Boundary Layer Governing Equations

4.3.1 Thin Shear Layer approximations

As discussed in Section 1.5.4, at high Reynolds number the viscous layers are thin compared to their stream – wise length. This allows making the following Thin Shear Layer (TSL) approximations in the locally – cartesian s, n surface coordinates (see Figure 3.2).

Approximation (4.15) is a geometric consequence of the streamlines having only a small angle away from the wall and the s axis, as shown in Figure 3.2. Approximation (4.16) follows from the relatively rapid variation of the velocity across the layer. Together with (4.15) this allows dropping all but the du/dn term in the 2D version of the full viscous stress tensor (1.22), so that only the off-diagonal shear-stress terms Tsn = Tns = t are significant.

(4.18)

Approximation (4.17) follows from the streamlines being almost parallel within the layer. This was already used in Chapter 3 to give the wall pressure result (3.2), shown in Figure 3.1. Here this approximation allows replacing the n-momentum equation with the simple statement that the pressure across the boundary layer at any s location is constant, and equal to the inviscid flow’s edge pressure at that same s location.

P(s, n) ~ Pe(s) (4.19)

Consequently, the streamwise pressure gradient in the remaining s-momentum equation can be replaced by the edge velocity gradient using the inviscid streamwise momentum equation.