Injection into boundary layers: solutions beyond the classical form

Abstract

This theoretical and numerical study presents three-dimensional boundary-layer solutions for laminar incompressible flow adjacent to a semi-infinite flat plate, subject to a uniform free-stream speed and injection through the plate surface. The novelty in this case arises from a fully three-dimensional formulation, which also allows for slot injection over a spanwise lengthscale comparable to the boundary-layer thickness. This approach retains viscous effects in both the spanwise and transverse directions, and effectively results in a parabolised Navier-Stokes system (sometimes referred to as the 'boundaryregion equations'). Any injection profile can be described in this approach, but we restrict attention to three-dimensional states driven by a finite-width slot aligned with the flow direction and self-similar in their downstream development. The classical two-dimensional states are known to only exist up to a critical ('blow off') injection amplitude, but the three-dimensional solutions here appear possible for any injection velocity. These new states take the form of low-speed streamwise-aligned streaks whose geometry depends on the amplitude of injection and the spanwise width of the injection slot; intriguingly, although very low wall shear is typically obtained, streamwise flow reversal is not observed, however hard the blowing. Asymptotic descriptions are provided in the limit of increasing slot width and fixed injection velocity, which allows for classification of the solutions according to two bounding injection rates.

Micro-slot injection into a boundary layer driven by a favourable pressure gradient

“…The injection through the plate surface is concentrated in the region |ζ | < ζ 0 as defined by (8) and indicated by the arrows, whilst ζ = 0 is a symmetry line. For β = 0 (a, c, e) three distinct flow regimes are observed as described in [14]. However results for β = 0.1 (b, d, f) show a qualitatively different response (e) K > K I I , a much more prominent low-speed streak is observed which is separated from the outer flow by a semi-circular free shear layer.…”

“…The previous work of [14] showed that for K > 0 (injection) and no applied pressure gradient (β = 0), three distinct flow regimes are observed, as the injection-slot width ζ 0 is increased. These three regimes are delineated by the injection rates: 0 < K < K I ≈ 0.876, K I < K < K I I ≈ 1.95 and K > K I I .…”

“…Our computational scheme extends that of Hewitt et al [14,19], to accommodate the non-zero pressure gradient in the free stream. We employ a non-uniform computational mesh that concentrates nodes in the slot region and at the edge of the slot.…”

“…The recent work of Hewitt et al [14] extended the classical two-dimensional formulation, in the absence of a pressure gradient (i.e. n = 0 such that U e = 1), to a three-dimensional formulation by allowing injection over a spanwise length scale comparable to the boundary-layer thickness.…”

“…Generalising the three-dimensional boundarylayer injection solutions of [14] to include an applied pressure gradient is important not only because non-uniform external flows are commonplace, but also because we know that for macro-scale injection, the flow character is significantly affected by the pressure gradient.…”