A new time-dependent analysis of the global and local fluctuating velocity signals in grid turbulence is conducted to assess the scaling laws for non-equilibrium turbulence. Experimental datasets of static- and active-grid turbulence with different Rossby numbers $R_o({=}U/\varOmega M$: $U$ is the mean velocity, $\varOmega$ is the mean rotation rate and $M$ is the grid mesh size) are considered. Although the global (long-time-averaged) non-dimensional dissipation rate $C_\varepsilon$ is independent of the Reynolds number $Re_\lambda$ based on the global Taylor microscale, the local (short-time-averaged) non-dimensional dissipation rate $\left \langle C_\varepsilon (t_i) \right \rangle$ ($t_i$ is the local time) both in the static- and active-grid turbulence clearly show the non-equilibrium scaling $\left \langle C_\varepsilon (t_i)\right \rangle / \sqrt {Re_0} \propto \left \langle Re_\lambda (t_i) \right \rangle ^{-1}$ ($\left \langle Re_\lambda (t_i) \right \rangle$ and $Re_0$ are the Reynolds numbers based on the local Taylor microscale $\lambda (t_i)$ and the global integral length scale, respectively), which has only been confirmed for global statistics in the near field of grid turbulence. The local value of $\left \langle L(t_i) / \lambda (t_i) \right \rangle$ ($L(t_i)$ is the local integral length scale) shifts from the equilibrium to non-equilibrium scaling as $\left \langle Re_\lambda (t_i) \right \rangle$ increases, further confirming that the non-equilibrium scalings are recovered for local statistics both in the static- and active-grid turbulence. The local values of $\left \langle C_\varepsilon (t_i) \right \rangle$ and $\left \langle L(t_i) / \lambda (t_i) \right \rangle$ follow the theoretical predictions for global statistics (Bos & Rubinstein, Phys. Rev. Fluids, vol. 2, 2017, 022601).

Despite significant advances in the field of man-made micro- and nanomotors, it remains a challenge to precisely control their motion in bounded environments. Here, we present a theoretical analysis of a thermally activated micromotor near a plane wall under the action of a background linear temperature field. The coupling between the autonomous and field-directed motions has been resolved using a combined analytical–numerical framework comprising general solutions in bispherical coordinates and the reciprocal theorem for creeping flows. Results reveal giant augmentation in swimming speeds, the controlling parameter zones for positive and negative thermotaxes and the flexibility of steering perpendicular to the field gradient for an isolated micromotor. Boundary-instigated thermo-fluidic modulations at different levels of confinements and preferential orientations cause directional switching of both the vertical translation and rotation parallel to the wall, thereby drastically altering the phase portraits of the swimmer dynamics. Contrasting trajectory characteristics, e.g. escape, attraction, are partitioned by unstable separatrices in the phase portraits, while competitive repulsion (attraction) after attraction (repulsion) characteristics emerge for different relative field strengths $\mathcal {S}$ and gradient orientations $\theta _T$. Below $\mathcal {S}=0.25$, highly counter-intuitive trajectories result when the micromotor is initially launched from an overlapping escape zone. Moreover, external-field-assisted microswimming can uniquely tune the directionality of wall-parallel translation, broadening the scope of dynamic regulation of self-propulsion. Thus, providing insights into a precisely controlled fuel-free actuation of micromotors near a physical obstacle, the present study stands as a step toward addressing the increasing demand for successful implementation of micromotors in futuristic clinical and environmental applications.

We experimentally and numerically study the dynamics of a liquid jet issued from a rotating orifice, whose breakup is regulated by a vibrating piezo element. The helical trajectory of the spiralling jet yields fictitious forces varying along the jet whose longitudinal projections stretch and thin the jet, affecting the growth of perturbations. We show that by quantifying these fictitious forces, one can estimate the jet intact length and size distribution of drops formed at jet breakup. The presence of the locally varying fictitious forces may render high-frequency perturbations, that would otherwise be stable in the abscence of stretching, unstable, as observed similarly in the case of straight jets stretching under gravity. The perturbation amplitude then dictates how strong the perturbation is coupled to the jet compared with random noise that is inherently present in any experimental set-up. In the present study we exploit the slenderness of the jet to separate the calculation of the base flow and the growth of perturbations. The fictitious forces calculated from the base flow trajectory are then used in a nonlinear slender-jet model, which treats the spiralling jet as a quasi-straight jet with locally varying body forces. We show both experimentally and numerically that jet breakup characteristics (e.g. intact length and drop size distribution) can be controlled by finite-amplitude perturbations created by mechanically induced pressure modulations. Finally, we revisit the integrated net gain approach developed for straight jets under gravity and we provide simple analogous relations for spiralling jets.

Well-resolved direct numerical simulations (DNS) have been performed of the flow in a smooth circular pipe of radius $R$ and axial length $10{\rm \pi} R$ at friction Reynolds numbers up to $Re_\tau =5200$ using the pseudo-spectral code OPENPIPEFLOW. Various turbulence statistics are documented and compared with other DNS and experimental data in pipes as well as channels. Small but distinct differences between various datasets are identified. The friction factor $\lambda$ overshoots by $2\,\%$ and undershoots by $0.6\,\%$ the Prandtl friction law at low and high $Re$ ranges, respectively. In addition, $\lambda$ in our results is slightly higher than in Pirozzoli et al. (J. Fluid Mech., vol. 926, 2021, A28), but matches well the experiments in Furuichi et al. (Phys. Fluids, vol. 27, issue 9, 2015, 095108). The log-law indicator function, which is nearly indistinguishable between pipe and channel up to $y^+=250$, has not yet developed a plateau farther away from the wall in the pipes even for the $Re_\tau =5200$ cases. The wall shear stress fluctuations and the inner peak of the axial turbulence intensity – which grow monotonically with $Re_\tau$ – are lower in the pipe than in the channel, but the difference decreases with increasing $Re_\tau$. While the wall value is slightly lower in the channel than in the pipe at the same $Re_\tau$, the inner peak of the pressure fluctuation shows negligible differences between them. The Reynolds number scaling of all these quantities agrees with both the logarithmic and defect-power laws if the coefficients are properly chosen. The one-dimensional spectrum of the axial velocity fluctuation exhibits a $k^{-1}$ dependence at an intermediate distance from the wall – also seen in the channel. In summary, these high-fidelity data enable us to provide better insights into the flow physics in the pipes as well as the similarity/difference among different types of wall turbulence.

We experimentally investigate the evaporation of very volatile liquid droplets (Novec 7000 Engineered Fluid, chemical name hydrofluoroethers HFE-7000) in a turbulent spray. Droplets with diameters of the order of a few micrometres are produced by a spray nozzle and then injected into a purpose-built enclosed dodecahedral chamber filled with air containing various amounts of water vapour. The ambient temperature and relative humidity in the chamber are carefully controlled. We observe water condensation on the rapidly evaporating droplet, both for the spray and for a single acoustically levitated millimetric Novec 7000 droplet. We further examine the effect of humidity, and reveal that a more humid environment leads to faster evaporation of the volatile liquid, as well as more water condensation. This is explained by the much larger latent heat of water as compared with that of Novec 7000. We extend an analytical model based on Fick's law to quantitatively account for the data.

The aeroacoustic characteristics of flying vehicles with pitch-fixed rotors differ from traditional helicopters with pitch-controlled rotor blades. Accurate predictions of rotor noise are still challenging because many uncertainty factors and unsteadinesses exist. This work investigates the aeroacoustic effects of rotational speed deviation, rotation speed fluctuation, blade vibration and blade geometric asymmetry. The analysis is based on the efficient computation of rotor noise under different working conditions. The mean aerodynamic variables are computed using the blade element moment theory, while small-amplitude fluctuations are introduced to account for the unsteadiness and uncertainty factors. It is shown that periodic rotation speed fluctuations and blade vibrations can produce significant extra tones. By contrast, if the fluctuations and vibrations are random, the noise level in a wide frequency range is increased. The intriguing result reminds us of the need to revisit the rotor broadband noise sources commonly attributed to turbulent flows. The influences are observer angle dependent, and the extra noise production is more significant in the upstream and downstream directions. The asymmetric blade geometry can cause extra tonal noise at the harmonics of the blade shaft frequency. The noise features of dual rotors are also investigated. Usually, the noise is sensitive to the initial phase difference and rotation directions due to the interference effect. However, the noise features are vastly altered if there are slight differences in the rotation speeds. Although the influences of some factors on rotor noise were already known, the present study provides a more comprehensive analysis of the problem. The results also highlight the need to consider these practical factors for accurate noise prediction of multi-rotor flying vehicles.

The dynamics of an evaporating droplet in an unsteady flow is of practical interest in many industrial applications and natural processes. To investigate the transport and evaporation dynamics of such droplets, we present a numerical study of an isolated droplet in an oscillating gas-phase flow. The study uses a one-way coupled two-phase flow model to assess the effect of the amplitude and the frequency of a sinusoidal external flow field on the lifetime of a multicomponent droplet containing a non-volatile solute dissolved in a volatile solvent. The results show that the evaporation process becomes faster with an increase in the amplitude or the frequency of the gas-phase oscillation. The liquid-phase transport inside the droplet also is influenced by the unsteadiness of the external gas-phase flow. A scaling analysis based on the response of the droplet under the oscillating drag force is subsequently carried out to unify the observed evaporation dynamics in the simulations under various conditions. The analysis quantifies the enhancement in the droplet velocity and Reynolds number as a function of the gas-phase oscillation parameters and predicts the effects on the evaporation rate.

Immiscible and incompressible liquid–liquid flows are considered in a Taylor–Couette geometry and analysed by direct numerical simulations coupled with the volume-of-fluid method and a continuum surface force model. The system Reynolds number $Re \equiv r_i \omega _i d / \nu$ is fixed to $960$, where the single-phase flow is in the steady Taylor vortex regime, whereas the secondary-phase volume fraction $\varphi$ and the system Weber number $We \equiv \rho r_i^2 \omega _i^2 d / \sigma$ are varied to study the interactions between the interface and the Taylor vortices. We show that different Weber numbers lead to two distinctive flow regimes, namely an advection-dominated regime and an interface-dominated regime. When $We$ is high, the interface is easily deformed because of its low surface tension. The flow patterns are then similar to the single-phase flow, and the system is dominated mainly by advection (advection-dominated regime). However, when $We$ is low, the surface tension is so large that stable interfacial structures with sizes comparable to the cylinder gap can exist. The background velocity field is modulated largely by these persistent structures, thus the overall flow dynamics is governed by the interface (interface-dominated regime). The effect of the interface on the global system response is assessed by evaluating the Nusselt number $Nu_{\omega }$ based on the non-dimensional angular velocity transport. It shows non-monotonic trends as functions of the volume fraction $\varphi$ for both low and high $We$. We explain how these dependencies are closely linked to the velocity and interfacial structures.

A mathematical model is developed to investigate seabed heat transfer processes under long-crested ocean waves. The unsteady convection–diffusion equation for water temperature includes terms depending on the velocity field in the laminar boundary layer, analogous to mass transfer near the seabed. Here we consider regular progressive waves and standing waves reflected from a vertical structure, which complicate the convective term in the governing equation. Rectangular and Gaussian distributions of seabed temperature and heat flux are considered. Approximate analytical solutions are derived for uniform and trapezoidal currents, and compared against predictions from a numerical solver of the full equations. The effects of heat source profile, location and strength on heat transfer dynamics in the thermal boundary layer are explained, providing insights into seabed temperature forced convection mechanisms enhanced by free-surface waves.

In two-dimensional decaying homogeneous isotropic turbulence, kinetic energy and enstrophy are respectively transferred to larger and smaller scales. In such spatiotemporally complex dynamics, it is challenging to identify the important flow structures that govern this behaviour. We propose and employ numerically two flow-modification strategies that leverage the inviscid global conservation of energy and enstrophy to design external forcing inputs that change these quantities selectively and simultaneously, and drive the system towards steady-state or other late-stage behaviour. One strategy employs only local flow field information, while the other is global. We observe various flow structures excited by these inputs and compare them with recent literature. Energy modification is characterized by the excitation of smaller wavenumber structures in the flow than enstrophy modification.

After decades of research efforts, wind–wave interaction mechanisms have been recognized as extremely elusive. The reason is the complex nature of the problem, which combines complex coupling mechanisms between turbulent wind and water waves with the presence of multiple governing parameters, such as the friction Reynolds number of the wind, the water depth and the wind fetch. As shown unequivocally here, the use of suitable flow settings allows us to reduce the complex problem of wind–wave interaction to its essential features, mainly as a function of the sole friction Reynolds number of the wind. The resulting numerical solution allows us to study the interactions between water and air layers with their own fluid properties, and to unveil very interesting features, such as an oblique wave pattern travelling upstream and a wave-induced Stokes sublayer. The latter is responsible for a drag reduction mechanism in the turbulent wind. Despite the simulated flow conditions being far from the intense events occurring at the ocean–atmosphere interface, the basic flow phenomena unveiled here may explain some experimental evidence in wind–wave problems. Among other things, the wave-induced Stokes sublayer may shed light on the large scatter of the drag coefficient data in field measurements where swell waves of arbitrary directions are often present. Hence the present results and the developed approach pave the way for the understanding and modelling of the surface fluxes at the ocean–atmosphere interface, which are of overwhelming importance for climate science.

The propagation of waves from a vertical uplift of a slender rectangular fault in a sea of constant depth is discussed, accounting for water compressibility, gravity and seabed elasticity. The compressed water column results in the generation of acoustic–gravity waves that travel at the speed of sound in water. Acoustic–gravity waves are found to terminate after a finite time, with the decay time most influenced by seabed rigidity, which is in contrast to the rigid stationary-phase model where signals persist indefinitely. At certain frequencies acoustic–gravity waves couple with the elastic seabed and travel at the shear velocity (speed of sound in an elastic solid). Improved estimates of the critical frequencies are derived. Moreover, besides the usual tsunami, a second – very small amplitude – surface wave mode travelling at the speed of sound arises under certain frequencies. We derive the cut-off frequency for this mode. The acoustic modes possess a frequency spectrum which depends on the time evolution and spatial properties of the rupture. We find that appropriate filtering of the acoustic–gravity wave signal can reveal characteristic peaks that encode information on the fault's geometry and dynamics.

This paper presents an analytic model for the analysis of co-planar turbine fences that partially span the width of a channel in which the flow is driven by a sinusoidally oscillating driving head. The thrust presented by the turbines reduces the flow rate through the channel leading to a solution for overall power that is dependent upon turbine resistance and flow blockage as well as on channel characteristics. We introduce a return parameter, in terms of power per turbine area, to assess optimum turbine fence deployment for a given channel. We find that the optimal deployment rests on a universal curve independent of the channel characteristics, and that these characteristics – namely the integrated channel bed friction and a modified channel Froude number – move the optimum along this curve. We find that blockage considerations play a large role in the performance of a tidal farm – its achievable power, optimal return, channel flow rate reduction and device thrust – and that the scales of blockage must be considered even when designing relatively unblocked farms. The impact of the channel characteristics on the optimal arrangement, alongside environmental constraints that may limit permissible flow blockage, are quantified and discussed.

‘Freeze-out’ of amplitude growth, i.e. the amplitude growth stagnation of a shocked helium–air interface, is realized through a reflected shock, which produces baroclinic vorticity of the opposite sign to that deposited by the first shock. Theoretically, a model is constructed to calculate the relations among the initial parameters for achieving freeze-out. In particular, if the amplitude growth is within the linear regime at the arrival of the reflected shock, the time interval between the impacts of two shock waves is linearly related to the initial perturbation wavelength, and is independent of the initial perturbation amplitude. Experimentally, an air–SF$_6$ (or air–argon) plane interface is adopted to produce a weak reflected shock. Seven experimental runs with specific initial conditions are examined. For all cases, freeze-out is achieved after the reflected shock impact under the designed conditions.

Density segregation of multi-component granular mixtures in a dense, gravity-driven flow over a rough and bumpy periodic chute surface is studied using theory and simulations. An existing theoretical model for predicting the steady-state concentration field of each species in a binary mixture using the forces acting on the particles is generalised for multi-component mixtures in this work. In addition, the rheological model for binary mixtures is also extended to multi-component mixtures. In contrast to the percolation velocity-based empirical segregation models that do not account for the rheology and need prior knowledge of the velocity field, the present approach accounts for the inter-coupling of rheology with segregation. The momentum balance equations are solved, along with the mixture rheological model as well as the multi-component density segregation model, to obtain concentration fields using an iterative numerical method. The theoretical predictions are compared with discrete element method (DEM) simulations for ternary, quaternary and quinary granular mixtures differing in density. The steady-state profiles for the concentration of different species as well as other flow properties predicted from the theory are in excellent agreement with the DEM simulation results for a variety of compositions over a range of inclination angles for different density ratios. Through this work, we take the first essential step towards proposing a generalised particle force-based segregation theory for multi-component mixtures differing in size and/or density.

Nonlinearity of density stratification modulates buoyancy effects. We report results from a body-inclusive large eddy simulation of a wake in nonlinear stratification, specifically for a circular disk at diameter-based Reynolds number (${\it Re} $) of $5000$. Five density profiles are considered; the benchmark has linear stratification and the other four have hyperbolic tangent profiles of the same thickness to model a pycnocline. The disk moves inside the central core of the pycnocline in two of those four cases and, in the other two cases with a shifted density profile, the disk moves partially/completely outside the pycnocline. The maximum buoyancy frequency ($N_{max}$) for all the profiles is the same. The first part of the study investigates the centred cases. Non-uniform stratification results in increasing wake turbulence relative to the benchmark owing to reduced suppression of turbulence production as well as wave trapping in the pycnocline. Steady lee waves are also quantified to understand the limitations of linear theory. The second part pays attention to the effect of a relative shift between the pycnocline and the disk. The wake defect velocity decays substantially faster in the cases with a shift and the wake has higher turbulence level. The effect of disk location on the Kelvin wake waves (a family of steady waves within the pycnocline) and its modal form is obtained and explained by solving the Taylor–Goldstein equation. The family of unsteady internal gravity waves that are generated by the wake is also studied and the effect of disk shift is quantified.

We extend the scaling relations of strongly (stably) stratified turbulence from the geophysical regime of unity Prandtl number to the astrophysical regime of extremely small Prandtl number applicable to stably stratified regions of stars and gas giants. A transition to a new turbulent regime is found to occur when the Prandtl number drops below the inverse of the buoyancy Reynolds number, i.e. $Pr\,Rb<1$, which signals a shift of the dominant balance in the buoyancy equation. Application of critical balance arguments then derives new predictions for the anisotropic energy spectrum and dominant balance of the Boussinesq equations in the $Pr\,Rb\ll 1$ regime. We find that all the standard scaling relations from the unity $Pr$ limit of strongly stratified turbulence simply carry over if the Froude number, $Fr$, is replaced by a modified Froude number, $Fr_M\equiv Fr/(Pr\,Rb)^{1/4}$. The geophysical and astrophysical regimes are thus smoothly connected across the $Pr\,Rb=1$ transition. Applications to vertical transport in stellar radiative zones and modification to the instability criterion for the small-scale dynamo are discussed.