Book cover for Fluid Mechanics

Fluid Mechanics

Frank M. White

ISBN #9789385965494

8th Edition

1,418 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Dimensional analysis is an essential tool in fluid mechanics that uses the principle of dimensional homogeneity to convert complex physical relations into simpler, dimensionless forms. By applying the Buckingham Pi theorem and related techniques, engineers can identify key parameters such as Reynolds, Froude, Mach, Weber, and Strouhal numbers. These dimensionless groups not only simplify the mathematical analysis but also provide a powerful method for scaling model data to predict prototype behavior in applications ranging from duct flow and pipe friction to wind tunnel testing and hydraulic model experiments. Maintaining proper similarity conditions is critical for the success of model testing and the accurate translation of experimental results to full-scale designs.

Learning Objectives

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Key Concepts

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Example Problems

Example 1

For axial flow through a circular tube, the Reynolds number for transition to turbulence is approximately 2300 [see Eq. $(6.2)]$, based on the diameter and average velocity. If $d=5 \mathrm{cm}$ and the fluid is kerosene at $20^{\circ} \mathrm{C}$, find the volume flow rate in $\mathrm{m}^{3} / \mathrm{h}$ that causes transition.

Example 2

A prototype automobile is designed for cold weather in Denver, $\mathrm{CO}\left(-10^{\circ} \mathrm{C}, 83 \mathrm{kPa}\right)$. Its drag force is to be tested on a one-seventh-scale model in a wind tunnel at $150 \mathrm{mi} / \mathrm{h}, 20^{\circ} \mathrm{C},$ and 1 atm. If the model and prototype are to satisfy dynamic similarity, what prototype velocity, in $\mathrm{mi} / \mathrm{h},$ needs to be matched? Comment on your result.

Example 3

The transfer of energy by viscous dissipation is dependent upon viscosity $\mu,$ thermal conductivity $k,$ stream velocity $U$ and stream temperature $T_{0} .$ Group these quantities, if possible, into the dimensionless Brinkman number, which is proportional to $\mu$

Example 4

When tested in water at $20^{\circ} \mathrm{C}$ flowing at $2 \mathrm{m} / \mathrm{s},$ an $8-\mathrm{cm}-$ diameter sphere has a measured drag of 5 N. What will be the velocity and drag force on a 1.5 -m-diameter weather balloon moored in sea-level standard air under dynamically similar conditions?

Example 5

An automobile has characteristic length and area of 8 ft and $60 \mathrm{ft}^{2},$ respectively. When tested in sea-level standard air, it has the following measured drag force versus speed: $$\begin{array}{l|l|l|r} \mathrm{V}, \mathrm{mi} / \mathrm{h} & 20 & 40 & 60 \\ \hline \text { Drag, lbf } & 31 & 115 & 249 \end{array}$$ The same car travels in Colorado at $65 \mathrm{mi} / \mathrm{h}$ at an altitude of $3500 \mathrm{m}$. Using dimensional analysis, estimate $(a)$ its drag force and $(b)$ the horsepower required to overcome air drag.
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