Yaw due to roll rate derivative

Preparing a series of lectures on Aerodynamic Derivatives, I came through one of the less common and maybe less well understood stability derivatives, the yaw due to roll rate C_{n_p}.

mig29_rollDifferent textbooks provide different explanations on the physical origins of this derivative, the main which are the following:

  1. Wing drag increase (+). This actually refers to the profile drag increase due to the angle of attack increase in the down-moving wing while a reduction takes place in the up-moving side. The sign of this contribution is positive, considering that a right roll rate gives a right yaw tendency. This effect should not be confused with C_{n_{\delta_\alpha}} which is the yaw moment due to aileron deflection (aileron adverse yaw) and is attributed to the increase in profile drag due to the deflected control surfaces.
  2. Leading edge suction or wing lift vector tilting (-). The down-moving wing seeing an increased angle of attack, experiences a forward tilting of the lift vector, while the opposite happens for the up-moving wing. It is encountered in subsonic leading edges only and the result is a negative (left) yaw contribution from a right roll.
    cnp1
  3. Wing tip suction (-/+). This effect is encountered when a lift generated wing experiences roll, where a side force appears due to the asymmetry of the lift distribution and the  resulting asymmetry of the tip vortexes and tip side pressures. It is mostly evident in low aspect ratio, thick tip wings and its contribution can be either negative or positive based on the relationship between the tip side force center and center of gravity – if CoG is forward than the tip suction center the yawing contribution is negative.
    cnp2
  4. Tail contribution, C_{n_{p_{v}}} (+). In a right roll, the vertical tail of a conventional airplane experiences an increase in its angle of attack which generates a negative side-force on the tail surface, resulting in a positive yawing moment around the airplane’s CoG.

Correct prediction of the C_{n_p} is difficult and many times even the sign of the derivative is miscalculated and generally wind tunnel tests will be need to run to get a valid estimate. However, the effect of C_{n_p} in the airplane’s stability is frequently weak and its accurate prediction is not a critical item.

References

  1. US Air Force Flight Test Center, Flying Qualities Testing-Stability Derivatives.
  2. Bernard Etkin and Lloyd Duff Reid, “Dynamics of Flight, Stability and Control”.
  3. Ian Roskam, “Airplane Flight Dynamics and Automatic Flight Controls –Part I”.
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