We develop data-driven models to predict the dynamics of a freely settling sphere in a quiescent Newtonian fluid using experimentally obtained trajectories. Particle tracking velocimetry was used to obtain a comprehensive dataset of settling motions, which we use to train neural networks that model the spatial evolution of a spherical particle without explicitly resolving the surrounding fluid dynamics. We employ deterministic neural ordinary differential equations (NODEs) and stochastic neural stochastic differential equations (NSDEs) to reconstruct the sphere's trajectory and capture key statistical features of the settling process. The models are evaluated based on short- and long-time dynamics, including ensemble-averaged velocity evolution, settling time distributions, and probability density functions of the final settling positions. We also examine the correlation between lateral displacement and streamwise velocity and assess the impact of dataset size on predictive accuracy. While NODEs excel in trajectory reconstruction and generalization across different initial conditions, NSDEs effectively capture statistical trends in the long-time behavior but are more sensitive to data availability. Acceleration profiles computed via second-order finite difference schemes confirm that both approaches accurately capture long-time dynamics, although short-time transients pose challenges.

We demonstrate the capability to experimentally measure fluctuations of the convective heat transfer coefficient at the wall in a turbulent boundary layer. To achieve this, we measure two-dimensional fields of wall-temperature fluctuations beneath a zero-pressure-gradient turbulent boundary layer at two moderate friction Reynolds numbers (${\text{Re}}_{\tau }\approx 990$ and ${\text{Re}}_{\tau }\approx 1800$). Spatiotemporal data of wall temperature are acquired by means of a heated-thin-foil sensor as sensing hardware and an infrared camera as a temperature detector. At the lower ${\text{Re}}_{\tau }$ condition, the fields of Nusselt number fluctuations (${\text{Nu}}^{\prime }$) exhibit elongated features comprising streamwise and spanwise length scales comparable to those of near-wall streaks. At higher ${\text{Re}}_{\tau }$, the effective width and length of the streaks of Nu fluctuations increase. These findings are based on two-point correlations as well as streamwise-spanwise energy spectra of Nu fluctuations at the wall. The convective velocities of the Nu fluctuations are also computed using the available temporal resolution. This allows for resolving the multiscale nature of convective footprints of wall-bounded turbulence: Our experimental data reflect that larger streaks in the footprint convect at velocities in the order of the free-stream velocity, whereas more energetic smaller-scale features move at velocities in the order of $10u_{\tau }$. Measurements of the kind presented here offer a promising method for wall-based sensing of turbulence and thus for usage as input to flow control systems.

Natural transition in high-speed boundary layers is initiated by the exponential amplification of infinitesimal instability waves. Once these waves reach appreciable amplitudes, their evolution becomes nonlinear and they subsequently break down to turbulence. Assuming the availability of wall-pressure signals within the nonlinear flow regime, we attempt to predict the boundary-layer instability waves that led to the wall-pressure data. This data-assimilation (DA) problem is formulated as a nonlinear optimization: We search for the upstream disturbance parameters that reproduce the measurements, by minimizing a cost function defined in terms of the difference between the estimated and true wall-pressure data. The optimization is challenging due to the sparsity of measurements and the computational cost of evaluating the wall-pressure data from each estimate of the instability waves during the iterative procedure. Each iteration of the Bayesian approach identifies a new estimates of the control vector that can potentially minimize the DA cost function. This procedure requires a statistical model, and therefore repeated evaluations of the cost function, for which direct numerical simulations (DNS) are not practical. We therefore adopt a deep operator network (DeepONet) surrogate for the compressible Navier-Stokes equations, which is efficient and sufficiently accurate. Using an ensemble of DeepONets, we can evaluate the expectation of the cost function and its uncertainty, to inform the Bayesian optimization. Once the new estimate of the control vector is identified, a single DNS is performed, the accuracy of the DA solution is evaluated against a convergence criterion, and the DNS data are added to the training data of the DeepONet model. This procedure is then repeated to convergence. We demonstrate the Bayesian optimization with DeepONet (BO-DeepONet) by predicting the frequency and amplitude of the incident primary instability wave, and its phase shift relative to the subharmonic mode, from wall-pressure data in the nonlinear flow regime.

The fluid dynamics community has increasingly adopted machine learning to analyze, model, predict, and control a wide range of flows. These methods offer powerful computational capabilities for regression, compression, and optimization. In some cases, machine learning has even outperformed traditional approaches. However, many fluid mechanics problems remain beyond the reach of current machine learning techniques. As the field moves from its current state toward a more mature paradigm, this article offers a critical assessment of the key challenges that must be addressed. Tackling these technical issues will not only deepen our understanding of flow physics but also expand the applicability of machine learning beyond fundamental research. We also highlight the importance of community-maintained datasets and open-source code repositories to accelerate progress in this area. Furthermore, the future success of machine learning in fluid dynamics will depend on effective training—not only for the next generation of researchers but also for established fluid mechanicians adapting to this evolving landscape. Data-driven fluid dynamics is in its critical transitional state over the next few years to shape its future. This perspective article aims to spark discussions and encourage collaborative efforts to advance the integration of machine learning in fluid dynamics.

We consider a simple thermodynamically consistent model that captures the self-organized chemomechanical coupling resulting from the interplay between autocatalytically reacting surfactants, the Marangoni effect, and wetting dynamics. An ambient bath of surfactant acts as a chemostat and provides the system with chemical fuel, thereby driving it away from thermodynamic equilibrium. We find that a positive feedback loop between the local reactions and the Marangoni effect induces surface tension gradients that allow for self-propelled droplets. Besides simple directional motion, we find crawling and shuttling droplets as well as droplets performing random walks, thus exploring the entire substrate. We study the occurring chemomechanical motility modes and show how the observed dynamic states emerge from local and global bifurcations. Due to the underlying generic thermodynamic structure, we expect that our results are relevant not only to directly related biomimetic droplet systems but also to structurally similar systems like chemically active phase-separating mixtures.

The particle behavior and turbulent modulation of wall-bounded turbulent flow are analyzed by comparing a new simulation of channel turbulence with older simulation of spatially developing turbulent boundary layer at almost identical friction Reynolds number and laden particles with identical inertia and bulk volume fraction. It is found that particle locally accumulate in the wake region of the external boundary layer flow due to more pronounced turbulent/nonturbulent interface effect. Therefore, the particle-turbulence interaction in the outer layer of boundary layer results in stronger turbulent modulation than that in particle-laden channel. This influence not only makes the outer-layer structures more different but may penetrate to the near-wall region of wall turbulence, generating opposite modulation of turbulent mean velocity of the two flows. However, the modulation in second-order statistics and turbulent structures by the presence of inertial particles are qualitatively similar below the logarithmic layer. The similarities/dissimilarities are further explained by comparing the average stress, the turbulent kinetic energy budget and uniform momentum zones/quiescent core.

Hall, in his Ph.D. thesis (University of London, 1973), demonstrated that the viscous flow arising in a slender channel of slowly varying cross section subject to a purely oscillatory pressure difference between its open ends contains a steady streaming component. This study extends the analysis to channels with a porous interior. Using a homogenized flow model that incorporates a Darcy resistance term proportional to the local velocity, a closed-form solution for the streaming motion is derived in the asymptotic limit of small stroke-to-channel length ratios $\varepsilon \ll 1$. Consistent with Hall's findings, a net flow rate is seen to arise only when the channel ends have unequal widths. The presence of a porous medium significantly attenuates the streaming-flow magnitude, with the differences between porous and nonporous channels becoming more pronounced for large Womersley numbers, a limiting case considered separately. In nonporous channels with unequal end widths the streaming motion exhibits large velocities that are comparable in magnitude to those of the oscillatory flow, whereas in porous channels the streaming velocities remain a factor $\varepsilon$ smaller. Channels with equal end widths feature streaming recirculating vortices within near-wall Stokes boundary layers. Nonporous channels display additional recirculating vortices in the central core region, which are absent in porous configurations. These results provide insights into transport processes in biomedical and technological applications involving oscillatory flow in wall-bounded porous media.

In the past 15 years, flow instabilities reminiscent of the Taylor-like instabilities driven by hoop stresses have been observed in wormlike micelles based on surfactant molecules. In particular, purely elastic instabilities and turbulence have been shown to develop on top of shear banding, a type of flow specific to the semidilute and concentrated regimes. These instabilities have been identified as the origin of the large body of data showing complex spatiotemporal fluctuations, collected in shear-banded systems using multiple experimental techniques. Different categories of banding have been suggested depending on their stability, which involve intrinsic properties of the system and streamline curvature. It has been shown qualitatively that instabilities are promoted by an increase in the surfactant concentration or in the curvature of the flow geometry, while an increase in temperature stabilizes the flow. Here, using benchmark shear banding micellar systems, we quantify the effect of the streamline curvature on these flow instabilities, focusing more specifically on the transition toward purely elastic turbulence. Using various optical visualizations, we identify two transitional pathways to elastic turbulence. We construct a generic state diagram in a parameter space based on the curvature ratio and the Weissenberg number. The nature—supercritical vs subcritical—of the transition to elastic turbulence is discussed. The stress evolution is in favor of a change of nature from subcritical to supercritical transition as the curvature ratio increases. However, we show that finite size effects cannot be neglected and may artificially smooth the stress response. Furthermore, each domain of this diagram is characterized using velocimetry measurements. Finally, a scaling for the onset of elastic turbulence is determined.

Drying of fluids undergoing sol-gel transition in porous media, a process crucial for the consolidation of damaged porous structures in cultural heritage, often leads to skin formation at the surface. This phenomenon significantly hinders evaporation, yet its precise impact on drying kinetics remains poorly understood. To uncover the governing mechanisms, we develop a novel pore network model that closely replicates quasi-two-dimensional experimental porous media in a previous paper [Le Dizès Castell et al., J. Phys. Chem. Lett. 15, 628 (2024)], incorporating spatial gradients in pore size distribution, with smaller pores near the evaporation side. We demonstrate that this pore distribution dictates the air invasion path and extends the constant-rate period of drying in Newtonian liquids, reproducing the experimental drying curves for water. We further extend our model to capture the interplay of capillary-driven liquid flows, space-dependent viscosity increases, and localized skin formation. To incorporate skin formation, we implement a viscosity-dependent vapor pressure rule derived from experimental data on evaporation-induced sol-gel transition within a capillary tube. We identify a simple relationship: vapor pressure decreases once the meniscus fluid viscosity exceeds a critical threshold. By accounting for localized skin formation through reduced evaporation pressure at high-viscosity throats, our model successfully captures the slowdown of drying kinetics, achieving remarkable agreement with experimental drying curve.

Ice melting is a complex, dynamic, and multiscale process involving many feedback effects. An essential but poorly understood factor for floating ice is the potential to overturn, as dramatically demonstrated by capsizing icebergs. Here we study the melting-induced reshaping of ice coupled to its free rotation dynamics and find that periods of stable equilibrium are punctuated by spontaneous capsize events. Laboratory experiments on floating ice show that melting produces faceted and angular shapes whose tendency to orient to the waterline is repeatedly upset by rolling over. Similar geometries and motions are reproduced in simulations that drive the shape development via differential melt rate over the surface and which solve for the rotational dynamics subject to shape-dependent forcing from gravity, buoyancy, and hydrodynamics. Melting gradually drives the structure toward postural instability while also priming other orientations that prove stable when reached by abrupt flipping. These findings show how the shape and motions of ice are inextricably linked at fundamental scales, and they reveal mechanisms that may be operative for natural conditions.

We propose a geometry-to-flow diffusion model that utilizes obstacle shape as input to predict a flow field around an obstacle. The model is based on a learnable Markov transition kernel to recover the data distribution from the Gaussian distribution. The Markov process is conditioned on the obstacle geometry, estimating the noise to be removed at each step, implemented via a U-Net. A cross-attention mechanism incorporates the geometry as a prompt. We train the geometry-to-flow diffusion model using a dataset of flows around simple obstacles, including circles, ellipses, rectangles, and triangles. For comparison, two convolutional neural network (CNN)-based models and a variational autoencoder (VAE) model are trained on the same dataset. Tests are carried out on flows around obstacles with simple and complex geometries, representing interpolation and generalization on the geometry condition, respectively. To evaluate performance under demanding conditions, the test set incorporates scenarios including crosses and the characters “PKU.” Generated flow fields show that the geometry-to-flow diffusion model is superior to the CNN-based models and the VAE model in predicting instantaneous flow fields and handling complex geometries. Quantitative analysis of the accuracy and divergence demonstrates the model's robustness.

A comprehensive Darcy-type law for viscoplastic fluids is proposed. Different regimes of yield-stress fluid flow in porous media can be categorized based on the Bingham number (i.e., the ratio of the yield stress to the characteristic viscous stress). In a recent study [Chaparian, J. Fluid Mech. 980, A14 (2024)], we addressed the yield/plastic limit (infinitely large Bingham number), namely, the onset of flow when the applied pressure gradient is just sufficient to overcome the yield-stress resistance. A purely geometrical universal scale was derived for the nondimensional critical pressure gradient, which was thoroughly validated against computational data. In the present work, we investigate the Newtonian limit (infinitely large pressure difference compared to the yield stress of the fluid—ultralow Bingham number) both theoretically and computationally. We then propose a Darcy-type law applicable across the entire range of Bingham numbers by combining the mathematical models of the yield/plastic and Newtonian limits. The Exhaustive computational data generated in this study (using an augmented Lagrangian method coupled with anisotropic adaptive mesh at the pore scale) confirm the validity of the theoretical proposed law.

Temperature changes as small as $3{}^{\circ }\mathrm{C}$ have been observed to significantly impact how self-propelled organisms move through their environment, especially for those inhabiting the transitional flow regime in which both viscous and inertial effects are important. Nonetheless, many oceanic species can successfully migrate across temperature changes in the order of $20{}^{\circ }\mathrm{C}$, corresponding to $40\%$ differences in viscosity, via metachronal propulsion, suggesting that this propulsion mechanism is resilient to drastic changes in water column properties. We investigate marsh grass shrimp (Palaemon vulgaris) as a model organism to explore the combined physical and physiological effects on their locomotion at natural seasonal temperature extremes ($6^{\circ }\text{–}20{}^{\circ }\mathrm{C}$). Experimentally, we manipulate temperature and viscosity independently to isolate physical and physiological effects. We then use the shrimp morphology and gait data to inform a computational fluid dynamics parametric study to estimate the force-to-power ratios of varying viscosity and beat frequencies through naturally occurring extremes. Our research demonstrates that shrimp do not modify their gait parameters to naturally occurring viscosity changes, and their swimming performance is impacted by less than $9\%$. The robustness of the metachronal gait is evidence of the ecological success of shrimp-like organisms in all climates, from the tropics to pole waters and inland freshwater.

We investigate the settling hydrodynamics of a sphere confined along the centerline of a square tube filled with active nematic fluid. The sphere is restricted from rotation and subjected to either planar or homeotropic anchoring conditions on its surface. Our study focuses on the regime where direct active forces exerted on the sphere are much weaker than the hydrodynamic or chemical forces. We find that activity can still significantly alter the settling velocity by modifying the flow structure and nematic field. We first examine the fundamental settling behavior in the absence of activity, where the sphere's velocity varies nonmonotonically with the elastic constant ($K$) but increases with the collective rotational diffusion constant (Г). Introducing activity by activity number (${\tilde{K}}^{-1/2}$) while fixing $K=0.05$, $\mathrm{\Gamma }=0.5$, and Galileo number $\left(\text{Ga}\right)=5.0$, we observe that the axisymmetric flow structure emerges at low ${\tilde{K}}^{-1/2}$, where the extensile activity squeezes the vortex and enhances settling, whereas the contractile activity expands the vortex and suppresses settling. In addition, active nematics can alter the “effective viscosity” of the local flow, which in turn affects the settling velocity. As activity increases, we found that transitions between distinct flow patterns also play a critical role in altering the settling velocity. Specifically, the flow with nonzero helicity emerges at moderate activity in extensile fluids and leads to an abrupt reduction in the sphere's settling velocity. This reduction is related to the transition in the flow pattern and a corresponding shift in the deformation of the nematic field from bend to twist. The effect is particularly pronounced with homeotropic anchoring owing to the stronger helicity of the flow.

The evaluation of added mass holds practical significance in various engineering applications dealing with flow-induced vibration and fluid-structure interaction problems. A notable situation arises when bodies oscillate in an unsteady flow, leading to a coupling mass that deviates from the conventional added mass used for assessing inertial forces on a body oscillating in a steady fluid. In this paper, we revisit a comprehensive theoretical approach initially proposed by Batchelor in 1967 for quantifying the inertial force acting on a body in an accelerating fluid. To enhance this approach, we incorporate an exact and straightforward superposition of the acceleration fields along with numerical evaluation based on the kinetic energy of potential flow. Our refined approach encompasses a wide range of body shapes, ranging from simple two-dimensional primitive shapes to intricate three-dimensional bluff bodies. The paper presents a simplified concept to clarify the relationship between added mass for an accelerating body in a stationary fluid and the acceleration coefficient of a motionless object in an unsteady flow. This method benefits fluid-structure interaction studies, including seismic design, underwater systems, and multiphase flow modeling.

This paper presents a theoretical investigation of lock-released gravity currents propagating in the inertia-buoyancy regime through channels with diverse cross-sectional shapes, incorporating the effects of entrainment and drag. The study develops a generalized model based on shallow-water equations that accounts for the effects of entrainment and drag in channels with various cross-sectional shapes, extending previous models that primarily focused on rectangular channels. The finite-difference solution of the governing equations, using realistic initial and boundary conditions, effectively predicts the evolution of current height, velocity, and concentration over time. The results indicate that entrainment and drag significantly reduce current propagation speed while increasing its height and volume, particularly in channels with nonrectangular geometries. Furthermore, a high drag-to-entrainment ratio notably decreases current speed and distance of propagation, affecting the concentration profile near the current's nose, while keeping it stable at the tail.

The dynamics of a single vortex ring impinging on the concave hemicylindrical shell with different radii at $\mathrm{Re}=1500$ are explored experimentally. Five different ratios of hemicylindrical cavity radii to vortex ring are studied: $D_m/D_e=15,10,5,2,1$. Planar laser-induced fluorescence and two-dimensional particle image velocimetry techniques are adopted for flow measurements, and the finite-time Lyapunov exponent field based on the Lagrangian framework is employed to identify the Lagrangian coherent structures of the vortex dynamics. The experimental results indicate that for $D_m/D_e=15,10$ conditions, the impinging dynamics of the vortex ring are similar to the vortex ring interacting with a flat plate. The subvortex cluster consisting of secondary vortex rings and tertiary vortex rings, dominates the motion of the induced vortex rings after the primary vortex ring impinges with concave surfaces. For $D_m/D_e=5,2$ conditions, the collision of the vortex ring on the concave surfaces is quite different. The concave surface effectively constrains the primary vortex ring after impinging it. The concave secondary vortex ring rotates earlier than the straight-edged secondary vortex ring. Eventually, the straight-edged secondary vortex ring is lifted in the straight-edged direction, moving diagonally towards the collision axis, away from the solid wall. Subvortex rings in the final dominant flow field change from the tertiary to the secondary as the model radius decreases. As the diameter decreases to $D_m/D_e=1$, the primary vortex ring deforms farther in the straight-edged direction. After the straight-edged secondary vortex ring is formed, it moves vertically away from the wall. Finally, a dynamic model is proposed to illustrate the effects of cylindrical cavity radii on the vortex-ring impingement.

Laminar jets in crossflow are analyzed using direct numerical simulation to identify the source of vorticity leading to the formation of the counter-rotating vortex pair. A vorticity transport analysis conducted on steady cases shows that all counter-rotating vortex pair vorticity is generated in the nozzle walls for large velocity ratio jets ($R$), but that additional boundary layer vorticity is entrained at low $R$. The transport mechanisms are quantified and discussed in detail, including the influence of having a square jet cross section. The study also examines the effect of varying the velocity ratio in the range $0.25\le R\le 5$ and the jet Reynolds number in the range $100\le {\text{Re}}_{\text{jet}}\le 300$. Increasingly complex flow patterns are revealed, even in the steady regime, including the onset of bubble vortex breakdown of the counter-rotating vortex pair. Breakdown onset is quantified and analyzed in detail, revealing that it is a gradual process that depends on the jet vorticity and the adverse pressure gradient formed downstream of the jet inlet. Additionally, breakdown is discovered to precede the onset of unsteady flow for decreasing $R$ and fixed ${\text{Re}}_{\text{jet}}$ for jets with both cross sections. A newly discovered asymmetric instability is identified at ${\text{Re}}_{\text{jet}}=300$ and characterized, where each of the counter-rotating vortices undergoes breakdown while oscillating laterally at low frequency. Hairpin vortex shedding, typical of low $R$ laminar jets in crossflow, is captured in our simulations. Breakdown bubbles are identified for $R>0.4$, where the flow remains approximately steady in the near field. For $R\le 0.4$, the increasing fluctuation amplitude in the near field results in the rollup of fluid and the destruction of the breakdown bubbles in the instantaneous flow field.

High-order methods, such as the discontinuous Galerkin method, have gained increasing attention in the computational fluid dynamics community because they are expected to deliver high-accuracy results on complex flows and geometries. However, in realistic high Reynolds number flows, the significant computational cost and lack of robustness are unavoidable concerns when compared to low-order methods. The present paper develops a detached eddy simulation model based on the $k\text{−}g$ equations, where the near-wall scaling of the new transport variables suits the polynomial nature of the discontinuous Galerkin method. The hybrid modeling framework of “${\ell }^2\times \frac{1}{T}$” mimics traditional Smagorinsky and dynamic subgrid models in the eddy-resolving regions, and provides seamless hybrid modeling of the unresolved scales in coarse-mesh simulations of wall turbulence and separated flows. Robustness is achieved by both proper near-wall scaling of turbulence variables and the scale-dependent hybrid modeling of unresolved turbulence. For the latter, no extra cutoff filtering, overintegration, or additional viscosity is needed regardless of the mesh resolution. Our model can perform both traditional delayed detached eddy simulation and wall-modeled large eddy simulation for high Reynolds number turbulent flows like its predecessor, the adaptive ${\ell }^2\text{−}\omega$ model [Yin and Durbin, Int. J. Heat Fluid Flow 62, 499 (2016)]. By implementing the method in a spectral element discontinuous Galerkin solver, the proposed model is tested in a transonic airfoil with shock-turbulent boundary layer interaction.

Lagrangian acceleration has been investigated both experimentally and numerically in the past, and it has been shown to exhibit extreme fluctuations, which have been rationalized as events in which tracer particles get trapped into vortical structures such as vortex tubes or filaments. Here, we consider the statistics of acceleration within the multifractal framework, as in Biferale et al., Phys. Rev. Lett. 93, 064502 (2004), and investigate the statistics of Lagrangian acceleration using shell models of turbulence, as in Boffetta et al., Phys. Rev. E 66, 066307 (2002), that—by construction—do not contain vortical structures. Our analysis reveals that, despite not accounting for specific coherent vortex structures, the multifractal model accurately captures the extreme intermittent fluctuations observed in the acceleration, with predictions that remain robust across a wide range of Reynolds numbers.