Zero Mystery

Copyright 2016 by Paul Niquette. All rights reserved.

Looking head-on at the Corsair and the Zero we see only differences in dimensions and distinctions in wing configuration.  With the propeller rotating clockwise as viewed from the cockpit, the so-called 'prop-wash' will be given a rotating component of motion, which manifests pressures on all structures along the fuselage. 
For example, prop-wash applies a downward force on the top of the right wing and an upward force on the bottom of the left wing, both tending to roll the plane to the right.  Same for the elevator.  The moment-arms for those forces are short, thereby yielding mechanical advantage to the ailerons located near the tips of the wings.  Lateral forces from prop-wash acting on the external fuel tanks or ordnance generally toward the left cannot have an explanatory effect -- even if they were different between the two aircraft -- since their longitudinal locations are close to the airframe's center of gravity, thus their moment-arms are negligible.  The rudder is a different story...
As shown in the sketch below, the spiraling wind from the propeller strikes the rudder on the left side, which tends to move the tail to the right and thus to yaw the aircraft toward the left.  Since the rudder is situated only atop the fuselage, there is no opposition to that lateral force apart from the rudder's control surface.  In straight and level flight, the pilot makes adjustments in yaw by rudder-pedals, using the rudder's trim tab as necessary to balance out pedal forces.  At combat speeds the rudder trim setting to the right would need to be increased, and during dogfight maneuvers there would be little time available to adjust rudder trim. With acceleration during a diving chase, as described in the puzzle, the yaw-to-the-left force becomes stronger according as the square of airspeed, so the pilot must press more firmly on the right rudder-pedal, while being unable to neutralize pedal forces with trim adjustments.  The effect -- let's call it the 'R-Factor' -- is the same for both Corsair and Zero. 
Well, not quite the same, as we shall see.

Head-on Comparison 
From the side views in the illustration below, we see confirmation of the hint in the puzzle: That yaw control is located ahead of pitch control on the Corsair and behind pitch control in the Zero.  Locating the rudder forward would seem to reduce the Corsair's yaw authority compared to the Zero's -- and remember: Any coordinated turn in an airplane requires the application of both yaw and roll controls simultaneously. 
There must be more to the story...  

Side View

To provide an explanation
for why the Zero does not turn right as quickly as the Corsair, we will make numerical comparisons between the two aircraft for this tabulation using estimates from the illustrations...

Parameter Estimations

Symbol

F4U Corsair

A6M Zero

Ratio (C/Z)

Fuselage Length ft

xL

33.3

29.7

1.12

Empennage Height ft

hE

16.1

10.0

1.61

Center of Gravity ft

xCG

8.5

8.3

1.02

Propeller Diameter ft

pD

13.0

10.0

1.30

Rotating Speed rpm

rpm

1,655

2,150

0.77

Propeller Tip Speed mph

pV

768

768

1.00

Combat Speed mph

cV

332

332

1.00

Prop-Wash Entry Angle deg

ePW

24

24

1.00

Prop-Wash Area ft2

aPW

132.7

78.5

1.69

Prop-to-Rudder Distance ft

xPR

26.6

28.2

0.94

Trim for Prop-Wash ft3

yPW

3,530

2,214

1.60

Rudder Height ft

hR

6.0

5.0

1.20

Rudder Area ft2

aR

24.0

15.0

1.60

Rudder Control Area ft2

aC

14.4

6.0

2.40

Moment Arm to CG ft

xM

13.9

12.9

1.07

Yaw Authority ft-ft2

yA

200

77

2.58

Excess Trim Factor ft3/ft-ft2

yTF

17.7

28.8

0.61




 











Fuselage length
xL for the Corsair is slightly larger than the Zero, and the Corsair also has a larger propeller diameter pD producing a cross-sectional area for its prop-wash aPW some 70% larger than the Zero's.  It has a higher empennage at the rudder station hE by about 60%.  Setting trim for prop-wash yPW must be proportional to the product of prop-to-rudder distance xPR and the cross-sectional area of the prop-wash aPWIn both aircraft, the top of the rudder is slightly above the top of the projected prop-wash.  Although the prop-to-rudder distance xPR is slightly shorter for the Corsair compared to the Zero, the moment arm to the center of gravity CG from the yaw control surface xM is slightly longer. 

The size of the rudder aR on the Corsair is considerably larger than that for the Zero.  More significantly, the rudder control area aC is about 2.4 times larger.  Yaw authority yA is proportional to the product of the moment arm xM and aC.  Because of the differences in rudder shape -- 'trapezoidal' versus 'triangular' -- the yaw authority yA for the Corsair is nearly 2.6 times that of the Zero.  Oh, but that difference would apply to turns in either direction.  We must look closer at prop-wash.

Let us make explicit assumptions: That in both aircraft, the propeller tip speed is limited to the speed of sound pV = 768 mph and that the maximum speed of the A6M Zero (provided in the puzzle) will be used for the combat speed cV = 332 mph for both aircraft in a dogfight.  Difference in pD allows the rotating speed of the Corsair engine rpm to be 23% slower than that for the Zero, but a parcel of air coming off the propeller tip of either aircraft would make a prop-wash entry angle ePW of about 24 degrees relative to the plane of propeller rotation.  After about 60 ms, that parcel will arrive at the aircraft's rudder where it exerts an increment of leftward yaw force. 
For combat flight, while aiming weapons at an enemy aircraft, it would be necessary for variable yaw forces to be provided by the rudder control surface independent of trim.  We will use the dimensionless parameter yTF = yPW / yA to characterize that requirement, and yTF for the Corsair is only about 60% of yTF for the Zero.  The required force on the rudder-pedals also depends on the aerodynamic features of the rudder control surface, which is determined by its hinged attachment to the vertical stabilizer.  Solvers will notice that much of the Corsair rudder moves outward into the slipstream mostly above prop-wash, which partially balances the required force on the rudder-pedals for yawing in either direction.  Such is not the case on the Zero. 
One might imagine a Zero trimmed for 332 mph chasing a Corsair in a dive, when suddenly...
The Corsair commences a coordinated turn to the right.  The pilot initiates a roll to the right.  At that instant, the left wing is developing more lift than the right wing and therefore more drag.  That unbalance near the wingtips might inadvertently cause the Corsair to yaw to the left -- exactly opposite to the intended maneuver.  That's adverse yaw.  However, the pilot of the Corsair, with its superior yaw authority yA by a factor of 2.58, readily prevents adverse yaw by simultaneously applying a force on the right rudder-pedal.  By the way, the plane has not yet started the turn.  For a second or so, it continues to fly straight ahead with the wings going steadily more crooked in the sky.  When the Corsair has established the intended 'angle-of-bank', its pilot neutralizes the ailerons to stop the roll and relieves the force on the right rudder-pedal.  The plane starts its turn to the right.  Yaw trim is still adjusted to neutralize the force on the right pedal attributable to prop-wash.

The pilot of the Zero sees the beginning of the Corsair maneuver and starts to set up a right turn.  However, with its inferior yaw authority yA by a factor of 1/2.58 = 0.39, the rudder trim tab has already been set toward its right extreme, leaving a reduced range for increasing yaw forces toward the right by rudder-pedal.  With limited ability to prevent leftward adverse yaw, the pilot of the Zero must roll more slowly toward the right -- hey, while 'standing on the right rudder-pedal', earnestly applying whatever control force is still available beyond the right trim setting needed to correct for prop-wash at combat airspeed.  Perhaps as much as a second is lost by the Zero in rolling into the same 'angle-of-bank' as that being used by the Corsair.  But that won't be enough. The Zero needs to get into a steeper turn in order to re-establish its aim at the Corsair ahead.

Let the 'R-Factor' be offered to the world as our solution the Zero Mystery...

The 'R-Factor'
The Vought F4U Corsair has a huge advantage in yaw authority over the Mitsubishi A6M Zero, which becomes especially pronounced in right turns at high airspeed due to the asymmetry of prop-wash forces on the rudder.



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