The boundary-layer equations

To fix ideas it is helpful to think about the flow over a flat plate. This is a particularly simple flow, although like much else in aerodynamics the more one studies the details the less simple it becomes. If we consider the case of infinite Reynolds number,

i. e. ignore viscous effects completely, the flow becomes exceedingly simple. The stream­lines are everywhere parallel to the flat plate and the velocity uniform and equal to Uqo, the value in the free stream infinitely far from the plate. There would, of course, be no drag, since the shear stress at the wall would be equivalently zero. (This is a special case of d’Alembert’s paradox that states that no force is generated by irrota – tional flow around any body irrespective of its shape.) Experiments on flat plates would confirm that the potential (i. e. inviscid) flow solution is indeed a good approximation at high Reynolds number. It would be found that the higher the Reynolds number, the closer the streamlines become to being everywhere parallel with the plate. Furthermore, the non-dimensional drag, or drag coefficient (see Section 1.4.5), becomes smaller and smaller the higher the Reynolds number becomes, indicating that the drag tends to zero as the Reynolds number tends to infinity.

But, even though the drag is very small at high Reynolds number, it is evidently important in apphcations of aerodynamics to estimate its value. So, how may we use this excellent infinite-Reynolds-number approximation, i. e. potential flow, to do this? Prandtl’s boundary-layer concept and theory shows us how this may be achieved. In essence, he assumed that the potential flow is a good approximation everywhere except in a thin boundary layer adjacent to the surface. Because the boundary layer is very thin it hardly affects the flow outside it. Accordingly, it may be assumed that the flow velocity at the edge of the boundary layer is given to a good approximation by the potential-flow solution for the flow velocity along the surface itself. For the flat plate, then, the velocity at the edge of the boundary layer is Ux. In the more general case of the flow over a streamlined body like the one depicted in Fig. 7.1, the velocity at the edge of boundary layer varies and is denoted by Ut. Prandtl went on to show, as explained below, how the Navier-Stokes equations may be simplified for application in this thin boundary layer.