Equilibrium similarity, effects of initial conditions and local Reynolds number on the axisymmetric wake

Abstract

Equilibrium similarity considerations are applied to the axisymmetric turbulent wake, without the arbitrary assumptions of earlier theoretical studies. Two solutions for the turbulent flow are found: one for infinite local Reynolds number which grows spatially as x1/3; and another for small local Reynolds number which grows as x1/2. Both solutions can be dependent on the upstream conditions. Also, the local Reynolds number diminishes with increasing downstream distance, so that even when the initial Reynolds number is large, the flow evolves downstream from one state to the other. Most of the available experimental data are at too low an initial Reynolds number and/or are measured too near the wake generator to provide evidence for the x1/3 solution. New results, however, from a laboratory experiment on a disk wake and direct numerical simulations (DNS) are in excellent agreement with this solution, once the flow has had large enough downstream distance to evolve. Beyond this the ratio of turbulence intensity to centerline velocity deficit is constant, until the flow unlocks itself from this behavior when the local Reynolds number goes below about 500 and the viscous terms become important. When this happens the turbulence intensity ratio falls slowly until the x1/2 region is reached. No experimental data are available far enough downstream to provide unambiguous evidence for the x1/2 solution. The prediction that the flow should evolve into such a state, however, is confirmed by recent DNS results which reach the x1/2 solution at about 200 000 momentum thicknesses downstream. After this the turbulence intensity ratio is again constant, until box-size affects the calculation and the energy decays exponentially.