechanism of strain-induced frontogenesis. For linear solutions, the secondary circulation and frontogenesis are essentially independent of wind direction, but in nonlinear solutions ageostrophic advection provides a strong intensification of the peak vertical velocity, while generally preserving the ageostrophic circulation pattern, when the Rossby number is order one and the wind orientation relative to the frontal axis is favourable. At large Rossby number the solution procedure fails to converge, with an implication of a failure of existence of wholly balanced circulations.

In this work the quantitative and qualitative ability of a kinetic-theory-based two-fluid model (KT-TFM) is assessed in a state of fully periodic sedimentation (fluidization), with a focus on statistically steady, unstable (clustered) states. The accuracy of KT-TFM predictions is evaluated via direct comparison to direct numerical simulation (DNS) data. The KT-TFM and DNS results span a rather wide parameter space: mean-flow Reynolds numbers on the order of 1 and 10, mean solid volume fractions from 0.1 to 0.4, solid-to-fluid density ratios from 10 to 1000 and elastic and moderately inelastic (restitution coefficient of 0.9) conditions. Data from both KT-TFM and DNS display a rich variety of statistically steady yet unstable structures (clusters). Instantaneous snapshots of KT-TFM and DNS demonstrate remarkable qualitative agreement. This qualitative agreement is quantified by calculating the critical density ratio at which the structure transitions from a chaotic, dynamic state to a regular, plug-flow state, with good overall comparisons. Further quantitative assessments of mean and fluctuating velocities show good agreement at high density ratios but weaker agreement at intermediate to low density ratios depending on the mean-flow Reynolds numbers and solid fractions. Deviations of the KT-TFM results from the DNS data were traced to a breakdown in one of the underlying assumptions of the kinetic theory derivation: high thermal Stokes number. Surprisingly, however, even though the low Knudsen number assumption, also associated with the kinetic theory derivation, is violated throughout most of the parameter space, it does not seem to affect the good quantitative accuracy of KT-TFM simulations.

The Greek aperitif Ouzo is not only famous for its specific anise-flavoured taste, but also for its ability to turn from a transparent miscible liquid to a milky-white coloured emulsion when water is added. Recently, it has been shown that this so-called Ouzo effect, i.e. the spontaneous emulsification of oil microdroplets, can also be triggered by the preferential evaporation of ethanol in an evaporating sessile Ouzo drop, leading to an amazingly rich drying process with multiple phase transitions (Tan et al., Proc. Natl Acad. Sci. USA, vol. 113 (31), 2016, pp. 8642–8647). Due to the enhanced evaporation near the contact line, the nucleation of oil droplets starts at the rim which results in an oil ring encircling the drop. Furthermore, the oil droplets are advected through the Ouzo drop by a fast solutal Marangoni flow. In this article, we investigate the evaporation of mixture droplets in more detail, by successively increasing the mixture complexity from pure water over a binary water–ethanol mixture to the ternary Ouzo mixture (water, ethanol and anise oil). In particular, axisymmetric and full three-dimensional finite element method simulations have been performed on these droplets to discuss thermal effects and the complicated flow in the droplet driven by an interplay of preferential evaporation, evaporative cooling and solutal and thermal Marangoni flow. By using image analysis techniques and micro-particle-image-velocimetry measurements, we are able to compare the numerically predicted volume evolutions and velocity fields with experimental data. The Ouzo droplet is furthermore investigated by confocal microscopy. It is shown that the oil ring predominantly emerges due to coalescence.

The scaling behaviour of the longitudinal velocity structure functions $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$ (where $2p$ represents the order) is studied for various wall-bounded turbulent flows. It has been known that for very large Reynolds numbers within the logarithmic region, the structure functions can be described by $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}/U_{\unicode[STIX]{x1D70F}}^{2}\approx D_{p}\ln (r/z)+E_{p}$ (where $r$ is the longitudinal distance, $z$ the distance from the wall, $U_{\unicode[STIX]{x1D70F}}$ the friction velocity and $D_{p}$, $E_{p}$ are constants) in accordance with Townsend’s attached eddy hypothesis. Here we show that the ratios $D_{p}/D_{1}$ extracted from plots between structure functions – in the spirit of the extended self-similarity hypothesis – have further reaching universality for the energy containing range of scales. Specifically, we confirm that this description is universal across wall-bounded flows with different flow geometries, and also for both the longitudinal and transversal structure functions, where previously the scaling has been either difficult to discern or differences have been reported when examining the direct representation of $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$. In addition, we present evidence of this universality at much lower Reynolds numbers, which opens up avenues to examine structure functions that are not readily available from high Reynolds number databases.

Gas–surface interactions play important roles in internal rarefied gas flows, especially in micro-electro-mechanical systems with large surface area to volume ratios. Although great progress has been made to solve the Boltzmann equation, the gas kinetic boundary condition (BC) has not been well studied. Here we assess the accuracy of the Maxwell, Epstein and Cercignani–Lampis BCs, by comparing numerical results of the Boltzmann equation for the Lennard–Jones potential to experimental data on Poiseuille and thermal transpiration flows. The four experiments considered are: Ewart et al. (J. Fluid Mech., vol. 584, 2007, pp. 337–356), Rojas-Cárdenas et al. (Phys. Fluids, vol. 25, 2013, 072002) and Yamaguchi et al. (J. Fluid Mech., vol. 744, 2014, pp. 169–182; vol. 795, 2016, pp. 690–707), where the mass flow rates in Poiseuille and thermal transpiration flows are measured. This requires that the BC has the ability to tune the effective viscous and thermal slip coefficients to match the experimental data. Among the three BCs, the Epstein BC has more flexibility to adjust the two slip coefficients, and hence for most of the time it gives good agreement with the experimental measurements. However, like the Maxwell BC, the viscous slip coefficient in the Epstein BC cannot be smaller than unity but the Cercignani–Lampis BC can. Therefore, we propose to combine the Epstein and Cercignani–Lampis BCs to describe gas–surface interaction. Although the new BC contains six free parameters, our approximate analytical expressions for the viscous and thermal slip coefficients provide useful guidance to choose these parameters.

This study investigates the shock transformation in an underexpanded jet in a confined duct when the jet total pressure is increased. Experimental study reveals that the Mach reflection (MR) in the fully underexpanded jet transforms to a regular reflection (RR) at a certain jet total pressure. It is observed that neither the incident shock angle nor the upstream Mach number varies during the MR–RR shock transformation. This is in contradiction to the classical MR–RR transformations in internal flow over wedges and in underexpanded open jets. This transformation is found to be a total pressure variation induced transformation, which is a new kind of shock transformation. The present study also reveals that the critical jet total pressures for MR–RR and RR–MR transformations are not the same when the primary pressure is increasing and decreasing, suggesting a hysteresis in the shock transformations.

The aeroacoustic feedback loop establishing in a supersonic round jet impinging on a flat plate normally has been investigated by combining compressible large-eddy simulations and modelling of that loop. At the exit of a straight pipe nozzle of radius $r_{0}$, the jet is ideally expanded, and has a Mach number of 1.5 and a Reynolds number of $6\times 10^{4}$. Four distances between the nozzle exit and the flat plate, equal to $6r_{0}$, $8r_{0}$, $10r_{0}$ and $12r_{0}$, have been considered. In this way, the variations of the convection velocity of the shear-layer turbulent structures according to the nozzle-to-plate distance are shown. In the spectra obtained inside and outside of the flow near the nozzle, several tones emerge at Strouhal numbers in agreement with measurements in the literature. At these frequencies, by applying Fourier decomposition to the pressure fields, hydrodynamic-acoustic standing waves containing a whole number of cells between the nozzle and the plate and axisymmetric or helical jet oscillations are found. The tone frequencies and the mode numbers inferred from the standing-wave patterns are in line with the classical feedback-loop model, in which the loop is closed by acoustic waves outside the jet. The axisymmetric or helical nature of the jet oscillations at the tone frequencies is also consistent with a wave analysis using a jet vortex-sheet model, providing the allowable frequency ranges for the upstream-propagating acoustic wave modes of the jet. In particular, the tones are located on the part of the dispersion relations of the modes where these waves have phase and group velocities close to the ambient speed of sound. Based on the observation of the pressure fields and on frequency–wavenumber spectra on the jet axis and in the shear layers, such waves are identified inside the present jets, for the first time to the best of our knowledge, for a supersonic jet flow. This study thus suggests that the feedback loop in ideally expanded impinging jets is completed by these waves.

This work reports an experimental investigation of the flow through a random array of fixed solid spheres. The volume fraction of the spheres is 2 %, and the Reynolds number $Re$ based on the sphere diameter and the average flow velocity is varied from 120 to 1040. Using time and spatial averaging, the fluctuations have been decomposed into two contributions of different natures: a spatial fluctuation that accounts for the strong inhomogeneity of the flow around each sphere, and a time fluctuation that comes from the instability of the flow at large enough Reynolds numbers. The evolutions of these two contributions with the Reynolds number are different, so that their relative importance varies. However, when each is normalized by using its own variance and the integral length scales of the fluctuations, their spectra and probability density functions (PDFs) are almost independent of $Re$ . The spatial fluctuation mostly comes from the velocity deficit in the wakes of the spheres, and is thus dominated by scales larger than one or two sphere diameters. It is found to be responsible for the asymmetry of the PDFs of the vertical fluctuations and of the major part of the anisotropy level between the vertical and the horizontal components of the fluctuations. The time fluctuation dominates at scales smaller than the integral length scale. It is isotropic and its PDFs, well described by an exponential distribution, are non-Gaussian. The spectra of the spatial and the time fluctuations both show an evolution as the power $-3$ of the wavenumber, but not exactly in the same subrange. All these properties are found in remarkable agreement with the results of both experimental investigations and large eddy simulations (LES) of a homogeneous bubble swarm. This confirms that the main mechanism responsible for the production of bubble-induced fluctuations is the interaction of the velocity disturbances caused by obstacles immersed in a flow and that the structure of this agitation is weakly dependent on the precise nature of the obstacles. The understanding and the modelling of the agitation generated by the motion of a dispersed phase, such as the bubble-induced agitation, therefore require one to distinguish between the roles of these two contributions.