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