This paper documents and discusses the structure and Reynolds number scaling of the viscous sublayer in front of a cylinder–wall junction. The data were obtained by large-eddy simulation and particle-image velocimetry in an open channel at cylinder Reynolds numbers of $Re_D=20\ 000$, $39\ 000$ and $78\ 000$. We distinguish between outer and inner flow regions. The outer region covers the downflow, the horseshoe vortex and an upstream-directed near-wall jet in front of the cylinder. When normalized with the bulk velocity and the cylinder diameter, the dynamics of the outer flow depends only weakly on the Reynolds number in the investigated range. The inner region is the thin laminar sublayer developing along the bottom wall under the near-wall jet. In the region of the maximum wall shear stress, the amplitude of the friction coefficient and the inner layer thickness both scale with the inverse of the square root of the cylinder Reynolds number. The velocity profiles in the inner layer can be approximated well by Falkner–Skan profiles, scaled by the thickness of the inner layer and the velocity at the interface between the inner and outer layers. Even when the pressure gradient parameter of the Falkner–Skan profiles was estimated by outer flow variables, a good match between velocity profiles in the inner layer and the corresponding Falkner–Skan solutions is observed in the region of maximal wall shear stress. In the whole region of the upstream directed wall jet, the wall shear stress can be estimated from these Falkner–Skan profiles with an accuracy between $10\,\%$ and $20\,\%$.

We present a fully analytical solution for the natural oscillation of an inviscid sessile drop with small Bond number (surface tension dominates gravity) and a fixed contact line on a flat horizontal plate. The governing equations are expressed in terms of the toroidal coordinate system which yields solutions involving hypergeometric functions. Resonant frequencies are identified for zonal, sectoral and tesseral vibration modes. The predictions show excellent agreement with experimental data reported in the literature, particularly for flatter drops (lower $\theta _{c}$, but not so low as to incur significant viscous dissipation) and higher modes of vibration.

The stability of the boundary layer generated by the harmonic oscillations of a plate in its own plane in a fluid otherwise at rest (Stokes boundary layer) is investigated by considering the time development of perturbations of small amplitude and introducing a momentary criterion of instability. The temporal scale of the perturbations is assumed to be much smaller than the period of the plate oscillations because transition takes place at values of the Reynolds number much larger than one. The results confirm that the Stokes boundary layer is linearly unstable when the Reynolds number is larger than a first critical value $R_{\delta ,c1}$ equal to $85$, which is almost coincident with that determined by Von Kerczek & Davis (J. Fluid Mech., vol. 62, issue 4, 1974, 753–773) for a Stokes boundary layer in a fluid domain which is bounded by a second fixed plate at a distance $h^{*}$ from the oscillating one. For values of the Reynolds number close to $R_{\delta ,c1}$, the instability is restricted to phases close to the inversion of the plate velocity. When the Reynolds number becomes larger than a second threshold value $R_{\delta ,c2}$ close to $200$, the instability rapidly pervades a large part of the cycle. However, only when the Reynolds number becomes larger than a third critical value $R_{\delta ,c3}$ equal to $850$, is the instability present during the whole cycle. Heuristically, these three critical values of the Reynolds number can be associated with the transition from the laminar regime to the disturbed laminar, the intermittently turbulent and fully turbulent regimes.

Wall roughness induces extra drag in wall-bounded turbulent flows. Mapping any given roughness geometry to its fluid dynamic behaviour has been hampered by the lack of accurate and direct measurements of skin-friction drag. Here, the Taylor–Couette (TC) system provides an opportunity as it is a closed system and allows direct and reliable measurement of the skin-friction. However, the wall curvature potentially complicates the connection between the wall friction and the wall roughness. Here, we investigate a highly turbulent TC flow, with a hydrodynamically fully rough, rotating inner cylinder, while the outer cylinder is kept smooth and stationary. We carry out particle image velocimetry (PIV) measurements in the Twente Turbulent Taylor–Couette (T3C) facility with Reynolds numbers in the range of $4.6\times 10^5 < Re_i < 1.77\times 10^6$. From these we find, while taking into account the influence of the curved walls on the turbulence, that the observed effects of a hydrodynamically fully rough surface are similar for TC turbulence and flat-plate turbulent boundary layer flows (BL). Hence, the equivalent sand grain height ks, that characterizes the drag properties of a rough surface, is similar for both flow geometries. Next, we obtain the dependence of the torque (skin-friction drag) on the Reynolds number for a given wall roughness, characterized by ks, and find agreement with the same results derived from PIV measurements within $5\,\%$. Our findings demonstrate that global torque measurements in the TC facility could be well suited to reliably deduce wall-drag properties for any rough surface.

Effects of fast-propagating water waves on the overlying wind are investigated using simulation and theoretical analysis. By performing a large eddy simulation (LES) of turbulent wind over water waves with high wave age, we observe that the perturbation to wind velocity and pressure by the waves, or the wave-induced airflow, is mainly induced by the vertical movement of the wave surface. We perform scaling analysis to show that the turbulent stress effects on fast wave-induced airflow are negligible. Moreover, we find that the curvilinear model developed for opposing wave effects on wind by Cao et al. (J. Fluid Mech., vol. 901, 2020, p. A27) provides predictions that agree well with the present LES results of wind following fast waves. Our analyses of the results indicate that the fast wave-induced airflow is a quasilinear process. To elucidate the mechanisms for fast wave effects, we split the curvilinear model into two equations corresponding to wave kinematics and forcing by wave elevation, respectively. Using these equations, we illustrate that the vertical component of wave orbital velocity induces a strong airflow perturbation, which produces the dominant components of fast wave-induced airflow and determines its overall spatial structure. Furthermore, we discover that the weak components of fast wave-induced airflow are forced by the dominant components via viscous stress and by the forcing induced by wave elevation, and generate the form drag on the wave surface.

An analytic estimate of the lower bounds on zonal enstrophy has been studied for the beta-plane model of two-dimensional barotropic flow. The estimate provides exact lower bounds on the zonal enstrophy, which hence must be satisfied regardless of the dynamics. The energy, impulse, circulation as well as the total enstrophy are invoked as constraints for the minimization of the zonal enstrophy. The corresponding variational principle has an unusual mathematical structure (primarily because the target functional is not a coercive form), by which the constraints work out in an interesting way. A discrete set of zonal enstrophy levels is generated by the energy constraint; each level is specified by an eigenvalue that represents the meridional mode number of zonal flow. However, the value itself of the zonal enstrophy level is a function of only impulse and circulation, being independent of the energy (and total enstrophy). Instead, the energy works in selecting the ‘level’ (eigenvalue) of the relaxed state. The relaxation occurs by emitting small-scale wavy enstrophy, and continues as far as the nonlinear effect, scaled by the energy, can create wavy enstrophy. Comparison to numerical simulations shows that the theory gives a consistent estimate of the zonal enstrophy in the relaxed state.

Turbulence in the flow of fluid through a pipe can be suppressed by buoyancy forces. As the suppression of turbulence leads to severe heat transfer deterioration, this is an important and undesirable phenomenon in both heating and cooling applications. Vertical flow is often considered, as the axial buoyancy force can help drive the flow. With heating measured by the buoyancy parameter $C$, our direct numerical simulations show that shear-driven turbulence may either be completely laminarised or it transitions to a relatively quiescent convection-driven state. Buoyancy forces cause a flattening of the base flow profile, which in isothermal pipe flow has recently been linked to complete suppression of turbulence (Kühnen et al., Nat. Phys., vol. 14, 2018, pp. 386–390), and the flattened laminar base profile has enhanced nonlinear stability (Marensi et al., J. Fluid Mech., vol. 863, 2019, pp. 50–875). In agreement with these findings, the nonlinear lower-branch travelling-wave solution analysed here, which is believed to mediate transition to turbulence in isothermal pipe flow, is shown to be suppressed by buoyancy. A linear instability of the laminar base flow is responsible for the appearance of the relatively quiescent convection driven state for $C\gtrsim 4$ across the range of Reynolds numbers considered. In the suppression of turbulence, however, i.e. in the transition from turbulence, we find clearer association with the analysis of He et al. (J. Fluid Mech., vol. 809, 2016, pp. 31–71) than with the above dynamical systems approach, which describes better the transition to turbulence. The laminarisation criterion He et al. propose, based on an apparent Reynolds number of the flow as measured by its driving pressure gradient, is found to capture the critical $C=C_{cr}(Re)$ above which the flow will be laminarised or switch to the convection-driven type. Our analysis suggests that it is the weakened rolls, rather than the streaks, which appear to be critical for laminarisation.

We theoretically and numerically investigate the instabilities driven by diffusiophoretic flow, caused by a solutal concentration gradient along a reacting surface. The important control parameters are the Péclet number $Pe$, which quantifies the ratio of the solutal advection rate to the diffusion rate, and the Schmidt number $Sc$, which is the ratio of viscosity and diffusivity. First, we study the diffusiophoretic flow on a catalytic plane in two dimensions. From a linear stability analysis, we obtain that for $Pe$ larger than $8{\rm \pi}$ mass transport by convection overtakes that by diffusion, and a symmetry-breaking mode arises, which is consistent with numerical results. For even larger $Pe$, nonlinear terms become important. For $Pe > 16{\rm \pi}$, multiple concentration plumes are emitted from the catalytic plane, which eventually merge into a single larger one. When $Pe$ is even larger ($Pe \gtrsim 603$ for Schmidt number $Sc=1$), there are continuous emissions and merging events of the concentration plumes. The newly found flow states have different flow structures for different $Sc$: for $Sc\geqslant 1$, we observe the chaotic emission of plumes, but the fluctuations of concentration are only present in the region near the catalytic plane. In contrast, for $Sc<1$, chaotic flow motion occurs also in the bulk. In the second part of the paper, we conduct three-dimensional simulations for spherical catalytic particles, and beyond a critical Péclet number again find continuous plume emission and plume merging, now leading to a chaotic motion of the phoretic particle. Our results thus help us to understand the experimentally observed chaotic motion of catalytic particles in the high $Pe$ regime.

This study is inspired by the velvety structures on an owl's upper wing surface. Anechoic wind tunnel experiments were conducted to study the effect of the velvety structures on trailing edge noise as well as the boundary layer flow of a flat plate model. The tests were conducted in The Hong Kong University of Science and Technology low-speed wind tunnel, ultra-quiet noise injection test and evaluation device (UNITED). It was found that the trailing edge noise spectra are significantly modified by the velvety structures. In general, the velvety structures increase the low-frequency noise below a cross-over Strouhal number $St_c$ but reduce the spectral level at higher frequencies. The velvety surface also changes the boundary layer characteristics in terms of the boundary layer thickness, non-dimensional velocity distribution and turbulence distribution. Vortex shedding is suppressed by the velvety coating despite the blunt trailing edge. An analytic model is proposed for the trailing edge noise of a flat plate, including the effect of finite trailing edge thickness and velvety structures on the flat plate surface. The model uses the near wake distribution of the mean and fluctuating velocities in the streamwise direction as the input. The predictions, which require no empirical corrections, match well with the experiments for both the baseline and velvet-coated configurations. With a detailed non-dimensional analysis, this study proposes a potential aeroacoustic function of velvet structures, i.e. noise control through manipulation of boundary layer statistics.

As a stiff polymer tumbles in shear flow, it experiences compressive viscous forces that can cause it to buckle and undergo a sequence of morphological transitions with increasing flow strength. We use numerical simulations to uncover the effects of these transitions on the steady shear rheology of a dilute suspension of stiff polymers. Our results agree with classic scalings for Brownian rods in relatively weak flows but depart from them above the buckling threshold. Signatures of elastoviscous buckling include enhanced shear thinning and an increase in the magnitude of normal stress differences. We discuss our findings in the light of past work on rigid Brownian rods and non-Brownian elastic fibres and highlight the subtle role of thermal fluctuations in triggering instabilities.

Rock–ice avalanches are liquid–granular flows that consist of a mixture of rock, ice and a liquid. The dynamics that distinguishes these types of flows from other geophysical flows is the ice melting. This process is responsible for mass and momentum transfers between the solid and liquid components of the mixture and for the effects of lubrication and fluidization that reduce the mixture strength. In this work, we analyse the problem from a mathematical point of view. Starting from the partial differential equations of a complete three-phase approach, we identify two basic assumptions that can be used to build a framework of classes of simplified models. The implications of these assumptions on the physical description of the flow are carefully analysed for each class, and particular attention is paid to the simplification of the melting process expressed in terms of mass and momentum transfers. Moreover, the derived framework allows us to classify the existing literature models and to identify a new class of models that can be considered a reasonable trade-off between simplicity and completeness. Finally, the mathematical nature of each class is investigated by performing an in-depth analysis of the eigenvalues. Results show that the most simplified models are strictly hyperbolic, while the most complete approaches are affected by a loss of hyperbolicity in given ranges of the model parameters. Further research is necessary to understand the reasons and the numerical implications of this feature.

The classical theory by Jeffery predicts that, in the absence of Brownian fluctuations, a thin rigid platelet rotates continuously in a shear flow, performing periodic orbits. However, a stable orientation is possible if the surface of the platelet displays a hydrodynamic slip length $\lambda$ comparable to or larger than the thickness of the platelet. In this article, by solving the Fokker–Plank equation for the orientation distribution function and corroborating the analysis with boundary integral simulations, we quantify a threshold Péclet number, ${Pe}_{c}$, above which such alignment occurs. We found that for ${Pe}$ smaller than ${Pe}_{c}$, but larger than a second threshold, a regime emerges where Brownian fluctuations are strong enough to break the platelet's alignment and induce rotations, but with a period of rotation that depends on the value of $\lambda$. For ${Pe}$ below this second threshold, slip has a negligible effect on the orientational dynamics. We use these thresholds to classify the dynamics of graphene-like nanoplatelets for realistic values of $\lambda$ and apply our results to the quantification of the orientational contribution to the effective viscosity of a dilute suspension of nanoplatelets with slip. We find a non-monotonic variation of this term, with a minimum occurring when the slip length is comparable to the thickness of the particle.

Hypersonic flow over a two-dimensional compression corner with a Mach number of 7.7 and unit Reynolds number of 4.2 × 106 m−1 is numerically investigated. Special emphasis is given to the onset of global instability with respect to three-dimensional perturbations. Global stability analysis is performed for various ramp angles and wall temperatures. It is found that the shock-induced separated flow system becomes unstable when the ramp angle is beyond a certain value. The critical ramp angle increases slightly with the wall temperature, although the length of the separation region is significantly enlarged. The global instability is shown to be closely linked with the occurrence of secondary separation beneath the primary separation bubble. A criterion is established based on a scaled ramp angle defined in the triple-deck theory to predict the global stability boundary, which depends on the free-stream conditions and geometries only.

We study the flow past a cylinder whose axis undergoes prescribed oscillations, translating uniformly in a direction transverse to the oncoming flow. We consider modest Reynolds numbers ($Re\leq 100$), for which the flow is two-dimensional; when the cylinder is fixed, vortices are shed periodically in a so-called 2S pattern. We choose the period of the prescribed oscillation to be identical to the period of the vortex shedding for a fixed cylinder. At a fixed Reynolds number of $Re=100$, an increase in the amplitude of the oscillations leads to a change in the topology of the shed vortices: the 2S pattern becomes a P+S pattern. We employ a space–time discretisation to directly compute time-periodic solutions of the Navier–Stokes equations and thus demonstrate that the transition between the two vortex shedding patterns arises through a spatio-temporal symmetry-breaking bifurcation of the time-periodic 2S solution. The P+S solution exists only for a finite range of amplitudes, however, and eventually reconnects with the 2S solution branch via a second symmetry-breaking bifurcation. There are ranges of amplitudes over which the system is bistable and both 2S and P+S could, in principle, be seen in experiments. As the Reynolds number is reduced, the 2S and P+S branches disconnect, but a bistable region remains until the isolated P+S solutions ultimately disappear, leaving only the 2S solution. The inferred stability of the various time-periodic solution branches is confirmed through time integration of the Navier–Stokes equations. Finally, we illustrate the evolution of the vorticity field along the solution branches.

Phase-averaged and cycle-to-cycle analysis of key contributors to sound production in phonation is examined in a scaled-up vocal-fold model. Simultaneous temporally and spatially resolved pressure and velocity measurements permitted examination of each term in the streamwise integral momentum equation. The relative sizes of these terms were used to address the issue of whether transglottal pressure is a surrogate for vocal-fold drag, a quantity directly related to sound production. Further, time traces of transglottal pressure and volume flow rate provided insight into the role of cycle-to-cycle variations in voiced sound production which affect voice quality. Experiments were conducted using a 10× scaled-up model in a free-surface water tunnel. Two-dimensional vocal-fold models with semi-circular ends inside a square duct were driven with constant opening and closing speeds. The time from opening to closed, To, was half the oscillation period. Time-resolved digital particle image velocimetry (DPIV) and pressure measurements along the duct centreline were made for 3650 ≤ Re ≤ 8100 and equivalent life frequencies from 52.5 to 97.5 Hz. Results showed that transglottal pressure does serve as a surrogate for the vocal-fold drag. However, smaller but non-negligible momentum flux and inertia terms, caused by the jet and vocal-fold motions, may also contribute to vocal-fold drag. Further, cycle-to-cycle variations including jet switching and modulation are inherent in flows of this type despite their high degrees of symmetry and repeatability. The origins of these variations and their potential role in sound production and voice quality are discussed.

A Boussinesq fluid inside a stably thermally stratified square container whose walls are inclined ${45}^{\circ }$ with respect to gravity, with two opposite walls kept at constant temperatures and the other two insulated is nearly isothermal in the regions above and below the horizontal diagonal. The flow is concentrated in the wall boundary layers and a shear layer centred about the horizontal diagonal. The equilibrium is maintained by the balance between dissipation in the shear and boundary layers, the heat fluxes at the constant temperature walls, and the induced flow resulting from the no-flux condition at the inclined insulated walls. The dynamical response of the fluid to vertical oscillations of the container is studied over a range of forcing frequencies. For a small forcing amplitude and below a viscosity-dependent cutoff forcing frequency, this response exhibits a modal cellular structure localized about the shear layer. With increasing forcing amplitude, the response experiences instabilities, studied here numerically at a forcing frequency above the cutoff frequency, that are similar to those encountered in the Faraday wave problem, such as parametric subharmonic instability, triadic resonance and resonant collapse.

Aquatic locomotion of a deformable body from rest up to its asymptotic speed is given by the unsteady motion which is produced by a series of periodic reactions dictated by the body configuration and by the style of swimming. The added mass plays a crucial role, not only for the initial burst, but also along each manoeuvre, to accelerate the surrounding fluid for generating the kinetic energy and to enable vortex shedding in the wake. The estimate of these physical aspects has been largely considered in most theoretical models, but not sufficiently deepened in many experimental and numerical investigations. As a motivation, while the vortical structures are easily detectable from the flow field, the added mass, on the contrary, is usually embedded in the overall forcing terms. By the present impulse formulation, we are able to separate and to emphasize the role of the added mass and vorticity release to evaluate in a neat way their specific contributions. The precise identification of the added mass is also instrumental for a well-posed numerical problem and for easily readable results. As a further point, the asymptotic speed is found to be guided either by the phase velocity of the prescribed undulation and by the unavoidable recoil motion induced by the self-propelled swimming. The numerical results reported in the present paper concern simplified cases of non-diffusing vorticity and two-dimensional flow.

Interaction of droplets with vortical structures is ubiquitous in nature, ranging from raindrops to a gas turbine combustor. In this work, we elucidate the mechanism of co-axial interaction of a droplet with a vortex ring of different circulation strengths ($\varGamma = 45\text {--}161\ \textrm {cm}^2\ \textrm {s}^{-1}$). We focus on both the droplet and the vortex dynamics, which evolve in a spatio-temporal fashion during different stages of the interaction, as in a two-way coupled system. Vortex rings of varying circulation strengths are generated by injecting a slug of water into a quiescent water-filled chamber. Experimental techniques such as high-speed particle image velocimetry, planner laser-induced fluorescence imaging and high-speed shadowgraphy are used in this work. In the droplet dynamics, different regimes of interaction are identified, including deformation (regime-I), stretching and engulfment (regime-II) and breakup of the droplet (regime-III). Each interaction regime is explained using existing theoretical models that closely match the experimental data. In the vortex dynamics, we compare the interaction's effect on different characteristics of the vortex rings, such as pressure and the vorticity distribution, circulation strength, total energy and enstrophy variation with time. It is found that the interaction leads to a reduction in these parameters.

Given the current lack of experimental data for shock waves interacting with incoming transitional boundary layers, the goal of this study was to characterize the dynamics of such an interaction to better understand the fundamental fluid physics of these complex phenomena. Here, the mean flow field and time-dependent characteristics of a three-dimensional Mach 5 cylinder-induced shock-wave/boundary-layer interaction where the upstream boundary layer is transitional, have been studied experimentally. The interactions were generated with a right circular cylinder mounted on a flat plate. Streamwise–spanwise planar laser scattering from a condensed alcohol fog and schlieren imaging were used to characterize the mean and instantaneous structure of the interaction, and fast-response wall-pressure measurements on the centreline upstream of the cylinder enabled characterization of the unsteadiness. The pressure measurements show a mean pressure profile that resembles a composite of an upstream laminar profile and a downstream turbulent profile. The upstream influence location of the transitional interaction was approximately 8.5 diameters ($d$) upstream of the cylinder leading edge, which is between that of a laminar and a turbulent interaction, and is followed by a plateau region to approximately $4d$ upstream of the cylinder. The plateau region is a region with a thicker boundary layer and possible flow separation. The plateau pressure was within 7 % of the value predicted by Hill's correlation for free-interaction phenomena. Furthermore, a statistical analysis of the pressure histories suggests that the entire interaction stretches and contracts in concert. Power spectral densities of the pressure fluctuations showed unsteadiness throughout the interaction with energy content primarily centred between a region defined by a separation-length-based Strouhal number $St_{L} = 0.05\text {--}0.2$, comparing well with other related studies of cylinder-induced interactions. Cross-correlations and coherence functions in the interaction suggest that the unsteadiness in the laminar region may be due to the entire ‘laminar’ region oscillating in response to the ‘turbulent’ unsteadiness of the intermittent region.

This paper presents the results of the time resolved flow field measurements around a realistic rowing oar blade that moves along a realistic path through water. To the authors’ knowledge no prior account of this complex flow field has been given. Simultaneously with the flow field measurements, the hydrodynamic forces acting on the blade were measured. These combined measurements allow us to identify the relevant flow physics that governs rowing propulsion, and subsequently use this information to adjust the oar blade configuration to improve rowing propulsion. Analysis of the instationary flow field around the oar blade during the drive phase indicated how the initial formation, and subsequent development, of leading-edge and trailing-edge vortices are related to the generation of instationary lift and drag forces, and how these forces contribute to rowing propulsion. It is shown that the observed individual flow mechanisms are similar to the flow mechanisms observed in bird flight, but that the overall propulsive mechanism for rowing propulsion is fundamentally different. To quantify the rowing propulsion efficiency, we introduced the energetic efficiency $\eta _E$ and the impulse efficiency $\eta _J$, where the latter can be interpreted as the alignment of the generated impulse with the propulsive direction. It is found that in the conventional oar blade configuration, the generated impulse is not aligned with the propulsive direction, indicating that the propulsion is suboptimal. By adjusting the angle at which the blade is attached to the oar, the generation of leading- and trailing-edge vortices is altered such that the generated impulse better aligns with the propulsive direction, thus increasing the efficiency.