Analytical solution for the submerged free jet

Laminar submerged free jet theory still falls short in the near-nozzle region and transition to Schlichting's self-similar jet. The author's recent solution, based on mass conservation, is found lacking beyond the near-nozzle jet-core region. Instead, it is here constrained to conserve momentum, resulting in a locally linearized convection-diffusion equation, valid over jet width and up to self-similarity, when compared to simulations. This new solution leads to profile-specific values of virtual-origin correction to Schlichting's solution. Additionally, extensive jet characteristics are examined: (1) curvature core, (2) radial inflection location, (3) radial velocity, (4) vorticity field, (5) issuing mass, and (6) jet width. All are well predicted, and new insights are gained for a variety of issuing profiles: from uniform, through a non-monotonous profile and up to fully developed. The issuing mass of all non-uniform profiles undergoes an initial contraction proportional to the profile's level of development. Interestingly, the submerged jet contracts identically to the free-surface jet in the very near-nozzle region, before significant influence of their differing boundary conditions. Moreover, unless the issuing profile contains a radial inflection point, the inflection always occurs in the entrained fluid, just beyond the bounds of the issuing mass. It also follows an initial contraction and only later a widening toward the self-similar trend. Despite this contraction, the entrained fluid causes monotonous total jet-widening, at a rate inversely proportional to the level of development. Finally, this new solution correctly captures additional jet features, such as the local radial velocity and decay of the primary vorticity.