Motivated by the need to characterize the spatio-temporal structure of turbulence in wall-bounded flows, we study wavenumber–frequency spectra of the streamwise velocity component based on large-eddy simulation (LES) data. The LES data are used to measure spectra as a function of the two wall-parallel wavenumbers and the frequency in the equilibrium (logarithmic) layer. We then reformulate one of the simplest models that is able to reproduce the observations: the random sweeping model with a Gaussian large-scale fluctuating velocity and with additional mean flow. Comparison with LES data shows that the model captures the observed temporal decorrelation, which is related to the Doppler broadening of frequencies. We furthermore introduce a parameterization for the entire wavenumber–frequency spectrum $E_{11}(k_{1},k_{2},{\it\omega};z)$ , where $k_{1}$ , $k_{2}$ are the streamwise and spanwise wavenumbers, ${\it\omega}$ is the frequency and $z$ is the distance to the wall. The results are found to be in good agreement with LES data.

We consider steady advection–diffusion about a slender body of revolution at arbitrary $O(1)$ Péclet numbers ( $\mathit{Pe}$ ). The transported scalar attenuates at large distances and is governed by axisymmetric (either Dirichlet or Neumann) data prescribed at the body boundary. The advecting field is assumed to be an axisymmetric Stokes flow approaching a uniform stream at large distances and satisfying impermeability at the boundary; otherwise, the interfacial distribution of tangential velocity is assumed to be arbitrary, irrotational and no-slip Stokes flows being particular cases. Employing the method of matched asymptotic expansions, we develop a systematic scheme for the calculation of the scalar concentration in increasing powers of $\ln ^{-1}(1/{\it\epsilon})$ , ${\it\epsilon}\ll 1$ being the body’s characteristic thickness to length ratio. The leading term in the inner expansion coincides with the pure diffusion case, the second term depends nonlinearly on the magnitude of the far-field stream and higher-order terms depend on the boundary distribution of tangential velocity. In the special case of irrotational flow and Neumann boundary conditions the logarithmic expansion terminates, leaving an algebraic error in ${\it\epsilon}$ . The general formulae developed can be directly applied to numerous physical scenarios. We here consider the classical problem of forced heat convection from an isothermal body, finding a two-term expansion for $\mathit{Nu}({\it\epsilon},\mathit{Pe})/\mathit{Nu}({\it\epsilon},0)$ , the ratio of the Nusselt number to its value at $\mathit{Pe}=0$ . This ratio is insensitive to the particle shape at the asymptotic orders considered; at moderately large $\mathit{Pe}$ ( $\ll {\it\epsilon}^{-1}$ ) its deviation from unity is $O[\ln (\mathit{Pe})/\text{ln}\,(1/{\it\epsilon})]$ , marking the poor effectiveness of advection about slender bodies. The expansion is compared with a numerical computation in the case of a prolate spheroid in both irrotational and no-slip Stokes flows.

The spiral vortex breakdown has been experimentally studied in the flow field generated by a lean premixing prevaporizing (LPP) combustor model. The vortex breakdown structures are characterized by a solid-body rotation. Hence, phase-locked laser Doppler velocimeter measurements, performed in a meridional plane, have allowed reconstruction of the periodic velocity and Reynolds stress fields within a time period and in a relative cylindrical frame of reference. Then, the turbulent kinetic energy (TKE) production term has been computed. The velocity distribution analysis indicates that a spiral vortex core wraps around an inner reverse flow zone. Both are characterized by a precessing motion. Then, the main contributions to the TKE production have been identified together with their spatial locations. Their distribution indicates that the maxima are placed in two regions: one located between the inner reverse flow and the spiral vortex and the second between the spiral vortex core and the external nearly quiescent fluid. In the latter, the TKE production term distributions have the features of a Kelvin–Helmholtz-like instability. In the former, the TKE production seems to be related to a Kelvin–Helmholtz-like instability only at the interface between the bubble and the spiral vortex core.

The transformation of small-amplitude internal waves on the oceanic shelf is studied numerically. The transmission and reflection coefficients are found for the simplified step-wise model of the oceanic shelf in a two-layer fluid. The approximate formulae are proposed for the transformation coefficients as functions of incident wavenumber, density ratio of layers, depth of the pycnocline and height of the bottom step. Results of direct numerical modelling of internal wave transformation are obtained and presented as functions of all aforementioned parameters. It is shown that there is a good agreement between the outcomes of approximate theory and numerical data. Both the theoretical and the numerical results agree well with the law of energy flux conservation.

We consider the inertia-dominated rise of a meniscus around a vertical circular cylinder. Previous experiments and scaling analysis suggest that the height of the meniscus, $h_{m}$ , grows with the time following the initiation of rise, $t$ , like $h_{m}\propto t^{1/2}$ . This is in contrast to the rise on a vertical plate, which obeys the classic capillary–inertia scaling $h_{m}\propto t^{2/3}$ . We highlight a subtlety in the scaling analysis that yielded $h_{m}\propto t^{1/2}$ and investigate the consequences of this subtlety. We develop a potential flow model of the dynamic problem, which we solve using the finite element method. Our numerical results agree well with previous experiments but suggest that the correct early time behaviour is, in fact, $h_{m}\propto t^{2/3}$ . Furthermore, we show that at intermediate times the dynamic rise of the meniscus is governed by two parameters: the contact angle and the cylinder radius measured relative to the capillary length scale, $t^{2/3}$ . This result allows us to collapse previous experimental results with different cylinder radii (but similar static contact angles) onto a single master curve.

We analyse the self-diffusiophoresis of a spherical particle animated by a non-uniform chemical reaction at its boundary. We consider two models of solute absorption, one with a specified distribution of interfacial solute flux and one where this flux is governed by first-order kinetics with a specified distribution of rate constant. We employ a macroscale model where the short-range interaction of the solute with the particle boundary is represented by an effective slip condition. The solute transport is governed by an advection–diffusion equation. We focus upon the singular limit of large Péclet numbers, $\mathit{Pe}\gg 1$ . In the fixed-flux model, the excess-solute concentration is confined to a narrow boundary layer. The scaling pertinent to that limit allows the problem governing the solute concentration to be decoupled from the flow field. The resulting nonlinear boundary-layer problem is handled using a transformation to stream-function coordinates and a subsequent application of Fourier transforms, and is thereby reduced to a nonlinear integral equation governing the interfacial concentration. Its solution provides the requisite approximation for the particle velocity, which scales as $\mathit{Pe}^{-1/3}$ . In the fixed-rate model, large Péclet numbers may be realized in different limit processes. We consider the case of large swimmers or strong reaction, where the Damköhler number $\mathit{Da}$ is large as well, scaling as $\mathit{Pe}$ . In that double limit, where no boundary layer is formed, we obtain a closed-form approximation for the particle velocity, expressed as a nonlinear functional of the rate-constant distribution; this velocity scales as $\mathit{Pe}^{-2}$ . Both the fixed-flux and fixed-rate asymptotic predictions agree with the numerical values provided by computational solutions of the nonlinear transport problem.

A formal relationship between the skewness and the correlation coefficient of large and small scales, termed the amplitude modulation coefficient, is established for a general statistically stationary signal and is analysed in the context of a turbulent velocity signal. Both the quantities are seen to be measures of phase in triadically consistent interactions between scales of turbulence. The naturally existing phase relationships between large and small scales in a turbulent boundary layer are then manipulated by exciting a synthetic large-scale motion in the flow using a spatially impulsive dynamic wall roughness perturbation. The synthetic scale is seen to alter the phase relationships, or the degree of modulation, in a quasi-deterministic manner by exhibiting a phase-organizing influence on the small scales. The results presented provide encouragement for the development of a practical framework for favourable manipulation of energetic small-scale turbulence through large-scale inputs in a wall-bounded turbulent flow.

Soft bio-microcapsules are drops bounded by a thin elastic shell made of cross-linked proteins. Their shapes and their dynamics in flow depend on their membrane constitutive law characterized by shearing and area-dilatation resistance. The deformations of such capsules are investigated experimentally in planar elongation flows and compared with numerical simulations for three bidimensional models: Skalak, neo-Hookean and generalized Hooke. An original cross-flow microfluidic set-up allows the visualization of the deformed shape in the two perpendicular main fields of view. Whatever the elongation rate, the three semi-axis lengths of the ellipsoid fitting the experimental shape are measured up to 180 % of stretching of the largest axis. The geometrical analysis in the two views is sufficient to determine the constitutive law and the Poisson ratio of the membrane without a preliminary knowledge of the shear elastic modulus $G_{s}$ . We conclude that the membrane of human serum albumin capsules obeys the generalized Hooke law with a Poisson ratio of 0.4. The shear elastic modulus is then determined by the combination of numerical and experimental variations of the Taylor parameter with the capillary number.

It is often assumed on the basis of single-parcel energetics that compressible effects and conversions with internal energy are negligible whenever typical displacements of fluid parcels are small relative to the scale height of the fluid (defined as the ratio of the squared speed of sound to the gravitational acceleration). This paper shows that the above approach is flawed, however, and that a correct assessment of compressible effects and internal energy conversions requires the consideration of the energetics of at least two parcels or, more generally, of mass-conserving parcel rearrangements. As a consequence, it is shown that it is the adiabatic lapse rate and its derivative with respect to pressure, rather than the scale height, that controls the relative importance of compressible effects and internal energy conversions when considering the global energy budget of a stratified fluid. Only when mass conservation is properly accounted for is it possible to explain why the available internal energy can account for up to 40 % of the total available potential energy in the oceans. This is considerably larger than the prediction of single-parcel energetics, according to which this number should be no more than approximately 2 %.

In boundary-layer flows, one may reduce skin-friction drag by delaying the onset of laminar-to-turbulent transition via the attenuation of small-amplitude Tollmien–Schlichting (TS) waves. In this work, we use numerical simulations and experiments to compare the robustness of adaptive and model-based techniques for reducing the growth of two-dimensional TS disturbances. In numerical simulations, the optimal linear quadratic Gaussian (LQG) regulator shows the best performance under the conditions it was designed for. However, it is found that the performance deteriorates linearly with the drift of the Reynolds number from its nominal value. As a result, an order-of-magnitude loss of performance is observed when applying the computation-based LQG controller in wind-tunnel experiments. In contrast, it is shown that the adaptive filtered-X least-mean-squares (FXLMS) algorithm is able to maintain an essentially constant performance for significant deviations of the nominal values of the disturbance amplitude and Reynolds number.

The translational and rotational dynamics of oblate spheroidal particles suspended in a directly simulated turbulent channel flow have been examined. Inertial disk-like particles exhibited a significant preferential orientation in the plane of the mean shear. The rotational inertia about the symmetry axis of the disk-like particles hampered the spin-up of the flattest particles to match the mean flow vorticity. The influence of the particle shape on the orientation and rotation diminished as the translational inertia increased from Stokes number 1 to 30. An isotropization of both orientation and rotation could be observed in the core region of the channel. The translational motion of the oblate spheroids had a weak dependence on the aspect ratio. We therefore concluded that inertial particles sample nearly the same flow field irrespective of shape. Nevertheless, the orientation and rotation of disk-like particles turned out to be qualitatively different from the dynamics of fibre-like particles.

We present a depth-integrated equation for the mechanics of generation, propagation and dissipation of low-frequency hydro-acoustic waves due to sudden bottom displacement in a weakly compressible ocean overlying a weakly compressible viscous sediment layer. The model is validated against a full 3D computational model. Physical properties of these waves are studied and compared with those for waves over a rigid sea bed, revealing changes in the frequency spectrum and modal peaks. The resulting model equation can be used for numerical prediction in large-scale domains, overcoming the computational difficulties of 3D models while taking into account the role of bottom dissipation on hydro-acoustic wave generation and propagation.

Flowing over erodible beds, channelized granular avalanches can alter their volume by entraining or detraining basal grains. In detail, entrainment results from a gradual adjustment of stress and velocity profiles over depth, bringing bed material past yield (and vice versa for detrainment). To capture this process, we propose new depth-integrated equations that balance kinetic energy in addition to mass and momentum. The equations require a local granular rheology, assumed viscoplastic, but no extra erosion law. Entrainment rates are instead deduced from the depth-integrated layer dynamics. To check the approach, we obtain solutions for non-equilibrium heap flows, and compare them with experiments conducted in a seesaw channel.

While fluid flows are known to promote dissolution of materials, such processes are poorly understood due to the coupled dynamics of the flow and the receding surface. We study this moving boundary problem through experiments in which hard candy bodies dissolve in laminar high-speed water flows. We find that different initial geometries are sculpted into a similar terminal form before ultimately vanishing, suggesting convergence to a stable shape–flow state. A model linking the flow and solute concentration shows how uniform boundary-layer thickness leads to uniform dissolution, allowing us to obtain an analytical expression for the terminal geometry. Newly derived scaling laws predict that the dissolution rate increases with the square root of the flow speed and that the body volume vanishes quadratically in time, both of which are confirmed by experimental measurements.

We examine experimentally the influence of flexibility on the side-to-side fluttering motion of passive wings settling under the influence of gravity. Our results demonstrate the existence of an optimal flexibility that allows flexible wings to remain airborne twice as long as their rigid counterparts of identical mass and size. Flow visualization and measurements allow us to elucidate the role of flexibility in generating increased lift and wing circulation by shedding additional vorticity at the turning point. Theoretical scalings are derived from a reduced model of the flight dynamics and yield quantitative agreement with experiments. These scalings rationalize the strong positive correlation between flexibility and flight time. Our experimental results and theoretical scalings represent an ideal system for the validation of computational approaches to model biologically inspired fluid–structure interaction problems.

For the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoulli’s constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli’s constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

Graetz and Nusselt studied heat transfer between a developed laminar fluid flow and a tube at constant wall temperature. Here, we extend the Graetz–Nusselt problem to dense fluid flows with partial wall slip. Its limits correspond to the classical problems for no-slip and no-shear flow. The amount of heat transfer is expressed by the local Nusselt number $\mathit{Nu}_{x}$ , which is defined as the ratio of convective to conductive radial heat transfer. In the thermally developing regime, $\mathit{Nu}_{x}$ scales with the ratio of position $\tilde{x}=x/L$ to Graetz number $\mathit{Gz}$ , i.e.  $\mathit{Nu}_{x}\propto (\tilde{x}/\mathit{Gz})^{-{\it\beta}}$ . Here, $L$ is the length of the heated or cooled tube section. The Graetz number $\mathit{Gz}$ corresponds to the ratio of axial advective to radial diffusive heat transport. In the case of no slip, the scaling exponent ${\it\beta}$ equals $1/3$ . For no-shear flow, ${\it\beta}=1/2$ . The results show that for partial slip, where the ratio of slip length $b$ to tube radius $R$ ranges from zero to infinity, ${\it\beta}$ transitions from $1/3$ to $1/2$ when $10^{-4}<b/R<10^{0}$ . For partial slip, ${\it\beta}$ is a function of both position and slip length. The developed Nusselt number $\mathit{Nu}_{\infty }$ for $\tilde{x}/\mathit{Gz}>0.1$ transitions from 3.66 to 5.78, the classical limits, when $10^{-2}<b/R<10^{2}$ . A mathematical and physical explanation is provided for the distinct transition points for ${\it\beta}$ and $\mathit{Nu}_{\infty }$ .

One of the simplest splashing scenarios results from the impact of a single drop on a deep pool. The traditional understanding of this process is that the impact generates an axisymmetric sheet-like jet that later breaks up into secondary droplets. Recently it was shown that even this simplest of scenarios is more complicated than expected because multiple jets can be generated from a single impact event and there are transitions in the multiplicity of jets as the experimental parameters are varied. Here, we use experiments and numerical simulations of a single drop impacting on a deep pool to examine the transition from impacts that produce a single jet to those that produce two jets. Using high-speed X-ray imaging methods we show that vortex separation within the drop leads to the formation of a second jet long after the formation of the ejecta sheet. Using numerical simulations we develop a phase diagram for this transition and show that the capillary number is the most appropriate order parameter for the transition.

In this numerical investigation we explore the possibility of applying free-stream vortices as a passive flow control method for delaying the transition to turbulence. The work is motivated by previous experimental studies demonstrating that stable streamwise boundary layer (BL) streaks can attenuate both two- and three-dimensional disturbances inside the BL, leading to transition delay, with the implication of reducing skin-friction drag. To date, successful control has been obtained using physical BL modulators mounted on the surface in order to generate stable streaks. However, surface mounted BL modulators are doomed to failure when the BL is subject to free-stream turbulence (FST), since a destructive interaction between the two is inevitable. In order to tackle free-stream disturbances, such as FST, a smooth surface is desired, which has motivated us to seek new methods to induce streamwise streaks inside the BL. A first step, in a systematic order, is taken in the present paper to prove the control idea of generating free-stream vortices for the attenuation of ordinary Tollmien–Schlichting waves inside the BL. In this proof-of-concept study we show that, by applying a spanwise array of counter-rotating free-stream vortices, inducing streamwise BL streaks further downstream, it is possible to alter the BL stability characteristics to such a degree that transition delay may be accomplished. For the demonstration we use direct numerical simulations along with stability analysis.

Analysis of bioconvection in dilute suspensions of bottom-heavy but randomly swimming micro-organisms is commonly based on a model introduced in 1990. This couples the Navier–Stokes equations, the cell conservation equation and the Fokker–Planck equation (FPE) for the probability density function for a cell’s swimming direction $\boldsymbol{p}$ , which balances rotational diffusion against viscous and gravitational torques. The results have shown qualitative agreement with observation, but the model has not been subjected to direct quantitative testing in a controlled experiment. Here, we consider a simple configuration in which the suspension is contained in a circular cylinder of radius $R$ , which rotates at angular velocity ${\rm\Omega}$ about a horizontal axis. We solve the FPE and calculate the cells’ mean swimming velocity, which proves to be horizontal when $B{\rm\Omega}\gg 1$ , where $B$ is the gyrotactic reorientation time scale. Then we compute the cell concentration distribution, which is non-uniform only in a thin boundary layer near the cylinder wall when ${\it\beta}^{2}={\rm\Omega}R^{2}/D\gg 1$ , where $D$ is the cells’ translational diffusivity. The fact that cells are denser than water means that this concentration distribution drives a perturbation to the underlying solid-body rotational flow which can be calculated analytically. The predictions of the theory are evaluated in terms of a proposed experimental realisation of the configuration, using suspensions of the alga Chlamydomonas nivalis or Chlamydomonas reinhardtii or the algal colony Volvox.