Higher approximations in boundary-layer theory Part 1. General analysis

Abstract

Prandtl's boundary-layer theory is embedded as the first step in a systematic scheme of successive approximations for finding an asymptotic solution for viscous flow at large Reynolds number. The technique of inner and outer expansions is used to treat this singular-perturbation problem. Only analytic semi-infinite bodies free of separation are considered. The second approximation is analysed in detail for steady laminar flow past plane or axisymmetric solid bodies. Attention is restricted to low speeds and small temperature changes, so that the velocity field is that for an incompressible fluid, the temperature field being calculated subsequently. The additive effects are distinguished of longitudinal curvature, transverse curvature, external vorticity, external stagnation enthalpy gradient, and displacement speed. The effect of changing co-ordinates is examined, and the behaviour of the boundary-layer solution far downstream discussed. Application to specific problems will be made in subsequent papers.

Exact Solution for Thermal Stagnation-Point Flow with Surface Curvature and External Vorticity Effects

“…This neglect is justifiable in laminar flows [4] [5] if 1 kδ  . On the other hand, the neglect of the effect of streamline curvature is not justifiable in the case of turbulent flows [6] even when 1 kδ  .…”

“…Effect of surface curvature on stagnation-point flow was first examined in [4] [5]. Assuming that 1 Re  and 1 kδ  , the analysis in [4] [5] proceeded to expand the stream function in terms of 1/Re.…”

“…Assuming that 1 Re  and 1 kδ  , the analysis in [4] [5] proceeded to expand the stream function in terms of 1/Re. The problem was then solved using the technique of matched asymptotic expansions and the results gave the second-order effect on heat transfer coefficient and skin friction due to surface curvature.…”

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“…The usual boundary-layer coordinates are adopted, where x denotes the coordinate along the surface and v is the coordinate normal to the surface. Curvature can be neglected in the analysis through first order as discussed by Gersten [5] and Van Dyke [15]. allows the boundary layer equations to be written as d2\p dip d2\p d\p d2xp _ dUe We 3ydt 9y dxdy dx sy2 9t e 9* a/'…”

“…Writing the stream function aŝ (f,Tj,r; a, e) = %(£,!]) + , tj, t; a) + 0(e2) (15) and substituting into (10)-(12) yields the following system of equations:1…”