Upstream influence in boundary layers 45 years ago

Abstract

My two–part paper ‘Boundary layers and upstream influence’, published in 1953, surveyed a wide range of experimental evidence on how a disturbance in supersonic flow, which in inviscid theory would affect only downstream conditions, is able to exercise an upstream influence through the agency of a boundary layer, either laminar or turbulent. Then, by systematically comparing the data with existing attempts to account for the phenomenon theoretically, it concluded that, essentially, two mechanisms of upstream influence exist. Mechanism (i), first suggested by Oswatitsch & Wieghardt in 1941, depended on a particular property of supersonic flow over a wall: that either wall curvature on inviscid theory, or (for a flat wall) curvature d 2 δ 1 /d x 2 of the displacement–thickness contour on boundary–layer theory, generates a proportional pressure gradient; which, in the latter case, is A 2 d 2 δ 1 /d x 2 being a known positive function of Mach number. Also, this positive pressure gradient might be expected to thicken the layer at a spatial rate d δ 1 /d x = A 1 ( A 2 d 2 δ 1 /d x 2 ), where A 1 , although far from precisely known, must be less for a turbulent than for a laminar layer; so that, finally, the e–folding distance of upstream influence would be A 1 A 2 . Mechanism (i) was compared, in part II of my paper, with a different proposal (see the work of Howarth in 1948) for a theoretical programme concerned with ‘propagation up the subsonic layer’, in which only the undisturbed boundary–layer distribution (including its subsonic part) would be taken, as influenced by viscosity, while disturbances to it would be treated inviscidly. The reason why attempts to carry out this programme had failed was explained in terms of earlier theories of boundary–layer instability, in which time–dependent disturbances had been found to be influenced by viscosity in two layers: a wall layer and a critical layer. For disturbances independent of time these would coincide into a single wall layer in which, however, the influence of viscosity still needed to be taken into account; in which case, the analysis could be satisfactorily completed but became in essence merely an expression of mechanism (i) with a relatively precise determination of A 1 . Mechanism (ii), identified in work by Lees in 1949 at Princeton and by Liepmann, Roshko & Dhawan in 1949 at Caltech, depended on the upstream spreading of a separation bubble till it became sufficiently slender to cause no further separation ahead of it. Part I of my paper was concerned to point out that, although mechanism (i) can work only when a well–defined coefficient A 2 exists (that is, for supersonic flow), mechanism (ii) is effective in both subsonic and supersonic flow. This was illustrated by analysing data on flow up a step at various Mach numbers (with various locations for transition to turbulence) in terms of boundary–layer separation studies. Those instructive examples, which may today be somewhat less known, and which included several interesting cases of both steady and also unsteady separated flows, can appropriately be recalled in a colloquium devoted to such phenomena.