Self-induced suction of fluid in a turbulent boundary layer on a permeable surface

A turbulent boundary layer of a viscous incompressible fluid developing past 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, while the thickness of the boundary layer is small. The problem is analyzed using the asymptotic method of multiple scales, which is used to find solutions of the Navier–Stokes equations. The velocity and pressure in the turbulent boundary layer are represented in the form of a sum of steady and perturbed terms, instead of traditional sums of time-averaged quantities and their fluctuations. It turns out that such a steady flow (called secondary) inside the turbulent boundary layer is determined by classical ideas and results of Reynolds and Kolmogorov without using any closure hypothesis. In terms of physics, this steady solution represents self-induced suction of the fluid from the outer flow to the turbulent boundary layer, which ensures kinetic energy transfer from the maximum-velocity zone to the basic part of the boundary layer. The found solution explains the concept of turbulent viscosity, since the solution is applicable on the scale of the boundary layer thickness. In the basic approximation, the zones of generation and dissipation of Kolmogorov vortices do not affect this steady solution. The features of solutions in the cases when the fluid flows into and out of the turbulent boundary layer through a permeable surface are studied. The obtained solutions are compared with available experimental data.