Conditions at transition

It is usually assumed for boundary-layer calculations that the transition from lam­inar to turbulent flow within the boundary layer occurs instantaneously. This is obviously not exactly true, but observations of the transition process do indicate that the transition region (streamwise distance) is fairly small, so that as a first approximation the assumption is reasonably justified. An abrupt change in momentum thickness at the transition point would imply that d0/dx is infinite. The
simplified momentum integral equation (7.66) shows that this in turn implies that the local skin-friction coefficient Cf would be infinite. This is plainly unacceptable on physical grounds, so it follows that the momentum thickness will remain constant across the transition position. Thus

ви = 0r, (7.89)

where the suffices L and T refer to laminar and turbulent boundary layer flows respectively and t indicates that these are particular values at transition. Thus

ви (^Jq «(1 – tf)dj^ = 6r, Qf m(1 – M)dj^

The integration being performed in each case using the appropriate laminar or turbulent profile. The ratio of the turbulent to the laminar boundary-layer thick­nesses is then given directly by

(7.90)

Using the values of / previously evaluated for the cubic and seventh-root profiles (Eqns (ii), Sections 7.6.1 and 7.7.3):

This indicates that on a flat plate the boundary layer increases in thickness by about 40% at transition.

It is then assumed that the turbulent layer, downstream of transition, will grow as if it had started from zero thickness at some point ahead of transition and developed along the surface so that its thickness reached the value <5rt at the transition position.