When a viscous fluid spreads underneath a deformable surface skin or crust, the peeling dynamics at the fluid front can control the rate of advance rather than bulk self-similar flow. For an elastic skin, this control results in a quasi-static interior blister held at constant pressure that is matched to a narrow peeling region behind the fluid front. In this paper, the analogous problem is considered for a skin that deforms either viscously or plastically, or both. In particular, the deformable surface is assumed to be a thin plate of material governed by the Herschel–Bulkley constitutive law. We examine how such a skin controls viscous flow underneath, fed at constant flux and spreading as either a planar or axisymmetric current. As for an elastic skin, the peeling dynamics at the viscous fluid front again controls the rate of spreading. However, contrary to that situation, the mathematical matching problem for viscoplastic peeling is simplified considerably as a result of an integral constraint. Despite this, the structure of the peeling region is complicated significantly by any plasticity in the skin, which can create a convoluted peeling wave ahead of the main blister that features interwoven yielded and plugged sections of the plate.

We study the term in the eddy energy budget of continuously stratified quasigeostrophic turbulence that is responsible for energy extraction by eddies from the background mean flow. This term is a quadratic form, and we derive Euler–Lagrange equations describing its eigenfunctions and eigenvalues, the former being orthogonal in the energy inner product and the latter being real. The eigenvalues correspond to the instantaneous energy growth rate of the associated eigenfunction. We find analytical solutions in the Eady problem. We formulate a spectral method for computing eigenfunctions and eigenvalues, and compute solutions in Phillips-type and Charney-type problems. In all problems, instantaneous growth is possible at all horizontal scales in both inviscid problems and in problems with linear Ekman friction. We conjecture that transient growth at small scales is matched by linear transfer to decaying modes with the same horizontal structure, and we provide simulations supporting the plausibility of this hypothesis. In Charney-type problems, where the linear problem has exponentially growing modes at small scales, we expect net energy extraction from the mean flow to be unavoidable, with an associated nonlinear transfer of energy to dissipation.

Power minimisation of fluid transport in branched fluidic networks has become of paramount importance for microfluidics, additive manufacturing and hierarchical functional materials. For fully developed laminar flow of Newtonian fluids, Murray's theory provides a solution for the channel and network dimensions that minimise power consumption. However, design and optimisation of networks that transport complex fluids is still challenging. Here, we generalise Murray's theory towards fluid rheologies, including non-Newtonian (power-law) and yield-stress fluids (Bingham, Herschel–Bulkley, Casson). A straightforward graphical approach is presented that provides the optimal radii in a branching network, and the angles between these branches. The wall shear stress is found to be uniform over the entire network, and the velocity profile is self-similar. Furthermore, the effect of non-optimal channel radii on the power consumption of the network is investigated. Finally, examples illustrate how this approach applies to a wide variety of systems.

A computational study was performed of the transport of both non-adhesive and adhesive particles in a porous bed with a body-centred cubic (BCC) structure. Pore-scale simulation of the flow within the porous bed was achieved through combining the immersed boundary method and the lattice Boltzmann method. Particle transport is computed using an adhesive discrete-element method based on a multi-time-scale soft-sphere model. The fluid flow results are validated by comparison with experimental data for dimensionless permeability of flow in a porous bed of spheres. For computations with non-adhesive particles, the particles are observed to drift to the centre of ‘channels’ in the BCC array, within which most of the fluid flow occurs. The mechanism of this inward drift was found to be related to the phenomenon of oscillatory clustering, which is an inertial drift mechanism observed for particles in a corrugated channel. A measure for particle drift into these channels was developed, and the time rate of change of this measure was found to compare closely with an approximate theoretical prediction based on oscillatory clustering theory. The drift measure was observed to be limited at long time by hold-up of outlier particles caught in long-duration collisions with the fixed bed particles in regions of low fluid velocity magnitude. Simulations with adhesive particles exhibited marked increase in collision duration, as well as inhibition of the tendency to drift toward the flow channels due to adhesive hold-up.

Helicity plays a key role in the evolution of vortex structures and turbulent dynamics. The helicity dynamics and vortex structures in streamwise-rotating channel turbulence are discussed in this paper using the helicity budget equation and the differentiated second-order structure function equation of helicity. Generally, rotation and Reynolds numbers exhibit opposing effects on the interscale helicity dynamics and the vortices. Under the buffer layer, the positions of the helicity peaks are proportional to the ratio between the Reynolds and rotation numbers. The mechanism is related to the opposing effects of convection and rotation. Rotation directly affects the helicity balance through the Coriolis term and corresponding pressure term. In the buffer layer, the scale helicity is negative at small scales but positive at large scales, which is mainly induced by the spatial effects (the production and the spatial turbulent convection) but reduced by interscale cascades. Examination of structures reveals the close association between scale helicity and streaks, with streak lift angles exhibiting an increase with rotation and a decrease with Reynolds numbers. In the log-law layer, the Coriolis terms and corresponding pressure terms are proportional to the rotation numbers but remain independent of the Reynolds numbers. The negative scale helicity is forward cascaded towards small scales. Generally, spanwise vortices in the log-law layer are related to sweep events and forward cascades. Our findings indicate that these spanwise vortices are suppressed by rotation but recover with increasing Reynolds numbers, aligning with the effects observed in the scale helicity balance.

Experiments and numerical simulations of inertial particles in underexpanded jets are performed. The structure of the jet is controlled by varying the nozzle pressure ratio, while the influence of particles on emerging shocks and rarefaction patterns is controlled by varying the particle size and mass loading. Ultra-high-speed schlieren and Lagrangian particle tracking are used to experimentally determine the two-phase flow quantities. Three-dimensional simulations are performed using a high-order, low-dissipative discretization of the gas phase while particles are tracked individually in a Lagrangian manner. A simple two-way coupling strategy is proposed to handle interphase exchange in the vicinity of shocks. Velocity statistics of each phase are reported for a wide range of pressure ratios, particle sizes and volume fractions. An upstream shift of the Mach disk in the presence of particles reveals significant two-way coupling even at low mass loading. A semi-analytic model that predicts the extent of the Mach disk shift is presented based on a one-dimensional Fanno flow that takes into account volume displacement by particles and interphase exchange due to drag and heat transfer. The per cent shift in Mach disk is found to scale with the mass loading, nozzle pressure ratio and interphase slip velocity and inversely with the particle diameter.

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.

Direct numerical simulations of spiral Poiseuille flows in a narrow gap geometry are performed with the aim of identifying the mechanisms governing the dynamics of the axial friction coefficient. The investigation has explored a small portion of the Reynolds number–Taylor number phase space ($600 \leq Re \leq 5766$ and $1500 \leq Ta \leq 5000$), for which reference experimental results are available. The study is focused on the mechanism leading to the enhancement of the axial friction coefficient with the Taylor number when the Reynolds number is kept constant. The analysis of the spatial distribution of the Reynolds stress tensor and of the turbulent energy budget has evidenced the key role of the pressure–strain correlation in the energy transfer from the azimuthal to the axial component. The latter eventually determines the increase of the axial friction coefficient through the enhanced radial mixing of axial momentum. Data have also shown that the flow dynamics is heavily dependent on the $Ta/Re$ ratio, and different regimes develop (ranging from laminar to turbulent), each with peculiar behaviours.

In this work, the shape of a bluff body is optimized to mitigate velocity fluctuations of turbulent wake flows based on large-eddy simulations (LES). The Reynolds-averaged Navier–Stokes method fails to capture velocity fluctuations, while direct numerical simulations are computationally prohibitive. This necessitates using the LES method for shape optimization given its scale-resolving capability and relatively affordable computational cost. However, using LES for optimization faces challenges in sensitivity estimation as the chaotic nature of turbulent flows can lead to the blowup of the conventional adjoint-based gradient. Here, we propose using the regularized ensemble Kalman method for the LES-based optimization. The method is a statistical optimization approach that uses the sample covariance between geometric parameters and LES predictions to estimate the model gradient, circumventing the blowup issue of the adjoint method for chaotic systems. Moreover, the method allows for the imposition of smoothness constraints with one additional regularization step. The ensemble-based gradient is first evaluated for the Lorenz system, demonstrating its accuracy in the gradient calculation of the chaotic problem. Further, with the proposed method, the cylinder is optimized to be an asymmetric oval, which significantly reduces turbulent kinetic energy and meander amplitudes in the wake flows. The spectral analysis methods are used to characterize the flow field around the optimized shape, identifying large-scale flow structures responsible for the reduction in velocity fluctuations. Furthermore, it is found that the velocity difference in the shear layer is decreased with the shape change, which alleviates the Kelvin–Helmholtz instability and the wake meandering.

Keeping a tube from being plugged by a fluid is an important process in applications. An interesting re-entrant phenomenon for the capillary state with the occluding state sandwiching the non-occluding state from both the high- and low-Bond-number regions can appear by inserting a rod into a horizontal tube at an eccentric position (Tan et al., J. Fluid Mech, vol. 946, 2022, A7). Containers with rounded corners are very common. We theoretically investigate a situation for a horizontal open tube with rounded corner(s). The results show that a re-entrant non-occlusion at a contact angle can also appear without the insertion of any object. The competition between the rounded corner wetting/non-wetting effect and gravity effect can lead to a re-entrant non-occlusion. The re-entrant non-occlusion is affected by the shape and orientation of the rounded corner(s). For a tube with only one rounded corner, the re-entrant non-occlusion exists when the rounded corner has a not-so-large corner radius and is not in a landscape orientation. For a tube with two (or more) rounded corners, the corner(s) with the strongest corner effect will determine the existence or non-existence of the re-entrant non-occlusion. This paper provides an effective scheme for designing a high-performance capillary with corners that are not easily occluded by a fluid and removing fluid blockage from a capillary in optofluidic/microfluidic applications.

Wind tunnel measurements of the incident turbulent velocity fields and axial forces on a horizontal axis turbine and porous disc analogues are reported. The models were tested in both a simulated atmospheric boundary layer (ABL) and in grid turbulence, allowing for a range of turbulence length scale to rotor diameter ratios to be considered. A theoretical framework to account for the combined effect of distortion and potential flow blocking in the induction zone is presented. In the case of very large length-scale turbulence to diameter ratios, where distortion effects are minimal, a quasi-steady approach is adopted for the effect of blocking. For the small length-scale ratio limit, the method is developed from the classical analyses for rapid distortion of turbulence and blockage from flow through a porous sheet of resistance. For general length-scale ratios, an efficient prediction method based on interpolation between the two length-scale ratio extremes is established. For very large length-scale ratios, a quasi-steady theory without distortion is appropriate for a rotor or disc in a simulated ABL. The small length-scale theory is applicable for tests conducted in grid turbulence. The results of the study can inform the prediction and interpretation of typical measurements of turbulence within the induction zone and the fluctuating loads on a rotor, at both prototype and full scale. This is of particular importance to fatigue load assessments.

This study investigates the coherence of turbulent fluctuations in a turbulent vertical natural convection boundary layer immersed in a stably stratified medium (turbulent buoyancy layer). A turbulent buoyancy layer of a fluid having a Prandtl number of $0.71$ at a Reynolds number of $800$ is numerically simulated using direct numerical simulation. The two-point correlations reveal that the streamwise velocity fluctuations are coherent over large streamwise distances, with the length scale of the streamwise coherence being greater than the boundary layer thickness. This is due to large-scale motions (LSMs), similar to the LSMs observed in canonical wall-bounded turbulence despite the stark differences in flow dynamics. Both high-speed (positive) and low-speed (negative) streamwise velocity fluctuations form LSMs, with their streamwise length scales increasing with increasing wall-normal distance. High-speed LSMs are composed of upwash flow with high temperatures, while low-speed LSMs are composed of downwash flow with low temperatures. Both high-speed and low-speed LSMs meander appreciably in the streamwise direction, with the degree of meandering being correlated with the sign of the spanwise velocity fluctuations. The LSMs exhibit coherence across significant wall-normal distances and contribute significantly to the turbulence production in the outer layer. Examining the one-dimensional energy spectra of the turbulent buoyancy layer shows that the LSMs are the dominant energy-containing motions, implying that the length scale of the energy-containing range is of the order of boundary layer thickness. Notably, wall-normal velocity, spanwise velocity and buoyancy fluctuations do not form LSMs with streamwise length scales comparable to streamwise velocity fluctuations.

The stability and postcritical behaviour of a horizontal flag undergoing gravity-induced deformation and periodic contact with a nearby horizontal rigid wall are experimentally investigated. The results elucidate the combined effects of gravity and contact on flutter, and reveal design principles for application to triboelectric energy harvesting. By varying the free-stream velocity, flag thickness and distance between the flagpole and the wall, the dynamics of the flag are classified into quasistatic equilibrium, flutter, partial contact and saturated contact modes. Considering the significance of gravitational effects, a new dimensionless flow velocity is proposed to identify the distribution of the dynamic modes, and its definition varies according to whether the wall is placed above or below the flag. The critical conditions for transitions between the dynamic modes are determined from the balance of fluid dynamic and gravitational effects. The distance from the flagpole to the wall is found to be more critical for transitions in the lower-wall configuration than in the upper-wall configuration. The peak contact force as well as the oscillation amplitude and frequency at postequilibrium exhibits remarkably different trends depending on the location of the wall. The peak contact force imposed on the wall by the fluttering flag weakens as the distance to the wall increases in the case of an upper wall, whereas it becomes stronger in the case of a lower wall.

The causal relevance of local flow conditions in open-channel turbulence is analysed using ensembles of interventional experiments in which the effect of perturbing the flow within a small cell is monitored at some future time. When this is done using the relative amplification of the perturbation energy, causality depends on the flow conditions within the cell before it is perturbed, and can be used as a probe of the flow dynamics. The key scaling parameter is the ambient shear, which is also the dominant diagnostic variable for wall-attached perturbations. Away from the wall, the relevant variables are the streamwise and wall-normal velocities. Causally significant cells are associated with sweeps that carry the perturbation towards the stronger shear near the wall, whereas irrelevant ones are associated with ejections that carry it towards the weaker shear in the outer layers. Causally significant and irrelevant cells are themselves organised into structures that share many characteristics with classical sweeps and ejections, such as forming spanwise pairs whose dimensions and geometry are similar to those of classical quadrants. At the wall, this is consistent with causally significant configurations in which a high-speed streak overtakes a low-speed one, and causally irrelevant ones in which the two streaks pull apart from each other. It is argued that this is probably associated with streak meandering.

Can the similarity between rotation and stratification provide quantitative predictions for complex turbulence? In this work, we focus on plane Couette turbulence as the background flow, which allows us to eliminate the key differences between Rayleigh–Bénard and Taylor–Couette turbulence, and facilitates a quantitative mapping across many complex turbulent systems involving rotation, stratification and curvature effects. To characterize the separated or coupled effects of rotation and stratification, we introduce an overall Richardson number ${Ri}_\chi$ which is the sum of the Coriolis Richardson number $Ri_\theta$ and the buoyancy Richardson number $Ri_T$. When the Prandtl number $Pr=1$, the heat and inertial-frame momentum transport almost coincide and mainly depend on ${Ri}_\chi$ and the Reynolds number, regardless of the specific ratio between $Ri_\theta$ and $Ri_T$. When $Pr$ varies, the weighted average ${Nu}_\chi$ of the transport coefficients can still remain approximately invariant in most cases. Furthermore, the large-scale structures in purely rotating and purely stratified cases exhibit strikingly similar features. These findings not only indicate a more reasonable analogy for the ultimate Taylor–Couette turbulence but also pave the way for developing new predictive models for natural and industrial processes.

The impact of a heavier droplet falling into a deep pool of lighter liquid is investigated using three-dimensional numerical simulations. We demonstrate that the heavier droplets can hang from the surface of a lighter liquid using surface tension. The impact phenomenon and the evolution of the heavier droplet as a function of its size and release height are explored. A theoretical model is also formulated to understand the role of different forms of energy associated with the hanging droplet. We further solve the force balance equations for the hanging droplets analytically, and demonstrate that the results obtained from our simulations match very well the analytical solution. This research offers opportunities in many areas, including drug and gene delivery, encapsulation of biomolecules, microfluidics, soft robots, and remediation of oil spills.

Radial unstable stratification is a potential source of turbulence in the cold regions of accretion disks. To investigate this thermal effect, here we focus on two-dimensional Rayleigh–Bénard convection in an annulus subject to radially dependent gravitational acceleration $g \propto 1/r$. Next to the Rayleigh number $Ra$ and Prandtl number $Pr$, the radius ratio $\eta$, defined as the ratio of inner and outer cylinder radii, is a crucial parameter governing the flow dynamics. Using direct numerical simulations for $Pr=1$ and $Ra$ in the range from $10^7$ to $10^{10}$, we explore how variations in $\eta$ influence the asymmetry in the flow field, particularly in the boundary layers. Our results show that in the studied parameter range, the flow is dominated by convective rolls and that the thermal boundary-layer (TBL) thickness ratio between the inner and outer boundaries varies as $\eta ^{1/2}$. This scaling is attributed to the equality of velocity scales in the inner ($u_i$) and outer ($u_o$) regions. We further derive that the temperature drops in the inner and outer TBLs scale as $1/(1+\eta ^{1/2})$ and $\eta ^{1/2}/(1+\eta ^{1/2})$, respectively. The scalings and the temperature drops are in perfect agreement with the numerical data.