Sn. Timoshin et Ft. Smith, SINGULARITIES ENCOUNTERED IN 3-DIMENSIONAL BOUNDARY-LAYERS UNDER AN ADVERSE OR FAVORABLE PRESSURE-GRADIENT, Philosophical transactions-Royal Society of London. Physical sciences and engineering, 352(1698), 1995, pp. 45-87
Singularities in solutions of the classical boundary-layer equations a
re considered, numerically and analytically, in an example of steady h
ypersonic flow along a flat plate with three-dimensional surface rough
ness. First, a wide parametric study of the breakdown of symmetry-plan
e flow is performed for two particular cases of the surface geometry.
Emphasis is put on the structural stability of the singularities' deve
lopment to local/global variation of the pressure distribution. It is
found that, as usual, the solution behaviour under an adverse pressure
gradient involves the Goldstein- or marginal-type singularity at a po
int of zero streamwise skin friction. As the main alternative, typical
of configurations with favourable or zero pressure forcing, an invisc
id breakdown in the middle of the flow is identified. Similarly to uns
teady flows, the main features of the novel singularity include infini
tely growing boundary-layer thickness and finite limiting values of th
e skin-friction components. Subsequent analytical extensions of the si
ngular symmetry-plane solution then suggest two different scenarios fo
r the global boundary-layer behaviour: one implies inviscid breakdown
of the flow at some singular line, the other describes the development
of a boundary-layer collision at a downstream portion of the symmetry
plane. In contrast with previous studies of the collision phenomenon
in steady flows, the present theory suggests logarithmic growth of bou
ndary-layer thickness on both sides of the discontinuity. Finally, an
example of numerical solution of the full three-dimensional boundary l
ayer equations is given. The flow regime chosen corresponds to invisci
d breakdown of a centreplane flow under a favourable pressure gradient
and development of the discontinuity/collision downstream. The numeri
cal results near the origin of the discontinuity are found to be suppo
rtive, producing quantitative agreement with the local analytical desc
ription.