Data-driven low-order modelling has been enjoying rapid advances in fluid mechanics. Arguably, Sirovich (Q. Appl. Maths, vol. XLV, 1987, pp. 561–571) started these developments with snapshot proper orthogonal decomposition, a particularly simple method. The resulting reduced-order models provide valuable insights into flow physics, allow inexpensive explorations of dynamics and operating conditions, and enable model-based control design. A winning argument for proper orthogonal decomposition (POD) is the optimality property, i.e. the guarantee of the least residual for a given number of modes. The price is unpleasant frequency mixing in the modes which complicates their physical interpretation. In contrast, temporal Fourier modes and dynamic mode decomposition (DMD) provide pure frequency dynamics but lose the orthonormality and optimality property of POD. Sieber et al. (J. Fluid Mech., vol. 792, 2016, pp. 798–828) bridge the least residual and pure frequency behaviour with an ingenious interpolation, called spectral proper orthogonal decomposition (SPOD). This article puts the achievement of the TU Berlin authors in perspective, illustrating the potential of SPOD and the challenges ahead.

The nonlinear stability of immiscible two-fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni effects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatio-temporal evolution of the interface and its local surfactant concentration. The system is non-local and arises by appropriately matching solutions of the linearised Navier–Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled Péclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two- and three-dimensional disturbances is investigated and a Squire’s type theorem is found to hold when inertia is absent. When inertia is present, three-dimensional disturbances can be more unstable than two-dimensional ones and so Squire’s theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evolution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two-dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stability via Hopf bifurcations to time-periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics becomes more complex and includes time-periodic, quasi-periodic as well as chaotic fluctuations. It is also found that one-dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three-dimensional flows with interfacial profiles that are two-dimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to three-dimensional disturbances but not two-dimensional ones. These are found to have a one-dimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form.

Partial cavitation in the separated region forming from the apex of a wedge is examined to reveal the flow mechanism responsible for the transition from stable sheet cavity to periodically shedding cloud cavitation. High-speed visualization and time-resolved X-ray densitometry measurements are used to examine the cavity dynamics, including the time-resolved void-fraction fields within the cavity. The experimentally observed time-averaged void-fraction profiles are compared to an analytical model employing free-streamline theory. From the instantaneous void-fraction flow fields, two distinct shedding mechanisms are identified. The classically described re-entrant flow in the cavity closure is confirmed as a mechanism for vapour entrainment and detachment that leads to intermittent shedding of smaller-scale cavities. But, with a sufficient reduction in cavitation number, large-scale periodic cloud shedding is associated with the formation and propagation of a bubbly shock within the high void-fraction bubbly mixture in the separated cavity flow. When the shock front impinges on flow at the wedge apex, a large cloud is pinched off. For periodic shedding, the speed of the front in the laboratory frame is of the order of half the free-stream speed. The features of the observed condensation shocks are related to the average and dynamic pressure and void fraction using classical one-dimensional jump conditions. The sound speed of the bubbly mixture is estimated to determine the Mach number of the cavity flow. The transition from intermittent to transitional to strongly periodic shedding occurs when the average Mach number of the cavity flow exceeds that required for the generation of strong shocks.

The scale interactions occurring within a turbulent boundary layer are investigated in the presence of free-stream turbulence. The free-stream turbulence is generated by an active grid. The free stream is monitored by a single-component hot-wire probe, while a second probe is roved across the height of the boundary layer at the same streamwise location. Large-scale structures occurring in the free stream are shown to penetrate the boundary layer and increase the streamwise velocity fluctuations throughout. It is speculated that, depending on the extent of the penetration, i.e. based on the level of free-stream turbulence, the near-wall turbulence production peaks at different wall-normal locations than the expected location of $y^{+}\approx 15$ for a canonical turbulent boundary layer. It is shown that the large scales dominating the log region have a modulating effect on the small scales in the near-wall region; this effect becomes more significant with increasing turbulence in the free stream, i.e. similarly increasing $Re_{\unicode[STIX]{x1D706}_{0}}$. This modulating interaction and its Reynolds-number trend have similarities with canonical turbulent boundary layers at high Reynolds numbers where the interaction between the large scales and the envelope of the small scales exhibits a pure amplitude modulation (Hutchins & Marusic, Phil. Trans. R. Soc. Lond. A, vol. 365 (1852), 2007, pp. 647–664; Mathis et al., J. Fluid Mech., vol. 628, 2009, pp. 311–337). This similarity has encouraging implications towards generalising scale interactions in turbulent boundary layers.

The interactions between homogeneous turbulence and a planar shock wave are analytically investigated using rapid distortion theory (RDT). Analytical solutions in the solenoidal modes are obtained. Qualitative answers to unsolved questions in a report by Andreopoulos et al. (Annu. Rev. Fluid Mech., vol. 524, 2000, pp. 309–345) are provided within the linear theoretical framework. The results show that the turbulence kinetic energy (TKE) is increased after interaction with a shock wave and that the contributions to the amplification can be interpreted primarily as the combined effect of shock-induced compression, which is a direct consequence of the Rankine–Hugoniot relation, and the nonlinear effect, which is an indirect consequence of the Rankine–Hugoniot relation via the perturbation manner. For initial homogeneous axisymmetric turbulence, the amplification of the TKE depends on the initial degree of anisotropy. Furthermore, the increase in energy at high wavenumbers is confirmed by the one-dimensional spectra. The enstrophy is also increased; its increase is more significant than that of the TKE because of the significant increase in enstrophy at high wavenumbers. The vorticity components perpendicular to the shock-induced compressed direction are amplified more than the parallel vorticity component. These results strongly suggest that a high resolution is needed to obtain accurate results for the turbulence–shock-wave interaction. The integral length scales ($L$) and the Taylor microscales ($\unicode[STIX]{x1D706}$) are decreased for most cases after the interaction. However, $L_{22,3}(=\,L_{33,2})$ and $\unicode[STIX]{x1D706}_{22,3}(=\,\unicode[STIX]{x1D706}_{33,2})$ are amplified. Here, the subscripts 2 and 3 indicate the perpendicular components relative to the shock-induced compressed direction. The dissipation length and TKE dissipation rate are amplified.

Statistical properties of turbulent Rayleigh–Bénard convection at low Prandtl numbers $Pr$, which are typical for liquid metals such as mercury or gallium ($Pr\simeq 0.021$) or liquid sodium ($Pr\simeq 0.005$), are investigated in high-resolution three-dimensional spectral element simulations in a closed cylindrical cell with an aspect ratio of one and are compared to previous turbulent convection simulations in air for $Pr=0.7$. We compare the scaling of global momentum and heat transfer. The scaling exponent $\unicode[STIX]{x1D6FD}$ of the power law $Nu=\unicode[STIX]{x1D6FC}Ra^{\unicode[STIX]{x1D6FD}}$ is $\unicode[STIX]{x1D6FD}=0.265\pm 0.01$ for $Pr=0.005$ and $\unicode[STIX]{x1D6FD}=0.26\pm 0.01$ for $Pr=0.021$, which are smaller than that for convection in air ($Pr=0.7$, $\unicode[STIX]{x1D6FD}=0.29\pm 0.01$). These exponents are in agreement with experiments. Mean profiles of the root-mean-square velocity as well as the thermal and kinetic energy dissipation rates have growing amplitudes with decreasing Prandtl number, which underlies a more vigorous bulk turbulence in the low-$Pr$ regime. The skin-friction coefficient displays a Reynolds number dependence that is close to that of an isothermal, intermittently turbulent velocity boundary layer. The thermal boundary layer thicknesses are larger as $Pr$ decreases and conversely the velocity boundary layer thicknesses become smaller. We investigate the scaling exponents and find a slight decrease in exponent magnitude for the thermal boundary layer thickness as $Pr$ decreases, but find the opposite case for the velocity boundary layer thickness scaling. A growing area fraction of turbulent patches close to the heating and cooling plates can be detected by exceeding a locally defined shear Reynolds number threshold. This area fraction is larger for lower $Pr$ at the same $Ra$, but the scaling exponent of its growth with Rayleigh number is reduced. Our analysis of the kurtosis of the locally defined shear Reynolds number demonstrates that the intermittency in the boundary layer is significantly increased for the lower Prandtl number and for sufficiently high Rayleigh number compared to convection in air. This complements our previous findings of enhanced bulk intermittency in low-Prandtl-number convection.

We compute the hydrodynamic torque on a dumbbell (two spheres linked by a massless rigid rod) settling in a quiescent fluid at small but finite Reynolds number. The spheres have the same mass densities but different sizes. When the sizes are quite different, the dumbbell settles vertically, aligned with the direction of gravity, the largest sphere first. But when the size difference is sufficiently small, then its steady-state angle is determined by a competition between the size difference and the Reynolds number. When the sizes of the spheres are exactly equal, then fluid inertia causes the dumbbell to settle in a horizontal orientation.

The Richtmyer–Meshkov instability on a ‘V’ shaped air/SF$_{6}$ gaseous interface is experimentally studied in a shock tube. By the soap film technique, a discontinuous interface without supporting mesh is formed so that the initial conditions of the interface can be accurately controlled. Five ‘V’ shaped air/$\text{SF}_{6}$ interfaces with different vertex angles ($60^{\circ }$, $90^{\circ }$, $120^{\circ }$, $140^{\circ }$ and $160^{\circ }$) are created where the ratio of the initial interface amplitude to the wavelength varies to highlight the effects of initial condition on the flow characteristics. The wave patterns and interface morphologies are clearly identified in the high-speed schlieren sequences, which show that the interface deforms in a less pronounced manner with less vortices generated as the vertex angle increases. A regime change is observed in the interface width growth rate near a vertex angle of $160^{\circ }$, which provides an experimental evidence for the numerical results obtained by McFarland et al. (Phys. Scr. vol. T155, 2013, 014014). The growth rate of interface width in the linear phase is compared with the theoretical predictions from the classical impulsive model and a modified linear model, and the latter is proven to be effective for a moderate to large initial amplitude. It is found that the initial growth rate of the interface width is a non-monotone function of the initial vertex angle (amplitude–wavelength ratio), i.e. the interface width growth rate in the linear stage experiences an increase and then a decrease as the vertex angle increases. A similar conclusion was also reached by Dell et al. (Phys. Plasmas, vol. 22, 2015, 092711) numerically for a sinusoidal interface. Finally, the general behaviour of the interface width growth in the nonlinear stage can be well captured by the nonlinear model proposed by Dimonte & Ramaprabhu (Phys. Fluids, vol. 22, 2010, 014104).

We report experimental results for the power spectra, variance, skewness and kurtosis of temperature fluctuations in turbulent Rayleigh–Bénard convection (RBC) of a fluid with Prandtl number $Pr=12.3$ in cylindrical samples with aspect ratios $\unicode[STIX]{x1D6E4}$ (diameter $D$ over height $L$) of 0.50 and 1.00. The measurements were primarily for the radial positions $\unicode[STIX]{x1D709}=1-r/(D/2)=1.00$ and $\unicode[STIX]{x1D709}=0.063$. In both cases, data were obtained at several vertical locations $z/L$. For all locations, there is a frequency range of about a decade over which the spectra can be described well by the power law $P(f)\sim f^{-\unicode[STIX]{x1D6FC}}$. For all $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D6E4}$, the $\unicode[STIX]{x1D6FC}$ value is less than one near the top and bottom plates and increases as $z/L$ or $1-z/L$ increase from 0.01 to 0.5. This differs from the finding for$Pr=0.8$ (He et al., Phys. Rev. Lett., vol. 112, 2014, 174501) and the expectation for the downstream velocity of turbulent wall-bounded shear flow (Rosenberg, J. Fluid Mech., vol. 731, 2013, pp. 46–63), where $\unicode[STIX]{x1D6FC}=1$ is found or expected in an inner layer ($0.01\lesssim z/L\lesssim 0.1$) near the wall but in the bulk. The variance is described better by a power law $\unicode[STIX]{x1D70E}^{2}\sim (z/L)^{-\unicode[STIX]{x1D701}}$ than by the logarithmic dependence found or expected for $Pr=0.8$ and for turbulent shear flow. For both $\unicode[STIX]{x1D6E4}$, we found that, independent of Rayleigh number, $\unicode[STIX]{x1D701}\simeq 2/3$ near the sidewall ($\unicode[STIX]{x1D709}=0.063$), where plumes primarily rise or fall and the large-scale circulation (LSC) dynamics is most influential. This result agrees with a model due to Priestley (Turbulent Transfer in the Lower Atmosphere, 1959, University of Chicago Press) for convection over a horizontal heated surface. However, we found $\unicode[STIX]{x1D701}\simeq 1$ along the sample centreline ($\unicode[STIX]{x1D709}=1.00$), where there are relatively few plumes moving vertically and the LSC dynamics is expected to be less important; that result is consistent with one of two possible interpretations by Adrian (Intl J. Heat Mass Transfer, vol. 39, 1996, pp. 2303–2310) of a model due to Libchaber et al. (J. Fluid Mech., vol. 204, 1989, pp. 1–30). We discuss the composite nature of fluctuations in turbulent RBC, with contributions from intrinsic background fluctuations, plumes, the stochastic dynamics of the LSC, and the sloshing and torsional mode of the LSC. None of the models advanced so far explicitly consider all of these contributions.

We systematically study the shape and dynamics of a Newtonian ferrofluid drop immersed in an immiscible, Newtonian and non-magnetic viscous fluid under the action of a uniform external magnetic field. We obtain the exact equilibrium drop shapes for arbitrary ferrofluids, characterize the extent of deviations of the exact shape from the commonly assumed ellipsoidal shape, and analyse the smoothness of highly curved tips in elongated drops. We also present a comprehensive study of drop deformation for a Langevin ferrofluid. Using a computational scheme that allows fast and accurate simulations of ferrofluid drop dynamics, we show that the dynamics of drop deformation by an applied magnetic field is described up to a numerical factor by the same time scale as drop relaxation in the absence of any magnetic field. The numerical factor depends on the ratio of viscosities and the ratio of magnetic to capillary stresses, but is independent of the nature of the ferrofluid in most practical cases.

Fast liquid jets, called micro-jets, are produced within cavitation bubbles experiencing an aspherical collapse. Here we review micro-jets of different origins, scales and appearances, and propose a unified framework to describe their dynamics by using an anisotropy parameter $\unicode[STIX]{x1D701}\geqslant 0$, representing a dimensionless measure of the liquid momentum at the collapse point (Kelvin impulse). This parameter is rigorously defined for various jet drivers, including gravity and nearby boundaries. Combining theoretical considerations with hundreds of high-speed visualisations of bubbles collapsing near a rigid surface, near a free surface or in variable gravity, we classify the jets into three distinct regimes: weak, intermediate and strong. Weak jets ($\unicode[STIX]{x1D701}<10^{-3}$) hardly pierce the bubble, but remain within it throughout the collapse and rebound. Intermediate jets ($10^{-3}<\unicode[STIX]{x1D701}<0.1$) pierce the opposite bubble wall close to the last collapse phase and clearly emerge during the rebound. Strong jets ($\unicode[STIX]{x1D701}>0.1$) pierce the bubble early during the collapse. The dynamics of the jets is analysed through key observables, such as the jet impact time, jet speed, bubble displacement, bubble volume at jet impact and vapour-jet volume. We find that, upon normalising these observables to dimensionless jet parameters, they all reduce to straightforward functions of $\unicode[STIX]{x1D701}$, which we can reproduce numerically using potential flow theory. An interesting consequence of this result is that a measurement of a single observable, such as the bubble displacement, suffices to estimate any other parameter, such as the jet speed. Remarkably, the dimensionless parameters of intermediate and weak jets ($\unicode[STIX]{x1D701}<0.1$) depend only on $\unicode[STIX]{x1D701}$, not on the jet driver (i.e. gravity or boundaries). In the same regime, the jet parameters are found to be well approximated by power laws of $\unicode[STIX]{x1D701}$, which we explain through analytical arguments.

A standard approach to propulsion velocities of autophoretic colloids with thin interaction layers uses a reciprocity relation applied to the slip velocity although the surface flux (chemical, electrical, thermal, etc.), which is the source of the field driving the slip, is often more accessible. We show how, under conditions of low Reynolds number and a field obeying the Laplace equation in the outer region, the slip velocity can be bypassed in velocity calculations. In a sense, the actual slip velocity and a normal field proportional to the flux density are equivalent for this type of calculation. Using known results for surface traction induced by rotating or translating an inert particle in a quiescent fluid, we derive simple and explicit integral formulas for translational and rotational velocities of arbitrary spheroidal and slender-body autophoretic colloids.

Detonations usually form through either direct initiation or deflagration-to-detonation transition (DDT). In this work, a detonation initiation process is introduced that shows attributes from each of these two processes. Energy is deposited into a finite volume of fluid in an amount of time that is similar to the acoustic time scale of the heated fluid volume. Two-dimensional simulations of the reactive Euler equations are used to solve for the evolving detonation initiation process. The results show behaviour similar to both direct initiation and DDT. Localized reaction transients are shown to be intimately related to the appearance of a detonation. Thermomechanical concepts are used to provide physical interpretations of the computational results in terms of the interaction between compressibility phenomena on the acoustic time scale and localized, spatially resolved, chemical energy addition on a heat-addition time scale.

We consider the motion of a single ellipsoidal vortex with uniform potential vorticity in a rotating stratified fluid at finite Rossby number $\unicode[STIX]{x1D716}$. Building on previous solutions obtained under the quasi-geostrophic approximation (at first order in $\unicode[STIX]{x1D716}$), we obtain analytical solutions for the balanced part of the flow at $O(\unicode[STIX]{x1D716}^{2})$. These solutions capture important ageostrophic effects giving rise to an asymmetry in the evolution of cyclonic and anticyclonic vortices. Previous work has shown that, if the velocity field induced by an ellipsoidal vortex only depends linearly on spatial coordinates inside the vortex, i.e. $\boldsymbol{u}=\unicode[STIX]{x1D64E}\boldsymbol{x}$, then the dynamics reduces markedly to a simple matrix equation. The instantaneous vortex shape and orientation are encapsulated in a symmetric $3\times 3$ matrix $\unicode[STIX]{x1D63D}$, which is acted upon by the flow matrix $\unicode[STIX]{x1D64E}$ to provide the vortex evolution. Under the quasi-geostrophic approximation, the flow matrix is determined by inverting the potential vorticity to obtain the streamfunction via Poisson’s equation, which has a known analytical solution depending on elliptic integrals. Here we show that higher-order balanced solutions, up to second order in the Rossby number, can also be calculated analytically. However, in this case there is a vector potential that requires the solution of three Poisson equations for each of its components. The source terms for these equations are independent of spatial coordinates within the ellipsoid, depending only on the elliptic integrals solved at the leading, quasi-geostrophic order. Unlike the quasi-geostrophic case, these source terms do not in general vanish outside the ellipsoid and have an inordinately complicated dependence on spatial coordinates. In the special case of an ellipsoid whose axes are aligned with the coordinate axes, we are able to derive these source terms and obtain the full analytical solution to the three Poisson equations. However, if one considers the homogeneous case, whereby the outer source terms are neglected, one can obtain an approximate solution having a compact matrix form analogous to the leading-order quasi-geostrophic case. This approximate solution proves to be highly accurate for the general case of an arbitrarily oriented ellipsoid, as verified through comparisons of the solutions with solutions obtained from numerical simulations of an ellipsoid using an accurate nonlinear balance model, even at moderate Rossby numbers.

Direct numerical simulations of particle-laden spatially developing turbulent boundary layers over a flat plate have been performed to investigate the effect of inertial particles on turbulence modulation, using the Eulerian–Lagrangian point-particle approach with two-way coupling. The particles are smaller than the Kolmogorov length scale of the dilute flow, and inter-particle collisions are not considered. The simulation results show that the addition of small solid particles increases the mean streamwise fluid velocity, which in turn leads to a reduction in the boundary layer integral parameters and an increase in the skin-friction drag. These effects become more pronounced as the particle Stokes number and mass loading increase. The streamwise turbulence intensity is slightly enhanced in the close vicinity of the wall but damped in the outer layer. In contrast, the Reynolds stress and the turbulence intensities in the wall-normal and spanwise directions are substantially attenuated across the entire boundary layer, and the levels of attenuation increase monotonically with both particle Stokes number and mass loading. The exchange of kinetic energy between particles and fluid indicates that particle–fluid interactions cause extra energy dissipation, which plays a crucial role in turbulence modulation.

The Rayleigh–Taylor instability induced turbulence is studied under the Boussinesq approximation focusing on vorticity and mixing. A direct numerical simulation has been carried out with an auto-adaptive multidomain Chebyshev–Fourier–Fourier numerical method. The spatial resolution is increased up to $(24\times 40)\times 940^{2}=848\,M$ collocation points. The Taylor Reynolds number is $\mathit{Re}_{\unicode[STIX]{x1D706}_{zz}}\approx 142$ and a short inertial range is observed. The nonlinear growth rate of the turbulent mixing layer is found to be close to $\unicode[STIX]{x1D6FC}_{b}=0.021$ . Our conclusions may be summarized as follows.

(i) The simulation data are in agreement with the scalings for the pressure ( $k^{-7/3}$ ) and the vertical mass flux ( $k^{-7/3}$ ).

(ii) Mean quantities have a self-similar behaviour, but some inhomogeneity is still present. For higher-order quantities the self-similar regime is not fully achieved.

(iii) In the self-similar regime, the mean dissipation rate and the enstrophy behave as $\langle \overline{\unicode[STIX]{x1D700}}\rangle \propto t$ and $\langle \overline{\unicode[STIX]{x1D714}_{i}\,\unicode[STIX]{x1D714}_{i}}^{1/2}\rangle \propto t^{1/2}$ , respectively.

(iv) The large-scale velocity fluctuation probability density function (PDF) is Gaussian, while vorticity and dissipation PDFs show large departures from Gaussianity.

(v) The pressure PDF exhibits strong departures from Gaussianity and is skewed. This is related to vortex coherent structures.

(vi) The intermediate scales of the mixing are isotropic, while small scales remain anisotropic. This leaves open the possibility of a small-scale buoyancy. Velocity intermediate scales are also isotropic, while small scales remain anisotropic. Mixing and dynamics are therefore consistent.

(vii) Properties and behaviours of vorticity and enstrophy are detailed. In particular, equations for these quantities are written down under the Boussinesq approximation.

(viii) The concentration PDF is quasi-Gaussian. The vertical concentration gradient is both non-Gaussian and strongly skewed.

We present a three-region model for the time-averaged behaviour of established turbulent axisymmetric fountains at high source Froude numbers $(Fr_{0})$ in which we uniquely account for entrainment of ambient fluid both laterally and at the fountain top. High-$Fr_{0}$ ‘forced’ fountains, as originally investigated experimentally by Turner (J. Fluid Mech., vol. 26 (4), 1966, pp. 779–792), are characterised by an upflow, a counterflow and a fountain top where the flow reverses direction. Through the inclusion of the flow-reversal region and by accounting for fountain-top entrainment, which is neglected in all existing models, close agreement is achieved between our solutions and existing experimental data. Moreover, our predictions of the fluxes within the fountain are in accord with scaling arguments deduced in recent studies. Our model reveals five key ratios that characterise the fountain asymptote to constant values in the high-$Fr_{0}$ limit. These are the ratios of the (1) initial and mean rise heights, (2) vertical extents of the fountain top and upflow regions, (3) fluxes of volume entrained into the fountain top and entrained laterally into the counterflow, (4) forces of inertia and buoyancy acting on the counterflow at the level of the source and (5) average times taken for fluid to rise through the upflow and fall through the counterflow. Attributing the invariance of these ratios to the global self-preserving behaviour of the fountain, we propose a threshold source Froude number for which a continuous negatively buoyant release may be regarded as giving rise to a ‘forced’ fountain.

A field observation array for the atmospheric surface layer (ASL) was built on a dry flat bed of Qingtu Lake in Minqin (China) as the Qingtu Lake Observation Array (QLOA) site, which is similar to the Surface Layer Turbulence and Environmental Science Test (SLTEST) site in the Utah (USA) Western desert. The present observation array can synchronously perform multi-point measurements of wind velocity and temperature at different vertical and streamwise positions. In other words, three-dimensional turbulent ASL flows can be measured at the QLOA station and Reynolds numbers as high as $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{6})$ can be achieved with steady wind conditions. By careful selection and pretreatment for measured data of more than 1200 h, the QLOA data have been validated to be reliable for high Reynolds number turbulent boundary layer research. Results from correlation and spectral analysis confirm that very large scale motions (VLSMs) exist in the ASL at a Reynolds number up to $Re_{\unicode[STIX]{x1D70F}}\approx 4\times 10^{6}$. Through premultiplied spectral analysis, it is revealed that the spectral energy in the high-wavenumber region decreases with height, similar to turbulent boundary layers at low or moderate Reynolds numbers, while it increases with height in the low-wavenumber region resulting in a log–linear increase of VLSMs energy with height, which is different from turbulent boundary layers at low or moderate Reynolds numbers. The present analyses support the view that the evolution of the VLSMs cannot be fully attributed to a ‘bottom-up’ mechanism alone, and probably other mechanisms, including a ‘top-down’ mechanism, also play a role.

The velocity fields induced by regular breaking waves past a fixed bar on a 1 : 10 rigid plane slope were measured and analysed using a volumetric particle-tracking velocimetry system. Under specific conditions, the interaction between waves and morphological features steepens the waves, which eventually break. The geometry of the boundaries of the present experiments is common in natural environments, where reefs, sand and gravel bars, and submerged coastal structures, interact with the incoming wave field, ‘affecting’ the transport budget of substances (sediment, nutrients and pollutants), with relevant consequences on the water quality. The aims of the present work are the analysis of the flow field in the breaker, and the quantification of the terms in the equations usually adopted for modelling the flow and the turbulence. Two sets of attacking monochromatic wave trains with different periods and heights were used to generate a data set of instantaneous velocity, which was further analysed to extract turbulence. The measurement volume extended from the wave crest to a portion of the domain below the wave trough. The balance of linear momentum for the average field and the balance of turbulence were scrutinized, and included all the terms in a three-dimensional (3D) approach. The analysed data and results are original and novel because they include all the contributions derived from the 3D structure of a real flow field, and constitute a huge data set for the calibration of numerical codes.

Panel flutter is a dangerous aeroelastic instability of the skin panels of supersonic flight vehicles. Though the linear stability of panels in uniform flow has been studied in detail, the influence of the boundary layer is still an open question. Most studies of panel flutter in the presence of the boundary layer are devoted to the ($1/7$)th-power velocity law and yield a stabilising effect of the boundary layer. Recently, Vedeneev (J. Fluid Mech., vol. 736, 2013, pp. 216–249) considered arbitrary velocity and temperature profiles and showed that, for a generalised convex boundary layer profile, a decrease of the growth rates of ‘supersonic’ perturbations (responsible for single-mode panel flutter) is accompanied by destabilisation of ‘subsonic’ perturbations that are neutral in uniform flow. However, this result is not self-consistent, as the long-wave expansion for solutions of the Rayleigh equation was used, whereas subsonic perturbations, generally speaking, cannot be considered as long waves. More surprising results are obtained for the boundary layer profile with a generalised inflection point, where the effect of the layer is destabilising even for ‘supersonic’ perturbations, and such waves can also have short lengths. In order to overcome this inconsistency, in this paper, we solve the Rayleigh equation numerically and investigate the stability of short-wave perturbation of the elastic plate in the presence of the boundary layer. As before, two problem formulations are investigated. First, we study running waves in an infinite plate. Second, we analyse eigenmodes of the plate of large finite length and use Kulikovskii’s global instability criterion. Based on the results of calculations, we confirm that the effect of the boundary layer with a generalised inflection point can be essentially destabilising. On the other hand, for generalised convex boundary layers, calculations show that, unlike the prediction of the long-wave approximation, the finite plate is fully stabilised for sufficiently thick boundary layers.