Direct numerical simulations have been performed for turbulent thermal convection between horizontal no-slip, permeable walls with a distance $H$ and a constant temperature difference $\Delta T$ at the Rayleigh number $Ra=3\times 10^3\text {--}10^{10}$. On the no-slip wall surfaces $z=0$, $H$, the wall-normal (vertical) transpiration velocity is assumed to be proportional to the local pressure fluctuation, i.e. $w=-\beta p'/\rho$, $+\beta p'/\rho$ (Jiménez et al., J. Fluid Mech., vol. 442, 2001, pp. 89–117). Here $\rho$ is mass density, and the property of the permeable wall is given by the permeability parameter $\beta U$ normalised with the buoyancy-induced terminal velocity $U=(g\alpha \Delta TH)^{1/2}$, where $g$ and $\alpha$ are acceleration due to gravity and volumetric thermal expansivity, respectively. The critical transition of heat transfer in convective turbulence has been found between the two $Ra$ regimes for fixed $\beta U=3$ and fixed Prandtl number $Pr=1$. In the subcritical regime at lower $Ra$ the Nusselt number $Nu$ scales with $Ra$ as $Nu\sim Ra^{1/3}$, as commonly observed in turbulent Rayleigh–Bénard convection. In the supercritical regime at higher $Ra$, on the other hand, the ultimate scaling $Nu\sim Ra^{1/2}$ is achieved, meaning that the wall-to-wall heat flux scales with $U\Delta T$ independent of the thermal diffusivity, although the heat transfer on the wall is dominated by thermal conduction. In the supercritical permeable case, large-scale motion is induced by buoyancy even in the vicinity of the wall, leading to significant transpiration velocity of the order of $U$. The ultimate heat transfer is attributed to this large-scale significant fluid motion rather than to transition to turbulence in boundary-layer flow. In such ‘wall-bounded’ convective turbulence, a thermal conduction layer still exists on the wall, but there is no near-wall layer of large change in the vertical velocity, suggesting that the effect of the viscosity is negligible even in the near-wall region. The balance between the dominant advection and buoyancy terms in the vertical Boussinesq equation gives us the velocity scale of $O(U)$ in the whole region, so that the total energy budget equation implies the Taylor dissipation law $\epsilon \sim U^3/H$ and the ultimate scaling $Nu\sim Ra^{1/2}$.

We revisit the dynamics of a thick liquid film flowing down a vertical fibre. Instead of deriving a long-wave model, we directly solve the Navier–Stokes equations using a domain mapping technique and the exact steady travelling wave solutions are explored using a dynamical systems theory approach. Three distinct flow regimes, labelled ‘a’, ‘b’ and ‘c’, observed in previous experiments (Kliakhandler et al., J. Fluid Mech., vol. 429, 2001, pp. 381–390) are investigated. Flow regime ‘a’ refers to a steady flow state in which large droplets are separated by a long thin film. Flow regime ‘b’ is a necklace-like flow. In flow regime ‘c’, a cyclic process of droplet coalescence and breakup was observed. By matching the mean flow rates of the travelling wave solutions and experimental data, our travelling wave solutions show an excellent agreement with flow regimes ‘a’ and ‘b’. The time-periodic flow regime ‘c’ is compared with direct numerical simulation of the Navier–Stokes equations. A snapshot of the simulation shows a remarkable similarity to an experimental image and the discrepancy of mean wave speed and maximal wave height between our numerical simulation and experimental data is negligible.

Rotating Rayleigh–Bénard convection is investigated numerically with the use of an asymptotic model that captures the rapidly rotating, small Ekman number limit, $Ek \rightarrow 0$. The Prandtl number ($Pr$) and the asymptotically scaled Rayleigh number ($\widetilde {Ra} = Ra Ek^{4/3}$, where $Ra$ is the typical Rayleigh number) are varied systematically. For sufficiently vigorous convection, an inverse kinetic energy cascade leads to the formation of a pair of large-scale vortices of opposite polarity, in agreement with previous studies of rapidly rotating convection. With respect to the kinetic energy, we find a transition from convection dominated states to a state dominated by large-scale vortices at an asymptotically reduced (small-scale) Reynolds number of $\widetilde {Re} \approx 6$ ($\widetilde {Re} = Re Ek^{1/3}$, where $Re$ is the Reynolds number associated with vertical flows) for all investigated values of $Pr$. The ratio of the depth-averaged kinetic energy to the kinetic energy of the convection reaches a maximum at $\widetilde {Re} \approx 24$, then decreases as $\widetilde {Ra}$ is increased. This decrease in the relative kinetic energy of the large-scale vortices is associated with a decrease in the convective correlations with increasing Rayleigh number. The scaling behaviour of the convective flow speeds is studied; although a linear scaling of the form $\widetilde {Re} \sim \widetilde {Ra}/Pr$ is observed over a limited range in Rayleigh number and Prandtl number, a clear departure from this scaling is observed at the highest accessible values of $\widetilde {Ra}$. Calculation of the forces present in the governing equations shows that the ratio of the viscous force to the buoyancy force is an increasing function of $\widetilde {Ra}$, that approaches unity over the investigated range of parameters.

The phenomena of self-sustained shock-wave oscillations over conical bodies with a blunt axisymmetric base subject to uniform high-speed flow are investigated in a hypersonic wind tunnel at Mach number $M = 6$. The flow and shock-wave dynamics is dictated by two non-dimensional geometric parameters presented by the three length scales of the body, two of which are associated with the conical forebody and one with the base. Time-resolved schlieren imagery from these experiments reveals the presence of two disparate states of shock-wave oscillations in the flow, and allows for the mapping of unsteadiness boundaries in the two-parameter space. Physical mechanisms are proposed to explain the oscillations and the transitions of the shock-wave system from steady to oscillatory states. In comparison with the canonical single-parameter problem of shock-wave oscillations over spiked-blunt bodies reported in literature, the two-parameter nature of the present problem introduces distinct elements to the flow dynamics.

The dynamics of intense vorticity is investigated by means of synchronisation experiments in direct numerical simulations of isotropic turbulence. By imposing similar dynamics above the dissipative range, the same structures of intense vorticity appear in two independent turbulent flows, showing that intense vorticity synchronises to large-scale dynamics. Remarkably, this synchronisation takes place despite the presence of chaos, and affects mostly the intense vorticity, but not so much the weak vorticity background, which remains comparatively asynchronous. These results pinpoint the role of large-scale dynamics in the formation of intense vorticity structures, the so-called ‘worms’, and rule out the possibility that they emerge primarily due to interactions within the dissipative range, and then grow or coalesce into elongated structures. The stretching of the vorticity vector by the large-scale rate-of-strain tensor is identified as the mechanism responsible for the synchronisation of intense vorticity, supporting the extended view of vortex stretching as a fundamental inter-scale mechanism in turbulence.

This study investigates deterministic wave forecasting from the perspective of the Zakharov equation. Forecasts based on linear dispersion, weakly nonlinear amplitude dispersion and the Zakharov equation are compared for reference ocean surfaces generated from the Joint North Sea Wave Observation Project and Pierson–Moskowitz spectra. This approach allows for the role of nonlinearity to be isolated, and demonstrates the success of simple frequency corrections in forecasting wave fields up to moderate steepness. The role of second-order bound waves is investigated by means of analytical formulae, and their impact on forecast accuracy is illustrated for a range of forecast times.

Conservation laws that relate the local time-rate-of-change of the spatial integral of a density function to the divergence of its flux through the boundaries of the integration domain provide integral constraints on the spatio-temporal development of a field. Here we show that a new type of conserved quantity exists that does not require integration over a particular domain but which holds locally at any point in the field. This is derived for the pseudo-energy density of non-divergent Rossby waves where local invariance is obtained for (i) a single plane wave, and (ii) waves produced by an impulsive point source of vorticity. The definition of pseudo-energy used here consists of a conventional kinetic part, as well as an unconventional pseudo-potential part, proposed by Buchwald (Proc. R. Soc. Lond. A, vol. 328, issue 1572, 1972, pp. 37–48). The anisotropic nature of the energy flux that appears in response to the point source further clarifies the role of the beta plane in the observed western intensification of ocean currents.

Solidifying ternary systems can exhibit complex natural convection phenomena, particularly due to the presence of two porous zones (cotectic and primary mush), and the rejection of two differently dense solutes. The primary objectives of this study are to investigate the following: (i) the natural convection patterns in various compositional regimes of a typical ternary system, and (ii) the role of the combined existence of the microstructure (facets and dendrites) in the porous zone on natural convection, with a motivation to enhance the current understanding of the microstructure–convection relationships. A ternary mixture is chosen such that different compositions of the three primary solidifying components lead to the formation of distinct ice, dendritic and faceted solid structures that cover the complete span of microstructure–convection relationships. The observations of flow in different compositional regimes show convection occurring in the form of plumes, random mixing and double-diffusive layering, as well as combinations of these, which are governed by the type of coexisting microstructures. The study reveals the occurrence of Rayleigh–Taylor instability with varying amounts of the heavier component. The bulk liquid composition showed a tendency to cross the cotectic line, and thus also change the nature of primary solidifying structure from faceted to dendritic in cases where facets and dendrites were present in cotectic mush, and facets in primary mush. These insights are believed to elucidate the complex mechanisms of ternary solidification, as well as provide important real-time data for direct numerical simulations.

The three-dimensional turbulence structure of the canonical daytime atmospheric boundary layer (ABL) reflects the balance between shear and buoyancy turbulence production characterized by $-z_i/L>0$, where $z_i$ is the boundary layer height and $L<0$ is the Obukhov length. In the shear-driven neutral ABL ($-z_i/L=0$) the surface layer is characterized by coherent low-speed streaks, while the ‘moderately convective’ state ($-z_i/L\sim 1\text {--}10$) contains streamwise-elongated sheet-like updraughts. Using large-eddy simulation, we analyse the transition in ABL turbulence structure in response to systematic increases in surface heat flux and $-z_i/L$ from neutral to moderately convective. We discover a sudden change in turbulence structure at the ‘critical’ state $-z_i/L=0.433$ with nearly three-fold increase in the streamwise coherence length of ‘streaks’ at the upper surface layer and four-fold increase in the streamwise scale of updraughts in the mixed layer – as well as the initiation of spatial correlations between the two that contribute to the local generation of thermal updraughts as surface heating increases. At subcritical stability states, buoyancy impacts only the vertical thickness of the streaks. ABL receptivity to buoyancy changes when supercritical and updraughts now grow in the vertical with increasing surface heat flux; the previously amalgamated updraughts extend further in the streamwise direction. The post-critical regime is highlighted by a ‘maximum coherence state’ at $-z_i/L=1.08$ and maximally coherent ‘large-scale rolls’. At increasingly unstable states ($-z_i/L>1.08$), roll coherence decreases and the vertical scales asymptote to the canonical moderately convective ABL.

The closure of a human lung airway is modelled as a pipe coated internally with a liquid that takes into account the viscoelastic properties of mucus. For a thick-enough coating, the Plateau–Rayleigh instability blocks the airway by the creation of a liquid plug, and the preclosure phase is dominated by the Newtonian behaviour of the liquid. Our previous study with a Newtonian-liquid model demonstrated that the bifrontal plug growth consequent to airway closure induces a high level of stress and stress gradients on the airway wall, which is large enough to damage the epithelial cells, causing sublethal or lethal responses. In this study, we explore the effect of the viscoelastic properties of mucus by means of the Oldroyd-B and FENE-CR model. Viscoelasticity is shown to be very relevant in the postcoalescence process, introducing a second peak of the wall shear stresses. This second peak is related to an elastic instability due to the presence of the polymeric extra stresses. For high-enough Weissenberg and Laplace numbers, this second shear stress peak is as severe as the first one. Consequently, a second lethal or sublethal response of the epithelial cells is induced.

Wave shape (e.g. wave skewness and asymmetry) impacts sediment transport, remote sensing and ship safety. Previous work showed that wind affects wave shape in intermediate and deep water. Here, we investigate the effect of wind on wave shape in shallow water through a wind-induced surface pressure for different wind speeds and directions to provide the first theoretical description of wind-induced shape changes. A multiple-scale analysis of long waves propagating over a shallow, flat bottom and forced by a Jeffreys-type surface pressure yields a forward or backward Korteweg–de Vries (KdV)–Burgers equation for the wave profile, depending on the wind direction. The evolution of a symmetric, solitary-wave initial condition is calculated numerically. The resulting wave grows (decays) for onshore (offshore) wind and becomes asymmetric, with the rear face showing the largest shape changes. The wave profile's deviation from a reference solitary wave is primarily a bound wave and trailing, dispersive, decaying tail. The onshore wind increases the wave's energy and skewness with time while decreasing the wave's asymmetry, with the opposite holding for offshore wind. The corresponding wind speeds are shown to be physically realistic, and the shape changes are explained as slow growth followed by rapid evolution according to the unforced KdV equation.

In a theoretical analysis, we generalise well-known asymptotic results to obtain expressions for the rate of transfer of material from the surface of an arbitrary, rigid particle suspended in an open pathline flow at large Péclet number, ${Pe}$. The flow may be steady or periodic in time. We apply this result to numerically evaluate expressions for the surface flux to a freely suspended, axisymmetric ellipsoid (spheroid) in Stokes flow driven by a steady linear shear. We complement these analytical predictions with numerical simulations conducted over a range of ${Pe} = 10^1\text {--}10^4$ and confirm good agreement at large Péclet number. Our results allow us to examine the influence of particle shape upon the surface flux for a broad class of flows. When the background flow is irrotational, the surface flux is steady and is prescribed by three parameters only: the Péclet number, the particle aspect ratio and the strain topology. We observe that slender prolate spheroids tend to experience a higher surface flux compared to oblate spheroids with equivalent surface area. For rotational flows, particles may begin to spin or tumble, which may suppress or augment the convective transfer due to a realignment of the particle with respect to the strain field.

Recent simulations indicate that streamwise-preferential porous materials have the potential to reduce drag in wall-bounded turbulent flows (Gómez-de-Segura & García-Mayoral, J. Fluid Mech., vol. 875, 2019, pp. 124–172). This paper extends the resolvent formulation to study the effect of such anisotropic permeable substrates on turbulent channel flow. Under the resolvent formulation, the Fourier-transformed Navier–Stokes equations are interpreted as a linear forcing–response system. The nonlinear terms are considered the endogenous forcing in the system that gives rise to a velocity and pressure response. A gain-based decomposition of the forcing–response transfer function – the resolvent operator – identifies response modes (resolvent modes) that are known to reproduce important structural and statistical features of wall-bounded turbulent flows. The effect of permeable substrates is introduced in this framework using the volume-averaged Navier–Stokes equations and a generalized form of Darcy's law. Substrates with high streamwise permeability and low spanwise permeability are found to suppress the forcing–response gain for the resolvent mode that serves as a surrogate for the energetic near-wall cycle. This reduction in mode gain is shown to be consistent with the drag reduction trends predicted by theory and observed in numerical simulations. Simulation results indicate that drag reduction is limited by the emergence of spanwise rollers resembling Kelvin–Helmholtz vortices beyond a threshold value of wall-normal permeability. The resolvent framework also predicts the conditions in which such energetic spanwise-coherent rollers emerge. These findings suggest that a limited set of resolvent modes can serve as the building blocks for computationally efficient models that enable the design and optimization of permeable substrates for passive turbulence control.

The interaction of fluids with surface-mounted obstacles in canopy flows leads to strong turbulence that dominates dispersion and mixing in the neutrally stable atmospheric surface layer. This work focuses on intermittency in the Lagrangian velocity statistics in a canopy flow, which is observed in two distinct forms. The first, small-scale intermittency, is expressed by non-Gaussian and not self-similar statistics of the velocity increments. The analysis shows an agreement in comparison with previous results from homogeneous isotropic turbulence (HIT) using the multifractal model, extended self-similarity and velocity increments’ autocorrelations. These observations suggest that the picture of small-scale Lagrangian intermittency in canopy flows is similar to that in HIT and, therefore, they extend the idea of universal Lagrangian intermittency to certain inhomogeneous and anisotropic flows. Second, it is observed that the root mean square of energy increments along Lagrangian trajectories depends on the direction of the trajectories’ time-averaged turbulent velocity. Subsequent analysis suggests that the flow is attenuated by the canopy drag while leaving the structure function's scaling unchanged. This observation implies the existence of large-scale intermittency in Lagrangian statistics. Thus, this work presents a first empirical evidence of intermittent Lagrangian velocity statistics in a canopy flow that exists in two distinct senses and occurs due to different mechanisms.

We present several time-dependent analytical solutions for the incompressible Euler system with a free surface. These analytical solutions give quantitative descriptions of some physical phenomena, such as water motion or waves on a space variable topography, and can be used as reference solutions when validating numerical simulation codes. They concern fluid flows governed by Euler equations with or without the hydrostatic hypothesis including a wet/dry interface, variable density and a wide variety of boundary conditions.

This work addresses the effect of travelling thermal waves applied at the fluid layer surface, on the formation of global flow structures in two-dimensional (2-D) and 3-D convective systems. For a broad range of Rayleigh numbers ($10^3\leq Ra \leq 10^7$) and thermal wave frequencies ($10^{-4}\leq \varOmega \leq 10^{0}$), we investigate flows with and without imposed mean temperature gradients. Our results confirm that the travelling thermal waves can cause zonal flows, i.e. strong mean horizontal flows. We show that the zonal flows in diffusion dominated regimes are driven purely by the Reynolds stresses and end up always travelling retrograde. In convection dominated regimes, however, mean flow advection, caused by tilted convection cells, becomes dominant. This generally leads to prograde directed mean zonal flows. By means of direct numerical simulations we validate theoretical predictions made for the diffusion dominated regime. Furthermore, we make use of the linear stability analysis and explain the existence of the tilted convection cell mode. Our extensive 3-D simulations support the results for 2-D flows and thus provide further evidence for the relevance of the findings for geophysical and astrophysical systems.

The probability density function (p.d.f.) of the streamwise velocity has been shown to indicate the presence of uniform momentum zones in wall-bounded turbulent flows. Most studies on the topic have focused on the instantaneous characteristics of this p.d.f. In this work, we show how the use of time-resolved particle image velocimetry data highlights robust features in the temporal behaviour of the p.d.f. and how these patterns are associated with the change of the number of zones present in the flow over time. The use of a limited resolvent model provides a clear link between this experimentally observed behaviour and the underlying velocity structures and their phase characteristics. This link is further supported by an extended resolvent model consisting of self-similar hierarchies centred in the logarithmic region, with triadically consistent members, yielding much more complex patterns in the p.d.f. Results indicate that the geometric similarity of these members instantaneously, as well as their relative evolution in time (dictated by their wall-normal varying wave speed), both inherent to the model, can reproduce many experimentally identified features.

Jumps in granular chute flow are obtained as continuous solutions of the properly adapted Saint–Venant equations. We elucidate their internal structure via a dynamical systems approach and show that the jumps in phase space manifest themselves as trajectories organized around the stable/unstable manifold of the fixed point representing the uniform flow. Based on this, we derive an analytic approximate expression for the jump length. The paper concludes with a numerical experiment confirming the stability of the jumps.

It is commonly accepted that the breakup criteria of drops or bubbles in turbulence is governed by surface tension and inertia. However, also buoyancy can play an important role at breakup. In order to better understand this role, here we numerically study two-dimensional Rayleigh–Bénard convection for two immiscible fluid layers, in order to identify the effects of buoyancy on interface breakup. We explore the parameter space spanned by the Weber number $5\leqslant We \leqslant 5000$ (the ratio of inertia to surface tension) and the density ratio between the two fluids $0.001 \leqslant \varLambda \leqslant 1$, at fixed Rayleigh number $Ra=10^8$ and Prandtl number $Pr=1$. At low $We$, the interface undulates due to plumes. When $We$ is larger than a critical value, the interface eventually breaks up. Depending on $\varLambda$, two breakup types are observed. The first type occurs at small $\varLambda \ll 1$ (e.g. air–water systems) when local filament thicknesses exceed the Hinze length scale. The second, strikingly different, type occurs at large $\varLambda$ with roughly $0.5 < \varLambda \leqslant 1$ (e.g. oil–water systems): the layers undergo a periodic overturning caused by buoyancy overwhelming surface tension. For both types, the breakup criteria can be derived from force balance arguments and show good agreement with the numerical results.

This paper presents an analytical solution of the Burnett equations for gaseous flow in a long microchannel. A non-dimensional analysis is first undertaken to reduce the governing equations into somewhat simplified differential equations, which are solved to obtain the pressure and velocity fields. The exact solution for pressure has been obtained by solving the cross-stream momentum equation, while the solution for velocity is obtained from the streamwise momentum equation. The required boundary conditions are obtained from the direct simulation Monte Carlo (DSMC) method. The obtained analytical solution is compared with the available DSMC and perturbation based solutions, and found to agree well. The work is particularly significant because analytical solutions of the Burnett equations are currently known in only a very few cases. The present analytical solution opens the possibility for further analysis by employing the expressions for pressure and velocity provided here.