Closed-form expressions for aerodynamic force on an accelerating aerofoil were presented in the 1930s, relating instantaneous force to geometric and kinematic parameters under the following assumptions: a thin aerofoil, small-amplitude motions, planar wake development, and a flow that is inviscid, incompressible and two-dimensional. The present work is a step towards analogous closed-form expressions for large-amplitude motions of thick foils when the flow remains attached and boundary-layer thickness approaches (but does not equal) zero. A mathematical framework is derived from vortical flow theory to highlight the finite degrees of freedom that must be solved or predicted in order to yield a predictive aerodynamic model under the stated conditions. The special case of periodic motion is further considered, and an equation is derived to calculate mean forces from known or assumed time histories of circulation, vorticity-weighted mean wake convection velocity and trailing-edge velocity.

We present a unified framework derived from the total heat flux equation, enabling the direct formulation of the relationship between mean temperature and velocity fields, as well as the development of mean temperature scalings in compressible turbulent channel flows. The proposed mean temperature–velocity relationship, combined with a simple damping function model for the mixed Prandtl number, demonstrates high efficacy in channels with both symmetric and asymmetric thermal boundary conditions across a range of Mach and Reynolds numbers. In contrast, the state-of-the-art generalised Reynolds analogy (GRA) relation (Zhang et al., 2014, J. Fluid Mech., vol. 739, pp. 392–420) is shown to be insufficient for asymmetric cases due to mismatched boundary conditions at the effective boundary layer edge. By introducing a mean temperature decomposition, we clarify that while the GRA relation effectively characterises the component associated with turbulence production and viscous dissipation, it fails to account for the contribution arising from non-zero edge total heat flux. Furthermore, we rigorously derive mean temperature transformations compatible with arbitrary velocity scalings for the first time. These findings provide some physical insights into the mean momentum and heat transport in compressible wall-bounded turbulence, and may be helpful for developing near-wall models.

The skin-friction coefficient is a dimensionless quantity defined by the wall shear stress exerted on an object moving in a fluid, and it decreases as the Reynolds number increases for wall-bounded turbulent flows over a flat plate. In this work, a novel transformation, based on physical and asymptotic analyses, is proposed to map the skin-friction relation of high-speed turbulent boundary layers (TBLs) for air described by the ideal gas law to the incompressible skin-friction relation. Through this proposed approach, it has been confirmed theoretically that the transformed skin-friction coefficient $C_{f,i}$, and the transformed momentum-thickness Reynolds number $Re_{\theta ,i}$ for compressible TBLs with and without heat transfer, follow a general scaling law that aligns precisely with the incompressible skin-friction scaling law, expressed as $ (2/C_{f,i} )^{1/2}\propto \ln Re_{\theta ,i}$. Furthermore, the reliability of the skin-friction scaling law is validated by compressible TBLs with free-stream Mach number ranging from $0.5$ to $14$, friction Reynolds number ranging from $100$ to $2400$, and the wall-to-recovery temperature ratio ranging from $0.15$ to $1.9$. In all of these data, $ (2/C_{f,i} )^{1/2}$ and $\ln Re_{\theta ,i}$ based on the present theory collapse to the incompressible relation, with a squared Pearson correlation coefficient reaching an impressive value $0.99$, significantly exceeding $0.85$ and $0.86$ based on the established van Driest II and the Spalding–Chi transformations, respectively.

We present the measurements of the decay of stationary turbulence at Reynolds numbers based on the Taylor microscale $Re_{\lambda }=493, 599, 689$ produced in a large-scale von Kármán flow using stereoscopic particle image velocimetry. First, steady-state conditions were established, after which the impellers were simultaneously and abruptly stopped, and the turbulent decay was measured over 10–20 impeller rotation periods. A total of 258 decay experiments were performed. The temporal evolution of the ensemble-averaged turbulent kinetic energy (TKE) showed excellent agreement over all $Re_{\lambda }$ and exhibited two distinct phases: a short, initial transition phase where the TKE remained almost constant due to the inertia of the flow and lasted approximately $0.4$ impeller rotations, followed by a classical power-law decay. To extract the decay exponent $n$, a curve-fitting function based on a one-dimensional energy spectrum was used, and successfully captured the entire measured decay process. A value $n=1.62$ was obtained based on ensemble-averaged TKE. However, different decay exponents were found for individual velocity components: $n=1.38$ for the axial component consistent with various reports in the literature and Loitsiansky’s prediction ($n=1.43$), and $n=1.99$ for the radial and circumferential components indicating saturation/confinement effects. Similarly, the longitudinal integral length scale in the axial direction grew as $L\propto t^{2/7}$, whereas it remained nearly constant in the radial direction. Finally, the evolution of the ensemble-averaged velocity gradients showed that after the impellers were stopped, the mean flow pattern persisted for a short time before undergoing a large-scale reversal before the onset of the turbulent decay.

We study the homogeneous isotropic turbulence of a shear-thinning fluid modelled by the Carreau model, and show how the variable viscosity affects the multiscale behaviour of the turbulent flow. We show that Kolmogorov theory can be extended to such non-Newtonian fluids, provided that the correct choice of average is taken when defining the mean Kolmogorov scale and dissipation rate, to properly capture the effect of the variable viscosity. Thus the classical phenomenology à la Kolmogorov can be observed in the inertial range of scale, with the energy spectra decaying as $k^{-5/3}$, with $k$ being the wavenumber, and the third-order structure function obeying the $4/5$ law. The changing viscosity instead strongly alters the small scale of turbulence, leading to an enhanced intermittent behaviour of the velocity field.

Turbulent Taylor–Couette flow displays traces of axisymmetric Taylor vortices even at high Reynolds numbers. With this motivation, Feldmann & Avila (2025) J. Fluid Mech, 1008, R1, carry out long-time numerical simulations of axisymmetric high-Reynolds-number Taylor–Couette flow. They find that the Taylor vortices, using the only degree of freedom that remains available to them, carry out Brownian motion in the axial direction, with a diffusion constant that diverges as the number of rolls is reduced below a critical value.

The quasi-geostrophic two-layer model is a widely used tool to study baroclinic instability in the ocean. One instability criterion for the inviscid two-layer model is that the potential vorticity (PV) gradient must change sign between the layers. This has a well-known implication if the model includes a linear bottom slope: for sufficiently steep retrograde slopes, instability is suppressed for a flow parallel to the isobaths. This changes in the presence of bottom friction as well as when the PV gradients in the layers are not aligned. We derive the generalised instability condition for the two-layer model with non-zero friction and arbitrary mean flow orientation. This condition involves neither the friction coefficient nor the bottom slope; even infinitesimally weak bottom friction destabilises the system regardless of the bottom slope. We then examine the instability characteristics as a function of varying slope orientation and magnitude. The system is stable across all wavenumbers only if friction is absent and if the planetary, topographic and stretching PV gradients are aligned. Strong bottom friction decreases the growth rates but also alters the dependence on bottom slope. In conclusion, the often mentioned stabilisation by steep bottom slopes in the two-layer model holds only in very specific circumstances, thus probably plays only a limited role in the ocean.

Removing liquid from a channel is an important process. In a horizontal slit in the presence of a downward gravity field, two distinct liquid states were commonly observed: gravity-driven liquid non-occlusion and liquid plug (Parry et al. 2012 Phys. Rev. Lett. 108, 246101). A wetting-driven non-occlusion at some contact angles was induced by insertion of a rod into a horizontal tube at an eccentric position (Tan et al. 2022 J. Fluid Mech. 946, A7). Insertion of a plate into a horizontal slit may enhance the capacity of removing liquid. This situation is theoretically investigated, and the theoretical results are mutually verified by a computational fluid dynamics (CFD) numerical method that is first employed to determine the critical non-occlusion conditions. Four types of liquid states are observed. The effects of contact angles, plate position and Bond number (measured by downward gravitational force relative to surface tension force) on different types of liquid states are analysed. This paper additionally provides a CFD numerical method for understanding the conditions for the stability and existence of the liquid plugs in complex situations (e.g. considering the effect of the sidewalls, or when a rod or plate is inserted into a circular, elliptical or polygonal tube) in the future.

The function of aortic heart valves is to prevent regurgitant flow from the aorta into the left ventricle. A higher regurgitant flow is observed in bileaflet mechanical heart valves (BMHVs) compared with bioprosthetic heart valves (BHVs) because of their delayed closure. Here, we investigate this behaviour through fluid–structure interaction simulations of a BMHV compared with a trileaflet mechanical heart valve (TMHV) and a BHV under similar conditions. We find that the TMHV and BHV begin to close during the systolic deceleration, whereas BMHV only begins to close when the flow reverses. We found this to be related to hemodynamics as the TMHV and BHV, when fully opened, generate a central jet-dominant flow, whereas the BMHV generates triple jets with lateral jets being wider than its central jet. The flow deceleration of the central jet during late systole is higher than that of the sinuses, which results in a lower pressure in the central region than the sinuses to drive the leaflets of the TMHV and BHV towards the centre for closure. Conversely, the pressure on the sinus- and central flow-sides of the BMHV leaflets is nearly the same until the end of systole. We, contrary to what classically believed, did not find any evidence of sinus vortices generating high pressure or viscous stresses to initiate valve closure. Overall, the results suggest that the generation of a strong central jet and the direction of the leaflets’ closure towards the centre are the design principles to ensure an early valve closure and minimise regurgitation.

The transport of a passive scalar at unity Schmidt number in a turbulent flow over a random sphere pack is investigated by direct numerical simulation. A bed-normal scalar flux is introduced by prescribed scalar concentration values at the bottom and top domain boundaries, whereas sphere surfaces are impermeable to scalar fluxes. We analyse eight different cases characterised by friction Reynolds numbers $Re_\tau \in [150, 500]$ and permeability Reynolds numbers $Re_K \in [0.4, 2.8]$ at flow depth-to-sphere diameter ratios of $h/D \in \{ 3, 5, 10 \}$. The dimensionless roughness heights lie within $k_s^+ \in [20,200]$. The free-flow region is dominated by turbulent scalar transport and the effective diffusivity scales with flow depth and friction velocity. Near the interface, dispersive scalar transport and molecular diffusion gain importance, while the normalised near-interface effective diffusivity is approximately proportional to $Re_K^2$. Even without a macroscopic bed topography, local hotspots of dispersive scalar transport are observed (‘chimneys’), which are linked to strong spatial variations in the time-averaged scalar concentration field. The form-induced production of temporal scalar fluctuations, however, goes along with a homogenisation of those spatial variations of the scalar concentration field due to turbulent fluid motion. Accordingly, form-induced production determines the interaction of turbulent and dispersive scalar transport at the interface. With increasing $Re_K$, momentum from the free-flow region entrains deeper into the sediment bed, such that the form-induced production intensifies and peaks at lower positions. As a result, the transition from dispersive to turbulent scalar transport is observed deeper inside the sphere pack.

In this study, we experimentally examine the behaviour of a free-falling rigid sphere penetrating a quiescent liquid pool. Observations of the sphere trajectory in time are made using two orthogonally placed high-speed cameras, yielding the velocity and acceleration vectors through repeated differentiation of the time-resolved trajectories. The novelty of this study is twofold. On the one hand, a methodology is introduced by which the instantaneous forces acting on the sphere can be derived by tracking the sphere trajectory. To do this, we work in a natural coordinate system aligned with the pathline of the sphere. In particular, the instantaneous lift and drag forces can be separately estimated. On the other hand, the results reveal that when decelerating, the sphere experiences a very high drag force compared with steady flow. This is attributed to an upstream shift of the mean boundary-layer separation. The sphere also experiences significant lift force fluctuations, attributed to unsteady and asymmetric wake fluctuations. The trajectories can be reduced to three stages, common in duration for all initial Reynolds numbers and density ratios when expressed in dimensionless time. In addition, the sphere velocity and deceleration magnitude for different initial parameters exhibit a high degree of uniformity when expressed in dimensionless form. This offers prediction capability of how far a sphere penetrates in time and the forces acting on it.

Here, we show that the thrust force of oscillating airfoils calculated within the linearised potential flow approach by means of the vortex impulse theory coincides with the one resulting from the integration of the unsteady pressure distribution around the solid obtained by Garrick (1936) when the vertical component of the wake velocity is calculated self-consistently and the analysis retains the contribution of the flux of horizontal momentum induced by the starting vortex. The limitations of the self-consistent linearised potential flow approach for predicting the thrust force of airfoils oscillating periodically with small amplitudes but large values of the reduced frequency are also discussed, as well as the reasons behind the ability of other results in the literature to approximate measurements better than Garrick’s theory. In fact, for those cases in which the airfoil oscillates periodically, the flux of horizontal momentum induced by the starting vortex is negligible and the vortices in the wake are convected parallel to the free-stream velocity, we have deduced an equation for the mean thrust coefficient which differs from previously published results and is in agreement with experimental and numerical results. In addition, for those cases in which the airfoil is suddenly set into motion, we have also deduced an equation that retains the effect of the starting vortex and correctly quantifies the transient thrust force.

The dynamics of flow over an isolated surface-mounted hemisphere are investigated with tomographic particle image velocimetry (PIV). The 10 mm height hemisphere is completely submerged in the laminar boundary layer, and the height-based Reynolds number is 1530. The evolution of typical coherent structures around the hemisphere are discussed, with emphasis on the hairpin vortex (HV) and side hairpin vortex (SHV) formed periodically in the middle and both sides of the wake, respectively. Proper orthogonal decomposition (POD) analysis is conducted to explore the vortex dynamics. The shedding processes of the HV and SHV are each dominated by two different POD modes with correspondingly different characteristic frequencies, which has not been reported before in the literature. Furthermore, the coexistence of symmetric and asymmetric shedding patterns is explored for the first time in the shedding process of the HV at such a low Reynolds number. The asymmetric behaviour is controlled by the asymmetric shedding POD mode, whose dominant frequency is exactly half of the symmetric mode. In addition, SHVs on both sides of the wake are throughout formed and shed alternately, and the streamwise extensions of a horseshoe vortex also oscillate asymmetrically, which are responsible for the formation of the asymmetric shedding pattern of the HV. These findings help to fill the gaps in the related field and contribute to studies on the vortex dynamics of the flow over a hemisphere.

Embedding the intrinsic symmetry of a flow system in training its machine learning algorithms has become a significant trend in the recent surge of their application in fluid mechanics. This paper leverages the geometric symmetry of a four-roll mill (FRM) to enhance its training efficiency. Stabilising and precisely controlling droplet trajectories in an FRM is challenging due to the unstable nature of the extensional flow with a saddle point. Extending the work of Vona & Lauga (Phys. Rev. E, vol. 104(5), 2021, p. 055108), this study applies deep reinforcement learning (DRL) to effectively guide a displaced droplet to the centre of the FRM. Through direct numerical simulations, we explore the applicability of DRL in controlling FRM flow with moderate inertial effects, i.e. Reynolds number $\sim \mathcal{O}(1)$, a nonlinear regime previously unexplored. The FRM’s geometric symmetry allows control policies trained in one of the eight sub-quadrants to be extended to the entire domain, reducing training costs. Our results indicate that the DRL-based control method can successfully guide a displaced droplet to the target centre with robust performance across various starting positions, even from substantially far distances. The work also highlights potential directions for future research, particularly focusing on efficiently addressing the delay effects in flow response caused by inertia. This study presents new advances in controlling droplet trajectories in more nonlinear and complex situations, with potential applications to other nonlinear flows. The geometric symmetry used in this cutting-edge reinforcement learning approach can also be applied to other control methods.

Numerous studies showed that the flow and transport phenomena in angstrom channels are different from existing understandings. In this work, we investigate the electrokinetic phenomena in a charged angstrom channel, including homogeneous and heterogeneous charge distributions at the wall to mimic the charging mechanisms of electrified metal-like surfaces and deprotonated dielectric surfaces, respectively. Our results show that both the streaming current and the flow velocity linearly increase as the applied pressure increases in a homogeneously charged system. However, in a heterogeneously charged system, the streaming current is activated only when the applied pressure exceeds a critical threshold. This behaviour arises from the strong Coulomb interactions between counterions and the surface charge, manifesting as an obvious nonlinear feature. The dissociation of counterions from the surface charge may not only cause pressure-dependent streaming conductance but also reduce the friction coefficient of the system, thus the flow resistance, when the system friction is governed by the bound ions. We found that such pressure-dependent streaming conductance gradually weakens as the channel size increases and reaches the regime of classical nanofluidic theories. Taking one-dimensional non-equilibrium statistics and Markov chains for the sequence evolution of bound-ion dissociation, our theory can well explain the pressure-dependent streaming conductance and water permeability in angstrom charged channels. Voltage-driven nonlinear ionic transport and electro-osmosis were also observed in heterogeneously charged systems. Our findings will be helpful for understanding the ionic transport in angstrom-scale channels and possibly useful in ion separations.

Flow over bluff bodies encounters instability at supercritical Reynolds numbers, exhibiting the periodic vortex shedding that leads to structural vibrations and acoustic noise. In this paper, a new aerodynamic shape optimisation strategy based on resolvent analysis is proposed to passively control the vortex shedding over two-dimensional cylinders. Firstly, we show that when the flow satisfies the rank-1 approximation, minimizing the maximal resolvent gain enhances flow stability. Secondly, we formulate the geometry-constrained resolvent-based optimisation problem that can be solved by the nonlinear conjugate gradient algorithm. Compared with conventional stability-based optimisation, the proposed approach is more effective as it avoids the cumbersome eigendecomposition of the high-dimensional Jacobian matrix. The efficacy of the proposed resolvent-based optimisation is validated through improving the stability of the one-dimensional Ginzburg–Landau equation. Thirdly, this approach is applied to suppress the vortex shedding of bluff bodies, initialised by a circular cylinder. Although the optimisation is performed at a subcritical state $Re = 40$, reduced vortex shedding and drag forces can be achieved at supercritical Reynolds numbers, while the critical Reynolds number is extended from $47$ to $60$. Dynamic mode decomposition is then performed to reveal that the optimised system becomes more stable and satisfies the rank-1 approximation. Finally, we demonstrate that the combined effects of the flattened surface and the Coanda effect delay flow separation, keeping the separation point nearly unchanged at supercritical Reynolds numbers (e.g. between 80 and 140) for the optimised geometry. This results in a substantial reduction in the strength of vortex shedding, which in turn leads to decreased drag forces. The optimised shape still achieves drag reduction in turbulent flows at a relatively high Reynolds number.

We examined theoretically, experimentally and numerically the origin of the acoustothermal effect using a standing surface acoustic wave-actuated sessile water droplet system. Despite a wealth of experimental studies and a few recent theoretical explorations, a profound understanding of the acoustothermal mechanism remains elusive. This study bridges the existing knowledge gap by pinpointing the fundamental causes of acoustothermal heating. Theory broadly applicable to any acoustofluidic system at arbitrary Reynolds numbers, going beyond the regular perturbation analysis, is presented. Relevant parameters responsible for the phenomenon are identified and an exact closed-form expression delineating the underlining mechanism is presented. We also examined the impact of viscosity on acoustothermal phenomena by modelling temperature profiles in sessile glycerol–water droplets, underscoring its crucial role in modulating the acoustic field and shaping the resulting acoustothermal profile. Furthermore, an analogy between the acoustothermal effect and the electromagnetic heating is drawn, thereby deepening the understanding of the acoustothermal process.

A spherical capsule (radius $R$) is suspended in a viscous liquid (viscosity $\mu$) and exposed to a uniaxial extensional flow of strain rate $E$. The elasticity of the membrane surrounding the capsule is described by the Skalak constitutive law, expressed in terms of a surface shear modulus $G$ and an area dilatation modulus $K$. Dimensional arguments imply that the slenderness $\epsilon$ of the deformed capsule depends only upon $K/G$ and the elastic capillary number ${Ca}=\mu R E/G$. We address the coupled flow–deformation problem in the limit of strong flow, ${Ca}\gg 1$, where large deformation allows for the use of approximation methods in the limit $\epsilon \ll 1$. The key conceptual challenge, encountered at the very formulation of the problem, is in describing the Lagrangian mapping from the spherical reference state in a manner compatible with hydrodynamic slender-body formulation. Scaling analysis reveals that $\epsilon$ is proportional to ${Ca}^{-2/3}$, with the hydrodynamic problem introducing a dependence of the proportionality prefactor upon $\ln \epsilon$. Going beyond scaling arguments, we employ asymptotic methods to obtain a reduced formulation, consisting of a differential equation governing a mapping field and an integral equation governing the axial tension distribution. The leading-order deformation is independent of the ratio $K/G$; in particular, we find the approximation $\epsilon ^{2/3} {Ca}\approx 0.2753\ln (2/\epsilon ^2)$ for the relation between $\epsilon$ and $Ca$. A scaling analysis for the neo-Hookean constitutive law reveals the impossibility of a steady slender shape, in agreement with existing numerical simulations. More generally, the present asymptotic paradigm allows us to rigorously discriminate between strain-softening and strain-hardening models.

Noise source identification has been a long-standing challenge for decades. Although it is known that sound sources are closely related to flow structures, the underlying physical mechanisms remain controversial. This study develops a sound source identification method based on longitudinal and transverse process decomposition (LTD). Large-eddy simulations were performed on the flow around a cylinder at a Reynolds number of 3900. Using the new LTD method, sound sources in the cylinder flow were identified, and the mechanisms linking flow structures with noise generation were discussed in detail. Identifying the physical sound sources from two levels, low-order theory and high-order theory, the physical mechanism of wall sound sources was also analysed. Results indicate that the sound sources in the flow field mainly come from the leading edge, shear layer and wake region of the cylinder. The high-order theory reveals that sound sources are correlated with the spatio-temporal evolution of enstrophy, vortex stretching and surface deformation processes, this reflecting the coupling between transversal and longitudinal flow fields. The boundary thermodynamic flux and boundary dilatation flux distribution of the cylinder were analysed. Results indicate that the wall sound sources mainly come from the separation point and have a disorderly distribution on the leeward side of the cylinder, which is the main region where longitudinal variables enter the fluid from the wall surface, and the wall sound source is related to the boundary enstrophy flux.

A ground vortex engendered by the interaction of uniform flow over a plane surface with suction into a cylindrical conduit whose axis is normal to the cross-flow and parallel to the ground plane is investigated in wind tunnel experiments. The formation and evolution of the columnar vortex and its ingestion into the conduit’s inlet are explored using planar/stereo particle image velocimetry over a broad range of formation parameters that include the speeds of the inlet and cross-flows and the cylinder’s elevation above the ground plane with specific emphasis on the role of the surface vorticity layer in the vortex initiation and sustainment. The present investigations show that the appearance of a ground vortex within the inlet face occurs above a threshold boundary of two dimensionless formation parameters, namely the inlet’s momentum flux coefficient and its normalised elevation above the ground surface. Transitory initiations of wall-normal columnar vortices are spawned within a countercurrent shear layer that forms over the ground plane within a streamwise domain on the inlet’s leeward side by the suction flow into the duct. At low suction speeds, these wall-normal vortices are advected downstream with the cross-flow but when their celerity is reversed with increased suction, they are advected towards the cylinder’s inlet, gain circulation and stretch along their centrelines and become ingested into the inlet at a threshold defined by the formation parameters. Finally, the present investigations demonstrated that reduction of the countercurrent shear within the wall vorticity layer by deliberate, partial bypass of the inlet face flow through the periphery of the cylindrical duct can significantly delay the ingestion of the ground vortex to higher level thresholds of the formation parameters.