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.

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.

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.

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.

The high-Reynolds-number axisymmetric wake of a slender body with a turbulent boundary layer is investigated using a hybrid simulation. The wake generator is a $6:1$ prolate spheroid and the Reynolds number based on the diameter $D$ is $Re=10^5$. The transition of the wake to a state of complete self-similarity is investigated by looking for the first time into the far field of a slender-body wake. Unlike bluff-body wakes, here the flow is not dominated by vortex shedding in the near wake. Instead, the recirculation region is very small, the near wake is quasi-parallel and is characterised by the presence of broadband turbulence. Until $x/D\approx 20$, the wake decay of a slender body with turbulent boundary layer is very similar to the classic high-$Re$ behaviour, $U_d\sim x^{-2/3}$. Extrapolation of this observation to larger $x/D$ has led to the belief that these wakes decay following the asymptotic $-2/3$ decay law. Our results show, however, that this is not the case and the wake transitions to a faster decay rate once complete self-similarity is achieved. In this later region ($20 < x/D < 80$), mean and turbulence profiles are self-similar. Furthermore, despite the high global and local Reynolds numbers, the classic hypotheses that lead to the well-known decay exponents are not fulfilled. Instead, turbulent dissipation follows a non-equilibrium scaling and a new decay rate $U_d\sim x^{-6/5}$ is observed. The transition from $U_d\sim x^{-2/3}$ to $U_d\sim x^{-6/5}$ is preceded by the dominance of the azimuthal $|m|=1$ mode and the emergence of a large-scale helical structure.

We investigate evanescent inertial waves, both theoretically and experimentally, in a fluid subject to a background rotation of $\varOmega$. We predict that there is a smooth transition from conventional inertial waves to evanescent disturbances at a frequency of $\varpi = 2\varOmega$, and that at this cross-over frequency the evanescent disturbances are spatially extensive, having a horizontal extent which is limited only by viscosity, or by the size of the domain. These findings are confirmed by our experiments, which, to the best of our knowledge, represent the first quantitative experimental investigation of evanescent inertial waves.

We report on an investigation of bubble-induced turbulence. Bubbles of a size larger than the dissipative scale cannot be treated as pointwise inclusions, and generate important hydrodynamic fields in the carrier fluid when in motion. Furthermore, bubble motions may induce a collective agitation due to hydrodynamic interactions which display some turbulent-like features. We tackle this complex phenomenon numerically, performing direct numerical simulations with a volume-of-fluid method. In the first part of the work, we perform both two-dimensional and three-dimensional tests in order to determine appropriate numerical and physical parameters. We then carry out a highly resolved simulation of a three-dimensional bubble column, with a set-up and physical parameters similar to those used in laboratory experiments. This is the largest simulation attempted for such a configuration and is only possible thanks to adaptive grid refinement. Results are compared both with experiments and previous coarse-mesh numerical simulations. In particular, the one-point probability density function of the velocity fluctuations is in good agreement with experiments. The spectra of the kinetic energy show a clear $k^{-3}$ scaling. The mechanisms underlying the energy transfer and notably the possible presence of a cascade are unveiled by a local scale-by-scale analysis in physical space. The comparison with previous simulations indicates to what extent simulations not fully resolved may yet give correct results, from a statistical point of view.

A Lagrangian experimental study of an axisymmetric turbulent water jet is performed to investigate the highly anisotropic and inhomogeneous flow field. Measurements are conducted within a Lagrangian exploration module, an icosahedron apparatus, to facilitate optical access of three cameras. Stereoscopic particle tracking velocimetry results in three-component tracks of position, velocity and acceleration of the tracer particles within the vertically oriented jet with a Taylor-based Reynolds number ${\textit {Re}}_\lambda \simeq 230$. Analysis is performed at seven locations from 15 diameters up to 45 diameters downstream. Eulerian analysis is first carried out to obtain critical parameters of the jet and relevant scales, namely the Kolmogorov and large (integral) scales as well as the energy dissipation rate. Lagrangian statistical analysis is then performed on velocity components stationarised following methods inspired by Batchelor (J. Fluid Mech., vol. 3, 1957, pp. 67–80), which aim to extend stationary Lagrangian theory of turbulent diffusion by Taylor to the case of self-similar flows. The evolution of typical Lagrangian scaling parameters as a function of the developing jet is explored and results show validation of the proposed stationarisation. The universal scaling constant $C_0$ (for the Lagrangian second-order structure function), as well as Eulerian and Lagrangian integral time scales, are discussed in this context. Constant $C_0$ is found to converge to a constant value (of the order of $C_0 = 3$) within 30 diameters downstream of the nozzle. Finally, the occurrence of finite particle size effects is investigated through consideration of acceleration-dependent quantities.

We investigate the dynamics of inertial particles in homogeneous isotropic turbulence, under one-way momentum coupling, using a new computational approach that incorporates the effect of long-range many-body aerodynamic interactions along with the short-range lubrication forces. The implementation couples hybrid direct numerical simulations (HDNS) with the analytical solutions of two rigid spheres moving in an unbounded fluid. Concerning the velocity field seen by the particles, the algorithm switches from the flow solution in terms of HDNS to analytical formulae when the separation distance between particles becomes comparable to their average radius. Standard HDNS is unable to correctly represent the short-range interactions since this method is based on the superposition of the Stokes solutions for single spheres. Our results show that for the turbulent kinetic energy dissipation rates typical of atmospheric clouds, the radial relative velocities (RRVs) of the droplets increase, and the radial distribution function (RDF) decreases in the near-contact region if the lubrication forces are taken into account. These changes are more pronounced when the effect of gravity is considered. Away from the contact region, there is not much change in RRVs and RDFs. For turbulent clouds with lower dissipation rates lubrication forces significantly enhance the average RRV in the limit of low Stokes number. This enhancement, however, is statistically insignificant because the number of particle pairs at close proximity is very small. The effect of mass loading on the collision statistics is also investigated, demonstrating an increase in RRV and a reduction in RDF with the droplet concentration.

This paper investigates the motion of a liquid blister, trapped between an elastic sheet and a rigid substrate. The blister is driven by a frictionless blade moving at a constant velocity, forcing a constant gap that causes fluid to bleed from the blister. The sheet adheres to the substrate ahead of the blister. The main goal of the study is to assess the magnitude and orientation of the force applied by the blade on the moving blister. The solution is constructed for the asymptotic case of a long blister. Thanks to a separation of scales, the asymptotic solution is obtained by matching the boundary layers at the front end and at the back end of the blister to an outer solution characterised by a uniform pressure in the bulk. Both boundary layers are formulated as travelling-wave equations for the gap between the sheet and the substrate. The formulation accounts for a moving fluid front, distinct from the separation edge, and for a tail with a gap tending to an a priori unknown value far behind the blister. Scaling of the governing equations indicates that the solution depends on two numbers: a dimensionless toughness $\mathcal {K}$ and a scaled gap $\mathcal {W}$ imposed by the moving blade. The key result concerns the dependence of the scaled force on the two numbers controlling the solution of the moving liquid blister. There are two asymptotic solutions: for small gaps at the blade, the force on the blade is dominated by viscous dissipation at the back end and only depends on aperture $\mathcal {W}$; for large gaps, the horizontal force $H$ only depends on toughness $\mathcal {K}$, a function of both fluid viscosity and energy of separation at the front end, whereas the vertical force $V$ depends on both $\mathcal {K}$ and $\mathcal {W}$.

The coating of rotating discrete objects with surface topography is a problem commonly encountered in manufacturing processes. Surface topography may induce undesired disturbances in the coating, leading to coatings of non-uniform thickness. To study this problem, we model the flow of thin liquid coatings in three dimensions on topographically patterned cylinders that rotate about their horizontal axes. An evolution equation describing variations in the coating thickness as a function of the axial coordinate, the angular coordinate, and time is solved numerically using a variable time-step finite-difference scheme. In the limit of a rapidly rotating cylinder, we neglect the effects of gravity and find that liquid accumulates at either pattern crests or pattern troughs. Using a long-wave analysis, we derive an expression for the critical Weber number that separates these regimes. If gravity is reincorporated, the accumulation of liquid at crests or troughs may cause the coating to sag under its weight, leading to the formation of droplets or rings whose spacing at large rotation rates is controlled by the balance between centrifugal and surface-tension forces. At lower rotation rates, where gravitational forces dominate, simulation results indicate that cylinder topography tends to alter the rate at which droplets form, but does not necessarily systematically affect the spacing between droplets. Flow visualization experiments yield results that agree quantitatively with predictions of the simulations and long-wave analysis. We observe the most uniform coatings in experiments at moderate rotation rates, where disturbances in the coating thickness develop slowly. This indicates that to obtain nearly uniform coatings in practice, the coating must be solidified faster than disturbances can develop.

Through ensemble-based data assimilation, we address one of the most notorious difficulties in phase-resolved ocean wave forecast, regarding the deviation of numerical solution from the true surface elevation due to the chaotic nature of and underrepresented physics in the nonlinear wave models. In particular, we develop a coupled approach of the high-order spectral (HOS) method with the ensemble Kalman filter (EnKF), through which the measurement data can be incorporated into the simulation to improve the forecast performance. A unique feature in this coupling is the mismatch between the predictable zone and measurement region, which is accounted for through a special algorithm to modify the analysis equation in EnKF. We test the performance of the new EnKF–HOS method using both synthetic data and real radar measurements. For both cases (though differing in details), it is shown that the new method achieves much higher accuracy than the HOS-only method, and can retain the phase information of an irregular wave field for an arbitrarily long forecast time with sequentially assimilated data.

The turbulent boundary layer over a Gaussian-shaped bump is computed by direct numerical simulation of the incompressible Navier–Stokes equations. The two-dimensional bump causes a series of strong pressure gradients alternating in rapid succession. At the inflow, the momentum thickness Reynolds number is approximately $1000$ and the boundary layer thickness is $1/8$ of the bump height. Direct numerical simulation results show that the strong favourable pressure gradient (FPG) causes the boundary layer to enter a relaminarization process. The near-wall turbulence is significantly weakened and becomes intermittent, however, relaminarization does not complete. The streamwise velocity profiles deviate above the standard logarithmic law and the Reynolds shear stress is reduced. The strong acceleration also suppresses the wall-shear normalized turbulent kinetic energy production rate. At the bump peak, where the FPG switches to an adverse gradient (APG), the near-wall turbulence is suddenly enhanced through a partial retransition process. This results in a new highly energized internal layer which is more resilient to the strong APG and only produces incipient flow separation on the downstream side. In the strong FPG and APG regions, the inner and outer layers become largely independent of each other. The near-wall region responds to the pressure gradients and determines the skin friction. The outer layer behaves similarly to a free shear layer subject to pressure gradients and mean streamline curvature effects. Results from a RANS simulation of the bump are also discussed and clearly show the lack of predictive capacity of the near-wall pressure gradient effects on the mean flow.