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A laboratory wind tunnel has a square test section with sides of width \(W=1\) ft and length \(L=2\) ft. When the freestream air speed at the test section entrance is \(U_{1}=80 \mathrm{ft} / \mathrm{s}\), the head loss from the atmosphere is 0.3 in. \(\mathrm{H}_{2} \mathrm{O}\). Turbulent boundary layers form on the top, bottom, and side walls of the test section. Measurements show the boundary-layer thicknesses are \(\delta_{1}=0.8\) in at the entrance and \(\delta_{2}=1\) in at the outlet of the test section. The velocity profiles are of \(\frac{1}{7}\) -power form. Evaluate the freestream air speed at the outlet from the test section. Determine the static pressures at the test section inlet and outlet.

Step by step solution

01

Calculate the average velocity at the inlet of the test section

First, determine \(\bar{U}_1\) - the average velocity at the inlet. For \(\frac{1}{7}\) -power velocity profiles, \(\bar{U}\) is about \(0.82U\), so \(\bar{U}_1 = 0.82U_1 = 0.82 \times 80 \,\mathrm{ft/s} = 65.6\,\mathrm{ft/s}\).

02

Calculate the flow rate at the inlet

Then, calculate the flow rate \(Q_1\) at the inlet using the equation \(Q=\bar{U}A\), where \(A\) is the area of the flow normal to the velocity. Since the flow is bounded by boundary layers, the effective area is \((W-2\delta_1)\times (W)\) where \(\delta_1 = 0.8/12\) feet (converted from in to ft). Hence \(Q_1 = \bar{U}_1 \times (1-2\delta_1) \times 1 = 65.6\,\mathrm{ft/s} \times (1-2 \times 0.8/12) \times 1 = 63\,\mathrm{ft}^3/\mathrm{s}\).

03

Calculate the average and freestream velocity at the outlet

By conserving the flow, the flow rate at the outlet \(Q_2\) is equal to \(Q_1\). Using a similar approach as in Step 2, we can figure out the area at the outlet \(A_2 = 1 \times (1-2\delta_2)\) with \(\delta_2 = 1/12\) feet. Then, we calculate the average outlet velocity from \(\bar{U}_2 = Q_2 / A_2 = 63\,\mathrm{ft}^3/\mathrm{s} / (1 - 2 \times 1/12) = 72\,\mathrm{ft/s}\). And, then we can pose \(U_2 = \bar{U}_2 / 0.82 = 72\,\mathrm{ft/s} / 0.82 = 88\, \mathrm{ft/s}\).

04

Compute the static pressures at the inlet and outlet

Lastly, the static pressures at the inlet and outlet can be obtained using Bernoulli's equation: \(P_{1,\text{static}} = P_{\text{atm}} - \text{head loss} - \frac{1}{2} \cdot \rho \cdot U_1^2\) and \(P_{2,\text{static}} = P_{\text{atm}} - \frac{1}{2} \cdot \rho \cdot U_2^2\). Here, you should note that the head loss will convert to pressure (using the fact that 1 in. H2O is about 249 Pa), and the density \(\rho\) of air has to be known or assumed (typically \(\rho_{\text{air}} \approx 1.2 \,\mathrm{kg/m}^3\)).

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