Integral solution for the mean flow profiles of turbulent jets, plumes, and wakes

Integral methods are used to derive similarity solutions for several quantities of interest including the cross-stream velocity Reynolds stress, the dominant turbulent kinetic energy production term, and eddy diffusivities of momentum and heat for axisymmetric and planar turbulent jets, plumes, and wakes. A universal constant is evaluated for axisymmetric and planar plumes. The cross-stream velocity profiles show that jets and axisymmetric plumes experience an outflow near the axis and an inflow far away from it. The outflow is attributed to the decay of the centerline velocity with downstream distance, and the extent and magnitude of outflow correlates with the streamwise decay of the centerline velocity. It is also shown that the entrainment velocity should not in general be equated to the product of the entrainment coefficient and the centerline velocity. It is found that, due to similar governing equations, profiles for jets and plumes are qualitatively similar. Our results show that the derived quantities are strong functions of streamwise and cross-stream positions, in contrast to previous approaches that assumed constant (in the cross-stream direction) eddy viscosity and thermal diffusivity. The turbulent Prandtl number is approximately equal to unity which matches the value quoted in the literature.