8. Laminar Boundary Layer

In Sect. 3.​3 we have seen that when a viscous fluid flows past a submerged object at high Reynolds number, \(Re \gg 1\) then its convective momentum flux greatly exceeds its diffuse counterpart, i.e., inertial forces are much larger than viscous forces. This is true, however, provided that the fluid points that we are considering are not too close to the outer surface of the object, where the fluid velocity is null, due to the no-slip boundary condition. Accordingly, near the surface of the object, we define a small region of thickness δ, denoted boundary layer, such that, when the distance from the wall, y, is larger than δ, i.e. when \(y > \delta\) inertial forces prevail while when \(y < \delta\) viscous forces are dominant. Then, as we saw in Sect. 3.​3, the boundary layer thickness can be determined by imposing that at the edge of this region, when y ≈ δ, inertial forces balance viscous forces. At the end, we found that the boundary layer thickness decreases, proportionally to the size, L, of the object, as the inverse of the square root of the Reynolds number, i.e., \({\delta \mathord{\left/ {\vphantom {\delta L}} \right. \kern-0pt} L} \simeq Re^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}}.\) This relation reveals why boundary layers are so important, although they occupy only a small region of space: the drag of the object depends on the shear stress at the wall, τ _w , which is determined by the velocity profile at the wall, and so it is inversely proportional to the boundary layer thickness; therefore, as Re increase, δ decreases and τ _w increases. In this chapter, after analyzing the scaling of the problem in Sect. 8.1, in Sect. 8.2 we will study the classical Blasius self-similar solution of the flow past a flat plate, and then in Sect. 8.3 consider more general cases, leading to flow separation. Finally, in Sect. 8.4, we analyze the approximate von Karman–Pohlhausen integral method.