18 - Viscous effects of the irrotational flow outside boundary layers on rigid solids

Summary

High-Reynolds-number flows may be approximated by an outer irrotational flow and small layers on the boundary and narrow wakes where vorticity is important. The irrotational flow gives rise to an extra viscous dissipation over and above the dissipation in the boundary layer. At high Reynolds numbers the viscous dissipation in the irrotational flow outside is a very small fraction of the total that vanishes asymptotically as the Reynolds number tends to infinity.

Prandtl's boundary-layer theory is asymptotic and does not account for the viscous effects of the outer irrotational flow. Viscous effects on the normal stresses at the boundary of a solid cannot be obtained from Prandtl's theory. It is very well known and easily demonstrated that, as a consequence of the continuity equation, the viscous normal stress must vanish on a rigid solid. The only way that viscous effects can act on a boundary is through the pressure, but the pressure in Prandtl's theory is not viscous. It is determined by Bernoulli's equation in the irrotational flow and is imposed unchanged on the wall through the thin boundary layer. Therefore the important pressure drag cannot be calculated from Prandtl's theory. In addition, the mismatch between the irrotational shear stress and the shear stress at the outer edge of the boundary layer given by Prandtl's theory is not resolved.