The effect of three-dimensional obstacles on marginally separated laminar boundary layer flows

Abstract

We consider the steady viscous/inviscid interaction of a two-dimensional, nearly separated, boundary layer with an isolated three-dimensional surface-mounted obstacle, for example in the large Reynolds number flow around the leading edge of a slender airfoil at a small angle of attack. An integro-differential equation describing the effect of the obstacle on the wall shear stress valid within the interaction regime is derived and solved numerically by means of a spectral method, which is outlined in detail. Typical solutions of this equation are presented for different values of the spanwise width B of the obstacle including the limiting cases B → 0 and B → ∞. Special emphasis is placed on the occurrence of non-uniqueness. On the main (upper) solution branch the disturbances to the flow field caused by the obstacle decay in the lateral direction. Conversely a periodic flow pattern, having no decay in the spanwise direction, was found to form on the lower solution branch. These branches are connected by a bifurcation point, which characterizes the maximum (critical) angle of attack for which a solution of the strictly plane interaction problem exists. An asymptotic investigation of the interaction equation, in the absence of any obstacle, for small deviations of this critical angle clearly reflects the observed behaviour of the numerical results corresponding to the different branches. As a result we can conclude that the primarily local interaction process breaks down in a non-local manner even in the limit of vanishing (three-dimensional local) disturbances of the flow field.