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.

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 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.

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.

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.

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.

Wind tunnel experiments are performed to investigate stall and reattachment transients for an aerofoil and wing model at low chord Reynolds numbers ($8\times 10^4\leqslant {{Re}}_c\leqslant 1\times 10^5$) where a laminar separation bubble (LSB) may form on the suction surface. Direct force measurements and particle image velocimetry (PIV) are employed simultaneously to characterise the transient aerodynamic loading and flow field development. The imposed changes in operating conditions leading to stall and reattachment include changes in angle of attack at multiple pitch rates and changes in Reynolds number. The evolution of the lift coefficient is consistent with dynamic stall at higher Reynolds numbers, with a reduction in time delay between the passing of the static stall condition and the loss of lift for increasing pitch rate. During an increase in angle of attack, the separation bubble moves upstream prior to rapidly bursting, whereas for a decrease of Reynolds number, the LSB undergoes a more gradual monotonic increase in length prior to bursting. In contrast to notable differences in the aerodynamic loading and flow field development for different types of transients leading to LSB bursting, the process of LSB formation is less sensitive to the type of imposed change in operating conditions. Spanwise PIV measurements on the aerofoil and wing models indicate that the spanwise flow development is also insensitive to the type of imposed transient during LSB bursting and formation.

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.

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.

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.