On the role of the hidden integral of motion in the theory of non-self-similar submerged jets of a viscous incompressible fluid

The role of the hidden integral of motion for correct description of the far field of velocities and pressures is discussed based on the complete Navier-Stokes equations for the case of non-self-similar submerged jets of an incompressible viscous fluid when the source of motion has a non-zero characteristic size. It is shown that the emergence of the hidden conservation integral is due to the fact that the coordinates of the point of the effective momentum source and the point of the effective mass source may not coincide for real spatially extended sources of jet flow. Using special functions, an exact analytical solution for all terms of the asymptotic expansion of the far field of a non-self-similar submerged jet was obtained that is described by all integrals of motion: conservation of the total momentum flux, conservation of the total angular momentum flux, conservation of the total mass flux, and the additional hidden conservation integral associated with the conservation of the total angular momentum flux. It is shown that the hidden integral was actually first obtained by L. G. Loitsyansky in studying a non-self-similar solution for a submerged jets within the boundary layer approximation, but it was mistakenly interpreted as the integral of conservation of the mass flux flowing from source of the jet. The obtained exact solution was used to perform calculations of velocity and pressure fields at different Reynolds numbers and different values of the hidden integral for the model of the jet outflow from a circular tube of finite size. The influence of the hidden integral of motion on the flow pattern is analyzed.