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Chapter 7: Problem 64

The drag characteristics for a newly designed automobile having a maximum characteristic length of \(20 \mathrm{ft}\) are to te determined through a model study, The characteristics at both low speed (approximately \(20 \mathrm{mph}\) ) and high speed \((90 \mathrm{mph})\) are of interest. For a series of projected model tests, an unpressurized wind tunnel that will accommodate a model with a maximum characteristic length of \(4 \mathrm{ft}\) is to be used. Determine the range of air velocities that would be required for the wind tunnel if Reynolds number similarity is desired. Are the velocities suitable? Explain.

Short Answer

Expert verified
The wind tunnel would need to produce air speeds of 100 mph and 450 mph to simulate the car moving at low speed (20 mph) and high speed (90 mph), respectively. The low speed characteristics can be studied using a regular wind tunnel but the high speed conditions cannot due to the excessively high required speed.

Step by step solution

01

Derive the formula for velocity ratio

Since the Reynolds number for the model and the real car should be the same, we have \(Re_{\text{car}} = Re_{\text{model}} \) which leads to \(\frac{ρV_{\text{car}}d_{\text{car}}}{μ} = \frac{ρV_{\text{model}}d_{\text{model}}}{μ}\). After simplification, we can see that the ratio of the velocities, \(V_{\text{model}} / V_{\text{car}}\), should be equal to the ratio of the lengths, \(d_{\text{car}} / d_{\text{model}}\). This equation can then be solved for the model's velocity, \(V_{\text{model}}\), to get \(V_{\text{model}} = V_{\text{car}} * (d_{\text{car}} / d_{\text{model}})\).
02

Calculate wind tunnel velocities needed for low speed and high speed characteristics

To determine the velocities required for the wind tunnel, we use the formula derived in Step 1. The characteristic dimension ratio \(d_{\text{car}} / d_{\text{model}}\) is 20ft/4ft, which is 5. Using this, the model velocities for low speed (\(20 \, \text{mph}\)) and high speed (\(90 \, \text{mph}\)) conditions can be calculated as such: \(V_{\text{model, low speed}} = 20 \, \text{mph} * 5 = 100\, \text{mph}\) and \(V_{\text{model, high speed}} = 90 \, \text{mph} * 5 = 450 \, \text{mph}\)
03

Evaluate feasibility of the calculated velocities

Typically, wind tunnels can generate air speeds up to about 100 mph. We find that the calculated velocity for the low speed characteristics, \(100 \, \text{mph}\), is possible to achieve in a wind tunnel. However, the value of \(450 \, \text{mph}\) for the high speed conditions is not achievable in regular wind tunnels and could, realistically, only be achieved in a high-speed wind tunnel. Therefore, while the low speed characteristics can be studied through the model study, the high speed characteristics would likely not be feasible.

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