摘要:
In at least one case where the potential flow in the neighborhood of the outer edge of the boundary-layer varies rapidly, the boundary-layer theory has not been well understood. The interest in this problem led the author to the study of the laminar flow of a viscous incompressible fluid over a curved surface whose curvature, as has been found previously, displays rather large effects on the nature of boundary-layer flow. The specific point to be investigated here is the question as to how to join the boundary-layer with the potential flow. On the basis of the fact that the viscous effects decay exponentially in the lateral direction, it has been intuitively suggested that, in the neighborhood of the outer edge of the boundary-layer, the velocity in the main stream direction ui should asymptotically approach that in the main stream. Namely,where x and y are the coordinates shown in Fig. 1. Rc( = 1/ε2, K and R denoteReynolds number, the curvature of the surface and a large constant respectively.By an order-of-magnitude analysis, Murphy obtained the following differential equation of the motion in a boundary-layer over a particular curved surface with curvature K = A/εα~1/2 ,along which the surface pressure gradient is zero. where A and f are respectively a small constant and the stream function; ηis definedas y/(2εx~1/2), and f is a function of nnnnnn only.In the case of zero pressure gradient, Murphy gave an approximate representation, of the potential flow as follows:which, by virtue of (1), loads toThe no-slip condition ad the wall yields:Herotofore the problem may be further simplified to a steaightforward numerical com-pulation. Since A is very small, assume a solution of ( 2 ) of this formBy expanding both sides of (8) into power serics of A, there result the following joining conditions:when η≤1From the no-slip condition at wall it followsSituilarly the differential equations for various orders may bo deduced from (2) and(5): The first equation of ( 7 ) appears in the case of the flat plate problem is well-known Blasius solution. It can be shown at the present stage that the solution of (7 ) and conditions given in ( 6) are consistent. Substituting ( 6 ) in ( 7 ) and neglecting terms of order e-η3 it indicates that expressions in ( 6 ) are the solutions of ( 7 ) in the region of large η. Consequently there is no difficulty in satisfying the joining conditions. With the joining conditions given in ( 6) and no-slip conditions, the equation of (7) can be easily solved numerically. For small 17, the following solutions are found: order A0order A1 whereorder A2 wherewhere a=1.32824, C1 and C2 are numerically found to be-5.767 and -2.3 respectively. It is noticed that Murphy's solution misses the part C1h1, and that a is not a constant, but a function of A. Similarly, the solutions of higher order can be obtained. In Fig. 3 the tangential velocities are calculated to order of A2, In conclusion, the following points may be noted:1. the boundary layer flow joins the potential flow asymptotically, that is, with an order of o(e-η3).2. the curvature has little effect on boundary layer, the effect on shearing stress τ0 is where μis the measure of viscosity.3. the curvature has larger effect on the velocity profile in the neighbourhood of the outer edge of the layer. The velocity for convex surface is greater than for a flat-plate, whereas for concave surface it is smaller. In the region of small η, the velocities may be approximately represented by the power series given above; however,in the region of large η, the velocity profiles can bo approximated by 1/(1+2Aη).