Impinging Jet Flow and Hydraulic Jump of Newtonian and Viscoplastic Liquids
Abstract
The steady laminar incompressible flow of an axisymmetric impinging jet of either a Newtonian fluid or a viscoplastic fluid of the Heschel-Bulkley type and the hydraulic jump of either a circular or polygonal shape on a solid disk is analyzed. The polygonal jump is induced by azimuthal dependence edge conditions: a non-circular disk or a circular disk with a variable edge film thickness. The thin-film and Kármán–Pohlhausen approaches are utilized as theoretical tools.
To cross the jump smoothly, a composite mean-field thin-film approach is proposed. The stress singularity for a film freely draining at the disk edge is found to be equivalent to an infinite film slope. The flow in the supercritical region is insensitive to the gravity strength, but is greatly affected by the viscosity. The existence of the jump is not necessarily commensurate with the presence of recirculation.
The disk size is found to can affect the film thickness in the subcritical region, vortex size and jump length significantly. The jump is relatively steeper with a stronger recirculation zone for a higher obstacle. Scaling laws for the jump properties, such as the jump radius and length, and edge film thickness, are proposed. The surface scaling separating the regions of existence/non-existence of the recirculation is found through numerical results.
The non-circular jump originated from the disk non-circularity or periodic edge film thickness is found. The balance of mass and momentum is established in the radial and azimuthal directions. The geometry of a non-circular disk has little influence on the jump shape. A small azimuthal variation in the edge thickness for a circular disk leads to a significant loss of axial symmetry. An increase in the number of peaks and valleys appears as the disk radius decreases.
The viscoplastic jump is found to occur closer to impingement, with growing height, as the yield stress increases; the subcritical region becomes invaded by the pseudo-plug layer. The viscosity does not influence sensibly the jump location and height except for small yield stress; only the yielded layer is found to remain sensitive to the power-law rheology for any yield stress.