# How Those “Award-Winning” Lab Videos Came into Being

Last Thursday evening the College of Engineering of the University of Tennessee at Chattanooga presented me with a couple of awards.

I find myself disparaging some of my activities as “award-losing,” and they certainly are, but this time is different, and some explanation is in order.

In presenting this award, my department head, Dr. James C. Newman III, specifically noted that it was given because of the series of Fluid Mechanics Laboratory videos which I produced in response to the COVID crisis. He commented that the quality of these were “professional” (not quite) and perhaps I paid to have them produced (which I certainly did not.)

Coming to produce videos of any kind (along with other media projects, such as books and websites) was a long process which involves many experiences outside of the engineering and technical career. The primary “school” for learning this kind of thing was my years working in the Church of God Lay Ministries department. In that capacity I first was introduced to web design. This site’s “ancestor” was started on Geocities; the suggestion for same came from a local church men’s leader in Wisconsin.

I did that for many of my activities: the church, myself and Pile Buck, for whom I was now a consultant and a writer. The problem with video, however, was that to produce it took a great deal of both hardware and software that was scarce and expensive in the late 1990’s and early 2000’s. A substitute for that were the musically synchronised PowerPoint presentations I did for the Church of God Chaplains Commission. The first (and IMHO the best) of these was the one I did in the wake of 9/11; you can see that below:

From there my work for the church and for Pile Buck took on a symbiotic nature, especially with the next stage: print layout and production. That was the genesis for my print and (then) CD projects, which underwrite the basic expenses of these sites.

In 2008 I started my YouTube channel. I would be the first to admit that the effort has been an off again, on again business. I dabbled in video production and using video software for a long time. In 2018, however, I began posting albums from my Music Pages effort on YouTube. Since these people had cleverly discontinued the ability to post audio files, that meant that I had to wrap them in a video file for upload. To make this happen I turned to the kdenlive software on Linux. Getting these albums ready for YouTube wasn’t very challenging but it got me used to using the timeline and organising the content.

When COVID hit in 2020 we had to go online over Spring Break. The way Fluid Mechanics Laboratory is done, it’s more than just watching the computer taking data; it requires the student to actually look at something and write (or type) something down. This makes it easier to run the experiment in front of a camera and put it together.

Things got more complicated when it became obvious that in Summer 2020 the lab would have to be completely virtual. So in May and early June 2020 I went into the lab and made my videos. It’s hard to explain how I organised my videos, put together a timeline (usually in my head) recorded and then assembled them. It’s one of those things where God had been preparing me for this and, when it happened, it clicked. The virtual lab worked.

The utility of the videos has continued because I still make them watch the initial lecture on video (the lab is an acoustical disaster, something one of my students pointed out a long time ago and which became evident recording the earliest videos.) After that they can watch the video of the experiment to get an idea of what to do, one which is better than just reading the verbiage of a lab procedure. The videos can still be used to “conduct” the experiment virtually. This remains important because COVID has turned academic life into a revolving door, with faculty, staff and students going in and out of quarantine and requiring a virtual environment.

I went on to make videos for my lecture classes in Soil Mechanics and Foundations, but talking head videos are much easier to make than the lab ones. I think we rely too much on talking heads to fill air time (and I mean that in every sense.)

I am grateful to Dr. Newman, Dean Daniel Pack and the College of Engineering and Computer Science for giving me the opportunity to teach in this way and for having recognised its benefits. As I say at the end of all of my videos, “Thanks for watching, and God bless.”

# Chet’s Propellers: How Did They Work?

All of Chet’s planes–and those of the people around him–were propeller-driven aircraft. With the advent of jet aircraft after World War II, it seemed that propeller-driven craft were relegated to private use or “puddle jumpers” for commercial aviation. But the advent of drone technology has put propellers back into the spotlight, although, like the auto-gyros of Chet’s day and the helicopters that succeeded them, it applies them in a different way. But the principles are the same. (In reverse, it’s the same technology that’s used with wind turbines.)

## How Aircraft Stay Aloft

Let’s start by looking at the diagram on the right, the top panel. It’s really a matter of static and/or dynamic equilibrium: as the propeller or jet engine produces thrust, the plane moves forward, inducing upward lift in the wings to overcome the downward force of gravity. The thrust also overcomes the drag which is inevitable with bodies moving through the air. By definition the thrust must be greater than the drag for forward motion of the aircraft, and the lift must be equal to the force of gravity to maintain level flight (that relationship changes as the aircraft pitches, rolls and yaws.)

Concerning the pitch, if we look at the second panel, as the aircraft pitches upward, the component of the gravity in the direction of thrust and drag increases. The lift does not necessarily increase, so one would expect the aircraft to slow down with a constant thrust as it ascends, which in fact they do. The opposite effect is shown in the third panel, which shows a descending aircraft.

The last panel links conventional propeller-driven aircraft of Chet’s day and now with drones, as it shows an extreme upward pitch. With a drone the angle between the horizontal and the thrust becomes 90 degrees or nearly so. When the drone is stationary the drag is zero as the velocity is zero; for the drone to maintain a constant altitude the thrust and gravity must be equal. To move the drone up and down the thrust can be varied. Additional sources of thrust must be used to move the drone from side to side.

For more information on the airfoils that produce lift and related topics, please see our monograph Wind Tunnel Testing.

## The Basics of Propellers

Propellers antedate aircraft; they first find themselves on marine craft, something which Chet, his ancestors and descendants were well familiar with. It is the reason why someone like Froude, who has been mentioned elsewhere, developed the momentum theory for propeller operation, which we will discuss later. An example of a propeller on a ship, in this case a steam yacht, is shown below.

In any case some of the basic terminology of aircraft propellers is shown below.

Note that, as the sections are taken further out from the centre, the blade’s orientation changes. This is to maintain a constant angle of attack from centre to edge of the blades due to the linear increase of the speed as the distance increases from the centre. This innovation was done by the Wright brothers and was crucial in their successful pioneering flights.

Basically propellers act as a screw in the fluid they’re immersed in, driving the ship or plane forward in a similar fashion as a screw does in a solid. (I can remember that, during my yachting years, the ships’ propellers were referred to as “screws.”) If the fluid is incompressible and everything is “efficient,” then the screw will propel the ship or plane like its solid counterpart. Unfortunately that seldom happens, so there is slip, and the actual travel per revolution of the propeller is less than ideal.

The interaction between the advancing propeller and the air it travels through is shown at the right. A propeller is basically an airfoil, as should be evident by looking at these diagrams. Airfoils, as discussed in Wind Tunnel Testing, generate lift by passing the air over them, generating differential pressure between the upper surface and the lower surface. Drag is an inevitable result of a body passing through a viscous fluid (and if the airfoil is not symmetric about the chord, an inviscid one too.)

Let’s start by assuming that the propeller profile is not symmetric (and most are this way.) If there is no slip, the angle of attack between the propeller blade and the airflow produced by the propeller is zero, and the blade and effective helix angles are the same.

As slip increases, the angle of attack increases, which means that we have to watch for stall in the design of the propeller. When the craft isn’t moving forward at all, the blade angle and angle of attack are the same.

In Chet’s day, the way you got to the last state was to put chocks under the wheels and run the engine and propeller, as you can see below. Now producing thrust against a motionless craft is part of the design objective. It’s not the most efficient use of propeller technology, but it’s useful and it works, especially now with the advances in electric motors and batteries and control systems. It’s a good example of how applications of technology aren’t always the most efficient but produce desirable results.

## Momentum Theory and Propeller Performance

With the preliminaries out of the way, we’ll get to the heart of the matter: how do we estimate the performance of propellers? The “classic” way to do this is momentum theory, which was first derived by Froude. There are other theories out there (blade-element theory comes to mind) but momentum theory, crude as it is, is a good place to start. The derivation we present is a standard derivation; this presentation is based on Streeter with some modification.

Let us start with presenting a diagram of the system and its control volume, shown below.

We’ll also make a few assumptions:

• The fluid is inviscid, so Bernoulli’s Equation can be used.
• The fluid is incompressible. This may seem unbelievable with air, but the pressure differences–and by extension the density differences–make this assumption possible. (After all, we did say this theory was crude…)
• The fluid velocity is uniform at any x-axis point in the control volume, similar to the assumption we make with piping
• As it passes across the propeller, the fluid’s pressure changes but its velocity does not.
• The shape of the control volume, but we’ll justify that in a minute

That being the case, there are two ways we can analyse this. The first is to use momentum theory of fluids. A detained discussion of this is beyond the scope of this presentation, but it can be shown that the net force (thrust) exerted by the control volume on the surrounding fluid is

$F = \dot{m} (V_4 - V_1) = \rho VA (V_4 - V_1)$ (1)

since

$\dot{m} = \rho Q$ (2)

and

$Q = VA$ (3)

The nomenclature is at the end.

Another way is to note that the thrust is, by static equilibrium, equal to the differential pressure across the propeller times the area the propeller sweeps during rotation, or

$F = A (p_3 - p_2)$ (4)

we can equate Equations (1) and (4) yield

$\rho V (V_4 - V_1) = p_3 - p_2$ (5)

To arrive at values for the two pressures, we turn to Bernoulli’s Equation, which we have discussed elsewhere. Dropping the gravity term out entirely from our considerations, from point 1 to 2 we have

$p_1 + \frac{\rho V_1^2}{2} = p_2 + \frac{\rho V^2}{2}$ (6)

and from point 3 to 4,

$p_3 + \frac{\rho V^2}{2} = p_4 + \frac{\rho V_4^2}{2}$ (7)

Combining the two, and assuming that both $p_1$ and $p_4$ are the same (ambient pressure,) we have

$p_3 - p_2 = \frac{\rho (V_4^2 - V_1^2)}{2}$ (8)

Equating the left hand side of Equation (5) with the right hand side of Equation (8) and rearranging, we have at last

$V = \frac{V_4+V_1}{2}$ (9)

From this the fluid velocity at the propeller is the average of the velocities a the boundary of the control volume. This is a nice result; some professors will confidently assert that it is “intuitively obvious,” but we’ll try to spare you from such conceit.

## Power and Energy Losses

The useful work (power) done by the propeller in moving the aircraft forward is given by the equation

$N = FV_1 = \rho Q (V_4 - V_1) V_1$ (10)

Obviously all of the energy put into rotating the propeller does not end up moving the aircraft forward; the total power input is given by the equation

$N_{tot} = \rho Q (V_4 - V_1) V_1 + \frac{\rho*Q*(V_4 - V_1)^2}{2}$ (11)

The efficiency, i.e., the ratio of the useful power to the total power input, is thus

$\eta_i = \frac{2*V_1}{V_4 + V_1}$ (12)

## Reducing the Control Volume

Up until now we’ve assumed a control volume which goes from Points 1 to 4 in the diagram above. We do this because it’s usually easier to measure the velocities $V_1$ and $V_4$. What would happen if we attempted to measure the velocity at the propeller (or as near to it as possible) between Points 2 and 3? We start by rewriting Equation (1) as

$F = \dot{m} V\times V = \rho \pi R^2 V^2$ (13)

The usefulness of this will become apparent.

## Test Stand Case and Calculations

We start with a test stand, shown below.

At this point we’ll use the traverse to record a series of velocity readings across the output of the propeller. There is more than one way to do this; the method we’ll employ divides up the area facing the propeller into annuli as follows:

Let’s use the centimetres as an index. With apologies to my fellow FORTRAN programmers, that makes the centre index zero. Using the tall grid marks as the points where we take our data, the area of the annulus around the data point is given by the equation

$A_n = \pi (R_{n+1}^2 - R_{n-1}^2)$ (14)

The only exception to this is the centre circle, whose area is

$A_0 = \pi R_1^2$ (15)

Thus, for each circle/annulus, Equation (13) becomes

$F_n = \rho A_n V^2$ (16)

The areas are obviously in square centimetres and unfortunately will have to be converted to square meters to use.

Now consider the sample case. The recorded positions of the Pitot-static tube and the velocity are given as follows:

At this point we need to do three things: a) average the pairs of air speeds for the radii, b) compute the areas of the annuli, and c) compute the thrust on the annulus using Equation (13). This is done in the table below

The thrusts are then summed to 1.18 N, which is within 9% of the instrumented total.

## Acknowledgements

I would like to thank Dr. Kidambi Sreenivas and Mr. Jacob Jenkins of the University of Tennessee at Chattanooga, without whose efforts this experiment would not have come into being.

## Nomenclature

• $A = area,\,m^2$
• $D = diameter,\,m$
• $F = thrust,\,N$
• $\dot{m} = mass\,flow\,rate,\,\frac{kg}{sec}$
• $N = power,\,W$
• $Q = volume\,flow\,rate,\,\frac{m^3}{sec}$
• $R = radius\,at\,propeller,\,m$
• $V = velocity,\,\frac{m}{sec}$
• $\rho = density,\,\frac{kg}{m^3}$

## References

• Elger, D.F., LeBret, B.A., Crowe, C.T. and Roberson, J.A. (2016) Engineering Fluid Mechanics. Eleventh Edition. Hoboken, NJ: John Wiley & Sons Inc.
• Hibbler, R.C. (2018) Fluid Mechanics. Second Edition. New York: Pearson.
• Lusk, H.F. (1940) General Aeronautics. New York: Ronald Press Company
• Jordanoff, A. (1936) Your Wings. New York: Funk and Wagnalls Company
• Streeter, V.L. (1966) Fluid Mechanics. Fourth Edition. New York: McGraw-Hill Book Company
• Vennard, J.K. 1940) Elementary Fluid Mechanics. New York: John Wiley & Sons Inc.
• Wood, K.D. (1935) Technical Aerodynamics. New York: McGraw-Hill Book Company

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