Compressibility transformations have received considerable attention for extending well-established incompressible wall models to high-speed flows. While encouraging progress has been made in mean velocity scalings, research on temperature transformations has lagged behind. In this study, we rigorously derive a general framework for both velocity and temperature transformations directly from the compressible Reynolds-averaged Navier–Stokes (RANS) equations and their ‘incompressible’ counterparts, elucidating how these transformations guide the development of compressible algebraic RANS models in the inner layer. The introduction of the mixed Prandtl number further links the mean momentum and energy transport, facilitating the formulation of novel temperature transformations through integration with arbitrary mean velocity scalings, thereby unifying existing transformation methods while providing a systematic approach for further improvement. A detailed evaluation using direct numerical simulation databases of canonical compressible wall-bounded turbulent flows (CWBTFs) demonstrates that temperature transformations based on the Griffin–Fu–Moin and our recently proposed velocity scalings exhibit superior accuracy and robustness across a wide range of Reynolds and Mach numbers, as well as varying wall thermal boundary conditions. We also perform a preliminary investigation into the applicability of the proposed integral mean temperature–velocity relation and inverse temperature transformations for near-wall temperature modelling in cold-wall boundary layer flows, where discontinuities caused by non-monotonic temperature distributions are effectively avoided. Although the omission of higher-order terms in deriving the total heat flux equation enables closed-form wall modelling, it remains a key limitation to the model’s accuracy at the current stage. Future work may therefore need to address this issue to achieve further advances. These findings enhance the physical understanding of mean momentum and energy transport in canonical CWBTFs, and offer promising prospects for advancing near-wall temperature modelling within RANS and wall-modelled large eddy simulation frameworks.

To address the possible occurrence of a finite-time singularity during the oblique reconnection of two vortex rings, (Moffatt and Kimura 2019, J. Fluid Mech., vol. 870, R1) developed a simplified model based on the Biot–Savart law and claimed that the vorticity amplification $\omega _{{max}}/\omega _0$ becomes very large for vortex Reynolds number $Re_{\varGamma } \geqslant 4000$. However, with direct numerical simulations (DNS), Yao and Hussain (2020a, J. Fluid Mech.vol. 888, pp. R2) were able to show that the vorticity amplification is in fact much smaller and increases slowly with $Re_{\varGamma }$. This suppression of vorticity was linked to two key factors – deformation of the vortex core during approach, and formation of hairpin-like bridge structures. In this work, a recently developed numerical technique called log-lattice (Campolina & Mailybaev, 2021, Nonlinearity, vol. 34, 4684), where interacting Fourier modes are logarithmically sampled, is applied to the same oblique vortex ring interaction problem. It is shown that the log-lattice vortex reconnection displays core compression and formation of bridge structures, similar to the actual reconnection seen with DNS. Furthermore, the sparsity of the Fourier modes allows us to probe very large $Re_{\varGamma } = 10^8$ until which the peak of the maximum norm of vorticity, while increasing with $Re_{\varGamma }$, remains finite, and a blow-up is observed only for the inviscid case.

Turbulence is an out-of-equilibrium flow state that is characterised by non-zero net fluxes of kinetic energy between different scales of the flow. These fluxes play a crucial role in the formation of characteristic flow structures in many turbulent flows encountered in nature. However, measuring these energy fluxes in practical settings can present a challenge in systems other than the case of unrestricted turbulence in an idealised periodic box. Here, we focus on rotating Rayleigh–Bénard convection, being the canonical model system to study geophysical and astrophysical flows. Owing to the effect of rotation, this flow can yield a split cascade, where part of the energy is transported to smaller scales (direct cascade), while another fraction is transported to larger scales (inverse cascade). We compare two different techniques for measuring these energy fluxes throughout the domain: one based on a spatial filtering approach and an adapted Fourier-based method. We show how one can use these methods to measure the energy flux adequately in the anisotropic, aperiodic domains encountered in rotating convection, even in domains with spatial confinement. Our measurements reveal that in the studied regime, the bulk flow is dominated by the direct cascade, while significant inverse cascading action is observed most strongly near the top and bottom plates, due to the vortex merging of Ekman plumes into larger flow structures.

We join the theories that describe the orientation, treated as a tensor, of liquid crystals and the agitation of inelastic grains to obtain a mathematical model of non-spherical particles contained in a quasi-2D square box and driven into dissipative collisions through the vibration of two of the four flat walls, in the absence of gravity and mean flow. The particle agitation induces spatial inhomogeneities in the density and the isotropic–nematic transition to take place somewhere inside the box, if the particle shape is sufficiently far from spherical. We show quantitative agreement between the theory and discrete numerical simulations of ellipsoids of different length-to-diameter ratio. We need to fit two dimensionless parameters that were not previously available or determined in different configurations. These parameters, of order unity and weakly dependent on the shape of the particles, are indicative of the resistance to alignment distortion associated with entropic elasticity.

We investigate a short-wave instability mode recently identified via temporal stability analysis in weakly inclined falling liquid films sheared by a confined turbulent counter-current gas flow (Ishimura et al. J. Fluid Mech. vol. 971, 2023, p. A37). We perform spatio-temporal linear stability calculations based on the Navier–Stokes equations in the liquid film and the Reynolds-averaged Navier–Stokes equations in the gas, and compare these with our own experiments. We find that the short-wave instability mode is always upward-convective. The range of unstable group velocities is very wide and largely coincides with negative values of the wave velocity. Turbulence affects this mode both through the level of gas shear stress imparted and via the shape of the primary-flow gas velocity profile. Beyond a critical value of the counter-current gas flow rate, the short-wave mode merges with the long-wave Kapitza instability mode. The thus obtained merged mode is unstable for group velocities spanning from large negative to large positive values, i.e. it is absolute. The onset of the short-wave mode is precipitated by decreasing the channel height and inclination angle, and by increasing the liquid Reynolds number or the gas-to-liquid dynamic viscosity ratio. For vertically falling liquid films, merging occurs before the short-wave mode can become unstable on its own. Nonetheless, the ability to generate upward-travelling ripples is endowed to the merged mode. Preliminary calculations neglecting the linear perturbation of the turbulent viscosity suggest that three-dimensional perturbations could be more unstable than two-dimensional ones.

Recently, data-driven methods have shown great promise for discovering governing equations from simulation or experimental data. However, most existing approaches are limited to scalar equations, with few capable of identifying tensor relationships. In this work, we propose a general data-driven framework for identifying tensor equations, referred to as symbolic identification of tensor equations (SITE). The core idea of SITE – representing tensor equations using a host–plasmid structure – is inspired by the multidimensional gene expression programming approach. To improve the robustness of the evolutionary process, SITE adopts a genetic information retention strategy. Moreover, SITE introduces two key innovations beyond conventional evolutionary algorithms. First, it incorporates a dimensional homogeneity check to restrict the search space and eliminate physically invalid expressions. Second, it replaces traditional linear scaling with a tensor linear regression technique, greatly enhancing the efficiency of numerical coefficient optimization. We validate SITE using two benchmark scenarios, where it accurately recovers target equations from synthetic data, showing robustness to noise and flexible expressive capability. Furthermore, SITE is applied to identify constitutive relations directly from molecular simulation data, which are generated without reliance on macroscopic constitutive models. It adapts to both compressible and incompressible flow conditions and successfully identifies the corresponding macroscopic forms, highlighting its potential for data-driven discovery of tensor equation.

We present an acoustic characterisation of a model-scale wind turbine using large eddy simulation and the acoustic analogy. The analysis is representative of medium-sized turbines with low tip Mach number (${\sim} 0.10$). The fluid dynamic analysis revealed: a turbulent boundary layer over the blades, together with a trailing edge vortex sheet; a complex near-wake structure, including tip and root vortices; an intermediate wake with vortex instabilities triggering leap-frogging and vortex grouping mechanisms; and a far wake characterised by fully developed turbulence. Two primary noise generation mechanisms were identified. The unsteady pressure field over the turbine surface generates tonal noise at the blade passing frequency and a high-frequency broadband noise, associated with the trailing edge vortex sheet (linear-noise contribution). The turbulent wake generates broadband low-frequency noise, driven by the complex fluid-dynamic processes outlined previously (nonlinear noise contribution). The linear part of the noise was found to dominate over the nonlinear one in the acoustic far field, while the opposite is true in the acoustic near field. As a composition of the two contributions to the noise, the directivity exhibits a non-symmetric dipole shape oriented along the flow direction, with lobes recovering symmetry moving from the near to the far field. Finally, analysis of the acoustic decay rates reveals that the linear term in the near field decays according to an $r^{-(n+1)}$ law within the rotor plane, where n is the number of blades, consistent with recent findings on the acoustics of rotating sources.

One of the challenges with modelling subsurface flows is the uncertainty in measurements of geological properties, mostly due to limited resolution in observation methods. Many subsurface flows can be modelled as a gravity current, which, for uniform material properties and power-law injection rate, has a well-characterised similarity solution. The similarity solution forms a dynamical attractor that is typically approached rapidly from a host of initial conditions. Here, we consider the impact of transverse variations to the permeability field by performing a perturbation analysis of the self-similar spreading. This treats the response as perturbations to the self-similar flow. We restrict our focus to permeability fields that vary laterally, or across the flow, starting with the simple case of a sinusoidal perturbation to a uniform permeability. At early times, the height and nose position of the current are determined by the local permeability, and deviations to the height and nose grow at the same rate as the mean, and proportional to the amplitude, of the permeability variation. The transition between the early and late time regimes is governed by the wavelength of the permeability. At late times, lateral spreading between high and low permeability streaks is dominant; the height deviations decay, and the nose deviations approach a steady state. The magnitudes of both depend on the product of the wavelength and amplitude of the permeability. The single mode sets the groundwork for examining more complex, multimodal permeabilities, which are more representative of real geological structures.

Using pore-resolved direct numerical simulation (DNS), we investigate passive scalar transport at a unit Schmidt number in a turbulent flow over a randomly packed bed of spheres. The scalar is introduced at the domain’s free-slip top boundary and absorbed by the bed, which maintains a constant and uniform scalar value on the sphere surfaces. Eight cases are analysed, which are characterised by friction Reynolds numbers of ${\textit{Re}}_\tau \in [150, 500]$ and permeability Reynolds numbers of ${\textit{Re}}_{{\kern-1pt}K} \in [0.4, 2.8]$, while flow depth-to-sphere-diameter ratios vary within $h/D \in \{ 3, 5, 10 \}$ and the roughness Reynolds numbers lie within $k_s^+ \in [20,200]$. For cases with comparable ${\textit{Re}}_\tau$, the permeable wall behaviour enhances scalar absorption, as indicated by increases in the Sherwood number and the scalar roughness function $\Delta c^+$ over ${\textit{Re}}_{{\kern-1pt}K}$. At progressively higher ${\textit{Re}}_{{\kern-1pt}K}$, the scalar absorption diverges increasingly from the momentum absorption, as its distribution peaks deeper below the crests of the sphere pack and spreads over a wider vertical region. The fixed-value scalar boundary condition emphasises certain similarities between the scalar and velocity fields. Near-interface scalar fluctuations are correlated with streamwise velocity fluctuations, and the turbulent Schmidt number remains close to its value in the free-flow region. Compared with zero-flux scalar boundary conditions, prescribing a uniform scalar value on the sphere surfaces reduces spatial heterogeneity within the pore space, thereby limiting both dispersive transport and the form-induced production of temporal scalar fluctuations.

Fully resolving turbulent flows remains challenging due to a turbulent systems’ multiscale complexity. Existing data-driven approaches typically demand expensive retraining for each flow scenario and struggle to generalize beyond their training conditions. Leveraging the universality of small-scale turbulent motions (Kolmogorov’s K41 theory), we propose a scale-oriented zonal generative adversarial network (SoZoGAN) framework for high-fidelity, zero-shot turbulence generation across diverse domains. Unlike conventional methods, SoZoGAN is trained exclusively on a single dataset of moderate-Reynolds-number homogeneous isotropic turbulence (HIT). The framework employs a zonal decomposition strategy, partitioning turbulent snapshots into subdomains based on scale-sensitive physical quantities. Within each subdomain, turbulence is synthesized using scale-indexed models pretrained solely on the HIT database. A SoZoGAN demonstrates high accuracy, cross-domain generalizability and robustness in zero-shot super-resolution of unsteady flows, as validated on untrained HIT, turbulent boundary layer and channel flow. Its strong generalization, demonstrated for homogeneous and inhomogeneous turbulence cases, suggests potential applicability to a wider range of industrial and natural turbulent flows. The scale-oriented zonal framework is architecture-agnostic, readily extending beyond generative adversarial networks to other deep learning models.

This paper presents an experimental investigation focusing on the impact of structural damping on the flow-induced vibration (FIV) of a set of generic three-dimensional bodies, in this case, elastically mounted oblate spheroids. The objective is to identify and analyse the two primary FIV responses: vortex-induced vibration (VIV) and galloping, and how these vary with structural damping ratio. The VIV response has similarities to that observed for a sphere, reaching a maximum amplitude of approximately one major diameter. However, and not seen in the sphere case, a galloping-like response exhibits a linear amplitude growth as the reduced velocity is increased beyond the normal resonant range, akin to the transverse galloping response seen for a D-section or elliptical cross-section cylinder. By increasing the damping ratio, this aerodynamic-instability-driven response is effectively suppressed. However, increased damping also significantly reduces the VIV response, decreasing its maximum amplitude and contracting the VIV synchronisation, or lock-in, region. These results suggest that three-dimensional spheroids, as for two-dimensional cylindrical bodies such as D-section and elliptical cylinders, can encounter asymmetric aerodynamic forces that support movement-induced vibration, resulting in substantial body oscillation – beyond that expected under VIV alone. The study indicates that modifying the structural damping ratio can facilitate a transition between the VIV and galloping responses. These findings offer novel insights into the dynamics of fluid–structure interactions and have potential implications for designing structures and devices that can experience resonant flow conditions. Additionally, the energy harvesting performance of oblate spheroids has been evaluated, revealing that the afterbody significantly influences energy harvesting capabilities. Notably, an oblate spheroid can extract up to $50\,\%$ more power from the fluid flow than a sphere.

Analytical expressions for the mean wall-normal velocity and wall shear stress in compressible boundary layers are derived by integrating the mean continuity and momentum equations. In the constant-density limit, the momentum integral formulation recovers the classical Kármán–Pohlhausen equation for incompressible boundary-layer flows. In compressible regimes, particularly under strong pressure gradients, streamwise density gradients are shown to play a crucial role in shaping boundary-layer dynamics. The derived analytical equations are validated against high-fidelity direct numerical simulation data, demonstrating both accuracy and robustness. Furthermore, the analytical equations offer insights into the physical mechanisms of compressible boundary layers, particularly the influence of density gradients. The effect of compressibility on the wall-normal velocity is explicitly demonstrated, highlighting the distinct behaviour of compressible boundary layers compared with incompressible flows. Finally, an analytical expression for the skin-friction coefficient is developed, revealing its close connection to the mean wall-normal velocity at the boundary-layer edge.

We study the force exerted by the uniform flow of a Bingham fluid around two- and three-dimensional particles in the regime of slow creeping flow and relatively weak yield stress. Matched asymptotic expansions are employed to couple a viscously dominated Stokes flow close to the particle with a far field in which the yield stress and viscous stresses are comparable. The far-field region is therefore modelled as a Bingham fluid driven by a point force at the origin (i.e. a viscoplastic Stokeslet). It features the full nonlinearity of the viscoplastic rheology, and its solution is computed through direct numerical simulation. Asymptotic matching then leads to a quasi-analytical expression for the drag force in terms of the dimensionless Bingham number ${\textit{Bi}}$, which measures the magnitude of the yield stress relatively to viscous effects at the particle scale. We deploy this methodology to determine the drag force on a sphere in three dimensions, and circular and elliptic cylinders in two dimensions, confirming our asymptotic predictions by comparison with full numerical simulations of the motion. We also generalise the three-dimensional result to arbitrary particles. The viscoplastic correction to the Newtonian drag in three dimensions scales as ${\textit{Bi}}^{1/2}$. In two dimensions, however, the effects of viscoplasticity are non-negligible at leading order. The drag varies with $[\ln (1/{\textit{Bi}})]^{-1}$, but this asymptotic result is only approached very slowly. Instead, an accurate representation of the drag is derived in terms of a single algebraic relation between the drag and the Bingham number.

We investigate the scale-by-scale transfers of energy, enstrophy and helicity in homogeneous and isotropic polymeric turbulence using direct numerical simulations. The study relies on the exact scale-by-scale budget equations, derived from the governing model equations, that fully capture the back-reaction of polymers on the fluid dynamics. Polymers act as dynamic sinks and sources and open alternative routes for interscale transfer whose significance is modulated by their elasticity, quantified through the Deborah number (${\textit{De}}$). Polymers primarily deplete the nonlinear energy cascade at small scales, by attenuating intense forward and inverse transfer events. At sufficiently high ${\textit{De}}$, a polymer-driven flux emerges and dominates at small scales, transferring on average energy from larger to smaller scales, while allowing for localised backscatter. For enstrophy, polymers inhibit the stretching of vorticity, with fluid–polymer interactions becoming the primary enstrophy source at high ${\textit{De}}$. Accordingly, an analysis of the small-scale flow topology reveals that polymers promote two-dimensional straining states and enhance the occurrence of shear and planar extensional flows, while suppressing extreme rotation events. Helicity, injected at large scales, exhibits a transfer mechanism analogous to energy, being dominated by nonlinear dynamics at large scales and by polymer-induced fluxes at small scales. Polymers enhance the breakdown of small-scale mirror symmetry, as indicated by a monotonic increase in relative helicity with ${\textit{De}}$ across all scales.

A large-scale parametric study of the flow over the prolate spheroid is presented to understand the effect of Reynolds number and angle of attack on the separation, the wake formation and the loads. Large-eddy simulation is performed for six Reynolds numbers ranging from ${\textit{Re}} = 0.15\times 10^6$ to $4 \times 10^6$ and for eight angles of attack ranging from $\alpha = 10^\circ$ to $\alpha = 90^\circ$. For all the cases considered, the boundary layer separates symmetrically and forms a recirculation region. Several distinct flow topologies are observed that can be grouped into three categories: proto-vortex, coherent vortex and recirculating wake. In the proto-vortex state, the recirculation does not have a distinct centre of rotation, instead, a two-layer detached flow structure is formed. In the coherent vortex state, the separated shear layer rolls into a three-dimensional vortex that is aligned with the axis of the spheroid. This vortex has a clear centre of rotation corresponding to a minimum of pressure and transforms the transverse momentum from the separated shear layer into axial momentum. In the recirculating wake regime, the recirculation is incoherent and the primary separation forms a dissipative shear layer that is convected in the direction of the free stream. This symmetric pair of shear layers bounds a low-momentum recirculating cavity on the leeward side of the spheroid. The properties of these states are not constant, but evolve along the axis of the spheroid and are dictated by the characteristics of the boundary layer at separation. The variation of the flow with Reynolds number and angle of attack is described, and its connection to the loads on the spheroid are discussed.

Contact between fluctuating, fluid-lubricated soft surfaces is prevalent in engineering and biological systems, a process starting with adhesive contact, which can give rise to complex coarsening dynamics. One representation of such a system, which is relevant to biological membrane adhesion, is a fluctuating elastic interface covered by adhesive molecules that bind and unbind to a solid substrate across a narrow gap filled with a viscous fluid. This flow is described by the stochastic elastohydrodynamic thin film equation, which incorporates thermal fluctuations into the description of viscous nanometric thin-film flow coupled to elastic membrane deformation. The average time it takes the fluctuating elastic membrane to adhere is predicted by the rare event theory, increasing exponentially with the square of the initial gap height. When the forces arising from spring-like adhesive molecules are included in the simulations, thermal fluctuations initiate phase separation of domains of bound and unbound molecules. The coarsening process of these unbound pockets displays close similarities to classical Ostwald ripening; however, the inclusion of hydrodynamics affects power-law growth. In particular, we identify a new bending-dominated coarsening regime, which is slower than the well-known tension-dominated case.

Not all particulate matter carried by fluid flows has constant buoyancy. In some cases, the buoyancy of a particle can change dynamically based on the local flow. We refer to this phenomenon as ‘active buoyancy.’ Although actively buoyant particles are found throughout nature, their dynamics is not well understood, particularly when they are also highly inertial. Motivated by the problem of the transport of firebrands in wildfires, whose effective buoyancy is modulated by conductive and convective heat transfer to the surrounding fluid, we conducted a series of experiments to investigate the effects of active buoyancy on particle settling in quiescent fluid. We find that, depending on the control parameters, active buoyancy can either hinder or enhance settling, in some cases to a large extent. The details of this settling modulation, however, cannot be simply captured by any single control parameter. Analysis of the trajectories of the falling particles showed that they fall along nearly sinusoidal paths even though the particle Reynolds number is higher than expected for this regime, suggesting that active buoyancy may act to stabilise their wakes. Our results suggest both that models of actively buoyant particles such as firebrands must account for the effects of active buoyancy and that there is still much to be understood about the behaviour of these complex particles.

We investigate the convective stability of a thin, infinite fluid layer with a rectangular cross-section, subject to imposed heat fluxes at the top and bottom and fixed temperature along the vertical sides. The instability threshold depends on the Prandtl number as well as the normalized flux difference ($f$) and decreases with the aspect ratio ($\epsilon$), following a $\epsilon f^{-1}$ power law. Using a three-dimensional (3-D) initial value and two-dimensional eigenvalue calculations, we identify a dominant 3-D mode characterized by two transverse standing waves attached to the domain edges. We characterize the dominant mode’s frequency and transverse wavenumber as functions of the Rayleigh number and aspect ratio. An analytical asymptotic solution for the base state in the bulk is obtained, valid over most of the domain and increasingly accurate for lower aspect ratios. A local stability analysis, based on the analytical base state, reveals oscillatory transverse instabilities consistent with the global instability characteristics. The source term for this most unstable mode appears to be interactions between vertical shear and horizontal temperature gradients.

This paper aims to elucidate the physical mechanisms underlying airfoil–vortex gust interaction and mitigation. The vortex gust mitigation problem consists in finding the pitch rate sequence that minimises the gust-induced lift disturbance of an NACA0012 airfoil at Reynolds number 1000. The instantaneous flow fields and resulting lift are obtained from numerical resolution of the Navier–Stokes equations. The controller is modelled as an artificial neural network and trained to minimise the lift fluctuation using deep reinforcement learning (DRL). The paper shows that DRL-trained controllers are able to mitigate medium- and high-intensity vortex gusts by more than 80 % compared to the uncontrolled scenario. It then presents a comparative analysis of the controlled and uncontrolled lift generation mechanisms using the force partitioning method (FPM). The FPM provides a quantitative assessment of the amount of lift generated by each flow region. For medium-intensity gusts, the main phenomenon is the asymmetry in the airfoil boundary layer induced by the vortex. The control strategy mitigates the gust-induced lift by restoring the flow symmetry around the airfoil. For high-intensity gusts, the boundary layer asymmetry remains, but the gust interaction with the airfoil also triggers flow separation and the formation of a strong leading-edge vortex (LEV). Consequently, the control command balances several aerodynamic phenomena such as boundary layer asymmetry, flow detachment, LEV, and secondary recirculation regions to produce a net quasi-zero lift fluctuation. Thus this work highlights the potential of DRL control, enhanced by advanced post-processing such as FPM, to discover and interpret optimal flow control mechanisms.

An experimental study is performed to control flow separation from a two-dimensional curved ramp using a spanwise pulsed blowing slit jet placed near the separation point of the baseline flow. The momentum-thickness-based Reynolds number $ \textit{Re}_{\theta}$ is 5700. Four control parameters are investigated, including the velocity ratio $\overline{U_{J,c}^{*}}$, duty cycle dc, dimensionless excitation frequency $f_{e}^{{*}}$ and jet blowing angle $\alpha$. The control mechanisms are found to differ from small to large jet angle. Empirical scaling analysis for $\alpha \leq 55^{\circ}$ unveils that $\Delta \overline{C_{p,e}}=f_{1}(\overline{U_{J,c}^{*}}, { d}c, f_{e}^{*}, \alpha , Re_{\theta })$ may be reduced to $\Delta \overline{C_{p,e}}/\varPi (\tau )=f_{2}(\xi )$, where $f_{1}$ and $f_{2}$ are different functions, $\Delta \overline{C_{p,e}}$ is the variation in the pressure coefficient at the end of the ramp under control, $\varPi (\tau )$ is a function of dimensionless duration $\tau$ at which the jet is closed within one excitation period, $\Delta \overline{C_{p,e}}/\varPi (\tau )$ represents the control efficiency, and $\xi$ is a scaling factor that is physically the energy ratio per unit area of the blowing jet to the mainstream. This scaling law is also found to be valid for steady jet control. Several interesting inferences can be made from this scaling law, which provides important insight into the physics of flow separation control.