In Example 92, we provide expressions for u, v, and for flow through a compressible converging duct. Generate an expression for the compressible stream

In Example 9–2, we provide expressions for u, v, and ρ for flow through a compressible converging duct. 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.

Data from Example 9-2.

A two-dimensional converging duct is being designed for a high-speed wind tunnel. The bottom wall of the duct is to be flat and horizontal, and the top wall is to be curved in such a way that the axial wind speed u increases approximately linearly from u₁ = 100 m/s at section (1) to u₂ = 300 m/s at section (2) (Fig. 9–12). Meanwhile, the air density r is to decrease approximately linearly from ρ₁ = 1.2 kg/m³ at section (1) to ρ₂ = 0.85 kg/m³ at section (2). The converging duct is 2.0 m long and is 2.0 m high at section (1). 

(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) How high should the duct be at section (2), the exit of the duct?

FIGURE 9–12

2.0 m (1) X Ax = 2.0 m (2)