The accurate quantification of wall-shear stress dynamics is of substantial importance for various applications in fundamental and applied research, spanning areas from human health to aircraft design and optimization. Despite significant progress in experimental measurement techniques and postprocessing algorithms, temporally resolved wall-shear stress fields with adequate spatial resolution and within a suitable spatial domain remain an elusive goal. Furthermore, there is a systematic lack of universal models that can accurately replicate the instantaneous wall-shear stress dynamics in numerical simulations of multiscale systems where direct numerical simulations (DNSs) are prohibitively expensive. To address these gaps, we introduce a deep learning architecture that ingests wall-parallel streamwise velocity fields at $y^+ \approx 3.9 \sqrt {Re_\tau }$ of turbulent wall-bounded flows and outputs the corresponding two-dimensional streamwise wall-shear stress fields with identical spatial resolution and domain size. From a physical perspective, our framework acts as a surrogate model encapsulating the various mechanisms through which highly energetic outer-layer flow structures influence the governing wall-shear stress dynamics. The network is trained in a supervised fashion on a unified dataset comprising DNSs of statistically one-dimensional turbulent channel and spatially developing turbulent boundary layer flows at friction Reynolds numbers ranging from $390$ to $1500$. We demonstrate a zero-shot applicability to experimental velocity fields obtained from particle image velocimetry measurements and verify the physical accuracy of the wall-shear stress estimates with synchronized wall-shear stress measurements using the micro-pillar shear-stress sensor for Reynolds numbers up to $2000$. In summary, the presented framework lays the groundwork for extracting inaccessible experimental wall-shear stress information from readily available velocity measurements and thus, facilitates advancements in a variety of experimental applications.

Time-dependent fluid dynamics plays a crucial role in both natural phenomena and industrial applications. Understanding the flow instabilities and transitions within these dynamical systems is essential for predicting and controlling their unsteady behaviour. A classic example of time-dependent flow is the Stokes layer. To study the transition mechanism in this flow, we employ the finite-time Lyapunov exponent (FTLE) to demonstrate that a linear energy amplification mechanism may explain the intracyclic instability in the transitional Stokes layer, supported by favourable comparisons with experimental measurements of axial turbulence intensity. This complements existing theories applied to the Stokes layer in the literature, including the Floquet analysis and the instantaneous/momentary analyses, which have struggled to capture this experimental observation accurately. The FTLE analysis is closely related to the transient growth analysis, formulated as an optimisation problem of the disturbance energy growth over time. We found that the energy amplification weakens as the finite Stokes layer becomes more confined, and the oscillating frequency has a non-monotonic effect on the maximum transient growth. Based on these results, we recommend future experimental studies to validate this linear mechanism.

An experimental study was conducted in the CICLoPE long-pipe facility to investigate the correlation between wall-pressure and turbulent velocity fluctuations in the logarithmic region, at high friction Reynolds numbers ($4794 \lesssim Re_\tau \lesssim 47\,015$). Hereby, we explore the scalability of employing wall-pressure to effectively estimate off-the-wall velocity states (e.g. to be of use in real-time control of wall-turbulence). Coherence spectra for wall-pressure and streamwise (or wall-normal) velocity fluctuations collapse when plotted against $\lambda _x/y$ and thus reveals a Reynolds-number-independent scaling with distance-from-the-wall. When the squared wall-pressure fluctuations are considered instead of the linear wall-pressure term, the coherence spectra for the wall-pressure-squared and velocity are higher in amplitude at wavelengths corresponding to large-scale streamwise velocity fluctuations (e.g. at $\lambda _x/y = 60$, the coherence value increases from roughly 0.1 up to 0.3). This higher coherence typifies a modulation effect, because low-frequency content is introduced when squaring the wall-pressure time series. Finally, quadratic stochastic estimation is employed to estimate turbulent velocity fluctuations from the wall-pressure time series only. For each $Re_\tau$ investigated, the estimated time series and a true temporal measurement of velocity inside the turbulent pipe flow yield a normalised correlation coefficient of $\rho \approx 0.6$ for all cases. This suggests that wall-pressure sensing can be employed for meaningful estimation of off-the-wall velocity fluctuations and thus for real-time control of energetic turbulent velocity fluctuations at high-$Re_\tau$ applications.

In the dynamical systems approach to turbulence, unstable periodic orbits (UPOs) provide valuable insights into system dynamics. Such UPOs are usually found by shooting-based Newton searches, where constructing sufficiently accurate initial guesses is difficult. A common technique for constructing initial guesses involves detecting recurrence events by comparing past and future flow states using their $L_2$-distance. An alternative method uses dynamic mode decomposition (DMD) to generate initial guesses based on dominant frequencies identified from a short time series, which are signatures of a nearby UPO. However, DMD struggles with continuous symmetries. To address this drawback, we combine symmetry-reduced DMD (SRDMD) introduced by Marensi et al. (2023, J. Fluid Mech., vol. 954, A10), with sparsity promotion. This combination provides optimal low-dimensional representations of the given time series as a time-periodic function, allowing any time instant along this function to serve as an initial guess for a Newton solver. We also discuss how multi-shooting methods operate on the reconstructed trajectories, and we extend the method to generate initial guesses for travelling waves. We demonstrate SRDMD as a method complementary to recurrent flow analysis by applying it to data obtained by direct numerical simulations of three-dimensional plane Poiseuille flow at the friction Reynolds number $\textit{Re}_\tau \approx51$ ($\textit{Re}=802$), explicitly taking a continuous shift symmetry in the streamwise direction into account. The resulting unstable relative periodic orbits cover relevant regions of the state space, highlighting their potential for describing the flow.

In typical atomic force microscopy (AFM) measurements, the AFM probe, mounted on a compliant cantilever, is brought into close proximity to the test substrate. At this range, interfacial attractive van der Waals (vdW) forces can deflect the cantilever by pulling the probe, often causing the probe to suddenly jump into contact with the substrate. For deformable substrates such as gels or bio-tissues, the attraction-induced substrate deformation can further reduce the gap beneath the probe, which can increase the vdW force and hence trigger jump-to-contact prematurely. Since soft gels and tissues are frequently tested in liquid environments, where surface tension and the approaching dynamics of the probe can significantly influence deformation behaviour, this study examines the statics and dynamics of jump-to-contact on elastic substrates incorporating the effect of solid surface tension. We first discuss the theoretical setting for the static problem, deriving perturbation solutions for limiting cases of small and large solid surface tension. Notably, even under conditions of large solid surface tension, elasticity remains critical, as far-field elastic forces are required to smooth surface deformations in a convergent manner. Recognising that practical experiments are inherently dynamic, we also analyse the role of hydrodynamic pressure, which can delay the premature jump-to-contact. Our analysis focuses on identifying the conditions under which dynamic effects are negligible, enabling the simple analytical solutions in the static problem to reliably interpret AFM experimental results.

When an evaporating water droplet is deposited on a thermally conductive substrate, the minimum temperature will be at the apex due to evaporative cooling. Consequently, density and surface tension gradients emerge within the droplet and at the droplet–gas interface, giving rise to competing flows from, respectively, the apex towards the contact line (thermal-buoyancy-driven flow) and the other way around (thermal Marangoni flow). In small droplets with diameter below the capillary length, the thermal Marangoni effects are expected to dominate over thermal buoyancy (‘thermal Rayleigh’) effects. However, contrary to these theoretical predictions, our experiments show mostly a dominant circulation from the apex towards the contact line, indicating a prevailing of thermal Rayleigh convection. Furthermore, our experiments often show an unexpected asymmetric flow that persisted for several minutes. We hypothesise that a tiny amount of contaminants, commonly encountered in experiments with water/air interfaces, act as surfactants and counteract the thermal surface tension gradients at the interface and thereby promote the dominance of Rayleigh convection. Our finite element numerical simulations demonstrate that under our specified experimental conditions, a mere 0.5 % reduction in the static surface tension caused by surfactants leads to a reversal in the flow direction, compared to the theoretical prediction without contaminants. Additionally, we investigate the linear stability of the axisymmetric solutions, revealing that the presence of surfactants also affects the axial symmetry of the flow.

In this paper, we study the receptivity of non-modal perturbations in hypersonic boundary layers over a blunt wedge subject to free stream vortical, entropy and acoustic perturbations. Due to the absence of the Mack-mode instability and the rather weak growth of the entropy-layer instability within the domain under consideration, the non-modal perturbation is considered as the dominant factor triggering laminar–turbulent transition. This is a highly intricate problem, given the complexities arising from the presence of the bow shock, the entropy layer and their interactions with oncoming disturbances. To tackle this challenge, we develop a highly efficient numerical tool, the shock-fitting harmonic linearised Navier–Stokes (SF-HLNS) approach, which offers a comprehensive investigation on the dependence of the receptivity efficiency on the nose bluntness and properties of the free stream forcing. The numerical findings suggest that the non-modal perturbations are more susceptible to free stream acoustic and entropy perturbations compared with the vortical perturbations, with the optimal spanwise length scale being comparable with the downstream boundary-layer thickness. Notably, as the nose bluntness increases, the receptivity to the acoustic and entropy perturbations intensifies, reflecting the transition reversal phenomenon observed experimentally in configurations with relatively large bluntness. In contrast, the receptivity to free stream vortical perturbations weakens with increasing bluntness. Additionally, through the SF-HLNS calculations, we examine the credibility of the optimal growth theory (OGT) on describing the evolution of non-modal perturbations. While the OGT is able to predict the overall streaky structure in the downstream region, its accuracy in predicting the early-stage evolution and the energy amplification proves to be unreliable. Given its high-efficiency and high-accuracy nature, the SF-HLNS approach shows great potential as a valuable tool for conducting future research on hypersonic blunt-body boundary-layer transition.

A dual scaling of the second-order scalar structure function $\overline {{(\delta \theta )}^2}$, i.e. a scaling based on the Batchelor–Kolmogorov scales $\theta _B$, $\eta$ and another based on $\theta '$, $L$, representative of the large-scale motion, is examined in the context of the transport equation for $\overline {{(\delta \theta )}^2}$. Direct numerical simulation data over a relatively wide range of the Taylor microscale Reynolds number $Re_\lambda$ and a Schmidt number of order 1 in statistically stationary homogeneous isotropic turbulence with a uniform mean scalar gradient are used. It is observed that as $Re_\lambda$ increases, a dual scaling appears to emerge, where the scaling based on $\theta '$, $L$ extends to increasingly smaller values of $r/L$, where $r$ is the separation associated with the increment $ {{\delta \theta }}$, while the scaling based on $\theta _B$, $\eta$ extends to increasingly larger values of $r/\eta$. This suggests that both scalings should eventually overlap over a range of scales as $Re_\lambda$ continues to increase. Further, it is shown that such a dual scaling leads to the power-law relation $\overline {{(\delta \theta )}^2} \sim r^{\zeta _2}$, where $\zeta _2=2/3$ in the overlap region. The use of an empirical model for the local slope of $\overline {{(\delta \theta )}^2}$ (i.e. $\zeta _2$) shows that a value of $Re_\lambda$ of order $10^4$ is required for the slope to first reach the value $2/3$. Clearly, values larger than $10^4$ will be required before a $r^{2/3}$ inertial range is established.

Oscillations of a heated solid surface in an oncoming fluid flow can increase heat transfer from the solid to the fluid. Previous studies have investigated the resulting heat transfer enhancement for the case of a circular cylinder undergoing translational or rotational motions. Another common geometry, the flat plate, has not been studied as thoroughly. The flat plate sheds larger and stronger vortices that are sensitive to the plate’s direction of oscillation. To study the effect of these vortices on heat transfer enhancement, we conduct two-dimensional numerical simulations to compute the heat transfer from a flat plate with different orientations and oscillation directions in an oncoming flow with Reynolds number 100. We consider plates with fixed temperature and fixed heat flux, and find large heat transfer enhancement in both cases. We investigate the effects of the plate orientation angle and the plate oscillation direction, velocity, amplitude and frequency, and find that the plate oscillation velocity and direction have the strongest effects on global heat transfer. The other parameters mainly affect the local heat transfer distributions through shed vorticity distributions. We also discuss the input power needed for the oscillating-plate system and the resulting Pareto optimal cases.

Active wake control (AWC) has emerged as a promising strategy for enhancing wind turbine wake recovery, but accurately modelling its underlying fluid mechanisms remains challenging. This study presents a computationally efficient wake model that provides end-to-end prediction capability from rotor actuation to wake recovery enhancement by capturing the coupled dynamics of wake meandering and mean flow modification, requiring only two inputs: a reference wake without control and a user-defined AWC strategy. The model combines physics-based resolvent modelling for large-scale coherent structures and an eddy viscosity modelling for small-scale turbulence. A Reynolds stress model is introduced to account for the influence of both coherent and incoherent wake fluctuations, so that the time-averaged wake recovery enhanced by the AWC can be quantitatively predicted. Validation against large-eddy simulations (LES) across various AWC approaches and actuating frequencies demonstrates the model’s predictive capability, accurately capturing AWC-specific and frequency-dependent mean wake recovery with less than 8 % error from LES while reducing computational time from thousands of central-processing-unit hours to minutes. The efficiency and accuracy of the model makes it a promising tool for practical AWC design and optimization of large-scale wind farms.

Though ubiquitous in many engineering applications, including drug delivery, the compound droplet hydrodynamics in confined geometries have been barely surveyed. For the first time, this study thoroughly investigates the hydrodynamics of a ferrofluid compound droplet (FCD) during its migration in a microchannel under the presence of a pressure-driven flow and a uniform external magnetic field (UEMF) to manipulate its morphology and retard its breakup. Finite difference and phase-field multiple-relaxation time lattice Boltzmann approaches are coupled to determine the magnetic field and ternary flow system, respectively. First, the influence of the magnetic Bond number (${Bo}_m$) on the FCD morphology is explored depending on whether the core or shell is ferrofluid when the UEMF is applied along $\alpha =0^\circ$ and $\alpha =90^\circ$ relative to the fluid flow. It is ascertained that imposing the UEMF at $\alpha =0^\circ$ when the shell is ferrofluid can postpone the breakup. Intriguingly, when the core is ferrofluid, strengthening the UEMF enlarges the shell deformation. Afterwards, the effects of the capillary number (${Ca}$), density ratio, viscosity ratio, radius ratio and surface tension coefficients are scrutinised on the FCD deformation and breakup. The results indicate that augmenting the core-to-shell viscosity and density ratios accelerates the breakup process. Additionally, surface tension between the core and shell suppresses the core deformation. Moreover, increasing the ${Ca}$ intensifies the viscous drag force exerted on the shell, flattening its rear side, which causes a triangular-like configuration. Ultimately, by varying ${Bo}_m$ and ${Ca}$, five distinct regimes are observed, whose regime map is established.

This paper presents a novel machine learning framework for reconstructing low-order gust-encounter flow field and lift coefficients from sparse, noisy surface pressure measurements. Our study thoroughly investigates the time-varying response of sensors to gust–airfoil interactions, uncovering valuable insights into optimal sensor placement. To address uncertainties in deep learning predictions, we implement probabilistic regression strategies to model both epistemic and aleatoric uncertainties. Epistemic uncertainty, reflecting the model’s confidence in its predictions, is modelled using Monte Carlo dropout – as an approximation to the variational inference in the Bayesian framework – treating the neural network as a stochastic entity. On the other hand, aleatoric uncertainty, arising from noisy input measurements, is captured via learned statistical parameters, and propagate measurement noise through the network into the final predictions. Our results showcase the efficacy of this dual uncertainty quantification strategy in accurately predicting aerodynamic behaviour under extreme conditions while maintaining computational efficiency, underscoring its potential to improve online sensor-based flow estimation in real-world applications.

We explored the dynamics of Taylor–Couette flows within square enclosures, focusing primarily on the turbulence regime and vortex behaviour at varying Reynolds numbers. Laboratory experiments were conducted using particle image velocimetry for Reynolds numbers $Re_{\varDelta }\in [0.23, 4.6]\times 10^3$ based on the minimum gap $\varDelta /d = 1/16$, $1/8$ and $1/4$, where $d$ is the cylinder diameter, or $Re\in [1.8, 9.8]\times 10^3$ based on $d/2$. At lower $Re$, the flow was dominated by well-defined Taylor and Görtler vortices, while higher $Re$ led to a turbulent state with distinct motions. Space–time radial velocity analysis revealed persistent Taylor vortices at lower $Re$, with larger gaps but increased turbulence, and irregular motions at higher $Re$, with smaller gaps. Velocity spectra reveal that the energy distribution is maintained at frequencies lower than the integral-type frequency $f_I$ across varying $\varDelta$ due to the dominance of large vortices. However, there is a monotonic increase in energy at higher frequencies beyond $f_I$. The reduced characteristic frequency $f_I\varDelta /\omega _ir_i \sim 1/10$ indicates that these motions scale linearly with angular velocity, and inversely with the gap. Proper orthogonal decomposition (POD) and spectral POD were used to distinguish between Taylor and Görtler vortices, showing the effects of gap size and the associated energy cascade. Linear stability analysis included as complementary support revealed primary instability of the Taylor vortex, which is similar to the circular enclosure, along with multiple corner modes that are unique to the geometry.

Long-duration and time-resolved particle image velocimetry measurements were conducted in rough-wall open channel flows (OCFs), with the friction Reynolds number ranging from 642 to 2034. The primary objective is to investigate the impacts of various turbulent motions at different scales on the mean wall-shear stress ($\langle \tau _w \rangle$). To achieve this aim, a physical decomposition of $\langle \tau _w \rangle$ was initially performed utilizing the double-averaged methodology proposed by Nikora et al. (2019 J. Fluid Mech. 872, 626–664). This method enabled the breakdown of $\langle \tau _w \rangle$ into three distinct constituents: viscous, turbulent and dispersive stress segments. The findings underscore the substantial roles that turbulent and dispersive stresses play, accounting for over 75 % and 9 % of $\langle \tau _w \rangle$, respectively. Subsequently, a scale decomposition was further applied to analyse the contributions of coherent motions at different scales to $\langle \tau _w \rangle$. Adopting typical cutoff streamwise wavelengths ($\lambda _x = 3h$ and $10h$), the contribution of large-scale motions (LSMs) and very large-scale motions (VLSMs) to the overall wall-shear stress was quantified. It was revealed that turbulent motions with $\lambda _x \gt 3h$ and $\lambda _x \gt 10h$ contribute more than 40 % and 18 % of $\langle \tau _w \rangle$, respectively. The scale decomposition of the wall-shear stress and the contribution from LSMs and VLSMs exhibit evident dependencies on the Reynolds number. The contribution of LSMs and VLSMs to $\langle \tau _w \rangle$ is lower in rough OCFs compared with those of smooth counterparts. Secondary currents induced by the rough wall are hypothesised to be responsible for the reduced strength of LSMs and VLSMs and decreases in their contribution to $\langle \tau _w \rangle$.