When calculating the two-dimensional Stokes flow about two identical cylinders that counterrotate about fixed axes, it is impossible to obtain a solution that decays at large distances. When considering the corotation variant of that “Jeffery's paradox,” Watson [Mathematika 42, 105 (1995)] illustrated that the comparable impediment can be resolved by the introduction of an external torque on the cylinder pair. Both the corotation and counterrotation problems were recently revisited by Dormy and Moffatt [Phys. Rev. Fluids 9, 044102 (2024)] who focused upon the case where the separation $\varepsilon a$ between the cylinders is small compared with their radius $a$. In the corotation problem, they showed that the torque calculated using Watson's solution approaches a finite limit as $\varepsilon \to 0$. In the counterrotation problem, they introduced a remote boundary of radius $a{R}_0$, rendering the problem well-posed provided an external force is applied on the cylinder pair. Dormy and Moffatt employed a numerical scheme to solve the confined problem for finite values of $R_0$ and $\varepsilon$. Given the numerical results, they conjectured that the force decays inversely with $ln{R}_0$ as $R_0\to \infty$. We here address the zero-separation limit $\varepsilon =0$ from the outset, obtaining solutions for the unconfined problem via integral transforms. In the corotation problem, we obtain the torque as a closed-form quadrature, thus avoiding the need to extrapolate from a finite-separation solution. In the counterrotation problem we solve the flow field in the presence of an external force. This solution provides an asymptotic “inner” approximation for the respective confined problem in the limit $R_0\to \infty$, with the cylinder pair appearing as a point singularity in the “outer” approximation. Analyzing that problem using matched asymptotic expansions, we derive approximations for the force for both no-slip and shear-free boundaries, confirming the conjecture of Dormy and Moffatt.

The high-order surface quasigeostrophic $\left({\mathrm{SQG}}^{+1}\right)$ turbulence is investigated numerically in this work with a focus on how lateral divergence modifies cross-scale energy transfers. The ${\mathrm{SQG}}^{+1}$ model extends the classic SQG model by incorporating a velocity correction that introduces controlled divergence, whose strength is controlled by a characteristic Rossby number. We demonstrate that, at the minimal Rossby number, the system exhibits near-SQG behaviors. The potential energy cascades forward and converts to the inversely cascading kinetic energy through frontolysis, with negligible divergence effects. At higher Rossby numbers, however, divergence reorganizes the flow into cold cyclones embedded in expansive warm filaments and fundamentally alters the energy transfer dynamics. Our derived seven-term balance reveals that the compression of cold cyclones by large-scale background flows drives kinetic energy downscale, while the expansion of warm filaments sustains the upscale potential and kinetic energy transfers. These divergence-driven fluxes intensify with the increasing Rossby number, systematically energizing large-scale background flows. This work provides the first quantification of the divergence-driven energy fluxes in balanced submesoscale turbulence.

Here we show that passive scalars possess a hidden scaling symmetry when considering suitably rescaled fields. Such a symmetry implies (i) universal probability distribution for scalar multipliers and (ii) Perron-Frobenius scenario for the anomalous scaling of structure functions. We verify these predictions with high resolution simulations of a passive scalar advected by a 2D turbulent flow in inverse cascade.

Stochastic resonance (SR), the phenomenon discovered more than 40 years ago in bistable nonlinear dynamical systems in the presence of weak periodic forcing and external noise, results in strong amplification of the periodic signal, though at shifted frequency. Recently we reported the SR discovery in inertialess viscoelastic channel flow just above pure elastic instability onset in a limited lower subrange of the transition chaotic flow regime, where a sharp high-energy peak in the streamwise velocity chaotic power spectrum, $E_u$, is observed unexpectedly. The spanwise velocity power spectrum, $E_w$, remains flat, indicating white noise, and is accompanied by weak intensity elastic waves. These three necessary conditions for the SR emergence are consistent with those found in autonomous deterministic dynamical systems exhibiting deterministic stochastic resonance (DSR). Notably, DSR disappears in the upper subrange of the transition regime, where $E_w$ becomes chaotic followed by strongly increased elastic wave intensity. Here we clarify the mechanism of DSR emergence by experimentally verifying three key ingredients of the DSR mechanism: chaotic $E_u$, flat white noise $E_w$, and weak elastic waves at multiple channel locations. Finally, we present a DSR existence range as a function of $\mathrm{Wi}$ and $u_{\mathrm{rms}}$, the root mean square of streamwise velocity fluctuations along the channel length, as another control parameter.

The translation and shape deformations of a passive viscous Newtonian droplet immersed in an active nematic liquid crystal under circular confinement are analyzed using a linear stability analysis. We focus on the case of a sharply aligned active nematic in the limit of strong elastic relaxation in two dimensions. Using an active liquid crystal model, we employ the Lorentz reciprocal theorem for Stokes flow to study the growth of interfacial perturbations as a result of both active and elastic stresses. Instabilities are uncovered in both extensile and contractile systems, for which growth rates are calculated and presented in terms of the dimensionless ratios of active, elastic, and capillary stresses, as well as the viscosity ratio between the two fluids. We also extend our theory to analyze the inverse scenario, namely, the stability of an active nematic droplet surrounded by a passive viscous layer. Our results highlight the subtle interplay of capillary, active, elastic, and viscous stresses in governing droplet stability. The instabilities uncovered here may be relevant to a plethora of biological active systems, from the dynamics of passive droplets in bacterial suspensions to the organization of subcellular compartments inside the cell and the cell nucleus.

Heat transfer in a fluid can be greatly enhanced by natural convection, giving rise to the nuanced relationship between the Nusselt number and Rayleigh number that has been a focus of modern fluid dynamics. Our work explores convection in an annular domain, where the geometry reinforces the large-scale circulatory flow pattern that is characteristic of natural convection. The flow must match the no-slip condition at the boundary, leading to a thin boundary layer where both the flow velocity and the temperature vary rapidly. To understand the system's heat transfer characteristics, we derive a reduced model from the Navier-Stokes-Boussinesq equations, whereby the equations of flow and heat are transformed to a system of low-order partial differential equations (PDEs) that take the form of a reaction-diffusion system. Solutions to the reaction-diffusion system, though they fail to predict dynamic events, preserve the same boundary-layer structure seen in the direct numerical simulation (DNS). By matching the solutions inside and outside the boundary layer, asymptotic analysis predicts a power-law relationship $\text{Nu}\propto \text{Ra}{}^{1/4}$. Though difficult to distinguish from an exponent of $2/7$, the predicted power law agrees well with measurements from DNS over several decades of the Rayleigh number. Considering the model's deficiencies in describing turbulent fluctuations and reversal events, the agreement regarding heat transfer characteristics is encouraging and suggests that the methodology of systematically deriving low-order PDEs from the governing equations may provide a useful complement to existing theories.

The influence of Navier-slip boundary conditions on two-dimensional electroconvective flows in a dielectric liquid confined between two infinite parallel plates is investigated, highlighting their pivotal role in modulating flow stability and transport characteristics. The presence of Navier-slip conditions significantly reduces both the linear and nonlinear instability thresholds, offering a plausible explanation for the persistent discrepancies between experimental observations and numerical predictions of these criteria. A critical slip-length range, $l_s\in [{10}^{-3},10]$, is identified, within which instability thresholds and macroscopic flow features exhibit pronounced sensitivity to $l_s$. In the turbulent regime, the electric Nusselt number scales with the electric Rayleigh number as $\text{Ne}\propto T^{0.5}$ under partial- and free-slip boundary conditions, in contrast to the no-slip case, where a weaker scaling $\text{Ne}\propto T^{0.15}$ is observed. At sufficiently high electric Rayleigh numbers, a transition from convective states to zonal flow states, characterized by large-scale horizontal flows, emerges. Notably, this transition occurs at lower $T$ values as $l_s$ increases. In the zonal flow regime, the observed saturation of the electric Nusselt number provides a possible interpretation of the asymptotic behavior reported in former experimental studies. These results establish a comprehensive framework for understanding the interplay between Navier-slip boundary conditions and electroconvective dynamics, providing critical insights for future experimental validation and theoretical advancements.

The pair of fluids exhibiting temperature-sensitive miscibility gap and phase separation behavior are termed binary fluids with miscibility gap. The dynamic nature of the thermophysical properties allows deployment of such fluids in crucial microfluidic applications, such as enhancement of liquid-liquid mass transfer and flow pattern tuning, to name a few. Therefore, a thorough understanding is necessary to achieve the desired flow feature control. In the present study, a modified phase-field method is proposed to model the continuous transition of binary fluids from immiscible state to a miscible state via partially miscible states. The model is employed to investigate the isothermal single-mode Rayleigh-Taylor (RT) instability for binary fluids as the system temperature is varied. Assuming potential flow and utilizing Boussinesq approximation, we derived the dispersion relation for gravity-capillary waves and the RT instability. For early-time stages, three zones with distinct qualitative behavior for the growth rate are identified as a function of Atwood number and Weber number. Subsequently, the Boussinesq approximation is relaxed to obtain coupled Cahn-Hilliard-Navier-Stokes equations to perform numerical simulations. The results from the numerical simulations corroborate the findings from the dispersion relation at early stages, thereby establishing the validity of the proposed phase-field model. The emanation of secondary instability in the form of Kelvin-Helmholtz rolls is observed. Finally, we present the effect of the slow nature of the diffusion process as compared to the growth rate of the instability as the system is driven toward the state of complete miscibility, starting from a purely immiscible state.

Vortex crystals, ordered structures observed in superconductors and rotating superfluids, have also been hypothesized to form in classical fluids, based on numerical simulations and observations of the Jovian polar atmospheres. We perform direct numerical simulations of the Navier-Stokes equations in rotating frames, to investigate the spontaneous emergence of metastable vortex crystals. We analyze the energy spectrum, vortex morphology, and spatiotemporal dynamics to understand their roles in crystal formation and evolution. In addition, we explore domains with varying aspect ratios to examine their impact on the vortex lattice. Our results indicate a relationship between the crystal lifespan and dissipation, and we propose a scaling law linking the rotation rate, domain geometry, and vortex lattice periodicity. Finally, we identify a critical threshold in the control parameter, the Rossby number, suggesting a behavior similar to that found in phase transitions.

Fluid-fluid interfaces laden with discrete particles behave curiously like continuous elastic sheets, leading to their applications in emulsion and foam stabilization. Although existing continuum models can qualitatively capture the elastic buckling of these particle-laden interfaces, often referred to as particle rafts, under compression, they fail to link their macroscopic collective properties to the microscopic behaviors of individual particles. Thus, phenomena such as particle expulsion from the compressed rafts remain unexplained. Here, by combining systematic experiments with first-principle modeling, we reveal how the macroscopic mechanical properties of particle rafts emerge from particle-scale interactions. We construct a phase diagram that delineates the conditions under which a particle raft collapses via collective folding versus single-particle expulsion. Guided by this theoretical framework, we demonstrate control over the raft's failure mode by tuning the physicochemical properties of individual particles. Our study highlights the dual nature of particle rafts and exemplifies how collective dynamics can arise from discrete components with simple interactions.

This study employs high-fidelity fluid-structure interaction simulations to investigate design optimizations for wave-assisted propulsion (WAP) systems using flapping foils. Building on prior work that identified the leading-edge vortex as critical to thrust generation for these flapping foil propulsors, this work explores pitch control mechanisms and foil geometries to improve performance across varying sea states. Two pitch-limiting strategies—a spring limiter and an angle limiter—are evaluated. Results show that while both perform similarly at higher wave-amplitude sea states, the angle limiter yields superior thrust at the low-wave-amplitude sea state, making it the preferred mechanism due to its simplicity and effectiveness. Additionally, foil geometry effects are analyzed, with thin elliptical and flat plate foils outperforming the baseline NACA0015 shape. The elliptical foil offers marginally better performance and is recommended for WAP applications. A fixed pitch amplitude of $5^{\circ }$ provides thrust across all sea states, offering a practical alternative to more complex adaptive systems. These findings demonstrate how insights into the flow physics of flapping foils can inform the design of more efficient WAP systems and these findings could also inform other biological and bioinspired systems that employ flapping foils.

This study uses stereo-particle image velocimetry to describe the wake behind a multi-wake model at a Reynolds number of $5.64\times {10}^5$ based on the model length $L_x$. All three velocity components are measured in cross-stream planes at various streamwise locations starting from the trailing edge of the model. The time-averaged, three-dimensional flow is reconstructed from the planar measurements, and three important regions in the wake are identified. The near-wake is dominated by a separation bubble that forms immediately downstream of the model. Shear layers, which detach from the sides of the model, morph into a single connected ‘bound vortex’ that stays inside the separation bubble and dominates the flow up to $x/L_x\approx 0.4$. The far-wake is mainly influenced by four corner vortices that originate from the upper and lower rear slant edges. In the near-wake region, the corner vortices enhance the inward mass and momentum transfer in the vertical direction and oppose them in the spanwise direction. As such, they warp the shape of the separation bubble although their overall effect on the internal flow within the separation bubble is minor. In the far wake, they redistribute the momentum in the vertical-spanwise plane. Their induced flow reduces the momentum deficit in the central streamwise-vertical plane relative to off-center planes. It can be concluded that it is the corner vortices which dominate the far-wake and that they remain coherent several body lengths behind the vehicle. Similar to the Ahmed body, the slant angles play a major role on the evolution of the wake which has implications for reducing drag.

In a confined flow, the performance of a turbine and its near-wake fluid dynamics depend on the blockage ratio, defined as the ratio of the turbine projected area to the channel cross-sectional area. While blockage is understood to increase the power coefficient for turbine “fences” spanning a channel, most investigations at the upper range of practically achievable blockage ratios have been theoretical or numerical in nature. Furthermore, while linear momentum actuator disk theory is frequently used to model turbines in confined flows, as confinement increases, the ability of this idealized model to describe performance and flow fields has not been established. In this work, the performance and near-wake flow field of a pair of cross-flow turbines are experimentally evaluated at blockage ratios from $30\%$ to $55\%$. The fluid velocity measured in the bypass region is found to be well predicted by the open-channel linear momentum model developed by Houlsby et al. [Application of Linear Momentum Actuator Disk Theory to Open Channel Flow, Technical Report OUEL 2296/08 (University of Oxford, Oxford, 2008)], while the wake velocity is not. Additionally, self-similar power and thrust coefficients are identified across this range of blockage ratios when array performance is scaled by the modeled bypass velocity following Whelan et al. [J. Fluid Mech. 624, 281 (2009)] adaptation of the bluff-body theory of Maskell [A Theory of the Blockage Effects on Bluff Bodies and Stalled Wings in a Closed Wind Tunnel, Reports and Memoranda 3400 (Ministry of Aviation, 1963)]. This result demonstrates that, despite multiple nonidealities, relatively simple models can quantitatively describe highly confined turbines. From this, an analytical method for predicting array performance as a function of blockage is presented. Overall, this work illustrates turbine performance at relatively high confinement and demonstrates the suitability of analytical models for predicting and interpreting their hydrodynamics.

In this study, we report intermittent bursting of cyclonic columnar structures in direct numerical simulations of forced rotating turbulence in a three-dimensional ($3\text{D}$) periodic box. Columnar structure formation is associated with dominant inverse cascade of energy in addition to the typical forward cascade at large wave numbers. The forcing is random and isotropic, applied in two distinct wave-number bands ($k_f=[3,4]$ and [6,7]). Notably, bursting is observed only for small forcing wave number, $k_f=[3,4]$ and not at $k_f=[6,7]$, for the same rotation rate and initial conditions. These bursting events are associated with forward cascade of energy in contrast to the inverse cascade of rotating turbulence, as confirmed from the ring spectrum and time series of modal kinetic energy of modes with $k_z=2,3$. Identifying some of the triads involved in the energy transfer from modal analysis, we also calculate the mode-to-mode energy transfer. We find that, before the event, the circular cyclonic vortex becomes elliptical due to cyclone-anticyclone interactions leading to elliptical instability. We argue that energy transfer preceding the bursting events to modes with higher $k_z$ triggers the Crow instability. These instabilities ultimately lead to bursting of the columnar vortex, which subsequently forms again due to the external rotation. Our results portray a tug-of-war between the destabilizing effect of elliptical instability and the stabilizing effect of external rotation.

A phenomenon called thermal striping, consisting of quasiperiodic temperature oscillations, is of significant concern in liquid-metal-cooled nuclear reactors. While it is believed to be caused by a shear-flow instability, the physical mechanism is not fully understood, and is thus difficult to predict. We consider plane Poiseuille flow with stable density stratification in the wall-normal direction as a model for hydrodynamic instability in a heated, wall-bounded shear flow characteristic of thermal striping scenarios. The temporal linear stability eigenvalue problem is then solved, and the neutral curves are calculated for different stratification levels. Excellent collapse of the neutral curves is observed in the limit of small Péclet number, $\text{Pe}\lesssim 1$, when curves are shown as a function of a modified Richardson number, $\mathcal{R}=\text{Ri}·\text{Pe}$. Furthermore, the normal modes are stable for all Reynolds numbers when $\mathcal{R}$, is greater than some critical value ${\mathcal{R}}^{*}$, with ${\mathcal{R}}^{*}\doteq 0.3338$. We further develop a perturbation solution of the low-Péclet-number equations (LPNE) of Lignières [Astron. Astrophys 348, 933 (1999)], which contains $\mathcal{R}$ as the natural parameter controlling buoyancy forces. The perturbation eigenvalue problem is solved to $\mathcal{O}\left(\mathcal{R}\right)$. The perturbation of the growth rate is negative, indicating stratification has a strictly stabilizing effect. An analytical solution in the infinite Re limit is developed for the perturbation system, which also confirms stratification's stabilizing effect. The relevance of the results to thermal striping in liquid-metal-cooled nuclear reactors is also discussed.

Biogeochemical scalars, representing many different species of phytoplankton, zooplankton, nutrients, etc., play an important role in marine ecosystems. Here we examine a simplified two-species system of generic phytoplankton and nutrient populations, modeled here as reacting scalars governed by advection-diffusion equations with added terms governing the biological reaction (nutrient uptake, phytoplankton death/remineralization). We perform direct numerical simulations of an idealized scenario in which initial layers of phytoplankton and nutrients are separated by an unstable stratified shear layer susceptible to either Kelvin-Helmholtz or Holmboe instability (canonical examples of overturning and scouring mixing events, respectively). We further vary the ratio of the reaction and flow timescales, considering cases of fast, slow, and comparable reaction rates. We find that turbulent advective fluxes show large oscillations during the evolution of the shear instabilities, associated with reversible fluid motions. To isolate the irreversible turbulent scalar fluxes, we instead calculate diascalar fluxes, that is, diffusive fluxes across isoscalar surfaces. This concept has been previously used to quantify irreversible mixing of buoyancy in stratified mixing events; here, we extend this approach to also account for the effects of biogeochemical reactions. With this analysis, we show that both instabilities accelerate phytoplankton growth, that the KH-driven mixing tends to have a larger impact on the phytoplankton growth than the Holmboe flow, and that the largest impact of the flow on the scalar evolution occurs when reaction timescales are comparable to the buoyancy period.

Lagrangian particles in turbulence separate away from each other faster in the backward in time direction as compared to forward in time. This work shows that this time asymmetry can be explained on the grounds of kinematics: when viewed backward in time, the particles' relative velocities are more aligned with the separation vector as compared to when viewed forward in time. This makes the separation process backward in time more efficient, leading to the observed time asymmetry.

Today's weather forecast is based on a simulation of established laws of physics and mathematics. Despite important progress in computing power, observations, and mathematics, predicting the weather reliably for more than a few days has always been a major challenge. In the late 1950s and 1960s, Lewis Fry Richardson, and later, Edward Lorenz, set up the basis for the understanding of this challenge, using observations, phenomenological arguments, and low-order models. Here, we first summarize their findings, then discuss some of the recent advances in this domain, made possible through the advent of new numerical and theoretical tools.

This study investigates the linear stability of viscosity-stratified miscible two-layer flows in a channel, addressing the inherent complexity introduced by the time-dependent base state resulting from diffusion of solute concentration at the interface. Unlike previous studies that rely on the quasi-steady-state approximation (QSSA), which neglects the transient nature of the base state by assuming it remains frozen, we employ an initial value problem (IVP) approach to incorporate the unsteady base state into the stability analysis. Employing the IVP approach, we solve the linearized incompressible Navier-Stokes and convection-diffusion equations. We demonstrate that, unlike QSSA, the IVP method effectively provides critical insights into early-time perturbation growth, primarily driven by initial diffusion, and accurately predicts the onset time of instability. Through energy amplification analysis, we compute perturbation growth rates over time, revealing their nonmonotonic dependence on the log-mobility ratio ($R$) and mean interface location ($h^{\prime }$). We show that the growth rates increase monotonically with increasing Reynolds ($\text{Re}$) and Péclet ($\text{Pe}$) numbers, and decrease with increasing interface thickness ($\delta$). Systematic analyses further unveil a nonmonotonic dependence of onset time of instability ($t_{\text{on}}$) on $R$, identifying critical values of the log-mobility ratio for instability onset. These critical values, $R_l^c$ and $R_u^c$, are influenced by $\text{Re}, \text{Pe}, h^{\prime }$, and $\delta$. Through an energy budget analysis of concentration perturbations, we highlighted the interplay between convection and diffusion in determining the overall perturbation growth rates, thereby explaining the observed nonmonotonic variation with the log-mobility ratio $R$.

This study investigates the time evolution of vortex rings generated by the normal translation of a disk either toward or away from a wall. We systematically vary the control parameters, including the disk size, stroke length, travel time, and distance from the wall, to analyze their influence on vortex dynamics. Experiments are conducted with particle image velocimetry, while numerical simulations are performed using the flow solver Basilisk. The circulation and core radius are used as primary metrics to describe the properties of the vortex ring. A quantitative agreement is observed between the experimental and the numerical results. Both approaches reveal that the vortex circulation increases for disk motions in either direction—toward or away from the wall. However, the effect of the bottom wall on the vortex core radius differs depending on the motion of the disk. The presence of the wall increases the core radius only when the disk moves toward the wall, while no significant effect is observed for a translation away from the wall. Furthermore, we establish scaling laws to describe the maximum circulation and core size of the vortex ring as a function of the control parameters: the disk diameter, the typical time and stroke length of its motion, and the minimal distance from the wall. These scalings, which differ from the unbounded case, contribute to a deeper understanding of vortex dynamics in the vicinity of solid boundaries.