To date, a growing body of literature has documented the existence and impacts of coherent structures known as large- and very-large-scale motions within wall-bounded turbulent flows under neutral and unstable thermal stratification. These coherent structures can account for a considerable fraction of the overall turbulent transport and have been found to modulate small-scale turbulent fluctuations near the wall. In the context of stably stratified flows, however, the examination of such coherent structures has garnered relatively little attention. Stable stratification limits vertical transport and turbulent mixing within flows, which makes it unclear the extent to which previous findings on coherent structures under unstable and neutral stratification are applicable to stably stratified flows. In this study, we investigate the existence and characteristics of coherent structures under stable stratification with a wide range of statistical and spectral analyses. Outer peaks in premultiplied spectrograms under weak stability indicate the presence of large-scale motions, but these peaks become weaker and eventually vanish with increasing stability. Quadrant analysis of turbulent transport efficiencies (the ratio of net fluxes to their respective downgradient components) demonstrates dependencies on both stability and height above ground, which is evidence of morphological differences in the coherent structures under increasing stability. Amplitude modulation by large-scale streamwise velocity was found to decrease with increasing gradient Richardson number, whereas modulation by large-scale vertical velocity was approximately zero across all stability ranges. For sufficiently stable stratification, large eddies are suppressed enough to limit any inner–outer scale interactions.

Both experimental and theoretical studies of fast and microscale physical phenomena occurring during the growth of vapour bubbles in nucleate pool boiling are reported. The focus is on the liquid film of micrometric thickness (a ‘microlayer’) that can form between the heater and the liquid–vapour interface of a bubble. The microlayer strongly affects the macroscale heat transfer and is thus important to be understood. The microlayer appears as a result of the inertial forces that cause the hemispherical bubble shape. It is shown that the microlayer can be seen as the Landau–Levich film deposited by the bubble foot edge during its receding. Paradoxically, the deposition is controlled by viscosity and surface tension. The microlayer profile measured with white-light interferometry, the temperature distribution over the heater, and the bubble shape are observed with synchronised high-speed cameras. According to the numerical simulations, the microlayer consists of two regions: a dewetting ridge near the contact line, and a longer and flatter bumped part. It is shown that the ridge cannot be measured by interferometry because of its intrinsic limitation on the interface slope. The ridge growth is linked to the contact line receding. The simulated dynamics of both the bumped part and the contact line agrees with the experiment. The physical origin of the bump in the flatter part of microlayer is explained.

We study the behaviour of the streamwise velocity variance in turbulent wall-bounded flows using a direct numerical simulation (DNS) database of pipe flow up to friction Reynolds number ${{Re}}_{\tau } \approx 12000$. The analysis of the spanwise spectra in the viscous near-wall region strongly hints to the presence of an overlap layer between the inner- and the outer-scaled spectral ranges, featuring a $k_{\theta }^{-1+\alpha }$ decay (with $k_{\theta }$ the wavenumber in the azimuthal direction, and $\alpha \approx 0.18$), hence shallower than suggested by the classical formulation of the attached-eddy model. The key implication is that the contribution to the streamwise velocity variance $(\langle{u}^2\rangle)$ from the largest scales of motion (superstructures) slowly declines as ${{Re}}_{\tau }^{-\alpha }$, and the integrated inner-scaled variance follows a defect power law of the type $\langle u^2 \rangle ^+ = A - B \, {{Re}}_{\tau }^{-\alpha }$, with constants $A$ and $B$ depending on $y^+$. The DNS data very well support this behaviour, which implies that strict wall scaling is restored in the infinite-Reynolds-number limit. The extrapolated limit distribution of the streamwise velocity variance features a buffer-layer peak value of $\langle u^2 \rangle ^+ \approx 12.1$, and an additional outer peak with larger magnitude. The analysis of the velocity spectra also suggests a similar behaviour of the dissipation rate of the streamwise velocity variance at the wall, which is expected to attain a limiting value of approximately $0.28$, hence slightly exceeding the value $0.25$ which was assumed in previous analyses (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, R3). We have found evidence suggesting that the reduced near-wall influence of wall-attached eddies is likely linked to the formation of underlying turbulent Stokes layers.

Immersed nonlinear elements are prevalent in biological systems that require a preferential flow direction, such as the venous and the lymphatic system. We investigate here a certain class of models where the fluid is driven by peristaltic pumping and the nonlinear elements are ideal valves that completely suppress backflow. This highly nonlinear system produces discontinuous solutions that are difficult to study. We show that, as the density of valves increases, the pressure and flow are well approximated by a continuum of valves which can be analytically treated, and we demonstrate through numeric simulation that the approximation works well even for intermediate valve densities. We find that the induced flow is linear in the peristaltic amplitude for small peristaltic forces and, in the case of sinusoidal peristalsis, is independent of pumping direction. Despite the continuum approximation used, the physical valve density is accounted for by modifying the resistance of the fluid appropriately. The suppression of backflow causes a net benefit in adding valves when the valve density is low, but once the density is high enough, valves predominately suppress forward flow, suggesting there is an optimum number of valves per wavelength. The continuum model for peristaltic pumping through an array of valves presented in this work can eventually provide insights about the design and operating principles of complex flow networks with a broad class of nonlinear elements.

Three-dimensional direct numerical simulations of rotating Rayleigh–Bénard convection in the planar geometry with no-slip top and bottom and periodic lateral boundary conditions are performed for a broad parameter range with the Rayleigh number spanning in $5\times 10^{6}\leq Ra \leq 5\times 10^{13}$, Ekman number within $5\times 10^{-9}\leq Ek \leq 5\times 10^{-5}$ and Prandtl number $Pr=1$. The thermal and Ekman boundary layer (BL) statistics, temperature drop within the thermal BL, interior temperature gradient and scaling behaviours of the heat and momentum transports (reflected in the Nusselt $Nu$ and Reynolds numbers $Re$) as well as the convective length scale are investigated across various flow regimes. The global and local momentum transports are examined via the $Re$ scaling derived from the classical theoretical balances of viscous–Archimedean–Coriolis (VAC) and Coriolis–inertial–Archimedean (CIA) forces. The VAC-based $Re$ scaling is shown to agree well with the data in the cellular and columnar regimes, where the characteristic convective length scales as the onset length scale ${\sim } Ek^{1/3}$, while the CIA-based $Re$ scaling and the inertia length scale $\sim (ReEk)^{1/2}$ work well in the geostrophic turbulence regime for $Ek\leq 1.5\times 10^{-8}$. The examinations of $Nu$, global and local $Re$, and convective length scale as well as the temperature drop within the thermal BL and its thickness scaling behaviours, indicate that for extreme parameters of $Ek\leq 1.5\times 10^{-8}$ and $80\lesssim RaEk^{4/3}\lesssim 200$, we have reached the diffusion-free geostrophic turbulence regime.

Two-dimensional (2-D) and three-dimensional (3-D) direct numerical simulations are conducted for flow past rectangular cylinders with various cross-sectional aspect ratios. The primary focuses are the interactions between the 2-D wake transitions in the spanwise vortex street (with distance downstream) and the 3-D wake transitions in the streamwise vortices, and the influence of both 2-D and 3-D wake transitions on the hydrodynamic forces on the cylinder. The 2-D wake transitions generally move upstream with increasing Reynolds number and decreasing aspect ratio. The corresponding reasons are explained. The 2-D wake transitions emerging close to the cylinder may directly alter the hydrodynamic forces on the cylinder, e.g. the Strouhal number, time-averaged drag coefficient and root-mean-square lift coefficient. By using specifically designed numerical cases to decompose the effects of the two 2-D transitions, it is found that the first 2-D transition from the primary to the two-layered vortex street results in reductions in the hydrodynamic forces, while the second 2-D transition to the secondary vortex street results in increases in the forces. The reduction/increase in the hydrodynamic forces becomes more significant when the transition location moves closer to the cylinder. The physical mechanisms for the influence on the hydrodynamic forces are elucidated. The upstream movement of the 2-D wake transitions also induces complex interactions between the 2-D and 3-D wake transitions (which also depends on the type of the 3-D mode). Correspondingly, the 3-D hydrodynamic forces may be governed by both 2-D and 3-D wake transitions (and their mutual influence).

Flexible canopy flows are often encountered in natural scenarios, e.g. when crops sway in the wind or when submerged kelp forests are agitated by marine currents. Here, we provide a detailed characterisation of the turbulent flow developed above and between the flexible filaments of a fully submerged dense canopy and we describe their dynamical response to the turbulent forcing. We investigate a wide range of flexibilities, encompassing the case in which the filaments are completely rigid and standing upright as well as that where they are fully compliant to the flow and deflected in the streamwise direction. We are thus able to isolate the effect of the canopy flexibility on the drag and on the inner–outer flow interactions, as well as the two flapping regimes of the filaments already identified for a single fibre. Furthermore, we offer a detailed description of the Reynolds stresses throughout the wall-normal direction resorting to the Lumley triangle formalism, and we show the multi-layer nature of turbulence inside and above the canopy. The relevance of our investigation is thus twofold: the fundamental physical understanding developed here paves the way towards the investigation of more complex and realistic scenarios, while we also provide a thorough characterisation of the turbulent state that can prove useful in the development of accurate turbulence models for RANS and LES.

The scale-dependent variability of convective velocities and structure inclination angles in wall-bounded turbulence was studied experimentally via space–time energy spectrum measurements. We found that the variability of convection velocities for large-scale motions (LSMs) decreased inversely with streamwise wavenumbers, and that the variability trend was not fully explained by earlier applications of Kraichnan's ‘random-sweeping’ model of turbulence that assumed perfect scale separation. By analytically extending the random-sweeping model to allow for a dominant large scale in the random-sweeping signal that can interact with other LSMs, we showed how scale interactions can explain the variability trend in convection velocities for LSMs. The variability in convection velocities was also shown to correlate with the scale-dependent inclination angles of coherent structures that were obtained via cross-spectral analysis. Large-scale motions tended to exhibit shallower inclination to the wall with increasing convection velocity, while small-scale motions and those far from the wall exhibited the reverse behaviour. We proposed that these two opposite relationships between inclination angle and convection velocity can be explained in terms of a balance between opposing effects of the mean shear and the coherent structure geometry. Descriptions and models of convection velocity variability effects are useful both for modelling turbulence spectra and explaining the geometry of coherent structures.

We study the stationary Navier–Stokes equations in the region between two rotating concentric cylinders. We first prove that, for a small Reynolds number, if the fluid flow is axisymmetric and if its velocity is sufficiently small in the $L^\infty$-norm, then it is necessarily the Taylor–Couette–Poiseuille flow. If, in addition, the associated pressure is bounded or periodic in the $z$ axis, then it coincides with the well-known canonical Taylor–Couette flow. We discuss the relation between uniqueness and stability of such a flow in terms of the Taylor number in the case of narrow gap of two cylinders. The investigation in comparison with two Reynolds numbers based on inner and outer cylinder rotational velocities is also conducted. Next, we give a certain bound of the Reynolds number and the $L^\infty$-norm of the velocity such that the fluid is, indeed, necessarily axisymmetric. As a result, it is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the Taylor–Couette–Poiseuille flow with the exact form of the pressure.

We investigate Reynolds number effects in strong shock-wave/turbulent boundary-layer interactions (STBLI) by leveraging a new database of wall-resolved and long-integrated large-eddy simulations. The database encompasses STBLI with massive boundary-layer separation at Mach $2.0$, impinging-shock angle $40^{\circ }$ and friction Reynolds numbers ${\textit {Re}}_\tau$ $355$, $1226$ and $5118$. Our analysis shows that the shape of the reverse-flow bubble is notably different at low and high Reynolds number, while the mean-flow separation length, separation-shock angle and incipient plateau pressure are rather insensitive to Reynolds number variations. Velocity statistics reveal a shift in the peak location of the streamwise Reynolds stress from the separation-shock foot to the core of the detached shear layer at high Reynolds number, which we attribute to increased pressure transport in the separation-shock excursion domain. Additionally, in the high Reynolds case, the separation shock originates deep within the turbulent boundary, resulting in intensified wall-pressure fluctuations and spanwise variations associated with the passage of coherent velocity structures. Temporal spectra of various signals show energetic low-frequency content in all cases, along with a distinct peak in the bubble-volume spectra at a separation-length-based Strouhal number $St_{L_{sep}}\approx 0.1$. The separation shock is also found to lag behind bubble-volume variations, consistent with the acoustic propagation time from reattachment to separation and a downstream mechanism driving the shock motion. Finally, dynamic mode decomposition of three-dimensional fields suggests a Reynolds-independent statistical link among separation-shock excursions, velocity streaks and large-scale vortices at low frequencies.

Symmetry-breaking bifurcations, where a flow state with a certain symmetry undergoes a transition to a state with a different symmetry, are ubiquitous in fluid mechanics. Much can be understood about the nature of these transitions from symmetry alone, using the theory of groups and their representations. Here, we show how the extensive databases on groups in crystallography can be exploited to yield insights into fluid dynamical problems. In particular, we demonstrate the application of the crystallographic layer groups to problems in fluid layers, using thermal convection as an example. Crystallographic notation provides a concise and unambiguous description of the symmetries involved, and we advocate its broader use by the fluid dynamics community.

A new wall-wake law is proposed for the streamwise turbulence in the outer region of a turbulent boundary layer. The formulation pairs the logarithmic part of the profile (with a slope $A_1$ and additive constant $B_1$) to an outer linear part, and it accurately describes over 95 % of the boundary layer profile at high Reynolds numbers. Once the slope $A_1$ is fixed, $B_1$ is the only free parameter determining the fit. Most importantly, $B_1$ is shown to follow the same trend with Reynolds number as the wake factor in the wall-wake law for the mean velocity, which is tied to changes in scaling of the mean flow and the turbulence that occur at low Reynolds number.

We investigate flow of liquid which is partially filled in a cylindrical container horizontally rotating about its axis of symmetry. Even if the rotation is slow enough to keep the liquid–gas interface almost undeformed, convection cells whose circulation axis is perpendicular to the container's rotational axis can be sustained. We conduct experiments by particle image velocimetry and direct numerical simulations with the S-CLSVOF and immersed boundary methods to reveal the condition of the Reynolds number, the aspect ratio of the container and the filling ratio of liquid for the onset of these convection cells. When the filling ratio is not too large, as the Reynolds number increases, convection cells appear through a pitchfork bifurcation in an infinitely long cylinder. This bifurcation becomes imperfect in the case of a finite-length cylinder. In contrast, when the filling ratio is large enough, convection cells appear through a subcritical bifurcation. Through these investigations, it becomes evident that the axial wavelength of sustained convection cells is an increasing function of the filling ratio in an infinitely long cylinder. In practice, to sustain intense convection cells, we should use a cylinder with the length equal to an integer multiple of the wavelength of the most unstable mode in the infinite-length cylinder. Although we focus on the liquid-pool regime with small Froude numbers, the critical Reynolds number for the pitchfork bifurcation weakly depends on the Froude number. This dependence is explained by considering the changes in the effective filling ratio and the convection velocity.

Low Stokes number particles at dilute concentrations in turbulent flows can reasonably be approximated as passive scalars. The added presence of a drift velocity due to buoyancy or gravity when considering the transport of such passive scalars can reduce the turbulent dispersion of the scalar via a diminution of the eddy diffusivity. In this work, we propose a model to describe this decay and use a recently developed technique to accurately and efficiently measure the eddy diffusivity using Eulerian fields and quantities. We then show a correspondence between this method and standard Lagrangian definitions of diffusivity and collect data across a range of drift velocities and Reynolds numbers. The proposed model agrees with data from these direct numerical simulations, offers some improvement to previous models in describing other computational and experimental data and satisfies theoretical constraints that are independent of Reynolds number.

Dense mixtures of particles of varying size tend to segregate based on size during flow. Granular size segregation impacts many industrial and geophysical processes, but the development of coupled, continuum models capable of predicting the evolution of segregation dynamics and flow fields in dense granular media across different geometries remains a challenge. One reason is because size segregation stems from two driving forces: pressure gradients and shear-strain-rate gradients. Another reason is the challenge of integrating segregation models with rheological constitutive equations for dense granular flow. In this paper we develop a continuum model that accounts for pressure-gradient-driven and shear-strain-rate-gradient-driven segregation, coupled to rheological modelling of a dense granular medium across the quasi-static and dense inertial flow regimes. To calibrate and test the continuum model, we perform discrete element method (DEM) simulations of dense flow of bidisperse granular systems in two flow geometries in which both segregation driving forces are present: inclined plane flow and planar shear flow with gravity. Steady-state DEM data from inclined plane flow is used to determine the dimensionless material parameters in the pressure-gradient-driven segregation model for both spheres and disks. Then, predictions of the continuum model are tested against DEM data across different cases of inclined plane flow and planar shear flow with gravity, while varying parameters such as the size of the flow geometry, the flow speed and the initial conditions. We find that it is crucial to account for both driving forces to capture segregation dynamics across both flow geometries with a single set of parameters.

This paper gives, in the limit of infinite Froude number, a closed-form, analytical solution for steady, two-dimensional, irrotational, infinite-depth, free-surface, attached flow over a submerged tandem cascade of hydrofoils for arbitrary angle of attack, depth of submergence and interfoil separation. The multiply connected flow domain is conformally mapped to a concentric annulus in an auxiliary plane. The complex flow potential and its derivative, the complex velocity, are obtained in the auxiliary plane by considering their form at known special points in the flow and the required conformal mapping is determined by explicit integration, allowing accurate evaluation of various flow quantities including the lift on each foil. The circulation around the foils causes the foil array to act as a row of point vortices, or a shear layer, and so, for positive angles of attack, the flow speed at the free surface can substantially exceed the speed at depth, with the speeds simply related through the lift coefficient. Decreasing the interfoil separation decreases the disturbance to the free surface and greatly increases the lift per hydrofoil, thus allowing for the shallower operation of a hydrofoil array than of an isolated foil for a given lift requirement. Further, the flow over a hydrofoil array approaches its infinite depth form significantly more rapidly than that over an isolated foil. In contrast to the infinite-submergence case where a through-array flow can be imposed, in the finite submergence case, periodicity and the presence of the free surface mean that there is no net flow between the foils.