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For rudder deflection angles of 5, 10, 20, 30, and 45 degrees, at what (negative) angle-of-attack does a fin-rudder combination (including dorsal fin if present) have a lift coefficient of exactly zero?

For the purpose of this question, define angle-of-attack in relation to the chord line of the fin alone, not including the deflected rudder. If we ignore things like spiral slipstream, deflection of airflow around cabin, etc, it would be the same as the sideslip angle.

Loosely speaking, at this angle-of-attack we could say the force from the deflected rudder is being exactly cancelled by the force from the fin, though in reality each influences the airflow around the other and the fin-rudder combination acts as a single unit.

To allow for a definitive answer, I'll specify to use the fin (including dorsal fin) and rudder from current -production Cessna 172 S, but feel to post an answer addressing the fin and rudder from any other aircraft.

Note that an aircraft could never sustain a sideslip angle that caused the fin plus deflected rudder to have a lift coefficient of zero, unless some other yaw torque due to something like asymmetric thrust or extreme adverse yaw from deflected ailerons was acting to maintain the slip angle.

Approximations are fine-- obviously an exact answer would involve a complicated computer modelliing project, or actual wind tunnel experiments. My hope is that existing information relating to the lift curves of symmetrical airfoils with deployed flaps may be adapted to give an approximation of an answer to this question.

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  • $\begingroup$ Despite specification, this is a very broad question. Obviously, for each combination of airfoil, relative rudder chord and other minor factors there will be 'angle of zero lift' for each deflection; it's just a matter of mundane calculations or testing. What information each of these random points can give you? (Say, NACA 0012, 25% rudder, 10° deflection -> -5° $\alpha_0$. So?) Trends may be interesting, such as this angle per deflection vs. relative chord, but that's not what you're asking... $\endgroup$ – Zeus Aug 5 at 0:45
  • $\begingroup$ Besides, "an aircraft could never sustain a sideslip angle that caused the fin plus deflected rudder to have a lift coefficient of zero" - why, this always happens in a sustained sideslip (even without adverse ailerons, fuselage interference etc.) $\endgroup$ – Zeus Aug 5 at 0:49
  • $\begingroup$ @Zeus - why do you say that? I would say that in a sideslip in a Cessna 152, where you are pushing hard on the (let's say right) rudder pedal to keep the rudder deflected, it's likely that the NET contribution of the rudder + fin is a yaw torque toward the right, acting against the leftwards yaw torque created by the fuselage. (Basically I'm saying that the center of lateral area or effective center of aerodynamic yaw torque of the fuselage alone is likely well behind the CG, since the rear parts of the fuselage act at such a large moment-arm.) $\endgroup$ – quiet flyer Aug 5 at 16:37
  • $\begingroup$ Why do you include fuselage? You excluded more relevant things like "spiral slipstream, deflection of airflow around cabin, etc.", why should I assume that fuselage should be calculated? Its influence can be quite complicated. A slender body has its AC quite far forward actually, but when you include interference of other parts (wing, "deflection of airflow around cabin, etc"), there may be no AC as such. With fuselage, the exercise cannot be reduced to "lift curves of symmetrical airfoils with deployed flaps", so you either should exclude it or define exactly. $\endgroup$ – Zeus Aug 6 at 0:56
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Here are two links to Tom Speer's homepage. They show the performance of a couple of flapped airfoils. They aren't symmetrical, but it doesn't matter for what you are doing. What you want to look at is the change in the AoA between adjacent flap deflections.

For instance, it the first link, the there is a series for -3 degrees flap and one for 5 degrees flap. Holding the coefficient of lift constant, you'll see about a 4 degree difference in AoA. Thus deflecting a smallish 20% flap is about the same as rotating the whole foil by half as much.

http://www.tspeer.com/NACA23012/N23012wflap.htm
http://www.tspeer.com/NACA23012/N23012wflap.htm

For a 50% flap, it can be as much as 80% effective. See also Tom's analyses of wingsail performance with various sized flaps at the same site. These are symmetric foils, but you have to do some extrapolation to run the lift down to zero.

Tom worked for Boeing, and has worked on numerous high performance projects including the America's cup wingsails and landyacht projects.

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If you want an approximate answer, it is any time a steady heading sideslip is performed (rudder vs. resulting sideslip). It's approximate because the fuselage and ailerons also contribute to the yawing. But if the fuselage is only mildly destabilizing, the answer should be in the ball park.

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  • $\begingroup$ Agreed- I guess I would tend to think in a typical light plane the fuselage is rather strongly stabilizing. Maybe I am wrong. Maybe another question for ASE. (Example Zenair where there is virtually no fixed fin at all; obviously an extreme case.) $\endgroup$ – quiet flyer Aug 2 at 22:37

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