# Quantify the insensitivity of a rotor to gusts compared to a fixed wing.

Of course nothing wrong with your assumption of disc AOA increase due to gust, but maybe you can explain in a little more detail how you used that to arrive at instantaneous load/thrust increase. I suspect that will explain the discrepancy.
My spreadsheet uses 15 tables of 11 columns (blade parts) and 24 lines (azimuth positions).
table 1: Tangential airspeeds of each blade segment for each azimuth
Table 2 Multiplier coefficient for average induced speed (non-uniform)
Table 3 Air velocity component perpendicular to section trajectory (takes into account previous tables and conicity)
Table 4 Relative air velocity (combination of tangential and "axial" velocities)
Table 5 Angles of attack according to previous tables
Table 6 Slope correction dCL/di as a function of Mach Number
Table 7 CL calculation based on tables 5 and 6, taking into account possible stall and aspect ratio
Table 8 Lift calculation based on tables 4 and 7
Table 9 Cd0 calculation based on table 5 and cd min initially entered
Table 9 bis Calculation of Cd according to tables 7 and 9 and possible stall
Table 10 Drag calculation based on tables 9 and 4
Table 11 Angle of pressure resultant relative to axis of rotation based on tables 9,7,5 and pitch setting initially entered
Table 12 Resultant thrust based on tables 10 and 8
Table 13 Torques based on tables 12 and 11
Table 14 Flapping torques as a function of tables 12 and 10
Table 15 Axial thrust components based on tables 12 and 11

(I can send you the file if you give me your e-mail address)

After a few checks, I can't find any errors in my spreadsheet, and the results that seem surprising to you seems confirmed by the simple reasoning:

My spreadsheet finds that the steady rpm is around 400 when the rotor load is 2144 N at 34.2 m/s (i.e. mu = 0.25).

This calculation is not in doubt, given the numerous comparisons with accurate in-flight measurements that have already been published in the past by Naca.

For that, the indicated angle of attack of the disk is 7° or 0.122 rad, which should give a perpendicular disk speed of 34.2 m/s * 0.122 = 4.17 m/s (only for such small angles)

The induced velocity found of 0.78 m/s is not in doubt either, in this case of high Mu ratio, where the disk behaves almost like a circulare fixed wing. This means that the flow through the disc has a perpendicular component of 4.17 - 0.78 = 3.39 m/s.

At 3/4 of R, which is assumed to represent the mean effective radius, the circumferential speed is 100 m/s. So, the mean aerodynamic A.o.A. of blades with pitch setting of 3.5° (0.061 rad) is therefore 0.061 + 3.4/100 = 0.095 rad (or 5.4 degrees) when load factor=1

A vertical gust of 2.5 m/s now occurs. It increases the angle of attack of the disc by 0.073 rad, which becomes 0.122 + 0.073 = 0.195 rad, giving a speed component perpendicular to the disc of 34.2 m/s * 0.195 = 6.7 m/s (If we still confuse the angle with its tangent).

The average A.o.A of the blades would now be 0.061 + 6.7/100 = 0.128 rad, giving a load factor of 0.128 / 0.95 = 1.35

Except that this overload obviously implies an induced speed correction which increases by about 0.27 m/s (*) The speed of the flow perpendicular to the disc is therefore rather 6.7 m/s - 0.27 m/s = 6.43 m/s

And the average A.o.A of the blades is therefore 0.061 + 6.43/100 = 0.125 rad, giving a load factor of 0.125 / 0.95 = 1.33, a value close to that obtained from the spreadsheet

(*) 1.055 m/s is the induced speed with factor load 1.33

JC, this (post 21) makes sense and I have no issue with it. My surprise has to do with your post 13. The green dots (cl max 2) is almost twice as large as red dot- (cl max is 1.3) . I am finding hard to see how this happens.

Since you agree the reasoning of my post 21, you can see that it's valid only if lift is proportional to angle of attack, i.e. as long as the stall angle is not reached (green dots). Otherwise, lift does not increase, or even decreases (red dots).

At mu= 0.25, before the gust, the retreating blade is already stalled up to 0.5 R (if CL max = 1.3)
During the gust, it will stall beyond 0.8 R with a low delta g
Whereas with CL max = 2, it will only stall farther than 0.8 R at higher overloads.
(therefore little change in the area of the stalled zone on the disc despite higher CLmax and overload)

Since you agree the reasoning of my post 21, you can see that it's valid only if lift is proportional to angle of attack, i.e. as long as the stall angle is not reached (green dots). Otherwise, lift does not increase, or even decreases (red dots).
but the impact of the cl max is in a very small region were the tangential velocity is low (<0.35 of tip speed) and hence contribution to total thrust is low as well. You are showing ~70% increase (green dot versus red dot) at 0.25 mu.

As you can see in the contours below, in the case of the 12 deg stall, the stall boundary (red zone), as you say, is in fact larger but the rest of the area acts at a higher cl to compensate. In other words there should not be much difference (certainly not 70% difference) . Do your stalled regions look similar ? Maybe your induced flow model causes a higher region of stall.

as long as the stall angle is not reached (green dots). Otherwise, lift does not increase, or even decreases (red dots).
Maybe this is the confusion- the reason for flattening of the slope with speed has to do with the relative impact of induced flow being diminished with air speed not the increasing effect of stall. Stall only has a meaningful impact in this rotor case after mu >0.35. If you did not factor induced flow (orange line) the curve would be flat.

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Do your stalled regions look similar ? Maybe your induced flow model causes a higher region of stall.
The non-uniformity of the induced speed does not change the value of the blade angle of attack at azimuth 9h - 3h. Consequently, it does not change the extent of the stalled zone. Only transverse flapping b1 is lowly affected.

Your stalled areas are smaller than shown by my spreadsheet. I guess you don't take into account the longitudinal flapping angle a1
which is important and has a direct impact on blade angles of attack at 9h - 3h(increasing the angle of attack of the retreating blade and decreasing that of the advancing blade)
( )

but the impact of the cl max is in a very small region were the tangential velocity is low
You can see here that for CL max = 1.3, the stall extends to more than 0.8 R (due to a1 = 3.8 degrees), where the tangential velocity is already hight.
Whereas if CL max = 2 it extends to just 0.5 R, where the tangential velocity is still low.

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Thanks JC! I will look at the numbers in the table closely. Meanwhile, my calculations/ contour plots of AOA do take into account a1 but ignore other flapping modes that depend on blade mass constant gamma. The a1 I get though is 2.8 deg in the gust case.

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As I said, I compared my results with the in-flight measurements presented in Table 1 of NACA report n°475 for four situations covering a wide range of mu (from 0.11 to 0.52). Glide flights avoids the disruptive effects of propeller blast::
Rrpm calculated: 141.7 measured141.1
A.o.A disc calculated 27° measured 26
a1 calculated: 1.3°, measured: 1.1°

Rrpm calculated: 142.3 measured139.1
A.o.A disc calculated 13.2° measured 12°
a1 calculated: 2.2°, measured: 1.5°

Rrpm calculated: 140.8 measured142.2
A.o.A disc calculated 7.3° measured 7.6°
a1 calculated: 3.4°, measured : 3.1°

Rrpm calculated: 132.9 measured133.1
A.o.A disc calculated 7.2° measured 6.1°
a1 calculated: 5.8°, measured : 5°

Of course, the manufacturing torsion, elastic torsion under load, and planform indicated in the Nace 575 report have been taken into account, as well as air density, and the no-rotative lifting.

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As I said, I compared my results with the in-flight measurements presented in Table 1 of NACA report n°475 for four situations covering a wide range of mu (from 0.11 to 0.52). Glide flights avoids the disruptive effects of propeller
JC, It seems to me you assumed the measured AOA in table 1 is the AOA of the rotor (control plane) and you converted it to the measured disc AOA by adding a1 in table 1 to AOA. I suspect (based on the reading of ref 1) that the AOA in table 1 is the fuselage AOA and not rotor. This makes sense as it is fixed rotor with fuselage and wings. Am I missing something ?

I did a quick check check with the data on NACA 487 and plotted my points (blue dots) on it and got reasonably good match as well. One thing I noticed was I had to use a higher min drag coefficient (0.120 )and consequently this meant that the rotor operated at higher inflow and hence vulnerable to stall earlier. In my prior calculations here I have used min drag as 0.0087. I guess that may explain to a larger extent why you are seeing earlier stall impact- as your profile drag perhaps is higher.

Wonderful work, Raghu! I noticed, as you did, that the Cdmin = 0.012 should be used for these old rotors. I guess it's because of the drag from their many cables and shrouds. For our modern rotors, I use: profile Cd = 0.009 + 0.008*CL².

For my part, I made some unforgivable careless errors on the green dots (I entered a constant gust angle instead of one that decreases with forward speed). Corrected below

This obviously impacts the area of the unstalled zone. Your intuition that delta g in gusts seemed too highn when CL max =2, was correct.

That doesn't change my conclusion that the discrepancy (green Vs red) is only due to the expanded stalling, but I'm ashamed.

You haven't forgotten anything. The angle of attack of the disc is equal to the bearing angle added to a1

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Thanks JC! Yes these numbers seem a lot more likely and much closer (~10% rather than ~70%). Thanks also for helping me validate my model with this discussion and Pitcairn data.

One thing that I still am a little curious about is
1. your lift versus aoa function is very steep at stall (cl drops from 1.3 to 0.5 instantly at 12 deg).

2. and combine this with rather large evenly divided rotor segments that you sum the forces over

I wonder if small changes can result in big swings in results (based on were the boundaries land). Have you tried changing the total number of segments from 10 to say 20 and seen if there is any impact ? Or keeping the same number of segments but making the inboard segments larger and the outboard ones smaller (i.e finer resolution were it has a bigger impact)?

Perhaps you have done this before and it may well not be an issue. It is just something that stood out.

You're right to wonder about the effect of the sudden drop in CL between two neighboring segments at the stall limit, due to the overly simple model adopted.
But this discrepancy is only localized on the retreating blade, at < 0.5 R tip speed, and for a small DElta azimuth of 15°.
As you yourself have seen, even the choice of a very different CLMax has little impact on performance.

Have you tried changing the total number of segments from 10 to say 20 and seen if there is any impact ? Or keeping the same number of segments but making the inboard segments larger and the outboard ones smaller (i.e finer resolution were it has a bigger impact)?
Your suggestion is clever. I'll see if I can exploit it.

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