Attached two-dimensional coherent vortices in a turbulent boundary layer

The turbulent boundary layer (TBL) of a viscous incompressible fluid that develops past the surface of a flat plate at finite distances from the laminar–turbulent transition zone is studied. It is assumed that the characteristic Reynolds number of the flow is large, and that the boundary layer is thin. An asymptotic method of multiple scales is used to find solutions to the Navier–Stokes equations. The velocities and pressure in the TBL are presented as a sum of steady and perturbed terms instead of the traditional decomposition into time-averaged values and their fluctuations. This article describes the process of generation of “inviscid” two-dimensional coherent vortices at selected points on the plate surface. Such solutions relate to the well-known Kraichnan's theory of two-dimensional turbulence, although they are derived as a particular case from three-dimensional analysis. A countable spectrum of possible “elementary” eigensolutions in the zone of turbulence generation near the streamlined wall is described. The evolution of generated coherent vortices is calculated numerically against the background of a steady basic longitudinal velocity profile over the entire thickness of the TBL. It is found that longitudinal, time-averaged velocity perturbations have logarithmic behavior close to the wall. The coefficients of these logarithmic terms are calculated, which makes it possible to find the local coefficients of skin friction on the streamlined surface. A satisfactory comparison with classical experimental data is made.