Boundary-layer flows with short-scale spanwise disturbances

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Abstract

This work considers the three-dimensional laminar boundary layer on a semi-infinite plate aligned with a uniform oncoming stream or driven by a favourable pressure gradient of power-law type. The three dimensionality is induced by a wall injection that exists over a finite spanwise scale that is comparable to the boundary-layer thickness. These short-scale disturbances require the inclusion of spanwise and transverse diffusion in the boundary layer. As the spanwise extent of the disturbances maintains a constant ratio with the boundary-layer thickness a self-similar formulation is used. This self-similar formulation is later extended to allow for downstream variation in the injection profile. In the absence of a pressure gradient, classical two-dimensional theory predicts separation when an injection into the boundary layer is of sufficient strength; injection over short spanwise scales leads to qualitatively different flow responses where no separation occurs. The flow exists in one of three distinct regimes depending upon the magnitude of the surface injection rate. These regimes take the form of low-speed streaks where the amplitude and spanwise width of the injection determines the geometry of the streaks. An asymptotic description in the limit of a large injection slot width and a fixed injection velocity is presented, this provides information on critical injection rates which distinguish the three regimes. Applying a favourable pressure gradient reduces the extent of the streaks and removes the delineation between the three flow regimes. For sufficiently strong pressure gradients the Falkner--Skan solution is recovered throughout the domain, albeit with an injection boundary condition in the slot region. Near the centreline of the injection slot the asymptotic description does not agree with the full numerical calculations due to the presence of spatially unstable eigenmodes. These local spatial (cross-flow) eigenmodes are associated with a cross-flow collisional process at the centre of the injection slot. The stability of the streak flow to a travelling wave disturbance is considered, both in terms of the fixed Reynolds number viscous stability and the inviscid limit. The inviscid analysis produces a two-dimensional analogue of the Rayleigh stability equation which forms a critical layer where the streak velocity is equal to the wave speed. The inviscid equations are well approximated by a two-dimensional analogue of the Orr-Sommerfeld equation when the Reynolds number is sufficiently large. The stability of the streak flow at various injection rates, pressure gradients, wavenumbers and Reynolds numbers is examined.

Keyword(s)

Boundary-layer; Injection; Short spanwise scale; Stability

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