We determine the globally minimum time $T$ needed to translate a thin submerged flat plate a given distance parallel to its surface within a work budget. The Reynolds number for the flow is assumed to be large so that the drag on the plate arises from skin friction in a thin viscous boundary layer. The minimum is determined computationally using a steepest descent, where an adjoint formulation is used to compute the gradients. Because the equations governing fluid mechanics for this problem are nonlinear, multiple local minima could exist. Exploiting the quadratic nature of the objective function and the constraining differential equations, we derive and apply a ‘spectral condition’ to show the converged local optimum to be a global one. The condition states that the optimum is global if the Hessian of the Lagrangian in the state variables constructed using the converged adjoint field is positive semi-definite at every instance. The globally optimum kinematics of the plate starts from rest with speed $\propto t^{1/4}$ and comes to rest with speed $\propto (T-t)^{1/4}$ as a function of time $t$. For distances much longer than the plate, the work-minimizing kinematics consists of an optimum startup, a constant-speed cruising, and an optimum stopping.

Direct numerical simulations are performed to investigate the characteristics of a turbulent core–annular flow with water-lubricated high viscosity oil in a horizontal pipe. Six different superficial velocity ratios ($\,j_w/j_o = 0.057\unicode{x2013}0.41$) are examined by changing the water superficial velocity $j_w$ for a fixed oil superficial velocity $j_o$. The pressure drops in the pipe and the shapes of the phase interface agree well with those from previous experiments. The oil core flow is almost a plug flow, and the gaps between the phase interface and pipe wall are narrow and wide near the upper and lower surfaces of the pipe, respectively, due to the buoyancy. Within a narrow gap, water is confined mostly in a valley region of the wavy phase interface and hardly goes through its crest. On the other hand, water near the phase interface at a wide gap convects downstream almost at the core speed and the flow near the wall is similar to that of single-phase wall-bounded turbulent flow. The annular flow is characterized by three different regimes depending on the clearance Reynolds number ($Re_c$) based on the core velocity and local gap size: laminar Couette flow driven by the core for $Re_c \le 600$, transitional flow for $600 < Re_c < 2500$ and turbulent flow for $Re_c \ge 2500$. The minimum pressure drop occurs at $j_w/j_o = 0.11$ in the early transition regime. For all $j_w/j_o$ considered, the negative lift force acting on the core comes from the pressure force which balances the buoyancy.

We leverage the snap-through response of a bistable origami mechanism to induce a discontinuous evolution of drag with flow speed. The transition between equilibrium states is actuated passively by airflow, and we demonstrate that large shape reconfiguration over a small increment of flow velocity leads to a pronounced and sudden drop in drag. Moreover, we show that systematically varying the geometrical and mechanical properties of the origami unit enables the tuning of this drag discontinuity and the critical speed and loading at which it occurs. Experimental results are supported by a theoretical aeroelastic model, which further guides inverse design to identify the combination of structural origami parameters for targeted drag collapse. This approach sheds light on harnessing origami-inspired mechanisms for efficient passive drag control in a fluid environment, applicable for load alleviation or situations requiring swift transitions in aerodynamic performances.

This paper investigates the linear and nonlinear evolution of radiating modes under the influence of the spontaneously emitted Mach waves in a simple set-up of the supersonic boundary layers that develop in the entry region of a channel formed by two parallel semi-infinite flat plates. Two scenarios are considered. The first occurs in the boundary layers having identical wall conditions, where the Mach wave emitted by a radiating mode in one boundary layer influences the instability in the other. The second scenario takes place when the wall temperatures are different, in which case the spontaneously radiated Mach wave is reflected by the other boundary layer back to act on the radiating mode. Appropriate amplitude equations with the acoustic feedback effect being accounted for are derived. In each case, the effect of the spontaneously emitted sound contributes a linear term of delay type to the respective amplitude equation. For the first scenario, analytical and numerical studies of the amplitude equations show that due to the back action of the spontaneously radiated Mach wave, the amplitude exhibits rapid oscillations, and in the case of enhanced feedback effects, its envelope experiences near extinction followed by resurrection. The study of the coupled equations shows that the two modes with different initial amplitudes either undergo oscillations before attenuating, or terminate a finite-distance singularity at different locations. For the second scenario, the acoustic feedback produces similar effects in a broad range of wall temperature. The effects become pronounced, and the dependence on the wall temperature becomes more sensitive when the latter approaches the value corresponding to the resonance. Estimates suggest that such acoustic feedback is likely to be present in typical wind tunnel experiments and models for scramjet combustors.

Using laboratory experiments and numerical simulations, we examine the transfer of soluble material from small, spherical particles sinking in homogeneous turbulence at large Péclet number. A theoretical analysis predicts two distinct mechanisms of convective mass transfer: strain due to turbulence and slip due to gravitational settling. Their relative strength is parametrised by the sinking ratio, ${Sr} = w_0 \tau _\eta /a$, where $w_0$ is the quiescent settling velocity, $a$ is the particle radius and $\tau _\eta$ is the Kolmogorov time scale. This analysis predicts that the topology of the concentration wake changes from a symmetric topology at ${Sr} \ll 1$ to an asymmetric topology at ${Sr} \gg 1$ as the dominant mechanism of mass transfer changes. Particle tracking flow visualisations of small spheres releasing dye in turbulence confirm the existence of this change in mechanism at ${Sr} = O(1)$. We complement these experiments with numerical simulations of the mass transfer from sinking particles. The transfer rate predicted by the simulation is found to be in good agreement with literature data for mass transfer to turbulent suspensions of solid particles and is consistent with asymptotic expressions for mass transfer in uniform flow when ${Sr} \gg 1$. A decomposition of the convective fluxes confirms the transition in the transfer mechanism. At ${Sr} = O(1)$, both mechanisms provide comparable contributions to the transfer rate. Cross-correlation analysis reveals that particle-scale knowledge of both the recent strain and velocity history is required to predict the instantaneous transfer rate. Turbulence-induced particle rotation has a modest suppression effect upon convective transfer by sinking.

A surfactant-laden liquid film that lines the inside of an oscillating spherical cap is considered as a model of lung alveoli. Pulmonary surfactant solubility is described by Langmuir adsorption kinetics, modified by incorporating the intrinsic compressibility of the adsorbed monolayer. A novel boundary condition, supported by experimental data and scaling arguments, is applied at the rim. The condition enforces mass conservation of water and surfactant by matching the ‘large-scale’ dynamics of the alveolus to ‘small-scale’ equilibrium over mid-alveolar septa of small but finite thickness. Linear and weakly nonlinear analysis around the conditions in a non-oscillating cap indicates that the occurrence of shearing motion in the liquid is related to the non-zero film thickness over the rim, and shearing velocity at the interface is predicted an order-of-magnitude lower than the velocity of radial oscillation. Marangoni stresses dominate the interfacial dynamics, but capillary stresses affect significantly the interior flow field. In particular, they produce spatial modulations in flow rate, surface concentration of surfactant and wall shear stress, whose length scale varies with $Ca^{-1/3}$, i.e. is determined by a balance between capillary and viscous forces. Non-zero adsorption kinetics modifies at first order only the amplitude and phase of surface concentration, but affects all other variables at second order. In particular, it sets a steady drift of surfactant away from the alveolus and towards the rim. Finally, an attempt is made to relate the present predictions to physiological findings about air flow and particle deposition inside alveoli, and about shear stress-inflicted damage in diseased lungs.

As most mathematically justifiable Lagrangian coherent structure detection methods rely on spatial derivatives, their applicability to sparse trajectory data has been limited. For experimental fluid dynamicists and natural scientists working with Lagrangian trajectory data via passive tracers in unsteady flows (e.g. Lagrangian particle tracking or ocean buoys), obtaining material measures of fluid rotation or stretching is an active topic of research. To facilitate frame-indifferent investigations in unsteady and sparsely sampled flows, we present a novel approach to quantify fluid stretching and rotation via relative Lagrangian velocities. This technique provides a formal objective extension of quasi-objective metrics to unsteady flows by accounting for mean flow behaviour. For extremely sparse experimental data, fluid structures may be significantly undersampled and the mean flow behaviour becomes difficult to quantify. We provide a means to maintain the accuracy of our novel sparse flow diagnostics in extremely sparse sampling scenarios, such as ocean buoy data and Lagrangian particle tracking. We use data from multiple numerical and experimental flows to show that our methods can identify structures beyond existing limits of sparse, frame-indifferent diagnostics and exhibit improved interpretability over common frame-dependent diagnostics.

Vortex rings are critical for thrust production underwater. In the ocean, self-propelled mesozooplankton generate vortices while swimming within a weakly stratified fluid. While large-scale biogenic transport has been observed during vertical migration in the wild and lab experiments, little focus has been given to the evolution of induced vortex rings as a function of their propagation direction relative to the density gradient. In this study, the evolution of an isolated vortex ring crossing the interface of a stable two-layer system is examined as a function of its translation direction with respect to gravity. The vortex ring size and position are visualized using planar laser-induced fluorescence (PLIF) and the induced vorticity field derived from particle image velocimetry (PIV) is examined. It is found that the production of baroclinic vorticity significantly affects the propagation of vortex rings crossing the density interface. As a result, any expected symmetry between vortex rings travelling from dense to light fluids and from light to dense fluids breaks down. In turn, the maximum penetration depth of the vortex ring occurs in the case in which the vortex propagates against the density gradient due to the misalignment of the pressure and density gradients. Our results have far-reaching implications for the characterization of local ecosystems in marine environments.

The linear stability of the incompressible flow in an infinitely extended cavity with rectangular cross-section is investigated numerically. The basic flow is driven by a lid which moves tangentially, but at yaw with respect to the edges of the cavity. As a result, the basic flow is a superposition of the classical recirculating two-dimensional lid-driven cavity flow orthogonal to a wall-bounded Couette flow. Critical Reynolds numbers computed by linear stability analysis are found to be significantly smaller than data previously reported in the literature. This finding is confirmed by independent nonlinear three-dimensional simulations. The critical Reynolds number as a function of the yaw angle is discussed for representative aspect ratios. Different instability modes are found. Independent of the yaw angle, the dominant instability mechanism is based on the local lift-up process, i.e. by the amplification of streamwise perturbations by advection of basic flow momentum perpendicular to the sheared basic flow. For small yaw angles, the instability is centrifugal, similar as for the classical lid-driven cavity. As the spanwise component of the lid velocity becomes dominant, the vortex structures of the critical mode become elongated in the direction of the bounded Couette flow with the lift-up process becoming even more important. In this case the instability is made possible by the residual recirculating part of the basic flow providing a feedback mechanism between the streamwise vortices and the streamwise velocity perturbations (streaks) they promote. In the limit when the basic flow approaches bounded Couette flow the critical Reynolds number increases very strongly.

The present study offers a twofold contribution on counter-gradient transport (CGT) of turbulent scalar flux. First, by examining turbulent scalar mixing through synchronized particle image velocimetry and planar laser-induced fluorescence on an inclined jet in cross-flow, we clarify the previously unexplained phenomenon of CGT, revealing key flow structures, their spatial distribution and modelling implications. Statistical analysis identifies two distinct CGT regions: local cross-gradient transport in the windward shear layer and non-local effects near the wall after injection. These behaviours are driven by specific flow structures, namely Kelvin–Helmholtz vortices (local) and wake vortices (non-local), suggesting that scalar flux can be decomposed into a gradient-type term for gradient diffusion and a term for large-eddy stirring. Second, we propose a new approach for reconstruction of turbulent mean flow and scalar fields using continuous adjoint data assimilation (DA). By rectifying model-form errors through anisotropic correction under observational constraints, our DA model minimizes discrepancies between experimental measurements and numerical predictions. As expected, the introduced forcing term effectively identifies regions where traditional models fall short, particularly in the jet centreline and near-wall regions, thereby enhancing the accuracy of the mean scalar field. These enhancements occur not only within the observation region but also in unseen regions, underscoring present DA approach's reliability and practicality for reproducing mean flow behaviours from limited data. These findings lay a solid foundation for adjoint-based model-consistent data-driven methods, offering promising potential for accurately predicting complex flow scenarios like film cooling.

The interaction of steady free-surface flows of viscoplastic material with a surface-piercing obstruction of square cross-section on an inclined plane is investigated theoretically. The flow thickness increases upstream of the obstruction and decreases in its lee. The flow depends on two dimensionless parameters: an aspect ratio that relates the flow thickness, the obstruction width and the plane inclination; and a Bingham number that quantifies the magnitude of the yield stress relative to the gravitationally induced stresses. Flows with a non-vanishing yield stress always form a static ‘dead’ zone in a neighbourhood of the upstream and downstream stagnation points. For relatively wide obstructions, a deep ‘ponded’ region develops upstream with a small dead zone, while the deflected flow reconnects over relatively long distances downstream. The depth of the upstream pond increases with both the dimensionless yield stress and width of the obstruction, while the unyielded dead zone varies primarily with the yield stress. Both are predicted asymptotically by balancing the volume flux of fluid into and out of the ponded region. When the obstruction is narrow, the perturbation to the depth of the oncoming flow is reduced. It exhibits fore–aft antisymmetry, while the dead zone is symmetric to leading order. Increasing the yield stress leads to larger dead zones that eventually encompass all of the upstream- and downstream-facing boundaries of the obstruction and fully divert the flow. Results for obstructions with circular and rhomboidal cross-sections are also presented and illustrate the effects of boundary shape on the properties of the steady flow.

We consider a theoretical model for the settling of rod-shaped particles of a dilute, initially homogeneous, suspension in rapid rotation. The particle Reynolds number and the particle Taylor number of the detailed flow around the particles are assumed small, representing a relevant limit for an industrial centrifugal separation process. By applying a statistical approach using the Fokker–Planck equation, and neglecting particle–particle interactions, we obtain an explicit, analytical solution for the time dependent, spatially uniform particle orientation distribution function. Not only does the volume fraction in the bulk of the suspension decrease with time due to the divergent centrifugal field, as similarly described in the literature for suspensions of spherical particles, the orientation of the rod particles also changes with time from an initially uniform distribution to one where the particles tend to align with a plane perpendicular to the axis of rotation. The corresponding particle trajectories, as also influenced by first-order effects from the Coriolis acceleration and gyroscopic effects, are obtained numerically for different initial particle orientation angles.