We apply a physics-informed deep learning method to reconstruct the flow field from sparse and noisy particle tracks, focusing on accurately assimilating the global minimum pressure on a fine-grid domain from very sparse velocity data. The spatial resolution of the velocity data could be much larger than the mesh spacing of the reconstructed flow field. To address this problem, we first quantify the performance of the physics-informed neural network (PINN) model using a series of analytical vortex models. Synthetic particle data are generated using the analytical solutions of the velocities; the PINN model is trained on these sparse particle measurements, and the model then reconstructs the finer-resolution flow field within a fixed domain of interest and obtains the minimum pressure. We also investigate the effects of the reconstruction resolution, particle densities, particle distributions, incomplete measurements, velocity noise, and noise in spatial coordinates. The methodology is applied to a turbulent flow of interacting counterrotating vortices of unequal strength obtained from Large-Eddy Simulation, and the results are discussed.

Reliable description of the inflow boundary conditions of the fully developed turbulence in atmospheric boundary layers for large-eddy simulation (LES) has been a subject of significant research interest due to its considerable impact on simulation results. In existing synthetic random Fourier methods for generating inhomogeneous inflow turbulence, the inhomogeneous field can be obtained by superimposing multiple homogeneous turbulence fields, each weighted by an assigned weighting function. However, inappropriate mapping strategies may distort the spatial correlations and continuity of turbulence structures from the targeted values, which could further deteriorate the simulation results. In this study, the impacts of the weighting functions on the total-energy profiles, spatial correlations, and the continuity condition of turbulence are clarified, based on which an adaptive energy-preserving mapping (AEPM) strategy is developed. The results from a priori numerical test demonstrate that inflow turbulence fields generated by the established synthetic random Fourier framework in combination with the AEPM accurately preserve the target turbulence energy profiles while maintaining good agreement with spatial correlation properties. Furthermore, the robustness and effectiveness of the AEPM are demonstrated through three LES case studies. Case 1 and Case 2 simulate atmospheric boundary layers around an isolated rectangular building and are validated against experimental data under neutral and unstable thermal stratification, respectively. In these two cases, wall-modeled LES (WMLES) results are compared with wall-resolved LES to investigate the performance of WMLES in urban environment simulations. In Case 3, the building in Case 1 is removed to form an empty domain, in which downstream turbulence statistics are evaluated to further assess the AEPM-generated inflow.

A physics-based analysis framework is developed to characterize low-frequency unsteadiness (LFU) in separated wake flows. LFU is interpreted as an outcome of a temporal imbalance of kinetic energy within the mean recirculation region. A kinetic energy transport equation is derived from the rotational form of the Navier-Stokes equations using vorticity-based quantities, motivated by their direct connection to vortical dynamics and flow structures relevant to LFU. To isolate LFU from dominant wake motions, temporal low-pass filtering is applied together with conditioning based on a local intermittency measure, extracting intermittent fluctuations at frequencies much lower than the primary vortex shedding frequency. The framework is demonstrated for a two-dimensional laminar wake past a normal plate. The filtered drag coefficient exhibits a clear LFU signature and is strongly correlated with the net filtered kinetic energy within the mean recirculation region. Analysis of the kinetic energy budget identifies the Bernoulli energy flux and Lamb-vector transport as the dominant mechanisms governing kinetic-energy charging and discharging, respectively. Strong correlations are observed between the filtered drag and the net Lamb-vector divergence, attributed to vortex merging and intensified roll-up that redistribute the Lamb vector and reduce base pressure. The LFU frequency is estimated using dimensional scaling and a data-driven approach. The analysis highlights the central role of the Lamb-vector divergence in understanding and predicting LFU in separated wake flows. By providing a reduced, physics-based description of LFU in terms of kinetic energy dynamics, the proposed framework offers a useful tool for analyzing LFU in turbulent separated flows, where visual inspection is limited and modal decompositions remain high dimensional.

This paper studies the interrelation between the Rayleigh-Bénard and the electroconvective instability in concentration polarization during ionic dc current passage from an electrolyte solution into a charge-selective solid, such as a metal electrode, an ion-exchange membrane, or a set of nanochannels. We conclude that, whereas in the underlimiting current regime the effect of electroconvection on the onset of buoyant convection is insignificant, these two phenomena become strongly intertwined at the limiting current. In this regime, the development of the extended space charge and its related nonequilibrium electroconvective instability have a major effect upon the onset of the buoyant instability; and vice versa, buoyancy has a major effect on the features of the electroconvective instability. This includes lowering of the onset threshold and regularization of the shortwave singularity typical of electroconvective instability.

Turbulent puffs in pipe flow remain sustained for an extended period of time before abruptly transitioning to laminar flow through viscous exponential decay. During the statistically stable state, the topology of sectional streamlines reveals the presence of numerous saddle and nodal points in proximity to the wall. The number of saddle and node points decreases as relaminarization begins, and they migrate away from the near-wall region. In the final stage of relaminarization, direct numerical simulations reveal, in some cases, sequences of saddle-node bifurcations that ultimately lead to the annihilation of saddle-node pairs. The Riccati equation is proposed as a simple approach to model the motion of these pairs.

Controlling water motion on surfaces is critical for applications ranging from thermal management to passive water harvesting to self-cleaning coatings. Yet stabilizing continuous water films, desirable for their high surface coverage and drainage capacity, remains challenging with pure water, due to its high surface tension. Existing strategies rely on extreme wettability achieved by coating or fine-scale patterning, which are costly, fragile, or complex to scale. A robust, purely geometric solution is still lacking. We demonstrate that a pair of laser-engraved grooves on a moderately hydrophilic vertical substrate can laterally anchor water and stretch a continuous thin film, without any other surface treatment. Once stabilized, the film extends vertically over 100 capillary lengths ($>30\mathrm{cm}$), with thickness tunable via both flow rate and groove geometry. At the groove extremities, the end of anchoring triggers a cyclic instability, characterized by film rupture, retraction, and droplet release. The thickness of the film and retraction height obey predictive models, while droplet mass varies systematically with spacing and surface tension. This groove-based method offers a straightforward and scalable approach to creating, sustaining, and controlling thin water films. It opens new directions for passive liquid control in condensation, surface cleaning, and 2D millifluidic systems.

The ability to trap a single cell or microparticle in three dimensions is important for biomedical and microfluidic applications. Single-beam acoustic tweezers based on focused waves offer a compact and biocompatible solution with their high spatial resolution and intensity gradient. However, the three-dimensional (3D) trapping remains limited because the weak axial restoring radiation forces cannot always surpass the pushing drag force from the acoustic bulk streaming (ABS), particularly at high frequencies. The combined effect of the acoustic radiation force (ARF, $F_{\mathrm{rad}}$) and the drag force ($F_{\mathrm{drag}}$) induced by the ABS on a microparticle has not yet been studied in the free space. It is known that the ARF depends on the square of the pressure amplitude at the focus as $p_{\text{foc}}^2$, whereas the drag force induced by the ABS remains an open question with respect to $p_{\text{foc}}$ at different flow conditions. In this study, we propose a unified theoretical-numerical framework to compare the contribution of the ARF and the drag force by the ABS, and we systematacially derive an explicit scaling law for the streaming velocity $U_0\sim p_{\text{foc}}^n$ with the hydrodynamic flow from the viscous to the inertial regimes. It is found that the exponential power $n$ of $p_{\text{foc}}$ is bounded by $n=2$ for the viscous limit (the flow Reynolds number ${\text{Re}}_{\lambda }\ll 1$) and by $n=4/3$ for the inertial limit (${\text{Re}}_{\lambda }\gg 1$), and $n\in (4/3,2)$ throughout the transition regime (${\text{Re}}_{\lambda }\sim 1$). In addition, the Schiller-Naumann model is introduced to facilitate a more accurate estimation of the drag force compared to the Stokes model. Based on the combined effect, we find that the trap ratio of the axial ARF over the drag force can vary nonmonotonically with $p_{\text{foc}}$, which contradicts the conventional expectation of the monotone increase. This work provides a theoretical foundation for the 3D trapping of a single cell with single-beam acoustical tweezers and guidance for the optimized parameters.

We report the first direct numerical simulations using a recently introduced exact director-based formulation of the equations of motion for inertial spheroids in turbulence. The simulations consider homogeneous isotropic flow across a broad range of particle aspect ratios (from disklike to rodlike) and Stokes numbers (masses), both with and without gravity. The results reveal that particle shape and finite inertia induce complex, nonmonotonic trends in preferential clustering. Gravity markedly alters these dynamics, enhancing small-scale clustering for rod- and disklike particles while suppressing clustering for nearly spherical particles; we propose an explanation for this behavior. We confirm that at weak inertia, rodlike spheroids tend to align with the flow's major stretching direction, whereas disklike spheroids align with its major shrinking direction. We derive a generalization to the case of finite inertia, establishing preferred alignment directions that do not coincide with the flow's principal Lagrangian stretching or shrinking axes, and provide preferential alignment directions across all examined Stokes numbers and gravity conditions. We also confirm that gravity makes the symmetry axis of disklike particles orient preferentially horizontally. Finally, we introduce an efficient framework for computing Lyapunov exponents and their generalizations for inertial particles. Given its superior efficiency and ability to unify previously disparate data, we believe our approach should serve as a foundation for future simulations and a powerful tool for both theoretical and numerical studies of spheroids in turbulence.

Shear-thickening suspensions of non-Brownian polydisperse particles are simulated in two dimensions using a lubrication flow-discrete element method-based algorithm at high packing fractions ($\varphi$) and large nondimensional stresses ($\sigma$). Rigidity analysis of the stress-induced particle clusters is carried out using pebble game algorithm for polydisperse suspensions and compared with the statistically equivalent bidisperse systems. A critical value of the packing fraction, ${\varphi }_c$, close to the shear-jamming transition, ${\varphi }_J^{\mu }$, (${\varphi }_c<{\varphi }_J^{\mu }$) is obtained where rigid particle clusters begin to grow sharply. The growth is found to be characterized by a critical transition of an order parameter ($f_{\text{rig}}$), defined by the fraction of particles in rigid clusters, which scales as $f_{\text{rig}}\sim {(\varphi -{\varphi }_c)}^{\beta }$ for $\varphi >{\varphi }_c$, and by the susceptibility scaling, ${\chi }_{\text{rig}}\sim {|\varphi -{\varphi }_c|}^{-\gamma }$, with exponents having values consistent with the critical exponents in two-dimensional (2D) percolation transition. The variations of $f_{\text{rig}}$ and ${\chi }_{\text{rig}}$ in polydisperse suspensions are found to be identical to that of the statistically equivalent bidisperse suspensions. Finite-size-scaling analysis shows a divergence of correlation length near ${\varphi }_{c_{\infty }}$ following critical exponent $\nu \approx 1.33$, in agreement with the 2D percolation theory. Furthermore, ${\varphi }_c$ and ${\varphi }_J^{\mu }$ are found to vary nonmonotonically with polydispersity index and depend on the particle stiffness.

We develop a reduced-order theoretical framework for inertial spheroidal particles in turbulent Navier-Stokes flow, representing their orientation with a single unit vector (the director) instead of using a full rotation matrix. This symmetry-based simplification, analogous to Jeffery's classical description of neutrally buoyant spheroids, yields a compact set of evolution equations with fewer degrees of freedom. In the limit of small inertia, the director obeys Jeffery's equation which we consider as the equation on direction of distance between two infinitesimally close trajectories of a chaotic system. This implies that the particle's director is uniquely determined by its position, defining a director field that, in turn, induces a smooth, weakly compressible effective flow through which the spheroids move. This effective flow is distinct from the underlying incompressible turbulent flow, thereby generalizing Maxey's result for spheres. In homogeneous, anisotropic turbulence with zero mean fluid flow, the effective particle flow can acquire a nonzero mean; for example, in rotating turbulence, particles drift uniformly along the rotation axis. We show analytically that prolate spheroids align with the major Lagrangian stretching direction of the turbulent flow, whereas oblate spheroids align with the major Lagrangian compression direction. In spatially inhomogeneous flows (e.g., a wall-bounded channel flow), particle anisotropy can alter turbophoretic drift, possibly reversing the usual migration toward quiescent regions. We also show that inertial spheroids exhibit preferential concentration. Finally, when gravity is included, we derive analytic expressions for the mean settling velocity in the weak-gravity regime. This settling velocity increases monotonically with the particle aspect ratio, from $8/9$ of the Stokes settling velocity of an equivalent sphere for disklike particles to $4/3$ for rodlike particles. In the strong-gravity limit, the driving of spheroids by the Navier-Stokes turbulence can be effectively described as white noise. We show that very small oblate spheroids settle edge-first, but small ones settle broadside-on. The orientation-dependent settling velocity field is spatially varying and weakly compressible, providing a new mechanism for multifractal preferential concentration. Thus, this director-based framework provides a unified, physically transparent generalization of Jeffery's and Maxey's theories, yielding multiple new analytical predictions for the transport of nonspherical particles in isotropic, anisotropic, and stratified turbulent flows and environments.

Owing to the spatially local nature of the constitutive equations typically used to model polymeric stresses, the differential operators governing the linearized dynamics of bounded viscoelastic shearing flows have singular points. As a result, the eigenspectra of such shearing flows contain, in addition to discrete eigenvalues, continuous spectra (CS) comprising singular eigenfunctions. On account of the singularity, numerical methods capture these spectra only as extended balloons that slowly converge to the theoretical loci. A clear understanding of the theoretical CS loci is crucial in discriminating physically genuine (discrete) eigenvalues from the poorly approximated numerical CS. This is especially important because, when a parameter such as the Weissenberg number $\left(\text{Wi}\right)$ is varied, discrete eigenvalues can emerge from or disappear into the CS, as the latter are often branch cuts in the complex spectral plane. For rectilinear shear flows of Oldroyd-B fluids, the CS are a pair of line segments, with lengths equal to the base-state range of velocities. In this study, we provide the first comprehensive account of the nature of the CS for both rectilinear and curvilinear shearing flows of the FENE-P fluid. The FENE-P model is one of the most widely used constitutive equations for dilute polymer solutions, and predicts both shear thinning of the polymer viscosity and the first normal stress coefficient. In stark contrast to the CS for the Oldroyd-B fluid mentioned above, we show analytically that there are up to six distinct continuous spectra for shearing flows of FENE-P fluids. Further, owing to shear thinning, and due to the highly coupled nature of the linearized equations, these spectra possess nontrivial structures in the complex plane. When the finite extensibility parameter $L$ is greater than 50, as appropriate for large molecular weight polymers used in experiments, three of the CS are nearly identical, and independent of the solvent-to-solution viscosity ratio $\left(\beta \right)$. The other three CS are $\beta$-dependent, with one of them being the analog of the solvent (viscous) continuous spectrum in the Oldroyd-B fluid. The remaining two $\beta$-dependent CS are novel features of the FENE-P spectrum, and can have phase speeds outside the base range of velocities, including negative ones. For nonlinear shearing flows subjected to streamwise invariant disturbances, the latter two CS have a novel “winglike” structure in the complex plane, symmetric about the imaginary axis. All but two of the theoretical CS are shown to collapse onto master curves for different $(\text{Wi},L)$ pairs, with $\text{Wi}/L$ fixed, a feature not shared by the discrete spectrum. The location and structure of the theoretical CS are broadly in accord with their numerical approximations obtained using the spectral method. The complexity of the CS predicted here, for shearing flows of FENE-P fluids, is expected to carry over to other nonlinear viscoelastic models that exhibit a shear-thinning rheology. Our analytical predictions for the CS will therefore be helpful in identifying physically genuine, discrete eigenvalues in the numerically computed spectra.

We investigate analytically and numerically the dynamics of a two-dimensional superflow governed by the Gross-Pitaevskii equation passing over finite-size rectangular obstacles: an impenetrable wall and an impenetrable rectangular well. Extending classical studies of vortex nucleation around smooth obstacles, we focus on the role of sharp corners in determining the onset of vortex nucleation. Using a combination of analytical techniques based on the Schwarz-Christoffel methods for potential flow and on numerical simulations, we show that local velocity amplification near sharp corners crucially controls the critical flow velocity for vortex nucleation. For both wall and well configurations, we identify analytically and theoretically the critical velocities as a function of the obstacle width and its height or depth, finding excellent agreement between the theory and our numerical simulations. Our results provide a simple framework for understanding superflow stability past finite-size obstacles with sharp features, and they are directly relevant to experimentally realizable configurations in atomic Bose-Einstein condensates and related superfluid systems.

Slender floaters drifting in propagating gravity waves slowly rotate towards a preferential state of orientation with respect to the angle of incidence. This angular drift arises from a wave-induced, second-order mean yaw moment. We develop a diffractionless, hydroelastic theory to compute this mean yaw moment for a thin, flexible structure whose width and thickness are small compared with the wavelength. For floater lengths smaller than half the wavelength, we derive a simple, predictive criterion for the preferred orientation: Soft, short, and heavy floaters prefer the longitudinal state, while stiff, long, and light floaters prefer the transverse state. For floaters longer than the wavelength, the orientational dynamics become more intricate and may exhibit multiple equilibrium states. We discuss the implications of the model for flexible floating structures such as pontoons and inflatable structures.

We investigate active interfacial ion flux in droplet electrohydrodynamics with an emphasis on deformation, pinch-off, and recoalescence. A thermodynamically consistent phase-field framework is developed that couples Nernst–Planck–Poisson electrodiffusion with incompressible Navier–Stokes–Cahn–Hilliard flow and incorporates active interfacial ionic fluxes. In a baseline case without active interfacial transport, applied fields merely polarize the droplet, and deformation is negligible. In contrast, surface-localized active transport drives the accumulation of positive ions within the droplet, elevates the interior potential, and generates nonuniform electric fields. The resulting electric stresses stretch and displace the droplet, thin interfacial necks, and trigger pinch-off; the daughter droplets subsequently recoalesce, often after wall contact, yielding flattened remnants. In multiple-droplet configurations, charging induced by active interfacial transport produces lateral electrostatic repulsion and asymmetric deformation; under geometric confinement, crescent bending and star-like morphologies emerge. Poiseuille-flow tests further show that a droplet with active transport can be immobilized and ruptured, while a neighbor without active flux is advected downstream, suggesting a route to sorting. Together, the results establish active interfacial transport as an internal actuation mechanism that robustly controls droplet morphology and dynamics across configurations.

We study the collective sedimentation of cube-shaped particles in the Stokes regime using direct numerical simulations based on the smooth profile method. The particles, constructed as rigid assemblies of $3\times 3\times 3$ spherical beads, settle in a Newtonian fluid at Reynolds number $\mathrm{Re}=0.1$ over volume fractions $0.001\le \varphi \le 0.08$. Although orientation-dependent hydrodynamic forces at the single-particle level, the suspension recovers the classical Stokesian scaling laws traditionally associated with spherical particles. The radial distribution function exhibits a single dominant peak that grows monotonically with $\varphi$, indicating isotropic microstructural organization and progressive clustering. Orientation statistics show random alignment in dilute suspensions and a transition toward preferential alignment at ${45}^{\circ }$ with respect to gravity as concentration increases. The mean settling velocity follows the Richardson-Zaki law with exponent $n\simeq 6$, velocity fluctuations scale as ${\varphi }^{1/2}$, and relaxation timescale as ${\varphi }^{-1/2}$. Self-diffusion coefficients increase weakly with volume fraction and asymptotically approach the ${\varphi }^{1/2}$ scaling characteristic of spherical suspensions. These results demonstrate that geometric symmetry, rather than sphericity alone, governs the emergence of sphere-like collective sedimentation behavior in nonspherical particle suspensions under creeping-flow conditions.

Particle image velocimetry is the most widely used optical technique for measuring two-dimensional velocity fields in fluids. However, with the standard cross-correlation algorithm, improving the spatial resolution of instantaneous velocity fields and obtaining dense velocity fields in real time and at high frequencies remains challenging. Optical flow velocimetry (OFV) provides an alternative route to per-pixel velocimetry. In this study, we show that dense velocity fields (one vector per pixel) with high effective spatial resolution can be obtained in real time at frequencies up to kHz using an optical flow approach. First, using synthetic images on two cases (a well-defined Rankine vortex and a homogeneous isotropic turbulence DNS dataset), we show that resolving large displacement gradients down to small scales is feasible when the particle seeding is appropriately tuned. Second, we show that real-time processing rates can be achieved through algorithmic and GPU-oriented optimizations, together with appropriate choices of optical flow parameters. With this implementation, 21 Mp velocity fields are measured in real time (live) at 90 Hz, while 4 Mp velocity fields can be measured up to 460 Hz and 1 Mp velocity fields can be measured up to 1400 Hz. The approach is further validated on experimental measurements of the flow past a circular cylinder, where dense instantaneous fields are obtained and used to compute derived quantities in real time over long durations. These processing rates enable real-time computation of derived flow quantities during experiments, including long-duration monitoring, low-frequency dynamics captured from sustained high-rate acquisition, and closed-loop flow-control strategies based on OFV measurements. In addition to enabling live operation, the approach accelerates standard off-line postprocessing, reducing turnaround time and computational cost.

The multiscale structure of the respiratory organ encompasses a challenging environment to predict the transport and deposition of inhaled aerosols, whether relevant to therapeutic drug delivery or hazardous exposure assessment. While the traditional leading deposition mechanisms, foremost impaction, gravitational settling and diffusion, are generally well understood and extensively studied, the role of electrostatic charge present in aerosols generated with common inhalers remains in comparison poorly characterized. Here, we employ a so-called one-path airway model integrated into an airway-on-chip to explore the contribution of electrostatic forces to aerosol deposition along the respiratory tract. By comparing neutralized and charged particles (diameters 0.2 and 2 µm), we show that electrostatic forces significantly alter deposition patterns, foremost shifting deposition from the deep acinar region to more proximal airways in the small bronchial regions (i.e. bronchioles), in analogy to a screening phenomenon. Our experiments, complemented by nondimensional analysis, underscore conditions where electrostatics begin to dominate over traditional deposition mechanisms, thereby underlining the importance of recognizing the role of electrostatics towards pulmonary deposition.

The transient flow induced by an array of wall-mounted flexible blades performing impulsive metachronal rowing, i.e., the sudden onset of coordinated oscillations of wall-mounted flexible blades with phase differences among them in an initially quiescent fluid, is numerically investigated across a wide range of the Cauchy number ($\mathrm{Ca}$), which quantifies the ratio of fluid inertial to the blade elastic force. Two distinct flow regimes emerge depending on the ratio of the natural frequency ($f_n$) to the rowing frequency ($f$). For stiff blades ($f_n>f$), the mean deflection angle scales linearly with $\mathrm{Ca}$, and the rescaled flow rate remains high. In contrast, for soft blades ($f_n<f$), the deflection saturates, and the rescaled flow rate decays rapidly as ${\mathrm{Ca}}^{-2.35}$. The maximum fluid transport is achieved near $\mathrm{Ca}=1$ and at shorter rowing wavelength (six blades per wavelength and unit spacing), where tip-shed concentrated vortices are (1) optimally positioned to reinforce the flow around power-stroke blades and (2) subsequently captured by those blades, thereby enhancing thrust. These findings provide physical insights into the hydrodynamic optimization principles underlying bio-inspired metachronal propulsion.

Unstably stratified mixing layers laden with inertial, heated particles are investigated using Eulerian-Lagrangian direct numerical simulations, primarily aiming to account for interphasial energy transfer due to the momentum and thermal couplings between particle and fluid phases. The heated particles with Stokes numbers $\mathrm{St}=0.2$, 1.3, 8.0, 20.0, and 38.0 settling across unstably stratified mixing layers are considered. Compared with the unstably stratified mixing layers in the absence of particles, the addition of heated particles enhances turbulence and induces the faster transition from shear-driven to turbulent mixing affected by the combined buoyancy and shear. The integrated turbulent kinetic energy $E$ of fluid phase varies nonmonotonically as St increases. When the particles are initially distributed in the whole domain, the heated particles with $\mathrm{St}\approx 1$ maximize $E$ and its viscous dissipation due to the most energy injected from particles into fluid phase. Moreover, the mean kinetic energy transferred from heated particles to fluids varies nonmonotonically while the fluctuating counterpart increases with increasing St. In contrast, the interphasial heat transfer decreases with increasing St due to the larger thermal response time of large particles. Heated particles give rise to stronger flow instability and turbulence modulation when initially distributed in the upper stream, attributing to transfer of more gravitational potential energy and stronger particle-fluid interactions. Concerning the dynamics of heated particles settling across unstably stratified mixing layers, it is found that the drag force of the particles increases in the direction opposite to the shear and decreases in vertical direction, forming the vertically inclined particle streaks which contribute the enhanced buoyancy production and fluctuating power to fluid.

In this paper, we study the free streamline theory for the separating flow over a circular cylinder. The objective of this paper is twofold: (i) to demonstrate the validity of the condition of matching curvature and (ii) to obtain a reasonable estimate of the separation angle in the subcritical regime ($\text{Re}={10}^4–{10}^5$) without explicitly modeling the boundary layer, following Prandtl's conjecture in his seminal paper [L. Prandtl, in Proceedings of the Third International Mathematical Congress in Heidelberg (B. G. Teubner, Leipzig, 1904), pp. 484–491]. For the former goal, we study Roshko's free streamline model [Roshko, Tech. Rep. No. NACA-TN-3168 (NASA, Washington, D.C., 1954)]; it is an ideal flow model with sheets of discontinuities that represent the separating shear layers in the near-wake region. It is known that the model fails to predict the correct separation angle over a curved surface. Roshko attributed this discrepancy to the condition of matching curvature, deeming it invalid. This condition (known as the condition of smooth detachment) asserts that the curvature of the separating streamline at the separation point must match that of the cylinder. We show that the condition of matching curvature is legitimate and is not the real culprit for the failure of Roshko's model in predicting the correct separation angle. We also show that Roshko's model is nonunique. There are solutions in Roshko's family with matching curvature whose separation point agrees with experimental measurements. As for the second goal, we rely on the principle of minimum pressure gradient (PMPG), which asserts that an incompressible flow evolves from one instant to another in order to minimize the total magnitude of the pressure gradient over the domain. Encouraged by the fact that the flow characteristics in the range $\text{Re}={10}^4–{10}^5$ are fairly independent of $\text{Re}$, we aim to predict the separation angle in this regime without modeling the boundary layer—a task that may seem impossible, though anticipated in Prandtl's original paper that introduced the concept of boundary layer [Prandtl, Proc. Intl. Congress Math. (1904)]. Over the family of kinematically admissible equilibrium flows, we utilize the PMPG to single out the separating flow with the minimum pressure gradient cost. Interestingly, the obtained separation angles match the experimental measurements over the $\text{Re}={10}^4–{10}^5$ regime.