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## Q. 7.5

Flanged Slot in Streaming Flow
Air flows in the negative direction over a plane containing a slot (width 2w, w = 0.020 m) through which air is withdrawn at a volumetric flow rate (Q/L) equal to 30.0 m²/min. The far field air velocity $(U_0)$ is uniform and equal to 30.5 m/min (100 FPM). These values of Q/L and $U_0$ are identical to those of Example 7.2 for comparison.
To do: Compute and plot several streamlines using the algorithm, and compare to the results of Example 7.2.

## Verified Solution

The dividing streamline ($ψ = ψ_d$) is

$ψ_d=\frac{Q}{L} =30.0\frac{m^2}{min}$

The procedure described in Section 7.6.1 was used to generate the streamlines. The authors used Excel, and the file is available on the book’s web site. Figure E7.5a shows several streamlines in the vicinity of the slot.

Compared to the flanged slot without a streaming flow (Figure E7.3a), it is clear that the streamlines here are not symmetric about the slot, but are tilted to the right, reflecting the distortion in the slot streamlines due to the streaming flow. A view of the same streamlines from “further away,” and in dimensional coordinates (x and y) is shown in Figure E5.7b. These streamlines can be compared to those of Figures 7.6 and E7.2a.

The dividing streamline looks similar to the dividing streamline of Example 7.2 in which the air was withdrawn through a line sink of equal magnitude to that in the present problem, 30.0 m²/min. In fact all aspects of the dividing streamline correspond favorably, as indicated in the comparison below:

 parameter line sink (Example 7.2) finite slot (Example 7.5) $h_∞$ 0.984 m 0.984 m $h_o$ 0.492 m 0.490 m $r_0$ 0.313 m 0.314 m

Discussion: The streamlines of a line sink in streaming flow and those of a slot in streaming flow are very similar, especially when viewed from afar. Thus, at distances more than a few slot widths away from the inlet, users may assume that the finite slot can be replicated by a line sink. Since the mathematics for a line sink is far simpler than for a slot, the approximation affords a great deal of simplification.