Turbulent wake solutions of the Prandtl alpha equations

A derivation of the Navier-Stokes alpha equations for spatially dependent alpha is presented. It is shown that an extra term in the equation is necessary to ensure the conservation of momentum. The Prandtl form of these variable alpha equations are determined for both planar and axisymmetric pressure-gradient-driven boundary-layer flows correcting previous work on the subject. The Prandtl equations are then solved analytically for four different asymptotic wake flows: the classical planar wake, the classical axisymmetric wake, the planar dragless wake, and the axisymmetric dragless wake. Least-squares fits of the theoretical solutions with available turbulent mean-flow velocity data for classical planar and axisymmetric wakes are given. We point out that the dissipation coefficient does not have to be equal to the kinematic viscosity, and its numerical value may be estimated from experimental data.