Interaction between a dipole moving downstream and the boundary layer on a plate

Abstract The paper considers a two-dimensional laminar flow of a viscous incompressible fluid past a flat plate at high Reynolds numbers. In the framework of the asymptotic theory of viscous-inviscid interaction, the effect of a body moving downstream at a low velocity relative to the plate on the Blasius boundary layer is studied. A special case is investigated, where an external small body, modelled by a potential dipole, moves downstream at a constant speed. This classical problem is formally unsteady in the plate's reference frame; however, as a result of the transition to a moving coordinate system associated with the dipole, it is described by steady solutions of the interaction theory but on a wall moving upstream. The paper proposes a technique to solve this problem containing counterflows, i.e., a layer of fluid flowing upstream is near the surface, while above it, the fluid in the boundary layer flows downstream. It was possible to find an exact analytical solution to the linear problem in the near-wall viscous sublayer for small moment of the potential dipole. The solution contains closed and open separation regions near the line of zero streamwise velocity, even in the linear approximation. At wall velocities below a certain critical value, the disturbances in the viscous sublayer are concentrated mainly downstream of the dipole; at wall speeds above the critical value, the disturbances form an upstream ‘wake’ propagating in the near-wall upstream layer.