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Chapter 7: Problem 65

The drag characteristics of an airplane are to be determined by model tests in a wind tunnel operated at an absolute pressure of \(1300 \mathrm{kPa}\). If the prototype is to cruise in standard air at 385 \(\mathrm{km} / \mathrm{hr},\) and the corresponding speed of the model is not o differ by more than \(20 \%\) from this (so that compressibility effects may be ignored), what range of length scales may be used if Reynolds number similarity is to be maintained? Assume the viscosity of air is unaffected by pressure, and the temperature of air in the tunnel is equal to the temperature of the air in which the airplane will fly.

Short Answer

Expert verified
First, we must convert the speed of the prototype to m/s. Next, calculate the maximum and minimum speeds the model can use to avoid significant compressibility effects. After this, maintain Reynolds number similarity by setting up the Reynolds number equality and solving for the model length (D). The length scale of the model must be between D_min and D_max.

Step by step solution

01

Determine the Actual Flight Conditions

First, convert the cruise speed of the prototype from km/hr to m/s. This is because the SI unit of speed is m/s. 385 \(\mathrm{km/hr}\) is equivalent to \(107.22\ \mathrm{m/s}\).
02

Calculate the Maximum and Minimum Model Speeds

Next, it's mentioned that the model's speed shouldn't deviate more than 20% from the prototype's. We calculate the ranges: \[V_{max} = 107.22 \times 1.20 = 128.66 \,\mathrm{m/s} \] and \[V_{min} = 107.22 \times 0.80 = 85.776 \,\mathrm{m/s}.\] Thus, the model should maintain a speed between 85.776 m/s and 128.66 m/s to limit compressibility effects.
03

Maintain Reynolds Number Similarity

The Reynolds number for the prototype and the model must be the same in order to maintain dynamic similarity. The Reynolds number for the model is zero because the prototype is not yet designed; its size and shape are undefined. As a result, we may use a size (length) scale to keep the Reynolds numbers equivalent. This results in the following equations: \[D_{max} = V_{max}*D_{prototype}/v \] and \[D_{min} = V_{min}*D_{prototype}/v\] These equations give us the allowable range of the model length in order to maintain Reynolds number similarity.

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Most popular questions from this chapter

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