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Chapter 7: Problem 62

7.62 A thin rectangular plate is towed through seawater at an average velocity of 5 mph. The plate is held in a vertical position and projects above the undisturbed level of the water to a height \(z . A 1: 4\) scale model is to be used to predict the drag on the plate, and the model fluid is also seawater. (a) Assuming that Froude number similarity must be maintained, determine the required model velocity. (b) What is the required value of \(z_{m} / z ?\) (c) A measured drag of 1 lb on the model will correspond to what drag on the prototype?

Short Answer

Expert verified
(a) Model velocity is 3.67 ft/s. (b) The value of \(z_m/z\) is \(1/4\). (c) Prototype drag is 64 lb.

Step by step solution

01

Understand Froude Number Similarity

To ensure Froude number similarity between the model and the prototype, the Froude number (Fr) must remain constant between these two. The Froude number is defined as \(Fr = \frac{V}{\sqrt{gL}}\), where \(V\) is velocity, \(g\) is the acceleration due to gravity (32.2 ft/s"), and \(L\) is a characteristic length (such as \(z\), the height of the plate).
02

Set up Equation for Model and Prototype

For similarity using Froude number, we equate the Froude number of the prototype and the model: \( \frac{V_p}{\sqrt{gz}} = \frac{V_m}{\sqrt{gz_m}} \). Given that the model is a 1:4 scale, the model length \(z_m = \frac{1}{4}z\).
03

Calculate the Model Velocity

Plugging the scale factor into the Fr equation, we have: \[ \frac{5 \text{ mph}}{\sqrt{gz}} = \frac{V_m}{\sqrt{g \cdot \frac{1}{4}z}} \]. Converting 5 mph to ft/s: \(5 \times \frac{5280}{3600} = 7.33 \text{ ft/s}\). Solving for \(V_m\), \[ V_m = 7.33 \times \frac{1}{2} = 3.67 \text{ ft/s}\].
04

Determine the Required Ratio \(z_m/z\)

Given that the model is a 1:4 scale, the height of the model \(z_m\) is \(1/4\) of the prototype's height \(z\). Hence, the required value of \(z_m/z\) is \(1/4\).
05

Calculate Corresponding Prototype Drag

The drag force is proportional to the square of the velocity and the surface area. When Froude similarity is maintained, the drag force on the model and prototype is related by:\[ F_p = F_m \left( \frac{z}{z_m} \right)^3 \].Given \(F_m = 1 \text{ lb}\) and \(z/z_m = 4\),\[ F_p = 1 \times 4^3 = 1 \times 64 = 64 \text{ lb}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fluid Mechanics
Fluid mechanics is a branch of physics concerned with the behavior of fluids (liquids, gases, and plasmas) and the forces acting upon them. One of the fundamental principles in fluid mechanics is understanding how fluids move and interact with objects. Here are some key points:
  • Flow Dynamics: Examining how fluids flow around different surfaces is essential in predicting and managing forces such as drag.
  • Gravity鈥檚 Role: Gravity significantly impacts how fluids behave, influencing pressure distributions and wave formations.
  • Relevance to Modeling: Accurate fluid dynamic equations and principles are critical when creating scale models to predict real-world scenarios.
By comprehending fluid dynamics, we can rig models effectively to simulate and predict interactions that occur at full scale.
Scale Modeling
Scale modeling involves creating smaller, yet proportionate representations of larger objects. This technique is vital in fields like engineering, where predicting a prototype's behavior is crucial before full-scale construction.

Scale models offer several benefits:
  • Cost Efficiency: Testing on models saves both money and resources.
  • Feasibility Testing: Models allow for experimenting with different design variables efficiently.
  • Visualization: Smaller versions help visualize outcomes that are otherwise hard to predict.
The importance of maintaining proper scale ratios, like a 1:4 model in this exercise, ensures that tests reflect realistic conditions and provide credible data.
In the case of this exercise, ensuring the correct scaling (e.g., maintaining the Froude number) is crucial for accurate results.
Drag Force Prediction
Predicting drag force is a critical component of design for vehicles, aircraft, and even maritime structures.

Drag force is the resistance force caused by the motion of the body through a fluid, which involves:
  • Velocity Dependence: Drag force tends to increase with the square of velocity, meaning that higher speeds result in much higher forces.
  • Surface Area and Shape: Larger surface areas and less streamlined shapes lead to more drag.
  • Froude Number Similarity: As shown in the exercise, keeping the Froude number constant allows us to maintain accurate predictions between scaled models and their full-scale prototypes.
The calculation of prototype drag based on model measurements (like the 64 lb drag in this case) requires careful consideration of these factors, with the assumption that the scale and conditions are properly aligned.
Accurate drag predictions facilitate effective design optimizations and increased efficiency.

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Most popular questions from this chapter

If a large oil spill occurs from a tanker operating near a coastline, the time it would take for the oil to reach shore is of great concern. Design a model system that can be used to investigate this type of problem in the laboratory, Indicate alt assumptions made in developing the design and discuss any difficulty that may arise in satisfying the similarity requirements arising from your model design.

The time, \(t,\) it takes to pour a certain volume of liquid from a cylindrical container depends on several factors, including the viscosity of the liquid. (See Video \(V 1.1\) ) Assume that for very viscous liquids the time it takes to pour out \(2 / 3\) of the initial volume depends on the initial liquid depth, \(\ell\), the cylinder diameter, \(D,\) the liquid viscosity, \(\mu,\) and the liquid specific weight, \(\gamma\). The data shown in the following table were obtained in the laboratory. For these tests \(\ell=45 \mathrm{mm}, D=67 \mathrm{mm},\) and \(\gamma=9.60 \mathrm{kN} / \mathrm{m}^{3}\) (a) Perform a dimensional analysis and based on the data given, determine if variables used for this problem appear to be correct. Explain how you arrived at your answer. (b) If possible, determine an equation relating the pouring time and viscosity for the cylinder and liquids used in these tests. If it is not possible, indicate what additional information is needed. $$\begin{array}{l|l|l|l|l|l} \mu\left(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right) & 11 & 17 & 39 & 61 & 107 \\\\\hline t(\mathrm{s}) & 15 & 23 & 53 & 83 & 145\end{array}$$

A liquid flows with a velocity \(V\) through a hole in the side of a large tank. Assume that \\[V=f(h, g, \rho, \sigma)\\] where \(h\) is the depth of fluid above the hole, \(g\) is the acceleration of gravity, \(\rho\) the fluid density, and \(\sigma\) the surface tension. The following data were obtained by changing \(h\) and measuring \(V,\) with a fluid having a density \(=10^{3} \mathrm{kg} / \mathrm{m}^{3}\) and surface tension \(=0.074 \mathrm{N} / \mathrm{m}\) \begin{tabular}{l|l|l|l|l|l} \(V(\mathrm{m} / \mathrm{s})\) & 3.13 & 4.43 & 5.42 & 6.25 & 7.00 \\ \hline\(h(\mathrm{m})\) & 0.50 & 1.00 & 1.50 & 2.00 & 2.50 \end{tabular} Plot these data by using appropriate dimensionless variables. Could any of the original variables have been omitted?

For a certain model study involving a 1: 5 scale model it is known that Froude number similarity must be maintained. The possibility of cavitation is also to be investigated, and it is assumed that the cavitation number must be the same for model and prototype. The prototype fluid is water at \(30^{\circ} \mathrm{C},\) and the model fluid is water at \(70^{\circ} \mathrm{C}\). If the prototype operates at an ambient pressure of \(101 \mathrm{kPa}\) (abs), what is the required ambient pressure for the model system?

The drag on a small, completely submerged solid body having a characteristic length of \(2.5 \mathrm{mm}\) and moving with a velocity of \(10 \mathrm{m} / \mathrm{s}\) through water is to be determined with the aid of a model. The length scale is to be \(50,\) which indicates that the model is to be larger than the prototype. Investigate the possibility of using either an unpressurized wind tunnel or a water tunnel for this study. Determine the required velocity in both the wind and water tunnels and the relationship between the model drag and the prototype drag for both systems. Would either type of test facility be suitable for this study?
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