Here we evaluate the skin coefficient of friction for steady turbulent radial wall jets across smooth and rough surfaces. Although the Colebrook equation has been used successfully for many decades to evaluate friction factors for flows through smooth and rough pipes, how roughness affects the skin friction coefficient for steady turbulent radial wall jets remains unclear. Here we explore a Colebrook-type equation for skin friction coefficients associated with single-phase turbulent radial wall jets arising from orthogonally impinging circular jets. The fully iterative solution, based on well-established concepts of turbulent wall-bounded flow, is presented along with a power-law approximation and a non-iterative approximation for the friction coefficient derived therefrom. We find the skin coefficient of friction defined on the peak radial velocity to be a function of position over rough but not smooth surfaces in contrast to pipe friction factors that remain independent of axial position. These results follow expected trends, explain prior heterogeneity in power-law expressions for the skin friction coefficient and have significant implications for the industrial use of jets in mixing vessels.

In Rayleigh–Bénard convection and Taylor–Couette flow cellular patterns emerge at the onset of instability and persist as large-scale coherent structures in the turbulent regime. Their long-term dynamics has been thoroughly characterised and modelled for the case of turbulent convection, whereas turbulent Taylor rolls have received much less attention. Here we present direct numerical simulations of axisymmetric Taylor–Couette flow in the corotating regime and reveal a transition to spatio–temporal chaos as the system size increases. Beyond this transition, Taylor rolls suddenly undergo erratic drifts evolving on a very slow time scale. We estimate an effective diffusion coefficient for the drift and compare the dynamics with analogous motions in Rayleigh–Bénard convection and Poiseuille flow, suggesting that this spontaneous diffusive displacement of large coherent structures is common among different types of wall-bounded turbulent flows.

Wind turbines operate in the atmospheric boundary layer (ABL), where Coriolis effects are present. As wind turbines with larger rotor diameters are deployed, the wake structures that they create in the ABL also increase in length. Contemporary utility-scale wind turbines operate at rotor diameter-based Rossby numbers, the non-dimensional ratio between inertial and Coriolis forces, of $\mathcal {O}(100)$ where Coriolis effects become increasingly relevant. Coriolis forces provide a direct forcing on the wake, but also affect the ABL base flow, which indirectly influences wake evolution. These effects may constructively or destructively interfere because both the magnitude and sign of the direct and indirect Coriolis effects depend on the Rossby number, turbulence and buoyancy effects in the ABL. Using large eddy simulations, we investigate wake evolution over a wide range of Rossby numbers relevant to offshore wind turbines. Through an analysis of the streamwise and lateral momentum budgets, we show that Coriolis effects have a small impact on the wake recovery rate, but Coriolis effects induce significant wake deflections which can be parsed into two regimes. For high Rossby numbers (weak Coriolis forcing), wakes deflect clockwise in the northern hemisphere. By contrast, for low Rossby numbers (strong Coriolis forcing), wakes deflect anti-clockwise. Decreasing the Rossby number results in increasingly anti-clockwise wake deflections. The transition point between clockwise and anti-clockwise deflection depends on the direct Coriolis forcing, pressure gradients and turbulent fluxes in the wake. At a Rossby number of 125, Coriolis deflections are comparable to wake deflections induced by ${\sim} 20^{\circ }$ of yaw misalignment.

Lift and drag forces on moving intruders in flowing granular materials are of fundamental interest but have not yet been fully characterized. Drag on an intruder in granular shear flow has been studied almost exclusively for the intruder moving across flow streamlines, and the few studies of the lift explore a relatively limited range of parameters. Here, we use discrete element method simulations to measure the lift force, $F_{{L}}$, and the drag force on a spherical intruder in a uniformly sheared bed of smaller spheres for a range of streamwise intruder slip velocities, $u_{{s}}$. The streamwise drag matches the previously characterized Stokes-like cross-flow drag. However, $F_{{L}}$ in granular shear flow acts in the opposite direction to the Saffman lift in a sheared fluid at low $u_{{s}}$, reaches a maximum value and then decreases with increasing $u_{{s}}$, eventually reversing direction. This non-monotonic response holds over a range of flow conditions, and the $F_{{L}}$ versus $u_{{s}}$ data collapse when both quantities are scaled using the particle size, shear rate and overburden pressure. Analogous fluid simulations demonstrate that the flow around the intruder particle is similar in the granular and fluid cases. However, the shear stress on the granular intruder is notably less than that in a fluid shear flow. This difference, combined with a void behind the intruder in granular flow in which the stresses are zero, significantly changes the lift-force-inducing stresses acting on the intruder between the granular and fluid cases.

We study the dynamics of a thin liquid sheet that flows upwards along the sides of a vertically aligned, impacting plate. Upon impact of the vertical solid plate onto a liquid pool, the liquid film is ejected and subsequently continues to flow over the solid surface while the plate enters the water. With increasing impact velocity, the liquid film is observed to rise up faster and higher. We focus on the time evolution of the liquid film height and the thickness of its upper rim and discuss their dynamics in detail. Similar to findings in previous literature on sheet fragmentation during drop impact, we find the rim thickness to be governed by the local instantaneous capillary number based on gravity and the deceleration of the liquid sheet, showing that the retraction of the rim is primarily due to capillarity. In contrast, for the liquid film height, we demonstrate that the viscous dissipation in the thin boundary layer is the dominant factor for the vertical deceleration of the liquid sheet, by modelling the time evolution of the film height and showing that the influences of capillarity, gravity and deceleration due to the air phase are all negligible compared with the viscous term. Finally, we introduce characteristic viscous time and length scales based on the initial rim thickness and show that the maximum height of the film and the corresponding time can be determined from these viscous scales.

The real-fluid effect induced by large density variation at supercritical pressure (SCP) modulates the turbulent dynamics and heat transfer, and poses challenges to existing turbulence models that are based on ideal-gas conditions. This study conducts direct numerical simulations of fully developed channel flows at SCP, with the upper and lower channel walls being isothermally heated and cooled, respectively. Emphasis is placed on examining the effects of various levels of density variations on near-wall turbulence as well as turbulent heat transfer by changing wall temperatures. The results show that the density fluctuation significantly impacts both first-order and second-order turbulence statistics near the heated wall owing to the close vicinity of pseudo-boiling point. Such real-fluid impact increases substantially with increasing density ratio, and tends to weaken the turbulent kinetic energy by damping turbulence production, while simultaneously inducing an additional turbulent mass flux that partially offsets this reduction. Detailed quadrant analysis reveals that the ‘ejection’ events dominate diverse effects of density fluctuation on Reynolds shear stresses, with density fluctuation contributing positively on the cooled wall side, and negatively on the heated wall side. Regarding the turbulent heat transfer, density fluctuation enhances the enthalpy–pressure–gradient correlation, tending to weaken the turbulent heat flux, which is slightly compensated by additional terms induced by density fluctuations. The overall negative contribution of density fluctuation to turbulent heat flux stems primarily from ‘hot ejection’ motions. Instantaneous flow characteristics provide additional support for these findings. Additionally, the mechanisms by which density fluctuations affect Reynolds shear stress and turbulent heat flux could also be extended to the skin friction coefficient and Nusselt number, respectively.

Active fluids encompass a wide range of non-equilibrium fluids, in which the self-propulsion or rotation of their units can give rise to large-scale spontaneous flows. Despite the diversity of active fluids, they are commonly viscoelastic. Therefore, we develop a hydrodynamic model of isotropic active liquids by accounting for their viscoelasticity. Specifically, we incorporate an active stress term into a general viscoelastic liquid model to study the spontaneous flow states and their transitions in two-dimensional channel, annulus and disk geometries. We have discovered rich spontaneous flow states in a channel as a function of activity and Weissenberg number, including unidirectional flow, travelling-wave and vortex-roll states. The Weissenberg number acts against activity by suppressing the spontaneous flow. In an annulus confinement, we find that a net flow can be generated only if the aspect ratio of the annulus is not too large nor too small, akin to some three-dimensional active-flow phenomena. In a disk geometry, we observe a periodic chirality switching of a single vortex flow, resembling the bacteria-based active fluid experiments. The two phenomena reproduced in our model differ in Weissenberg number and frictional coefficient. As such, our active viscoelastic model offers a unified framework to elucidate diverse active liquids, uncover their connections and highlight the universality of dynamic active-flow patterns.

Many particles, whether passive or active, possess elongated shapes. When these particles settle or swim in shear flows, they often form regions of accumulation and depletion. Additionally, the density contrast between the particles and the fluid can further alter the flow by increasing the local suspension density, resulting in a two-way buoyancy–flow coupling mechanism. This study investigates the buoyancy–flow coupled dispersion of active spheroids, examining the effects of elongation, orientation-dependent settling and gyrotaxis in a vertical pipe subjected to either downwards or upwards discharge. While the concentration and velocity profiles of passive settling spheroids and spherical gyrotactic swimmers can be analysed similarly to a recent study, notable differences in dispersion characteristics emerge due to different streamline-crossing mechanisms. For suspensions of elongated swimmers, the interplay between orientation-dependent settling, gyrotaxis-induced accumulation and shear-induced trapping results in distinct concentration and velocity distributions compared to those of neutrally buoyant particles and extremely dilute suspensions with negligible coupling effect. These differences further impact drift velocity, dispersivity, and the time elapsed to steady dispersion under varying flow rates. Interestingly, low-shear trapping of non-settling elongated swimmers around the centreline, commonly observed in planar Poiseuille flow, is absent in the vertical pipe due to the change of confinement from reflectional to rotational symmetry. However, elongated settling swimmers show a non-trivial concentration response to strong downwelling discharge. This phenomenon, linked to the centreline accumulation of passive settling spheroids, bears similarities to low-shear trapping observed in planar Poiseuille flow.

We study the response of a flexible prism with a square cross-section placed in cross-flow through a series of experiments conducted at increasing flow velocities. We show that as the reduced velocity (a dimensionless flow velocity that also depends on the natural frequency of the structure) is increased, the prism undergoes vortex-induced vibration (VIV) in its first mode, which then transitions to VIV in the second mode and then third mode. In these ranges, the shedding frequency is synchronised with the oscillation frequency, and the oscillations are mainly in the transverse (cross-flow – CF) direction. As we keep increasing the reduced velocity, we observe a linear increase in the amplitude of the torsional oscillations of the prism, resembling a torsional galloping. This increase in the torsional oscillations then causes an increase in the amplitudes of the CF and inline (IL) oscillations while the third structural mode is still excited in the CF direction. A transition to oscillations in the fourth structural mode is observed at higher reduced velocities, which reduces the CF and IL amplitudes, while the torsional oscillations reach a plateau. After this plateau is reached in the torsional oscillations, galloping is observed in the CF oscillations of the response, which results in large-amplitude oscillations in both the CF and IL directions. The CF galloping response at these higher reduced velocities is accompanied by a torsional VIV response and the shedding frequency is synchronised with the frequency of the torsional oscillations.

An adaptable estimation technique is presented to reconstruct time-evolving three dimensional (3-D) velocity fields from planar particle image velocimetry measurements. The methodology builds on the multi-time-delay estimation technique of Hosseini et al. (2015) by implementing the finite-impulse-response spectral proper orthogonal decomposition (FIR-SPOD) of Sieber et al. (2016). The candidate flow is the highly modulated turbulent near wake of a cantilevered square cylinder with a height-to-width ratio $h/d=4$, protruding a thin laminar boundary layer ($\delta /d=0.21$ with $\delta$ being the boundary layer thickness) at the Reynolds number $Re=10600$, based on d. The novelty of the estimation technique is in using the modal space obtained by FIR-SPOD to better isolate the spatio-temporal scales for correlating velocity and pressure modes. Using FIR-SPOD, irregular coherent contributions at frequencies centred at $f_{ac1}=(1\pm 0.05)f_s$ and $f_{ac2}=(1\pm 0.1)f_s$ (with $f_s$ the fundamental shedding frequency) could be separated, which was not possible using proper orthogonal decomposition. With the FIR-SPOD bases, the quality of the estimation improved significantly using only linear terms, and the correct phase relationships between pressure and velocity modes are retained, as is required for synchronizing coherent motions along the height of the obstacle. It is shown that a low-dimensional reconstruction of the flow field successfully captures the cycle-to-cycle variations of the dominant 3-D vortex shedding process, which give rise to vortex dislocation events. Thus, the present methodology shows promise in 3-D reconstruction of challenging turbulent flows, which exhibit non-periodic behaviour or contain multi-scale phenomena.

We investigate the sliding dynamics of a millimetre-sized particle trapped in a horizontal soap film. Once released, the particle moves toward the centre of the film in damped oscillations. We study experimentally and model the forces acting on the particle, and evidence the key role of the mass of the film on the shape of the film and particle dynamics. Not only is the gravitational distortion of the film measurable, it completely determines the force responsible for the motion of the particle – the catenoid-like deformation induced by the particle has negligible effect on the dynamics. Surprisingly, this is expected for all film sizes as long as the particle radius remains much smaller than the film width. We also measure the friction force, and show that ambient air and the film contribute almost equally to the friction. The theoretical model that we propose predicts exactly the friction coefficient as long as inertial effects can be neglected in air (for the smallest and slowest particles). The fit between theory and experiments sets an upper boundary $\eta _s \leqslant 10^{-8}$ Pa s m for the surface viscosity, in excellent agreement with recent interfacial microrheology measurements.

The demand for separating and analysing rare target cells is increasing dramatically for vital applications such as cancer treatment and cell-based therapies. However, there remains a grand challenge for high-throughput and label-free segregation of lesion cells with similar sizes. Cancer cells with different invasiveness usually manifest distinct deformability. In this work, we employ a hydrogel microparticle system with similar sizes but varied stiffness to mimic cancer cells and examine in situ their deformation and focusing under microfluidic flow. We first demonstrate the similar focusing behaviour of hydrogel microparticles and cancer cells in confined flow that is dominated by deformability-induced lateral migration. The deformation, orientation and focusing position of hydrogel microparticles in microfluidic flow under different Reynolds numbers are then systematically observed and measured using a high-speed camera. Linear correlations of the Taylor deformation and tilt angle of hydrogel microparticles with the capillary number are revealed, consistent with theoretical predictions. Detailed analysis of the dependence of particle focusing on the flow rate and particle stiffness enables us to identify a linear scaling between the equilibrium focusing position and the major axis of the deformed microparticles, which is uniquely determined by the capillary number. Our findings provide insights into the focusing and dynamics of soft beads, such as cells and hydrogel microparticles, under confined flow, and pave the way for applications including the separation and identification of circulating tumour cells, drug delivery and controlled drug release.

Low-inertia pulsatile flows in highly distensible viscoelastic vessels exist in many biological and engineering systems. However, many existing works focus on inertial pulsatile flows in vessels with small deformations. As such, here we study the dynamics of a viscoelastic tube at large deformation conveying low-Reynolds-number oscillatory flow using a fully coupled fluid–structure interaction computational model. We focus on a detailed study of the effect of wall (solid) viscosity and oscillation frequency on tube deformation, flow rate, phase shift and hysteresis, as well as the underlying flow physics. We find that the general behaviour is dominated by an elastic flow surge during inflation and a squeezing effect during deflation. When increasing the oscillation frequency, the maximum inlet flow rate increases and tube distention decreases, whereas increasing solid viscosity causes both to decrease. As the oscillation frequency approaches either $0$ (quasi-steady inflation cycle) or $\infty$ (steady flow), the behaviours of tubes with different solid viscosities converge. Our results suggest that deformation and flow rate are most affected in the intermediate range of solid viscosity and oscillation frequency. Phase shifts of deformation and flow rate with respect to the imposed pressure are analysed. We predict that the phase shifts vary throughout the oscillation; while the deformation always lags the imposed pressure, the flow rate may either lead or lag depending on the parameter values. As such, the flow rate shows hysteresis behaviour that traces either a clockwise or counterclockwise curve, or a mix of both, in the pressure–flow rate space. This directional change in hysteresis is fully characterised here in the appropriate parameter space. Furthermore, the hysteresis direction is shown to be predicted by the signs of the flow rate phase shifts at the crest and trough of the oscillation. A distinct change in the tube dynamics is also observed at high solid viscosity which leads to global or ‘whole-tube’ motion that is absent in purely elastic tubes.

We report pattern formation in an otherwise non-uniform and unsteady flow arising in high-speed liquid entrainment conditions on the outer wall of a wide rotating drum. We show that the coating flow in this rotary dragout undergoes axial modulations to form an array of roughly vertical thin liquid sheets which slowly drift from the middle of the drum towards its sidewalls. Thus, the number of sheets fluctuates in time such that the most probable rib spacing varies ever so slightly with the speed, and a little less weakly with the viscosity. We propose that these axial patterns are generated due to a primary instability driven by an adverse pressure gradient in the meniscus region of the rotary drag-out flow, similar to the directional Saffman–Taylor instability, as is wellknown for ribbing in film-splitting flows. Rib spacing based on this mechanistic model turns out to be proportional to the capillary length, wherein the scaling factor can be determined based on existing models for film entrainment at both low and large capillary numbers. In addition, we performed direct numerical simulations, which reproduce the experimental phenomenology and the associated wavelength. We further include two numerical cases wherein either the liquid density or the liquid surface tension is quadrupled while keeping all other parameters identical with experiments. The rib spacings of these cases are in agreement with the predictions of our model.