Bluff-body wakes generally become three-dimensional (3-D) and then turbulent when the Reynolds number exceeds a few hundred. Other than an alternate shedding of the spanwise vortices behind the body and a gradual decay and annihilation of the vortices with distance downstream, whether a secondary vortex street would develop in the relatively far wake has been a long-standing argument in the literature. This argument is addressed in the present study. Specifically, direct numerical simulations and transient growth analysis are performed to examine the two-dimensional and 3-D wakes of different bluff bodies, including circular cylinder, square cylinder, diamond cylinder and rectangular cylinders with different cross-sectional aspect ratios. We found that a secondary vortex street is absent for most 3-D and turbulent wakes. The root cause is the weakening of spanwise vortices by 3-D wake instability modes and streamwise circulation/vorticity. The weakened spanwise vortices induce reduced mean shear in the intermediate wake, which then induces much smaller perturbation energy growth that is below the threshold for the emergence of a secondary vortex street. This finding suggests that the 3-D and turbulence characteristics, and the momentum, mass and heat transport in the relatively far wake of bluff bodies, would not be influenced by extra anisotropy or inhomogeneity caused by a secondary vortex street.

We present a systematic study on the effects of small aspect ratios $\varGamma$ on heat transport in liquid metal convection with a Prandtl number of $Pr=0.029$. The study covers $1/20\le \varGamma \le 1$ experimentally and $1/50\le \varGamma \le 1$ numerically, and a Rayleigh number $Ra$ range of $4\times 10^3 \le Ra \le 7\times 10^{9}$. It is found experimentally that the local effective heat transport scaling exponent $\gamma$ changes with both $Ra$ and $\varGamma$, attaining a $\varGamma$-dependent maximum value before transition-to-turbulence and approaches $\gamma =0.25$ in the turbulence state as $Ra$ increases. Just above the onset of convection, Shishkina (Phys. Rev. Fluids, vol 6, 2021, 090502) derived a length scale $\ell =H/(1+1.49\varGamma ^{-2})^{1/3}$. Our numerical study shows $Ra_{\ell }$, i.e. $Ra$ based on $\ell$, serves as a proper control parameter for heat transport above the onset with $Nu-1=0.018(1+0.34/\varGamma ^2)(Ra/Ra_{c,\varGamma }-1)$. Here $Ra_{c,\varGamma }$ represents the $\varGamma$-dependent critical $Ra$ for the onset of convection and $Nu$ is the Nusselt number. In the turbulent state, for a general scaling law of $Nu-1\sim Ra^\alpha$, we propose a length scale $\ell = H/(1+1.49\varGamma ^{-2})^{1/[3(1-\alpha )]}$. In the case of turbulent liquid metal convection with $\alpha =1/4$, our measurement shows that the heat transport will become weakly dependent on $\varGamma$ with $Ra_{\ell }\equiv Ra/(1+1.49\varGamma ^{-2})^{4/3} \ge 7\times 10^5$. Finally, once the flow becomes time-dependent, the growth rate of $Nu$ with $Ra$ declines compared with the linear growth rate in the convection state. A hysteresis is observed in a $\varGamma =1/3$ cell when the flow becomes time-dependent. Measurements of the large-scale circulation suggest the hysteresis is caused by the system switching from a single-roll-mode to a double-roll-mode in an oscillation state.

The practical implementation of machine learning in flow control is limited due to its significant training expenses. In the present study the convolutional neural network (CNN) trained with the data of the restricted nonlinear (RNL) model is used to predict the normal velocity on a detection plane at $y^+=10$ in a turbulent channel flow, and the predicted velocity is used as wall blowing and suction for drag reduction. An active control test is carried out by using the well-trained CNN in direct numerical simulation (DNS). Substantial drag reduction rates up to 19 % and 16 % are obtained based on the spanwise and streamwise wall shear stresses, respectively. Furthermore, we explore the online control of wall turbulence by combining the RNL model with reinforcement learning (RL). The RL is constructed to determine the optimal wall blowing and suction based on its observation of the wall shear stresses without using the label data on the detection plane for training. The controlling and training processes are conducted synchronously in a RNL flow field. The control strategy discovered by RL has similar drag reduction rates with those obtained previously by the established method. Also, the training cost decreases by over thirty times at $Re_{\tau }=950$ compared with the DNS-RL model. The present results provide a perspective that combining the RNL model with machine learning control for drag reduction in wall turbulence can be effective and computationally economical. Also, this approach can be easily extended to flows at higher Reynolds numbers.

We introduce a low-order dynamical system to describe thermal convection in an annular domain. The model derives systematically from a Fourier–Laurent truncation of the governing Navier–Stokes Boussinesq equations and accounts for spatial dependence of the flow and temperature fields. Comparison with fully resolved direct numerical simulations (DNS) shows that the model captures parameter bifurcations and reversals of the large-scale circulation (LSC), including states of (i) steady circulating flow, (ii) chaotic LSC reversals and (iii) periodic LSC reversals. Casting the system in terms of the fluid's angular momentum and centre of mass (CoM) reveals equivalence to a damped pendulum with forcing that raises the CoM above the fulcrum. This formulation offers a transparent mechanism for LSC reversals, namely the inertial overshoot of a forced pendulum, and it yields an explicit formula for the frequency $f^*$ of regular LSC reversals in the high-Rayleigh-number (Ra) limit. This formula is shown to be in excellent agreement with DNS and produces the scaling law $f^* \sim {Ra}^{0.5}$.

We report on an experimental study in which Lagrangian tracking is applied to millions of microscopic particles floating on the free surface of turbulent water. We leverage a large jet-stirred zero-mean-flow apparatus, where the Reynolds number is sufficiently high for an inertial range to emerge while the surface deformation remains minimal. Two-point statistics reveal specific features of the flow, deviating from the classic description derived for incompressible turbulence. The magnitude of the relative velocity is strongly intermittent, especially at small separations, leading to anomalous scaling of the second-order structure functions in the dissipative range. This is driven by the divergent component of the flow, leading to fast approaching/separation rates of nearby particles. The Lagrangian relative velocity shows strong persistence of the initial state, such that the ballistic pair separation extends to the inertial range of time delays. Based on these observations, we propose a classification of particle pairs based on their initial separation rate. When this is much smaller than the relative velocity prescribed by inertial scaling (which is the case for the majority of the observed particle pairs), the relative velocity transitions to a diffusive growth and the Richardson–Obukhov super-diffusive dispersion is recovered.

The influence of symmetry-breaking effects of ridge-type roughness on secondary currents in turbulent channel flow is investigated using direct numerical simulations. The ridges have triangular cross-section, which is systematically varied from isosceles to right-angled triangle, introducing an imbalance to the slopes of the ridges’ lateral surfaces while the streamwise homogeneity of the surfaces is maintained. In all cases, secondary current vortices are produced, but asymmetric ridge cross-sections break the symmetry of these vortices. As a result of the asymmetry-induced misalignment and imbalance in the secondary current vortices, net spanwise flow emerges. The magnitude of the spanwise flow increases with the slope ratio of the ridge lateral surfaces and significantly modifies the mean flow topology, leading to the merging of critical points in the case of the right-angled triangular ridge shape. Within the cavities, the net spanwise flow is accompanied by a non-zero mean spanwise pressure gradient, while from the perspective of the outer flow, the scalene ridge surfaces have a similar effect as a wall that is slowly moving in the spanwise direction. Overall, the present results suggest the existence of a special type of Prandtl's secondary currents of the second kind, namely those that result in net spanwise flow.

The scaling relations mapping the turbulence statistics in compressible turbulent boundary layers (TBLs) onto their incompressible counterparts are of fundamental significance for turbulence modelling, such as the Morkovin scaling for velocity fields, while for pressure fluctuation fields, a corresponding scaling relation is currently absent. In this work, the underlying scaling relations of pressure fluctuations about Mach number ($M$) contained in their generation mechanisms are explored by analysing a series of direct numerical simulation data of compressible TBLs over a wide Mach number range $(0.5\leq M \leq 8.0)$. Based on the governing equation of pressure fluctuations, they are decomposed into components according to the properties of source terms. It is notable that the intensity of the compressible component, predominantly originating from the acoustic mode, obeys a monotonic distribution about the Mach number and wall distance; further, the intensity of the rest of the pressure components, which are mainly generated by the vorticity mode, demonstrates a uniform distribution consistent with its incompressible counterpart. Moreover, the coupling between the two components is negligibly weak. Based on the scaling relations, semiempirical models for the fluctuation intensity of both pressure and its components are constructed. Hence, a mapping relation is obtained that the profiles of pressure fluctuation intensities in compressible TBLs can be mapped onto their incompressible counterparts by removing the contribution from the acoustic mode, which can be provided by the model. The intrinsic scaling relation can provide some basic insight for pressure fluctuation modelling.

The mean flow in a turbulent boundary layer (TBL) deviates from the canonical law of the wall (LoW) when influenced by a pressure gradient. Consequently, LoW-based near-wall treatments are inadequate for such flows. Chen et al. (J. Fluid Mech., vol. 970, 2023, A3) derived a Navier–Stokes-based velocity transformation that accurately describes the mean flow in TBLs with arbitrary pressure gradients. However, this transformation requires information on total shear stress, which is not always readily available, limiting its predictive power. In this work, we invert the transformation and develop a predictive near-wall model. Our model includes an additional transport equation that tracks the Lagrangian integration of the total shear stress. Particularly noteworthy is that the model introduces no new parameters and requires no calibration. We validate the developed model against experimental and computational data in the literature, and the results are favourable. Furthermore, we compare our model with equilibrium models. These equilibrium models inevitably fail when there are strong pressure gradients, but they prove to be sufficient for boundary layers subjected to weak, moderate and even moderately high pressure gradients. These results compel us to conclude that history effects in mean flow, which negatively impact the validity of equilibrium models, can largely be accounted for by the material time derivative term and the pressure gradient term, both of which require no additional modelling.

We consider an internally heated fluid between parallel plates with fixed thermal fluxes. For a large class of heat sources that vary in the direction of gravity, we prove that $\smash { \smash {{\langle {\delta T} \rangle _h}} } \geq \sigma R^{-1/3} - \mu$, where $\smash { \smash {{\langle {\delta T} \rangle _h}} }$ is the average temperature difference between the bottom and top plates, $R$ is a ‘flux’ Rayleigh number and the constants $\sigma,\mu >0$ depend on the geometric properties of the internal heating. This result implies that mean downward conduction (for which $\smash { \smash {{\langle {\delta T} \rangle _h}} }< 0$) is impossible for a range of Rayleigh numbers smaller than a critical value $R_0:=(\sigma /\mu )^{3}$. The bound demonstrates that $R_0$ depends on the heating distribution and can be made arbitrarily large by concentrating the heating near the bottom plate. However, for any given fixed heating profile of the class we consider, the corresponding value of $R_0$ is always finite. This points to a fundamental difference between internally heated convection and its limiting case of Rayleigh–Bénard convection with fixed-flux boundary conditions, for which $\smash {{\langle {\delta T} \rangle _h}}$ is known to be positive for all $R$.

A series of recent studies has indicated that the component of the bottom drag caused by irregular small-scale topography in the ocean varies non-monotonically with the flow speed. The roughness-induced forcing increases with the speed of relatively slow abyssal currents but, somewhat counterintuitively, starts to decrease when flows are sufficiently swift. This reduction in drag at high speeds leads to the instability of laterally uniform currents, and the resulting evolutionary patterns are explored using numerical and analytical methods. The drag-law instability manifests in the spontaneous emergence of parallel jets, aligned in the direction of the basic flow and separated by relatively quiescent regions. We hypothesize that the mechanisms identified in this investigation could play a role in the dynamics of zonal striations commonly observed in the ocean.

Liquid metal buoyant flow around two differentially heated horizontal cylinders in the presence of a uniform vertical magnetic field is investigated experimentally. While magneto-convection in pipes or ducts has been studied theoretically and experimentally in recent years, data for heat transfer at immersed obstacles are rare and, to our knowledge, detailed experimental investigations on this fundamental magnetohydrodynamic problem do not exist. In the present work, two horizontal cylinders inserted into an adiabatic rectangular cavity filled with gallium–indium–tin are kept at constant temperatures to establish a driving temperature gradient in the surrounding liquid metal. The buoyancy-driven flow, quantified by the Grashof number $Gr$, is varied in the range ${10^{6} \leq Gr \leq ~5\times 10^{7}}$. With increasing magnetic field, expressed via the Hartmann number $Ha$, different flow regimes are identified from measurements for $0 \leq Ha \leq ~3000$. The effect of the electromagnetic force primarily consists in suppressing turbulence and damping the convective flow. The heat transfer is quantified in terms of the non-dimensional Nusselt number $Nu$, and its dependence on $Gr/{Ha}^{2}$, which is identified as the important group governing the flow, is discussed.

This study proposes a new mechanism that can lead to layering or convection from the finite amplitude perturbation acting on the double diffusive convection with uniform background shear. We focus on the double diffusive convection in the diffusive regime with the cold fresh water laying above the warm salty water. We demonstrate that, although the unperturbed system is linearly stable, the finite amplitude perturbation can trigger the initial flow motions which subsequently obtain energy from the gravitational potential energy and from the uniform background shear, and evolve to layering or convection. By using the linear stability analysis for the initial growth stage and the energy analysis for the following transitional stage, the critical Richardson number can be predicted theoretically. Here the Richardson number measures the relative strength of stratification to the background shear. The dominant wavenumbers and the growth rates of the corresponding modes given by linear theory agree well with the two-dimensional direct numerical simulations, and so does the critical Richardson number predicted by the theoretical model. The layering state is dominated by the double diffusion process, while the convection state at smaller Richardson number exhibits stronger influences from shear and generates smaller heat and salinity fluxes. The theoretical model is further applied to the parameter range which is relevant to the real oceanic environments and reveals that for the typical density ratio observed in the staircase regions in the Arctic Ocean, the current mechanism can lead to layering for relatively weak shear.

Many biological fluids are composed of suspended polymers immersed in a viscous fluid. A prime example is mucus, where the polymers are also known to form a network. While the presence of this microstructure is linked with an overall non-Newtonian response of the fluid, swimming cells and microorganisms similar in size to the network pores and polymer filaments instead experience the heterogeneous nature of the environment, interacting directly with the polymers as obstacles as they swim. To characterise and understand locomotion in these heterogeneous environments, we simulate the motion of an undulatory swimmer through three-dimensional suspensions and networks of elastic filaments, exploring the effects of filament and link compliance and filament concentration up to 20 % volume fraction. For compliant environments, the swimming speed increases with filament concentration to values approximately 10 % higher than in a viscous fluid. In stiffer environments, a non-monotonic dependence is observed, with an initial increase in speed to values 5 % greater than in a viscous fluid, followed by a dramatic reduction to speeds just a fraction of its value in a viscous fluid. Velocity fluctuations are also more pronounced in stiffer environments. We demonstrate that speed enhancements are linked to hydrodynamic interactions with the microstructure, while reductions are due to the filaments restricting the amplitude of the swimmer's propulsive wave. Unlike previous studies where interactions with obstacles allowed for significant enhancements in swimming speeds, the modest enhancements seen here are more comparable to those given by models where the environment is treated as a continuous viscoelastic fluid.

Two-dimensional free-surface flow over localised topography is examined, with the emphasis on the stability of hydraulic-fall solutions. A Gaussian topography profile is assumed with a positive or negative amplitude modelling a bump or a dip, respectively. Steady hydraulic-fall solutions to the full incompressible, irrotational Euler equations are computed, and their linear and nonlinear stability is analysed by computing eigenspectra of the pertinent linearised operator and by solving an initial value problem. The computations are carried out numerically using a specially developed computational framework based on the finite-element method. The Hamiltonian structure of the problem is demonstrated, and stability is determined by computing eigenspectra of the pertinent linearised operator. It is found that a hydraulic-fall flow over a bump is spectrally stable. The corresponding flow over a dip is found to be linearly unstable. In the latter case, time-dependent simulations show that ultimately, the flow settles into a time-periodic motion that corresponds to an invariant solution in an appropriately defined phase space. Physically, the solution consists of a localised large-amplitude wave that pulsates above the dip while simultaneously emitting nonlinear cnoidal waves in the upstream direction and multi-harmonic linear waves in the downstream direction.

Single-flagellated bacteria are ubiquitous in nature. They exhibit various swimming modes using their flagella to explore complex surroundings such as soil and porous polymer networks. Some single-flagellated bacteria swim with two distinct modes, one with the flagellum extended away from its body and another with the flagellum wrapped around it. The wrapped mode has been observed when bacteria swim under tight confinements or in highly viscous polymeric melts. In this study we investigate the hydrodynamics of these two modes inside a circular pipe. We find that the wrapped mode is slower than the extended mode in bulk but more efficient under strong confinement due to a hydrodynamic increase of its flagellum translation–rotation coupling and an Archimedes’ screw-like configuration that helps to move the fluid along the pipe.

This study conducts particle-resolved direct numerical simulations to analyse how finite-size spherical particles affect the decay rate of turbulent kinetic energy in non-sustained homogeneous isotropic turbulence. The decaying particle-laden homogeneous isotropic turbulence is generated with two set-ups, i.e. (1) releasing particles into a single-phase decaying homogeneous isotropic turbulence and (2) switching off the driving force of a sustained particle-laden homogeneous isotropic turbulence. With both set-ups, the decay of turbulent kinetic energy follows a power-law when the flow is fully relaxed, similar to their single-phase counterparts. The dependence of the power-law decay exponent $n$ on the particle-to-fluid density ratio, particle size and volume fraction is also investigated, and a predictive model is developed. We find that the presence of heavier particles slows down the long-time power-law decay exponent.

Elastoinertial turbulence (EIT) is a chaotic flow resulting from the interplay between inertia and viscoelasticity in wall-bounded shear flows. Understanding EIT is important because it is thought to set a limit on the effectiveness of turbulent drag reduction in polymer solutions. Here, we analyse simulations of two-dimensional EIT in channel flow using spectral proper orthogonal decomposition (SPOD), discovering a family of travelling wave structures that capture the sheetlike stress fluctuations that characterise EIT. The frequency-dependence of the leading SPOD mode contains distinct peaks and the mode structures corresponding to these peaks exhibit well-defined travelling structures. The structure of the dominant travelling mode exhibits shift–reflect symmetry similar to the viscoelasticity-modified Tollmien–Schlichting (TS) wave, where the velocity fluctuation in the travelling mode is characterised by large-scale regular structures spanning the channel and the polymer stress field is characterised by thin, inclined sheets of high polymer stress localised at the critical layers near the channel walls. The travelling structures corresponding to the higher-frequency modes have a very similar structure, but are nested in a region roughly bounded by the critical layer positions of the next-lower-frequency mode. A simple theory based on the idea that the critical layers of mode $\kappa$ form the ‘walls’ for the structure of mode $\kappa +1$ yields quantitative agreement with the observed wave speeds and critical layer positions, indicating self-similarity between the structures. The physical idea behind this theory is that the sheetlike localised stress fluctuations in the critical layer prevent velocity fluctuations from penetrating them.

We study the behaviour of a particle bed immersed in water when a flow generated by an oscillating plate is induced above it. We first consider a rigid plate submerged and oscillated over a particle bed. During upward motion of the plate, a portion of the bed fails, allowing particle displacement, and the bed surface to deform into a heap. We have already determined the flow of the fluid above and within the bed. This work describes the particle motion within the failed region of the bed: when the particles are mobile, they follow the fluid. We depth average the balance of mass and obtain an evolution equation for the displacement of the bed surface. We solve this equation and compare the predictions with the measurements of surface displacement in earlier experiments on rigid square plates. We carry out new experiments to measure the surface displacements under elongated plates. Elongated rigid plates behave similarly to the rigid square ones. Flexible plates produce multiple heaps. We determine that the peaks of these heaps are correlated with the flexural modes of the plates and occur at points along the bed at which the fluid pressure has its extreme values. Different plate flexural modes, resulting in different numbers of heaps, are produced by driving the plate at different frequencies. The particle motion within the bed and heap evolution under a flexible plate can be roughly described by regarding it as two or more rigid plates. We test the predictions of the theory against experiments.

Consider the motion of a thin layer of electrically conducting fluid, between two closely spaced parallel plates, in a classical Hele-Shaw geometry. Furthermore, let the system be immersed in a uniform external magnetic field (normal to the plates) and let electrical current be driven between conducting probes immersed in the fluid layer. In the present paper, we analyse the ensuing fluid flow at low Hartmann numbers. Physically, the system is particularly interesting because it allows for circulation in the flow, which is not possible in the standard pressure-driven Hele-Shaw cell. We first elucidate the mechanism of flow generation both physically and mathematically. After formulating the problem using complex variables, we present mathematical solutions for a class of canonical multiply connected geometries in terms of the prime function framework developed by Crowdy (Solving Problems in Multiply Connected Domains, SIAM, 2020). We then demonstrate how recently developed fast numerical methods may be applied to accurately determine the flow field in arbitrary geometries.

We present evidence revealing that an object with specific properties can exhibit multiple stable falling postures at low Reynolds numbers. By scrutinizing the force equilibrium relationship of a fixed object at various attack angles and Reynolds numbers, we introduce a methodology that can obtain the stable falling postures of the object. This method saves computational resources and more intuitively presents the results in the full parameter domain. Our findings are substantiated by free-fall tests conducted through both physical experiments and numerical simulations, which validate the existence of multiple stable solutions in accordance with the interpolation results obtained with fixed objects. Additionally, we quantify the abundance and distribution patterns of stable falling postures for a diverse range of representative shapes. This discovery highlights the existence of multiple stable solutions that are universally present across objects of different shapes. The implications of this research extend to the design, stability control and trajectory prediction of all free and controlled flights in both air and water.