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

We investigate the effects of injection through a streamwise-aligned ‘micro-slot’ into a laminar boundary layer driven by a favourable pressure gradient of power-law type. The injection slot exists at all downstream locations, and is ‘micro’ in the sense that it has a finite spanwise width that is a fixed ratio of the local boundary-layer thickness. This approach is motivated by recent studies of micro-jets (of small spanwise and streamwise extents), which have indicated that for short spanwise scales, injection does not necessarily lead directly to separation. Injection in the absence of a free-stream pressure gradient has recently been analysed by Hewitt et al. (J Fluid Mech 822:617–639, 2017), and here we show that boundary layers in a favourable pressure gradient behave qualitatively differently. We present three-dimensional boundary-layer solutions affected by slot injection and contrast these with the corresponding zero pressure-gradient states. In the absence of a pressure gradient, injection results in low-speed streamwise-aligned streaks, where the amplitude and spanwise width of the injection determine the geometry of the streaks as one of the three possible types. The introduction of a favourable pressure gradient greatly reduces the spanwise extent of injection-driven streaks and removes the delineation between the three distinct flow regimes found in the zero pressure-gradient case. We present an asymptotic description in the limit of a large injection-slot width, thereby approaching the macro-slot limit from the micro-slot formulation. This description shows that not all injection rates and pressure gradients recover the expected Falkner–Skan solution at the centreline of the injection slot in the macro-slot limit. We explain this disagreement in terms of local spatial (cross flow) eigenmodes that are associated with a cross-flow collisional process at the centre of the injection slot.