**Begin your turn no later than the CDI dot corresponding to one-half of the angle to intercept divided by 3 and divided by the time to pass from one CDI dot to the next.**

This solution works for turning inbound & outbound for tracking a VOR radial, a LOC/ILS, or any situation where you can count intervals leading up to a desired track.

You are on a 45 degree intercept. The needle is alive and you count 4 seconds between CDI dots.

The mental math is: *45 divided by 3 is 15….. 15 divided by 4 is about 4…. Half of 4 is 2. Turn no later than 2 dots.*

The pilot needs a quicker and more intuitive solution. We can simplify this engineering-style intercept equation with a few steps.

Thus the pilot solution in words is: **Begin your turn no later than the CDI dot corresponding to one-half of the angle to intercept divided by 3 and divided by the time to pass from one CDI dot to the next**.

Full disclosure: I did not initially develop this intercept solution based on the mathematics. Rather, I experimentally flew many intercepts and developed the following rule of thumb. By a complete stroke of luck, the experimental solution happens to be exactly identical to the mathematical solution above.

This solution is only meant for intuition and the mathematical understanding of intercepts. You **must **follow ATC, the appropriate regulations and standard operating procedures when flying. Find and train with an instructor, preferably with a CFII. No exceptions!

**Requirements**: You need an indicated airspeed, a GPS with track and groundspeed readouts, a thermometer, and your altitude. You will need to fly three independent headings (approximately 120 degrees apart) for each data point. You will need to download and enter your data into the windclover program.

**Non-requirements**: You do NOT need any ground references. You do NOT need to know your precise heading or magnetic variation. You do NOT need accurate timing or any clock. You do NOT need to calculate your true airspeed. You do NOT need an aerospace engineering background or on-board flight test engineers or hardware.

**Cloverleaf Flight Profile:** You will need to fly three lines that are approximately 120 degrees apart (e.g. 100, 220, 320). Maintain a constant heading, altitude, and indicated airspeed. Using your GPS, record your ground speed and track. Enter these data values into heading columns #1, #2, and #3. The program determines the wind direction/speed and the calibration from KIAS to KCAS.

**Theory**: The theory and algorithm are summarized below as developed in a flight test engineering course’s class notes. It is important to note that there is a background routine for calculating true airspeed from calibrated airspeed and the atmospheric conditions. Also note that this theory is not necessary to use the above tool. For more information, contact Dr. O’Neill.

This is an engineering structures parody of the X Files from 1999 found in my class notes. April Fools… once every 20 years is about right for this engineering joke.

So why is this related to structural engineering? Well, there is a shortcut method using so-called singularity functions to calculate the moment and shear in beams. Refer to any classical engineering textbook. As my initial undergraduate instructor in a structural analysis course, Dr. Wolf Yeigh, would say, *it’s a good tool to have in your pocket*. FYI, that was an intense course and professor, but one that I’m really lucky and glad to have taken. And yes, I made an A.

Using a Turn-Twist strategy, once on the arc, the heading to fly is tangent to the arc. This makes the no-wind control law: Turn to heading = Radial plus 90 deg when CW or Radial minus 90 deg when CCW. Unfortunately, this control strategy contains inherent divergence; in other words, the aircraft always tracks outside the desired arc (Figure 2, left). With a Turn 10/Twist 10 step, the cross track error is 1.5%. For example, a 20 nm arc with 10 degree radial steps, would give 0.3 nm error every step.

*Is there a correction to exactly remain on the arc given a Turn-Twist step?* Yes, and amazingly enough, the result is exact and a trigonometric identity. The right portion of Figure 2 derives a correction angle (gamma) such that the exact track is from point A to point B, both on the same arc. The result is that exactly half the Turn/Twist angle is applied inside the normal +-90 heading.

For example, using a Turn 10/Twist 10 in a counter clockwise direction at the R-040 would require the heading be 305 degrees; this heading will precisely keep you on the exact DME arc at the R-030 radial.

Warning: The normal flying caveats apply: 1) This is only meant for insight and is not meant as instruction or as a change to your specific flight operations manual, 2) wind will require varying correction angles, and 3) aviate, navigate, communicate.

]]>As the (former) instructor of a course in PDEs, I reviewed classical solution techniques in a lecture titled A brief history of GES 554 PDE to prepare students for their final exam. This lecture makes an excellent refresher or rapid introduction.

If you want to review the entire 50 lecture course, visit here. Feel free to call it *The Brief History of the World of PDEs in 50 Parts*.

Topics covered are:

- Motivation, classification & canonical forms
- Diffusion, Elliptic, Hyperbolic, and Transport PDEs
- Solution methods: Series, Separation of variables, Monte Carlo, finite difference, Ritz / Galerkin and Transforms
- 1 page PDE toolbox
- Laplace vs Fourier transforms for PDEs
- Sturm Liouville Theory
- Wave Equations
- Strings, Beams, and Drums
- Characteristics in transport equations
- Systems of PDEs: eigenvalues & eigenvectors
- Green’s Functions
- Calculus of Variations for PDEs

- Nearly empty light-aircraft tanks in extreme hot and humid environments with extreme temperature swings theoretically could condense approximately a couple of fluid ounces a week.
- The generation rate linearly scales with empty tank volume and humidity, but exponentially with temperature.
- Normally vented tanks substantially reduce the water influx rate, but do have a breathing mode that can pump moist air during temperature and pressure swings.
- Condensation is more likely to be a long term storage threat; Large volumes of water are more likely to be ingress of liquid water.

Water is a key enabler of life and dramatically affects the behavior of air. We call “dry air” the mixture of mostly nitrogen (80%), oxygen (20%), and trace other constituents (Ar, carbon dioxide, etc). “Wet air” is what we normally encounter and is dry air + water. “Air” could also include particles + bugs + dirt. You can learn more at my course notes here for an introduction and here for non-standard atmospheres.

The important takeaways are:

- Adding water
**decreases**air density since water has a lower molecular mass than air. (Technical note: Water has 2 Hydrogen of mass 1 plus 1 Oxygen of mass 16 for a total of 18. Air on the other hand has 80% diatomic Nitrogen of mass 28 and 20% diatomic Oxygen at mass 32 for a total of 28.97.).

- Increasing temperature substantially
**increases**the absolute water carrying capacity of air. Water vapor at 100% relative humidity consists of 0.7% of the wet air mass at 50 F and 6% at 110 F, but a whopping 15% at 140 F. This is why pilots need to be much more concerned with high humidity at high temperatures and not so much at lower temperatures (This is in addition to the temperature effects on density altitude).

The saturation pressure (Ps) in Figure 2 is generated from the Arden-Buck approximation as an exponential function of temperature. This also indicates the pressure at which boiling occurs (e.g. 212 F at sea level pressures of 14.69 psi; and 200 F at 10000 ft with a pressure of approximately 10 psi).

Let’s pick a scenario where a nearly empty 25 gallon tank completely condenses the water vapor. Plus there is a complete air exchange/recharge of hot and humid (120 F and 100% relative humidity) air once per day. How much water is generated?

The answer is about 0.25 oz (7.5 ml) per day. Consider it one cupped-hand of water, or about a 2.5″ spot of water, or 0.75 seconds of fuel at 10 gal/hr. That’s enough to fully grab your attention.

The rate scales linearly with the tank’s air volume so keeping the tank 90% full reduces the generation rate by 90%. Being in 30% humidity air reduces the rate by 70%. Doubling the elapsed days doubles the generated volume. Notice that the rate exponentially scales with temperature (ps/T is exponential).

Critically, we can bound the generation rate. For example, at 120F and 100% humidity with a 20 gallon tank, the rate should be less than 0.20 oz per day.

If you are generating more than this, chances are extremely likely that there is another mechanism responsible. Go find it.

To be continued

To be continued

]]>The dB decibel scale can often be very intimidating to others, so here’s a quick way to simplify (i.e. no logs or powers) your explanation to two steps. The fundamental point to make is that a Bell is how many zeros. **A decibel is the number of zeros multiplied by 10.**

Let’s convert a ratio to dB. Pick 100. This number has a number 1 followed by two zeros before the decimal point.

**How many zeros?**“2”.**Multiply by 10.**“20”**Say that number**. “20 dB”

Let’s reverse the process and convert dB to a ratio. Pick 40 dB.

**Divide by 10.**“4”**Four zeros before the decimal place is:**“10000… ten thousand”

How about a more complicated case. Convert 25 dB to a ratio.

**Divide by 10.**“2.5”**Two and a half zeros before the decimal place is?**“more than 100 and less than 1000”**Yes, and half a decimal place is about 3.**“So 300?”**You got it.**“25 dB is about 300”

Now, convert 564 to dB.

**How many zeros?**“Almost the number 6 followed by two zeros. So 2”**Yes, but we had a 6 in front of the zeros. 6 is worth about 75% of a decimal place.**“So 2.75?”**Exactly, now multiply by 10.**“27.5”**Say that number.**“27.5 dB”

This approach is much easier to explain than defining dB = 10 log(R) and the inverse operation using pow(10) and gives much better intuition. So, in field work, I tend to just use this approach. This may seem trivial to experts, but any trick to increasing understanding and explain-ability is worth your consideration.

]]>Using a personal algorithm, the indicated to calibrated airspeed data points were reduced and plotted.

The trend is clear. At low speeds, the airspeed indicator reads too low (a common error). The errors at cruise are negative; the airspeed indicator reads too high. Only at around 70 MPH is the error near zero. Unfortunately, at the low end, there is more scatter than I hoped for. This scatter is likely resulting from the challenges of 1) precisely maintaining a specified indicated airspeed with an analog airspeed indicator, while 2) recording the average groundspeed and track. A future approach will use the raw GPS data points.

If we assume that the errors are solely resulting from errors in the static pressure (a reasonably good assumption), then we can determine the effective altitude errors associated with the static pressure error. These just barely meet the +-30 ft legal requirement at 100 kts.

For such a short flight, we were able to determine the overall character and approximate error curves of the airspeed indicator and altimeter. A more formal program would involve multiple people, multiple data points at the same test condition, and much more flight time.

]]>A: This is an excellent question. In this note, you will learn some fundamentals and tools of effective and robust communications.

First, listen to this lecture.

A brief aside: My experience as a professor taught me that its easy to talk about what I just spent hours preparing. In fact, 4 years of teaching made me **FAR **better at engineering than 4 years of engineering school. Why is that? I really believe the difference is engagement. I **had **to be ready to engage a conversation spontaneously. This meant that I **had **to know and understand.

If in the future I teach another class, I would have students make their own set of formal notes and example problems with solutions. No traditional homework.

There is a spectrum of communication effectiveness. Listen to this lecture.

Send me your comments. I look forward to hearing from you at oneill@aerofluids.com

]]>For the purposes of this engineering note, we will focus on the 180 hp versions: the PA-28 180 (constant), PA-28 180 Extended and PA-28 181 (tapered). The geometries taken from three POHs are in Figure 1:

The aircraft are essentially identical except for the wing planform, so for more insight, zoom into the wing shapes with Figure 2.

The tapered wing has a longer wingspan (35 ft vs 32.25 ft vs 30 ft) but the same root chord of 63 in. In fact, the PA-28 180 Extended and 181 have 170 ft^{2} wing areas whereas the 180 has 160 ft^{2}. The constant chord wings have no washout. The tapered wing has 3 degrees of washout at the tips.

The tapered and extended constant chord wing have about 6% more area than the shorter 180 wing. All things equal, the tapered wing can operate at a 6% lower lift coefficient compared to the constant chord wing. Since induced drag is proportional to k C_{L}^{2} , we might expect the tapered wing’s C_{L}^{2} to be about 12% lower than the constant chord wing.

Increasing aspect ratio, AR, tends to decrease induced drag for a given lift coefficient – all other things equal. The tapered wing has an aspect ratio of b^{2}/S = 420^{2}/170 = 7.2. The extended constant chord wing has an aspect ratio of 6.5. The shorter constant chord wing has an aspect ratio of 5.7. From Prandtl lifting line theory, we expect the induced drag to be about 10% lower for the higher aspect ratio wing (i.e. the ratio of AR), since induced drag is proportional to k C_{L}^{2} where k is approximately 1/(pi AR).

However, this does NOT tell all of the story. The tapered wing is operating closer to the optimal elliptical lift distribution such that the induced drag is lower still. Using my Prandtl lifting line software, the wings’ geometries and the airfoil properties, I computed the following lift distributions (blue curves) across an AOA range from -5 to 15 degrees in Figure 3.

Notice that the constant chord wing has significantly more downwash (red curves) at the tips. This lift distribution is not elliptical as the downwash is not constant. In contrast, the tapered wing has less downwash at the tips and more closely approaches an elliptical lift distribution, especially at cruise flight conditions.

But what does this mean? Plotting the induced drag indicates that the extended wing has an induced drag 17% lower than the constant chord wing. The tapered wing has an induced drag 32% lower than the constant chord wing.

Increasing span and carefully tapering a wing significantly decreases overall induced drag.

Ground effect strongly changes the magnitude of induced drag. Estimating the mean wing height gives a height/wingspan ratio of 0.086 for the constant wing, 0.084 for the extended wing and 0.0825 for the tapered wing. Using Hoerner’s estimate of the reduction of induced drag below, the short wing estimate is 48% of the free-air induced drag, the extended wing 47% and the tapered wing 46%.

Although the change in magnitude is substantial, the ratio delta between the wings is relatively minor. Surprisingly, the tapered wing’s dihedral coupled with a longer wingspan would tend to mitigate differences between the wings.

The PA-28 series uses the NACA 65(2)-415 airfoil, sometimes categorized as a laminar flow airfoil, although this is not precisely what happens in operation. A better characterization is that the airfoil naturally has a favorable pressure gradient across a substantial portion of the upper surface. The favorable pressure gradient (i.e. increasing speed and decreasing pressure) tends to keep the airfoil’s drag down and tends to form a “drag bucket”. The airfoil and pressure profile at CL = 0.2 using xfoil is in Figure 6.

Increasing AOA to CL=0.7 in Figure 7 shows the airfoil leaving the drag bucket region. The favorable pressure gradient on the upper surface is gone.

A typical airfoil shows an increase in profile drag with lift coefficient. This term has the same behavior as induced drag. Plotting the drag coefficient versus lift coefficient for a known wing area allows adding a rough estimate of the airspeed to the drag polar (Figure 8). These drag polar points for the 160 and 170 square foot wings are below. Notice that the 170 sq-ft wing has less airfoil drag for a given airspeed. This is particularly true at the lower airspeeds, such that at 60 knots the airfoil drag is about 30% lower with the 170 square foot PA-28 181. The critical airspeed for airfoil profile drag is about 80 knots.

Overall, Piper’s choice of the NACA 65(2)-415 is an excellent decision across structural and aerodynamic realms. The airfoil has a maximum thickness of 15% at 40% chord, which provides excellent depth for a high performance and light weight spar and excellent chordwise depth for fuel. The airfoil tends to provide consistent and smooth stall characteristics. The airfoil has a nice drag bucket in the aircraft’s effective operating region. The leading and trailing edges have a manageable radius and thickness. Overall, this was an excellent choice.

Gathering these results, the extended wing PA-28 has about 75% of the drag of the short wing PA-28 at touchdown. The tapered wing has about 60% of the drag of the short wing PA-28 at touchdown. Additionally, the overall induced drag for all variants of the Cherokee is reduced more than 50% when in groundeffect. Additionally, the effective geometric angle of attack for stall reduces in ground effect (c.f. Hoerner’s Fluid Dynamic Lift). These all contribute to a substantial difference during the final, flare and landing. Excess approach energy, especially when above 80 KTAS, will only slowly bleed away in ground effect.

Pilot reports that the tapered wing Piper Cherokee PA-28 floats more than the constant chord wings are completely substantiated with the engineering and physics analysis. Transitioning from a constant chord to a tapered wing Piper Cherokee has the potential to be particularly sensitive to the approach and flare speeds.

Comments? Contact me at oneill@aerofluids.com

Notice that this analysis is not meant to explore pilot techniques; you must refer to a qualified CFI for aircraft instruction. Fly safe.

There are limitations to this analysis. Some of these are:

- No dynamics
- No fuselage
- No flaps
- No trim or S&C
- Averaged CL for airfoil analysis
- Simple Prandtl Lifting Line model with Zero-Lift approximation