Although active flow control based on deep reinforcement learning (DRL) has been demonstrated extensively in numerical environments, practical implementation of real-time DRL control in experiments remains challenging, largely because of the critical time requirement imposed on data acquisition and neural-network computation. In this study, a high-speed field-programmable gate array (FPGA) -based experimental DRL (FeDRL) control framework is developed, capable of achieving a control frequency of 1–10 kHz, two orders higher than that of the existing CPU-based framework (10 Hz). The feasibility of the FeDRL framework is tested in a rather challenging case of supersonic backward-facing step flow at Mach 2, with an array of plasma synthetic jets and a hot-wire acting as the actuator and sensor, respectively. The closed-loop control law is represented by a radial basis function network and optimised by a classical value-based algorithm (i.e. deep Q-network). Results show that, with only ten seconds of training, the agent is able to find a satisfying control law that increases the mixing in the shear layer by 21.2 %. Such a high training efficiency has never been reported in previous experiments (typical time cost: hours).

A millimetric droplet may bounce and self-propel across the surface of a vertically vibrating liquid bath, guided by the slope of its accompanying Faraday wave field. The ‘walker’, consisting of a droplet dressed in a quasi-monochromatic wave form, is a spatially extended object that exhibits many phenomena previously thought exclusive to the quantum realm. While the walker dynamics can be remarkably complex, steady and periodic states arise in which the energy added by the bath vibration necessarily balances that dissipated by viscous effects. The system energetics may then be characterised in terms of the exchange between the bouncing droplet and its guiding or ‘pilot’ wave. We here characterise this energy exchange by means of a theoretical investigation into the dynamics of the pilot-wave system when time-averaged over one bouncing period. Specifically, we derive simple formulae characterising the dependence of the droplet’s gravitational potential energy and wave energy on the droplet speed. Doing so makes clear the partitioning between the gravitational, wave and kinetic energies of walking droplets in a number of steady, periodic and statistically steady dynamical states. We demonstrate that this partitioning depends exclusively on the ratio of the droplet speed to its speed limit.

A single particle representation of a self-propelled microorganism in a viscous incompressible fluid is derived based on regularised Stokeslets in three dimensions. The formulation is developed from a limiting process in which two regularised Stokeslets of equal and opposite strength but with different size regularisation parameters approach each other. A parameter that captures the size difference in regularisation provides the asymmetry needed for propulsion. We show that the resulting limit is the superposition of a regularised stresslet and a potential dipole. The model framework is then explored relative to the model parameters to provide insight into their selection. The particular case of two identical particles swimming next to each other is presented and their stability is investigated. Additional flow characteristics are incorporated into the modelling framework with in the addition of a rotlet double to characterise rotational flows present during swimming. Lastly, we show the versatility of deriving the model in the method of regularised Stokeslets framework to model wall effects of an infinite plane wall using the method of images.

The hydrodynamic analysis of motion of small particles (e.g. proteins) within lipid bilayers appears to be naturally suitable for the framework of two-dimensional Stokes flow. Given the Stokes paradox, the problem in an unbounded domain is ill-posed. In his classical paper, Saffman (J. Fluid Mech., vol. 73, 1976, pp. 593–602) proposed several possible remedies, one of them based upon the finite extent of the membrane. Considering a circular boundary, that regularisation was briefly addressed by Saffman in the isotropic configuration, where the particle is concentrically positioned in the membrane. We investigate here the hydrodynamic problem in bounded membranes for the general case of eccentric particle position and a rectilinear motion in an arbitrary direction. Symmetry arguments provide a representation of the hydrodynamic drag in terms of ‘radial’ and ‘transverse’ coefficients, which depend upon two parameters: the ratio $\lambda$ of particle to membrane radii and the eccentricity $\beta$. Using matched asymptotic expansions we obtain closed-form approximations for these coefficients in the limit where $\lambda$ is small. In the isotropic case ($\beta = 0$) we find that the drag coefficient is $4\pi /(\ln ({1}/{\lambda })- {1})$, contradicting the value $4\pi /(\ln ({1}/{\lambda })- {1}/{2})$ obtained by Saffman. We explain the oversight in Saffman’s argument.

We conducted a series of pore-scale numerical simulations on convective flow in porous media, with a fixed Schmidt number of 400 and a wide range of Rayleigh numbers. The porous media are modeled using regularly arranged square obstacles in a Rayleigh–Bénard (RB) system. As the Rayleigh number increases, the flow transitions from a Darcy-type regime to an RB-type regime, with the corresponding $Sh$–$Ra_D$ relationship shifting from sublinear scaling to the classical 0.3 scaling of RB convection. Here, $Sh$ and $Ra_D$ represent the Sherwood number and the Rayleigh–Darcy number, respectively. For different porosities, the transition begins at approximately $Ra_D = 4000$, at which point the characteristic horizontal scale of the flow field is comparable to the size of a single obstacle unit. When the thickness of the concentration boundary layer is less than approximately one-sixth of the pore spacing, the flow finally enters the RB regime. In the Darcy regime, the scaling exponent of $Sh$ and $Ra_D$ decreases as porosity increases. Based on the Grossman–Lohse theory (J. Fluid Mech. vol. 407, 2000, pp. 27–56; Phys. Rev. Lett. vol. 86, 2001, p. 3316), we provide an explanation for the scaling laws in each regime and highlight the significant impact of mechanical dispersion effects during the development of the plumes. Our findings provide some new insights into the validity range of the Darcy model.

The evolution of a Lamb–Oseen vortex is studied in a stratified rotating fluid under the complete Coriolis force. In a companion paper, it was shown that the non-traditional Coriolis force generates a vertical velocity field and a vertical vorticity anomaly at a critical radius when the Froude number is larger than unity. Below a critical non-traditional Rossby number $\widetilde {Ro}$, a two-dimensional shear instability was next triggered by the vorticity anomaly. Here, we test the robustness of this two-dimensional evolution against small three-dimensional perturbations. Direct numerical simulations (DNS) show that the two-dimensional shear instability then develops only in an intermediate range of non-traditional Rossby numbers for a fixed Reynolds number $Re$. For lower $\widetilde {Ro}$, the instability is three-dimensional. Stability analyses of the flows in the DNS prior to the instability onset fully confirm the existence of these two competing instabilities. In addition, stability analyses of the local theoretical flows at leading order in the critical layer demonstrate that the three-dimensional instability is due to the shear of the vertical velocity. For a given Froude number, its growth rate scales as $Re^{2/3}/\widetilde {Ro}$, whereas the growth rate of the two-dimensional instability depends on $Re/\widetilde {Ro}^2$, provided that the critical layer is smoothed by viscous effects. However, the growth rate of the three-dimensional instability obtained from such local stability analyses agrees quantitatively with those of the DNS flows only if second-order effects due to the traditional Coriolis force and the buoyancy force are taken into account. These effects tend to damp the three-dimensional instability.

Analytical expressions are derived for the velocity field, and effective slip lengths, associated with pressure-driven longitudinal flow in a circular superhydrophobic pipe whose boundary is patterned with a general arrangement of longitudinal no-shear stripes not necessarily possessing any rotational symmetry. First, the flow in a superhydrophobic pipe with $M$ different no-shear stripes in general position is found for $M=1, 2, 3$. The method, which is based on use of so-called prime functions, is such that with these cases covered, generalisation to any $M \geqslant 1$ follows in a straightforward manner. It is shown how any one of these solutions can be generalised to solve for flow along superhydrophobic pipes where that pattern of $M$ menisci is repeated $N \geqslant 1$ times around the boundary in a rotational symmetric arrangement. The work provides an extension of the canonical pipe flow solution for an $N$-fold rotationally symmetric pattern of no-shear stripes due to Philip (Angew. Math. Phys., vol. 23, 1972, pp. 353–372). The novel solution method, and the solutions that it produces, have significance for a wide range of mixed boundary value problems involving Poisson’s equation arising in other applications.

We analyse the steady viscoelastic fluid flow in slowly varying contracting channels of arbitrary shape and present a theory based on the lubrication approximation for calculating the flow rate–pressure drop relation at low and high Deborah ($De$) numbers. Unlike most prior theoretical studies leveraging the Oldroyd-B model, we describe the fluid viscoelasticity using a FENE-CR model and examine how the polymer chains’ finite extensibility impacts the pressure drop. We employ the low-Deborah-number lubrication analysis to provide analytical expressions for the pressure drop up to $O(De^4)$. We further consider the ultra-dilute limit and exploit a one-way coupling between the parabolic velocity and elastic stresses to calculate the pressure drop of the FENE-CR fluid for arbitrary values of the Deborah number. Such an approach allows us to elucidate elastic stress contributions governing the pressure drop variations and the effect of finite extensibility for all $De$. We validate our theoretical predictions with two-dimensional numerical simulations and find excellent agreement. We show that, at low Deborah numbers, the pressure drop of the FENE-CR fluid monotonically decreases with $De$, similar to the previous results for the Oldroyd-B and FENE-P fluids. However, at high Deborah numbers, in contrast to a linear decrease for the Oldroyd-B fluid, the pressure drop of the FENE-CR fluid exhibits a non-monotonic variation due to finite extensibility, first decreasing and then increasing with $De$. Nevertheless, even at sufficiently high Deborah numbers, the pressure drop of the FENE-CR fluid in the ultra-dilute and lubrication limits is lower than the corresponding Newtonian pressure drop.

Randomness is one of the most important characteristics of turbulence, but its origin remains an open question. By means of a ‘thought experiment’ via several clean numerical experiments based on the Navier–Stokes equations for two-dimensional turbulent Kolmogorov flow, we reveal a new phenomenon, which we call the ‘noise-expansion cascade’ whereby all micro-level noises/disturbances at different orders of magnitudes in the initial condition of Navier–Stokes equations enlarge consistently, say, one by one like an inverse cascade, to macro level. More importantly, each noise/disturbance input may greatly change the macro-level characteristics and statistics of the resulting turbulence, clearly indicating that micro-level noise/disturbance might have great influence on macro-level characteristics and statistics of turbulence. In addition, the noise-expansion cascade closely connects randomness of micro-level noise/disturbance and macro-level disorder of turbulence, thus revealing an origin of randomness of turbulence. This also highly suggests that unavoidable thermal fluctuations must be considered when simulating turbulence, even if such fluctuations are several orders of magnitudes smaller than other external environmental disturbances. We hope that the ‘noise-expansion cascade’, as a fundamental property of the Navier–Stokes equations, could greatly deepen our understandings about turbulence, and also be helpful for attacking the fourth millennium problem posed by the Clay Mathematics Institute in 2000.