             Radial wall jet flows across flat smooth surfaces have been studied for decades.  These studies show that the radial velocity of these jets decays inversely with distance from the nozzle with modest contribution from friction \[Poreh, et al., 1967; Rajaratnam, 1976\].  However, the extent to which flat surface results apply to curved surfaces remains unclear.  For example, many mixing vessels have curved bottoms, and slurried solids may settle to form particle beds with varying curvature.  Therefore, exploring the influence of surface curvature on attached radial wall jets is an important technological advance for a wide variety of industries. 

            Experimentally, flows emerging from circular nozzles near surfaces form radial wall jets.  When settled particle beds are present on these solid surfaces (e.g., concrete), the jet flow shears across the thin settled beds removing particles out to radial distances called effective clearing or cleaning radii (ECR) or alternatively zones of influence (ZOI).  Immediately prior to this distance, the settled solid bed such as a layer of sand is effectively absent.  Beyond this distance, the settled sand bed can be of approximately uniform thickness.  Between these two regions the sand profile is continuous.  Pease, et al., (2018) recently evaluated approaches for estimates of ECR and ZOI  using velocity profiles from flat radial wall jet expressions and analytical models of the Shields diagram and found quantitative failure of these models for both flat and curved surfaces.  The evaluation did show better agreement for thinner particle beds than thicker particle beds, suggesting the possibility that the slope of the particle bed at the edge of the radial extent of erosion may play a role.  However, quantitative expressions for jet flows across settled particle beds of varying height remain unavailable leaving important questions unanswered: Does the steepness of the particle beds affect jet flows?  Does the steepness of the particle beds affect the celerity of the jet front?  Can the jet flow be stopped by sufficiently steep surfaces?

            In this article we explore the influence of settled particle bed slope on these velocity profiles.  We model the step change in thickness as a sigmoidal curve of variable steepness and use conservation of momentum to evaluate the velocity profile.  We show that the velocity profile attenuates, not because of a normal force, but because of curvature.  For steady beds, the celerity of the jet front is also presented. 

               Next, we evaluate the influence of surface curvature on radial wall jet flow.  Jet flows over particle beds often introduce curvature in the particle bed profile but the influence of the developed curvature on the velocity profile has not been explored.  Here we develop a solution for steady fixed beds based on conservation of momentum.  We find that surface curvature has a significant influence on the velocity decay coefficients and celerity of radial wall jets, provided there is velocity slip in the vicinity of the particle bed interface, which is strictly true for particle surfaces.  Conservation of momentum predicts conditions where the forward momentum of the flow is directed completely upward.  The solution identifies two new dimensionless groups that determine whether a curved surface is sufficient to block radial flow and force flow vertically.


Radial Wall Jet Flow Over Sigmoidal Surfaces

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Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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