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

Abstract

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.

Buoyancy-driven algebraic (localised) boundary-layer disturbances

“…We seek spatially developing steady eigenmodes analogous to those found for the isothermal Blasius (n = 0) boundary layer by Hewitt and Duck [12] but extended to include buoyancy coupling with the energy equation and an external pressure gradient (n > 0, as formulated in [17]). These exist in the form…”

“…The eigenfunctions of Fig. 1 are symmetric about ζ = 0 (for the temperature field), hence in the limit k 1 we introduce the small wavenumber expansion Table 3 The first five greatest eigenvalues Λ = Λ i , with i = 1, 2, ..., 5, as determined from the one-dimensional eigenvalue problem (17) with the substitution (18) for a Blasius base flow (β = 0) with Prandtl number Pr = 0.71…”

“…For the one-dimensional eigenvalue equation (17), the explicit dependence of the integer m can be removed by introducing a new eigenvalue Λ, where…”

“…We now continue with the formulation (17) to determine the dependence of the spatial growth/decay exponent λ on the pressure gradient β and Prandtl number Pr. Fig.…”

Instability of isolated boundary-layer streaks to spatially-developing travelling waves

“…This general type of 'boundary-region' formulation has been previously employed in (for example) discussions of isolated streaks [16,19,20], spanwise-periodic streaks/vortices [21,22], their transient growth [23,24,25] and their self-sustained interaction with travelling waves [26].…”