The classical unsteady boundary layer: A numerical study

Abstract

The transition process from a laminar to turbulent flow near a solid wall (e.g., at the suction side of a slender airfoil) is still not fully understood. We focus on the early stage of the flow transition based on boundary layer (BL) theory, that is, high Reynolds number asymptotic expansions, which reduce the Navier–Stokes equations into simplified forms. While steady solutions of the resulting classical BL equations and corresponding regularisations based on viscous–inviscid interaction theory are studied quite extensively, there is only few information about the fully time-dependent scenario. Hence, we numerically study the classical unsteady BL and its breakdown structure, the Van Dommelen–Shen singularity, on the simplest example we can imagine, the incompressible planar fluid flow through a rectangular channel with suction. The depicted BL builds at the lower channel wall and follows initially a generic Blasius behaviour, which gets temporally modulated through suction by means of a slot located at the upper wall. As usual, the main part of the flow is inviscid and governed by Laplace's equation. Remarkably, the wall-slip velocity, or alternatively the pressure gradient imposed on the BL, can be calculated in closed form and we have full control of its temporal dependency through the variation of the applied suction strength. This specific flow configuration allows us to split off the initial Blasius solution and promises high-accuracy solutions of the remaining unsteady BL equation in the scaled stream function formulation. The Chebyshev collocation is used in the wall-normal direction, where the infinite domain is mapped onto [ − 1, 1]. On the contrary, we discretise the mainstream direction with a finite difference scheme of second-order accuracy, which proved to be more precise near the build-up of singularities. In order to trigger a dynamic breakdown of the unsteady BL equations, we increase the suction rate continuously from zero to a level slightly above the critical value (obtained from the steady case). As a consequence, the system cannot relax to a steady solution and instead blows up in finite time at some mainstream position. We try to resolve this process with high resolution and the final goal is to verify the inner structure (the two-dimensional blow-up profile) of the Van Dommelen–Shen singularity and its role in flow separation and transition.