A. The Continuity Equation
One reason I participated 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, the water will enter at a certain rate and leave at the same rate; if the quantity pushed in per unit time remains constant, an equal amount of water per unit time will be expelled. 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 that its flow is steady or unchanged with time. This reasoning is using the concept of conservation of mass, namely that 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? Let’s 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 centimeters 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 incompressible 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 (cm2) by 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
(figure available in print form)
where A is the cross sectional area at a point in the pipe, V is the average velocity of the water at he same point, and k is a constant, the rate of flow of the pipe, in our units cm
3
/sec. Since the equation is true for any two points in the pipe we could also write
(figure available in print form)
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 by thinking of a paper wrapper full 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. This is 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.
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? The ideas of geometric and dynamic similarity can be extended to three dimensional bodies, such as models of ships, of aircraft, of canals, or of river systems. We could do model testing.
Why would we want to do model testing? If we want to design a new ship, we could go ahead and build it and see what happens. This could be very expensive, especially if the design were poor or had some dangerous feature. Instead we can make scale models that are geometrically similar to the full size ship we will build. It is cheaper to run tests on the models since they are smaller and we can build more of them to test more features. We must have a way of transforming the measurements we take on the models to get the values we would have on the full size ship. To be able to transform the drag on a model, for example, to the drag on the full size ship, we need to have dynamic similarity.
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 prototype 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. Both lack of model testing and ignorance of model results have led to major loss 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,
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, 1940. 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 Aerodynamics to America
is a photograph of a model of the Tacoma Narrows Bridge in a wind tunnel at the University of Washington; the date, however, is 1941, too late to avoid the disaster. The picture does show the wind induced waves in the bridge deck. One reason to go to school is to learn from the experiences of those who can 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? Let us answer those questions. There are many similarity constants. Similarity constants are pure numbers, they are dimensionless. They are ratios of various kinds of forces. Three important ones are the Reynolds number, the Froude number, and the Mach 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. The following paragraphs should make this clearer.
The sources for the following are texts in Aerodynamics and Hydrodynamics, I make no claims to originality. Sources I found most helpful were von Kármán, Prandtl, Rouse, and Wegner, which are listed in the bibliography.
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 operated upon by another force. Some of the opposing force is due to the viscosity of the medium. 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 forces to viscous forces, which is called the Reynolds number, and the ration of inertial forces to gravitational forces, which is called the Froude number.
The word “constant” needs some comment. The numbers we are talking about have different values for different velocities, different media, and different prototypes. If we are doing testing, we must have the value of the number of the similarity parameter for the model situation equal to the value of the number for the full size situation. We must keep the numbers equal, keep them “constant.”
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 units are feet and the dimension is length. So the dimension length can be measured in many units: feet, inches, miles, meters,
etc
. In mechanics there are three basic dimensions: length, mass, and time, symbolized as [L], [M], and [T] respectively. 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], where 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 feet per sec per sec or 3 ft/sec
2
, the dimension of which would be [L/T2]. 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 rate of flow and density as mentioned in the previous section. Can you tell what is the dimension for area?
Before defining the Reynolds, Froude, and Mach numbers, we need to define some notation. The letter g stands for the acceleration due to gravity [L/T
2
]. It is a constant. Its value on earth is 9.8 m/sec
2
or 32 ft/sec
2
. That is the point of the story about Galileo and the leaning tower. The Greek letter (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 buoyancy 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 (nu) [L
2
/T]. Did you get the dimension for kinematic viscosity?
C. The Reynolds Number
This ratio known as the Reynolds number, Re, is
(figure available in print form)
where V is the velocity, L is a representative length such as the diameter of a pipe, the length of a ship, or the chord of an airfoil, and _ is the kinematic viscosity. The Reynolds number is named after Osborne Reynolds (1842-1912), a British engineer who did the first work on laminar versus turbulent flow in pipes.
What is the dimension of the Reynolds number? Substitute the dimensions for the defining variables of the number and reduce:
(figure available in print form)
The result is that the dimensions of the fraction cancel out, they reduce to one. The Reynolds number is said to be dimensionless, as would any other variable whose dimensions reduce to one.
Let us use the Reynolds number in an example. We want to test a 1/3 scale model of an automobile in a wind tunnel to determine its aerodynamic drag. How fast should the winds blow? The physical relationships are the same whether the car moves or the wind moves; it is easier to blow air against a fixed car model. Let the subscript m mean “of the model” and the subscript p mean “of the prototype.” If the Reynolds numbers of the model and the prototype are to be equal then
(figure available in print form)
Since the kinematic viscosity of the air is the same in the tunnel as on the highway we may cancel it. Substituting Lm = Lp/3 we get
(figure available in print form)
Solving for Vm, we get
(figure available in print form)
So the model velocity must be three times the velocity of the full size car. To model 55 mph the wind tunnel will be run at 165 mph. This is a reason to use large sized models when keeping the Reynolds numbers of the model and the prototype equal. Since the velocity is inversely proportional to its size, the smaller the model the faster the wind will have to be, perhaps too fast.
If the model were small enough we could be required to run the tunnel at supersonic speed. Even if we could (there are such tunnels) we would then have to worry about effects on the model that do not occur at subsonic speeds, such as the air being compressible and thermodynamic effects. The similarity constant for supersonic speeds is the Mach number. We will say more about the Mach number later. When there are forces that have significant effect on the model but negligible effect on the prototype, we have a scale effect. Working with larger scale models helps avoid scale effects.
D. The Froude Number
The Reynolds number is not the only similarity constant. After all, viscous forces are not the only forces that act on other systems. 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, William Froude. It is the ratio of the inertial forces to the gravitational forces.
William Froude (1810-1879), a nineteenth century English scientist, was one of the first to use a towing tank to test the designs of ships. Pictures of his apparatus can be found in Rouse and Ince,
History of Hydraulics
(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 of the same design 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.
(figure available in print form)
His experiments with the boards were to determine the proportionality constant (f), which he called the form factor, and the exponent (n) for velocity.
While experimenting with different size scale models he noticed that the wave patterns were the same in number along the hull when the ratios of speed to square root of length were the same. This leads to the dimensionless parameter
(figure available in print form)
where V is the velocity of the boat, g is the acceleration due to gravity, and L is the length of the boat. Fn is now called the Froude number, in his honor. Froude did his work before Osborne Reynolds, so he did not know about the Reynolds number.
Let us use dimensional analysis to show that the Froude number is dimensionless. Substitute the dimensions of the defining variables (look back) and simplify to get
(figure available in print form)
So the dimensions reduce to one and the Froude number is indeed dimensionless.
Froude’s program was to determine the frictional resistances from the wetted surface areas of the model and the ship, determine the residual resistance of the model by towing the model, scale the model’s residual resistance up to the full size ship, and add it to the ship’s frictional resistance to get the ship’s total resistance. The scaling up is where Froude’s number comes into the discussion. The residual resistances varied as the displacements when the Froude number for both the model and the prototype were equal. The displacement is the volume of water equal to the volume of the boat below its water line. The boat moves the water, it displaces the water.
If this theory is correct, it can be tested against a finished ship. That is what was done when the Admiralty made the battle ship
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
(figure available in print form)
If the model and the ship are both in the same kind of water (salt or fresh) the viscosity terms being equal divide out leaving
(figure available in print form)
and
(figure available in print form)
If the Froude numbers are equal
(figure available in print form)
Since g is a constant it may be canceled and
(figure available in print form)
Setting the two equations for Vm equal we get
(figure available in print form)
This equation says the only way the Reynolds and the Froude numbers can be equal is if the length of the model and the length of the ship are equal, that is, if the model is full sized. Does this mean that model testing is impossible? No, just that there are limitations to the modeling process; they are approximations in some aspects. The engineer needs to know what is being modeled and make corrections for the effects that are not being modeled.
Whether the Reynolds or the Froude numbers are kept equal depends on which effects are considered significant. Waves involve gravity, so if waves are involved the Froude numbers are kept equal. If we are submerged in a medium such as air for airplanes and cars or water for submarines and propellers, the Reynolds numbers are kept equal provided the speeds are moderate. If speeds are not moderate then we must consider the Mach number.
Froude leads us to the topic,
E. How to Test Model Ships
At the Sound School we try to integrate maritime topics into our curriculum, to use Long Island Sound as a theme for our teaching, hoping that students will be motivated to greater study of a topic if they see a use for it. Here is some history of naval architecture related to aerodynamics. Aerodynamics and hydrodynamics are both part of one field called fluid mechanics. Many of the principles of fluid mechanics were known before airplanes existed. These principles had been discovered in hydrodynamics. One apparatus of investigation was the towing tank.
Much of the hydrodynamics was motivated by the building and maintenance of canals. In fact Benjamin Franklin built a towing tank to test the observation he had made on the canals in Holland that boats go slower in shallower water. The tank was in fact a narrow wooden trough; the model was towed by a weight falling over a pulley at one end. An illustration of the apparatus is found in the Naval Historical Foundation booklet, The David Taylor Model Basin, A Brief History, which is my source for this section.
Even though Ben Franklin was a founder of the United States, the U. S. Navy did not get Congress to approve funding for a tank until 1896. The tank was built at the Washington Navy Yard under the supervision of Naval Constructor David Watson Taylor who directed it for the next fifteen years. Much significant work was done.
The claims of the advocates of model testing were substantiated early on. In 1902 two armored cruisers of 14,500 ton displacement were designed at the Model Basin. They were 820 tons heavier than similar predecessors but were able to cruise at 22 knots with less horsepower while consuming less fuel.
Taylor instituted the practice of using wooden models instead of wax models as used by other naval architects. This gave more accurate measurements, and avoided models melting in Washington, D.C. summers. It was much more expensive, however, $80 against fifty cents for wax that could be melted down and used again. He was responsible for the bulbous bow to dampen the bow wave, thus decreasing wave resistance. The bulbous bow is like a torpedo sticking forward from the bow just at or below the water line. This type of bow was first used on the USS
Delaware
in 1907 with great success. On field trips in New Haven harbor with Sound School students, we have observed the bow waves of freighters and tankers. The crest of the bow wave is in front of the bow, not at the bow. The bow does not “cut” the water. There is a trough, a depression in the water, at the bow.
Doing this project I had hoped to answer the question, “What does a towing tank measure?” Here is what I learned.
If we were to attach two models to the ends of a rod and tow the rod from a line attached to its center, what would happen? Most likely, one model would move forward and one would hold back. The model that held back would have more drag than the one that went forward. If the two models were equal in drag, the rod would be perpendicular to its tow line. We are back to Archimedes and a balance, the lever.
That is the idea of a towing tank. Instead of using two models, we put weights at one end of the rod to balance the force of the model. Furthermore, more balances are used because there are more forces to be measured. Our hypothetical example only measured the force in the direction of the towline. There are six motions for a boat, three linear and three rotational: surge, sway, heave, pitch, roll, and yaw. Surge is the linear motion fore and aft. Sway is linear motion in a sideways direction. Heave is the linear motion in a vertical direction. Pitch is rotational motion in which the bow goes down and the stern goes up, or vice versa. Roll is the sideways rotation. Yaw is rotational motion about the vertical axis.
So a system is designed to suspend the model with linkages for the six motions. Since the system is going to move, the balances will need to be dampened so the motion of the apparatus does not disturb them.
F. The Mach Number
The Froude number is of concern when we have gravity waves. When we deal with an elastic medium we have pressure waves. Sound is a pressure wave.
The ratio known as the Mach number is
(figure available in print form)
where V is the velocity of flow and a is the speed of sound in the medium. The Mach number, named after the Austrian physicist Ernst Mach (1838-1916), is the square root of the ratio of inertial forces to elastic forces. The Mach number is of concern when the medium is not considered incompressible, as when we assume that at high speed an object’s motion will compress the air.
The Mach number tells us when we have supersonic speed. If something is going faster than the speed of sound we have a Mach number greater than one. The object is moving faster than the sound it makes, so it arrives before it is heard. The speed of sound in air is approximately 1100 feet per second while in water it is approximately 4700 feet per second.
Supersonic motion appeared prior to the recent invention of high speed aircraft. Bullets and artillery shell can move at supersonic speed. In fact Benjamin Robbins (1707-1751), the inventor of the ballistic pendulum, found that increasing his powder charge increased his range, but only so far; at some point it became inefficient. His projectiles were approaching the speed of sound and the drag was increasing significantly.
