I have been spending a lot of time lately with TOWS chapter 1, NACA TR 572, Peery ch. 9, and Gudmundsson regarding lifting line theory and spanwise lift distribution. In particular, I am interested in wings that are elliptical (or nearly so) in planform.
In short, I am concerned that the addition of flaps to an elliptical wing may lead the tips to stall first when the flaps are deployed (not a good thing when you're in the flare with a crosswind). I believe this is unique to elliptical wings (and perhaps extremely tapered) and does not apply to rectangular or moderately tapered wings.
Now that my sound counterintuitive at first glance since the outboard portion of the wing obviously has less angle of attack than the flapped portion--but the c_l distribution of elliptical wings is different. Here is my thinking. (buckle up!)
...And is this really an issue on elliptical wings with flaps, or have I painted myself into a theoretical corner with invalid assumptions or poor interpretation of TR 572? I don't know. I can't seem to find much on the stalling manners with flaps deployed for Spitfires or P-47s or the more recent Silence Twister. (I'm guessing the crescent-winged F1Reno Roswell guys don't have flaps, although that is also another form of an elliptical wing.)
Background (just to make sure we're all on the same page--this is just a summary of TR 572):
Now for an elliptical wing, the principles are the same but the picture looks different. Next let's look at a an elliptical wing without twist. Without twist and with uniform airfoil sections, basic lift is zero. Prandtl and Munk demonstrated that spanwise lift distribution (following the circulation across the span) is elliptical. Since the lift distribution is chord length (which is also elliptical) times lift coefficient, we know that the entire wing is operating at the same c_l. Hence we know the whole span will let go at once when C_L reaches c_lmax. (In reality, the tip is operating at lower Re and therefore the c_lmax line droops near the tip and consequently the very tip stalls first.)
We can use a thicker or more cambered airfoil at the tip to help control tip stall. Next we'll just use a thicker airfoil at the tip which has a higher α_stall but the same α_ZL so there is zero twist and c_lb is still zero. Notice on the c_l distribution, c_lmax increases steadily from the root to the tip, and therefore at C_Lmax, the wing stalls first at the root. Although the lift distribution is unchanged* we've incurred a drag penalty with the thicker airfoil.
*Admittedly this is an assumption and I see no evidence in Anderson's multi-foil example in TR 572 that different root/tip foils change c_lal, but if I have missed something here please point it out!
The problem:
Now let's add a plain flap to the untwisted, single-airfoil elliptical planform from BL0 out to 60% of the semispan. As far as the air is concerned, this is just a discontinuity in the airfoil section and twist, occurring at 60% semispan. Let's say for the sake of discussion that this results in a discontinuity of 10° between α_ZL of the flapped portion and α_ZL of the outboard portion, and that Δc_lmax of the flapped portion is roughly 1.0.
Let's examine c_lb first. We know a) that c_lb = (1/2)(lift-curve slope)(α absolute) and b) the area under the curve of L_b, over the semispan, is zero. Also the discontinuity should be faired (as described by Peery in his fig. 9-15) since the air molecules can see their neighbors and aren't aware of Excel graphs and their sharp turns.
The thing that is troubling me is on the right-hand graph. The solid blue line, representing C_L, first touches the dashed gray line at the outermost 20% of the span, indicating the tips are sure to be the first thing to stall with the flaps in this configuration--no bueno. Adding a little twist or a different tip foil might help an elliptical wing (without flaps), but it doesn't appear to me that could fix this problem with the flaps deployed. (Or is it a problem at all in reality...?)
What this suggests to me is that full-span flaperons might be a better choice for an elliptical planform, in which case some stall margin could be maintained with a thicker and/or more cambered tip foil.
Why I don't think this is a problem for other planforms:
Applying the same method to a Hershey bar or moderately tapered wing with flaps, there is less of an issue because there is still some margin between c_lmax - c_lb and C_L*c_lal out near the wing tip. The tapered wing is of course more critical than the Hershey bar, so I'm only showing a tapered example here, but you can use your "engineering imagination" to visualize the rectangular version.
Worthy of note: This 0.5 TR wing with 1.75° of washout achieves nearly identical lift distribution and C_Lmax as the previous elliptical example, but with flaps deployed, it's stall behavior is much improved. More moderate taper would help in this regard (maybe 0.6 c_t/c_s or greater).
Finally...
If those lift distribution curves with flaps look familiar, maybe you've been reading Roncz's Sport Aviation design series from 1990. I didn't set out to recreate what he presented--I came about it using stuff published by the NACA in 1940--but it is interesting that the curves are coming out as he predicted. Just for grins, I measured these "rough" curves and came up with 92% vs. his 93%.
In short, I am concerned that the addition of flaps to an elliptical wing may lead the tips to stall first when the flaps are deployed (not a good thing when you're in the flare with a crosswind). I believe this is unique to elliptical wings (and perhaps extremely tapered) and does not apply to rectangular or moderately tapered wings.
Now that my sound counterintuitive at first glance since the outboard portion of the wing obviously has less angle of attack than the flapped portion--but the c_l distribution of elliptical wings is different. Here is my thinking. (buckle up!)
...And is this really an issue on elliptical wings with flaps, or have I painted myself into a theoretical corner with invalid assumptions or poor interpretation of TR 572? I don't know. I can't seem to find much on the stalling manners with flaps deployed for Spitfires or P-47s or the more recent Silence Twister. (I'm guessing the crescent-winged F1
Background (just to make sure we're all on the same page--this is just a summary of TR 572):
- c_l is the airfoil section's (2D) lift coefficient.
- c_lmax is the maximum lift coefficient the airfoil section can deliver before stall.
- C_L is the wing's (3D) lift coefficient at a given angle of attack.
- C_Lmax is the C_L at which some portion of the wing has reached its local c_lmax.
- Lift distribution (labeled w on the graphs below) at any spanwise station is the product of its chord c and the local lift coefficient c_l (well, and q also, but that's not important for this discussion).
- The lift distribution along the semispan is the sum of two functions:
- "Additional" lift distribution, whose quantity is proportional to the angle of attack and whose shape depends on aspect ratio and planform; and
- "Basic" lift distribution, which depends on twist and the selection of the root and tip airfoils, and whose shape and quantity does not vary with angle of attack.
- The additional lift and the basic lift have corresponding coefficients c_lal and c_lb.
- The additional lift coefficient curve corresponds to a wing acting with C_L=1.0. This line is scaled up (or down) as C_L increases or decreases.
- The basic lift coefficient shows the c_l distribution when the wing is at an angle of attack producing zero net lift. (The inboard portion may be producing some lift but the outboard portion is producing negative lift.)
- To determine the C_L at which some portion of the wing has reached cl_max, the additional lift coefficient is scaled until it touches the line c_lmax-c_lb. In other words, at that point, additional lift + basic lift = max lift coefficient and the wing has reached C_Lmax.
Now for an elliptical wing, the principles are the same but the picture looks different. Next let's look at a an elliptical wing without twist. Without twist and with uniform airfoil sections, basic lift is zero. Prandtl and Munk demonstrated that spanwise lift distribution (following the circulation across the span) is elliptical. Since the lift distribution is chord length (which is also elliptical) times lift coefficient, we know that the entire wing is operating at the same c_l. Hence we know the whole span will let go at once when C_L reaches c_lmax. (In reality, the tip is operating at lower Re and therefore the c_lmax line droops near the tip and consequently the very tip stalls first.)
We can use a thicker or more cambered airfoil at the tip to help control tip stall. Next we'll just use a thicker airfoil at the tip which has a higher α_stall but the same α_ZL so there is zero twist and c_lb is still zero. Notice on the c_l distribution, c_lmax increases steadily from the root to the tip, and therefore at C_Lmax, the wing stalls first at the root. Although the lift distribution is unchanged* we've incurred a drag penalty with the thicker airfoil.
*Admittedly this is an assumption and I see no evidence in Anderson's multi-foil example in TR 572 that different root/tip foils change c_lal, but if I have missed something here please point it out!
The problem:
Now let's add a plain flap to the untwisted, single-airfoil elliptical planform from BL0 out to 60% of the semispan. As far as the air is concerned, this is just a discontinuity in the airfoil section and twist, occurring at 60% semispan. Let's say for the sake of discussion that this results in a discontinuity of 10° between α_ZL of the flapped portion and α_ZL of the outboard portion, and that Δc_lmax of the flapped portion is roughly 1.0.
Let's examine c_lb first. We know a) that c_lb = (1/2)(lift-curve slope)(α absolute) and b) the area under the curve of L_b, over the semispan, is zero. Also the discontinuity should be faired (as described by Peery in his fig. 9-15) since the air molecules can see their neighbors and aren't aware of Excel graphs and their sharp turns.
The thing that is troubling me is on the right-hand graph. The solid blue line, representing C_L, first touches the dashed gray line at the outermost 20% of the span, indicating the tips are sure to be the first thing to stall with the flaps in this configuration--no bueno. Adding a little twist or a different tip foil might help an elliptical wing (without flaps), but it doesn't appear to me that could fix this problem with the flaps deployed. (Or is it a problem at all in reality...?)
What this suggests to me is that full-span flaperons might be a better choice for an elliptical planform, in which case some stall margin could be maintained with a thicker and/or more cambered tip foil.
Why I don't think this is a problem for other planforms:
Applying the same method to a Hershey bar or moderately tapered wing with flaps, there is less of an issue because there is still some margin between c_lmax - c_lb and C_L*c_lal out near the wing tip. The tapered wing is of course more critical than the Hershey bar, so I'm only showing a tapered example here, but you can use your "engineering imagination" to visualize the rectangular version.
Worthy of note: This 0.5 TR wing with 1.75° of washout achieves nearly identical lift distribution and C_Lmax as the previous elliptical example, but with flaps deployed, it's stall behavior is much improved. More moderate taper would help in this regard (maybe 0.6 c_t/c_s or greater).
Finally...
If those lift distribution curves with flaps look familiar, maybe you've been reading Roncz's Sport Aviation design series from 1990. I didn't set out to recreate what he presented--I came about it using stuff published by the NACA in 1940--but it is interesting that the curves are coming out as he predicted. Just for grins, I measured these "rough" curves and came up with 92% vs. his 93%.
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