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Can I have a critique of my explanation for induced drag
*Ignore this first post, it doesn't help explain anything!
The figure I've attached shows a Cl distribution along a hypothetical wing (y0 is a point along the wing span where the Cl changes). At 'y = y0' there is a pressure difference so air at 'y = y0 + dy' moves laterally inboard into the low pressure zone. This lowers the density at 'y0 + dy' thus lowering the pressure there. This in turn causes a pressure difference at 'y0 + 2dy' although a less significant one so the lateral motion induced at 'y0 + 2dy' is smaller than that induced at 'y0 + dy' so the pressure is still lowered overall at 'y0 + dy'. This process happens right up to the wing tip. A similar but opposite effect happens at 'y0  dy'. As the lateral flo w from 'y0 + dy' comes in, the density at 'y0  dy' increases thus the pressure increases there. Now the air at 'y0  dy' experiences a net pressure inducing lateral flow to go inboard. This lateral flow leaving 'y0  dy' is less than that entering hence the lift is decreased there. This can be modelled by a trailing vortex at 'y0'. Induced drag, induced thrust explanation (Not perfect, assuming constant pressure on lower surface of wing for simplicity) At the leading edge the spanwise pressure difference is the highest as no lateral flow has occurred yet. This results in the largest lateral motion from the high pressure zone to the low pressure zone. This decreases the pressure at the leading edge of the high pressure zone (as the density of air there has been decreased) and increases the pressure at the leading edge of the low pressure zone. That's my attempt at a physical explanation for why induced drag and a lift reduction occurs when a wing experiences a net downwash from the trailing vortices and why induced thrust and a lift increase occurs when a wing experiences a net upwash from the trailing vortices 

Last edited by Stazzo; Jan 03, 2017 at 08:22 AM.
Reason: Clarification





If you are trying to proof a mathematical equation that expresses induced drag, this is more technical than most modelers here would understand, including me. Perhaps you want to pose the question on an aeronautics board.
My understanding of induced drag is Neanderthal simple: it is drag caused by the airfoil generating lift at a given AOA or Cl. 





Thanks xlcrlee, so can we say that it(ID) is any drag other than the "native" drag of the airfoil at zero AOA that is, any drag induced by control inputs?






The wake behind a boat or airplane involves "roller" vortices and spray turbulence .... inherent in the very process of moving through a viscous fluid, water or air. In a wing, that includes spanwise flow, little and large vortices, etc. When one normally considers such extra motions forced on the flow it is for a fixed configuration, for ex., either for a wing with no control surface, a nondeflected control surface (the intersection creating turbulence) or with a deflected control surface.
I was referring to the physics involved in flows around a finite span wing where all motion is not 2dimensional (infinite span is a purely theoretical oversimplification). The induced drag is the realworld extra drag and consequent loss of lift caused by flows not totally redirected partially downward (the "deltarho vector" representing the change of momentum of the flow, which like jumping off a rowboat onto the dock, is the Newtonian reaction force that we call lift, etc.; in the rowboat case, the induced loss of energy would include your foot slipping, having to waste energy by jumping up to avoid falling into the water and even falling in the water, all not adding to moving the boat and being very inaccurate heuristic analogies, sorry ) 

Last edited by xlcrlee; Jan 02, 2017 at 07:00 AM.




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It seems to me that, for a flat plate at least, induced thrust has to come at the expense of lift if it is at a positive angle of attack. If it's at a negative angle of attack then induced thrust can be caused whilst increasing lift. Maybe it isn't a pressure drag, all I know is I'm confused! Quote:



Last edited by Stazzo; Jan 02, 2017 at 05:04 PM.





again, very rudimentary here, but I do not think a linear equation can express the relationships here because there are several variables. I do believe it can be solved with calculus, however. Func. Cl, dv/dA, >lim, where A denotes airflow at certain AOAs, and lim is stall angle. Then you could solve using airfoil data Cl/CD. Maybe...






I can more clearly explain why I don't understand induced drag now.
Using a flat plate, the resultant pressure force is normal to the surface of the flat plate. Lift is the vertical component of the resultant pressure force produced by low pressure on the upper surface and high pressure on the lower surface. If the airfoil is in the “downwash” of the wing vortex, this means it is accumulating air on the upper surface and losing air on the lower surface due to the lateral motion. This increases the pressure on the upper surface and decreases it on the lower surface. This means the net pressure difference has decreased meaning the resultant force is decreased so there is less lift but the drag has also reduced with it. This is why I don't understand induced drag, I always see it explained to me as though the downwash of the wing vortex increases the drag but I just physically can't see how this can happen unless its the following paragraph which explains it. To compensate for this loss in lift, the airfoil needs to be at a higher angle of attack to increase the pressure difference but now more pressure difference is required to get enough lift as the resultant force is more horizontal. The combined need for a higherpressure difference and angle of attack increase means the resultant force is larger and inclined further from vertical thus there is more pressure drag and this increase is induced drag. I don't know if this is true but it's what I'm going with for now 





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This is probably better explained in diagrams. The first one shows the direction of the freestream flow vector (red) and the lift, drag, and net force components if there is no downwash (in blue). You can think of this as the "infinite wing" or 2D airfoil case. The second diagram shows what happens when downwash is added (in pink). The local angle of attack is now decreased, and the lift vector is tilted back accordingly. The freestream flow vector and original lift vector are shown for reference. By tilting the lift vector backwards, an induced drag component appears (green), because the lift vector is no longer perpendicular to the freestream direction. (Note: in the second diagram, the skin friction drag component is not shown, and in both the vectors are not drawn to scale.) This is just how induced drag can be explained in air flow and force vectors. It actually shows up as a change in the pressure on the wing and is due to power being consumed in generating the tip vortices. 






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Thanks for being succinct! My generalizations (trying to cover ALL the notsimple "wiggly" movements) were obviously way too abstract 






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The wing vortices that arise because of a 3D wing cause lateral flow to occur over the wing. If the airfoil is in the downwash of the vortex as far as I'm aware the pressure on the upper surface will increase and the pressure on the lower surface will decrease (makes sense downwash forces air onto the upper surface of the wing and forces air away from the lower surface) thus reducing the resultant pressure force. This means there is less drag and less lift. The idea of the the lift vector rotating comes from the Kutta Joukowski theorem but this isn't a physical explanation as the theorem states that there is no pressure drag if the freestream is horizontal which is obviously false. I view at as being useful to estimate an induced drag value but it doesn't provide a physical explanation of induced drag. Quote:







I propose that your confusion comes from the fact that your concept of "pressure" grossly oversimplifies the actual situation. And what is happening in a viscous DYNAMIC fluid flow is that viscouslyconnected molecules (intramolecular forces!) are bouncing into and against the wing surface as well as each other .... and absolutely, in the aggregate not (as in NOT!) orthogonally.
Sorry to inject some reality into an otherwise pretty (oversimplified) "theory" 





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I made the mistake earlier in assuming you knew what you were talking about, won't be doing that again 






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To help understand it, consider the first three figures attached, which show the pressure distributions on NACA 00series airfoils as the thickness is reduced (12%, 8%, and 4%, respectively) at an angle of attack of 5 degrees. Each one has pressure vectors with components parallel to the chord line, particularly near the leading edge. As the thickness decreases, the surface area normal to the chord line gets smaller and smaller, but the suction pressure acting parallel to the chord line gets larger and larger. In the limiting case of a flat plate, the suction pressure would go to infinity and the associated area would go to zero. Multiplying the two together to get the suction force gives you an indeterminate answer. You don't know if it's infinite, zero, or some other finite value; you have to evaluate the limit to find out. The last figure shows the limiting behavior (from inviscid Xfoil calculations) of the leading edge suction force ("axial") and the normal force as thickness approaches zero. It turns out that at a constant angle of attack, the leading edge suction force remains almost constant and finite (in this case, the coefficient is ~0.05) as the thickness goes to zero. Thinking just about inviscid flow for now, the leading edge suction force is always just the right magnitude so that the resultant combination of the axial and normal components is the lift force predicted by the KuttaJoukowski theorem. This is the "missing" force that solves your conundrum of lift always acting normal to the freestream but pressure always acting normal to the surface. Quote:
It makes sense if you don't neglect the leading edge suction force, right? 


Last edited by Montag DP; Jan 03, 2017 at 09:13 AM.

