The role of symmetries in the axisymmetric jet mean velocity profile development

Abstract

The fact that physical conservation laws can be derived from symmetry properties of space and time, as shown by Emily N\"other, has been utilized in predicting the development of the round turbulent jet from the jet exit to the far field. In particular, the developing region has been described using an analytical approach in combination with using a numerical recursive program. Both approaches assume that the only forces acting on the flow are internal shear forces in a Newtonian constant density fluid. The analytical and numerical predictions both display excellent agreement with carefully conducted measurements. The jet spreading angle is observed to be directly coupled to the turbulent momentum diffusion, hence the spreading rate depends on the upstream, or initial, conditions. The jet entrainment and momentum rate are both observed to be constant, even across the developing jet. Since the solution of the jet development depends on the initial conditions, the total (molecular and turbulent) viscosity and the initial velocity profile must be input into the analytical or numerical solver to obtain the correct solution. The Reynolds number is observed to not enter into the analytical or numerical solution and experiments confirm that the jet spreading is independent of the Reynolds number in the tested range, $Re = 3\,200 - 32\,000$. We emphasize that our results do not rely on any assumptions of self-similarity or prior knowledge about the jet from experiments, only the Galilei symmetry properties, and that the results are valid throughout the jet.