The secondary flow in curved pipes

Abstract

The Navier-Stokes equations for the flow of a viscous incompressible fluid through curved pipes of different sections are solved in power series of the curvature of the pipe. The solution is given as far as the first power of the curvature for the case of an elliptic section and a discussion given of the effect of the aspect ratio of the pipe on the intensity of the secondary flow. It is shown that the axial velocity is modified by two curvature terms of opposite effect. For values of the aspect ratio near unity the first of these predominates and the resultant effect is an increase of velocity in the outer half of the bend and a decrease in the inner: for large values of the aspect ratio the second term is numerically much greater and there is a resultant decrease in axial velocity in the outer half of the bend and an increase in the inner half. The solution is also given to the first power of the curvature for the case of a square section. This shows that the intensity of the secondary flow in a pipe of square section is greater than that in a pipe of circular section. Finally the solution is given as far as the second power of the curvature for the case of flow through a curved pipe of circular section when suction proportional to the curvature is applied at the walls. The result shows that with the particular distribution of suction considered the diminution in flux through a curved pipe may be almost entirely eliminated.