# An Introduction to Aerodynamics

## The Continuity Equation, the Reynolds Number, the Froude Number

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## I. Physics

### A. The Continuity Equation

One reason to participate in the Aerodynamics Seminar was to find examples using high school mathematics that I could share with my students. The continuity equation is such an example. When trying to solve a problem we are told to look for things that stay the same. They then can be set equal to each other giving an equation to solve. Rules like that are easy to state, but examples are easier to understand.If we have water flowing through a pipe filled with water, the water will enter at a certain rate and leave at the same rate. The quantity pushed in per unit time will push out an equal amount. That seems like an obvious and reasonable observation. What are we taking for granted in the argument? We are assuming that water is incompressible, and its flow is steady it does not speed up or slow down during the discussion. This reasoning is using the concept of conservation of mass, namely in nature mass cannot be gained or lost in a system. A simple, characteristic example of scientific reasoning. So what can we do with it? Lets write a formula.

How would you express the rate of flow? If it were a bilge pump you would say so many gallons per hour. What do gallons measure? Volume. Volume is expressed as cubic units, cubic feet, cubic centimeters and so forth. Time can easily be changed from hours to minutes or seconds. So the rate of flow could be expressed as cubic centimeters per second. Cubic cenimeters per second could also be expressed as square centimeters times centimeters per second. What do square centimeters and centimeters per second each measure? Area and velocity, respectively. So the rate of flow could be expressed as volume per time equals area times velocity. The velocity would be an average velocity since we are using the total volume per unit time. The water in the center of the pipe goes faster than the water at the pipe wall. Can you tell what area and what velocity to use? The pipes have cross sectional areas and the water has an average velocity. So we could use the area of the entry or the exit opening. Does the argument make sense if we start with the area of the pipe and the average velocity of the water? If we multiply the cross sectional area of the pipe (cm
^{
2
}
) times the velocity of the water (cm / sec.) what do we get? Cubic centimeters per second, volume per unit time, a rate of flow. It makes sense both ways. Does it matter what cross sections we use? If our principle that the quantity of matter flowing in is the same as the amount flowing out then it must also be true everywhere in the pipe. Let us write this as an equation

AV = k where A is the cross sectional area at a point in the pipe, V is the average velocity of the water at the same point and k is a constant, the rate of flow of the pipe, in our units cm/sec. Since the equation is true for any two points in the pipe we could also write A1V1 = A2V2 where A1 and V1 are the area and velocity at one point in the pipe, and A2 and V2 are the area and velocity at some other second point in the pipe. This equation is called the continuity equation. The way I visualize the continuity equation is to think of a paper wrapper of pennies. Instead of closing it, fill it full to the ends so no more will fit in. Now try pushing three more pennies in at one end. What happens? Three pennies are pushed out the other end.

We have just shown that the velocity of water at some place in a full pipe is inversely proportional to the cross sectional area of the pipe at that point. That means when the pipe narrows down the velocity of the water goes up. Something you probably noticed when you put your thumb over your garden hose.

When fundamental principles are stated one should consider them for some time. Ask what the ramifications of the principle are, what else can be proven by it? Think on it long enough so you too can see that it is “self-evident”. After all, how often do you go around repeating the self-evident? If it is self-evident your listeners can figure it out for themselves. Let me share an experience where I was tripped up by this self-evident principle.

In the article “Kitchen-Sink Aerodynamics” in the book
*
The Amateur Scientist
*
by C. L. Stong the construction and use of a Hele-Shaw apparatus is explained. The apparatus may be described as a flow table. A glass plate over which a thin film of water is allowed to flow. Crystals of potasium permanganate are evenly spaced at one end to disolve setting up parallel flow lines over the table. The model, an airfoil cross section for example, is placed in the flow. We are told to put a large sheet of graph paper under the glass to make measuring the distances between the streamlines easier. We are told that the velocity of the flow is inversely proportional to the distance between the lines. I asked myself how could that be claimed? The continuity equation tells us so. The streamlines will not cross each other so when the distance between the model and the edge narrows down the flow, the streamlines will try to maintain even spacing. The distance between the lines is another way to measure the cross sectional area of the “pipe”.

In scientific reasoning we should always be on guard not to apply an argument for one case to a different situation. We talked about pipes, Hele-Shaw is open to the air, it is a chanel. There are different rules for flows in pipes and flows in open chanels. The article said that thin films of water gave the same results as water flowing between two closely spaced plates.

### B. Dynamic Similarity

In High School Geometry we study the concept of similarity. If two geometric figures are of the same type with corresponding angles congruent and corresponding sides proportional then the figures are said to be similar. In physics if the points in corresponding positions at corresponding times have proportional velocities and proportional accelerations then the systems are said to be dynamically similar. What could the corresponding points be? Models of ships or of canals or of river systems and their corresponding full-size prototypes. We could do model testing.Corresponding time should be explained. If the problem being modeled is cyclical in nature, such as the revolutions of propellers, the time for the model will be the fraction (number of prototype rpm over the number of model rpm) of the real time. If there are no cyclical features, find the times for the systems to trace out similar curves. The ratio of those times is the scale factor for time. If we can model time it is possible to tell how long a machine will last. If we run its model at 20 model cycles to one proptype cycle, the model will wear out in one twentieth the time the prototype will take to wear out.

Model testing is useful, it may save lives. Lack of model testing or ignorance of model results has led to major lose of life and property. Although there was no loss of life, one famous example is the Tacoma Narrows Bridge. It twisted itself apart, the wind pushed it one way, and each time the bridge reacted the wind continued to push it in the direction of reaction, making the effect bigger. This is known as resonance, the amplification of the effect. Mario Salvadori in his book
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Why Buildings
*
*
Stand Up
*
reports that on May 17, 1854, the Wheeling Bridge over the Ohio River collapsed in a wind storm in the same way as the Tacoma Narrows Bridge did on November 7, 1840. John Roebling who designed the Brooklyn Bridge knew of the failure and designed his bridges with diagonal stays so as to prevent the twist. Figure 10 in Paul A. Hanle’s
*
Bringing
*
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Aerodynamics to America
*
is a photograph of a model of the Tacoma Narrows Bridge in a wind tunnel at the University of Washington, however, the date is 1841, too late. The picture does show the wind induced hump. One reason to go to school is to learn from the experiences of those who came before us. Some models may be full size examples that are improved upon in the next version of the design.

How do we keep models dynamically similar? By running the model at the same value of the similarity constant as the prototype. What does that mean? What are similarity constants? There are many similarity constants. Similarity constants are pure numbers, they are dimensionless. Two very important ones are the Reynolds number and the Froude number. So if you want your model to be representative run it at the same Reynolds number or the same Froude number as the prototype.

There are various kinds of forces. There is the force of gravity which attracts things to the center of the earth. There are inertial forces. Inertia is the property of bodies in motion to stay in motion until opperated upon by an opposing force. One of the opposing forces is viscosity. When a bead is dropped into a jar of honey it does not fall as fast as it would in a jar of water. We say the honey is more viscous than water. We could form ratios of these forces in various ways. Two of the ratios are the ratio of inertial force to viscous force which is called the Reynolds number and the ratio of inertial force to gravitational force which is called the Froude number.

Dimensionless variables are a significant topic. Let us first learn what are units and what are dimensions. When we say a board is six feet long, the unit is feet and the dimension is length. So the dimension length can be measured in many units, feet, inches, miles, meters,
*
etc
*
. There are three basic dimensions: length, mass and time, symbolized as [L], [M] and [T] repectively. These three are combined algebraically to make the dimensions of other physical variables. For example, velocity has units such as feet per second so its dimension is [L/T], feet are units for length, per means division and seconds are units for time. Acceleration is how fast the speed changes, for example if the speed increased 3 feet per second every second we could say the acceleration was 3 ft. per sec. per sec. or 3 ft / sec
^{
2
}
whose dimension would be [L/T
^{
2
}
]. In Newtonian physics force is mass times acceleration so the dimension for force would be [ML/T
^{
2
}
]. So the dimensions are multiplied and divided as in Algebra I. When a new physical variable is presented the dimension may also be stated along with it. See if you can determine the dimensions for flow and density as mentioned in the continuity equation. Can you tell what is the dimension for area?

Before defining the Reynolds and Froude numbers we need some notation. The letter g stands for the acceleration due to gravity [L/T
^{
2
}
]. It is a constant. That is the point of the story about Galileo and the leaning tower. The Greek letter r (rho) stands for the density of the material under discussion. It is the mass of a unit volume of the material [M/L
^{
3
}
]. Density is a property we use with bouyancy to determine the water line of a ship. The Greek letter µ (mu) stands for the dynamic viscosity of the medium [M/(LT)]. The dynamic viscosity of a medium divided by its density is called its kinematic viscosity symbolized by the Greek letter u (nu)[L
^{
2
}
/T]. Did you get the dimension for kinematic viscosity?

### C. The Reynolds Number

The ratio known as the Reynolds number, Re is*(figure available in print form)*

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### D. The Froude Number

The Reynolds number is not the only similarity constant. After all viscosity is not the only force on a system. When the inertial and gravitational forces predominate then the similarity constant of concern is the Froude number. This number is also named after its discoverer. It is the ratio of the inertial forces to the gravitational forces. William Froude, a nineteenth century English scientist was one of the first to use a towing tank. Pictures of his apparatus can be found in Rouse and Ince,*History of Hydaulics*, see the bibliography. He wanted to discover the relationship between the forces on a model and the full size ship. By towing variously sized models and comparing them to each other, he found that there was no proportional relationship between the resistance of similar models and their sizes. He decided that the resistance was the sum of a frictional force and a residual force. His technique was to determine the frictional force, subtract it from the total force and get the residual force which he said was the wave making force. He determined the frictional resistance by towing boards of various lengths and finishes under water without making waves. He assumed that the frictional resistance (Rf) was proportional to the wetted surface area of the ship (S) and a power of the velocity (V) of the ship.

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If the theory makes sense it can be tested by using a finished ship. That is what was done. The Admiralty made the
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H.M.S. Greyhound
*
available, Froude took the measurements for a model and for the full size ship, the results matched.

If viscous and inertial forces are to be similar the Reynolds number of the model and the prototype must be equal. If the inertial forces and the gravitational forces are to be similar then the Froude number of the model and the prototype must be the same. Is it possible to have both numbers equal at the same time?

Let the subscript m mean “of the model” and the subscript p mean “of the prototype”. If the Reynolds numbers are equal then

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