Transformation of a Polynomial for a Contraction Wall Profile

Abstract

In the design of contractions for low-speed wind tunnels, several desirable characteristics of the wall profile are identified, primarily consisting of a wall profile described by a function having zero first- and second-order derivatives, and the radii of curvature at the inlet and outlet roughly proportional to the area 1. This condition would be the most favorable to achieve flow uniformity, thin boundary layers, and negligible losses 2. A contraction profile, described by a fifth degree polynomial, has been developed by Bell and Mehta 3 and has been repeatedly employed successfully in contraction design in two and three dimensions. This polynomial has proven so successful that it has largely become the design standard. However, because this polynomial is symmetrical and has identical radii of curvature at the inlet and outlet, an arrangement traditionally considered undesirable 1, many designers of wind tunnel contractions believe that a more qualitative, experimental design of contraction profiles, the so-called “by-eye” design, will produce better results. The purpose of this short note is to develop a transformation of Bell and Mehta’s 3 polynomial such that any curve characteristics selected by a designer can be described by an analytical function that will have vanishing first and second derivatives at the inlet and outlet, and will have continuous first and second derivatives throughout. It is important that this be achieved as simply as possible so that it would be of maximum utility to the designer. Systematic manipulation of as few parameters as possible should produce the curve desired.