Planform K Factor (Attention Igor)

Hi

Igor I moved to a new thread as we were diverging from the original.
Thanks very much for your charts.

I am interested in persuing this further. Not so much from CL
application but my interest in design has mostly related to RC
soaring and not so much to be serious about it all, just for some intellectual interest. Some years ago I wrote programs to assist
in optimising aspect ratio and wing taper for performance. Well there is arguably some folly in this but it sure was an interesting exercise and some of that interest remains.

You did not comment on my method of estimating K factor in the Cdi
equation. I would be very interested in you thoughts.

Mostly what I did was extrapolated from study. Basically I did not
have a method from texts but wanted to see if I could make some
reasonable conclusion on my own. It seemed to me that I did not
need anything very exacting. I was not trying to predict or model
accurate values, mearly compare one wing taper with another. If the values were proprotionatly relatable, that was enough.

So the basis of my method:

Looking at spanwise Cl diagrams from texts, constant chord wings show
an eliptical distribution of Cl across the span. As lift is a
function of area and Cl, It seemed that lift distripution would be a
function of local Cl and chord. In the above constant chord example,
the lift distrubition is therefore eliptical.

Next example, An eliptical wings Cl distribution is constant across the span while the chords are eliptical. This again results in an eliptical lift distribution.

Now I am assuming a great simplification that the wing has a constant section, is free of twist and is not swept. I know the issue of lift distribution is rather more complex and this model does not hold near the tips. I am looking for a simple model that will allow me to do some comparisons with simple maths.

My conclusion from this is that the lift force distribution is close to eliptical regardless of wing taper. Good enough for my purpose any way. What do you think.

If that is true, then I can calculate the Cl at any point along the span as simply the chord of an eliptical wing of the same span and area divided by the local chode of my wing. That is, where my wings chord is shorter than the chord of an elips, Cl will be higher average for my wing. Where My wings chord is longer than the chord of an elips, My Cl will be less than average for the wing.

Correct so far, your thoughts?

This immediatly gives some clue to the areas subject to early stall.

Further if Cl is varying along a geometrically straight wing, there must be variations in induced incidence along the span. This meand that the lift and drag vectors are inclined differently along the span. So taking the horizontal vector of our inclined lift and summing these gives an estimate for K.

Now if I remember correctly from years ago, I got values of K between 1 and 1.1 as you suggest is the likely range.

What do you think so far?

Your chart seems fine for single tapers wings but I was looking for something to handle double tapers. If I get a chance on the weekend, I'll dig out my old C128 and run the program to compare with your chart. That should be interesting, at least to me.

LAST EDITED ON Oct-16-04 AT 06:20 AM (CDT)
 
Hi Ken,

First of all I must tell you I never invested too much time to this problem, as it is really clear on our stunt models and very unclear on gliders.

>>> I did some estimates based around vectors of local induced incidence which I derived from local Cl. I based local Cl on the proportion of local chord to local eliptical chord as that seems to be roughly ture. However I do not know if that is really a valid approach. <<<

If you speak about regular conditions (see below) then it COULD be true. But I think it will need validation if it really matches reality. The induced drag comes from modified angle of attack and thus also modified direction of lift vector. So it makes a component oriented back and that is that induced drag. So proper way is to take small segment of wing, calculate chord difference to the next segment, calculate how much of sidewise leakage goes out of the area, that gives amount of lost air and that compared to kept air makes a component of vertical speed added to horizontally incoming air (it is oriented down). That little changes the incoming air DIRECTION – and that tilts whole coordinate system and makes that modified angle of attack. Now if you integrate over all span and if know that angle and overall lift, you can find also amount of that tilted lift vector as donation to the drag. It is not simple, but if you want only mental exercise, try to think if this matches what is happening on elliptical wing and on your wing and if that your system matches reality. I afraid that you will find hole somewhere at some goniometric function which translates that angle of lift vector to horizontal drag. I think that vertical component of incoming air is linear to spanwise flow, thus also angle of that vector is linear, but drag will not be linear because of nonlinear goniometric function which translates it, but I am not sure.

>>>Looking at spanwise Cl diagrams from texts, constant chord wings show an eliptical distribution of Cl across the span.<<<
OK

>>>As lift is a function of area and Cl, It seemed that lift distripution would be a function of local Cl and chord. In the above constant chord example,
the lift distrubition is therefore eliptical. <<<

It would be in regular conditions. That is the difference to elliptical planform. But you are speaking about soaring. It is typical that it flies at relatively high lift coefficient (or at max if you want minimal vertical speed). And now comes question – do you really know the lift coefficient on every particular place of wing? You have induced AoA, which can change condition in middle of wing and also at tips and run out of linear lift/alpha. It can happen that you will have higher effective AoA at tips and it will come to critical AoA and thus lift will be smaller that you expect and that means also lower induced drag. You can come also to problems with RE number. Smaller RE at tip can lead to smaller lift than you expect and thus also to smaller induced drag. … you know what I mean. … and if you combine it with different effective (geometrical) AoA on tip …

>>>I can calculate the Cl at any point along the span as simply the chord of an eliptical wing of the same span and area divided by the local chode of my wing. That is, where my wings chord is shorter than the chord of an elips, Cl will be higher average for my wing. Where My wings chord is longer than the chord of an elips, My Cl will be less than average for the wing.<<<

Yes, I agree that longer chord will make smaller lift, my question is if it is really linear.

>>>This immediatly gives some clue to the areas subject to early stall. <<<

That is true, that is why tapered wing needs negatives at tips.

>>>Further if Cl is varying along a geometrically straight wing, there must be variations in induced incidence along the span. This meand that the lift and drag vectors are inclined differently along the span. So taking the horizontal vector of our inclined lift and summing these gives an estimate for K. <<<

Yes, try your calculation and try to check it with that chard. You will see … however even if it does not match reality perfectly, it is still better than only guessing