Sheltering the perturbed vortical layer of electroconvection under shear flow

Sheltering of a perturbed vortical layer by a shear flow is a common method to control turbulence and transport in plasma physics. Despite the desire to exploit this phenomenon in wider engineering applications, shear sheltering has rarely been observed in general non-ionized fluids. In this study, we visualize this shear sheltering in a generic neutral-fluid situation in electromembrane desalination: electroconvection (EC) under the Hagen–Poiseuille flow initiated by ion concentration polarization. Our work is the first demonstration of shear sheltering in electrochemical systems. Experiment, numerical simulation and scaling analysis accurately capture the effect by pinpointing the threshold for shear suppression. Determined by balancing the velocity fluctuation (with EC vortices) and the flow shear (with no-slip walls), the threshold for shear suppression is scaled as the EC height. Stable EC with coherent unidirectional vortices occurs under the threshold height, whereas chaotic EC occurs beyond this height as the EC-induced vortical perturbation overwhelms the flow shear. Attractors in a time-delay phase space illustrate this sequence of steady–periodic (stable EC)–chaotic transitions precisely. Going one step further, the shear sheltering effect is decoupled from the shear-independent mechanism of vortex suppression, i.e. vortex sweeping by the mean flow. In the frequency domain, this shear-independent effect is negligible for stable EC (when shear sheltering dominates), whereas it can reduce the level of chaotic fluctuations of chaotic EC (when shear sheltering weakens). Lastly, taken together, we describe the EC-induced convective ion transport by the new scaling law for the electric Nusselt number as a function of the electric Rayleigh number and the Reynolds number. This work not only expands the scientific understanding of EC and the shear sheltering effect, but also affects a broad range of electrochemical applications, including desalination, energy harvesting and sensors.

The Formation of Parasitic Capillary Ripples on Gravity–Capillary Waves and the Underlying Vortical Structures

The evolution of moderately short, steep two-dimensional gravity–capillary waves, from the onset of the parasitic capillary ripples to a fully developed quasi-steady stage, is studied numerically using a spectrally accurate model. The study focuses on understanding the precise mechanism of capillary generation, and on characterizing surface roughness and the underlying vortical structure associated with parasitic capillary waves. It is found that initiation of the first capillary ripple is triggered by the fore–aft asymmetry of the otherwise symmetric carrier wave, which then forms a localized pressure disturbance on the forward face near the crest, and subsequently develops an oscillatory train of capillary waves. Systematic numerical experiments reveal that there exists a minimum crest curvature of the carrier gravity–capillary wave for the formation of parasitic capillary ripples, and such a threshold curvature (≈0.25 cm−1) is almost independent of the carrier wavelength. The characteristics of the parasitic capillary wave train and the induced underlying vortical structures exhibit a strong dependence on the carrier wavelength. For a steep gravity–capillary wave with a shorter wavelength (e.g., 5 cm), the parasitic capillary wave train is distributed over the entire carrier wave surface at the stage when capillary ripples are fully developed. Immediately underneath the capillary wave train, weak vortices are observed to confine within a thin layer beneath the ripple crests whereas strong vortical layers with opposite orientation of vorticity are shed from the ripple troughs. These strong vortical layers are then convected upstream and accumulate within the carrier wave crest, forming a strong “capillary roller” as postulated by Longuet-Higgins. In contrast, as the wavelength of the gravity–capillary wave increases (e.g., 10 cm), parasitic capillary ripples appear as being trapped in the forward slope of the carrier wave. The strength of the vortical layer shed underneath the parasitic capillaries weakens, and its thickness and extent reduces. The vortices accumulating within the crest of the carrier wave, therefore, are not as pronounced as those observed in the shorter gravity–capillary waves.

Fine-scale structure of thin vortical layers

A class of exact solutions of the Navier–Stokes equations is derived. Each of them represents the velocity field v=U+u of a thin vortical layer (a planar jet) under a uniform strain velocity field U in three-dimensional infinite space, and provides a simple flow model in which nonlinear coupling between small eddies plays a key role in small-scale vortex dynamics. The small-scale structure of the velocity field is studied by numerically analysing the Fourier spectrum of u. It is shown that the Fourier spectrum of u falls off exponentially with wavenumber k for large k. The Taylor expansion in powers of the coordinate (say y) in the direction perpendicular to the vortical layer suggests that the solution may be well approximated by a function with certain poles in the complex y-plane. The Fourier spectrum based on the singularities is in good agreement with that obtained numerically, where the exponential decay rate is given by the distance of the poles from the real axis of y.

The vortical layer on an inclined cone

The problem of flow over a circular cone inclined slightly to a uniform stream is solved using the technique of matched asymptotic expansions. The outer expansion is equivalent to Stone's solution of the problem. The inner expansion, valid in a thin layer near the body, represents Ferri's vortical layer. The solution to first order in angle of attack so obtained is uniformly valid everywhere in the flow field. In the second-order expansion an additional non-uniformity appears near the leeward ray. This defect is removed by inspection. The first-order solution is in agreement with that of Cheng, Woods, Bulakh and Sapunkov. Formulas are given that may be used to render Kopal's numerical result uniformly valid to second order in angle of attack.

Supersonic flow past cones of general cross-section

The differential equations representing the supersonic flow of a gas past a cone of any cross-section are integrated numerically, using a method similar to those used for bluff-body problems. A stream function is used as one of the independent variables and this is particularly suitable for determining the singular ‘vortical layer’. The method is here applied to the cases of elliptic cones at zero yaw and circular cones at incidence. The results are compared with experiment and with other numerical solutions.