Two-dimensional boundary layers in a free stream which oscillates without reversing

Abstract

The laminar boundary layer over the semi-infinite rigid plane y = 0, x > 0 is examined for the case when the free-stream velocity takes the form \[ U(x, t) = U_0(x) (1+\alpha_1 \sin \omega t), \] where 0 [les ] α₁ < 1 and U₀(x) ∝ xⁿ, 0 [les ] n [les ] 1. The corresponding steady solution is the Falkner-Skan boundary layer with zero (n = 0) or favourable pressure gradient (n = 1 corresponds to the stagnation-point boundary layer). The skin friction, and the heat transfer from the wall when that is maintained at a uniform temperature T₁ greater than the temperature T₀ of the oncoming fluid, are calculated by means of two asymptotic expansions: a regular one for small values of the frequency parameter ε₁(x) = ωx/U₀(x) and a singular one (requiring the use of matched asymptotic expansions) for large values of ε₁. The principal difference between this work and that of earlier authors is that here α₁ is not required to be small. Numerical computations are made for three values of n (0, $\frac{1}{3}$, 1), three values of α₁ (0·2, 0·5 and 0·8) and (in the case of the heat transfer) two values of the Prandtl number σ (0·72 and 7·1). It is demonstrated that for each n there is a value of ε₁ at which the small-and the large-ε₁ expansions for the skin friction overlap quite well, and that, near the overlap region, two terms of the small-ε₁ expansion provide a more accurate asymptotic representation of the solution than three. It is also shown that there is no region of overlap between the small- and the large-ε₁ heat-transfer expansions, except in one case (n = 1/3, σ = 0·72) where the overlap value of ε₁ (= 2·0) is the same as for the skin-friction expansions. The question of the existence of eigensolutions in the large-ε₁ expansion, where on physical grounds they can be expected to appear, is discussed in the appendix.