Some perturbation solutions in laminar boundary-layer theory

Abstract

The velocity fields associated with a variety of flows which may be described by perturbations of the Blasius solution are considered. These are flows which, for example, because of localized mass transfer, involve the initial-value problem of boundary-layer theory, or which involve a variable ratio of the viscosity-density product, or finally which involve mass transfer. The perturbation solutions are presented so that in accord with the usual linearization procedures further applications for the determination of first-order effects can be readily made. In addition, each of these perturbations involves a common differential operator whose eigenfunctions form a complete orthogonal set. Accordingly, a procedure for systematically improving each perturbation solution to obtain higher-order effects by quadrature is presented. The results of applications in several cases are given and are compared to more accurate solutions where available.

Modal description of internal optimal streaks

“…The streamwise, cross-flow velocity profiles and the //-profile (normalized with their máxima in 0 < f < oo) are plotted in figure 2, and the máxima of U, V, W and H are plotted versus x with solid lines in figure 3(a). As already noticed by Luchini (2000), figure 2{a) shows that the streamwise velocity of the optimal perturbation remains approximately constant, up to rescaling, and approximately equal to both the streamwise velocity of the first Libby-Fox mode, U = í;F" (Stewartson 1957;Libby & Fox 1964;and Luchini 1996) and the streamwise velocity of the first Luchini mode; these two are also plotted for comparison. Figure 2 also shows that the new variable H introduced above also remains approximately constant and approximately equal to U (when rescaled with its máximum), which suggests that a low-dimensional Ordinary Differential Equation (ODE) model of streaks should be possible.…”

Boundary-layer receptivity for a parabolic leading edge

“…The value of B 1 cannot be determined by the large-ξ analysis sinceξ −2 G 1 (η) is an eigensolution of the perturbation equation. The next term in the expansion is O(ξ −α ), α ≈ 3.774, the fractional power arising as the next eigensolution of an infinite sequence (Libby & Fox, 1963). It appears that B 1 , together with the set of similar constants appearing in higher-order terms, is dependent on conditions close to the nose of the body and hence can be determined only by numerical integration from ξ = 0.…”

Boundary-layer receptivity for a parabolic leading edge. Part 2. The small-Strouhal-number limit

“…For all γ > 0, G γ has solutions which decay algebraically as η → ∞, but only for certain γ do solutions exist which decay exponentially at infinity. Libby & Fox (1963) give the first 10 eigenvalues. The first such eigenvalue, γ 1 = 2 has already been discussed.…”