I am starting to research this question with respect to WWII aircraft - specifically the FM2 Wildcat, which was considered an excellent dogfighter.
To approximate an "apples to apples" comparison:
- For horizontal turns, I am using the maximum sustained turn rate that results in no net altitude loss. The aircraft will remain at this speed and altitude and turn rate for the entire time.
- For vertical loops, I computed the rate of turn for each speed that will return the aircraft to the same speed and altitude at the bottom of the loop. Since speed and rate of turn (both horizontal and vertical) vary throughout the loop, I am using the average rate of turn for the entire loop. For sake of simplicity, the loop is performed using the same angle of attack for the entire loop.
I created the above E-M Chart for the aircraft. As is typical, the performance envelope is bounded by the minimum speeds for the aircraft and the maximum g-load - which I assume to be around 7g's. In level sustained flight, the g-limit does not ever come into play because the horsepower of the aircraft is simply insufficient to push the aircraft around in high-g maneuvers. The maximum g's in a sustained turn are around 3 g's.
Within the chart, I plotted the sustained turn rate for level flight at various speeds (PS=0). As shown, as speed increases, the turn rate initially increases (along the Vmin line) and then decreases. At top speed, Thrust = Drag and Lift = Weight. As speed increases, less angle of attack is required to produce the Lift, which means that induced Drag also decreases. However, parasitic drag is also increasing - to the point that Parasitic Drag = Thrust - Induced Drag. At this speed, all of the Lift is used to keep the aircraft in level flight. There is no excess Lift which can be used in level turning flight. Any effort to turn the aircraft while maintaining altitude will result in the aircraft slowing down to the speed at which the desired rate of turn and altitude can be sustained.
The same kind of thing happens in a loop. When you increase the angle of attack to the desired amount, this creates more Lift and also more Induced Drag. This Induced Drag along with the fact that the aircraft is climbing and being slowed by gravity - will cause the aircraft to slow down. The aircraft will reach the lowest speed and fastest turn rate at the top of the loop. But, in contrast with a horizontal turn, this lost speed can be recovered in the last half of the loop, where the aircraft is descending.
As with the horizontal turn, the average turn rate for a loop decreases with speed. However, unlike a horizontal turn, the average turn rate never goes to zero. You can still perform a loop even when the aircraft is at maximum speed. The aircraft will lose speed during the first half of the loop and return to maximum speed at the bottom of the loop.
But, what was more surprising is that the vertical loop always yields a better average turn rate than the horizontal loop. This indicates that, if you are engaged in a turning contest with another aircraft, you should use vertical loops through the entire range at which they are available.
I have plotted the turn rates for loops on the E-M Chart. Note that the results appear linear.
My question is whether this result makes sense from a mathematical and real world sense.
I assume that others have made similar computations for other airplanes. The math uses the basic force equations for Thrust, Parasitic Drag, Induced Drag and Lift. I converted these forces to acceleration by dividing force by mass. I netted the acceleration from Thrust against Drag and the portion of Gravity working along the direction of flight (gravity X sin(pitch)). I netted Lift and the portion of gravity opposing Lift (gravity X cos(pitch)) and treated the net Lift as a flight path "deflector" where the angle of deflection = Net Lift X 360/(2*PI()*Speed). This last equation treats the Net Lift as operating along a circular path.
The turn rates for horizontal flight could be easily computed for each different speed. But because the speeds and turn rates for loops are constantly changing, I used a series of computations in 1/60 second intervals (on an excel spreadsheet). I tried different Angles of Attack until I found the one that resulted in the same speed for the beginning and end of each loop.
In real life, I have flown simulated air combat in Marchetti aircraft with F-16 safety pilots. We started using horizontal turns and quickly switched to using loops. We had been practicing 3-4 g horizontal turns. It seemed that the loops involved about the same g-load (except for the one case where I decided to extend the downward part of a loop before pulling up - that was 5.5 g's).
I also tend to use loops in computer air combat simulations.
Another factor that seems to tip the advantage in favor of the loop is that you have the advantage of being able to use bank to quickly change heading, especially when you are pointed straight up or straight down.
One of the comments below indicates that it would be helpful to provide a summary of my results for a particular speed, e.g. 250 mph:
deg frame dif mph cL 000.00 250.00 0.8181 090.00 309.8 309.8 168.08 0.8181 180.00 592.5 282.8 124.87 0.8181 270.00 852.3 259.8 201.65 0.8181 000.00 1127.7 275.5 250.00 0.8181 Recap: total time 18.796 secs average turn rate 19.153 deg/sec average speed 187.44 mph diameter of turn 1709 ft
This shows the speeds at various points on the loop and the time needed to reach each point (1 frame = 1/60 sec).
Regarding g-forces, my computations show that the cL for level flight at 250 mph is around 0.1696 (an Angle of Attack of around 1.696 deg). Thus, a cL of 0.8181 (an Angle of Attack of around 8.181 deg) should result in about 5 g's of force. At the top of the loop, the aircraft speed is 124.87 mph, well above stall. In real life, the pilot will likely vary the Angle of Attack, as suggested in the comments. However, it appears that (at least on the Fm2) a constant Angle of Attack does not break any limitations and should yield an approximation of the turn rate - possibly on the low side.
FM2 vs. ZERO (edited Jan 26, 2023)
The question of who would win a dogfight in a contest between an FM2 and Zero is interesting. For those who are interested there was an official comparison of these aircraft during WWII. The comparison concluded that: "The Zeke 52 could gain one turn in eight at 10,000 feet." and that the FM2 was more maneuverable at speeds above 200 kts (230 mph). The report recommended "DO NOT DOG-FIGHT WITH THE ZEKE 52." They made the same recommendation for the Hellcat.
I created an E-M Chart for the A6M5 Zero Model 52, a mid-war improved Zero. The FM2 and A6M5 are remarkably similar. Both have about the same top speed (around 300 mph) in level flight, about the same power to weight ratio and about the same wing area - although the A6M5 is lighter and, therefore, has a lower wing loading.
A comparison of the horizontal turn rates shows that, if both fighters are turning at the same G-loads, the A6M5 has a very slight advantage in turn rate. However, the A6M5 has a significant advantage if both are attempting to turn and maintain the same altitude. If you compare the charts, you will see that the level turn rate (PS=0) for the A6M5 is above the 3g line. The exact calculations are that, in a full circle, the A6M5 will gain 50.53 degrees, or one turn in seven, at sea level. The lower wing-loading seems to be the key factor in this difference.
But a comparison of vertical turn rates shows an interesting result. In this case, the FM2 now has a small advantage over the A6M5. Since lift plays less of a role in a vertical turn, this appears to be a result of the slightly higher power to weight ratio of the FM2.
The FM2 was a higher performance version of the F4F Wildcat. Thus, as with the Hellcat, unsuspecting Zero pilots might have been surprised by this improved vertical performance, to their detriment. This is reflected in the 32:1 K/D ratio of the FM2.
This has been an interesting discussion, although not what I expected when I asked the questions. Nevertheless, I have learned enough to tentatively conclude that my two questions have been answered.
First, with regard to real life experiences, it appears that vertical loops are not used tactically because of various limitations and complications. The limitations are that utilizing them requires that you complete at least 1/2 of the loop. The complications are that you now have to figure out how to re-engage with the enemy. It could be that vertical loops make more sense with smaller prop-driven aircraft, like the Marchetti or FM2. In the case of the Marchetti, we were in similar aircraft and the other person started doing vertical loops first - which meant we had to follow. In the case of the FM2, it has been pointed out that this would have been a losing strategy against the Zero.
Second, regarding the math, I have seen nothing to indicate that my computations are wrong - only that other factors outweigh any possible turn rate benefits. Vertical half-loops have been used to quickly change direction and escape from combat. Two examples are the Immelmann, a climbing vertical half-loop first used by the Germans in WWI and the Split-S, diving vertical half-loop first used in WWII and still used today.
So I will mark this question as answered.
Since writing this conclusion, an additional response has been received which focuses more on the math and is also very helpful pointing out that the reason the turn rate for the loop is faster is because all of the lift is directed towards creating the turn rate - in contrast with a horizontal turn, where some of the lift is used to offset gravity.
I have also replaced the E-M diagram with a diagram that uses the recognized term PS=0 to refer to a horizontal turn where vertical speed is zero. Also the description for the vertical axis has been modified to clarify that the E-M diagram measures the horizontal turn rate.