A group of students is designing a small, round (axisymmetric), low-speed wind tunnel

Chapter 9, Problem 9-115E

(choose chapter or problem)

A group of students is designing a small, round (axisymmetric), low-speed wind tunnel for their senior design project (Fig. P9115E). Their design calls for the axial component of velocity to increase linearly in the contraction section from uz, 0 to uz, L. The air speed through the test section is to be uz, L 5 120 ft/s. The length of the contraction is L 5 3.0 ft, and the entrance and exit diameters of the contraction are D0 5 5.0 ft and DL 5 1.5 ft, respectively. The air is at standard temperature and pressure. (a) Verify that the flow can be approximated as incompressible. (b) Generate an expression for the radial velocity component ur between z 5 0 and z 5 L, staying in variable form. You may ignore frictional effects (boundary layers) on the walls. (c) Generate an expression for the stream function c as a function of r and z. (d) Plot some streamlines and design the shape of the contraction, assuming that frictional effects along the walls of the wind tunnel contraction are negligible.