/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 76 A laboratory wind tunnel has a t... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 76

A laboratory wind tunnel has a test section that is square in cross section, with inlet width \(W_{1}\) and height \(H_{1}\) each equal to 1 ft. At freestream speed \(U_{1}=80 \mathrm{ft} / \mathrm{s},\) measurements show the boundary- layer thickness is \(\delta_{1}=0.4\) in. with a \(\frac{1}{7}\) -power turbulent velocity profile. The pressure gradient in this region is given approximately by \(d p / d x=\) -0.035 in. \(\mathrm{H}_{2} \mathrm{O} /\) in. Evaluate the reduction in effective flow area caused by the boundary layers on the tunnel bottom, top, and walls at section (1). Calculate the rate of change of boundary-layer momentum thickness, \(d \theta / d x,\) at section (1). Estimate the momentum thickness at the end of the test section, located at \(L=10\) in downstream.

Short Answer

Expert verified
The reduction in effective flow area, the rate of change of boundary-layer momentum thickness, and the momentum thickness at the end of the test section needs to be calculated using the provided measurements and formulas. Since we are supposing no initial momentum thickness, the final momentum thickness represents the momentum thickness at the end of the test section.

Step by step solution

01

Calculate the reduction of the effective flow area

Let's start by calculating the reduction of the effective flow area. The cross sectional area of the tunnel is given by \(W_{1} \times H_{1} = 1 \, \mathrm{ft}^2\). The boundary layer thickness of \(\delta_{1}\) = 0.4 in, which equals \(0.4/12 = 0.0333 \, \mathrm{ft}\). Given the boundary layer exists on all four sides of the tunnel, the effective width and height are \(W_{1} - 2\delta_{1}\) and \(H_{1} - 2\delta_{1}\) respectively. Then, the effective cross-sectional area \(A_{eff}\) can be calculated as \(A_{eff} = (W_{1} - 2\delta_{1}) \times (H_{1} - 2\delta_{1})\)
02

Calculate the rate of change of boundary layer momentum thickness

The rate of change of the boundary layer momentum thickness, \(d \theta / dx\), is evaluated via the Reynolds analogy, which can be written as: \(d \theta / dx = \left(\delta_{1}/U_{1}\right) \times (d p/d x)\). Given the pressure gradient \(-0.035 \, \mathrm{ in.} \, H_{2}O / \mathrm{in.}\) equals \(-0.035 \times (1/12) \times (62.4/144) = -0.0011 \, \mathrm{ lb/ft}^2 / \mathrm{in.}\), the rate of change of the boundary layer momentum thickness can thus be calculated.
03

Estimate the momentum thickness at the end of the test section

The momentum thickness \(\theta\) at the end of the test section can be found by integrating \(d \theta / dx\) over the length of the test section. Given that \(L=10 \, \mathrm{in}\) downstream, the final momentum thickness \(\theta_{final}\) can be calculated by integrating \(\theta_{initial} + \int_0^L d\theta / dx \, dx\), where \(\theta_{initial} = 0\) as we suppose no momentum thickness initially.

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Most popular questions from this chapter

Velocity profiles in laminar boundary layers often are approximated by the equations $$\begin{array}{cl} \text { Linear : } & \frac{u}{U}=\frac{y}{\delta} \\ \text { Sinusoidal : } & \frac{u}{U}=\sin \left(\frac{\pi}{2} \frac{y}{\delta}\right) \\ \text { Parabolic : } & \frac{u}{U}=2\left(\frac{y}{\delta}\right)-\left(\frac{y}{\delta}\right)^{2} \end{array}$$ Compare the shapes of these velocity profiles by plotting \(y / \delta\) (on the ordinate) versus \(u / U\) (on the abscissa).

A light airplane has 35 -ft effective wingspan and \(5.5-\mathrm{ft}\) chord. It was originally designed to use a conventional (NACA 23015 ) airfoil section. With this airfoil, its cruising speed on a standard day near sea level is 150 mph. A conversion to a laminar-flow (NACA \(66_{2}-215\) ) section airfoil is proposed. Determine the cruising speed that could be achieved with the new airfoil section for the same power.

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.

An approximation for the velocity profile in a laminar boundary layer is \\[ \frac{u}{U}=2 \frac{y}{\delta}-2\left(\frac{y}{\delta}\right)^{3}+\left(\frac{y}{\delta}\right)^{4} \\] Does this expression satisfy boundary conditions applicable to the laminar boundary-layer velocity profile? Evaluate \(\delta^{*} / \delta\) and \(\theta / \delta\)

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