Self-similarity of decelerating round jets

Self-similarity of a jet is a classic problem in fluid mechanics with a rich history. The axisymmetric Prandtl boundary layer equations for a laminar jet can be transformed to a self-similar coordinate for certain power-law forms of the jet's centerline velocity. However, as Squire hand pointed out for the case of spherical coordinates, the same self-similar transformation can be applied to the "full" Navier-Stokes momentum equation without the boundary-layer approximation. We revisit this observation for the cylindrical (round) jet, retaining the axial momentum diffusion, and incorporating Reynolds stresses in the turbulent regime using the Reynolds-averaged Navier-Stokes equations. We discover that the unsteady case does not allow for the same self-similar transformation as the steady case, and highlight the rebalancing of scales that must occur to obtain a self-similar transformation for an incompressible decelerating jet, keeping the axial shear stress components of the viscous and turbulent (Reynolds) stresses using Görtler's eddy viscosity model. Our formulation recovers the steady laminar round jet results under the boundary layer approximation, due to Schlichting, and provides updates to some transformations used in recent direct numerical simulation studies on turbulent round jets.