Asymmetric Thrust Explained

One of the very first things that people find out about when they start learning to fly is that it takes right rudder (sometimes a lot of right rudder) to keep the airplane going straight at the beginning of the takeoff roll and often after lift-off while the plane is slow and using a high angle of attack.  Three factors are all blamed for this requirement.  It depends on who you ask as to the effective importance of each one.  The three factors are “Corkscrewing slipstream”, “P-Factor” and Engine torque”. The physics of the situation is shown below.

 Corkscrewing Slipstream

It would be nice if the propeller would just take the air and throw it straight backwards, but it doesn't.  The propeller airfoil necessarily has some drag, so it drags the air in the direction of rotation to some extent.  Therefore the slipstream follows a corkscrew-like trajectory, rotating as it flows back over the craft.

The next thing to notice is that on practically all aircraft, the vertical fin and rudder stick up, not down, projecting well above the centerline of the slipstream.  That means the corkscrewing slipstream will strike the left side of the rudder, knocking the tail to the right, which makes the nose go to the left, which means you need right rudder to compensate.

 You don't notice the effect of the corkscrewing slipstream in cruise, because the aircraft designers have anticipated the situation.  The vertical fin and rudder have been installed at a slight angle, so they are aligned with the actual airflow, not with the axis of the aircraft.

In a high-airspeed, low-power situation (such as a power-off descent) the built-in compensation is more than you need, so you need to apply explicit left rudder (or dial in left-rudder trim) to undo the compensation and get the rudder lined up with the actual airflow.

 Conversely, in a high-power, low-airspeed situation (such as initial takeoff roll, or slow flight) the corkscrew is extra-tightly wound, so you have to apply explicit right rudder.

P-Factor

The term P-factor is defined to means asymmetric disk loading.  It is an extremely significant effect for helicopters. When the helicopter is in forward flight, the blade on one side has a much higher airspeed than the other.  If you tried to fly the blades at constant angle of attack, the advancing blade would produce quite a bit more lift than the retreating blade.

For airplanes, the same effect can occur, although it is very small. For the effect to occur at all, you need to have an angle between the propeller axis and the relative wind.  To be specific, imagine that the aircraft is in a nose-high attitude, but its direction of motion is horizontal (i.e. the relative wind is horizontal).  Then the downgoing blade will be going down and a little bit forward, while the upgoing blade will be going up and a little bit backward.  The downgoing blade will effectively have a slightly higher airspeed. Since it is on the right side of the airplane (In a typical American engine) it will tend to torque the airplane around to the left and you'll need right rudder to compensate.

The situation is depicted in the figure above.  The airplane is in level flight, with a 10 degree nose-up attitude.  The motion of the blade through the air is the resultant found by adding the rotational motion (perpendicular to the thrust line of the aircraft) and the motion of the airplane as a whole (horizontal).  The motion of the downgoing blade is shown with solid lines, while the motion of the upgoing blade is shown with dotted lines.  You can see that the downgoing resultant is longer than the upgoing resultant.

 This is the real story on P-factor: the advancing blade sees more relative wind, while the retreating blade sees less relative wind.

P-Factor Myth #1

There are quite a lot of myth surrounding P-factor.  For some reason, P-factor gets blamed for the fact that aircraft require right rudder on initial takeoff roll.  This is impossible for several reasons. 

 Nearly everybody these days learns to fly in nose-wheel type aircraft (tricycle gear arrangement) so the propeller disk is vertical throughout the takeoff roll.  P-factor obviously cannot occur.

 Now let's suppose, just for sake of argument, that you are flying a taildragger, in which the propeller disk is actually non-vertical during the initial takeoff roll.  Common experience is that the most right rudder is required at the very beginning of the takeoff, before much forward speed has been achieved.  The Flight Training Handbook says this is because P-factor is worst at low airspeeds. But wait a minute --- real P-factor is proportional to airspeed.  In the initial moments of the takeoff roll, there is no relative wind, so there can't possibly be any P-factor.  Of course, if you are taking off into a headwind, there could be a little bit of P-factor --- but does that mean if you take off with a slight tailwind there will be a negative amount of P-factor, requiring left rudder?  Don't bet on it.

The real reason that you need right rudder on initial takeoff roll is because of the corkscrewing slipstream.   P-factor exists in principle, but it cannot possibly explain the behavior we observe in real airplanes.

P-Factor Myth #2

Nearly all pilots have been taught that P-factor has something to do with the angle of attack of the right (downgoing) and left (upgoing) propeller blades.  I'm sorry to slaughter so many sacred cows, but this one has got to go.  The situation is analyzed in below.

P-Factor and Angle of Attack

The right panel shows the airplane in a 10 degree nose-up attitude as before, while for reference the level attitude is shown in the left panel. Also, since we want to discuss angle of attack, I have attached a "reference line'' pointer to each of the blades. The angle of attack of the propeller blade is just the angle between the reference line and the blade's motion through the air.

If you examine the figures, you will discover that even if the propeller disk is inclined to the direction of flight (so that P-factor really is occurring) the upgoing and downgoing blades have no significant difference in angle of attack.  P-factor occurs because of the difference in speeds, not any difference in angles.

Blowing air through the propeller disk does lower the angle of attack of the propeller blades, but even if the air crosses the disk at a reasonable angle, it lowers both blades equally, to an excellent approximation.

Many references point out that the downgoing blade makes a bigger angle to the vertical than does the upgoing blade.  Alas, that doesn't explain anything. The blade doesn't care which way is up --- all it cares about is where the relative wind is coming from.   You can incline the propeller disk as much as you want relative to vertical, but there will be no P-factor in the absence of relative wind.

P-Factor Myth #3

P-factor also gets blamed for "critical engine'' effects in multiengine aircraft.  I have two things to say about this:

Yes, P-factor really does occur.

Sorry, it is negligible compared to the corkscrewing slipstream effect.

I estimated the amount of P-factor for a Piper Apache:

I assumed the airplane was trimmed for zero yaw in cruising flight, and figured out what would happen when the airplane slowed down to “V_MC”.  I discovered that the center of effort of the propeller disk moved to the right by “one inch”.  When you compare that to how far off-center the engine is mounted, it seems rather negligible.

The corkscrewing slipstream may not seem like a big effect either, but remember that it operates by striking the rudder, which is large and has tremendous leverage. 

Since both P-factor and corkscrewing slipstream can both occur, I suggest the following experiment to see which is more significant. Take your favorite aerobatic airplane and paint the starboard rudder pedal green and the port rudder pedal red, just so we can keep straight which is which.  Now go to a safe altitude and set up for “inverted” slow flight.  In this high-power, low-speed situation, you need to push the port (red) pedal or the starboard (green) pedal?  If you believe in P-factor, the answer will be port, because that is now the downgoing, advancing blade. If you believe that the corkscrewing slipstream is more important, the answer is starboard, because the relationship between the propeller, rudder, and rudder pedals is unchanged by the inversion.

Engine Torque

This factor is the most easy to understand since it applies to everything that spins.  When an engine operates, how does it know whether to stand still and spin the propeller, or have the propeller stand still and spin the engine?  In some old aircraft rotary engines, the crank shaft, which normally holds the propeller and therefore spins, is actually held stationary to the aircraft frame and the engine body in its entirety spins.  The cylinders, the crankcase, everything.  The propeller is bolted to the crankcase.  In a conventional setup, the engine doesn’t spin because it is bolted to the aircraft frame which has much more mass than the propeller and crankshaft.  So it stays still and the propeller turns.  The mass is much greater, of course, but it is not infinite.  Therefore the engine does try to spin the plane.  Rev up your car engine.  Assuming it is a from to rear mounted (not sideways) engine, you will feel the car rock to the left – trying to spin – this is engine torque.  The car’s weight and the fact that it is standing on the ground will not allow the car to spin.  When a plane is in the air, it is more capable of spinning, Therefore, the engine torque will have more affect on a plane, especially at e slow speed, when it is in the air.

Generally you would use the ailerons to counter engine torque, but at low speeds the rudder has more authority and if the plane has sufficient dihedral the right rudder correction (for P-Factor and Corkscrewing slipstream) will also take care of engine torque.