Three-dimensional boundary layers with short spanwise scales

Abstract

We investigate three-dimensional (laminar) boundary layers that include a spanwise scale comparable to the boundary-layer thickness. A forcing of short spanwise scales requires viscous dissipation to be retained in the two-dimensional cross-section, perpendicular to the external flow direction, and in this respect the flows are related to previous work on corner boundary layers. We use two examples to highlight the main features of this category of boundary layer: (i) a flat plate of narrow (spanwise) width, and (ii) a narrow (spanwise) gap cut into an otherwise infinite flat plate; in both cases the plate is aligned with a uniform oncoming stream. We find that a novel feature arises in connection with the external flow; the presence of a narrow gap/plate (or indeed any comparable short-scale feature of long streamwise extent) necessarily modifies the streamwise mass flux in that vicinity, which in turn induces an associated boundary-layer transpiration on the same short spanwise length scale. This (short-scale) transpiration region leads to a half-line-source/sink correction to the outer inviscid, irrotational flow. Crucially, the volumetric flux associated with this line-source/sink must be explicitly included as part of the computational procedure for the leading-order boundary layer, and as such there is a weak interaction between the outer (inviscid) flow and the boundary layer. This is a generic feature of boundary layers that are forced through the presence of short-scale spanwise variations.

Injection into boundary layers: solutions beyond the classical form

“…This approach assumes that spanwise lengthscales are comparable to the transverse boundarylayer thickness, ensuring that diffusion in the cross section is retained in both directions, but the longer streamwise lengthscale leads to neglect of streamwise diffusion. Similar formulations have previously been employed in high Reynolds number descriptions of (for example) corner boundary regions (Dhanak & Duck 1997), wakes behind elongated roughness elements (Goldstein et al 2016), flow near small-scale surface gaps (Hewitt & Duck 2014), the influence of upstream vorticity on transition (Wundrow & Goldstein 2001) and the generation of laminar streaks by freestream vorticity (Ricco & Dilib 2010). The formulation of van Dommelen & Yapalparvi (2014) is also equivalent if one considers the zero-curvature limit of their equations, although in their case the short-scale spanwise forcing is also assumed to be periodic, which simplifies the far-field behaviour compared to the (finite-spanwise extent) our problem.…”

“…The decomposition (2.6) allows the far-field behaviour of (Ũ ,Φ,Ψ,Θ) to be explicitly enforced as part of the numerical solution procedure. The far-field asymptotic behaviour is discussed in detail in Hewitt & Duck (2014) and we only present the main results here. It is sufficient here to note that for ζ ≫ 1 and/or η ≫ 1,Φ andΨ both satisfy the harmonic equation, leading toΦ…”

“…A second-order (finite-difference) method for the numerical solution of (2.4) subject to the decomposition (2.6) is employed; this follows the method of Hewitt & Duck (2014). The computational mesh is non-uniformly spaced in theζ, η plane to ensure that more nodes are concentrated near the plate surface, within the injection region (ζ < 1) and at the edge of the injection region (ζ = 1) where the injection velocity changes rapidly to zero.…”

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

“…Far from the injection region, we expect that the flow decays back towards the Falkner-Skan base solution. The decay of (Φ,Ψ ) is algebraic in the far field [19], and it is best to impose far-field boundary conditions that are consistent with this asymptotic decay. An explicit inclusion of this (relatively slow, algebraic) decay avoids the necessity for excessively large computational domains that otherwise arise with a simple zero Dirichlet conditions.…”

“…with corresponding conditions forζ 1 and η = O(1) as discussed in [19]. The constant A is associated with a measure of the radial mass flux away from the injection region in the far field, with A > 0 being towards and A < 0 being away from the centreline (η =ζ = 0) of the injection slot.…”