A Study of the Effect of Curvature on Fully Developed Turbulent Flow

In aeronautics we are especially interested in the flow of air adjacent to surfaces, such as airfoils. There are two main types of flow of real fluids, laminar and turbulent and it is turbulent flow which is of practical importance in aeronautics. We should like to be able to predict the skin friction and flow conditions for any surface of any shape. There has recently been much success with the problem of predicting flow along a flat plate parallel to the direction of flow, and the problem was attacked by investigation of fully developed turbulent flow in straight channels, and direct application of the semi-empirical laws obtained, to the flow along a flat plate. However, surfaces met with in practice are, in general, curved, so that it would be important to be able to predict the effect of curvature on turbulent flow. Most of the previous work in curved flow, however, has been with curved pipes and channels where the behavior of the flow was complicated by secondary vortices. The present work had the purposes of isolating as far as possible the effect of curvature on a fully developed turbulent flow, with two dimensional mean motion. The curved channels used were 5 cm. in breadth and 90 cm. in depth, and had straight entrance sections over 60 x breadth in length to produce a fully developed straight flow before subjecting it to the effect of curvature. Channel I had inner radius 45 cm. and outer radius 50 cm., while channel II had inner radius 20 cm. and outer 25 cm. In addition, measurements were made in an appratus consisting of two concentric cylinders, the inner one of radius 20 cm. and rotating, the outer of radius 25.4 cm. and fixed. The curvature was made of the same order as channel II for purpose of comparison. Measurements on the channels consisted of pressure drop along the channel walls at several speeds, velocity distribution at 30° intervals around the curved portion, velocity distributions at several speeds, and for channel II, determination of the shearing stress at the walls of one of the curved sections. Measurements on the cylinders consisted of velocity distributions at two speeds and determination of shearing stress at the outer wall. Evaluation of results included: calculation of resistance law, calculation of the shearing stress, distribution in radial direction across the curved portion, determination of the exponential law for the velocity distribution near the walls in the various cases, calculation of the "mixing length" 1, from turbulent exchange theory, and several dimensionless methods of plotting velocity distributions to show similarity between measurements in the channels and in the concentric cylinders. Also included are calculations of the laminar flow distribution in a curved channel, and a discussion of Rayleigh's stability criterion. It appears that the distribution of centrifugal force has a strong influence on the stability of the flow, and affects materially the velocity distribution. The fact that similarity can be obtained for several cases by proper dimensionless reduction based on the effective breadth of the mixing region looks hopeful, and it remains for future investigations to determine more facts about the effective breadth of the mixing region.