This paper is a personal overview of the efforts over the past half century to understand fluid turbulence in terms of simpler coherent units. The consequences of chaos and the concept of coherence are first reviewed, using examples from free-shear and wall-bounded shear flows, and including how the simplifications due to coherent structures have been useful in the conceptualization and control of turbulence. It is remarked that, even if this approach has revolutionized our understanding of the flow, most of turbulence cannot yet be described by structures. This includes cascades, both direct and inverse, and possibly junk turbulence, whose role, if any, is currently unknown. This part of the paper is mostly a catalog of questions, some of them answered and others still open. A second part of the paper examines which new techniques can be expected to help in attacking the open questions and which, in the opinion of the author, are the strengths and limitations of current approaches, such as data-driven science and causal inference.

Upon the impact of a flat disk on a boiling liquid, i.e., a liquid that is in thermal equilibrium with its own vapor, a thin vapor layer is entrapped under the disk. Due to the tendency of vapor to undergo phase change under pressure variation upon impact, the dynamics of this entrapped vapor pocket are different from those of a noncondensable air pocket. In this work, we experimentally investigate the dynamics of the entrapped vapor pocket, more specifically its time evolution and its subsequent influence on the hydrodynamic loads at different equilibrium ambient temperatures and impact velocities. We find that the retraction of the vapor pocket at high ambient temperature and small impact velocity is slow, occurring from the disk edge, and driven by the dynamic pressure ${\rho }_L{U}_0^2$. In contrast, at lower ambient temperatures and large impact velocities, after a short initial stage, the vapor pocket will collapse rapidly due to condensation. This scenario is confirmed by conducting experiments where, by heating the disk, the vapor pocket collapse is observed to slow down. We attribute this to the vaporization of liquid near the three-phase contact line region that frustrates the condensation process and reduces the impact pressure on the disk. The violent collapse of the vapor pocket may impart additional instantaneous momentum, but the overall pressure and force impulses are still found to be closely associated with the liquid added mass. Finally, we found that at a high tilt angle of the disk, the three-phase contact line movement over the disk surface may hinder the proper entrapment and compression of the vapor pocket, which results in a lower central impact pressure as rapid condensation at the central disk region does not occur.

Solidification processes at the boundary between the Earth's inner and outer core are the primary source of buoyancy forces that drive convection and Geodynamo in the outer core. This study aims to numerically investigate the impact of different buoyancy sources on convection within the Earth's core and the resulting magnetic field. We analyze three models: thermally driven, chemically driven, and thermochemically driven dynamos. The results confirm that both the magnetic field and the convective velocity field depend on the sources of buoyancy. Additionally, dipole polarity reversals and the super-rotation of the Earth's inner core are influenced by different buoyancy sources. As the Ekman number decreases, this dependence weakens.

Input-output analysis has been widely used to predict the transition to turbulence in wall-bounded shear flows, but it typically does not capture the full nonlinear effects involved. This work employs the small-signal finite-gain (SSFG) ${\mathcal{L}}_p$ stability theorem to assess nonlinear input-output stability. This SSFG ${\mathcal{L}}_p$ stability theorem can predict permissible forcing amplitudes below which a finite nonlinear input-output gain can be maintained. Our analysis employs linear matrix inequalities and sum-of-squares as the primary tools to search for a quadratic Lyapunov function of an unforced nonlinear system. The resulting Lyapunov function can certify the SSFG ${\mathcal{L}}_p$ stability of a nonlinear input-output system. We demonstrate the applicability of the SSFG ${\mathcal{L}}_p$ stability theorem using a nine-mode shear flow model with a random body force. The predicted nonlinear input-output ${\mathcal{L}}_p$ gain (amplification) is consistent with numerical simulations; the ${\mathcal{L}}_p$ norm of the output from numerical simulations remains bounded by the theoretical prediction from the SSFG ${\mathcal{L}}_p$ stability theorem, with the gap between simulated and theoretical bounds narrowing as $p\to \infty$. The input-output gain obtained from the nonlinear SSFG ${\mathcal{L}}_p$ stability theorem is higher than the linear ${\mathcal{L}}_p$ gain, which suggests that nonlinearity can significantly amplify small disturbances. Both nonlinear ${\mathcal{L}}_p$ gain and linear ${\mathcal{L}}_p$ gain are valid for each $p\in [1,\infty ]$, and such generalizability leads to much higher upper bounds on input-output gain than those predicted by linear ${\mathcal{L}}_2$ gain (${\mathcal{H}}_{\infty }$ norm of transfer function). The SSFG ${\mathcal{L}}_p$ stability theorem requires the input forcing to be smaller than a permissible forcing amplitude to maintain finite input-output gain, which is an inherently nonlinear behavior that cannot be predicted by linear input-output analysis. We also identify such permissible forcing amplitude using numerical simulations and bisection search, where below such forcing amplitude, the output norm at any time will be lower than a given threshold value. The permissible forcing amplitude identified from the SSFG ${\mathcal{L}}_p$ stability theorem is conservative but also consistent with that obtained by numerical simulations and bisection search. This study demonstrates that the nonlinear SSFG ${\mathcal{L}}_p$ stability theorem is a powerful tool for analyzing nonlinear input-output systems such as transitional shear flows with external disturbance.

In this work, a liquid jet in supersonic crossflow (LJSC) was simulated by solving the quasi-conservative five-equation model with surface tension and viscosity terms. The jet trajectory and deformation were detailedly validated against experimental data, showing excellent agreement. Detailed turbulent vortices were identified in the flow field, further confirming the accuracy of the numerical schemes. The influence of Mach number on jet morphologies and trajectories was investigated under constant injection velocity conditions. As the Mach number increases, the liquid jet exhibits reduced penetration and faster fragmentation. The column breakup mechanism shifts from the development of axial instability waves at lower Mach numbers to intense aerodynamic drag at higher ones. By analyzing the relationship between the bow shock wave and jet deflection, a bow shock model was proposed and validated using simulation results. Furthermore, a theoretical model was developed to describe the primary breakup process of LJSC, incorporating the effects of critical forces, jet deformation, mass stripping, and the bow shock wave. The proposed model demonstrated significantly higher accuracy in near-field trajectory prediction compared to existing empirical correlations and proved applicable across a wide range of Mach numbers.

Axisymmetric direct numerical simulations are carried out to study the hydrodynamics of a laminar, stationary, incompressible, Newtonian, single-phase flow in an Annular Plane Couette (APC) channel. This configuration is that of a Straight Plane Couette (SPC) flow but curved around itself to form an annulus. These simulations are validated by Particle Image Velocimetry measurements at different Reynolds numbers. The flow is analyzed using three dimensionless parameters: the channel aspect ratio $A_c$, which controls the effects of sidewall confinement, the channel curvature ratio $C_r$, which affects the centrifugal forces due to curvature, and the Reynolds number $\mathrm{Re}$. The rotation of the top annular plate generates a main flow in the azimuthal direction, while generating a secondary recirculation flow in the plane of the channel cross section due to the presence of a centrifugal force difference across the radius. As a result, the vertical profile of the longitudinal velocity deviates from the classical linear profile of the SPC flow, adopting an unexpected S shape similar to that observed in a turbulent SPC flow. Depending on the values of $A_c$ and $C_r$, the flow exhibits a wide range of behaviors, from a quasi-2D flow with a homogeneous shear rate at moderate $\mathrm{Re}$ with appropriate geometrical parameters ($A_c\ge 5$, $C_r\le 0.1$) to a complex 3D flow otherwise. Whereas it is laminar, the APC flow shares a strong analogy with a turbulent Taylor-Couette (TC) flow. At large Reynolds numbers, the velocity gradients concentrate near the wall and the flow reaches an asymptotic regime where the torque scales as ${\mathrm{Re}}^{\alpha }$ and the flow structure becomes independent of $\mathrm{Re}$. Such properties make the APC flow an interesting configuration for fundamental investigations as a complement of the TC flow when shear and gravity are required to be in the same plane. In particular, it can be used to investigate the rheology of a dispersed two-phase mixture similarly to previous work by Yi et al. in the TC device.

This study conducts a systematic investigation into the effects of particle shape on granular flow dynamics through discrete element method (DEM) simulations, with particular emphasis on rheological properties of superballs in plane shear with gravity. Comparative analysis reveals that superballs exhibit distinct flow characteristics relative to spherical particles, including pronounced boundary effects, increased equivalent bulk viscosity, and modified velocity profiles. A generalized rheological framework is established through the development of particle shape-dependent constitutive model that quantitatively correlates particle shape parameters with effective friction coefficient. This model incorporates a dimensionless granular temperature to account for nonlocal effects, enabling comprehensive rheological characterization across both quasistatic and inertial flow regimes. Validation through continuous simulations, achieved by implementing rheology-dependent viscosity in the Navier-Stokes equations, demonstrates excellent agreement with corresponding DEM results in predicting transient flow behavior during granular collapse. These findings establish a robust theoretical foundation for predicting flow behaviors in granular systems with complex particle geometries, providing significant advancements in the modeling and analysis of natural and industrial granular materials.

A solvable model for space-time correlations of passive scalars advected by turbulent flows is proposed in the present paper. The model is a combination of Taylor's frozen-flow model and Kraichnan's random-sweeping and white-noise models, namely the Taylor–Kraichnan (TK) model, in which the advection velocity is the sum of a constant mean velocity, a random-sweeping velocity and a white-noise velocity. We find the exact solution of the TK model and consequently obtain analytical expressions for the space-time correlation. These results are used to demonstrate the scale invariance of space-time correlations and the self-similarity of iso-correlation contours in the inertial-convective subrange. In particular, the elliptic approximation to space-time correlations is justified and thus used to rescale spatial and temporal separations, collapsing a family of space-time correlations into a single curve.

The gap between two solid, coaxial cylinders is filled with two ferrofluids of differing viscosity and magnetic susceptibility. Axial motion is driven by a pressure gradient and/or translation of the inner cylinder. An axial magnetic field is imposed externally, while current flowing in the inner cylinder generates an azimuthal field. It is known that the axisymmetric capillary instability of the cylindrical fluid interface may be controlled either by hydrodynamic shear or magnetic effects, but each of these can give rise to other instabilities, often nonaxisymmetric. This paper investigates the 10-dimensional parameter space to determine when combined action of shear and magnetic effects can provide overall stability, essentially using shear to stabilize the troublesome helical "tearing" mode, while magnetic stresses control the hydrodynamic instability. Stability is easier to obtain when the inner fluid is more magnetic.

This paper focuses on designing a versatile state-space representation for parallel shear flows to facilitate both modal analysis and closed-loop control. Our objective in designing a control law is to regulate shear flows around a stationary desired state—a laminar flow—despite disturbances while ensuring the physical interpretability of the decision-making process. Our closed-loop control implementation involves linearizing the Navier-Stokes equations around the desired state, spatially discretizing the resulting linear system, and computing the feedback gain using an optimal control formulation. We assume that the actuators operate at the boundaries and model their effects as a forcing term using the elimination method rather than the lifting method, the former being standard in partial differential equations solvers but less common in flow control. Furthermore, we assume that the flow state is reconstructed from image sensors. To illustrate the efficiency of our approach, we derive a state-space representation for both the 2D periodic channel flow and the 2D spatially developing mixing layer flow. Hydrodynamic stability analysis confirms the representativeness of the state matrix and identifies the primary natural modes with exponential growth. The resolvent analysis highlights forcing modes, analogous to Orr structures in 2D shear flows, that induce misaligned response modes, analogous to natural modes, with the most significant transient energy growth. These observations provide insights for designing controllers that mitigate disturbances and enhance flow stability. Balanced truncation corroborates these observations and suggests an opposing control mechanism in which the control signal is derived by projecting the flow state onto the adjoint balanced modes. This approach targets dominant dynamic features, particularly the Orr structures, for effective control. For the two flows, linear quadratic regulator controllers are deduced from the full linear state-space model and implemented within a nonlinear Navier-Stokes solver. Additionally, for the periodic channel flow case, pole placement controllers are deduced from the reduced-order state-space model obtained via balanced truncation, as this reduced model naturally lends itself to effective pole placement. All simulation results confirm that these control laws stabilize the 2D periodic channel flow, preventing the development of unstable modes. Furthermore, they delay the onset of large eddies in the 2D spatially developing mixing layer by attenuating relatively slow endogenous disturbances in the control signal.

We investigate the dynamics of small inertial particles in a two-dimensional, steady Taylor-Green vortex flow. A classic study by Taylor (1940) showed that infinitely heavy inertial point particles (having density parameter $R=1$) are trapped by the flow separatrices when the particle Stokes number $\text{St}$, which measures the particle's inertia, is less than $1/4$. Here, we consider finitely heavy particles, incorporating the previously neglected effects of added mass and the Boussinesq-Basset history force. Using linear stability analysis near fixed points, we determine the critical parametric conditions in the $\text{St}\text{−}R$ plane that lead to particle trapping within vortex cells. Including the added mass effect identifies additional fixed points beyond the traditional ones located at the vortex cell corners, and we analyze their stability. Numerical analysis of the full nonlinear system confirms the existence of distinct particle behaviors—trapped, diffusive, and ballistic—depending on initial conditions, consistent with Nath et al. [Phys. Rev. Fluids 9, 014302 (2024)], with modifications due to added mass effect. We delineate the regions in the $\text{St}\text{−}R$ plane where these behaviors dominate based on the prominent particle dynamics. However, when both the history force and added mass effect are included, all particles exhibit ballistic motion regardless of $\text{St}$ and $R$.

Concentrated noncolloidal particles under nonuniform shear can migrate across streamlines even at low Reynolds numbers, breaking the well-known reversibility of Stokes flow. We conduct systematic experiments to study this phenomenon by oscillating a finite suspension plug comprising neutrally buoyant noncolloidal particles inside a cylindrical tube with varying strain amplitudes $\gamma$. We visualize the particles on the central axial plane of the tube via refractive index matching and track the motion of individual particles. Consistent with the previous findings, our experiments show that suspension plugs under oscillatory flows dilute to a critical volume fraction and become reversible in overall shape. However, our measurements reveal persistent particle motion even after this critical volume fraction threshold is reached, which points to a previously unreported disconnect between particle-scale and system-scale reversibility. In addition, we build a simplified two-dimensional model of particles under nonuniform shear and capture the gradual dispersion of the suspension plug, in qualitative agreement with the experiments.

Extreme wall shear stress (WSS) events, usually conditioned to the streamwise WSS, are examined with a further constraint of the azimuthal WSS through direct numerical simulation of turbulent pipe flow at a frictional Reynolds number ${\text{Re}}_{\tau }\approx 500$. In light of the segmented characteristics identified in the statistical distribution of the WSS ${\tau }_{\theta }^{+}$, both extreme positive (EP) events and backflow (BF) events are further categorized into two distinct types, with the help of the proposed local extremum intersection method. The four events each have their own unique probability of occurrence, and those with larger ${\tau }_{\theta }^{+}$ are found to be more intense through the joint probability density function of ${\tau }_x^{+}$ and ${\tau }_{\theta }^{+}$. By conducting a conditional average on the regions of these events, significant differences in their vortical structures and related characteristics have been identified, distinguishing them from previous research findings. On the one hand, the different formation mechanisms of the subdivided EP events' structures are clearly revealed, with the presence of the second vortex on the opposite side of the dominant streamwise vortex in EP leading to a weakening of the spanwise motion. On the other hand, the research on the subdivided BF events demonstrates that the flow characteristics show a close correlation with tilt angle of the dominant oblique vortex, with larger angles tending to induce stronger backflow phenomena. Moreover, a relationship between angles and azimuthal shear stress is established, which roughly follows a power-law distribution. This study highlights that introducing additional new constraints could refine or expand the currently widely accepted classification framework for extreme WSS events, enhancing current understanding of them.

In boundary-layer turbulence, when the distance to the wall is within the logarithmic regime, the streamwise velocity structure function can be divided into three regimes by the horizontal scale in the order of distance to the wall, $z$, and the boundary-layer thickness, $\delta$. In the inertial range, when the streamwise two-point distance $r$ is less than $z$, the velocity structure function satisfies a power-function dependence on $r/z$. In the logarithmic range when $z<r<\delta$, the velocity structure function has a logarithmic dependence on $r/z$. In the outer range when $r>\delta$, the function approaches a constant value. This work shows that the above three regimes for the velocity structure function also apply to the mixed third-order structure function constituted by the velocity and potential temperature differences $\langle \delta u\delta {\theta }^2\rangle$. Here, $\delta u$ and $\delta \theta$ are the streamwise differences in streamwise velocity and potential temperature, respectively. We derive the logarithmic dependence of the mixed third-order structure function from the Kármán-Howarth-Monin equation for the potential energy with the help of the attached eddy hypothesis. The inertial range and logarithmic range are identified by analyzing data from the Qingtu Lake Observation Array. The transition scale between the two ranges decreases with an increase of $z/L$ in weak stable conditions where $L$ is the Obukhov length.

Liquid films coating vertical cylinders can form annular liquid collars which translate downwards under gravity. We investigate the dynamics of a thin viscoplastic liquid film coating the interior or exterior of a vertical cylindrical tube, quantifying how the yield stress modifies both the Rayleigh-Plateau instability leading to collar formation and the translation of collars down the tube. We use thin-film theory to derive an evolution equation for the layer thickness, which we solve numerically to examine the nonlinear dynamics. Instability and collar formation occur when gravity is sufficiently strong to make the fluid yield initially. We use matched asymptotics to derive a model describing the quasisteady translation of a slender liquid collar when the Bond number is small. The structure of the asymptotic solution for a viscoplastic collar shares some features with the Newtonian version, but there are several novel asymptotic regions that emerge at the two ends of the collar. The global force balance, which determines the collar's speed, is modified by a leading-order contribution from viscous drag in the collar when the liquid is viscoplastic. We use the asymptotic model to describe slow changes in collar volume when the film thicknesses ahead of, and behind, the collar are unequal. When the film thickness ahead of the collar is less than a critical value that we determine, viscoplastic collars adjust their volume and reach a steadily translating state. This contrasts with the Newtonian problem, where the only state in which steady translation occurs is unstable to small changes in the film thickness.

Understanding the friction forces acting on droplets moving across non-wetting surfaces is essential for advancing applications in fluid transport and microfluidics. In this study we quantify droplet friction on both superhydrophobic and superheated (Leidenfrost) substrates using two complementary experimental approaches: inclined plane dynamics and cantilever force sensing. We identify distinct friction regimes governed by contact-line pinning, viscous dissipation, and aerodynamic drag over a broad range of velocities and viscosities. A force-based scaling law captures the transition between these regimes as a function of the capillary number, integrating contact-line effects, internal dissipation, and aerodynamic contributions. Our model is further validated through white light interferometry and high-speed reflective interference contrast microscopy, which reveal the structure and evolution of the thin air or vapor layers beneath the droplets. These findings offer a comprehensive framework for predicting droplet behavior on non-wetting surfaces and inform the design of surfaces for reduced friction in engineering and microfluidic systems.

We study locomotion of a model crawler corresponding to a deforming upper boundary of finite length above a thin Newtonian fluid film whose viscosity varies spatially. We first derive a general locomotion velocity formula with fluid viscosity variations via the lubrication theory. For further analysis, the surface of the crawler is described by a combination of transverse and longitudinal traveling waves and we find that under a uniform viscosity a transverse wave results in a retrograde crawler, while a longitudinal wave leads to a direct crawler. We then analyze the time-averaged locomotion behaviors under two scenarios: (i) a sharp viscosity interface and (ii) a linear viscosity gradient. Using the asymptotic expansions of small surface deformations and the method of multiple timescale analysis, we derive an explicit form of the average velocity that captures nonlinear, accumulative interactions between the crawler and the spatially varying environment. (i) In the case of a viscosity interface, the time-averaged speed of the crawler is always slower than that in the uniform viscosity, for both the transverse and longitudinal wave cases. Notably, the speed reduction is most significant when the crawler's front enters a more viscous layer and the crawler's rear exits from the same layer. (ii) In the case of a viscosity gradient, the crawler's speed becomes slower for the transverse wave, while for the longitudinal wave, the locomotion speed does not change significantly. Our analysis illustrates the fundamental importance of interactions between a locomotor and its environment, and separating the timescale behind the locomotion.

This experimental study examines the dynamics of the shock-induced breakup of a liquid drop within the shear-strip breakup regime. The interaction of the shock wave with the interface leads to its reflection and transmission. The reflected shock wave from the windward side of the drop, combined with the merging of the Mach stem at the downstream of the leeward side, creates an uneven pressure distribution across the drop. The compression of the windward face results in the formation of a surface wave, which is intensified by the shearing of flow. This process redistributes the liquid across the drop, resulting in the formation of a dense core and a peripheral meridional zone. With increasing strength of the Mach stem, a surface wave develops on the leeward side. The evolution of the surface wave is found to be governed by a shear-dominated cascading instability mechanism, suggesting the presence of multiscale instability. The surface wave on the windward and leeward sides collides during the drop deformation, transferring its momentum in the cross-stream direction. The acceleration of flow into the azimuthal ring causes corrugations to appear in the transverse direction. These corrugations are found to be governed by the Rayleigh-Taylor instability mechanism. These corrugations are further developed into fingers. These dynamics result in the identification of two submodes: ligament and cellular. In the ligament breakup mode, the radial fingers elongate streamwise to form a ligament that breaks due to surface tension. In cellular breakdown, the fingers evolve into U-shaped structures that combine with the azimuthal ring to form cellular structures. The azimuthal ring detaches from the deforming drop as the flow accelerates through it. The coupled Rayleigh-Taylor and Rayleigh-Plateau mechanisms govern the breakup of the azimuthal ring. An additional instability in the deforming drop leads to the formation of a secondary azimuthal ring. The core deforms as the flow accelerates into it. This deformation is governed by a Rayleigh-Taylor instability mechanism.

In hydrodynamic (HD) and magnetohydrodynamic (MHD) turbulence an energy cascade can occur when there is a substantial range of scales—the inertial range (IR)—over which the energy transfer process is self-similar. An elementary question is the following: What are the minimum Reynolds numbers needed for a turbulent flow to have a substantial IR, that is, one of a given bandwidth. The effect of finite Reynolds numbers on the IR has been well studied in HD turbulence, which provides a roadmap for analogous study in MHD turbulence. To help identify the minimum Reynolds number associated with a substantial IR, we conduct a series of incompressible MHD turbulence simulations with Taylor Reynolds numbers ${\mathrm{Re}}_{\lambda }\approx 200\text{–}400$. These are analyzed in terms of the MHD third-order law, one of the few rigorous results concerning MHD turbulence energy transfer within a well-defined IR. Specifically, we use simulations and modelled structure functions to estimate the IR boundaries as a function of ${\mathrm{Re}}_{\lambda }$. Following studies in HD turbulence, a semiempirical power-law model is applied to structure function behavior, which allows us to extrapolate to ${\mathrm{Re}}_{\lambda }$ that are much higher than those accessible with experiments or simulations. The present results estimate the minimum Reynolds number needed to establish an IR spanning a specified number of decades. For MHD flows whose Reynolds numbers satisfy the criteria, this provides a justification for the use of the energy cascade scenario and the third-order law.

We present direct numerical simulations (DNS) of particle deposition in a turbulent channel flow, incorporating a viscoelastic soft-sphere collision model with van der Waals adhesion. Particle-wall contact is governed by an adhesion number that varies with temperature, enabling exploration of a wide range of deposition behaviors. Deposition is strongly heterogeneous, especially for inertial particles, where rolling and sliding enhance nonuniformity. Spanwise radial distribution functions reveal that deposited particles form streaks with characteristic spacing set by near-wall two-point velocity correlations. A clustering metric confirms that high-inertia, low-adhesion particles deposit in elongated, anisotropic patterns due to spanwise particle creep driven by velocity fluctuations. Finally, it is shown that this heterogeneity in deposition leads to localized wall wear rates exceeding 10 times the mean, with the most severe wear associated with particles that are carried to the wall in clusters by sweep events.