Resolvent analysis is performed to identify the origin of two-dimensional transonic buffet over an airfoil. The base flow for the resolvent analysis is the time-averaged flow over a NACA 0012 airfoil at a chord-based Reynolds number of 2000 and a free-stream Mach number of 0.85. We reveal that the mechanism of buffet is buried underneath the global low-Reynolds-number flow physics. At this low Reynolds number, the dominant flow feature is the von Kármán shedding. However, we show that with the appropriate forcing input, buffet can appear even at a Reynolds number that is much lower than what is traditionally associated with transonic buffet. The source of buffet is identified to be at the shock foot from the windowed resolvent analysis, which is validated by companion simulations using sustained forcing inputs based on resolvent modes. We also comment on the role of perturbations in the vicinity of the trailing edge. The present study not only provides insights on the origin of buffet but also serves a building block for low-Reynolds-number compressible aerodynamics in light of the growing interests in Martian flights.

We consider the size spectrum of entrained bubbles under strong free-surface turbulence (SFST). We investigate the entrainment bubble-size spectrum per unit (mean) interface area, ${\mathcal{N}}_{a}^{E}(r)$, with dimension length$^{-3}$, and develop a physical/mechanistic model for ${\mathcal{N}}_{a}^{E}(r)$ through energy arguments. The model obtains two distinct regimes of ${\mathcal{N}}_{a}^{E}(r)$, separated by bubble-size scale $r_{0}$. For bubble radius $r>r_{0}$, the effects of gravity $g$ dominate those of the surface tension force $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}$, and ${\mathcal{N}}_{a}^{E}(r)\propto g^{-1}\unicode[STIX]{x1D716}^{2/3}r^{-10/3}$, where $\unicode[STIX]{x1D716}$ is the turbulence dissipation rate. For $r<r_{0}$, surface tension is more important and ${\mathcal{N}}_{a}^{E}(r)\propto (\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C})^{-1}\unicode[STIX]{x1D716}^{2/3}r^{-4/3}$. From the model, we show that $r_{0}\approx r_{c}=1/2\sqrt{\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}g}$, the capillary length scale, and not the generally assumed Hinze scale $r_{H}$. For an air–water interface and Earth gravity, $r_{c}\approx$ 1.5 mm. The model provides an $\unicode[STIX]{x1D716}$–$r$ entrainment regime map that identifies a critical dissipation rate $\unicode[STIX]{x1D716}_{cr}$ (constant for given $g$ and $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}$) above which there is appreciable air entrainment, thus separating SFST and weak FST. We confirm the theoretical model and its predictions using two-phase, high-fidelity direct numerical simulations of a canonical FST flow using the conservative volume-of-fluid method: the respective power laws of ${\mathcal{N}}_{a}^{E}(r)\propto r^{-10/3}$ and $r^{-4/3}$ for $r>r_{0}$ and $r<r_{0}$; the value $r_{0}=r_{c}$; the scaling ${\mathcal{N}}_{a}^{E}(r)\propto \unicode[STIX]{x1D716}^{2/3}$; and the predictions of the $\unicode[STIX]{x1D716}$–$r$ entrainment regime map.

High-resolution direct numerical simulations (DNS) were carried out to investigate the nonlinear breakdown process of a three-dimensional wave packet initiated by a short-duration pulse in a flared cone boundary layer at Mach 6 and zero angle of attack. For these simulations the cone geometry of the flared cone experiments conducted in the Boeing/AFOSR Mach 6 Quiet Tunnel (BAM6QT) at Purdue University was considered. The computational domain covered a large extent of the cone in the azimuthal direction to allow for a wide range of azimuthal wavenumbers ($k_{c}$) and to include shallow instability waves with small azimuthal wavenumbers. The simulation results indicated that the wave packet development was dominated by axisymmetric and shallow (small $k_{c}$) second-mode waves for a large downstream extent. Towards the downstream end of the computational domain a rapid broadening of the disturbance amplitude spectra was observed, which is an indication that the wave packet reached the strongly nonlinear stages. The disturbance spectra of the nonlinear regime, and the downstream amplitude development of the dominant disturbance wave components, provided conclusive evidence that the so-called fundamental breakdown was the dominant nonlinear mechanism. Furthermore, contours of the time-averaged Stanton number exhibited ‘hot’ streaks within the wave packet on the surface of the cone. Hot streaks have also been observed in the Purdue flared cone experiments using temperature sensitive paint (TSP) and in numerical investigations using DNS. The azimuthal streak spacing obtained from the wave packet simulation agrees well with that observed in the Purdue quiet tunnel experiments.

We prove a new rigorous upper bound on the vertical heat transport for Bénard–Marangoni  convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number $Ma\gg 1$ the Nusselt number $Nu$ is bounded asymptotically by $Nu\leqslant \text{const.}\times Ma^{2/7}(\ln Ma)^{-1/7}$. Key to our proof are a background temperature field with a hyperbolic profile near the fluid’s surface and new estimates for the coupling between temperature and vertical velocity.

Turbulent flow over a surface with streamwise-elongated rough and smooth stripes is studied by means of direct numerical simulation (DNS) in a periodic plane open channel with fully resolved roughness. The goal is to understand how the mean height of roughness affects the characteristics of the secondary flow formed above a spanwise heterogeneous rough surface. To this end, while the statistical properties of roughness texture as well as the width and spacing of the rough stripes are kept constant, the elevation of the smooth stripes is systematically varied in different simulation cases. Utilizing this variation, three configurations – representing protruding, recessed and an intermediate type of roughness – are analysed. In all cases, secondary flows are present and the skin friction coefficients calculated for all the heterogeneous rough surfaces are meaningfully larger than what would result from the area-weighted average of those of homogeneous smooth and rough surfaces. This drag increase appears to be linked to the strength of the secondary flow. The rotational direction of the secondary motion is shown to depend on the relative surface elevation. The present results suggest that this rearrangement of the secondary flow is linked to the spatial distribution of the spanwise-wall-normal Reynolds stress component, which carries opposing signs for protruding and recessed roughness.

The interaction of a shock wave and a water droplet embedded with a vapour cavity is experimentally investigated in a shock tube for the first time. The vapour cavity inside the droplet is generated by decreasing the surrounding pressure to the saturation pressure, and an equilibrium between the liquid phase and the gas phase is obtained inside the droplet. Direct high-speed photography is adopted to capture the evolution of both the droplet and the vapour cavity. The formation of a transverse jet inside the droplet during the cavity-collapse stage is clearly observed. Soon afterwards, at the downstream pole of the droplet, a water jet penetrating into the surrounding air is observed during the cavity-expansion stage. The evolution of the droplet is strongly influenced by the evolution of the vapour cavity. The phase change process plays an important role in vapour cavity evolution. The effects of the relative size and eccentricity of the cavity on the movement and deformation of the droplet are presented quantitatively.

The general exotic capillary tube is a non-uniform capillary tube which permits an entire continuum of equilibrium menisci if applying a pressure $p=-\unicode[STIX]{x1D700}z$ at the tube inlet. The shapes of general exotic capillary tubes under positive and negative loads are determined mathematically. Lowering the pressure at the tube inlet slightly from the value $p=-\unicode[STIX]{x1D700}z$ causes the tube to completely drain out, while raising the pressure slightly forces the tube to fill up, which implies that the general exotic capillary tube is sensitive to pressure. The general exotic capillary tube is also related to meniscus stability. It is found that the boundary parameters $\unicode[STIX]{x1D712}_{1}$ of general exotic cylinders with arbitrary contact angle are equal to the critical values $\unicode[STIX]{x1D712}_{1}^{\ast }$ for determining the meniscus stability. Then, a convenient alternative to solving the Jacobi equation for determining $\unicode[STIX]{x1D712}_{1}^{\ast }$ is proposed based on the ‘exotic’ property.

We consider the unbounded settling dynamics of a circular disk of diameter $d$ and finite thickness $h$ evolving with a vertical speed $U$ in a linearly stratified fluid of kinematic viscosity $\unicode[STIX]{x1D708}$ and diffusivity $\unicode[STIX]{x1D705}$ of the stratifying agent, at moderate Reynolds numbers ($Re=Ud/\unicode[STIX]{x1D708}$). The influence of the disk geometry (diameter $d$ and aspect ratio $\unicode[STIX]{x1D712}=d/h$) and of the stratified environment (buoyancy frequency $N$, viscosity and diffusivity) are experimentally and numerically investigated. Three regimes for the settling dynamics have been identified for a disk reaching its gravitational equilibrium level. The disk first falls broadside-on, experiencing an enhanced drag force that can be linked to the stratification. A second regime corresponds to a change of stability for the disk orientation, from broadside-on to edgewise settling. This occurs when the non-dimensional velocity $U/\sqrt{\unicode[STIX]{x1D708}N}$ becomes smaller than some threshold value. Uncertainties in identifying the threshold value is discussed in terms of disk quality. It differs from the same problem in a homogeneous fluid which is associated with a fixed orientation (at its initial value) in the Stokes regime and a broadside-on settling orientation at low, but finite Reynolds numbers. Finally, the third regime corresponds to the disk returning to its broadside orientation after stopping at its neutrally buoyant level.

We study the term in the eddy energy budget of continuously stratified quasigeostrophic turbulence that is responsible for energy extraction by eddies from the background mean flow. This term is a quadratic form, and we derive Euler–Lagrange equations describing its eigenfunctions and eigenvalues, the former being orthogonal in the energy inner product and the latter being real. The eigenvalues correspond to the instantaneous energy growth rate of the associated eigenfunction. We find analytical solutions in the Eady problem. We formulate a spectral method for computing eigenfunctions and eigenvalues, and compute solutions in Phillips-type and Charney-type problems. In all problems, instantaneous growth is possible at all horizontal scales in both inviscid problems and in problems with linear Ekman friction. We conjecture that transient growth at small scales is matched by linear transfer to decaying modes with the same horizontal structure, and we provide simulations supporting the plausibility of this hypothesis. In Charney-type problems, where the linear problem has exponentially growing modes at small scales, we expect net energy extraction from the mean flow to be unavoidable, with an associated nonlinear transfer of energy to dissipation.

We study experimentally the motion and deformation of individual capsules transported by a constant volume-flux flow of low Reynolds number, through the T-junction of a channel with rectangular cross-section. We use millimetric ovalbumin-alginate capsules which we characterise independently of the flow experiment. Centred capsules travel at constant velocity down the straight channel leading to the T-junction, where they decelerate and expand in the spanwise direction before turning into one of the two identical daughter channels. There, non-inertial lift forces act to re-centre them and relax their shape until they reach a steady state of propagation. We find that the dynamics of fixed-size capsules within our channel geometry is governed by a capillary number $Ca$ defined as the ratio of viscous shear forces to elastic restoring forces, which we quantify by statically compressing the capsule between parallel plates to 50 % of its initial diameter, in order to account for different membrane thickness, pre-inflation and nonlinear elastic deformation. We show that the maximum extension in the T-junction of capsules of different stiffness collapses onto a master curve in $Ca$. This provides a sensitive measure of the relative stiffness of capsules at constant flow rate, particularly for softer capsules. We also find that the T-junction can sort fixed-size capsules according to their stiffness because the position in the T-junction from which capsules are entrained into the daughter channel depends uniquely on $Ca$. We demonstrate that a T-junction can be used as a sorting device by enhancing this initial capsule separation through a diffuser.

Large-eddy simulation (LES) of a wind- and wave-forced water column based on the Craik–Leibovich (C–L) vortex force is used to understand the structure of small-scale Langmuir circulation (LC) and associated Langmuir turbulence. The LES also serves to understand the role of the turbulence in determining molecular diffusive scalar flux from a scalar-saturated air side to the water side and the turbulent vertical scalar flux in the water side. Previous laboratory experiments have revealed that small-scale LC beneath an initially quiescent air–water interface appears shortly after the initiation of wind-driven gravity–capillary waves and provides the laminar–turbulent transition in wind speeds between 3 and $6~\text{m}~\text{s}^{-1}$. The LES reveals Langmuir turbulence characterized by multiple scales ranging from small bursting eddies at the surface that coalesce to give rise to larger (centimetre-scale) LC over time. It is observed that the smaller scales account for the bulk of the near-surface turbulent vertical scalar flux. Although the contribution of the larger (centimetre-scale) LC to the near-surface turbulent flux increases over time as these scales emerge and become more coherent, the contribution of the smaller scales remains dominant. The growing LC scales lead to increased vertical scalar transport at depths below the interface and thus greater scalar transfer efficiency. Simulations were performed with a fixed wind stress corresponding to a $5~\text{m}~\text{s}^{-1}$ wind speed but with different wave parameters (wavelength and amplitude) in the C–L vortex force. It is observed that longer wavelengths lead to more coherent, larger centimetre-scale LC providing greater contribution to the turbulent vertical scalar flux away from the surface. In all cases, the molecular diffusive scalar flux at the water surface relaxes to the same statistically steady value after transition to Langmuir turbulence occurs, despite the different wave parameters in the C–L vortex force across the simulations. This implies that the small-scale turbulence intensity and the molecular diffusive scalar flux at the surface scale with the wind shear and not with the wave parameters. Furthermore, it is seen that the Langmuir (wave) forcing (provided by the C–L vortex force) is necessary to trigger the turbulence that induces elevated molecular diffusive scalar flux at the water surface relative to wind-driven flow without wave forcing.

The flow within an infinitely long cylinder exhibiting solid-body rotation (SBR) is impulsively stopped. The complete decay of the initial SBR is captured by means of direct numerical simulations for a wide range of Reynolds numbers ($Re$). Five distinct stages are identified during the decay process according to their flow structure and their underlying mechanisms of kinetic-energy dissipation. Initially, the laminar boundary layer undergoes a primary centrifugal instability, which causes the formation of coherent Taylor rolls. The flow then becomes turbulent, once the Taylor rolls are corrupted by secondary instabilities. Within the turbulent stage, two phases are distinguished. In the first turbulent phase, the SBR core is still intact and turbulence is sustained. The mean velocity profile is well described by the superposition of a near-wall region, a retracting SBR core and an intermediate region of constant angular momentum. In the latter region, the magnitude of angular momentum in viscous units $l^{+}(Re)$ is approximately constant in time. In the second turbulent phase, the SBR core breaks down, turbulence starts to decay exponentially and the kinetic energy of the mean flow decays logarithmically. Eventually, the flow relaminarises and the velocity profile of the analytical solution for purely laminar decay is recovered, albeit at an earlier temporal instant due to the net effect of transition and turbulence.

Two aspects of homogeneous rotating turbulence are quantified through forced direct numerical simulations in an elongated domain, which, in the direction of rotation, is approximately 340 times larger than the typical initial eddy size. First, by following the time evolution of the integral length scale along the axis of rotation $\ell _{\Vert }$, the growth rate of the columnar eddies and its dependence on the Rossby number $Ro_{\unicode[STIX]{x1D700}}$ is determined as $\unicode[STIX]{x1D6FE}=3.90\exp (-16.72\,Ro_{\unicode[STIX]{x1D700}})$ for $0.06\leqslant Ro_{\unicode[STIX]{x1D700}}\leqslant 0.31$, where $\unicode[STIX]{x1D6FE}$ is the non-dimensional growth rate. Second, a scaling law for the energy dissipation rate $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D708}}$ is sought. Comparison with current available scaling laws shows that the relation proposed by Baqui & Davidson (Phys. Fluids, vol. 27(2), 2015, 025107), i.e. $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D708}}\sim {u^{\prime }}^{3}/\ell _{\Vert }$, where $u^{\prime }$ is the root-mean-square velocity, approximates well part of our data, more specifically the range $0.39\leqslant Ro_{\unicode[STIX]{x1D700}}\leqslant 1.54$. However, relations proposed in the literature fail to model the data for the second and most interesting range, i.e. $0.06\leqslant Ro_{\unicode[STIX]{x1D700}}\leqslant 0.31$, which is marked by the formation of columnar eddies. To find a similarity relation for the latter, we exploit the concept of a spectral transfer time introduced by Kraichnan (Phys. Fluids, vol. 8(7), 1965, p. 1385). Within this framework, the energy dissipation rate is considered to depend on both the nonlinear time scale and the relaxation time scale. Thus, by analysing our data, expressions for these different time scales are obtained that result in $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D708}}\sim (u^{\prime 4}Ro_{\unicode[STIX]{x1D700}}^{0.62}\unicode[STIX]{x1D70F}_{nl}^{iso})/\ell _{\bot }^{2}$, where $\ell _{\bot }$ is the integral length scale in the direction normal to the axis of rotation and $\unicode[STIX]{x1D70F}_{nl}^{iso}$ is the nonlinear time scale of the initial homogeneous isotropic field.

Flow separation and its control have been the subject of intensive research for decades. Flow separation occurs when the boundary layer loses contact with the associated confining wall, which is usually caused by a pressure gradient acting against the local flow direction. Numerous strategies exist to control flow separation, and in this study we demonstrate experimentally that vertical flow separation over steep slopes in shallow free-surface flows may be suppressed by contracting the flow horizontally upstream of the slope. We found that, unexpectedly, introducing lateral non-uniformity in the upstream flow field could suppress vertical flow separation for steep slopes up to 1 in 2. This study reveals the possibility of two different flow states over steep slopes; (i) a vertically attached flow combined with horizontal convergence, and (ii) a vertically detached flow combined with horizontal divergence. A detailed analysis of the dynamics of the two different flow states is presented. Although a predictive relation determining the transition point between the two flow states was not found in the current study, the observed phenomena were shown to be strongly related to the magnitude of the lateral gradient at the upstream edge of the slope. The results demonstrate a significant influence of the vertical flow state – separated or attached – on the shear stress at the confining boundaries of the flow.

Significant insights in computational fluid dynamics have been obtained in recent years by adopting the data assimilation methods developed in the meteorology community. We apply the four-dimensional variational method to reconstruct the small scales of three-dimensional turbulent velocity fields with a moderate Reynolds number, given a time sequence of measurement data on a coarse set of grid points. The problem presents new challenges because the evolution of the flow is dominated by the nonlinear vortex stretching and the energy cascade process, which are absent from two-dimensional flows. The results show that, reconstruction is successful when the resolution of the measurement data, given in terms of the wavenumber, is of the order of the threshold value $k_{c}=0.2\unicode[STIX]{x1D702}_{K}^{-1}$, where $\unicode[STIX]{x1D702}_{K}$ is the Kolmogorov length scale of the flow. When the data are available over a period of one large eddy turnover time scale, the filtered enstrophy and other small-scale quantities are reconstructed with a 30 % or smaller normalized pointwise error, and a 90 % pointwise correlation. The spectral correlation between the reconstructed and target fields is higher than 80 % for all wavenumbers. Minimum volume enclosing ellipsoids (MVEEs) and MVEE trees are introduced to quantitatively compare the geometry of non-local structures. Results show that, for the majority samples, errors in the locations and the sizes of the reconstructed structures are within 15 %, and those in the orientations are within $15^{\circ }$. Overall, for this flow, satisfactory reconstruction of the scales two or more octaves smaller is possible if data at large scales are available for at least one large eddy turnover time. In comparison, a direct substitution scheme results in three times bigger pointwise discrepancy in enstrophy. The spectral difference between the reconstructed and target velocity fields is more than ten times higher than what is obtained with the four-dimensional variational method. The results show that further investigation is warranted to verify the efficacy of the method in flows with higher Reynolds numbers.

The flow-induced vibration of a tethered sphere was investigated through numerical simulations. A determination of the different modes of sphere vibration was made with simulations conducted at fixed Reynolds numbers (500, 1200 and 2000) with a sphere of mass ratio 0.8 over the reduced velocity range $U^{\ast }\in [3,32]$. The flow was governed by the incompressible Navier–Stokes equations, while the dynamic motion of the sphere was governed by coupled Newtonian mechanics. A new fluid–structure interaction (FSI) solver was implemented to efficiently solve the coupled FSI system. The effect of Reynolds number was found to be significant in the mode I and II regimes. A progressive increase in the response amplitude was observed as the Reynolds number was increased, especially in the mode II regime. The overall sphere response at the highest Reynolds number was relatively close to the observed behaviour of previous higher-$Re$ experimental studies. An aperiodic mode IV response was observed at higher reduced velocities beyond the mode II range in each case, without the intervening mode III regime. However, as the mass ratio increased from 0.8 to 80, the random response of the sphere (mode IV) gradually became more regular, showing a mode III response (characterized by a near-periodic sphere oscillation) at $U^{\ast }=30$. Thus, if the inertia of the system is low, mode IV appears at lower $U^{\ast }$ values, while for high-inertia systems, mode IV appears at high $U^{\ast }$ values beyond a mode III response.

We study the unsteady flow of a viscous fluid passing through a circular aperture in a plate characterized by a non-zero thickness. We investigate this problem by solving the incompressible linearized Navier–Stokes equations around a laminar base flow, in both the forced case (allowing us to characterize the coupling of the flow with acoustic resonators) and the autonomous regime (allowing us to identify the possibility of purely hydrodynamic instabilities). In the forced case, we calculate the impedances and discuss the stability properties in terms of a Nyquist diagram. We show that such diagrams allow us to predict two kinds of instabilities: (i) a conditional instability linked to the over-reflexion of an acoustic wave but requiring the existence of a conveniently tuned external acoustic resonator, and (ii) a purely hydrodynamic instability existing even in a strictly incompressible framework. A parametric study is conducted to predict the range of existence of both instabilities in terms of the Reynolds number and the aspect ratio of the aperture. Analysing the structure of the linearly forced flow allows us to show that the instability mechanism is closely linked to the existence of a recirculation region within the thickness of the plate. We then investigate the autonomous regime using the classical eigenmode analysis. The analysis confirms the existence of the purely hydrodynamic instability in accordance with the impedance-based criterion. The spatial structure of the unstable eigenmodes are found to be similar to the structure of the corresponding unsteady flows computed using the forced problem. Analysis of the adjoint eigenmodes and of the adjoint-based structural sensitivity confirms that the origin of the instability lies in the recirculation region existing within the thickness of the plate.

The three-dimensional clusters of streamwise velocity fluctuations ($u$) in turbulent boundary layers (TBLs) are explored from the perspective of the attached-eddy model, which provides a basis for understanding the asymptotic behaviours of high-Reynolds-number wall turbulence in terms of coherent structures. We extract the $u$ clusters from the direct numerical simulation data of a TBL subjected to an adverse pressure gradient ($\unicode[STIX]{x1D6FD}=1.43$). For comparison, the direct numerical simulation data of a zero-pressure-gradient TBL are included. The identified structures are decomposed into attached self-similar, attached non-self-similar, detached self-similar and detached non-self-similar motions with respect to the minimum distance from the wall ($y_{min}$) and height ($l_{y}$). The attached structures ($y_{min}\approx 0$) are the main energy-containing motions and carry approximately half of the streamwise Reynolds stress and the Reynolds shear stress in the logarithmic and outer regions. The sizes of the attached self-similar structures scale with $l_{y}$, and their population density has an inverse-scale distribution over the range $0.4\unicode[STIX]{x1D6FF}<l_{y}<0.58\unicode[STIX]{x1D6FF}$ ($\unicode[STIX]{x1D6FF}$ is the 99 % boundary layer thickness). They also contribute to the logarithmic variation of the streamwise Reynolds stress and to the presence of the $k_{z}^{-1}$ region in the pre-multiplied energy spectra ($k_{z}$ is the spanwise wavenumber), i.e. these structures are universal wall motions in the logarithmic region. The tall attached structures with $l_{y}=O(\unicode[STIX]{x1D6FF})$ are non-self-similar and responsible for the enhancement of the outer large scales under the adverse pressure gradient. They extend beyond $6\unicode[STIX]{x1D6FF}$ in the streamwise direction and penetrate deeply into the near-wall region, which is reminiscent of very-large-scale motions or superstructures. The detached self-similar structures ($y_{min}>0$ and $l_{y}>100\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$) are geometrically isotropic and mainly arise in the outer region, whereas the sizes of the detached non-self-similar structures ($y_{min}>0$ and $l_{y}<100\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$) scale with the Kolmogorov length scale. Here, $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $u_{\unicode[STIX]{x1D70F}}$ the friction velocity. The present study provides a new perspective on the analysis of turbulence structures in the view of the attached-eddy model.

In the Prandtl model for anabatic slope flows, a uniform positive buoyancy flux at the surface drives an upslope flow against a stable background stratification. In the present study, we conduct linear stability analysis of the anabatic slope flow under this model and contrast it against the katabatic case as presented in Xiao & Senocak (J. Fluid Mech., vol. 865, 2019, R2). We show that the buoyancy component normal to the sloped surface is responsible for the emergence of stationary longitudinal rolls, whereas a generalised Kelvin–Helmholtz (KH) type of mechanism consisting of shear instability modulated by buoyancy results in a streamwise-travelling mode. In the anabatic case, for slope angles larger than $9^{\circ }$ to the horizontal, the travelling KH mode is dominant whereas, at lower inclination angles, the formation of the stationary vortex instability is favoured. The same dynamics holds qualitatively for the katabatic case, but the mode transition appears at slope angles of approximately $62^{\circ }$. For a fixed slope angle and Prandtl number, we demonstrate through asymptotic analysis of linear growth rates that it is possible to devise a classification scheme that demarcates the stability of Prandtl slope flows into distinct regimes based on the dimensionless stratification perturbation number. We verify the existence of the instability modes with the help of direct numerical simulations, and observe close agreements between simulation results and predictions of linear analysis. For slope angle values in the vicinity of the junction point in the instability map, both longitudinal rolls and travelling waves coexist simultaneously and form complex flow structures.

We apply the weak formalism on the Boussinesq equations to characterize scaling properties of the mean and the standard deviation of the potential, kinetic and viscous energy fluxes in very well-resolved numerical simulations. The local Bolgiano–Oboukhov (BO) length is investigated and it is found that its value may change by an order of magnitude through the domain, in agreement with previous results. We then investigate the scale-by-scale averaged terms of the weak equations, which are a generalization of the Kármán–Howarth–Monin and Yaglom equations. We have not found the classical BO picture, but evidence of a mixture of BO and Kolmogorov scalings. In particular, all the energy fluxes are compatible with a BO local Hölder exponent for the temperature and a Kolmogorov 41 for the velocity. This behaviour may be related to anisotropy and to the strong heterogeneity of the convective flow, reflected in the wide distribution of BO local scales. The scale-by-scale analysis allows us also to compare the theoretical BO length computed from its definition with that empirically extracted through scalings obtained from weak analysis. Scalings are observed, but over a limited range. The key result of the work is to show that the analysis of local weak formulation of the problem is powerful to characterize the fluctuation properties.