Question

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 u_{1} = 300 m/s at section 1 to u_{2} = 100 m/s at section 2 . Meanwhile, the air density ? is to increase approximately linearly from ?_{1} = 0.85 kg/m^{3} at section 1 to ?_{2} = 1.2 kg/m^{3} 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. P9–36; the halfheight at section 1 is 0.80 m). (a) Predict the y-component of velocity, ?(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?

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