Modeling of the turbulent Poiseuille–Couette flow in a flat channel by asymptotic methods

Developed turbulent flow of a viscous incompressible fluid in a channel of small width at high Reynolds numbers is considered. The instantaneous flow velocity is represented as the sum of a stationary component and small perturbations, which are generally different from the traditional averaged velocity and fluctuations. The study is restricted to the search for and consideration of stationary solution components. To analyze the problem, an asymptotic multiscale method is applied to the Navier–Stokes equations, rather than to the RANS equations. As a result, a steady flow in the channel is found and investigated without using any closure hypotheses. The basic phenomenon in the Poiseuille flow turns out to be a self-induced fluid flow from the channel center to the walls, which ensures that kinetic energy is transferred from the maximum-velocity zone to the turbulence generation zone near the walls, although the total averaged normal velocity is, of course, zero. The stationary solutions for the normal and streamwise velocities turn out to be viscous over the entire width of the channel, which confirms the well-known physical concept of large-scale “turbulent viscosity”.