Local Cl values for a wing during a turn...

[SoarNeck]

Hi folks, need some more help. I somehow need to rationalize what
happens in a turn to a wing with variations in local Cl.

My thoughts:

For a given radius turn, it's relatively straightforward to
calculate the amount of lift required to get through the turn...
that's just mainly a function of mass and radius. Drag
should really only determine the speed the model has coming out of the
turn, assuming that it's flown to the given radius.

We then get into the plot of local Cl that can be obtained through
liftroll, which is really just a measure of how hard each bit of the
wing is actually working. An example plot can be seen on this link:

http://www.geocities.com/jebbushell/supplied.htm

For a wing with a flat local CL plot, each section of the wing
will be operating at the same Cl.

For a non-uniform distribution, however, this changes. If the tips
aren't carrying their share of the load and operate at a lower local
Cl, the root will have to pick up the slack and operate
at a higher Cl for the wing to generate a given amount of lift.

How do I determine what the numerical value for local Cl would be
at any given point knowing the required amount of lift and having
the LiftRoll plot? I sense the need to integrate something...

[Ollie]

The local angle of attack determines the local lift coefficient. The local lift is proportional to the local lift coefficient times the square of the local airspeed.

The local air speed and the sinking speed produce a local angle of attack variation along the span because the local air speed varies along the span according to the distance of the local spanwise station from the center of the turn. So the local lift varies because of the local airspeed squared and also because of the changing direction along the span of the vector sum of the local airspeed and sinking speed. If that were not complicated enough, the angle of attack component due to dihedral and yaw must be included along with any effects due to aileron deflection.

At his point I would ask myself if I really needed to know the lift distribution of the wing in a turn, why and how accurately I need the calculations to be. Unless you can limit the general case, make some simplifing assumptions and justify them, you are letting yourself in for some very complex calculations.

[Sparky Paul]

Considering how the inflow across the span of a 5-panel wing, say, would vary across each panel alone, not even considering merging all of them...
What Ollie says! :)

[markdrela]

Originally posted by Ollie
At his point I would ask myself if I really needed to know the lift distribution of the wing in a turn, why and how accurately I need the calculations to be.

Predicting the spanwise Cl in a turn is the most reliable way to design against tip stall. The extensions to a standard Vortex-Lattice method to allow it to do this are actually pretty simple. There is a bunch of example plots showing steady-turn Cl distributions at
http://groups.yahoo.com/group/Allegro-Lite/files/Vortex_Lattice/

It should be doable to put this capability into LiftRoll, since it's basically a bare-bones VL method.

The things which must be added are:

1) Roll rate, yaw rate, and sideslip angle contributions to local velocity.
2) Local dihedral angle effect on the local Cl relations.
3) Control-deflection contributions to the local zero-lift angle
4) Roll moment calculation.
5) Iteration scheme to converge to zero roll moment, varying either
a) sideslip for a poly glider
b) aileron deflection for an aileron glider

Items 1,2,3,4 are fairly straightforward extensions to the basic VL equations -- just a bunch of additional terms get added. But the iteration may get awkward in LiftRoll's spreadsheet implementation, dunno.

[Ollie]

Thanks, again, Mark. My education continues.

[SoarNeck]

Thanks to all three of you for your help so far. If you can bear with me through a few more questions, I'd really appreciate it. I've designed some pretty nice models using rules of thumb (a Thornburg disciple ;) ), but now I'd like to understand more of the theory.

Originally posted by Ollie
[B]The local angle of attack determines the local lift coefficient. The local lift is proportional to the local lift coefficient times the square of the local airspeed.


If I understand correctly, the local angle of attack is determined by the main wing angle of attack, plus/minus any geometric twist that might be added. To model this in LiftRoll, a compensation for differences in the zero-lift condition for two airfoils may be necessary.

To what effect will spanwise flow affect the local airspeed at a given station? How innaccuarate is it to say that the local airspeed is the same for all stations along the wing in level flight? (= models airspeed?) Are we just concerned with the chordwise component of velocity in order to calculate RE?

Finally, specifically to LiftRoll now, how is it coming up with a number for local Cl? It's not asking for any airfoil parameters anywhere, so how do I figure out what the exact local Cl value is for a "real wing". Is the output for local Cl relative to some generic max value?

What I'm trying to get my mind around is this:

Say I have an F3J model in a level flight condition (or constant sink rate really...just not turning for now). I want to figure out what the local Cl conditions are at each point on the wing, so I can start to choose proper airfoils for each portion of said wing.

I know the airspeed the model is flying at, so (unless there is a spanwise correction factor) I also know the local airspeed at each
station. I can calculate the local Reynolds number from air density, local velocity and local chord. This is where I start to sense too many unknowns.

Can I calculate an angle of attack at this point? That would allow me to translate that into a local angle of attack and such, which I could then look up on a plot of Cl vs alpha, and then get Cd from a plot of Cl vs Cd (at the appropriate Re).

I don't know if I can know what the sink rate of the model will be without knowing the total lift generated by the wing (dependant on alpha), which would be the vertical components of a force balance on the system (force of gravity acting on a given mass - lifting force).

Help please...the self-taught nature of my aerodynamic knowledge is rearing its ugly head :)

[shoe]

Originally posted by SoarNeck
I've designed some pretty nice models using rules of thumb (a Thornburg disciple ;) ), but now I'd like to understand more of the theory....
...Help please...the self-taught nature of my aerodynamic knowledge is rearing its ugly head :)

I'm in the same situation.

*grabs some popcorn and gets comfortable*

This is going to get interesting... :cool:

[SoarNeck]

Okay, I think I can partially answer my own question. I think I can get local Cl from:

V= (2x(w/s)/(rhoxCl))^0.5

I just need to know what I should be using for local velocity (just model airspeed....or is that too easy?).

[Sparky Paul]

"This is going to get interesting... "
Looking at Mark's reply, the definition of "simple" just went up a notch or three. :)

[SoarNeck]

Okay, I was never much good at math, so I need some help with the implementation.

Given that for a single panel with constant taper:

h1 = chord at left of panel
h2 = chord at right of panel
c1 = drag coefficient at h1
c2 = drag coefficient at h2

p = rho, air density
D = drag of a panel
v = local air velocity
s = area of the panel.
w = total panel span
Cd= coefficient of drag

D= Cd*s*p*v^2/2

Assumption: drag coefficient varies linearly as the airfoils blend.
Seems okay for airfoils that are likely to be similar anyway (no
S1223 -> S7037, for example).

Thus, if drag varies linearly from c1 -> c2, and x varies from 0 -> w

c(x) = c1+(c2-c1)x/w, where: c(0) = c1, c(w) = c2

Similarly, chord varies by the same relation, so:

h(x) = h1+ + (h2-h1)x/w, where: h(0) = h1, h(w) = h2

Itegrating h(x) should give an expression for panel area S as a function
of x, so:

s(x) = h1*x + (h2-h1)*x^2/w

Subsituting into the equation for drag:

D= Cd*s*p*v^2/2

D(x) = [c1+(c2-c1)x/w]*[h1*x + (h2-h1)*x^2/w]*p*v^2/2

Can I now integrate D(x) from 0 -> w to find total lift generated by the panel?

Remember, math was never my strong suit...and I'm sensing the last step is flawed.

[Sparky Paul]

"If I understand correctly, the local angle of attack is determined by the main wing angle of attack, plus/minus any geometric twist that might be added. "
.
Yes.
That's simple. :)
When you turn though, the inboard wing is going slower than the outboard, the di-poly-hedral for each section determines the -local- alpha for -that- part of the wing outboard and inboard...
Pretty much what you've outlined.
You'll have a bunch of panels to consider when blending.
Yup, it's gonna get interesting... :)

[shoe]

lemme know when I should whip out the C++ compiler and put this into a program. :D

[markdrela]

Originally posted by SoarNeck

Can I now integrate D(x) from 0 -> w to find total lift generated by the panel?


I'm not sure what you're trying to calculate here. Lift or drag? You don't get lift by integrating the local drag distribution.

The equation for the local cl at spanwise location y, in level flight is:

cl(y) = 2 pi [ alpha - aZL(y) + ag(y) - ai(y) ]

where

alpha = overall aircraft AoA
aZL(y) = local airfoil zero-lift angle
ag(y) = local twist angle, + for washin, - for washout
ai(y) = local induced angle

ai(y) is the most difficult to calculate, and LiftRoll uses a simple Vortex-Lattice method to get it. It depends on the overall cl distribution, so things actually get rather involved -- a linear system has to be set up and solved.

In any case, if you want to compute the lift and rolling moment on a wing in turning flight, the extended equation now is:

cl(y) = 2 pi [ (alpha+py/V)*cos(D) + beta*sin(D) - aZL(y) + ag(y) - ai(y) + a_d*delta ]

The new stuff is:

p = airplane roll rate, rad/sec
V = flight speed
beta = airplane sideslip angle
D(y) = local dihedral angle from horizontal
a_d(y) = change in effective local AoA per unit aileron deflection
delta = aileron deflection


Once you have the local cl(y), the local loading (force/span) is:

L'(y) = 0.5 rho (V - ry)^2 c(y) cl(y)

where

r = airplane yaw rate, rad/sec


To get total lift and roll moment, you sum or integrate across the span:

L = sum[ L'(y) * dy ]
Mroll = sum[ -L'(y) * y * dy ]

To compute a trimmed turning-flight case with turn radius R, you have to:

1) vary alpha to make L=weight*N
2) vary either beta or delta to make Mroll=0

where N is the load factor. Also, you have to set

phi = specified bank angle
N = sqrt[ 1 + tan(phi)^2 ]
r = (V/R)*cos(phi)
p = 0

For a steady roll rate (turn entry), you instead set
N = 1
r = 0
p = specified roll rate

The following are calculation by-products in each case:

1) alpha
2) beta or delta
3) spanwise cl(y) distribution


OK, so it wasn't "simple".

[markdrela]

BTW, here are the symbols I recommend using. They are pretty much standard.

b = span
y = spanwise coordinate, runs from -b/2 to +b/2

cl = local cl
CL = overall airplane CL

Avoid using "Cl" when you mean cl or CL. This is because the following standard symbols are also used:

Cl = roll moment coefficient
Cm = pitch moment coefficient
Cn = yaw moment coefficient

And while we're at it:

p = roll rate, + right tip down
q = pitch rate, + nose up
r = yaw rate, + to right

sideslip beta is + with wind on right cheek.

[SoarNeck]

Hi Mark,

Thanks a lot, I appreciate the help.

Sorry that I'm a little bit all over the place, but I'm doing a couple of things at once. In the above I was trying to get toal drag from integrating the local drag distribution...that's wrong I guess? I'm simultaneously trying to help a friend develop a program to simulate the difference between two models during a pylon race, so given that thrust (and mass) stays constant between two models with the same driveline, and lift is really only a factor in determining drag for a given turn diameter, drag losses can be used as the the determining factor as to which pylon racer is "better". I plan to neglect the effects of the fuselage for this purpose, simply because it should be a constant between two models for purposes of comparison.

While we're on the topic of pylon racing, could you give me your thoughts about this one (photo attached). Is there some advantage to this planform that would justify the extreme taper ratio, or is it a throwback to the Schuman planform discussions of a few years ago?

[SoarNeck]

Sorry I see the confusion now (typo). I went back and re-read what I posted and it should have read:

Can I now integrate D(x) from 0 -> w to find total DRAG generated by the panel?