Particulate flows at moderate particle Reynolds numbers are important in critical engineering and geological applications. This experimental study explores neutrally buoyant suspensions in an outer-rotating coaxial rheometer for solid fractions, $\phi$, from 0.1 to 0.5, and particle Reynolds number, $Re$, from 0.5 to 800, covering laminar, transitional and turbulent regimes; $Re$ is defined in terms of the square of the particle diameter and the shear rate. For $0.1 < \phi < 0.4$ and $0.5 < Re <10$, the direct torque measurements normalised by the laminar flow torque, $M/M_{lam}$, are independent of $Re$, but depend on $\phi$. For the same range of $\phi$ and for $10< Re<100$, the normalised torques depend on both $\phi$ and $Re$, and show an increasing dependence on $Re$. As $Re$ increases, the flow transitions to turbulence. Small particles delay the turbulent transition for $\phi \leqslant 0.3$, while large particles augment the transition. A modified Reynolds number, $Re^\prime$, that depends linearly on the particle diameter and the maximum velocity, $U_{o}$, is introduced for both laminar and turbulent flows and shows a better correlation of the results as compared with $Re$. For $\phi = 50\,\%$, the normalised torque minus the torque at zero rotational speed is nearly independent of $Re^\prime$. Rheological models based on $Re^\prime$ and the Krieger–Dougherty relative viscosity are proposed in the laminar regime for $10< Re^\prime <500$; in the turbulent regime, a correlation is proposed in terms of $Re^\prime$ and $\phi$ for $1000< Re^\prime < 6000$.

We investigate by direct numerical simulations the fluid–solid interaction of non-dilute suspensions of spherical particles moving in triperiodic turbulence, at the relatively large Reynolds number of $Re_\lambda \approx 400$. The solid-to-fluid density ratio is varied between $1.3$ and $100$, the particle diameter $D$ is in the range $16 \le D/\eta \le 123$ ($\eta$ is the Kolmogorov scale) and the volume fraction of the suspension is $0.079$. Turbulence is sustained using the Arnold–Beltrami–Childress cellular-flow forcing. The influence of the solid phase on the largest and energetic scales of the flow changes with the size and density of the particles. Light and large particles modulate all scales in an isotropic way, while heavier and smaller particles modulate the largest scales of the flow towards an anisotropic state. Smaller scales are isotropic and homogeneous for all cases. The mechanism driving the energy transfer across scales changes with the size and the density of the particles. For large and light particles the energy transfer is only marginally influenced by the fluid–solid interaction. For small and heavy particles, instead, the classical energy cascade is subdominant at all scales, and the energy transfer is essentially driven by the fluid–solid coupling. The influence of the solid phase on the flow intermittency is also discussed. Besides, the collective motion of the particles and their preferential location in relation to properties of the carrier flow are analysed. The solid phase exhibits moderate clustering; for large particles the level of clustering decreases with their density, while for small particles it is maximum for intermediate values.

An ultrasonic phased array system is introduced to study the three-dimensional (3-D) movement of a single bubble in a GaInSn alloy under a transverse magnetic field (MF), which is verified by bubble experiments in water. The 3-D motion trajectories of individual bubbles in the GaInSn are obtained under a horizontal MF. As the MF becomes stronger, the bubble successively oscillates in random directions (R mode), a direction perpendicular to the MF (V mode), a direction parallel to the MF (P mode) and finally it rises straight (S mode). The significant anisotropy of the oscillation directions at a moderate MF intensity may be due to the anisotropy of the vortex structure around the bubble. Furthermore, the oscillation amplitude gradually declines with increasing MF intensity until the bubble trajectory finally becomes a straight line. Our measurements allow us to specify the characteristic regions for the observed bubble modes in the $N-Eo-Re$ parameter space (N is the magnetic interaction parameter, Eo is the Eötvös number and Re is the Reynolds number). In addition, more detailed characteristics of bubble terminal velocity are revealed, showing that the bubble velocities are closely related to the motion modes. The increase in bubble velocity at a moderate MF intensity is caused by the weakening oscillation. At a high strength, the MF monotonically suppresses the rise velocity of the bubble with a fixed scaling law.

We consider two-dimensional flows above topography, revisiting the selective decay (or minimum enstrophy) hypothesis of Bretherton and Haidvogel. We derive a ‘condensed branch’ of solutions to the variational problem where a domain-scale condensate coexists with a flow at the (smaller) scale of the topography. The condensate arises through a supercritical bifurcation as the conserved energy of the initial condition exceeds a threshold value, a prediction that we quantitatively validate using direct numerical simulations. We then consider the forced–dissipative case, showing how weak forcing and dissipation select a single dissipative state out of the continuum of solutions to the energy-conserving system predicted by selective decay. As the forcing strength increases, the condensate arises through a supercritical bifurcation for topographic-scale forcing and through a subcritical bifurcation for domain-scale forcing, both predictions being quantitatively validated by direct numerical simulations. This method provides a way of determining the equilibrated state of forced–dissipative flows based on variational approaches to the associated energy-conserving system, such as the statistical mechanics of two-dimensional flows or selective decay.

Although the roll-streak (R-S) is fundamentally involved in the dynamics of wall turbulence, the physical mechanism responsible for its formation and maintenance remains controversial. In this work we investigate the dynamics maintaining the R-S in turbulent Poiseuille flow at $R=1650$. Spanwise collocation is used to remove spanwise displacement of the streaks and associated flow components, which isolates the streamwise-mean flow R-S component and the second-order statistics of the streamwise-varying fluctuations that are collocated with the R-S. This partition of the dynamics into streamwise-mean and fluctuation components facilitates exploiting insights gained from the analytic characterization of turbulence in the second-order statistical state dynamics (SSD), referred to as S3T, and its closely associated restricted nonlinear dynamics (RNL) approximation. Symmetry of the statistics about the streak centreline permits separation of the fluctuations into sinuous and varicose components. The Reynolds stress forcing induced by the sinuous and varicose fluctuations acting on the R-S is shown to reinforce low- and high-speed streaks, respectively. This targeted reinforcement of streaks by the Reynolds stresses occurs continuously as the fluctuation field is strained by the streamwise-mean streak and not intermittently as would be associated with streak-breakdown events. The Reynolds stresses maintaining the streamwise-mean roll arise primarily from the dominant proper orthogonal decomposition (POD) modes of the fluctuations, which can be identified with the time average structure of optimal perturbations growing on the streak. These results are consistent with a universal process of R-S growth and maintenance in turbulent shear flow arising from roll forcing generated by straining turbulent fluctuations, which was identified using the S3T SSD.

Pressure-driven flows of viscoelastic fluids in narrow non-uniform geometries are common in physiological flows and various industrial applications. For such flows, one of the main interests is understanding the relationship between the flow rate $q$ and the pressure drop $\Delta p$, which, to date, is studied primarily using numerical simulations. We analyse the flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, contracting channels and present a theoretical framework for calculating the $q-\Delta p$ relation. We apply lubrication theory and consider the ultra-dilute limit, in which the velocity profile remains parabolic and Newtonian, resulting in a one-way coupling between the velocity and polymer conformation tensor. This one-way coupling enables us to derive closed-form expressions for the conformation tensor and the flow rate–pressure drop relation for arbitrary values of the Deborah number ($De$). Furthermore, we provide analytical expressions for the conformation tensor and the $q-\Delta p$ relation in the high-Deborah-number limit, complementing our previous low-Deborah-number lubrication analysis. We reveal that the pressure drop in the contraction monotonically decreases with $De$, having linear scaling at high Deborah numbers, and identify the physical mechanisms governing the pressure drop reduction. We further elucidate the spatial relaxation of elastic stresses and pressure gradient in the exit channel following the contraction and show that the downstream distance required for such relaxation scales linearly with $De$.

Analysing the dynamics of velocity gradients is useful for understanding various nonlinear turbulence processes. This work focuses on how the vibrational non-equilibrium of the constituent molecules in a gaseous medium affects the dynamics of the velocity gradient and the pressure Hessian tensors. We first derive the exact evolution equation of the pressure-Hessian tensor in the presence of the vibrational non-equilibrium process. Subsequently, we perform several direct numerical simulations of compressible isotropic turbulence, including the vibrational relaxation process therein. Using flow fields extracted from these simulations, we conduct several parametric studies over a range of the Damköhler number (ratio of the relevant fluid time scale to that of the mean vibrational relaxation process) and the initial ratio of the vibrational temperature to the mean local temperature. We find that a variation in the initial Damköhler number does influence the evolution of the pressure-Hessian and the velocity gradient tensors. As the vibrational relaxation process becomes more rapid (an increase in the value of the initial Damköhler number), it causes a decrease in the strength of the pressure-Hessian tensor and simultaneous suppression of dilatational fluctuations in the flow field. On the other hand, a variation in the initial value of the ratio of the vibrational temperature to the local temperature does not seem to affect the pressure-Hessian or the velocity gradient tensor. These findings are expected to aid in the development of closure models for the pressure-Hessian tensor in compressible flows under vibrational non-equilibrium conditions.

The Holton–Lindzen–Plumb (HLP) model of the quasi-biennial oscillation (QBO) is investigated in order to assess the impact of introducing intermittency in the wave forcing. Intermittency is introduced to HLP by allowing the amplitude of the waves which force the QBO to evolve according to a stationary random process, driven by a stochastic differential equation (SDE) with an associated time scale $\tau$. Provided that $\tau$ is much shorter than the QBO period, it is shown that the impact on the QBO of the intermittent forcing is captured by a single intermittency parameter $\lambda$, and the value of $\lambda$ is proportional to $\tau$ and otherwise depends upon the details of the SDE. Numerical simulations, using a family of mean-reverting Ornstein–Uhlenbeck processes as the choice of SDE, show that the effect of increasing the intermittency parameter is invariably to decrease the QBO amplitude and increase its period. Changes to the QBO amplitude and period are indeed found to collapse onto a single curve controlled by $\lambda$, as predicted by the theory, provided that $\tau$ is small enough for the approximations used to be valid. The extension to broadband forcing is discussed in the context of stochastic gravity wave parameterisation, with the eventual goal of developing a representation of source intermittency in the most general situation with close fidelity to the physics.

We investigate the effects of bacterial activity on the mixing and transport properties of a passive scalar in time-periodic flows in experiments and in a simple model. We focus on the interactions between swimming Escherichia coli and the Lagrangian coherent structures (LCSs) of the flow, which are computed from experimentally measured velocity fields. Experiments show that such interactions are non-trivial and can lead to transport barriers through which the scalar flux is significantly reduced. Using the Poincaré map, we show that these transport barriers coincide with the outermost members of elliptic LCSs known as Lagrangian vortex boundaries. Numerical simulations further show that elliptic LCSs can repel elongated swimmers and lead to swimmer depletion within Lagrangian coherent vortices. A simple mechanism shows that such depletion is due to the preferential alignment of elongated swimmers with the tangents of elliptic LCSs. Our results provide insights into understanding the transport of micro-organisms in complex flows with dynamical topological features from a Lagrangian viewpoint.

Droplet impact on oscillating substrates is important for both natural and industrial processes. Recognizing the importance of the dynamics that arises from the interplay between droplet transport and substrate motion, in this work, we present an experimental investigation of the spreading of a droplet impacting a sinusoidally oscillating hydrophobic substrate. We focus particularly on the maximum spread of droplets as a function of various parameters of substrate oscillation. We first quantify the maximum spreading diameter attained by the droplets as a function of frequency, amplitude of vibration, and phase at the impact for various impact velocities. We highlight that there can be two stages of spreading. Stage I, which is observed at all impact conditions, is controlled by the droplet inertia and affected by the substrate oscillation. For certain conditions, a Stage II spreading is also observed, which occurs during the retraction process of Stage I due to additional energies imparted by the substrate oscillation. Subsequently, we derive scaling analyses to predict the maximum spreading diameters and the time for this maximum spread for both Stage I and Stage II. Furthermore, we identify the necessary condition for Stage II spreading to be greater than Stage I spreading. The results will enable optimization of the parameters in applications where substrate oscillation is used to control the droplet spread, and thus heat and mass transfer between the droplet and the substrate.

In this work, formal asymptotic solutions of a problem for linear water waves in a bounded basin are constructed. The solutions have the form of asymptotic quasimodes and are used for the description of standing water waves localised near the shoreline. Such short-wavelength quasimodes exist only for a discrete set of frequencies, which are determined by means of a quantisation-type condition. Some numerical results are also addressed.