On the initial phase of the triple deck stage in marginally separated flows

Abstract

The method of matched asymptotic expansions is used to investigate marginally separated boundary layer flows (laminar or alternatively transitional separation bubbles) at high Reynolds numbers. Typical examples include, among others, the flow past slender airfoils at small to moderate angels of attack and channel flows with suction. As is well-known, classical (hierarchical) boundary layer computations usually break down under the action of an adverse pressure gradient on the flow, a scenario associated with the appearance of the Goldstein separation singularity. If, however, the parameter controlling the strength of the pressure gradient (the angle of attack or the relative suction rate in the examples mentioned above) is adjusted accordingly, the application of a local viscous-inviscid interaction strategy is capable of describing localized boundary layer separation. Moreover, taking into account unsteady effects and flow control devices allows the investigation of the conditions leading to forced or self-sustained vortex generation and the subsequent evolution process culminating in bubble bursting. Within the asymptotic formulation of this stage bubble bursting is associated with the formation of finite time singularities in the solution of the underlying equations and a corresponding break down. The distinct blow-up structure gives rise to a fully non-linear triple deck interaction stage featuring shorter spatio-temporal scales characteristic of the successive vortex evolution process. The paper will focus on the numerical treatment of the initial phase of the latter stage. In suitably nondimensionalized and scaled form the spatio-temporal evolution of the stream function ψ(x, y, t) for planar incompressible flow is given by a boundary layer type equation with both, prescribed adverse, and induced pressure gradientsupplemented with the interaction law which relates the induced pressure P(x, t) and the displacement function A(x, t). Here x, y, and t denote the streamwise and wall normal coordinates and the time. Equations (1) are subject to the no slip boundary conditions ψ = ∂ψ/∂y = 0 at the solid wall y = 0, the far-field behaviors ψ ∼ (y + A) 3 /6 + · · · as y → ∞, and ψ ∼ y 3 /6 , A, P → 0 as x → ±∞. The connection to the triggering blow-up event in the preceding marginal separation stage is established via the matching, or equivalently, initial conditionwith ∼ 1 (x) + |t|2 (x) + · · · andψ ∼ψ 2 + |t| (2) into (1) recovers the blow-up structure of the preceding stage in form of the linearized boundary layer equationswith vanishing leading order right hand side b 1 = 0. Application of the adjoint operator approach to (3) and making use of Fredholm's alternative then leads to equations (solvability conditions, not repeated here) determining the unique blowup profile 1 (x), the eigenfunctionsê 1 (x) = E 1 ′ 1 ,ê 2 = E 2 ( 1 + 2 3x ′ 1 ) (the arbitrary amplitudes E 1 , E 2 enable the embedding into a global solution), the perturbation stream functionψ 2 (x,ŷ) and the next order correction of the displ...

The triple-deck stage of marginal separation

“…Although we started the investigation of transitional separation bubbles with the analysis of formally steady classical boundary layer flows on the verge of separation, rather weak unsteady forcing (blowing) in the first, large-scale, interaction stage of marginal separation eventually led us (via a second, shorter-scale triple-deck interaction) to a scenario that in literature is commonly referred to as unsteady separation, see the reviews [79,80] and references therein. In sharp contrast to the usual setting underlying the model problems studied in the context of unsteady separation, the outer inviscid flow field is not given in advance in the current formulation (63), (64). The spatio-temporal evolution of the flow is entirely self-induced once kicked by the initial condition (65).…”

“…More precisely, the explicit quotation of ψ 00 in (63) leads to a slightly modified formulation of the blow-up profiles, but without loss of generality. Current efforts aim, at first, to compute (at least) the next-order corrections in (65) in order for the associated initial value problems to be defined properly, and, secondly, to construct a suitable numerical scheme able to capture the vortex wind-up process governed by (63) and (64). As in the triple-deck stage, the infinite spatial and temporal domains and the initial similarity structure require special treatment, cf.…”

“…We essentially follow the comfortable differentiation matrices approach introduced by Gajjar [ 61 ] for the treatment of viscous–inviscid interaction problems. To this end, the (semi-)infinite physical domains are mapped to the Chebyshev domain , see [ 62 , 63 ]. Specifically, any (unknown) field variable f is interpolated by a polynomial p ( s ) in Lagrange form, where the basis polynomials are represented by the barycentric formula, see the very instructive article by Berrut and Trefethen [ 64 ] for details: Here , and denote the intrinsic unknowns of the problem, the unequally spaced Gauss–Lobatto grid points (nodes) for the interpolation process to be well conditioned and the weights with indices , respectively.…”