A group of students is designing a small, round (axisymmetric), low-speed wind tunnel for their senior design project (Fig. P9–115E). Their design calls for the axial component of velocity to increase linearly in the contraction section from u_{z, 0} to u_{z, L}. The air speed through the test section is to be u_{z, L} = 120 ft/s. The length of the contraction is L = 3.0 ft, and the entrance and exit diameters of the contraction are D₀ = 5.0 ft and DL = 1.5 ft, respectively. The air is at standard temperature and pressure.

FIGURE P9–115E

(a) Verify that the flow can be approximated as incompressible.

(b) Generate an expression for the radial velocity component u_r between z = 0 and z = L, staying in variable form. You may ignore frictional effects (boundary layers) on the walls. 

(c) Generate an expression for the stream function ψ 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.

Transcribed Image Text:

1 1 Do ir Contraction U₂ 0 z = 0 DL Test section 1 Uz L Z=L