Choking and hydraulic jumps in laminar flow

Abstract

<p> The (steady) viscous hydraulic jump still represents research in progress rather than a finalised edifice. The rigorous approaches of the last decades showed how this phenomenon is intrinsically associated with a bifurcation of the upstream flow adjacent to the guiding rigid plate, aligned perpendicularly to the direction of gravity. This initiates a process of viscous-inviscid interaction and reflects the likewise essential upstream influence, originating in suitable downstream conditions that trigger the transition from super- to subcritical flow (these notations have a well-defined meaning). However, the challenge of a self-consistent formulation involving the latter mechanism has only been mastered conclusively for relatively weak jumps, connecting states slightly detuned from choking conditions over a correspondingly short streamwise length in boundary-layer flows. The smooth jump in developed flow, however, is terminated by locally choked flow, as predicted by a marching singularity in the solution to the underlying shallow-water problem. Its localisation is associated with the trailing edge of the plate, but how this flow at critical conditions passes the latter and is finally transformed into a downfall is a topic under consideration yet. </p>

<p> We present recent advances in the establishment of a closed asymptotic theory for a developed jump. Currently, the global Froude number expressed in terms of the slenderness parameter of the flow is taken as so large that deviations from the parabolic shallow-water limit are predominantly due to streamline curvature. Then the flow only "chokes weakly", i.e. in its near-plate portion. A novel canonical interaction problem provides the regularisation of the accordingly weak form of the trailing-edge singularity on a streamwise length scale much smaller than the global one describing the full jump and a context to the aforementioned transcritical boundary-layer flow. As a side aspect, this scenario completes a long-standing debate on an analogous singularity and its smoothing occurring in interactive hypersonic boundary layers. We then demonstrate how that interactive flow regime gives way to a further one encompassing the edge on a reduced scale and accounting for the downfall process. </p>

<p> This analysis captures the upstream influence and how it controls the bifurcation process in a correct manner. Furthermore, it paves the way for a complete rational description of the hydraulic jump for moderately large Froude numbers on a global length scale, measuring the distance from the virtual origin of the supercritical flow to the plate edge. Differences to the axisymmetric jump over a spinning disc are addressed in brief. </p>