Characterization of entrainment through turbulent diffusion in a self-similar jet

A large-scale experimental Lagrangian study based on particle tracking velocimetry has been completed in an incompressible self-similar water jet. The jet is seeded with tracers only through the nozzle: inhomogeneous seeding called nozzle seeding. The Lagrangian flow of tracers therefore does not contain any contribution from particles entrained into the jet from the surrounding fluid. The mean velocity field based on tracer trajectories〈U_φ〉is essentially indistinguishable from the mean velocity field of the jet〈U〉for the axial velocity while important discrepancies are experimentally found for the radial velocity. Even if ▽ ·〈U〉= 0, we have ▽ ·〈U_φ〉≠ 0 due to the absence of entrained particles. By considering the mean tracer concentration field〈φ〉, a new mass conservation equation is proposed: ▽ · (〈φ〉〈U_φ〉) = 0, which leads to a correct description of the experimental radial velocity field and gives new analytical results about entrainment. Finally, to connect entrainment to turbulent diffusion, a classical advection-diffusion equation with a turbulent diffusivity K_T is proposed: ▽ · (〈φ〉〈U〉- K_T ▽〈φ〉) = 0. We theoretically and experimentally determine K_T, which can be linked to turbulent viscosity ν_T through turbulent Prandtl number σ_T determined as well.