Effects of upstream perturbations on the solution of the laminar and fully turbulent boundary layer equations with pressure gradients

Abstract

The aim of this work is to contribute to the understanding of sensitivity of boundary layers to the upstream boundary condition and history effects for both laminar and fully turbulent states in equilibrium conditions as well as some nonequilibrium turbulent boundary layers. Solutions of the two-dimensional boundary layer equations are obtained numerically for this study together with the Reynolds-averaged Navier-Stokes approach for turbulence modeling. The external pressure gradient is imposed through an evolution of the external velocity of the form Ue∝(x−x0)m, and boundary layers are initialized from a profile giving a perturbed shape factor. It is found that laminar boundary layers require very long distances for convergence toward the nondisturbed profiles in terms of the initial boundary layer thickness (∼104δin) and that this distance is dependent on m. In turbulent boundary layers, much shorter distances, although still large (∼102δin), are observed and they are also dependent on m. The maximum adverse pressure gradient for which convergence to a reference solution is possible is also studied finding that there is no limit for attached laminar boundary layers, whereas turbulent boundary layers do not converge once they are out of equilibrium. The convergence distances in turbulent boundary layers are also studied in terms of the turnover length (δUe+) because it has been shown to be more appropriate to refer the convergence distance to this length rather than the boundary layer thickness. The values for convergence using this criterion are extended to pressure gradient boundary layers. Moreover, an equivalent criterion is proposed and studied for laminar boundary layers based on the viscous characteristic time.