Expert: Paul Soderman - 9/20/2007

Question
QUESTION: Hello!
I know the explanation of induced drag as the energy wasted in the vortices, which manifests itself as a distortion of the pressure distribution on the airfoil.  But, I could also look at it from the standpoint that if the aircraft slows down (loses momentum) as a result of these vortices, where does the lost momentum appear?  Do the trailing wing tip vortices also get "dragged" forward?  Does the distorted pressure distribution  on the airfoil also cause the air to have forward momentum imparted to it?  (which is the lost momentum of the aircraft)

ANSWER: Joseph
I don't understand your questions completely, so let me explain what I think are the relationships of the various phenomenon you mention.  Start with drag.  Aircraft drag has many components, but they fall into viscous drag and induced drag.  Viscous drag includes friction, flow separation, form drag, etc. and induced drag is drag caused by lift.  Induced drag results from a rotation of the lift vector aft because shed vorticies create downwash at the wing to give the onset flow an induced flow angle.  The result is a rotation of the lift vector aft such that a component is in the drag direction.  So, I don't know what you mean by "energy wasted in the vortices".  The vortical flow is a key part of the picture, but is not considered a drag component itself unless you are talking about flow separation, which causes drag and also sheds vorticies.

Now, suppose we increase the aircraft drag without changing thrust.  From Newton's law we know that the aircraft momentum will decrease.  The lost momentum doesn't appear anywhere, the momentum just changes.

I also don't know what you mean by "vortices dragged forward".  The trailing vortices are always connected to the wing.  As for the pressure distribution, it will change but it does not impart forward momentum to the air other than the usual distortion of the flow field by a body moving through it.  Your thought process is unclear to me.
Paul

---------- FOLLOW-UP ----------

QUESTION: Paul:
Well, from the way I understand momentum conservation, the "force" that the air applies to the airplane, slowing it down, must have an equal but opposite force on the air "dragging" it forward.  For instance, friction drag, I have a flat plate approaching a stationary fluid element.  The friction of the fluid element against the plate causes the plate to feel a "retarding force".  The equal but opposite force on the fluid element drags the fluid forward.  The momentum that the "fluid" has gained, has come from the aircraft.  Since wingtip vortices are a form of drag, causing the aircraft to slow down (lose momentum) the air that makes up the vortices must "feel" a forward tugging force.  I hope that makes my question clearer?

Answer
Joseph
Most aerodynamic problems are analyzed with a coordinate system attached to the aircraft, and the air flowing past as in a wind tunnel.  Consider a flat plate in a wind tunnel perpendicular to the flow with a small element of air approaching with mass m and speed v.  If the air element hits the center of the plate it's velocity will be zero, and it will give up it's momentum to create high pressure on the plate, which results in drag.  Because of viscosity, the neighboring elements will draw our little element around the plate and it will accelerate back to full speed (thanks to momentum being put into the flow by the wind tunnel fans).  It is true that the fluid element felt a pressure gradient or force as it approached the plate, but at no time was it dragged forward.  The same can be said for a fluid element that encounters a wing tip and is swept into a vortex.  Depending on location, it will slow down or accelerate and follow the local stream lines but not get dragged forward.

A coordinate system attached to the fluid would result in the same analysis from a different point of view.

To summarize, you are correct that drag of a body is equivalent to a loss of momentum of the flow.  That momentum loss occurs in the entire wake behind the aircraft.  I believe that thinking that vortices are being tugged forward is confusing the issue.
Paul