任意截面细长弯曲管道内的可压缩势流解

COMPRESSIBLE POTENTIAL FLOW IN A SLENDER CURVED CHANNEL WITH ARBITRARY CROSS SECTIONS

利用摄动法给出了任意截面细长弯曲管道内的可压缩定常势流的解。引入了一个表示管道细长程度的小参数ε,并将流场的解对小参数进行渐近展开。文中给出了对密度和轴向速度具有两项展开式、对横向速度具有一项展开式的结果。理论分析表明,在细长管道的情况下,问题的解可降低一维。同时,在上述近似条件下,求横向速度分量相当于求解一个平面Laplace方程的Neumann边值问题。

Using asymptotic expansion method, the solution of the steady compressible potential flow in a slender curved channel with arbitrary cross sections is given in the paper. This is the extension of the theory of Ref. [10] given by the present author. In this paper, more complicated geometry in which the curved axis is assumed to be an arbitrary spatial curve is considered, whereas in Ref. [10], the axis is only assumed to be an arbitrary planar curve. Similar to Ref. [10], a small parameter e denoting the slenderness of the channel is introduced, and the dependent variables of the flow field are expanded as the ascending power series of s. However, according to the complexity of the geometry, the method of tensor analysis is used instead in the development of the theory. The solutions of two-term expansions for the density and the axial velocity component, and one-term expansions for the cross-sectional velocity components are given. Also similar to Ref. [10], the result shows that, apart from some elementary algorithm, the solution of the complex three-dimensional flow field reduces to the solution of two-dimensional harmonic functions with boundary conditions of Neumann's type.