{'fraction_non_alphanumeric': 0.01896551724137931, 'fraction_numerical': 0.0, 'mean_word_length': 4.822742474916388, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Hydraulic jump on the surface of a cone

Abstract This paper addresses several aspects of the axisymmetric flow of a liquid film over the surface of a downward-sloping cone. The study is rooted on a validated computational tool the results of which are interpreted with the help of a hyperbolic time-dependent reduced-order model also derived in the paper. The steady version of the model demonstrates the weakening and ultimate disappearance of the circular hydraulic jump as the cone surface transitions from planar to downward sloping. Mathematically, this evolution is reflected in a change of the model's critical point from spiral to node. A significant advantage of the time-dependent model is that, when it is integrated in time, the flow regions upstream and downstream of the critical point are connected. Due to this feature, when a hydraulic jump exists, its position can be sharply captured automatically with a good agreement with Navier–Stokes simulations. Surface-tension effects are properly accounted for and, in steady conditions, are shown to have a marginal effect on the flow, including the position of the hydraulic jump. A correlation is obtained for the jump radius as a function of the flow rate, liquid viscosity, gravitational acceleration and the angle of inclination of the cone surface. In a suitable limit, the model reduces to the optimal two-dimensional, first-order model for liquid film flow down an inclined plane and, in a different limit, it describes an axisymmetric thin liquid film falling down the surface of a vertical cylinder. Some results are also presented for the waves induced by a pulsating jet on the surface of the liquid film and for a jet impinging on the surface of a cone from below.