Recently, Nagib et al. (Phys. Fluids, vol. 36, no. 7, 2024, 075145) used indicator functions of streamwise normal stress profiles to identify the valid wall-distance and Reynolds number ranges for two models in direct numerical sumulation (DNS) of channel and pipe flows. Since such functions are challenging to construct from experimental data, we propose a simpler, more robust method better suited to experiments. Applied to the two leading models – logarithmic and power-law – for normal stresses in the ‘fitting region’ of wall-bounded flows, this method is tested on prominent experimental data sets in zero-pressure-gradient (ZPG) boundary layers and pipe flows across a wide Reynolds number range ($Re_\tau$). Valid regions for the models appear only for $Re_\tau \gtrapprox 10{\,}000$, with a lower bound $y^+_{in} \sim (Re_\tau )^{0.5}$ and $y^+_{in} \gtrapprox 400$. The upper bound is a fixed fraction of the boundary layer thickness or pipe radius, independent of $Re_\tau$. The power-law model is found to hold over a broader range, up to $Y \approx 0.4$ in ZPG and $Y \approx 0.5$ in pipe flows, compared with the logarithmic trend, which is formulated to be coincident with the classical logarithmic region for the mean flow ($Y \lessapprox 0.15$). A slightly higher exponent ($0.28$) than that of Chen & Sreenivasan (J. Fluid Mech. vol. 933, 2022, A20; J. Fluid Mech. vol. 976, 2023, A21) extends the power-law model’s validity and correcting for outer intermittency in ZPG flows further broadens it. Projections to the near-wall region of both models yield nearly identical predictions of near-wall peak stress across the highest available $Re_\tau$. These findings, alongside results from Monkewitz & Nagib (J. Fluid Mech. vol. 967, 2023, A15) and Baxerras et al. (J. Fluid Mech. vol. 987, 2024, R8), highlight the importance of nonlinear eddy growth and residual viscous effects in wall-bounded flow modelling, informing potential refinements to the logarithmic model, such as those proposed by Deshpande et al. (J. Fluid Mech. vol. 914, 2021, A5).

Rayleigh–Taylor instability (RTI) caused by rarefaction waves not only features variable acceleration but also incorporates time-dependent density, which introduces great challenges in predicting the finger growth behaviours. In this work, we propose a model for predicting the single-mode finger behaviours by extending the Layzer potential-flow framework to account for time-dependent acceleration and density. Relative to the previous models, the present model can evaluate the effect of time-dependent density on finger growth, and can describe the growth behaviours of both bubbles and spikes in rarefaction-driven RTI flows. In addition, the time-dependent curvature of the finger tip as it evolves from its initial value to the quasi-steady value is quantified. To validate the model, rarefaction-tube experiments and numerical simulations are conducted across a wide range of initial conditions. The results show that the present model can accurately capture the amplitude growth and curvature evolution of bubbles and spikes across various density ratios. Moreover, both the present model and experiments demonstrate that the continuous density reduction in rarefaction-driven flows causes larger asymptotic velocities of bubbles and spikes, leading to higher Froude numbers relative to those under constant or time-dependent acceleration.

This study uses the diffusion analogy (Miyake, Sci. Rep., 5R-6, 1965, Univ. of Washington, Seattle, USA) to predict the full growth behaviour of internal boundary layers (IBLs) induced by a roughness change for neutrally – and especially stably – stratified boundary layers with finite thickness. The physics of the diffusion analogy shows that the streamwise variation of the IBL thickness is dictated by $\sigma _w/U$ at the interface, where $\sigma _w$ and $U$ represent wall-normal Reynolds stress and mean streamwise velocity, respectively. The existing variants of the model, summarised by Savelyev & Taylor (2005, Boundary-Layer Meteorol., vol. 115, pp. 1–25), are tailored to IBLs confined within the constant shear stress layer. To extend the applicability of the model to the outer region, we investigate the relation between $\sigma _w/U$ and $U/U_\infty$ in the outer region across varying stratification, where $U_\infty$ is the free-stream velocity. Our analysis reveals that wind tunnel data from a number of facilities collapse onto a master curve when $\sigma _w/U$ is premultiplied by a height-independent parameter, which is a function of the ratio of Monin–Obukhov length to the boundary layer thickness. The scaled $\sigma _w/U$ decreases inversely with $U/U_\infty$ in the surface layer, transitioning to a linear decrease as $U/U_\infty$ increases. The new model, which integrates these findings, along with the effects of streamline displacement and acceleration, captures the complete characteristics of IBLs as they develop within turbulent boundary layers of finite thickness.

Turbulent flows exhibit large intermittent fluctuations from inertial to dissipative scales, characterised by multifractal statistics and breaking the statistical self-similarity. It has recently been proposed that the Navier–Stokes turbulence restores a hidden form of scale invariance in the inertial interval when formulated for a dynamically (nonlinearly) rescaled quasi-Lagrangian velocity field. Here we show that such hidden self-similarity extends to the large-eddy-simulation (LES) approach in computational fluid dynamics (CFD). In particular, we show that classical subgrid-scale models, such as implicit or explicit Smagorinsky closures, respect the hidden scale invariance at all scales – both resolved and subgrid. In the inertial range, they reproduce the hidden scale invariance of Navier–Stokes statistics. These properties are verified very accurately by numerical simulations and, beyond CFD, turn LES into a valuable tool for fundamental turbulence research.

We present a theoretical framework for porous media gravity currents propagating over rigid curvilinear surfaces. By reducing the flow dynamics to low-dimensional models applicable on surfaces where curvature effects are negligible, we demonstrate that, for finite-volume releases, the flow behaviour in both two-dimensional and axisymmetric configurations is primarily governed by the ratio of the released viscous fluid volume to the characteristic volume of the curvilinear surface. Our theoretical predictions are validated using computational fluid dynamics simulations based on a sharp-interface model for macroscopic flow in porous media. In the context of carbon dioxide geological sequestration, our findings suggest that wavy cap rock geometries can enhance trapping capacity compared with traditional flat-surface assumptions, highlighting the importance of incorporating realistic topographic features into subsurface flow models.

In this work, we study the effectiveness of the time-localised principal resolvent forcing mode at actuating the near wall cycle of turbulence. This mode is restricted to a wavelet pulse and computed from a singular value decomposition of the windowed wavelet-based resolvent operator (Ballouz et al. 2024b, J. Fluid Mech. vol. 999, A53) such that it produces the largest amplification via the linearised Navier–Stokes equations. We then inject this time-localised mode into the turbulent minimal flow unit at different intensities, and measure the deviation of the system’s response from the optimal resolvent response mode. Using the most energetic spatial wavenumbers for the minimal flow unit – i.e. constant in the streamwise direction and once-periodic in the spanwise direction – the forcing mode takes the shape of streamwise rolls and produces a response mode in the form of streamwise streaks that transiently grow and decay. Though other works such as Bae et al. (2021 J. Fluid Mech. vol. 914, A3) demonstrate the importance of principal resolvent forcing modes to buffer layer turbulence, none instantaneously track their time-dependent interaction with the turbulence, which is made possible by their formulation in a wavelet basis. For initial times and close to the wall, the turbulent minimal flow unit matches the principal response mode well, but due to nonlinear effects, the response across all forcing intensities decays prematurely with a higher forcing intensity leading to faster energy decay. Nevertheless, the principal resolvent forcing mode does lead to significant energy amplification and is more effective than a randomly generated forcing structure and the second suboptimal resolvent forcing mode at amplifying the near-wall streaks. We compute the nonlinear energy transfer to secondary modes and observe that the breakdown of the actuated mode proceeds similarly across all forcing intensities: in the near-wall region, the induced streak forks into a structure twice-periodic in the spanwise direction; in the outer region, the streak breaks up into a structure that is once-periodic in the streamwise direction. In both regions, spanwise oscillations account for the dominant share of nonlinear energy transfer.

The breakup dynamics of viscous liquid bridges on solid surfaces is studied experimentally. It is found that the dynamics bears similarities to the breakup of free liquid bridges in the viscous regime. Nevertheless, the dynamics is significantly influenced by the wettability of the solid substrate. Therefore, it is essential to take into account the interaction between the solid and the liquid, especially at the three-phase contact line. It is shown that when the breakup velocity is low and the solid surface is hydrophobic, the dominant channel of energy dissipation is likely due to thermally activated jumping of molecules, as described by the molecular kinetic theory. Nevertheless, the viscous dissipation in the bulk due to axial flow along the bridge can be of importance for long bridges. In view of this, a scaling relation for the time dependence of the minimum width of the liquid bridge is derived. For high viscosities, the scaling relation captures the time evolution of the minimum width very well. Furthermore, it is found that external geometrical constraints alter the dynamic behaviour of low and high viscosity liquid bridges in a different fashion. This discrepancy is explained by considering the dominant forces in each regime. Lastly, the morphology of the satellite droplets deposited on the surface is qualitatively compared with that of free liquid bridges.

A model is proposed for the one-dimensional spectrum and streamwise Reynolds stress in pipe flow for arbitrarily large Reynolds numbers. Constructed in wavenumber space, the model comprises four principal contributions to the spectrum: streaks, large-scale motions, very-large-scale motions and incoherent turbulence. It accounts for the broad and overlapping spectral content of these contributions from different eddy types. The model reproduces well the broad structure of the premultiplied one-dimensional spectrum of the streamwise velocity, although the bimodal shape that has been observed at certain wall-normal locations, and the $-5/3$ slope of the inertial subrange, are not captured effectively because of the simplifications made within the model. Regardless, the Reynolds stress distribution is well reproduced, even within the near-wall region, including key features of wall-bounded flows such as the Reynolds number dependence of the inner peak, the formation of a logarithmic region, and the formation of an outer peak. These findings suggest that many of these features arise from the overlap of energy content produced by both inner- and outer-scaled eddy structures combined with the viscous-scaled influence of the wall. The model is also used to compare with canonical turbulent boundary layer and channel flows, and despite some differences being apparent, we speculate that with only minor modifications to its coefficients, the model can be adapted to these flows as well.

Yaw control can effectively enhance wind farm power output, but the vorticity distribution and coherent structures in yawed turbine wakes remain poorly understood. We propose a physical model capable of accurately predicting tip vortex dynamics from their generation to destabilisation. This model integrates a point vortex framework with advanced blade element momentum theory and vortex cylinder theory for yawed turbines. Comparisons with large eddy simulations demonstrate that the model effectively predicts the vorticity distribution of tip vortices and the wake profile of yawed turbines. Finally, we employ sparsity-promoting dynamic mode decomposition to analyse the dynamics of the far wake. Our analysis reveals four primary mode types: (i) the averaged mode; (ii) shear modes; (iii) harmonic modes; and (iv) merging modes. Under yawed conditions, these modes become asymmetric, leading to interactions between the tip and root vortex modes. This direct interaction plays a critical role during the formation process of the counter-rotating vortex pair observed in yawed wakes.

Linear hydroelastic waves propagating in a frozen channel are investigated. The channel has a rectangular cross-section, finite depth and infinite length. The liquid in the channel is an inviscid, incompressible liquid and covered with ice. The ice is modelled as a thin elastic plate of variable thickness clamped to the channel walls. The thickness is constant along the channel length and varying across it. The flow induced by ice deflections is potential. The problem reduced to a problem of wave profiles across the channel and was solved using a piecewise linear approximation of a shape of the thickness. Normal modes are calculated to ensure continuous deflections, slopes, bending stresses and shear forces in the ice plate. Two thickness distributions are studied: in Case I, the thickness is constant at a middle segment and linearly increases at edge segments over the channel’s width; in Case II, the thickness linearly decreases at the edge segments. In Case I, there is one segment with the thin part of the ice cover where, as expected, the oscillations of the ice plate will be concentrated. In Case II, there are two such areas, separated by the middle segment with the thick part of the ice cover. Dispersion relations, phase and group velocities, wave profiles and strain distributions in the ice plate are studied. Results show that the properties of periodic hydroelastic waves are significantly influenced by the ice thickness distribution across the channel.

We investigate the effect of streamwise and transverse rotation on the wake behind an elastically mounted sphere. Simulations are performed at a Reynolds number $Re=500$ over a range of reduced velocity $2\le U^{\ast }\le 12$, considering a low and high rotational speed (0.2 and 1), keeping the mass ratio $m^{\ast }=2$. Streamwise rotation yields a structural response akin to the non-rotating case, while transverse rotation triggers induced vibration at lower $U^{\ast }$ and sustains it across a wider range. Like the non-rotating case, the streamwise rotating sphere exhibits synchronous, high-amplitude vibration across the entire $U^\ast$ range, whereas for low transverse rotation, it is confined to $5\le U^{\ast }\le 6$. Cross-stream displacement of the sphere remains unaffected by streamwise rotation with increasing $U^{\ast }$. In contrast, it monotonically increases due to transverse rotation, driven by the Magnus force, as supported by our theoretical and numerical estimations. While the spiral shedding mode dominates at $\Omega _{x}=0.2$, twisted hairpin and twisted spiral modes emerge as the rotation rate is increased. On the other hand, we observe the hairpin (HP) mode, as seen in the non-rotating case, for low transverse rotation. The HP mode gives rise to the ring vortical mode at the far wake, and with an increase in $U^\ast$, the wake shows small-scale stretched threads and reconnected bridgelets. Wake fluctuations increase with a streamwise rotation that saturates at higher $U^{\ast }$ during synchronisation, while desynchronisation at dimensionless transverse rotation rate $\Omega _{z}=1$ induces intermittent low-amplitude vibration via the Magnus effect. Space–time reconstruction at the near wake shows an undisturbed helical vortex core at $\Omega _{x}=0.2$ and $U^{\ast }=5$, which bifurcates at $\Omega _{x}=1$ owing to the centrifugal-induced distortion. At $\Omega _{x}=1$ and $U^{\ast }=5$, the phase difference between $(y, C_{y})$ and $(z, C_{z})$ exhibits in-phase synchrony with occasional phase slips. The wake vortex remains unaffected by the transverse rotation of the sphere; however, a streamwise rotating sphere couples the wake, leading to a rotational lock-in. The wake rotation shifts from anti-clockwise to clockwise sense earlier (in $U^\ast$) at a lower rotation rate. The reduced velocity is seen to have a favourable effect on the transfer of the sphere’s rotational inertia onto the wake as the measured penetration depth increases with $U^{\ast }$. Insights from the present research will aid in understanding complex flow interactions in rotational systems, improving efficiency, stability and control in modern engineering applications.

Phoretic particles are often used as a simple model for experimental and theoretical studies of active matter. We develop a computational framework to resolve hydrodynamic and chemical interactions of multiple self-propelling phoretic particles suspended in two-dimensional Stokes flow. The proposed method is precise enough to resolve correctly the subtle transitions between different modes of spontaneous locomotion for a single particle, and fast and versatile enough to study multiparticle dynamics in periodic or confined domains. The particles are modelled as chemically active rigid circles, which can emit or absorb a solute into surrounding fluid. The interaction between particles and solute induces a slip flow on particle surfaces, and the solute is advected by the fluid flow and diffuses with a constant diffusivity. A fast boundary integral method is proposed to solve fluid–structure interaction in Stokes flow. Acceleration of this method is provided by splitting the velocity field due to a set of point forces into a short-range part with singularity and a long-range part which is sufficiently smooth, thanks to an Ewald-like decomposition. An overlapping mesh method is employed for advection–diffusion of the solute with moving boundaries. The idea is to decompose the computational domain into several overlapping subdomains, and body-fitted meshes are used to ensure sharp resolution of boundary conditions. The framework is validated separately for the Stokes problem and the advection–diffusion problem, reaching relatively high order of accuracy. We apply our framework to several practical problems, such as a single particle in a channel and particle suspensions, showing rich sets of behaviours.

The relationship between salinity-driven (SD) and particle-driven (PD) gravity currents has long been a focal point of geophysical research. This study investigates salinity–particle dual-driven gravity currents using a direct numerical simulation discrete element method. The transition regime from SD to PD currents is explored. The results show that the transition is related to interfacial instability and material transport dynamics. During this transition, the enhancement of particle sedimentation weakens the interfacial stratification and heightens its susceptibility to shear instability. Consequently, the instability generates a series of billows that encourages fluid dilution, further amplifying the particle sedimentation effect. The transition regime is closely associated with this positive closed-loop feedback mechanism. It supplies sufficient energy at the slumping stage to maintain the front velocity of particle-dominated currents comparable to that of salinity-dominated currents. The interfacial vortices will expand spatially by the centrifugal forces on the particles, leading to a reduction in detrainment.

Fourier analysis is the standard tool of choice for quantifying the distribution of kinetic energy amongst the eddies in a turbulent flow. The resulting spectral energy-density function is the well-known energy spectrum. And yet, because eddies are distinct from waves, alternative approaches to finding energy-density functions have long been sought. Townsend (1976) outlined a promising approach to finding a spatial energy-density function, $V\!(r)$, where $r$ is the eddy size. Notably, this approach led to two distinct and mutually inconsistent formulations of $V\!(r)$ in homogeneous, isotropic turbulence. We revisit Townsend’s proposal and derive the corresponding three-dimensional $V\!(r)$ as well as introduce its one-dimensional variants (which, to our knowledge, have not been explicitly discussed before). By training our focus on the associated dimensionality of the function, we resolve the discrepancies between the previous formulations. Additionally, we generalise our analysis to include anisotropic flows. Finally, by means of concrete examples, we illustrate how one-dimensional spatial energy-density functions are useful for analysing empirical data. Some notable findings include new insights into the $k_1^{-1}$ scaling (where $k_1$ is the streamwise wavenumber) and a possible resolution of the enigmatic sizes of organised motions at large scales.

Shock interactions on a V-shaped blunt leading edge (VBLE) that are commonly encountered at the cowl lip of an inward-turning inlet are investigated at freestream Mach numbers ($ M_\infty$) 3–6. The swept blunt leading edges of the VBLE generate a pair of detached shocks with varying shapes due to the changes in $ M_\infty$ and $L/r$ (i.e. the ratio of the leading-edge length $L$ to the leading-edge blunt radius $r$), which causes intriguing shock interactions at the crotch of the VBLE. Three subtypes of regular reflection (RR) and a Mach reflection (MR) are produced successively with increasing $ M_\infty$ for a given $L/r$, which appear in the opposite order to those with increasing $L/r$ for a given $ M_\infty$. These shock interactions identified in numerical simulations are verified in supersonic and hypersonic wind tunnel experiments. It is demonstrated that the relative position of the shocks is crucial in determining the transitions of shock interactions by varying either $L/r$ or $ M_\infty$. Transition criteria between subtypes of RR and from RR to MR are theoretically established in the parameter space $(M_\infty,L/r)$ by analysing the shock structures, showing good agreement with the numerical and experimental results. Interactions between either immature or fully developed detached shocks are embedded in these criteria. Specifically, the transition criteria asymptotically approach the corresponding critical $ M_\infty$ when $L/r$ is sufficiently large. These transition criteria provide guidelines for improving the design of the cowl lip of an inward-turning inlet in supersonic and hypersonic regimes.

Experiments were performed that (i) document the effect of the steady spanwise buffer layer blowing on the mean characteristics of the turbulent boundary layer for a range of momentum thickness Reynolds numbers from 4760 to 10 386, and (ii) document the effect of the buffer layer blowing on the unsteady characteristics and coherent vorticity in a boundary layer designed to provide sufficiently high spatial resolution. The spanwise buffer layer blowing of the order of $u_{\tau }$ is produced by a surface array of pulsating direct current (pulsed-DC) plasma actuators. This was found to substantially reduce the wall shear stress that was directly measured with a floating element coupled with a force sensor. The direct wall shear measurements agreed with values derived using the Clauser method to within $\pm 0.85$ %. The degree to which the buffer layer blowing affected $\tau _w$ was found to primarily depend on the inner variable spanwise spacing between the pulsed-DC actuator electrodes, i.e. ‘blowing sites’. Utilizing pairs of $[u,v]$ and $[u,w]$ hot-wire sensors, the latter experiments correlated significant reductions in the $\omega _y$ and $\omega _x$ vorticity components that resulted from the buffer layer blowing and translated into lower Reynolds stresses and turbulence production. The time scale to which these observed changes in the boundary layer characteristics would return to the baseline condition was subsequently documented. This revealed a recovery length of $x^+ \approx 86\,000$ that translated to a streamwise fetch of $x \approx 66\delta$. Finally, a comparison with the recent work by Cheng et al. (2021, J. Fluid Mech. vol. 918, A24) and Wei & Zhou (2024 in TSFP13, June 25–28, 2024) that followed our experimental approach to achieve comparable wall shear stress (drag) reductions has led to a new scaling based on the baseline boundary layer $\textit{Re}_{\tau }$ and buffer layer blowing velocity.

The Taylor–Melcher leaky dielectric (LD) model is often used to study the physics of electrosprays operating in the cone-jet mode. Despite its success, there are electrospraying conditions in which the ion concentration fields must be retained, which requires an electrokinetic model. This article reproduces cone-jets with two electrokinetic formulations: the standard Poisson–Nernst–Planck (PNP) equations, and a modified electrokinetic (MEK) model that accounts for overscreening and overcrowding of electrolytes, which is important in fluids with high electrical conductivities such as ionic liquids (Kilic et al. 2007 Phys. Rev. E vol. 75, no. 2, 021502, 021503; Bazant et al. 2011 Phys. Rev. Lett. vol. 106, no. 4, 46102). In the case of liquids with low electrical conductivities, it is observed that the LD and PNP models agree under certain limiting conditions, but they are less restrictive than previously proposed (Baygents & Saville 1990 AIP Conf. Proc. vol. 197, 7–17; Schnitzer & Yariv 2015 Fluid Mech. vol. 773, 1–33); the effects of dissimilar ion diffusivities are also investigated. In the case of liquids with high electrical conductivities, in particular ionic liquids, overscreening and overcrowding effects are important, resulting in significant differences between the solutions of the PNP, MEK and LD models. In particular, the electrokinetic models yield increased dissipation and self-heating, leading to higher temperature variations and currents, in agreement with measurements. Furthermore, the MEK formulation describes the ion concentration fields with higher fidelity than the PNP equations.

In this study, we demonstrate, for the first time, the existence of a short-wave instability in a Lamb–Oseen vortex subjected to a triangular strain field generated by three satellite vortices, which we term the triangular instability. We identify this instability by numerically integrating the linearised Navier–Stokes equations around a quasi-steady base flow to capture the most unstable mode and validate it by comparing results with theoretical predictions. We evaluate this instability by calculating the growth rates associated with the parametric resonant coupling of two Kelvin waves with the triangular strain field in the limit of small strain rate and large Reynolds number. Our analysis reveals that resonance occurs only for combinations of the azimuthal wavenumbers $m = 1$ and $m = - 2$ (or their symmetric counterparts with opposite signs). We observe several unstable modes with positive growth rates for a moderate viscous Reynolds number $10^4$ and straining parameter value $\epsilon = 0.008$, defined as the cube of the ratio of the core size to the distance from the satellite vortices. The most unstable mode, dominant at typically high Reynolds numbers, has $k \approx 5.18/a$ and $\omega \approx - 0.312\Omega$ (where $a$ and $\Omega$ denote the core size and central angular velocity). It exhibits negligible critical layer damping and remains the most unstable mode over a wide range of ${Re}$ and $\epsilon$. At lower Reynolds numbers, another mode with $k \approx 1.76/a$ and $\omega \approx - 0.407\Omega$, despite significant critical layer damping, becomes the most unstable.