We revisit the vortex gas scaling theory for heat transport by baroclinic turbulence based on the empirical observation that the vortex core radius departs from the Rossby deformation radius for very low bottom drag coefficient. We derive a scaling prediction for the vortex core radius. For linear bottom drag this scaling dependence for the vortex core radius does not affect the vortex gas predictions for the eddy diffusivity and mixing length, which remain identical to those in Gallet and Ferrari [Proc. Natl. Acad. Sci. USA 117, 4491 (2020)]. By contrast, for quadratic drag the scaling dependence of the core radius induces new scaling laws for the eddy diffusivity and mixing length when the quadratic-drag coefficient becomes asymptotically low. We validate the modified scaling predictions through numerical simulations of the two-layer model with very low quadratic-drag coefficient.

Reaction fronts sustained by fluid mixing play a central role in a large range of porous media systems and applications. In many cases, pointwise continuous injection of a reactant that displaces a resident reactant in three dimensions leads to a growing spherical reaction front. While such configurations have until now been studied under the assumption of a constant diffusion coefficient, in porous media the dominant diffusive process at the continuum scale is hydrodynamic dispersion, which depends linearly on the flow velocity. Here we analyze the impact of this nonuniform and time-varying hydrodynamic dispersion on reactive transport occurring in such a spherically advected reaction front under point injection and at constant flow rate. During an initial transient regime, dispersion leads to a more advanced reaction front and a larger global reaction rate than when molecular diffusion is the only mixing process, as well as to different temporal scalings for the reaction front properties. At larger times, the reaction front eventually reaches a steady state, characterized by a static position and time-independent reactant concentrations and reaction rate, regardless of the presence and strength of dispersion. When dispersion is weak, the steady-state front is positioned in a region where dispersion is negligible compared to diffusion. Conversely, when dispersion is large, the steady-state front is positioned in the transition zone where dispersion and diffusion are comparable. Under this condition, hydrodynamic dispersion permanently affects the reaction front's transport by altering the steady state itself and augmenting the global reaction rate.

The present series of papers studies the coupling between transport phenomena and the dynamic response of droplets in vortical flows. We derive the nonlinear relations between the vortical flow field, droplet relaxation time, drag forces, and transport phenomena and analyze the dynamics using a Lagrangian particle tracking method. The present work, paper I of the series, aims to obtain a more thorough understanding of the role of mass transport and phase change in the dynamics of droplets in the presence of Burgers vortices. Such an analytically described flow structure serves here as a model that may capture the essence of turbulent flows. Our theoretical study highlights the role of three dimensionless parameters, i.e., the vortex Euler number, the vortex Reynolds number, and the droplet initial Stokes number, characterizing the droplet transport phenomena in the vicinity of Burgers vortices. Under certain thermodynamic and hydrodynamic conditions, droplets may undergo evaporation and condensation when circulating the vortex core due to sharp changes in the environmental conditions induced by the vortex. The thermodynamic gradients give rise to complex dynamics. Droplets may stabilize around the vortex, revealing a periodic solution that emerges only when the pressure drop is high enough to generate a condensation core. This periodic behavior of the dynamic system may suggest droplet clustering induced by phase change, while the emergence of the condensation region reveals a distinct bifurcation point. Furthermore, the response of solid particles significantly differs from the droplets' response; mass transport to and from the droplets inevitably alters their trajectories relative to particles. The model derived here suggests a distinctive outlook on the role of mass transport in governing the dynamic response of droplets in vortical flow; the following paper in the series [O. Avni and Y. Dagan, following paper, Droplet dynamics in Burgers vortices. II. Heat transfer, Phys. Rev. Fluids 8, 083605 (2023)] focuses on the role of heat transfer in the equilibrium and transient response of droplet-vortex systems.

The preceding paper [O. Avni and Y. Dagan, preceding paper, Droplet dynamics in Burgers vortices. I. Mass transport, Phys. Rev. Fluids 8, 083604 (2023)] presented a theoretical formulation and a numerical model aiming to obtain a more thorough understanding of the role of mass transport and phase change in the dynamics of droplets in the presence of Burgers vortices. Such vortical environments abruptly change the thermodynamic state within the laminar vortex core, giving rise to a unique droplet orbital clustering mechanism on the verge of the vortex condensation core. Here we provide an extension of the previous model to account for the role of heat transfer in the equilibrium and transient response of droplet-vortex systems. Such systems might serve as a case model for a broader range of interactions between droplets and vortical flows. A scaling analysis is provided to elucidate the coupled transport mechanism and uncover the timescale characteristic of each. Finally, we analyze the extent to which nonlinear transport processes play a significant role in the droplet's response dynamic response. Although simplified, our analysis may serve as an estimation for the thermodynamic conditions in which the incorporation of nonlinear effects could be substantial.

The prolonged temporal evolution of a magnetohydrodynamic (MHD) vortex influenced by a steady magnetic field along its axis is investigated both numerically and experimentally within a cubic domain. We directly validate the theory proposed by Davidson [J. Fluid Mech. 299, 153 (1995)] through numerical analysis: the angular momentum parallel with a magnetic field of a single vortex is conserved, whereas the perpendicular angular momentum decayed exponentially during a free-decay evolution. Moreover, as observed by Sreenivasan and Alboussière [J. Fluid Mech. 464, 287 (2002)], the initial linear phase, characterized by the dominant influence of the Lorentz force over the inertial force ($N_t\gg 1$, where $N_t$ represents the true interaction parameter), and the subsequent nonlinear phase, marked by an equilibrium between the Lorentz force and the inertial force ($N_t\sim 1$), are successfully corroborated through numerical and experimental means, particularly at a substantial initial interaction parameter ($N_0>1$, where $N_0$ denotes the initial interaction parameter). The transition time from linear phase to nonlinear phase varies with the square of $N_0$. The relative magnitude of the Lorentz force and the inertial force plays a pivotal role during the free-decay evolution, and we propose the governing equations for such a flow. Nevertheless, numerical simulations and experiments indicate that these two phases and the ensuing transition, as depicted by the velocity decay curve, are primarily limited to the vicinity of the vortex's periphery, exhibiting a certain degree of locality. By considering the scaling of the global kinetic energy $[\sim {(t/\tau )}^{-0.8}$, where $t$ denotes the physical evolution time normalized by the Joule time $\tau ]$, the Joule dissipation $[\sim {(t/\tau )}^{-2}]$, and the parallel length $[\sim {(t/\tau )}^{3/5}]$, it becomes evident that the global behavior of a vortex bears greater resemblance to the nonlinear phase rather than undergoing a direct transformation from one phase to another, even when subjected to a substantial initial interaction parameter.

We study the role of steric effects on the induced-charge electro-osmosis phenomenon using a recently developed mesoscale fluid model. A hybrid Eulerian-Lagrangian method is used to simulate the dynamics of discrete immersed ions in a thermally fluctuating solvent near a metallic plate embedded in the dielectric interface. We observe that the characteristic velocity scales almost linearly with electric field when the generated $\zeta$ potentials exceed the order of the thermal voltage, as opposed to a quadratic scaling predicted by Helmholtz-Smoluchowski equation, although qualitative agreement with experiments and theories is obtained at low electric fields. Our simulations reveal that the steric effects play a crucial role at strong electric fields, which is observed from the aggregation of ions towards the center of the metal plate instead of at the edges, and the overcharging of co-ions to the surface charge near the electric double layer. A comparison to a continuum electrolyte model also highlights significant differences in charge distribution and flow field that are attributed to the steric repulsion between ions.

We report flow statistics and visualizations from molecular-gas-dynamics simulations using the direct simulation Monte Carlo (DSMC) method for turbulent Couette flow in a minimal domain where the lower wall is replaced by an idealized permeable fibrous substrate representative of thermal-protection-system materials for which the Knudsen number is $O\left({10}^{-1}\right)$. Comparisons are made with smooth-wall DSMC simulations and smooth-wall direct numerical simulations (DNS) of the Navier-Stokes equations for the same conditions. Roughness, permeability, and noncontinuum effects are assessed. In the range of Reynolds numbers considered herein, the scalings of the skin friction on the permeable substrate and of the mean flow within the substrate suggest that they are dominated by viscous effects. While the regenerative cycle characteristic of smooth-wall turbulence remains intact for all cases considered, we observe that the near-wall velocity fluctuations are modulated by the permeable substrate with a wavelength equal to the pore spacing. Additionally, the flow within the substrate shows significant rarefaction effects, resulting in an apparent permeability that is 13% larger than the intrinsic permeability. In contrast, the smooth-wall DSMC and DNS simulations exhibit remarkably good agreement for the statistics examined, despite the Knudsen number based on the viscous length scale being as large as $O\left({10}^{-1}\right)$. This latter result is at variance with classical estimates for the breakdown of the continuum assumption and calls for further investigations into the interaction of noncontinuum effects and turbulence.

The heat transport in rotating Rayleigh-Bénard convection (RBC) can be significantly enhanced for moderate rotation, i.e., for an intermediate range of Rossby numbers $\mathrm{Ro}$, compared to the nonrotating case. At Rayleigh numbers $\mathrm{Ra}\lesssim 5\times {10}^8$, the largest enhancement is achieved when the thicknesses of kinetic and thermal boundary layer are equal. However, experimental and numerical observations show that, at larger $\mathrm{Ra}$ ($\gtrsim 5\times {10}^8$), the maximal heat transport starts to deviate from the expected optimal boundary layer ratio and its enhancement amplitude decreases drastically. We present data from direct numerical simulations of rotating RBC in a periodic domain in the range of ${10}^7\le \mathrm{Ra}\le {10}^{10}$ and $0\le {\mathrm{Ro}}^{-1}\le 40$ for Prandtl number $\mathrm{Pr}=4.38$ and 6.4 (corresponding to Ekman numbers $\mathrm{Ek}\gtrsim {10}^{-6}$) to identify the reason for the transition to this large-$\mathrm{Ra}$ regime of heat transport enhancement. Our analysis reveals that the transition occurs once the bulk flow at the optimal boundary layer ratio changes to geostrophic turbulence for large $\mathrm{Ra}$. In that flow state, the vertically coherent vortices, which support heat transport enhancement by Ekman pumping at smaller $\mathrm{Ra}$, dissolve into vertically decorrelated structures in the bulk such that the enhancing effect of Ekman pumping and the influence of the boundary layer ratio become small. Additionally, more heat leaks out of the Ekman vortices as the fraction of thermal dissipation in the bulk increases. We find that the rotation-induced shearing at the plates helps to increase the thermal dissipation in the bulk and thus acts as a limiting factor for the heat transport enhancement at large $\mathrm{Ra}$: Beyond a certain ratio of wall shear stress to vortex strength, the heat transport decreases irrespectively of the boundary layer ratio. This $\mathrm{Pr}$-dependent threshold, which roughly corresponds to a bulk accounting for $\approx 1/3$ of the total thermal dissipation, indicates the maximal heat transport enhancement and the optimal rotation rate ${\mathrm{Ro}}_{\mathrm{opt}}^{-1}$ at large $\mathrm{Ra}$.

The Lagrangian statistics of a time-dependent ABC flow are considered, with time dependence introduced via harmonic oscillation with frequency $\mathrm{\Omega }$. By calculating the finite-time Lyapunov exponents (FTLEs), the Lagrangian statistics of the system are determined for a range of values of $\mathrm{\Omega }$. These statistics are calculated for the kinematic regime where the flow remains an ABC flow, the nonlinear regime with dynamo action present, and a second hydrodynamic state reached through instability of the original ABC flow. It is found that there are significant differences between these three states, with most cases showing a decrease in their FTLEs as the flow deviates from its original ABC form. Furthermore, these changes are highly dependent on $\mathrm{\Omega }$, with lower frequencies leading to higher FTLEs in the nonlinear regime, and unstable regimes. By examining the Lagrangian statistics with respect to the dynamo behavior observed, we discuss their potential relevance to nonlinear saturation, self-killing dynamos, and the importance of the initial hydrodynamic state. The numerical code developed for this project is also available.

Collision of two dispersed drops in the matrix of a suspending liquid is the first step toward coalescence. However, to quantify the rate of coalescence, the configuration of the collision should be definable and the force that induces the collision should be measurable. We present a strategy to use the hydrodynamic force in a six-port microfluidic channel to steer two drops toward collision in extensional flow. By implementing the analytical solution in the control loop, the flow rates that are required to steer the drops toward their respective target points can be determined using a single control parameter. This parameter, ${\chi }^{*}$, is a dimensionless timescale that can manipulate the drops in two limits: (1) by engaging all six ports to create a flow field with two stagnation points $({\chi }^{*}\ll 1)$, or (2) by deactivating some of the ports and creating a linear extensional flow through the remaining active ports $({\chi }^{*}\gg 1)$. We determine specific orientations that are more suitable for the collision of Hele-Shaw drops in the six-port microfluidic channel. Based on the above strategy, we design and perform controlled head-on and glancing collisions for $\sim 100\mu \mathrm{m}$ radius Hele-Shaw perfluorodecalin drops in silicone oil. Using the conformal mapping technique, the analytical solution of the flow field accounting for the perturbation of the flow due to the hydrodynamic interactions between the Hele-Shaw drops was used in the control code. Coalescence time between two Hele-Shaw drops undergoing a head-on collision in a dimpled mode was found to be independent of the strain rate of the hydrodynamic flow.

The long-time solutions of the Galerkin-truncated three-dimensional, incompressible Euler equation relax to an absolute equilibrium as a consequence of phase space and kinetic-energy conservation in such a finite-dimensional system. These thermalized solutions are characterized by a Gibbs distribution of the velocity field and kinetic-energy equipartition among its (finite) Fourier modes. We now show, through detailed numerical simulations, the early stage triggers for the inevitable thermalization in physical space. Furthermore, some aspects of this process are shown to be reduced to an effective one-dimensional problem, making comparisons with the more studied Burgers equation feasible. Finally, we discuss how our understanding of the mechanism of thermalization can be exploited to numerically obtain dissipative solutions of the Euler equations and evidence for or against finite-time blowup in computer simulations.

A series of direct numerical simulations (DNSs) of forced compressible isothermal turbulence in a periodic box are conducted by using an eighth-order compact finite-difference scheme to study the Reynolds number and resolution-level ($k_{\max }\eta$) dependence of the statistics of the compressible turbulence, where $k_{\max }=\sqrt{3}N/2$, $\eta$ the Kolmogorov length scale. The number of grid points $N^3$ and the Taylor microscale Reynolds number $R_{\lambda }$ are up to ${4096}^3$ and 853, respectively, and the turbulent Mach number and the ratio of dilatational to solenoidal root-mean-square velocities are approximately 0.3 and 0.4, respectively. The DNSs have shown that the energy spectrum for the compressible isothermal turbulence increases to a higher wave-number range of $k\eta >1$ with increasing resolution levels suggesting a $k^{-3}$ scaling of its dilatational component; however, the energy spectrum and its solenoidal and dilatational components for $k\eta <1$ are not sensitive to resolution levels provided that $k_{\max }\eta \gtrsim 2$. When the solenoidal quantities are used for normalization, the solenoidal component of the energy spectrum and the solenoidal dissipation rate agree well with those gathered from the DNSs of incompressible turbulence. DNS studies indicate that the normalized dilatational component of energy dissipation is still finite nonzero for large $R_{\lambda }$ values (as opposed to an expectation from the result by John et al. [J. Fluid Mech. 920, A20 (2021)]). The PDFs of the solenoidal component of pressure fluctuations agree with those of incompressible turbulence. However, a close study shows that the solenoidal pressure and enstrophy fluctuations in compressible isothermal turbulence are consistently less intermittent than those in incompressible turbulence. The impact of bulk viscosity on the energy spectrum is examined using the DNSs of compressible isothermal turbulence with nonzero bulk viscosity.

We propose a phenomenological description of the asymptotic near-wall momentum-exchange mechanism of flat plate zero-pressure gradient turbulent boundary-layer flows at the extreme Reynolds number regime, based on a simple model of attached eddies scaling with the location of the mesolayer, ${\delta }_m^{+}\sim {\text{Re}}_{\tau }^{1/2}$, and satisfying Kolmogorov's inertial similarity scaling for their turnover velocities. This yields an asymptotic power-law formula for the skin friction, $C_f\sim {\text{Re}}_x^{-2/15}$, or, equivalently, $C_f\sim {\text{Re}}_{\delta }^{-2/13}$. We also derive a formula for the asymptotic thickness of the boundary layer. We show that these asymptotic scaling laws are in excellent agreement with experimental data, and are consistent with classical semiempirical formulas. For moderately large Re number flows, we argue that the intermediate asymptotic Blasius scaling, $C_f\sim {\text{Re}}_{\delta }^{-1/4}$, or, equivalently, $C_f\sim {\text{Re}}_x^{-1/5}$, is more appropriate. The asymptotic model relies on an asymptotic analysis of the bulk normalized axial mean-momentum equation, and it is related to a phenomenology previously proposed for pipe and channel flows, which suggests the existence of a universal transition of the turbulent momentum-exchange mechanism of wall-bounded flows in the asymptotically large Re number regime.

The coherent structures play a significantly important role in the spatial evolution of turbulent wakes. The spatial distribution and the organization of the coherent structures in a self-similar axisymmetric turbulent wake are numerically discussed by using the data from the previous simulation [Zhou and Vassilicos, Related self-similar statistics of the turbulent/non-turbulent interface and the turbulence dissipation, J. Fluid Mech. 821, 440 (2017).], in which it was found that the radial positions of the turbulent/nonturbulent (T/NT) interface at different downstream locations exhibits a self-similar form normalized by the wake width $\delta$. The Liutex method is used to automatically identify the vortex core center and compute the direction of the vortex axis. The probability density functions (PDFs) of the radial distance of the vortex core center $R_c$ at different streamwise locations can be superimposed on the same curve, and the PDFs of $R_c$ scale with $\delta$. There is a discernible preference for the vortex axis to be normal to the mean direction of the flow, and the PDFs of the polar angle $\alpha$ at different downstream locations collapse onto a single curve. The PDFs of the separation distance $R_{ct}$ between the T/NT interface and the nearest vortex core center are also self-similar and scale with $\delta$. The averaged radial distance $\overline{R_{ct}}$ is of the order of the Taylor microscale. The results reported in this work suggest that the coherent structures are orderly distributed and also regularly organized in a self-similar axisymmetric turbulent wake.

Turbulence modeling within the Reynolds-averaged Navier-Stokes (RANS) equations' framework is essential in engineering due to its high efficiency. Field-inversion and machine-learning (FIML) techniques have attempted to improve RANS turbulence models' predictive capabilities for separated flows. However, FIML-generated models often lack interpretability, limiting physical understanding and manual improvements based on prior knowledge. Additionally, these models typically struggle with generalization in flow fields distinct from the training set. This study addresses these issues by employing symbolic regression (SR) to derive an analytical relationship between the correction factor of the baseline turbulence model and local flow variables, enhancing the baseline model's ability to predict separated flow across diverse test cases. The shear-stress-transport (SST) model undergoes field inversion on a curved backward-facing step case to obtain the corrective factor field $\beta$, and SR is used to derive a symbolic map between local flow features and $\beta$. The SR-derived analytical function is integrated into the original SST model, resulting in the SST-SR model. The SST-SR model's generalization capabilities are demonstrated by its successful predictions of separated flow on various test cases, including 2D-bump cases with varying heights, periodic hill case where separation is dominated by geometric features, and the three-dimensional Ahmed-body case. In these tests, the model accurately predicts flow fields, showing its effectiveness in cases completely different from the training set. The Ahmed-body case, in particular, highlights the model's ability to predict the three-dimensional massively separated flows. When applied to a turbulent boundary layer with $R{e}_L=1.0\times {10}^7$, the SST-SR model predicts wall-friction coefficient and log layer comparably to the original SST model, maintaining the attached boundary-layer prediction performance.

We study three types of homogeneous, shear-forced, stably stratified turbulence in a triply periodic domain over a wide range of stratification strengths. The shear forcings correspond to three different turbulence generation scenarios: (i) vertically sheared mean horizontal flow, (ii) horizontally sheared mean vertical flow, and (iii) horizontally sheared mean horizontal flow. At all stratification strengths, i.e., from weak to strong, we observe a persistent ordering of the mixing coefficient $\mathrm{\Gamma }$ values such that the second set of simulations exhibit the largest values and the third set of simulations exhibit the smallest values, implying that the turbulence generation mechanism has a significant effect on $\mathrm{\Gamma }$. For very strong stratification, we observe that $\mathrm{\Gamma }$ begins to decrease, and this shift is associated with the difference between the vertical component of turbulent kinetic energy and turbulent potential energy becoming negative and the pressure scrambling term increasing in magnitude to keep the vertical buoyancy flux positive. Our findings demonstrate that the turbulence generation mechanism strongly affects the mixing characteristics of stably stratified turbulence and parametrizations of $\mathrm{\Gamma }$ for climate applications will need to distinguish how the turbulence is generated, which will require knowing more information than just the buoyancy Reynolds number ${\text{Re}}_b$ and the turbulent Froude number ${\text{Fr}}_k$.

Instabilities generated in Mach-6 quiet flow about an axisymmetric cone-cylinder-flare geometry with a $5^{\circ }$ half-angle sharp cone were studied in the Boeing/AFOSR Mach-6 Quiet Tunnel (BAM6QT). A small, $3.5{}^{\circ }$ flare angle was initially tested to provide a baseline with a model that introduces a compression to the flow but does not induce a separation. A ${10}^{\circ }$ flare was then used to make measurements with an axisymmetric separation bubble. Surface heat flux and pressure fluctuation measurements were made with IR thermography and PCB and Kulite pressure sensors, respectively. Off-the-surface density gradient fluctuations were observed using focused laser differential interferometry for the separated flow. Quiet-flow measurements resulted in laminar boundary layers through the downstream edge of both models. The shear layer was found to be mostly steady in space under quiet flow, as opposed to large spatial fluctuations seen in imagery from conventional noise tunnels. In addition to the second mode, a traveling instability was observed which, based on computational analysis, corresponds to oblique waves that are amplified by the shear layer above the separation bubble. This instability was found to phase lock with the second mode as it propagates downstream. Artificial perturbations were then introduced to the ${10}^{\circ }$ flare model upstream of the separation to study the resulting wave packets as they moved through the shear and boundary layers. This paper provides measurements that may be comparable to hypersonic flight due to the low-disturbance environment of the test facility.

In this research, we apply the single-step deep reinforcement learning (DRL) algorithm to optimize the spatial location and length of a downstream splitter plate in order to suppress vortex shedding behind a cylinder. This algorithm is rare in optimization problems as it differs significantly from the traditional decision-making models. After a case with only one optimization parameter (streamwise position) is validated, the study is then extended to more complex cases with additional parameters (lateral position and plate length) to explore the full potential of the DRL-based optimization algorithm in high-dimensional design spaces. The final results are well explained by analyzing the local stability of the wake flow. In this context, we incorporate techniques commonly used in the DRL field, such as adaptive learning rates and reward shaping, which crucially complements and extends the emerging single-step DRL algorithms. By balancing exploration and exploitation phases and embedding additional stability information obtained through dynamic mode decomposition, the resulting method not only achieves improved global convergence but also reduces the computation time for each fluid environment. The specifics and infeasibility of another type of single-step DRL algorithm are detailed in the Appendix, and the comparison with Bayesian optimization is discussed by benchmarking on the same optimization problem. These results provide insights into the potential of single-step DRL algorithms and suggest that, with further fine-tuning for specific problems, it may outperform existing advanced optimization methods.

We study the drag on a centimetric sphere in a uniform flow in the presence of a free surface as a function of submergence depth. Through direct force measurements in a custom benchtop recirculating flume, we demonstrate that the drag can significantly exceed the corresponding drag in a single-phase flow and achieves a peak at submergence depths just prior to complete immersion. The additional drag in the partially immersed state is rationalized by considering hydrostatic effects associated with the asymmetric surface height profile induced by the obstacle in the flow direction which persists for flow speeds below the minimum capillary-gravity wave speed. At these scales, the sphere's wettability plays a pronounced role in determining the maximum possible drag and results in hysteretic behaviors near touchdown and complete immersion. The influence of flow speed, sphere size, and surface tension on the drag characteristics are additionally explored through a combination of experiments and numerical simulations.