Influence of viscous flow relaxation time on self-similarity in free-surface jet impingement

Self-similar behavior in laminar free-surface jets regarding the velocity profile, its evolution, and the influence on stagnation-region heat transfer is presented. Approximate theoretical analysis shows that there are two regions with different scaling: a free-jet region where viscous relaxation dominates and an impingement region where a potential solution applies. Using the dimensionless time scale found for the first region, H/(D·Re), self-similarity is obtained for the axial velocity and its development. Fully resolved numerical simulations are performed to identify the crossover between the regions. It is further shown that the velocity profile at the cross-over point directly dictates the heat transfer distribution in the stagnation zone. The conditions under which this distribution has a central or off-center peak are found to be related to specific velocity profiles. Numerical results are used to find approximate closed-form relations tying the jet emergence velocity to the heat transfer distribution. A new comprehensive correlation is subsequently formulated for the heat transfer incorporating this new time scale, covering the range of velocity profiles emerging from pipe-type (parabolic) to orifice-type nozzles (uniform). A well-known analytical Prandtl dependency for stagnation flows is employed, yielding strong agreement with the numerical results. Agreement is observed for a very wide range of liquid and laminar flow conditions (i.e., 0.07<Pr<1307,73<Re<2000, and 4<H/D<110).