We investigate how compressibility affects the turbulent statistics from a Lagrangian point of view, particularly in the parameter range where the flow transits from the incompressible type to a state dominated by shocklets. A series of three-dimensional simulations were conducted for different types of driving and several Mach numbers. For purely solenoidal driving, as the Mach number increases a new self-similar region first emerges in the Lagrangian structure functions at sub-Kolmogorov time scale and gradually extends to larger time scale. In this region the relative scaling exponent saturates and the saturated value decreases as the compressibility becomes stronger, which can be attributed to the shocklets. The scaling exponent for the inertial range is still very close to that of incompressible turbulence for small Mach number, and discrepancy becomes visible when the Mach number is high enough. When the driving force is dominated by the compressive component the shocklet-induced self-similar region occupies a much wider range of time scales than that in the purely solenoidal driving case. Regardless of the type of driving force, the probability density functions of the velocity increment collapse onto one another for the time scales in the new self-similar region after proper normalization.

A simple geometry that exhibits near motion trapping is tested experimentally, along with perturbed versions of the structure. The motion of the freely floating structure and the surrounding wave field is tracked and the near-motion-trapped mode is found, characterised by a slowly decaying heave motion with very small linear radiation of energy. It is found that the latter property is a better discriminator of the perturbed geometries as viscous damping masks fine differences in radiation damping as far as the motion of the structure is concerned. The magnitude of this viscous damping is reasonably well predicted by a simple Stokes oscillatory boundary layer analysis.

The maximum spreading of drops impacting on smooth and rough surfaces is measured from low to high impact velocity for liquids with different surface tensions and viscosities. We demonstrate that dynamic wetting plays an important role in the spreading at low velocity, characterized by the dynamic contact angle at maximum spreading. In the energy balance, we account for the dynamic wettability by introducing the capillary energy at zero impact velocity, which relates to the spreading ratio at zero impact velocity. Correcting the measured spreading ratio by the spreading ratio at zero velocity, we find a correct scaling behaviour for low and high impact velocity and, by interpolation between the two, we find a universal scaling curve. The influence of the liquid as well as the nature and roughness of the surface are taken into account properly by rescaling with the spreading ratio at zero velocity, which, as demonstrated, is equivalent to accounting for the dynamic contact angle.

Suspended particles migrate towards inertial focusing positions close to walls and align into trains in finite inertia conduit flow. The relative contribution of inertial and viscous forces at the particle length scale, defined by the particle Reynolds number ( $\mathit{Re}_{p}$ ), is a key parameter, where $\mathit{Re}_{p}=\langle \dot{\unicode[STIX]{x1D6FE}}\rangle D^{2}/\unicode[STIX]{x1D708}$ depends on the mean shear rate $\langle \dot{\unicode[STIX]{x1D6FE}}\rangle$ , particle diameter $D$ and fluid kinematic viscosity  $\unicode[STIX]{x1D708}$ . Controlling the location of inertial focusing positions and the interparticle distance is critical in applications such as flow cytometry, imaging and cell entrapment in droplets. By using experimental observations in rectangular microchannels and lattice Boltzmann numerical simulations of dilute suspension flow, the spacing between particles aligned in trains is measured. From the modes of the probability density function of interparticle spacing, preferred spacings at $5D$ and $2.5D$ are observed. At lower $\mathit{Re}_{p}$ , the preferred spacing forms around $5D$ , and with increasing $\mathit{Re}_{p}$ the spacing at $2.5D$ becomes more pronounced. With increasing concentration of the suspension the spacing is influenced by particle crowding effects until stable trains are no longer observed.

The onset of monami – the synchronous waving of seagrass beds driven by a steady flow – is modelled as a linear instability of the flow. Unlike previous works, our model considers the drag exerted by the grass in establishing the steady flow profile, and in damping out perturbations to it. We find two distinct modes of instability, which we label modes 1 and 2. Mode 1 is closely related to Kelvin–Helmholtz instability modified by vegetation drag, whereas mode 2 is unrelated to Kelvin–Helmholtz instability and arises from an interaction between the flow in the vegetated and unvegetated layers. The vegetation damping, according to our model, leads to a finite threshold flow for both of these modes. Experimental observations for the onset and frequency of waving compare well with model predictions for the instability onset criteria and the imaginary part of the complex growth rate respectively, but experiments lie in a parameter regime where the two modes can not be distinguished.

Prior modal stability analysis (Kojima et al., Phys. Fluids, vol. 27, 1984, pp. 19–32) predicted that a rising or sedimenting droplet in a viscous fluid is stable in the presence of surface tension no matter how small, in contrast to experimental and numerical results. By performing a non-modal stability analysis, we demonstrate the potential for transient growth of the interfacial energy of a rising droplet in the limit of inertialess Stokes equations. The predicted critical capillary numbers for transient growth agree well with those for unstable shape evolution of droplets found in the direct numerical simulations of Koh & Leal (Phys. Fluids, vol. 1, 1989, pp. 1309–1313). Boundary integral simulations are used to delineate the critical amplitude of the most destabilizing perturbations. The critical amplitude is negatively correlated with the linear optimal energy growth, implying that the transient growth is responsible for reducing the necessary perturbation amplitude required to escape the basin of attraction of the spherical solution.

The wake behind a circular cylinder in Mach 4 flow is examined experimentally in the Reynolds number range $2\times 10^{4}$ to $5\times 10^{5}$ . Periodic oscillations of the sliplines in the wake are observed. The Strouhal number of the oscillations based on the diameter of the cylinder is found to increase monotonically from 0.30 to 0.50 with increasing Reynolds number. If the Strouhal number is formed using the length of the sliplines, however, it has a constant value of approximately 0.48 for all Reynolds numbers studied. This scaling indicates that the oscillations in supersonic flow are likely driven by acoustic signals propagating back and forth through the subsonic region between the separation points on the cylinder and the neck where the sliplines converge, unlike in subsonic flow where oscillations are caused by vortices shed from the cylinder surface.

When a drop impacts onto a solid surface, the lubrication pressure in the air deforms its bottom into a dimple. This makes the initial contact with the substrate occur not at a point but along a ring, thereby entrapping a central disc of air. We use ultra-high-speed imaging, with 200 ns time resolution, to observe the structure of this first contact between the liquid and a smooth solid surface. For a water drop impacting onto regular glass we observe a ring of microbubbles, due to multiple initial contacts just before the formation of the fully wetted outer section. These contacts are spaced by a few microns and quickly grow in size until they meet, thereby leaving behind a ring of microbubbles marking the original air-disc diameter. On the other hand, no microbubbles are left behind when the drop impacts onto molecularly smooth mica sheets. We thereby conclude that the localized contacts are due to nanometric roughness of the glass surface, and the presence of the microbubbles can therefore distinguish between glass with 10 nm roughness and perfectly smooth glass. We contrast this entrapment topology with the initial contact of a drop impacting onto a film of extremely viscous immiscible liquid, where the initial contact appears to be continuous along the ring. Here, an azimuthal instability occurs during the rapid contraction at the triple line, also leaving behind microbubbles. For low impact velocities the nature of the initial contact changes to one initiated by ruptures of a thin lubricating air film.

We present a general theory for determining the force (and torque) exerted on a boundary (or body) in active matter. The theory extends the description of passive Brownian colloids to self-propelled active particles and applies for all ratios of the thermal energy $k_{B}T$ to the swimmer’s activity $k_{s}T_{s}={\it\zeta}U_{0}^{2}{\it\tau}_{R}/6$ , where ${\it\zeta}$ is the Stokes drag coefficient, $U_{0}$ is the swim speed and ${\it\tau}_{R}$ is the reorientation time of the active particles. The theory, which is valid on all length and time scales, has a natural microscopic length scale over which concentration and orientation distributions are confined near boundaries, but the microscopic length does not appear in the force. The swim pressure emerges naturally and dominates the behaviour when the boundary size is large compared to the swimmer’s run length $\ell =U_{0}{\it\tau}_{R}$ . The theory is used to predict the motion of bodies of all sizes immersed in active matter.

Laboratory experiments involving downward air–helium fountains are presented. The large density differences between these releases and the ambient allow us to investigate how non-Boussinesq effects modify fountain heights and fountain fluctuations in comparison with the Boussinesq case (i.e. marginal density differences). In these experiments, the source Froude number is varied over a wide range covering (i) the very weak, (ii) the weak and (iii) the forced fountain regimes. It is shown that the classical Boussinesq correlations can be extended to the non-Boussinesq case provided that the Froude number is multiplied by the square root of the ratio between the released fluid density and that of the ambient. In the range investigated, no influence of the source Reynolds number is observed.

Enhancement of heat transport across a fluid layer is of fundamental interest as well as great technological importance. For decades, Rayleigh–Bénard convection has been a paradigm for the study of convective heat transport, and how to improve its overall heat-transfer efficiency is still an open question. Here, we report an experimental and numerical study that reveals a novel mechanism that leads to much enhanced heat transport. When vertical partitions are inserted into a convection cell with thin gaps left open between the partition walls and the cooling/heating plates, it is found that the convective flow becomes self-organized and more coherent, leading to an unprecedented heat-transport enhancement. In particular, our experiments show that with six partition walls inserted, the heat flux can be increased by approximately 30 %. Numerical simulations show a remarkable heat-flux enhancement of up to 2.3 times (with 28 partition walls) that without any partitions.

We numerically investigate the effects of non-Newtonian fluid properties, including shear thinning and elasticity, on the locomotion of Taylor’s swimming sheet with arbitrary amplitude. Our results show that elasticity hinders the swimming speed, but a shear-thinning viscosity in the absence of elasticity enhances the speed. The combination of the two effects, modelled using a Giesekus constitutive equation, hinders the swimming speed. We find that the swimming speed of an infinitely long waving sheet in an inelastic shear-thinning fluid has a maximum, whose value depends on the sheet undulation amplitude and the fluid rheological properties. The power consumption, on the other hand, follows a universal scaling law.

The influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is investigated for the case of stress-free mechanical boundary conditions. It is shown that whereas the leading-order system satisfies fixed temperature boundary conditions implicitly, a double boundary layer structure is necessary to satisfy the fixed heat flux thermal boundary conditions. The boundary layers consist of a classical Ekman layer adjacent to the solid boundaries that adjust viscous stresses to zero, and a layer in thermal wind balance just outside the Ekman layers that adjusts the normal derivative of the temperature fluctuation to zero. The influence of these boundary layers on the interior geostrophically balanced convection is shown to be asymptotically weak, however. Upon defining a simple rescaling of the thermal variables, the leading-order reduced system of governing equations is therefore equivalent for both boundary conditions. These results imply that any horizontal thermal variation along the boundaries that varies on the scale of the convection has no leading-order influence on the interior convection, thus providing insight into geophysical and astrophysical flows where stress-free mechanical boundary conditions are often assumed.

We present experimental evidence that a minute amount of polymer additives can significantly enhance heat transport in the bulk region of turbulent thermal convection. The effects of polymer additives are found to be the enhancement of coherent heat fluxes and suppression of incoherent heat fluxes. The enhanced heat transport is associated with the increased coherency of thermal plumes, as a result of the suppression of small-scale turbulent fluctuations by polymers. The incoherent heat flux, arising from turbulent background fluctuations, makes no net contribution to heat transport. The fact that polymer additives can increase the coherency of thermal plumes is supported by the measurements of a number of local quantities, such as the extracted plume amplitude and width, the velocity autocorrelation functions and the velocity–temperature cross-correlation coefficient. The results from local measurements also suggest the existence of a threshold value for the polymer concentration, only above which significant modification of the plume coherent properties and enhancement of the local heat flux can be observed. Estimation of the plume emission rate suggests a stabilization of the thermal boundary layer by polymer additives.

Many micro-organisms find themselves immersed in fluids displaying non-Newtonian rheological properties such as viscoelasticity and shear-thinning viscosity. The effects of viscoelasticity on swimming at low Reynolds numbers have already received considerable attention, but much less is known about swimming in shear-thinning fluids. A general understanding of the fundamental question of how shear-thinning rheology influences swimming still remains elusive. To probe this question further, we study a spherical squirmer in a shear-thinning fluid using a combination of asymptotic analysis and numerical simulations. Shear-thinning rheology is found to affect a squirming swimmer in non-trivial and surprising ways; we predict and show instances of both faster and slower swimming depending on the surface actuation of the squirmer. We also illustrate that while a drag and thrust decomposition can provide insights into swimming in Newtonian fluids, extending this intuition to problems in complex media can prove problematic.

Backwater curves denote the depth profiles of steady flows in a shallow open channel. The classification of these curves for turbulent regimes is commonly used in hydraulics. When the bottom slope $I$ is increased, they can describe the transition from fluvial to torrential regimes. In the case of an infinitely wide channel, we show that laminar flows have the same critical height $h_{c}$ as that in the turbulent case. This feature is due to the existence of surface slope singularities associated to plug-like velocity profiles with vanishing boundary-layer thickness. We also provide the expression of the critical surface slope as a function of the bottom curvature at the critical location. These results validate a similarity model to approximate the asymptotic Navier–Stokes equations for small slopes $I$ with Reynolds number $Re$ such that $Re\,I$ is of order 1.

We present new experiments to measure the rate of entrainment of ambient fluid into a high Reynolds number, axisymmetric, turbulent gravity current. The current is produced by the rapid release of a finite volume of aqueous salt solution from a lock of length $r_{o}$ into a diverging channel, $r>0$ , of angle $9.5^{\circ }$ , filled with a finite depth, $H$ , of fresh water. Using light attenuation we measure the evolving density of the flow, and using dye studies we illustrate the process of mixing between the current and ambient fluid. After an initial adjustment, a circulation develops in the head of the flow: current fluid reaches the nose of the flow, rises up and moves backwards relative to the nose. We find that, owing to the mixing, the volume of the current increases as $V\sim 0.2r_{n}^{7/4}r_{o}^{1/4}H$ while the maximum depth of the head decreases as $h_{n}\sim 0.5H(r_{o}/r_{n})^{1/4}$ , where $r_{n}$ is the location of the front of the current. Combining these results, we estimate that the recirculating current fluid mixes with a fraction $E=0.33\pm 0.09$ of the ambient fluid that is directly ahead of the current and displaced upwards by it. Some of the mixed fluid supplies the tail of the flow, while the remainder recirculates into the head, which becomes progressively more dilute. In accord with Huppert & Simpson (J. Fluid Mech., vol. 99, 1980, pp. 785–799), we find that the position of the front increases with time as $r_{n}\approx (1.28\pm 0.05)B^{1/4}t^{1/2}$ , where $B$ is the total buoyancy of the flow. We also find that the maximum value of the vertical integral of the buoyancy $(\overline{g^{\prime }}h)_{n}$ decreases with the position of the nose according to the relation $(\overline{g^{\prime }}h)_{n}\approx (0.89\pm 0.12)Br_{n}^{-2}$ , consistent with a Froude number $0.86\pm 0.07$ . We compare our measurements with a new idealised self-similar solution of the depth-averaged equations that accounts for the mixing at the nose, the vertical shear in the velocity and the lateral stratification of the buoyancy within the current.

The deformation of an initially circular miscible blob in a rectilinear displacement is investigated numerically for porous media when the blob is more viscous than the displacing fluid. We find in the parameter space spanned by the Péclet number and log-mobility ratio the existence of a new lump-shaped instability zone between two distinct regimes of comet and viscous fingering (VF) deformations. The more viscous circular blob is destabilized by VF only over a finite window of log-mobility ratio, contrary to the displacement of a more viscous finite slice with planar interfaces. This difference is attributed to the initial curvature of the miscible blob.

It is shown theoretically how the scaling of coherent structures in shear flows changes their asymptotic development at large Reynolds number. Based on the theory a family of nonlinear self-sustained states at Kolmogorov microscale is numerically identified on the laminar–turbulent boundary of shear flows. Theoretically and numerically the states connect to known asymptotic states existing at larger scale. The asymptotically very small amplitude of the new states may explain why strongly sheared, linearly stable laminar flows can cause a turbulent transition by small disturbances. The numerically obtained Kolmogorov-scale solutions can be used to describe the theoretically minimal self-sustained structures appearing in various shear flows.

We perform direct numerical simulations of an unstably stratified turbulent channel flow to address the effects of buoyancy on the boundary layer dynamics and mean field quantities. We systematically span a range of parameters in the space of friction Reynolds number ( $\mathit{Re}_{{\it\tau}}$ ) and Rayleigh number ( $\mathit{Ra}$ ). Our focus is on deviations from the logarithmic law of the wall due to buoyant motion. The effects of convection in the relevant ranges are discussed, providing measurements of mean profiles of velocity, temperature and Reynolds stresses as well as of the friction coefficient. A phenomenological model is proposed and shown to capture the observed deviations of the velocity profile in the log-law region from the non-convective case.