OK Knegel, I set aside some time to put the following together to explain in more detail. I hope you appreciate it!!
The Physics: Specific Excess Power (Ps) governs an aircraft's performance in a steady climb and a sustained turn.
(Don't let this elegant form of this equation trick you though! It looks simple but it's not and also not easy to calculate requiring knowledge of quite a few different curves and polars for a specific aircraft.)
So if a steady climb and a sustained turn are a function of the specific excess power of an airplane, why can't we extrapolate the performance of a steady climb to that of a sustained turn?
Let's breakdown just the power-required portion (Drag * Velocity) of the equation to demonstrate the reason why:
So when we breakdown power required, we see that it is a combination of the PARASITE power required and the INDUCED power required of the airplane. Assuming weight is constant, induced power required is a function of g-load, velocity, altitude, and oswald/span efficiency (e).
Excess power then depends on load-factor (squared), airspeed, altitude, and e. Ignoring altitude, the values of these variables are not the same between a steady climb vs. a sustained turn. The variables are also non-linear. Power required (overall) varies with cube of velocity and the square of load factor. Also e is no longer constant at higher angles of attack and varies non-linearly with increasing angle of attack.
The result is that the conditions for specific excess power for a steady climb are different from that of a sustained turn so you can't extrapolate steady climb performance to sustained turn performance.
Demonstration of Variation in Specific Excess Power - Steady Climbs vs. Turns Let's create a basic model of the F4U to illustrate the physics above.
For our basic F4U model let's use the following static parameters to compare a Corsair with and without flaps:
For propeller efficiency we'll use a generalized prop efficiency based on the following curve:
For key aerodynamic coefficients different between an F4U with and without flaps let's use the following values to compare the F4U’s:
CD0 come from a NACA wind tunnel F4U1-A test for flaps and no flaps. CL is calculated based on the 1g stall speeds listed. A couple of assumptions in my model: 1) it doesn’t factor in the effect of propwash on drag, and 2) constant e because I don’t have a good drag polar for the F4U to estimate the variation at higher aoa from.
Sparing everyone the gory math, using these parameters let’s see the results for specific excess power (Ps) for a steady climb vs. a sustained turn.
STEADY RATE OF CLIMB:
This is a comparison of the rates of climb at constant velocity for our F4U at 0 degrees and 50 degree flap settings. Rate of climb is specific excess power of the aircraft at 1g. Its obvious here that the excess power required due to flaps is much greater than that of the F4U without flaps deployed. The F4U with 50 degrees flaps deployed has a worse climb rate across the board compared to that of the F4U without flaps deployed.
Now let’s take a look at the Ps calculations for the F4U in a sustained turn, flaps and no flaps.
SUSTAINED TURN RATE AND Ps:The following are the calculations comparing turn rates of the F4U with and without flaps as well as the associated specific excess power. The performance is based on turns made at Clmax at the various velocities.
The Ps curves represent the specific excess power of the F4U with flaps and no flaps in a turn at Clmax. Notice how differently they are shaped compared to the Ps curves for the same aircraft in a steady climb at 1g. This is because of the impact on specific excess power due to variation in g-load and velocity in a turn vs. a steady climb.
Note the Ps axis and Ps curves. There is a point where Ps=0 for a turn at Clmax. Turns at Clmax above this airspeed result in Ps values <0 which would result in loss of altitude or airspeed. Turns at Clmax below this velocity result in Ps values >0 which mean the aircraft would gain altitude or airspeed. Point 1 is where Ps=0 for the F4U with 50 degree flaps. Point 2 is where Ps=0 for the F4U with no flaps.
Ps=0 is a key part of the aircraft’s envelope. This is where power-available of the aircraft directly equals power-required of the aircraft in a turn. This is the point where the best sustained turn occurs for the aircraft.
Plotted along with Ps is the rate of turn associated for that aircraft turning at Clmax at a given velocity for that specific excess power. If we draw a vertical straight line at the velocity for Ps=0, where the line intersects the turn-rate curve is the best sustained turn-rate for that aircraft. For our basic F4U model these are point 3 (for 50 degree flaps) and point 4 (for 0 degree flaps).
Notice the best sustained turn rates for our basic F4U model. For 50 degree flaps vs. no flaps they are essentially the same for all intents and purposes at ~18.5 dps which translates to ~19 seconds to complete 360 degrees!
So there you have it! Steady climb performance with flaps down is worse than that with flaps up, yet the best sustained turn rates are the same! Why? Because specific excess power varies with 1)load factor, 2)velocity, 3)altitude, 4)e – and these are different in a steady climb vs. a sustained turn. (Note also that in my basic model that I even left e constant and the results still yielded what they did.)
NACA F2A-3 Turn Performance ReportWhat about the NACA F2A-3 report referenced? It demonstrates the physics principles above at work.
The report talks about two types of turns tested: 1) constant speed turns and 2) sustained level turns. These turns were tested at various flap settings.
Let’s look at figures 32 and 34. These are diagrams show performance of the F2A-3 at 13,000 ft with 0 degrees of flaps vs. 56 degrees of flaps. Below I’ve taken figure 34 (56 degrees of flaps) and overlaid the “angle of climb” curve from figure 32 (0 degrees of flaps) onto the diagram.
The angle of straight climb curves basically give us a way to gauge the excess thrust of the F2A-3 in 1g flight with flaps and without flaps.
Sin(theta) = (T – D) / W
(theta) = sin^-1 [ (T – D) / W ]
(Theta) is the angle of climb. Knowing this gives us a way to calculate steady rate of climb for that velocity.
rate of climb = V * sin (theta)
So based on this equation we can visually see that the steady rate of climb of the F2A-3 is better without flaps vs. the rate of climb with 56 degrees of flaps. Power required is greater with flaps vs. without flaps. This relationship holds true except just under 80mph.
So now let’s take a look at sustained turn performance of the F2A-3 with and without flaps and see how that compares. Figure 22 and 24 in the report show the sustained turn radius and turn rates of the F2A-3 at 13,000 ft. Below are just the turn-rate portions of the graphs.
Notice how similar the best sustained turn rates are for 0 degrees of flaps vs. 56 degrees of flaps ~25-27 seconds to complete 360 degrees of a sustained turn.
So here we have it again. The F2A’s rate of climb without flaps is better than the rate of climb with 56 degrees of flaps. Yet the best sustained turn rates are virtually the same with flaps at 56 degrees vs. no flaps.
Why? Because specific excess power varies with load-factor, velocity, altitude and e which are not the same in a 1g steady climb vs. a >1g sustained turn.
A key note about all this. My calculations above aren't intended to measure the accuracy of how closely HTC has modeled the F4U. The intention was to illustrate the factors that govern specific excess power.
Hope this helps!
Tango, XO
412th FS Braunco Mustangs
