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Propeller Dynamics

Essential reading for model aircraft contest fliers. This is the only book on the market explaining propeller theory in non-mathematical terms. A rattling good read, I know, I wrote it.


Meredith Effect: Fact or Fantasy

By Joe Supercool

For those who came in late, or are too young to remember, the North American P51 Mustang was one of the great fighters produced on the allied side in World War 2. Perhaps the most surprising aspect of this machine was that it easily surpassed the speed of the highly developed Supermarine Spitfire, when using the same engine: even though the P51 was a least 1000lb heavier!

This achievement was claimed by N.A. designer J.Leland Attwood to be due to the low drag of the engine cooling system. Not just low drag, but very nearly zero drag as a result of thrust produced BY the cooling system: the design principle for this ducted cooling system was laid out by Meredith in 1935. If you want to read Meredith's paper, it is easy to get. Just open your web browser at naca.central.cranfield.ac.uk/reports/arc/rm/1683.pdf The report is entitled "Cooling of aircraft engines, with special reference to Ethylene Glycol radiators enclosed in ducts".

If you want to read Attwoods account, you need Aeroplane, May 1999. But first a warning: there is evidence of creative editing in that article. Also, the diagrams while very pretty are misleading. I do not believe the errors in that article came from Attwood's pen.

Now I have a motivation for addressing this topic. Trolling the internet on "Meredith effect", one finds a full spectrum of believers and doubters. Many of the doubters have clearly not read Meredith's paper. Many of the believers have no credibility because they have their physics wrong, often in their opening sentence. So how does one go about showing that Attwood was indeed successful in understanding and applying the Meredith effect to the P51?

Enter Gruenhagen's book "Mustang. The story of the P51 fighter", 1969. This text gives a rather full account of the cooling duct development. This was a very comprehensive series of trials to determine the design parameters and performance for the duct. Consider this: your boss comes in and says "we have mounted a P51 in the wind tunnel. Pick your best mate, go and sit in it while we run the tunnel up to 500 MPH, and do some measurements for us. Don't bother the life insurance companies, when we told them you are upwind of a 5000 HP fan drawing air through the tunnel, they declined your application". But that is what they did, and not only did they survive, but the data they measured is on page 79 of my edition.

Now at first glance, the only interesting part of the data are the sketches showing various arrangements of ducts for coolant and oil cooling. The numerical data is hard to read: the print is so small. But being half-blind anyway, I thought I better put on my 2.5 binocular magnifier and see what the column headings were. I was immediately revolted by the unit "slug": but as this turned out to be 1 slug equals 32.172 pounds, then that was not so bad. After all, the number 32.172 seemed to be related to the acceleration due to gravity, so some nameless old fossil was trying to separate the concepts of mass and weight. Of course, I am not yet a fossil, as I have not yet gone underground!

Above the word "slug" were the words "Mass flow". Now this demanded my interest, something to do with fluid dynamics, a pet interest as I am so ignorant of the topic. The mass flow was given for both the coolant and oil, for various openings of the nozzle duct. Meredith mentioned the need to control the position of the nozzle flap. Also given were the areas of the inlet apertures for the coolant and oil ducts, shown on the lowest sketch labelled "Modified Divided Duct". The speed of the tunnel flow was also given, to be precise 430 MPH.

This was starting to look like a comprehensive set of measured data. Not theoretical calculations, but actual, real, physical, entirely indisputable, measurements of the performance of the duct. The reason for the measurements in the first instance was related to tests for determining the cause of unacceptable duct "rumble".

These data did much more than show that the diffuser rumble could be eliminated. Recall that we started this discussion by arguing that the duct could produce THRUST. Remember that the sceptics were arguing that the duct could not produce thrust: these data show that for the proper setting of the nozzle exit flap, the duct could produce plenty of thrust.

Here is how the physics works. The wonderful "mass flow" is first of interest. If we put one kilogram of air in the inlet duct, then one kilogram of air MUST come out of the exit nozzle. The "mass flow" is a measured quantity which includes the effect of duct shape and radiator properties. Would it not be wonderful if we could use the measured mass flow and aperture sizes to get the thrust?

Now don't second guess me, this is my yarn. Of course we can get the thrust. But first we need to remember just what thrust is. Here is Newton's second law of motion.

"The time rate of change of momentum is proportional to the impressed force, and takes place in the direction of that force."

This is the law of forces. In words, the statement can be re-written: Force is mass times acceleration. Now thrust is a force. The acceleration is the rate of change of momentum. To find the thrust of the duct, we need to find the momentum of the air entering the duct and the momentum of the air leaving the duct. The difference of these latter two is just the value of thrust we seek.

We have the mass flow. To get the momentum flow, we need to recall that momentum flow is just the mas flow times the velocity of the air at the same position. Now this could be tricky, as there is a pressure field travelling ahead of the inlet duct, with air spilling out around the inlet aperture as well as entering the inlet aperture. We proceed as follows.

First consider that all the air goes into the inlet duct. Then the mass flow into the aperture is the density times the area of the duct times the air velocity. We have the density at something like 1.226 Kgm^3 (0.00238 slug/ft^3), the inlet area is 0.08948 m^2 (38.6 sq. in.) and the air velocity is 192.23 m/s (430 MPH). Forming the product yields a mass flow of 20.975 kg/s (1.44 slug/s)

However, the measured mass flow is 0.412 slug/s. This shows that a lot of the mass flow goes around outside the duct. The airspeed ahead of the duct is thus divided between going around the duct and into the duct. By continuity of flow, the velocity of the flow entering the duct must be the airspeed head of the duct (192.23 m/s) times the measured mass flow (0.412 slug/s) divided by the possible mass flow (1.44 slug/s), giving an inlet flow velocity of 55 m/s.

Since the exit mass flow rate must be the same as the inlet flow rate, and with the outlet flap closed down to only 0.01929 m^2, the exit velocity is 55 times the inlet area 0.089 m^s divided by the nozzle exit area we get for the nozzle velocity 258 m/s.

Why this business of calculating the inlet and exit flow speeds? Well we wanted the momentum of the flow at the entrance and the exit, as the difference in momenta is just the thrust.

The inlet momentum is the product of the density (1.226(kg/m^3) times the inlet area (0.089 m^2) times the square of the air velocity at the inlet (55 m/s) which gives 339 Newton.s. For the exit momentum, we have 1.226 times 0.019 times 258 squared which gives 1550 Newton.s.

The thrust is then the difference in momenta which is 1550 less 339 or 1211 Newtons. Now 4.45 Newton is 1 pound force so the thrust in more traditional units is 270 lb.

This is a wonderful result. The claim by Atwood of 300lb thrust generated by the duct appears to be entirely valid. Also validated are Meredith's design principles, as these are the basis for the P51 duct design. One can believe the believers, and doubt the doubters.

Worthwhile also is knowing the amount of power carried past the exit nozzle. The thrust generated by the air moving at 258 m/s is 1211 Newtons: with power being the product of thrust times velocity, this represents 312 kW, or 419 horsepower. Cooling drag in an arm waving way is about 30% of the total drag, so if the installed engine power is of order 1400 horsepower then the cooling drag would require 420 horsepower just for cooling. By the figures above, the thrust of air supplying 419 horsepower matches the power required for cooling of 420 horsepower. Nice result for North American Aviation!

Post Script

In researching this topic, I looked at hundreds of photos of Mustangs and other aircraft with liquid cooled engines. I found one only photo from the rear showing the exit from the P51 cooling duct: the exit was in shadow so you could not see the size of the closed duct. For the record, this photo was on page of 61 of "Mustangs International", Volume No. 33 - Issue No. 3. Worth looking at, the article was on the operation of the airscoop actuator of the P51. My point here is that I wish to emphasize just how small was the exit aperture setting at high speed. The exit area was 0.019 m^2, just 29.5 square inches, a square of side 5.5 inches. Spread the fingers of one hand and you have it!

The airscoop operation was fully automatic. The operating principle was that of a thermostat, the exit opening area varying the mass flow to keep the glycol coolant temperature within allowed limits. Cleverly, the opening at high speed both kept the coolant at the right temperature while also optimising thrust!

The role of magazine editors appears to be that of censor: the photos had all to be from pretty angles rather than in the nether regions of the cooling system. Thirties era biplane fighters literally had the coolant radiator slung under the fuselage, often between the undercarriage legs: the designers approach seemed to be "out of sight, out of mind". Some had no fairing around them at all, while later aircraft, such as the Hurricane, still had the radiator hung under the fuselage, but prettied-up with a fairing.

I found the SE5A cooling system a mystery. There was a system of louvres in front of the blunt, flat radiator, which let the cooling air in, but no sign of anywhere for the heated air to escape. Actually it was under the fuselage toward the lower wing: no pretence of ducting or reduction of drag at all.

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