On transcritical states in viscous flow passing the edge of a horizontal plate

Abstract

<p> As proposed by Higuera (<i>J. Fluid Mech.</i>, vol. 274, 1994), the shallow-water problem describing a steady planar developed liquid layer under the action of gravity and capillarity and driven by jet impingement over a horizontal plate is to be closed by a specific singular behaviour prescribed at its trailing edge. This singularity can be interpreted as the classical Burns-Lighthill criterion for vanishing phase speed being met locally. It therefore expresses criticality of developed flow in the usual sense of hampering long waves from making their way upstream. However, this hypothesis must be viewed as unsatisfactory and thus the description of this classical problem as incomplete: it admittedly anticipates the presence of short-sale perturbations anticipating the strong vertical acceleration (by gravity) which the layer undergoes when passing the edge but not their genuine form. Hence, neither is its existence conclusive nor can an alternative expansive singularity, well-known from hypersonic interactive boundary layers, be excluded as long if the flow is scrutinised solely in the long-wave limit. Moreover, it is inevitably associated with a smoothed hydraulic jump and pronounced flow reversal upstream of the edge. It thus does not allow for a gradual transition towards globally supercritical flow howsoever large the jet Froude number is. On the other hand, taking the Reynolds number as arbitrarily large suggests a much more involved asymptotic splitting of the layer without the need to impose that singularity or, equivalently, criticality at the edge. Our completion of the theory puts forward the correct structure of the flow close to the edge which not only renders the shallow-water problem weakly elliptic but also allows the layer to pass the edge in a self-consistent manner. We elucidate the circumstances implying Higuera´s singularity, how criticality vanishes in the above limits, and how the edge topography affects the local viscous-inviscid interaction. </p>