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). 

(a) Predict the y-component of velocity, v(x, y), in the duct. 

(b) Plot the approximate shape of the duct, ignoring friction on the walls. 

(c) What should be the half-height of the duct at section 2?

FIGURE P9–38

1 1 10.8 m y (1) X -Ar- 2.0 m- (2)