Feeding by marine invertebrate larvae depends critically on the low-Reynolds-number fluid mechanics of the feeding currents they generate. Yet the relative importance of gravity-driven versus drag-driven mechanisms in small invertebrate larvae has remained unresolved. Here, high-speed microscale imaging, micro-particle image velocimetry (μPIV), and an analytical axisymmetric Stokes-flow model are combined to quantify the feeding currents of free-swimming veliger larvae of the eastern oyster Crassostrea virginica. Despite their small body size ($<330$µm), the larvae possess a high excess density ($\sim 181\mathrm{kg}/{\mathrm{m}}^3$), shifting the force balance such that gravity-driven feeding currents dominate. μPIV measurements reveal flow fields with first-order (Stokeslet-type) spatial decay, yielding ecologically sufficient body-volume-specific maximal clearance rates of $\sim 2.5\times {10}^6/\mathrm{day}$. These values agree with predictions from the Stokes-flow model, which decomposes the clearance into gravity-driven and drag-driven components and shows that the former consistently outweighs the latter across observed swimming speeds. The model further explains how slow swimming modulates clearance by $\pm 44\%$ relative to hovering. Together, these results demonstrate that C. virginica veligers operate in a gravity-dominated feeding regime similar to that of larger calanoid copepods, and that their feeding strategy is shaped by an interplay of excess weight, swimming kinematics, and ciliary propulsion. This work highlights the fundamental role of gravity in larval feeding mechanics and underscores the need to incorporate excess weight when evaluating form-and-function relationships in marine invertebrate larvae, particularly for shelled bivalve and gastropod larvae with high excess density.

We investigate cavitation of a vortex ring in an impulsively starting jet induced by an in-tube expanding bubble, combining experimental, numerical, and theoretical approaches. As the jet cavitation number decreases, the cavitation evolves from discrete cavitation bubbles in the vortex core to a cavitating vortex ring in which the vapor phase fully envelops the vortex core azimuthally. A simplified one-dimensional model that approximates the in-tube bubble as a piston captures the time-dependent velocity of the starting jet, which features a remarkably high initial acceleration. This velocity history is then used as the inlet condition for nozzle-flow simulations, enabling us to address the following two scientific questions: (i) why vortex cavitation occurs at an exceptionally high cavitation number and (ii) why the incipient cavitation number increases with nozzle radius, known as the size-scale effect. To clarify the relationship between flow dynamics and cavitation, we analyze the vorticity flux and identify a previously underexplored contribution—the over-pressure component. This component, which strongly depends on the nozzle radius, dominates vortex ring circulation when the initial nozzle flow experiences an intense and brief acceleration. Notably, it may outweigh the shearing component predicted by conventional slug models. The impulsive nozzle flow provides a plausible mechanism for the unusually high incipient cavitation number and the observed size-scale effect. These results shed new light on the mechanisms governing cavitation onset in submerged jets and offer guidance for promoting or suppressing cavitation in practical applications.

We investigate the effects of horizontal oscillations on the drafting-kissing-tumbling (DKT) dynamics of two monodisperse spherical particles settling under gravity in a viscous fluid. Applying particle-resolved direct numerical simulations, we systematically vary the oscillation frequency and amplitude to assess their impact on the behavior of individual particles, their mutual interaction, and the orientation of the particle arrangement. The results demonstrate that the oscillatory effects on DKT become significant only when the particle Reynolds number ${\text{Re}}_p$, defined as the ratio of oscillation-induced inertial to viscous forces, exceeds unity. In this regime, oscillations alter the temporal characteristics of the DKT process with moderate amplitudes tending to prolong and larger amplitudes to reduce the kissing phase. Moreover, oscillations affect particle reorientation. At low ${\text{Re}}_p$, the particles maintain their initial orientation throughout the interaction, whereas an increasing ${\text{Re}}_p$ promotes a preferential alignment perpendicular to the direction of oscillation. We explain these findings by analyzing the oscillation-induced pressure fields surrounding the individual particles, which develops increasingly pronounced lateral anisotropy with increasing ${\text{Re}}_p$. The corresponding lateral hydrodynamic forcing likewise becomes increasingly anisotropic, providing a consistent physical basis for the observed modification of particle interactions and reorientation. These findings provide a physical framework for understanding how horizontal oscillations govern binary particle-particle interactions and orientation during gravitational settling.

A mathematical model for the evolution of, and deposition from, a thin particle-laden droplet on an infinitely thick, isotropic, fully flooded, porous substrate with interconnected pores undergoing simultaneous evaporation and imbibition is formulated and analyzed. In particular, analytical expressions for the evolution of the droplet, as well as for the flow within the droplet and the substrate, and for the transport and deposition onto the substrate of the particles are obtained for droplets evolving in four different modes. While the physical mechanisms driving evaporation (namely the diffusion-driven flow of vapor in the atmosphere away from the free surface of the droplet) and imbibition (namely the pressure-driven flow of liquid in the substrate away from the base of the droplet) are rather different, perhaps rather unexpectedly, it is found that there are a number of qualitative and quantitative similarities as well as differences in the resulting behavior of the droplet as it loses mass to its environment. For example, it is shown that a droplet undergoing pure imbibition in the constant contact radius mode never completely imbibes (i.e., it has an infinitely long lifetime), but if it evaporates (either with or without imbibition also occurring), then it has a finite lifetime, and increasing the strength of evaporation and/or imbibition shortens its lifetime. It is also shown that in the regime in which diffusion of particles is faster than axial advection but slower than radial advection of particles, the final deposit of particles left behind on the substrate after the droplet has completely evaporated and/or imbibed is independent of both the nature and the strength of the physical mechanism(s) driving the mass loss from the droplet. Not only are these results of theoretical interest, but they are also relevant to a wide variety of practical applications, including ink-jet printing, DNA chip manufacturing, and disease diagnostics, that would benefit from an improved ability to predict and/or control the final deposit pattern from a droplet undergoing simultaneous evaporation and imbibition.

We investigate the thermocapillary and solutocapillary instabilities in a self-rewetting liquid film that has a nonmonotonic variation of surface tension with temperature containing an insoluble surfactant at the interface. The dynamics of such a thin liquid film flowing axisymmetrically on a heated vertical cylinder is examined by deriving a set of coupled evolution equations for the film thickness and the surfactant concentration under the lubrication approximation. We perform a linear stability analysis of the steady film solution with a uniform surfactant concentration at the interface. The effects of solutocapillarity and quadratic thermocapillarity are analyzed in depth using $T_m$, the temperature at which surface tension is nearly minimum as the cutoff point, and $T_{0i,s}$, the interface temperature of the steady film. Beyond a critical value of the solutal Marangoni number ${\text{Ma}}_{\text{s}}$, the dispersion curve, featuring two extrema exhibits midwave instability that is characterized by two cutoff wavenumbers. The conventional thermal Marangoni effect switches to the anomalous/reverse Marangoni effect as $T_{0i,s}$ takes values greater than $T_m$. We perform a detailed parametric study to scrutinize the effect of thermal Marangoni number ${\text{Ma}}_{\text{T}}$, solutal Marangoni number ${\text{Ma}}_{\text{s}}$, Biot number Bi, Bond number Bo, and surface Péclet number ${\text{Pe}}_{\text{s}}$ on the stability of the base state. The Orr-Sommerfeld problem is formulated and solved numerically under the creeping flow condition for arbitrary wavenumbers. A comparison of the neutral curves obtained from the Orr-Sommerfeld problem and the lubrication model reveal excellent agreement for small wavenumber perturbations. The nonlinear simulations of the coupled set of partial differential equations (PDEs) for the interface and surfactant concentration demonstrate the modulation of interface deformation in the presence of temperature gradients and concentration gradients in the regular Marangoni regime, whereas such deformations are suppressed in the anomalous Marangoni regime.

We consider the relationship between Eulerian modal decompositions and Lagrangian coherent structures. The model sensitivity framework developed by Kaszás and Haller [Chaos 30, 113144 (2020)] is used to express data-driven modal representations of fluid flow in a Lagrangian space. The method, based on the computation of the finite-time Lyapunov exponent, computes the amplitude perturbations experienced by fluid particles due to specific modal components of the flow. Demonstrations of the method are presented for both periodic and turbulent flows, including direct numerical simulation (DNS) data of the classical cylinder wake flow, experimental data from the wake past an oscillating foil, and DNS data of a turbulent channel flow. This method provides a way to understand how Eulerian mode structures interact dynamically with features of the Lagrangian coherent structure across scales, offering additional physical insight into modal decompositions.

Turbulent flows can exhibit extreme events, which are characterized by sudden bursts in the system observables. These events pose significant challenges for prediction and control owing to their intermittent, high-dimensional, and strongly nonlinear nature. In this work we present a predominantly data-driven control framework for the suppression of extreme events in turbulent flows, leveraging reduced-order modeling for nonlinear compression of high-dimensional flow fields and a data-driven clustering algorithm for the identification of precursors to extreme events. A control law is defined in the low-dimensional latent space and is optimized to efficiently mitigate extreme events through actuation values associated with the preidentified clusters. To address issues in the latent space structure that arise from invariance transformations present in many nonlinear systems, leading to an inflated latent space, we employ symmetry-aware autoencoders to establish a more structured and compact latent space. By analyzing high-fidelity simulations of a canonical chaotic flow (Kolmogorov flow), we demonstrate how the resulting closed-loop dynamics exhibit a substantial reduction (up to ca. 99.4%) in the frequency and intensity of extreme events. The framework exhibits scalability to a more chaotic flow regime characterized by higher Reynolds number without encountering numerical instabilities, achieving a 96.9% reduction in the occurrence of extreme events. The ability to incorporate controller limitations, including actuator latency relevant to practical applications of the framework, has also been demonstrated. This study highlights the efficacy of data-driven methods that require little to no prior knowledge of the underlying system dynamics to achieve effective flow control, thereby providing a pathway for general real-time suppression of extreme events in turbulent flows.

Gas transport in nanoporous media is fundamentally governed by confinement effects and surface roughness, which challenge the applicability of conventional continuum theories. In this study, a molecular-based apparent permeability model for nanoporous flows is established by incorporating confinement effects, surface roughness, and wettability using a bottom-up approach. The roughness effect is incorporated through a correction function calibrated from molecular simulations, which characterizes the suppression of slip behavior induced by surface roughness. A generalized hydrodynamic model is employed to describe pore-scale transport behavior in smooth nanochannels, from which a semi-empirical scaling law for slip length is developed as a function of fluid-solid interactions, pore size, finite molecular size, and Knudsen number. The resulting slip-length scaling law is then incorporated into a slip-corrected permeability relation to obtain the apparent permeability model for rough nanoporous media. The new model is validated through systematic comparisons with molecular simulation results, experimental data, and existing theoretical models, demonstrating its capability to accurately capture nanoscale transport behavior under varying surface roughness conditions. Parametric analyses further show that the apparent permeability increases with hydrophobicity while decreases with surface roughness, both of which have been largely overlooked in previous apparent permeability models. Overall, the proposed model provides a physically grounded framework linking molecular-scale interactions with fluid transport in nanoporous media.

Fractal flocs are highly porous aggregates whose hydrodynamic response differs fundamentally from that of solid particles of comparable size and shape. In particular, the fluid trapped within the porous structure alters the effective added inertia and rotational coupling experienced by the aggregate, thereby influencing its translational and rotational dynamics in unsteady flows. In this work we investigate the added-mass and added-moment-of-inertia tensors of fractal flocs generated over a broad range of floc sizes and fractal dimensions using a binary-superposition-based computational framework. Large statistical ensembles are used to characterize both the mean behavior and the morphology-induced variability of the inertia tensors. The results show that the principal added-mass and added-moment-of-inertia coefficients exhibit systematic scaling with floc size and fractal morphology while displaying substantial deviations from equivalent solid ellipsoids. Despite significant geometric irregularity at the level of individual particles, the variability among different realizations decreases with increasing aggregate size, revealing an emergent self-averaging behavior for large flocs. Due to their highly anisotropic and porous structure, flocs with smaller fractal dimension exhibit substantially larger added-moment-of-inertia coefficients, which could influence their orientation dynamics in accelerating or turbulent environments. Based on these observations, reduced-order stochastic representations are developed to model the tensor statistics using a small number of physically meaningful parameters. The statistical framework developed here provides a foundation for incorporating morphology-dependent added-inertia effects into large-scale simulations of particle-laden and multiphase flows, including applications involving sediment transport, marine aggregates, aerosols, and industrial particulate systems.

High temporal resolution is essential for resolving the unsteady dynamics of three-dimensional velocity fields in fluid mechanics. However, achieving temporal super-resolution (SR) at large temporal intervals remains challenging for complex turbulent flows. To address this limitation, a residual network that integrates attention mechanisms with convolutional long short-term memory (ConvLSTM) is proposed to reconstruct three-dimensional velocity fields at intermediate time instants between two input flow fields under various temporal SR factors, corresponding to progressively larger temporal separations between the input flow fields. The model is trained and evaluated using direct numerical simulation data of turbulent channel flows spanning different friction Reynolds numbers. The reconstruction accuracy is found to be strongly correlated with the ratio between the temporal separation of the two input snapshots and the Kolmogorov timescale of the flow. A dimensionless temporal parameter ${\mathrm{\Pi }}_t$ is introduced, under which the reconstruction error exhibits an approximately linear dependence. A threshold of ${\mathrm{\Pi }}_t\le 4$ is identified for the proposed model to achieve accurate reconstruction of Reynolds stresses and coherent vortical structures. In contrast, a convolutional neural network based model attains comparable reconstruction accuracy only for ${\mathrm{\Pi }}_t\le 0.8$. To assess applicability to experimental measurements, the noise robustness of the proposed model is examined under different noise intensities and spectral characteristics. The model effectively suppresses noise and extracts physically meaningful flow structures, benefiting from spatial feature extraction enabled by attention mechanisms and temporal dependency modeling provided by ConvLSTM. Nevertheless, the reconstruction error increases monotonically with noise intensity due to the progressive loss of recoverable flow information irreversibly obscured by noise.

Film flow on vertical fibers has been studied for the past 30 years. However, it is only recently that the influence of the nozzle on the film and bead dynamics has been demonstrated. Consequently, similar to the regimes observed with a faucet, both a dripping and a jetting regime have been reported for film flow on fibers. This study investigates the dynamics of thin liquid films on vertical fibers near the nozzle outlet using both experimental and numerical approaches, drawing insights from dripping faucet studies. This allows for a clear definition of the recently observed dripping and jetting regimes, with a particular focus on gaining a deeper understanding of the flow and formation of beads in the dripping regime, as well as the transition to jetting.

We investigate the bursting dynamics of dense granular films punctured at imposed controlled liquid pressure. Using high-speed imaging, we reveal a two-stage opening process. At early times, the hole expands in a liquidlike manner with a nearly circular shape and a constant velocity, though significantly lower than the Taylor-Culick speed. This velocity is modeled by a balance between surface tension and effective surface viscosity, which depends on particle size and liquid depression. We show that as liquid pressure decreases, the initial opening slows down progressively until it ultimately reaches a jamming state. As the opening progresses, the film undergoes a transition to a solidlike regime, where faceting occurs due to fracture propagation within the granular skeleton. We show that the characteristic fracture size follows a $t^{3/4}$ scaling, consistent with a balance between surface energy release and viscous dissipation induced by liquid redistribution within the 2D granular network. These findings advance our understanding of the mechanics of granular films and their transition between fluidlike and solidlike behaviors.

This study examines the effective thermal diffusivity of sheared particle-fluid suspensions with non-Brownian neutrally buoyant particles in a thin gap Taylor-Couette cell. A steady canonical shear flow is generated with Taylor instabilities suppressed by outer cylinder rotation. Spherical acrylic particles of 1.93 mm diameter were used with a density-matched propylene glycol-glycerol solution. Thermal diffusivity of the medium is deduced by applying a thermal pulse to the stationary inner cylinder and monitoring temporal temperature decay on the inner cylinder surface. The study documents the effects of the particle Peclet number $(<1400)$ for four different particle volume fractions of 0.14, 0.22, 0.30, and 0.36. Here the Peclet number is defined as the ratio of timescales of thermal diffusion in the liquid over the particle scale to the timescale of the imposed shear. The applied shear corresponded to particle Reynolds numbers ranging from Stokes flow $({\mathrm{Re}}_p\ll 1)$ to a maximum of 0.85, where weak inertial effects are likely present. At low Peclet number $\mathrm{Pe}<100$, the enhancement due to shear is roughly linear with the Peclet number. At higher $\mathrm{Pe}$ $(100<\mathrm{Pe}<700)$, the enhancement displays power-law behavior with an exponent of 1/2, which has been predicted by some models for concentrated suspensions, but is also consistent with the emergence of a particle-free fluid layer near one wall. At still higher $\mathrm{Pe}$, the 1/2 power-law behavior is retained for all but the highest volume fractions, with $\varphi =0.35$ exhibiting a 1/11 power-law behavior, which has been previously predicted, albeit for dilute suspensions, for the high-$\mathrm{Pe}$ limit. The thermal diffusivity measurements are complemented by limited measurements of particle velocity through particle tracking, which suggest that particle motion does not contribute significantly to the observed enhancement at high Peclet numbers.

This study introduces a vortex gust generation method for isolated vortices impacting a downstream airfoil that is applicable to both numerical simulations and experiments. The vortex gust is generated by a symmetric airfoil undergoing a rapid pitching maneuver during a prescribed heaving motion. The resulting vortices propagate along trajectories nearly parallel to the incoming flow, while the associated wake extends obliquely from the vortex core. Despite differences in Reynolds number, rapid pitching duration, and detailed vortex structure between simulations and experiments, consistent trends are observed in how the vortex rotation orientation, strength, and position vary with the prescribed motion parameters. Analysis of the lift response of the downstream airfoil shows that the aerodynamic influence associated with the wake does not persist over extended timescales. These results demonstrate that the proposed method enables the controlled generation of vortex gusts with prescribed characteristics, providing a flexible approach for systematic studies of vortex-airfoil interaction.

The swirling flow in a Francis type hydropower turbine is known to be susceptible to the formation of a large helical structure, commonly referred to as a vortex rope. This vortex rope can be interpreted as an unstable mode associated with vortex breakdown. This perspective is adopted here in a simplified laminar flow setting. The helical vortex rope mode is shown to bifurcate supercritically from an axisymmetric base flow in a Hopf bifurcation within a turbine draft tube. When wall friction effects are neglected, a large recirculation region at the axis can form and a range of subcritical solutions is identified for a flow regime corresponding to partial load of the turbine. The existence of these subcritical solutions promotes the emergence of a hysteresis loop. We further describe a regular dynamics of a formation of recirculation bubble at the axis and its destruction due to the emergence of a helical vortex rope at its periphery. Increasing the axial flow discharge towards the regime corresponding to nominal turbine load leads to an unfolding of the steady solutions branch in a transcritical bifurcation. This bifurcation takes place at finite Reynolds number and complements existing evidence of transcritical bifurcation of the swirling jet flows, previously reported in the inviscid limit.

The surface tension of a liquid, which drives most free-surface flows at small scales, is often measured with the pendant drop method due to its simplicity and reliability. When the drop is suspended in air, controlling the ambient temperature and humidity is usually an afterthought, resulting in evaporation of the drop during the measurement. Here, we investigate the effect of evaporation on the measured surface tension using experiments and numerical simulations. In the experiments, we measured the evolution of the droplet temperature, which can be drastically reduced by $\mathrm{\Delta }T\approx {10}^{\circ }\mathrm{C}$ due to evaporative cooling, thereby altering the measured surface tension by more than 1 mN/m. This finding can be reproduced by numerical simulations, which additionally allow for controlled investigations of the individual influences of further effects on the pendant drop method, namely, shape deformations by evaporation-driven flows in the gas-phase and in the liquid-phase, including the resulting Marangoni flow. We provide a simple passive method to control the relative humidity without requiring additional instrumentation. Our findings are particularly pertinent to Marangoni flows, which are driven by surface tension gradients and which are consequently highly sensitive to measurement inaccuracies. We apply our method with different aqueous mixtures of glycerol and various diols. Our results and insights have implications for various applications, ranging from inkjet printing to agricultural sprays. Finally, we meticulously documented our setup and procedure for future reference.

The impact of localized surface roughness on the development of a short laminar separation bubble (LSB) forming on the suction side of a NACA 0018 airfoil at an angle of attack of $6^{\circ }$ and a chord-based Reynolds number of ${\text{Re}}_c=100000$ is investigated experimentally. Direct force measurements and two-component particle image velocimetry (PIV) in multiple planes are used to characterize the effect on flow development compared to the baseline flow over a smooth airfoil and fully tripped model. Results show that localized roughness not only eliminates the downstream LSB but also induces significant three-dimensional modifications to bubble topology over a substantial spanwise region ($\sim \mathcal{O}\left(c\right)$) extending beyond the roughness element. This leads to a more substantial impact on aerodynamic performance than that expected from the relative spanwise extent of the localized roughness. The results suggest that bypass transition leads to mean flow reattachment in the spanwise proximity of the localized roughness. The produced three-dimensional, open LSB is characterized by the presence of notable spanwise flow in the separated flow region and spanwise variations of mean separation, transition, and reattachment. Consequently, changes are observed in the formation and development of shear layer vortices along the span, despite the dominant streamwise wavelength and frequency of perturbations remaining nearly unchanged. Furthermore, the modified LSB exhibits relatively low-frequency spanwise “breathing,” which induces mean flow deformations that modulate the stability of the LSB and contribute to the more complex vortex dynamics observed in this flow region.

We experimentally investigate the impingement of a gas jet onto a flowing liquid metal film, focusing on the resulting free-surface deformation and the subsequent evolution of surface waves. Our results show that the bulk flow significantly influences both the deformation of the free surface and the characteristics of the generated surface waves, which is different from the classical gas jet impingement on static liquid surfaces. Under steady gas jet impingement, the periodicity of free surface waves intensifies with a dominant frequency around 10 Hz at high gas Reynolds number ${\mathrm{Re}}_g$ for a stationary liquid surface. Critically, the interaction of the bulk flow with the gas flow establishes asymmetric coupling conditions, countercurrent flow upstream and cocurrent flow downstream, which cause the momentum exchange between the gas and the surface waves to exhibit opposite trends upstream and downstream as the liquid Reynolds number ${\mathrm{Re}}_l$ increases. In addition, under the impingement of a pulsating gas jet, we use the attenuation coefficient $c$ to correlate the wave amplitude with the jet momentum at different locations. We also find that the nonlinear interaction between the bulk flow and the forced wave accelerates the nonlinear evolution and energy dissipation of the wave. These findings can help engineers design liquid metal divertors to prevent excessive liquid evaporation, dry spots, and droplet splashing, which could damage the divertor substrate.

The fresh water equation of state exhibits a fundamentally important nonlinearity; the temperature at which it attains its maximum density occurs at roughly $4^{\circ }\mathrm{C}$ as opposed to at $0^{\circ }\mathrm{C}$. The form of the equation of state in this regime is quadratic and has the well-known implication that two fluid parcels may mix and produce a child parcel with a higher density; a process known as cabbeling. In this work we report on direct numerical simulations of the Rayleigh-Taylor instability in the cabbeling regime. We demonstrate that the mature, three-dimensional state of the instability leads to cabbeling, which, in turn, serves to sustain an instability that would otherwise saturate outside of the cabbeling regime. We present a new quantitative measure of strong cabbeling potential and use it, along with the mechanical energy, to demonstrate that the Rayleigh-Taylor instability is arrested in two ways that are unique to the cabbeling regime. Our results have implications for radiatively driven, under-ice flows in late winter during which ice prevents mechanical driving by the wind. While our focus is on the freshwater setting, we extend the derivation of the strong cabbeling potential to heat-salt systems, thereby extending its usefulness to the broader community.

Relaxation of the turbulent momentum and passive scalar fields generated by a specific staggered roughness distribution in the entrance region of an initially laminar channel flow is studied through direct numerical simulations. The computational domain is necessarily large, and a new and fast algorithm is introduced to efficiently resolve the flow characteristics. Staggered roughness elements of height 0.27 times the half width of the channel recover the entry zone either entirely or partly, prior to a smooth channel labeled as the control channel (CC) wherein both the turbulent flow and scalar fields decay, because the Karman number is significantly low and it is close to the subcritical limit. Despite the low Reynolds numbers, the roughness elements are able to trigger the bypass transition mechanism leading to massive clusters of coherent quasistreamwise vortices entering into the smooth channel. The characteristics of the turbulence decay at the CC centerline collapse reasonably well with those of a homogeneous isotropic turbulence. The inner layer responds relatively rapidly to the rough-smooth step change in the control channel while the outer layer recovery is gradual. The decays of the fluctuating streamwise velocity and of the passive scalar intensities are particularly slow. Despite the fact that the turbulent activity decreases and that the flow approaches the relaminarization, the Nusselt number averaged over large streamwise distances in the smooth channel remains close to an equivalent fully developed turbulent smooth-channel flow.