Abstract
Recently, the authors considered a thin steady developed viscous free-surface flow passing the sharp trailing edge of a horizontally aligned flat plate under surface tension and the weak action of gravity, acting vertically, in the asymptotic slender-layer limit (J. Fluid Mech., vol. 850, 2018, pp. 924–953). We revisit the capillarity-driven short-scale viscous–inviscid interaction, on account of the inherent upstream influence, immediately downstream of the edge and scrutinise flow detachment on all smaller scales. We adhere to the assumption of a Froude number so large that choking at the plate edge is insignificant but envisage the variation of the relevant Weber number of$O(1)$. The main focus, tackled essentially analytically, is the continuation of the structure of the flow towards scales much smaller than the interactive ones and where it no longer can be treated as slender. As a remarkable phenomenon, this analysis predicts harmonic capillary ripples of Rayleigh type, prevalent on the free surface upstream of the trailing edge. They exhibit an increase of both the wavelength and amplitude as the characteristic Weber number decreases. Finally, the theory clarifies the actual detachment process, within a rational description of flow separation. At this stage, the wetting properties of the fluid and the microscopically wedge-shaped edge, viewed as infinitely thin on the larger scales, come into play. As this geometry typically models the exit of a spout, the predicted wetting of the wedge is related to what in the literature is referred to as the teapot effect.
Highlights
We continue to analyse a flow problem of fundamental importance as started in our forerunner study (Scheichl, Bowles & Pasias (2018), hereafter referenced as SBP18).Let a nominally steady and two-dimensional, developed, slender stream of a Newtonian liquid having uniform properties and at constant flow rate in an inertial frame of reference detach from a horizontal, solid, impenetrable, perfectly smooth plate with a trailing edge that is initially considered as abrupt and sharp
As an unexpected extension of the interactive flow structure around flow detachment at the free plate edge, we report neutral capillary Rayleigh modes on the upper free surface solely and immediately upstream of the edge
This is provided by a delta functional describing the transition of the vertical flow component over the geometric discontinuity formed by the trailing edge but tied in with an additional non-compact excitation by the displacement of the HRW, necessary to provoke the non-trivial Rayleigh state
Summary
We continue to analyse a flow problem of fundamental importance as started in our forerunner study (Scheichl, Bowles & Pasias (2018), hereafter referenced as SBP18). Let a nominally steady and two-dimensional, developed, slender stream of a Newtonian liquid having uniform properties and at constant flow rate in an inertial frame of reference detach from a horizontal, solid, impenetrable, perfectly smooth plate with a trailing edge that is initially considered as abrupt and sharp. Downstream, the resulting fluid jet divides its gaseous environment, fully at rest and under constant pressures, into two parts. This picture is relaxed insofar as the upper one still defines the zero pressure level but we allow for a non-zero, constant support pressure prescribed at the downside of the detached layer. We refer to the sketch in figure 1 throughout, illustrating the different flow regions considered when viewed on the global vertical scale defined by the height of the detaching layer. We critically review the prevailing, rather phenomenological view on this effect and its previous modelling
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