The influence of a single roughness element on the evolution of two-dimensional (2-D) Tollmien–Schlichting (TS) waves is investigated experimentally. Experiments are carried out in a region of zero pressure gradient of an airfoil section. Downstream from the disturbance source, TS waves interact with a cylindrical roughness element with a slowly oscillating height. The oscillation frequency of the roughness was approximately 1500 times lower than the wave frequency and approximately 250 times slower than the characteristic time of flow passing the region of transition development. Therefore, the roughness behaved as a quasi-steady disturbance. The set-up enabled us to perform hot-wire measurements phase locked to the waves and to the roughness movement. Experimental results show a scattering of the 2-D waves into oblique ones and a relatively weak distortion of the mean flow for roughness heights as large as 0.2 times the boundary layer displacement thickness ($\unicode[STIX]{x1D6FF}^{\ast }$). Transfer functions for TS wave scattering at the roughness are obtained. Results show an unexpected coincidence in shape with acoustic receptivity functions found in Würz et al. (J. Fluid Mech., vol. 478, 2003, pp. 135–163) for the problem of excitation of TS waves by scattering of acoustic waves at surface roughness. In the present work, the ratio between the incoming 2-D wave amplitude to the amplitude of the scattered oblique waves scaled linearly with the roughness height only for very shallow roughness. For roughness elements higher than $0.08\unicode[STIX]{x1D6FF}^{\ast }$ and below $0.2\unicode[STIX]{x1D6FF}^{\ast }$, the wave scattering exhibited a quadratic variation with respect to the roughness height. In addition, this feature did not vary significantly with respect to TS wave frequency. An analysis of the weakly nonlinear interactions triggered by the roughness element is also carried out, assisted by numerical solution of nonlinear parabolized stability equations, performed for a two-dimensional Blasius boundary layer. A comparison between experiments and simulations reveals that the weakly nonlinear interactions observed are not substantially affected by mean flow distortions that could be produced in the wake of the small and medium sized roughness elements ($h<0.2\unicode[STIX]{x1D6FF}^{\ast }$). From a practical perspective, results suggest that scattering coefficients might be employed to include the effect of isolated and medium sized roughness elements in transition prediction tools developed for smooth surfaces.

We experimentally study the iso-viscous displacement flow of two immiscible Newtonian fluids in an inclined pipe. The fluids have the same viscosity but different densities. The displacing fluid is denser than the displaced fluid and is placed above the displaced fluid (i.e. a density-unstable configuration) in a pipe with small diameter-to-length ratio ($\unicode[STIX]{x1D6FF}\ll 1$). In the limit considered, six dimensionless groups describe these flows: the pipe inclination angle, $\unicode[STIX]{x1D6FD}$, an Atwood number, $At$, a Reynolds number, $Re$, a densimetric Froude number, $Fr$, a capillary number, $Ca$, and the fluids static contact angle, $\unicode[STIX]{x1D703}$. Our experiments, carried out in an acrylic pipe using wetting salt-water solutions displacing non-wetting oils, cover a fairly broad range of these parameters. Completely different patterns than those of miscible flows have been observed, governed by distinct dynamics. The wetting properties of the displacing liquid and fluids immiscibility are found to significantly increase the efficiency of the displacement. During the early stage of the displacement, strong shearing is observed between the heavy and light layers, promoting Kelvin–Helmholtz instabilities. At later stages, the intensity of Kelvin–Helmholtz instabilities is reduced. However, surface-tension-driven Rayleigh-type instabilities will remain active causing droplet shedding (pearling) at displaced fluid receding contact lines. The speed of the advancing displacing front (inversely related to the displacement efficiency) is measured and characterized in dimensionless maps suggesting high values at low ranges of $Re$ and $Ca$. Depending on the degree of flow stability and droplet formation, three major flow regimes namely viscous, transitionary and dispersed are characterized and classified in dimensionless maps. In the absence of a mean imposed velocity (exchange flow), it is found that capillary blockage may occur hindering Rayleigh–Taylor instabilities.

Liquid-infused surfaces present a novel, passive method of turbulent drag reduction. Inspired by the Nepenthes Pitcher Plant, liquid-infused surfaces utilize a lubricating fluid trapped within structured roughness to facilitate a slip at the effective surface. The conceptual idea is similar to that of superhydrophobic surfaces, which rely on a lubricating air layer, whereas liquid-infused surfaces use a preferentially wetting liquid lubricant to create localized fluid–fluid interfaces. Maintaining the presence of these slipping interfaces has been shown to be an effective method of passively reducing skin friction drag in turbulent flows. Given that liquid-infused surfaces have only recently been considered for drag reduction applications, there is no available framework to relate surface and lubricant characteristics to any resulting drag reduction. Here we use results from direct numerical simulations of turbulent channel flow over idealized, liquid-infused grooves to demonstrate that the drag reduction achieved using liquid-infused surfaces can be described using the framework established for superhydrophobic surfaces. These insights can be used to explain drag reduction results observed in experimental studies of lubricant-infused surfaces. We also demonstrate how a liquid-infused surface can reduce drag even when the viscosity of the lubricant exceeds that of the external fluid flow, which at first glance can seem counter-intuitive.

The structure of a trailing vortex from a wing undergoing small amplitude, low frequency heaving motion is investigated using space–time representations determined from stereo particle image velocimetry. The evolution of the vortex shows large fluctuations of axial velocity deficit and circulation during the oscillation cycle. Correspondingly, large variations of swirl ratio occur and onset of pronounced azimuthal vorticity arises. At a given cross-section of the vortex, the pattern of azimuthal vorticity moves around its axis in an ordered fashion as both it and the pattern of velocity defect increase in magnitude and scale. When the swirl ratio attains its minimum value during the oscillation cycle, and this value lies below the theoretically established critical threshold for amplification of azimuthal modes, the magnitude and scale of the pattern of azimuthal vorticity is maximized. Subsequent increase of the swirl ratio yields attenuation of the azimuthal vorticity. Onset of pronounced azimuthal vorticity when the swirl ratio decreases involves rapid amplification, then disruption, of axial vorticity fluctuation.

Direct numerical simulation of open-channel flow over a bed of spheres arranged in a regular pattern has been carried out at bulk Reynolds number and roughness Reynolds number (based on sphere diameter) of approximately 6900 and 120, respectively, for which the flow regime is fully rough. The open-channel height was approximately 5.5 times the diameter of the spheres. Extending the results obtained by Chan-Braun et al. (J. Fluid Mech., vol. 684, 2011, pp. 441–474) for an open-channel flow in the transitionally rough regime, the present purpose is to show how the flow structure changes as the fully rough regime is attained and, for the first time, to enable a direct comparison with experimental observations. Different statistical tools were used to investigate the flow field in the roughness sublayer and in the logarithmic region. The results indicate that, in the vicinity of the roughness elements, the average flow field is affected both by Reynolds number effects and by the geometrical features of the roughness, while at larger wall distances this is not the case, and roughness concepts can be applied. Thus, the roughness function is computed which in the present set-up can be expected to depend on the relative submergence. The flow–roughness interaction occurs mostly in the region above the virtual origin of the velocity profile, and the effect of form-induced velocity fluctuations is maximum at the level of sphere crests. In particular, the root mean square of fluctuations about the streamwise component of the average velocity field reflects the geometry of the spheres in the roughness sublayer and attains a maximum value just above the roughness elements. The latter is significantly weakened and shifted towards larger wall distances as compared to the transitionally rough regime or the case of a smooth wall. The spanwise length scale of turbulent velocity fluctuations in the vicinity of the sphere crests shows the same dependence on the distance from the wall as that observed over a smooth wall, and both vary with Reynolds number in a similar fashion. Moreover, the hydrodynamic force and torque experienced by the roughness elements are investigated and the footprint left by vortex structures on the stress acting on the sphere surface is observed. Finally, the possibility either to adopt an analogy between the hydrodynamic forces associated with the interaction of turbulent structures with a flat smooth wall or with the surface of the spheres is also discussed, distinguishing the skin-friction from the form-drag contributions both in the transitionally rough and in the fully rough regimes.

Motivated by anomalous entrainment behaviour in cumulus clouds, Bhat et al. (Exp. Fluids, vol. 7, 1989, pp. 99–102) pioneered a laboratory experiment to study turbulent jets subjected to a volumetric heating away from the momentum source. The study concluded that the use of a constant entrainment coefficient was insufficient for the flow, and that the results did not confirm the analysis of Hunt (Recent Research Advances in the Fluid Mechanics of Turbulent Jets and Plumes, 1994, pp. 309–334, Kluwer Academic), which suggested that an increase in relative turbulent transport of streamwise momentum could lead to a decrease in entrainment. The present paper re-evaluates theoretical aspects of both studies, and includes a decomposition of the factors contributing to entrainment. The reworked analysis is then used to examine three-dimensional numerical simulations of turbulent jets with off-source heating. The data are consistent with previous work, but give deeper insight not easily obtainable through experiment. Specifically, direct measurement of flux integrals shows that previous inference from experimental measurements of centreline velocity and profile widths under the assumption of self-similarity can lead to underestimation of the mass flux by over 50 % in some cases. Radial profiles of temperature, radial velocity and turbulent correlations show significant departures from self-similarity. The flux measurements show that there is actually an increase in the entrainment coefficient with heating, and that it is locally enhanced by positive forcing and decreased by an increase in turbulent transport of streamwise momentum, thereby confirming the essence of the original proposal of Hunt.

An elegant model for passive scalar mixing and decay was given by Kraichnan (Phys. Fluids, vol. 11, 1968, pp. 945–953) assuming the velocity to be delta correlated in time. For realistic random flows this assumption becomes invalid. We generalize the Kraichnan model to include the effects of a finite correlation time, $\unicode[STIX]{x1D70F}$, using renewing flows. The generalized evolution equation for the three-dimensional (3-D) passive scalar spectrum $\hat{M}(k,t)$ or its correlation function $M(r,t)$, gives the Kraichnan equation when $\unicode[STIX]{x1D70F}\rightarrow 0$, and extends it to the next order in $\unicode[STIX]{x1D70F}$. It involves third- and fourth-order derivatives of $M$ or $\hat{M}$ (in the high $k$ limit). For small-$\unicode[STIX]{x1D70F}$ (or small Kubo number), it can be recast using the Landau–Lifshitz approach to one with at most second derivatives of $\hat{M}$. We present both a scaling solution to this equation neglecting diffusion and a more exact solution including diffusive effects. To leading order in $\unicode[STIX]{x1D70F}$, we first show that the steady state 1-D passive scalar spectrum, preserves the Batchelor (J. Fluid Mech., vol. 5, 1959, pp. 113–133) form, $E_{\unicode[STIX]{x1D703}}(k)\propto k^{-1}$, in the viscous–convective limit, independent of $\unicode[STIX]{x1D70F}$. This result can also be obtained in a general manner using Lagrangian methods. Interestingly, in the absence of sources, when passive scalar fluctuations decay, we show that the spectrum in the Batchelor regime at late times is of the form $E_{\unicode[STIX]{x1D703}}(k)\propto k^{1/2}$ and also independent of $\unicode[STIX]{x1D70F}$. More generally, finite $\unicode[STIX]{x1D70F}$ does not qualitatively change the shape of the spectrum during decay. The decay rate is however reduced for finite $\unicode[STIX]{x1D70F}$. We also present results from high resolution ($1024^{3}$) direct numerical simulations of passive scalar mixing and decay. We find reasonable agreement with predictions of the Batchelor spectrum during steady state. The scalar spectrum during decay is however dependent on initial conditions. It agrees qualitatively with analytic predictions when power is dominantly in wavenumbers corresponding to the Batchelor regime, but is shallower when box-scale fluctuations dominate during decay.

A new framework is presented for estimation and control of instabilities in wall-bounded shear flows described by the linearised Navier–Stokes equations. The control design considers the use of localised actuators/sensors to account for convective instabilities in an ${\mathcal{H}}_{2}$ optimal control framework. External sources of disturbances are assumed to enter the control domain through the inflow. A new inflow disturbance model is proposed for external excitation of the perturbation modes that contribute to transition. This model allows efficient estimation of the flow perturbations within the localised control region of a conceptually unbounded domain. The state-space discretisation of the infinite-dimensional system is explicitly obtained, which allows application of linear control theoretic tools. A reduced-order model is subsequently derived using exact balanced truncation that captures the input/output behaviour and the dominant perturbation dynamics. This model is used to design an ${\mathcal{H}}_{2}$ optimal controller to suppress the instability growth. The two-dimensional non-periodic channel flow is considered as an application case. Disturbances are generated upstream of the control domain and the resulting flow perturbations are estimated/controlled using point wall shear measurements and localised unsteady blowing and suction at the wall. The controller is able to cancel the perturbations and is robust to both unmodelled disturbances and sensor inaccuracies. For single-frequency and multiple-frequency disturbances with low sensor noise a nearly full cancellation is achieved. For stochastic forced disturbances and high sensor noise an energy reduction in perturbation wall shear stress of 96 % is shown.

Recent experiments with droplets impacting macro-textured superhydrophobic surfaces revealed new regimes of bouncing with a remarkable reduction of the contact time. Here we present a comprehensive numerical study that reveals the physics behind these new bouncing regimes and quantifies the roles played by various external and internal forces. For the first time, accurate three-dimensional simulations involving realistic macro-textured surfaces are performed. After demonstrating that simulations reproduce experiments in a quantitative manner, the study is focused on analysing the flow situations beyond current experiments. We show that the experimentally observed reduction of contact time extends to higher Weber numbers, and analyse the role played by the texture density. Moreover, we report a nonlinear behaviour of the contact time with the increase of the Weber number for imperfectly coated textures, and study the impact on tilted surfaces in a wide range of Weber numbers. Finally, we present novel energy analysis techniques that elaborate and quantify the interplay between the kinetic and surface energy, and the role played by the dissipation for various Weber numbers.

The linear stability of two incompressible coaxial jets, separated by a thick duct wall, is investigated by means of both a modal and a non-modal approach within a global framework. The attention is focused on the range of unitary velocity ratios for which an alternate vortex shedding from the duct wall is known to dominate the flow. In spite of the inherent convective nature of jet flow instabilities, such behaviour is shown to originate from an unstable global mode of the dynamics linearised around the axisymmetric base flow. The corresponding wavemaker is located in the recirculating-flow region formed behind the duct wall. At the same time, the transient-growth analysis reveals that huge amplifications (up to $20$ orders of magnitude) of small flow perturbations at the nozzle exit can occur in the subcritical regime, especially for high ratios between the outer and the inner velocities.

When a rigid body collides with a liquid surface with sufficient velocity, it creates a splash curtain above the surface and entrains air behind the sphere, creating a cavity below the surface. While cavity dynamics has been studied for over a century, this work focuses on the water entry characteristics of deformable elastomeric spheres, which has not been studied. Upon free surface impact, an elastomeric sphere deforms significantly, giving rise to large-scale material oscillations within the sphere resulting in unique nested cavities. We study these phenomena experimentally with high-speed imaging and image processing techniques. The water entry behaviour of deformable spheres differs from rigid spheres because of the pronounced deformation caused at impact as well as the subsequent material vibration. Our results show that this deformation and vibration can be predicted from material properties and impact conditions. Additionally, by accounting for the sphere deformation in an effective diameter term, we recover previously reported characteristics for time to cavity pinch off and hydrodynamic force coefficients for rigid spheres. Our results also show that velocity change over the first oscillation period scales with the dimensionless ratio of material shear modulus to impact hydrodynamic pressure. Therefore, we are able to describe the water entry characteristics of deformable spheres in terms of material properties and impact conditions.

This work reports the formation and characterization of antipleptic and symplectic metachronal waves in 3D cilium arrays immersed in a two-fluid environment, with a viscosity ratio of 20. A coupled lattice Boltzmann–immersed-boundary solver is used. The periciliary layer is confined between the epithelial surface and the mucus. Its thickness is chosen such that the tips of the cilia can penetrate the mucus. A purely hydrodynamical feedback of the fluid is taken into account and a coupling parameter $\unicode[STIX]{x1D6FC}$ is introduced, which allows tuning of both the direction of the wave propagation and the strength of the fluid feedback. A comparative study of both antipleptic and symplectic waves, mapping a cilium interspacing ranging from 1.67 up to 5 cilium lengths, is performed by imposing metachrony. Antipleptic waves are found to systematically outperform symplectic waves. They are shown to be more efficient for transporting and mixing the fluids, while spending less energy than symplectic, random or synchronized motions.

We examine the onset of turbulence in Waleffe flow – the planar shear flow between stress-free boundaries driven by a sinusoidal body force. By truncating the wall-normal representation to four modes, we are able to simulate system sizes an order of magnitude larger than any previously simulated, and thereby to attack the question of universality for a planar shear flow. We demonstrate that the equilibrium turbulence fraction increases continuously from zero above a critical Reynolds number and that statistics of the turbulent structures exhibit the power-law scalings of the (2 + 1)-D directed-percolation universality class.

The cascading process of turbulent kinetic energy from large-scale fluid motions to small-scale and lesser-scale fluid motions in isotropic turbulence may be modelled as a hierarchical random multiplicative process according to the multifractal formalism. In this work, we show that the same formalism might also be used to model the cascading process of momentum in wall-bounded turbulent flows. However, instead of being a multiplicative process, the momentum cascade process is additive. The proposed multifractal model is used for describing the flow kinematics of the low-pass filtered streamwise wall-shear stress fluctuation $\unicode[STIX]{x1D70F}_{l}^{\prime }$, where $l$ is the filtering length scale. According to the multifractal formalism, $\langle {\unicode[STIX]{x1D70F}^{\prime }}^{2}\rangle \sim \log (Re_{\unicode[STIX]{x1D70F}})$ and $\langle \exp (p\unicode[STIX]{x1D70F}_{l}^{\prime })\rangle \sim (L/l)^{\unicode[STIX]{x1D701}_{p}}$ in the log-region, where $Re_{\unicode[STIX]{x1D70F}}$ is the friction Reynolds number, $p$ is a real number, $L$ is an outer length scale and $\unicode[STIX]{x1D701}_{p}$ is the anomalous exponent of the momentum cascade. These scalings are supported by the data from a direct numerical simulation of channel flow at $Re_{\unicode[STIX]{x1D70F}}=4200$.

In this paper, drop impact onto a sphere is numerically investigated at moderate Reynolds and Weber numbers. It is naturally expected that the aspect ratio of the sphere to the drop, $\unicode[STIX]{x1D706}_{r}$, would make a big difference to drop spreading and retraction on the sphere, compared with drop impact onto a flat substrate. To quantitatively assess the effect of $\unicode[STIX]{x1D706}_{r}$, a diffuse-interface immersed-boundary method is adopted after being validated against experiments. With the help of numerical simulations, we identify the key regimes in the spreading and retraction, analyse the results by scaling laws, and quantitatively evaluate the effect of $\unicode[STIX]{x1D706}_{r}$ on the impact dynamics. We find that the thickness of the liquid film spreading on the sphere can be well approximated by $h_{L,\infty }(1+3/4\unicode[STIX]{x1D706}_{r}^{-3/2})$, where $h_{L,\infty }$ represents the film thickness of drop impact on a flat substrate. At the early stage of spreading, the temporal variation of the wetted area is independent of $\unicode[STIX]{x1D706}_{r}$ when the time is rescaled by the thickness of the liquid film. Drops are observed to retract on the sphere at a roughly constant speed, and the predictions of theoretical analysis are in good agreement with numerical results.

Long-term marginal stability of a new family of isolated oceanic vortices is analysed. Sign reversal of the radial gradient of the potential vorticity anomaly, as implied by the isolation requirement, leads to vortex unsteadiness but does not break the coherence of the vortex, which remains marginally stable even for high absolute Rossby numbers $Ro\simeq 0.8$. The marginally stable vortices are characterized by a zero amount of potential vorticity anomaly on every isopycnal. The marginally stable final state is an unsteady vortex whose inner one-signed potential vorticity anomaly experiences revolution, rotation, precession and nutation.