Oscillatory flows have become an indispensable tool in microfluidics, inducing inertial effects for displacing and manipulating fluid-borne objects in a reliable, controllable and label-free fashion. However, the quantitative description of such effects has been confined to limit cases and specialized scenarios. Here we develop an analytical formalism yielding the equation of motion of density-mismatched spherical particles in oscillatory background flows, generalizing previous work. Inertial force terms are systematically derived from the geometry of the flow field together with analytically known Stokes number dependences. Supported by independent, first-principles direct numerical simulations, we find that these forces are important even for nearly density-matched objects such as cells or bacteria, enabling their fast displacement and separation. Our formalism thus consistently incorporates particle inertia into the Maxey–Riley equation, and in doing so provides a generalization of Auton's modification to added mass, as well as recovering the description of acoustic radiation forces on particles as a limiting case.

We extend the resolvent framework to two-phase flows with low-inertia particles. The particle velocities are modelled using the equilibrium Eulerian model. We analyse the turbulent flow in a vertical pipe with Reynolds number of $5300$ (based on diameter and bulk velocity), for Stokes numbers $St^+=0-1$, Froude numbers $Fr_z=-4,-0.4,0.4,4$ and $1/Fr_z = 0$ (gravity omitted). The governing equations are written in input–output form and a singular value decomposition is performed on the resolvent operator. As for single-phase flows, the operator is low rank around the critical layer, and the true response can be approximated using one singular vector. Even with a crude forcing model, the formulation can predict physical phenomena observed in Lagrangian simulations, such as particle clustering and gravitational effects. Increasing the Stokes number shifts the predicted concentration spectra to lower wavelengths; this shift also appears in the direct numerical simulation spectra and is due to particle clustering. When gravity is present, there are two critical layers, one for the concentration field, and one for the velocity field. For upward flow, the peak of concentration fluctuations shifts closer to the wall, in agreement with the literature. We explain this with the aid of the different locations of the two critical layers. Finally, the model correctly predicts the interaction of near-wall vortices with particle clusters. Overall, the resolvent operator provides a useful framework to explain and interpret many features observed in Lagrangian simulations. The application of the resolvent framework to higher $St^+$ flows in combination with Lagrangian simulations is also discussed.

This paper extends the work in Tee et al. (Intl J. Multiphase Flow, vol. 133, 2020, 103462) to investigate the effect of turbulent fluid motions on the translation and rotation of lifting and wall-interacting spheres in boundary layers. Each sphere was released from rest in smooth-wall boundary layers with $Re_\tau =670$ and 1300 ($d^+=56$ and 116, respectively) and allowed to propagate with the incoming fluid. Sphere and surrounding fluid motions were tracked simultaneously via three-dimensional particle tracking velocimetry and stereoscopic particle image velocimetry in streamwise–spanwise planes. Two-point correlations of sphere and fluid streamwise velocities yielded long positive regions associated with long fast- and slow-moving zones that approach and move over the spheres. The related spanwise correlations were shorter due to the shorter coherence length of spanwise fluid structures. In general, spheres lag the surrounding fluid. The less-dense lifting sphere had smaller particle Reynolds numbers varying from near zero up to 300. Its lift-offs coincided with oncoming fast-moving zones and fluid upwash. Wall friction initially retarded the acceleration of the denser sphere. Later, fluid torque associated with approaching high-velocity regions initiated forward rotation. The rotation, which was long-lived, induced sufficient Magnus lift to initiate repeated small lift-offs, reduce wall friction, and accelerate the sphere to higher sustained velocity. Particle Reynolds numbers remained above 200, and vortex shedding was omnipresent such that the spheres clearly altered the fluid motion. Spanwise fluid shear occasionally initiated wall-normal sphere rotation and relatively long-lasting Magnus side lift. Hence the finite sphere size contributed to multiple dynamical effects not present in point-particle models.

We study the motion and deformation of a single bubble rising inside a cylindrical container filled with a viscoelastic material. We solve numerically the mass and momentum balances along with the constitutive equation for the viscoelastic stresses, without assuming axial symmetry to allow the growth of three-dimensional disturbances. Hence, we may predict the emergence of the notorious knife-edge shape of the bubble, which is a result of a purely elastic instability triggered in the locality of the trailing edge. Our results compare well with existing experiments. We visualize, for the first time to the best of our knowledge, the flow kinematics and dynamics that arise downstream of the bubble. We propose two quantities, one kinematical and one geometrical, for the determination of the onset of the instability. We demonstrate that extension-rate thinning in the constitutive law is necessary for the emergence of the instability. Moreover, our results indicate that increasing (a) the deformability of the bubble and (b) the initial extension rate hardening of the viscoelastic material, prior to thinning, triggers the instability earlier. These novel findings help us formulate and propose a mechanism that controls the onset of the instability and explain why the knife-edge shape is not encountered as frequently.

Through a combination of laboratory experiments and theoretical models, we investigate the interaction of a mean upwelling through a closed basin with a vertical buoyancy flux. The fluid is mixed by a horizontally oscillating rake, which either traverses the whole basin or which oscillates just near one vertical boundary. We first review the steady state and demonstrate that, in both mixing regimes, the vertical density profile across the basin is controlled by the steady-state balance between the upward advective and diffusive fluxes of salinity as described by the classical model introduced by Munk (Deep-Sea Res., vol. 13, issue 4, 1966, pp. 707–730). However, with boundary mixing, we show that both the upwelling and the buoyancy transport are localised to the mixing zone near the boundary, and the interior fluid is stagnant. We then develop a model to describe the transient evolution of the system if there is either a discrete increase or gradual decrease to the buoyancy flux. In the boundary mixing case, the change in the buoyancy flux at the lower boundary leads to a change in the buoyancy of the fluid in the boundary mixing region, and this induces a transient, buoyancy-driven flow in the boundary region in addition to the steady upwelling. In turn, an equal and opposite vertical flow develops in the interior, and this leads to a change in the density stratification of the interior fluid as the system adjusts to a new equilibrium. However, in our experiments, there is no vertical mixing in the interior and interior fluid may upwell or downwell dependent on the change to the buoyancy forcing. We discuss the implications of our results for the transport and mixing in the deep ocean, and the associated interpretation of field experiments.

The influence of the boundary layer (BL) thickness on the tonal noise generated by subsonic impinging jets is investigated. For that, initially laminar jets at Mach numbers $0.6$ and $0.9$ with BL thicknesses $0.05r_0$, $0.1r_0$ and $0.2r_0$, where $r_0$ is the pipe-nozzle radius, impinging on a plate at $6r_0$ from the nozzle, are simulated. For Mach number $0.9$, acoustic tones due to feedback loops of axisymmetric nature between the nozzle and the plate are generated at frequencies that do not vary with the BL thickness. The two dominant tones are, however, $17$ and $26\ \mathrm {dB}$ stronger for the thickest BL compared with the thinnest one. For Mach number $0.6$, for the thinnest BL no acoustic peaks appear, as observed in the experiments of the literature, but narrow peaks resulting from axisymmetric feedback loops emerge for thicker BLs. Therefore, low subsonic impinging jets can be resonant for specific nozzle-exit conditions. The increase in tone amplitude for Mach number $0.9$, and the establishment of feedback loops for Mach number $0.6$ with increasing BL thickness, are found to result from two changes in the jet flow. The first change is that the shear-layer laminar–turbulent transition occurs farther downstream for a thicker BL, leading to a greater predominance of the axisymmetric aerodynamic fluctuations near the plate. The second change is that the amplification of the flow fluctuations between the nozzle and the plate at the tone frequencies is stronger for thicker BLs.

The coherent dynamics of bubble clusters are of fundamental and industrial importance, and are elusive due to the complex interactions of disordered bubble oscillations. Here we introduce and demonstrate a method for decomposition of the Lagrangian time series of bubble dynamics data by combining theory and principal component analysis. The decomposition extracts coherent features of bubble oscillations based on their energy, in a way similar to proper orthogonal decomposition of Eulerian flow field data. This method is applied to a dataset of spherical clusters under harmonic excitation at different amplitudes, with various nuclei density and polydispersity parameters. Results indicate that the underlying correlated mode of oscillations is isolated in a single dominant feature in cavitating regimes, independent of the nuclei's parameters. A systematic data analysis procedure further suggests that this feature is globally controlled by the dynamic cloud interaction parameter of Maeda and Colonius (J. Fluid Mech., vol. 862, 2019, pp. 1105–1134) that quantifies the mean-field interactions, regardless of initial polydispersity or nonlinearity. The method provides a simplified and comprehensive representation of complex bubble dynamics as well as a new path to reduced-order modelling of cavitation and nucleation.

Particle trapping and manipulation have a wide range of applications in biotechnology and engineering. Recently, a flow-based, particle-trapping device called the Stokes trap was developed for trapping and control of small particles in the intersection of multiple branches in a microfluidic channel. This device can also be used to perform rheological experiments to determine the viscoelastic response of an emulsion or suspension. We show that besides these applications, the various flow modes produced by the Stokes trap are able to manipulate drop shapes and induce active mixing inside droplets. To this end, we analyse the dynamics of a droplet in a Stokes trap through boundary-integral simulations. We also explore the dynamic response of drop shape with respect to distinct external flow modes, which allows us to perform numerical experiments such as step strain and oscillatory extension. A linear controller is used to manipulate drop position, and the drop deformation is characterized by a spherical-harmonic decomposition. For small drop deformations, we observe a linear superposition of harmonics, which, surprisingly, seems to hold even for moderate deformations. This result indicates that such a device can be used for shape control of droplets. We also investigate how the different flow modes may be combined to induce mixing inside the droplets. The transient combination of modes produces an effective chaotic mixing, which is characterized by a mixing number. The mixing inside the droplet can be further enhanced for lower viscosity ratios and low, but non-zero capillary number and flow frequencies.

This experimental work explores the flow field around a three-dimensional expansion–compression geometry on a slender cone at Mach 8 using high-frequency pressure sensors, high-frame-rate schlieren, temperature-sensitive paint, shear-stress measurements and oil-flow visualizations. The $7^\circ$ cone geometry has a hyperbolic slice which acts as an expansion corner and suppresses the disturbances present in the upstream boundary layer. Downstream of the cone-slice corner, high-frequency boundary-layer disturbances attenuate in all cases. Under laminar conditions, second-mode instabilities from the cone diminish and lower-frequency second-mode waves develop on the slice at a frequency commensurate with the increased boundary layer thickness. For fully turbulent cases, the boundary layer over the slice shows evidence of a two-layered nature with a turbulent outer region and a near-wall region with strong attenuation of high-frequency disturbances and reappearance of lower-frequency instability waves. When a downstream compression ramp is added to the slice, the expanded boundary layer shows enhanced susceptibility to separation such that separation is observed at a $10^{\circ }$ deflection, which is smaller than expected for turbulent conditions. For a $30^{\circ }$ ramp, boundary-layer separation occurs further upstream where the heat flux contours show a decrease in heating that is characteristic of a transitional separation. These results demonstrate the effect of relaminarization caused by an upstream expansion on a subsequent shock-wave/boundary-layer interaction.

This study investigated the turbulent wake flow behind a flat-back Ahmed body using a combination of time-resolved tomographic particle image velocimetry measurements and spectral proper orthogonal decomposition (SPOD). The experiments were conducted at a Reynolds number of ReH = 10 000, which is defined as U∞H/ν, where U∞ represents the free-stream velocity, H is the height of the body and ν represents the viscosity. The SPOD analysis revealed four distinct flow motions present in the wake of the Ahmed body, each occupying a specific range of Strouhal number, StH. Here, StH is defined as f × H/U∞, where f is the frequency of the motion. At the lowest resolved StH of 0.007, the system exhibited a bi-stability mode, in which the wake switched between asymmetric states consisting of a tilted toroidal vortex and a streamwise vortex. In the next StH range of 0.014 to 0.123, the flow demonstrated swinging/flapping motions, characterized by small spanwise and vertical movements of the wake barycentre. These movements were attributed to the tilting of the toroidal vortex. The third category included the vortex-shedding motions and consisted of quasi-streamwise vortices. These vortices advected in the downstream direction, causing oblique displacements of the barycentre in the streamwise-vertical plane. The peak energy of the vortex shedding was observed at StH = 0.164. Finally, shear layer instabilities induced small vertical and spanwise velocity fluctuations along the shear layers at a high StH of 1.147.

We conducted experiments in a laboratory to study turbulent flow over wind generated water waves. The experiments were performed in a wind-wave-current flume with three free stream wind speeds of Uref = 6.0, 8.0 and 10.0 m s−1, corresponding to 10 m equivalent wind speed of U10 = 10.2, 12.2 and 14.1 m s−1 and the root-mean-square wave height of 0.7, 1.1 and 1.7 cm, respectively, at a fetch of 6.2 m. The instantaneous velocity fields above the waves were obtained by using a particle image velocimetry (PIV) technique. The velocity fields were decomposed into the mean, wave-induced and turbulent velocity components. The tested wind waves were primarily dissipated by capillaries and microscale breaking waves. The Bond number and the shear velocity-fetch based Reynolds number were found to correlate with the wind wave regimes well. The turbulent dissipation rates above the water surface were determined based on resolved spatial gradient of instantaneous velocities, where the time-averaged dissipation rate values were calibrated using those estimated from the one-dimensional velocity spectrum in the temporal space. Subsequently, the turbulent kinetic energy (TKE) budget including its production, dissipation, advection and turbulent transport was presented. In addition, conditional averaging analysis of the TKE budgets over leeward, windward sides and all phases was performed. The results showed a strong dependency with the wave phase in the TKE budget terms except for the dissipation. The production-dissipation ratio increased significantly as the wind speed increased, likely attributed to the increased roughness over the substantial coverage of micro-breaking waves.

This paper investigates the transport of drugs delivered by direct injection into the cerebrospinal fluid (CSF) that fills the intrathecal space surrounding the spinal cord. Because of the small drug diffusivity, the dispersion of neutrally buoyant drugs has been shown in previous work to rely mainly on the mean Lagrangian flow associated with the CSF oscillatory motion. Attention is given here to effects of buoyancy, arising when the drug density differs from the CSF density. For the typical density differences found in applications, the associated Richardson number is shown to be of order unity, so that the Lagrangian drift includes a buoyancy-induced component that depends on the spatial distribution of the drug, resulting in a slowly evolving cycle-averaged flow problem that can be analysed with two-time scale methods. The asymptotic analysis leads to a nonlinear integro-differential equation for the spatiotemporal solute evolution that describes accurately drug dispersion at a fraction of the cost involved in direct numerical simulations of the oscillatory flow. The model equation is used to predict drug dispersion of positively and negatively buoyant drugs in an anatomically correct spinal canal, with separate attention given to drug delivery via bolus injection and constant infusion.

Equations relating the pressure at a horizontal seabed, the free-surface profile and the surface pressure are derived for two-dimensional irrotational steady water waves with arbitrary pressure at the free surface. Special cases include gravity, capillary, flexural and wind waves. Without approximations, we show that the free-surface recovery from the bottom pressure requires the resolution of only one first-order ordinary differential equation independent of the surface pressure, thus providing a new general recovery method valid for a broad class of water waves. Another equation provides an explicit expression for the surface pressure as a function of the bottom pressure and of the free surface. Thus, if unknown, the surface pressure can also be recovered if one extra measurement is available. This new recovery procedure is illustrated analytically for the linear approximation of a flexural–capillary–gravity wave, and numerically for fully nonlinear capillary–gravity waves.