Taylor’s swimming sheet is a classical model of microscale propulsion and pumping. Many biological fluids and substances are fibrous, having a preferred direction in their microstructure; for example, cervical mucus is formed of polymer molecules which create an oriented fibrous network. Moreover, suspensions of elongated motile cells produce a form of active oriented matter. To understand how these effects modify viscous propulsion, we extend Taylor’s classical model of small-amplitude zero-Reynolds-number propulsion of a ‘swimming sheet’ via the transversely isotropic fluid model of Ericksen, which is linear in strain rate and possesses a distinguished direction. The energetic costs of swimming are significantly altered by all rheological parameters and the initial fibre angle. Propulsion in a passive transversely isotropic fluid produces an enhanced mean rate of working, independent of the initial fibre orientation, with an approximately linear dependence of the energetic cost on the extensional and shear enhancements to the viscosity caused by fibres. In this regime, the mean swimming velocity is unchanged from the Newtonian case. The effect of the constant term in Ericksen’s model for the stress, which can be identified as a fibre tension or alternatively a stresslet characterising an active fluid, is also considered. This stress introduces an angular dependence and dramatically changes the streamlines and flow field; fibres aligned with the swimming direction increase the energetic demands of the sheet. The constant fibre stress may result in a reversal of the mean swimming velocity and a negative mean rate of working if it is sufficiently large relative to the other rheological parameters.

We analyse the global flow structures in horizontal convection systems, where the heat supply and removal takes place through separated parts of a lower horizontal surface of a fluid layer. The results are based on direct numerical simulations for the length-to-height aspect ratio of the convection cell $\unicode[STIX]{x1D6E4}=10$, Rayleigh number $\mathit{Ra}$ from $3\times 10^{8}$ to $3\times 10^{11}$ and Prandtl number $\mathit{Pr}$ from 0.05 to 50. The structure of the mean flows in horizontal convection is described in terms of time-averaged spatial distributions of the temperature, velocity, kinetic energy, thermal and kinetic dissipation rates. A possible scenario of transition to turbulent horizontal convection in the whole convection cell of a large aspect ratio is discussed.

The spatial and temporal development of shear-induced overturning billows associated with breaking internal solitary waves is studied by means of a combined laboratory and numerical investigation. The waves are generated in the laboratory by a lock exchange mechanism and they are simulated numerically via a contour-advective semi-Lagrangian method. The properties of individual billows (maximum height attained, time of collapse, growth rate, speed, wavelength, Thorpe scale) are determined in each case, and the billow interaction processes are studied and classified. For broad flat waves, similar characteristics are seen to those in parallel shear flow, but, for waves not at the conjugate flow limit, billow characteristics are affected by the spatially varying wave-induced shear flow. Wave steepness and wave amplitude are shown to have a crucial influence on determining the type of interaction that occurs between billows and whether billow overturning can be arrested. Examples are given in which billows (i) evolve independently of one another, (ii) pair with one another, (iii) engulf/entrain one another and (iv) fail to completely overturn. It is shown that the vertical extent a billow can attain (and the associated Thorpe scale of the billow) is dependent on wave amplitude but that its value saturates once a given amplitude is reached. It is interesting to note that this amplitude is less than the conjugate flow limit amplitude. The number of billows that form on a wave is shown to be dependent on wavelength; shorter waves support fewer but larger billows than their long-wave counterparts for a given stratification.

A general theory for predicting the distribution of scalar gradients (or concentration differences) in heterogeneous flows is proposed. The evolution of scalar fields is quantified from the analysis of the evolution of elementary lamellar structures, which naturally form under the stretching action of the flows. Spatial correlations in scalar fields, and concentration gradients, hence develop through diffusive aggregation of stretched lamellae. Concentration levels at neighbouring spatial locations result from a history of lamella aggregation, which is partly common to the two locations. Concentration differences eliminate this common part, and thus depend only on lamellae that have aggregated independently. Using this principle, we propose a theory which envisions concentration increments as the result of a deconstruction of the basic lamella assemblage. This framework provides analytical expressions for concentration increment probability density functions (PDFs) over any spatial increments for a range of flow systems, including turbulent flows and low-Reynolds-number porous media flows, for confined and dispersing mixtures. Through this deconstruction principle, scalar increment distributions reveal the elementary stretching and aggregation mechanisms building scalar fields.

We experimentally investigate mixing in sheared particulate suspensions by measuring a crucial kinematic quantity of the flow: the stretching laws of material lines in the suspending liquid. High-resolution particle image velocimetry (PIV) measurements in the fluid phase are performed to reconstruct, following the Diffusive Strip Method (Meunier & Villermaux, J. Fluid Mech., vol. 662, 2010, pp. 134–172), the stretching histories of the fluid material lines. In a broad range of volume fractions $20\,\%\leqslant \unicode[STIX]{x1D719}\leqslant 55\,\%$ , the nature of the elongation law changes drastically from linear, in the absence of particles, to exponential in the presence of particles: the mean and the standard deviation of the material line elongations are found to grow exponentially in time and the distribution of elongations converges to a log-normal. A multiplicative stretching model, based on the distribution of local shear rates and on their persistence time, is derived. This model quantitatively captures the experimental stretching laws. The presence of particles is shown to accelerate mixing at large Péclet numbers ( ${\gtrsim}10^{5}$ ). However, the wide distribution of stretching rates results in heterogeneous mixing and, hence, broadly distributed mixing times, in qualitative agreement with experimental observations.

In this paper, we address the problem of how to account for second-order statistics of turbulent flows using low-complexity stochastic dynamical models based on the linearized Navier–Stokes equations. The complexity is quantified by the number of degrees of freedom in the linearized evolution model that are directly influenced by stochastic excitation sources. For the case where only a subset of velocity correlations are known, we develop a framework to complete unavailable second-order statistics in a way that is consistent with linearization around turbulent mean velocity. In general, white-in-time stochastic forcing is not sufficient to explain turbulent flow statistics. We develop models for coloured-in-time forcing using a maximum entropy formulation together with a regularization that serves as a proxy for rank minimization. We show that coloured-in-time excitation of the Navier–Stokes equations can also be interpreted as a low-rank modification to the generator of the linearized dynamics. Our method provides a data-driven refinement of models that originate from first principles and captures complex dynamics of turbulent flows in a way that is tractable for analysis, optimization and control design.

Using a nonlinear evolution equation we examine the dependence of the dispersion of directional surface gravity waves on the Benjamin–Feir index (BFI) and crest length. A parameter for describing the deviation between the dispersion of simulated waves and the theoretical linear dispersion relation is proposed. We find that for short crests the magnitude of the deviation parameter is low while for long crests the magnitude is high and depends on the BFI. In the present paper we also consider laboratory data of directional waves from the Marine Research Institute of the Netherlands (MARIN). The MARIN data confirm the simulations for three cases of BFI and crest length.

The dynamics of vortical structures in flow over a circular cylinder in the vicinity of a flat plate is investigated using particle image velocimetry (PIV). The cylinder is placed above the flat plate with its axis parallel to the wall and normal to the flow direction. The Reynolds number $Re_{D}$ based on the cylinder diameter $D$ is 1072 and the gap $G$ between the cylinder and the flat plate is varied from gap-to-diameter ratio $G/D=0$ to $G/D=3.0$. The flow statistics and vortex dynamics are strongly dependent on the gap ratio $G/D$. Statistics show that as the cylinder comes close to the wall ($G/D\leqslant 2.0$), the cylinder wake becomes more and more asymmetric and a boundary layer separation is induced on the flat plate downstream of the cylinder. The wake vortex shedding frequency increases with decreasing $G/D$ until a critical gap ratio (about $G/D=0.25$) below which the vortex shedding is irregular. The deflection of the gap flow away from the wall and its following interaction with the upper shear layer may be the cause of the higher shedding frequency. The vortex dynamics is investigated based on the phase-averaged flow field and virtual dye visualization in the instantaneous PIV velocity field. It is revealed that when the cylinder is close to the wall ($G/D=2.0$), the cylinder wake vortices can periodically induce secondary spanwise vortices near the wall. As the cylinder approaches the wall ($G/D=1.0$) the secondary vortex can directly interact with the lower wake vortex, and a further approaching of the cylinder ($G/D=0.5$) can result in more complex interactions among the secondary vortex, the lower wake vortex and the upper wake vortex. The breakdown of vortices into filamentary debris during vortex interactions is clearly revealed by the coloured virtual dye visualizations. For $G/D<0.25$, the lower shear layer is strongly inhibited and only the upper shear layer can shed vortices. Investigation of the vortex formation, evolution and interaction in the flow promotes the understanding of the flow physics for different gap ratios.

We report on new experiments designed to investigate bed destabilization processes in a two-dimensional wave flume physical model of a beach. The mobile bed consists of non-cohesive granular material of low density. The wave conditions are provided by repeating a cycle of waves made of two bichromatic groups of different period. The horizontal and vertical velocities are acoustically profiled vertically from free-stream elevation down to the still bed level in the surf zone. Additional measurements of the fluid pressure at positions closely aligned horizontally and vertically in and slightly above the sediment bed are undertaken. Mobile bed interfaces, still bed and top interface, are detected via acoustic and optical methods. Both methods are cross-compared and give similar results. Flow turbulence over the bed is analysed, the Reynolds turbulent shear stress is found negligible compared to the orbital flow induced momentum diffusion. The shear stress and the horizontal pressure gradient are computed at near-bed elevation and used in the bed incipient plug flow model of Sleath (Cont. Shelf Res., vol. 19 (13), 1999, pp. 1643–1664). Both the model and the measurements confirm that destabilization occurs when the non-dimensional pressure gradient (or Sleath number) exceeds the threshold value of 0.3 which is simultaneous with strong flow acceleration. The near-bottom fluid shear stress detected during these flow accelerations at steep wave fronts is found experimentally to be negative, which is retrieved with an unsteady plug flow model. This is suggesting that the fluid above the bed resists the sediment layer motion at these particular phases.

A decomposition of the energy and helicity fluxes is used to analyse turbulent hydrodynamic flows. The decomposition is based on the projection of the flow to a helical basis. This allows the roles of interactions among modes of different helicities to be investigated separately. The proposed formalism is applied to large-scale numerical simulations of non-helical and helical flows, where the decomposed fluxes are explicitly calculated. It is shown that the total energy flux can be split into three fluxes that independently remain constant in the inertial range. One of these fluxes which corresponds to the interactions of fields with the same helicity is negative, implying the presence of an inverse cascade that is ‘hidden’ inside the forward cascade. Similarly to the energy flux, it is also shown that the helicity flux can be decomposed into two fluxes that remain constant in the inertial range. Implications of these results as well as possible new directions for investigations are discussed.

For moderate-to-high Reynolds numbers, aerofoils are known to produce substantial levels of acoustic radiation, known as tonal noise, which arises from a complex interplay between laminar boundary-layer instabilities, trailing-edge acoustic scattering and upstream receptivity of the boundary layers on both aerofoil surfaces. The resulting acoustic spectrum is commonly characterised by distinct equally spaced peaks encompassing the frequency range of convectively amplified instability waves in the pressure-surface boundary layer. In this work, we assess the receptivity and sensitivity of the flow by means of global stability theory and adjoint methods which are discussed in light of the spatial structure of the adjoint global modes, as well as the wavemaker region. It is found that for the frequency range corresponding to acoustic tones the direct global modes capture the growth of instability waves on the suction surface and the near wake together with acoustic radiation into the far field. Conversely, it is shown that the corresponding adjoint global modes, which capture the most receptive region in the flow to external perturbations, have compact spatial support in the pressure surface boundary layer, upstream of the separated flow region. Furthermore, we find that the relative spatial amplitude of the adjoint modes is higher for those modes whose real frequencies correspond to the acoustic peaks. Finally, analysis of the wavemaker region points at the pressure surface near 30 % of the chord as the preferred zone for the placement of actuators for flow control of tonal noise.

The purpose of this work is to explore in detail the structure of the interior flow generated by periodic surface waves on a fluid with constant vorticity. The problem is mapped conformally to a strip and solved numerically using spectral methods. Once the solution is known, the streamlines, pressure and particle paths can be found and mapped back to the physical domain. We find that the flow beneath the waves contains zero, one, two or three stagnation points in a frame moving with the wave speed, and describe the bifurcations between these flows. When the vorticity is sufficiently strong, the pressure in the flow and on the bottom boundary also has very different features from the usual irrotational wave case.

The phenomenon of drag crisis for uniform flow past a smooth sphere is investigated via experiments in a low-turbulence wind tunnel for $1.5\times 10^{5}\leqslant Re\leqslant 5.0\times 10^{5}$. The Reynolds number, $Re$, is based on the free-stream speed and the diameter of the sphere. Based on the activity related to the laminar separation bubble (LSB), the critical regime for the occurrence of drag crisis ($3.4\times 10^{5}<Re<4.4\times 10^{5}$) is further divided into three subregimes. The gradual decrease of mean drag coefficient ($\overline{C}_{D}$) with $Re$, in subregime I, is due to the increase of base pressure and suction near the shoulder of the sphere. The flow is devoid of an LSB in this regime. The coefficient $\overline{C}_{D}$ decreases very rapidly with increase in $Re$ in subregime II primarily due to the increase in mean base pressure ($\overline{C}_{P,b}$). This subregime is characterized by intermittent switching of $C_{D}$ and $C_{P,b}$ between bistable states. Statistical analysis of the surface-pressure and force coefficients relates this behaviour to the intermittent appearance/disappearance of the LSB. The two states of the flow are referred to as the LSB and non-LSB states. The frequency of appearance of the LSB and the duration of its stay increase with increase in $Re$. An intermittency factor $I_{f}$, defined as the fraction of time during which the LSB exists in the flow, is estimated at each $Re$. The value of $I_{f}$ is zero in subregime I and increases from zero to one, with increase in $Re$, in subregime II. The variation of $\overline{C}_{D}$ with $Re$ is found to follow the variation of ($1-I_{f}$) with $Re$. This shows that the decrease of $\overline{C}_{D}$ with increase in $Re$, during drag crisis, is primarily due to the increased probability of the LSB state as opposed to the non-LSB state. A spatio-temporal analysis of the surface pressure measured at various polar locations on the surface of the sphere confirms the axisymmetric nature of the intermittent LSB. In subregime III of the critical regime, the LSB exists at all time instants ($I_{f}=1$). The $\overline{C}_{D}$ value continues to decrease with $Re$ in this subregime due to increase in $\overline{C}_{P,b}$. Unlike the general belief that the decrease in $\overline{C}_{D}$ with increase in $Re$ is due only to the increase in $\overline{C}_{P,b}$, it is found that the increase in suction upstream of the shoulder of the sphere, with increase in $Re$, also plays an important role. In particular, in the high subcritical regime ($2.5\times 10^{5}<Re<3.4\times 10^{5}$), the gradual decrease in $\overline{C}_{D}$ with increase in $Re$ is due solely to the increase in suction in a region upstream of the shoulder of the sphere.

The ability to create dynamic deformations of micron-sized structures is relevant to a wide variety of applications such as adaptable optics, soft robotics and reconfigurable microfluidic devices. In this work, we examine non-uniform lubrication flow as a mechanism to create complex deformation fields in an elastic plate. We consider a Kirchhoff–Love elasticity model for the plate and Hele-Shaw flow in a narrow gap between the plate and a parallel rigid surface. Based on linearization of the Reynolds equation, we obtain a governing equation which relates elastic deformations to gradients in non-homogeneous physical properties of the fluid (e.g. body forces, viscosity and slip velocity). We then focus on a specific case of non-uniform Helmholtz–Smoluchowski electro-osmotic slip velocity, and provide a method for determining the zeta-potential distribution necessary to generate arbitrary static and quasi-static deformations of the elastic plate. Extending the problem to time-dependent solutions, we analyse transient effects on asymptotically static solutions, and finally provide a closed form solution for a Green’s function for time periodic actuations.

Interfacial boundary conditions determined from empirical or ad hoc models remain the standard approach to model fluid flows over porous media, even in situations where the topology of the porous medium is known. We propose a non-empirical and accurate method to compute the effective boundary conditions at the interface between a porous surface and an overlying flow. Using a multiscale expansion (homogenization) approach, we derive a tensorial generalized version of the empirical condition suggested by Beavers & Joseph (J. Fluid Mech., vol. 30 (01), 1967, pp. 197–207). The components of the tensors determining the effective slip velocity at the interface are obtained by solving a set of Stokes equations in a small computational domain near the interface containing both free flow and porous medium. Using the lid-driven cavity flow with a porous bed, we demonstrate that the derived boundary condition is accurate and robust by comparing an effective model to direct numerical simulations. Finally, we provide an open source code that solves the microscale problems and computes the velocity boundary condition without free parameters over any porous bed.

We consider rotating convection in a shallow, cylindrical domain. We examine the conditions under which the resulting vortex develops an eye at its core; that is, a region where the poloidal flow reverses and the angular momentum is low. For simplicity, we restrict ourselves to steady, axisymmetric flows in a Boussinesq fluid. Our numerical experiments show that, in such systems, an eye forms as a passive response to the development of a so-called eyewall, a conical annulus of intense, negative azimuthal vorticity that can form near the axis and separates the eye from the primary vortex. We also observe that the vorticity in the eyewall comes from the lower boundary layer, and relies on the fact the poloidal flow strips negative vorticity out of the boundary layer and carries it up into the fluid above as it turns upward near the axis. This process is effective only if the Reynolds number is sufficiently high for the advection of vorticity to dominate over diffusion. Finally we observe that, in the vicinity of the eye and the eyewall, the buoyancy and Coriolis forces are negligible, and so although these forces are crucial to driving and shaping the primary vortex, they play no direct role in eye formation in a Boussinesq fluid.

The first and higher harmonic components of the resonant fluid response in the gap between two identical fixed rectangular boxes are experimentally investigated in a wave basin. Gap response is excited by transient wave groups (being based on scaled versions of the autocorrelation function of sea-state spectra, representing NewWaves, the average shape of large waves in a sea state). Several different wave groups with different maximum surface elevations, spectral peak frequencies and bandwidths are used, while the bilge shape of the boxes and approach angle of the waves are also varied. Unlike a simple regular wave, it is complicated to separate the harmonic components for a transient wave group due to nonlinear wave–wave and wave–structure interactions. A four-phase combination methodology is used to separate the first four harmonic components, and this also allows higher harmonic components to be isolated with simple digital frequency filtering. Harmonic components up to 14th order in the incident wave amplitude have been extracted. It is shown that for an incident group with appropriate frequency content, the linear gap response may be substantially smaller than the second harmonic component, which is strongly driven via quadratic coupling of the linear terms from the incident wave and occurs in the gap resonant modes. Double frequency excitation may have important practical implications for offshore operations. Fourth and zeroth (long-wave) harmonics in the gap are further driven via quadratic coupling of the second harmonic itself. Linear damping coefficients for the first few modes of the gap resonant response are derived from measured time series using a numerical fit and shown to be higher than those from linear diffraction calculations.

Wake vortex evolution of a square cylinder with a slot synthetic jet issuing from the cylinder’s rear surface has been experimentally investigated using the time-resolved particle image velocimetry technique. The Reynolds number based on the side length of the square cylinder is $Re=836$. The excitation frequency normalized by the natural shedding frequency $f_{e}/f_{0}$ varies from 0 to 6 at the dimensionless stroke length $L_{0}/w=72.6$. The distributions of the time-averaged Reynolds stresses present significant differences as the excitation frequency increases. With control, the mean streamwise velocity deficit of the wake recovers more quickly in comparison with the natural case, and the vertical velocity fluctuation intensity becomes weaker. Moreover, a drag reduction can be achieved for the control cases, especially, for $f_{e}/f_{0}=4$ and $f_{e}/f_{0}=6$, a thrust instead of drag reduction can be obtained. The profiles of the mean streamwise velocity tend to have jet-like distributions. The wake vortex dynamics and its evolution with the excitation frequency are revealed. (i) For the low excitation frequency cases ($f_{e}/f_{0}=0.5$, 1, 2), no significant changes in the dominant frequency and the spanwise vortex structures are observed in comparison with the natural case. (ii) For the moderate excitation frequency case ($f_{e}/f_{0}=3$), the wake vortex shedding frequency is locked on half of the control frequency. In this case, the shear layer is divided into two parts by the synthetic jet vortex, and the wake vortices with smaller scales still shed asymmetrically and appear closer to the square cylinder. (iii) For the high excitation frequency case ($f_{e}/f_{0}=6$), the flow is governed by the synthetic jet. As a result of strong perturbations of the synthetic jet, the wake vortex shedding becomes symmetric with the shedding frequency consistent with the control frequency. And the separation is suppressed effectively. The different control effects of the slot synthetic jet on a square cylinder and a circular cylinder are also compared in detail. Generally speaking, the circular cylinder is easier to be controlled due to its non-fixed separation points.

Kinetic theory and acoustic measurements have proven that the bulk viscosity associated with the expansion or compression effect cannot be ignored in compressible fluids except for monatomic gases. A new theoretical formula for the bulk viscosity coefficient (BVC) $\unicode[STIX]{x1D701}$ is derived by the continuum medium methodology, which provides a further understanding of the bulk viscosity, i.e. $\unicode[STIX]{x1D701}$ is equal to the product of the bulk modulus $K$ and the relaxation time $\unicode[STIX]{x1D70F}$ ($\unicode[STIX]{x1D701}=K\unicode[STIX]{x1D70F}$). The continuum and kinetic theories present consistent results from macro- and microperspectives respectively, only differing in terms of a coefficient. The theoretical predictions of the BVC in diatomic molecules, such as $\text{N}_{2}$, $\text{O}_{2}$ and CO, show good agreement with the experimental data over a wide range of temperature. In addition, the vibrational contributions to $\unicode[STIX]{x1D701}$ are controlled by a rapid exponential decrease at high temperatures, while at low temperatures a slow linear increase proceeds for the rotational cases. The relaxation time $\unicode[STIX]{x1D70F}$, collision number $Z$, BVC $\unicode[STIX]{x1D701}$ and ratio of bulk-to-shear viscosities $\unicode[STIX]{x1D701}/\unicode[STIX]{x1D707}$ in the vibrational mode are found to be several orders of magnitude larger than those in the rotational mode.

We have performed interface-resolved direct numerical simulations of forced homogeneous-isotropic turbulence in a dilute suspension of spherical particles in the Reynolds number range $Re_{\unicode[STIX]{x1D706}}=115{-}140$. The solid–fluid density ratio was set to $1.5$, gravity was set to zero and two particle diameters were investigated corresponding to approximately $5$ and $11$ Kolmogorov lengths. Note that these particle sizes are clearly outside the range of validity of the point-particle approximation, as has been shown by Homann & Bec (J. Fluid Mech., vol. 651, 2010, pp. 81–91). At the present parameter points the global effect of the particles upon the fluid flow is weak. We observe that the dispersed phase exhibits clustering with moderate intensity. The tendency to cluster, which was quantified in terms of the standard deviation of Voronoï cell volumes, decreases with the particle diameter. We have analysed the relation between particle locations and the location of intense vortical flow structures. The results do not reveal any significant statistical correlation. Contrarily, we have detected a small but statistically significant preferential location of particles with respect to the ‘sticky points’ proposed by Goto & Vassilicos (Phys. Rev. Lett., vol. 100 (5), 2008, 054503), i.e. points where the fluid acceleration field is acting such as to increase the local particle concentration in one-way coupled point-particle models under Stokes drag. The presently found statistical correlation between the ‘sticky points’ and the particle locations further increases when focusing on regions with high local concentration. Our results suggest that small finite-size particles can be brought together along the expansive directions of the fluid acceleration field, as previously observed only for the simplest model for sub-Kolmogorov particles. We further discuss the effect of density ratio and collective particle motion upon the basic Eulerian and Lagrangian statistics.