In Prob. 938 we developed expressions for u, v, and for flow through the compressible, two

Question:

In Prob. 9–38 we developed expressions for u, v, and ρ for flow through the compressible, two dimensional, diverging duct of a high-speed wind tunnel. Generate an expression for the compressible stream function ψ_ρ that describes this flow field. For consistency, set ψ_ρ = 0 along the x-axis. Plot several streamlines and verify that they agree with those you plotted in Prob. 9–38. What is the value of ψ_ρ at the top wall of the diverging duct?

Data from problem 9–38

A two-dimensional diverging duct is being designed to diffuse the high-speed air exiting a wind tunnel. The x-axis is the centerline of the duct (it is symmetric about the x-axis), and the top and bottom walls are to be curved in such a way that the axial wind speed u decreases approximately linearly from u₁ = 300 m/s at section 1 to u₂ = 100 m/s at section 2 (Fig. P9–38). Meanwhile, the air density r is to increase approximately linearly from ρ₁ = 0.85 kg/m³ at section 1 to ρ₂ = 1.2 kg/m³ at section 2. The diverging duct is 2.0 m long and is 1.60 m high at section 1 (only the upper half is sketched in Fig. P9–38; the half height at section 1 is 0.80 m).

FIGURE P9–38