Turbulent Poiseuille Flow in a Circular Pipe as Superposition of a Steady-State Solution and Perturbations

The turbulent flow of a viscous incompressible fluid in a round pipe driven by a pressure difference is investigated. It is assumed that the characteristic Reynolds number calculated using the maximum averaged-flow velocity and the pipe length is large, while the pipe radius is small compared to the pipe length. Solutions to the Navier–Stokes equations are sought by applying an asymptotic multiscale method in which the velocities and pressure are represented as series consisting of steady and perturbed terms, instead of using the traditional expansion of the solution into time-averaged quantities and their fluctuations. A viscous self-sustaining steady flow developing in the pipe against the background of rapid turbulent fluctuations is found. The connection of this solution with Prigogine’s theory of dissipative structures for open nonlinear parabolic systems is indicated. A solution for the steady-state radial velocity is found that describes self-induced fluid transport from the flow core to the solid/permeable wall. As a result, the solutions obtained for streamwise velocity differ significantly from laminar regimes. A qualitative comparison of the results with available experiments and direct numerical simulations is made.