I'm sure many people wonder how Oswald e is found. I'm also sure many people use the CDi formula CL^2 / pi*AR*e to find CDi. Well, here is the dilemma for FS FDE designers.... if I need Oswald e to find CDi, how on earth is e determined in the first place and what should I set it to in my aircraft.cfg??? If you don't have real flight test data (and even if you do) how do you find the real world induced drag???
Below is a discussion of how to do those two things. It is advanced, but explains things in a way that even beginners will get enough to figure it out. Consequently, all formulas and definitions are explained thoroughly.
The mechanics:
When a wing is installed on an actual airplane, there is an additional force applied to it in flight that doesn't happen in a wind tunnel. As air spills around the wingtip it causes a downward load on the rear of the wing. Because a wing in flight has nothing solid to support it, the only way for the wing to counter this force is by allowing the downward force until the weathervane effect of the airflow can balance it out. This causes the wing to fly at a higher angle of attack than needed. It also increases the downwash angle.
This is half of the problem. The other half is the airflow begins to turn before it reaches the leading edge of the wing, so the wing is flying through airflow that is already turned some. The average of this effect is half of the downwash angle. This is shown by the middle green arrow in the image below.
This new angle of attack is called the induced angle of attack or ai for Alpha induced. It is probably the most important variable in aircraft wing performance, but it is never published. The induced angle of attack increases the wing angle of attack, and so changes the lift direction angle rearward. This is because the angle of attack required to produce a specific amount of lift at a given airspeed is constant. So the wing angle of attack becomes the induced angle of attack + required angle of attack to produce current lift. This required angle of attack is called the Effective AoA, or ae for Alpha effective. Alpha effective is half of the downwash angle. If you remember from the previous paragraph, the effective AoA is caused by the airflow already starting to turn and is present even in a wind tunnel. The effective angle of attack is based from the effective airflow angle. The top of the induced angle of attack is the base of the effective angle of attack, and matches the effective airflow angle (which is averaged). The is shown in red by the image below.
Induced drag is created by the difference between the effective airflow and the relative airflow, and is perpendicular to the induced AoA (which is equal to the effective airflow angle). Mathematically this looks like:
Actual alpha = Alpha induced + Alpha effective
Because of the lack of detailed information available, and far too many so called 'internet experts', there are several popular misconceptions about induced drag. One of them is that the wingtip vortex creates an actual drag force. This is not true. The force the vortex applies to the wing is dissipated by the counter balance of the wing. In other words, as the vortex tries to rotate the wing, nose up... the airflow is countering it and trying to rotate the airplane nose down. The wing wants to weathervane into the wind. The two forces find a balance point which is the new induced angle. At this point the vortex force is dissipated, but not without a consequence. The wing angle and lift angle are now tilted rearward. Induced drag is solely the product of the rearward lift that is produced when the lift angle is tilted slightly rearward. This also produces a higher Lift coefficient when measured against the wing AoA, but an equal or lesser Lift coefficient when measured against the fuselage AoA. This is because the higher downwash angle will produce more lift, but the rear tilt of the lift transmits less lift upwards.
note: Contrary to popular belief the vortex force actually increases parasite drag by increasing the frontal area of the wings and entire fuselage. This is accounted for in CDparasite so we can forget about it for this discussion.
Finding Oswald e:
Now that we understand ae and ai we can find Oswald e, but in order to find Oswald e we need to find Alpha induced or CDi first; or 2D Cl_alpha. This is why CL^2 / pi AR e is not useable in the real world (until after someone charts e the first time or finds CDi or ai.)
To make this easier to understand let's use a real airplane, and real specs. I am going to show you how to do this with the absolute minimal of data. But first, a simple chart...
For the lazy ones out there, use this:
CDp can be assumed to be = total drag at minimum drag speed.
For the rest of you, you will need:
Aircraft weight in level flight: 100,000 pounds
Aircraft pitch in degrees: 3.28 degrees
Zero Lift alpha to body, aZL_body: -2.05
250KIAS speed in Mach at sea level: 0.3779
wing span b = 93.333
wing area S = 1000.7
Aspect ratio AR = b^2/S = 8.705
These numbers are based on a Douglas DC9-30.
Finding e:
Step 1
Find pressure Q for your speed.
1481.4 * Mach^2 = 1481.4 * 0.3779^2 = 211.55 Q
Step 2
Find CL
weight / (Q * S) = 100,000 / (211.55 * 1000.7) = 0.47237 CL measured from weight
Step 3
Find Cla
2pi * { AR / [2 + sqrt(4 + AR^2)] } = 6.28318531 * { 8.705 / [2 + sqrt( 4 + 8.705^2)] } = 5.0033 Cla
Step 4
Find CLa
Cla * [ AR / (AR + 2) ] = 5.0033 * [8.705 / (8.705 + 2)] = 4.0685 CLa
CLa / Cla = very close to e. There are a few things that will cause small errors.
Step 5
Find Cl and CL for Cla method
(Aircraft pitch in degrees - aZL_body) * Cla = (3.28 - -2.05) * 5.0033 = 26.667589 Cl
(Aircraft pitch in degrees - aZL_body) * CLa = (3.28 - -2.05) * 4.0685 = 21.685105 CL
(if those Cl and CL look weird, don't worry...they are in radians...they relate directly to pi*AR, which for this airplane is 27.3476.)
-OR-
Find alphas for alpha_induced method (much more accurate)
(CL measured from weight / Cla) * 57.2957 = (0.47237 / 5.0033) * 57.2957 = 5.4094 degrees alpha2D. (We can compare this to our pitch - zero lift value of 5.33 and see our zero lift alpha has changed a bit.)
(CL measured from weight / CLa) * 57.2957 = (0.47237 / 4.0685) * 57.2957 = 6.6523 degrees alpha3D. (We can compare this to our pitch - zero lift value of 5.33 and see the effect of alpha_induced.)
Step 6
Find e using Cla method...
CL / Cl = 21.685105 / 26.667589 = 0.81316 e
This method is much more prone to errors. See step 7 for more accuracy.
-OR-
Find ai_radians for ai method...
[alpha3D - alpha2D] / 57.2957 = (6.6523 - 5.4094) / 57.2957 = 0.021693 ai_radians
Step 7
Final calculations:
Method #1
Here's a good way to tell how far off you are.
Compare:
CL / Cl against CL / (pi*AR)
CL / Cl = 21.685105 / 26.667589 = 0.81316 e
CL / (pi*AR) = 21.685105 / (pi * 8.705) = 0.79294 e
0.81316 e against 0.79295 e. There is no way that I know of to confirm if the error is valid or not. It could be depending on the environment.
Method #2
e = CL / (pi AR * ai_radians) = 0.47237 / (pi * 8.7049 * 0.021693 ai_radians) = 0.79625 e
This is the most accurate one.
0.81316 e
0.79294 e
0.79625 e
The others will vary in error from 100% to +/- 5% drag error.
Pretty easy. If you have questions or need clarifications, feel free to ask.
Note: In applying this in FS, CDi will be low at high altitude and high mach due to the CDi bug. I would suggest repeating these steps using different scenarios to find the e for the phase of flight you use most often. For airliners and biz jets, it would be cruise altitude at cruise mach. For fighters, find their combat envelope and apply the accuracy there. Most everything else is low and slow and won't vary much.
================================
For those who are interested in the process and extended formulas, the lift force produced by the wing is found by:
F = density * (pi*b^2/4) * Velocity * (Velocity * sin(downwash angle))
Of course, starting from here we already have the downwash angle so Alpha induced = downwash angle / 2.
So in the real world we only need b=wingspan, an airspeed indicator, and a sensor (or string) to measure the wash angle. From these three things we easily find ai; and all the aerodynamic coefficients of the wing.
Here's a quick diagram for reference.
Below is a discussion of how to do those two things. It is advanced, but explains things in a way that even beginners will get enough to figure it out. Consequently, all formulas and definitions are explained thoroughly.
The mechanics:
When a wing is installed on an actual airplane, there is an additional force applied to it in flight that doesn't happen in a wind tunnel. As air spills around the wingtip it causes a downward load on the rear of the wing. Because a wing in flight has nothing solid to support it, the only way for the wing to counter this force is by allowing the downward force until the weathervane effect of the airflow can balance it out. This causes the wing to fly at a higher angle of attack than needed. It also increases the downwash angle.
This is half of the problem. The other half is the airflow begins to turn before it reaches the leading edge of the wing, so the wing is flying through airflow that is already turned some. The average of this effect is half of the downwash angle. This is shown by the middle green arrow in the image below.
This new angle of attack is called the induced angle of attack or ai for Alpha induced. It is probably the most important variable in aircraft wing performance, but it is never published. The induced angle of attack increases the wing angle of attack, and so changes the lift direction angle rearward. This is because the angle of attack required to produce a specific amount of lift at a given airspeed is constant. So the wing angle of attack becomes the induced angle of attack + required angle of attack to produce current lift. This required angle of attack is called the Effective AoA, or ae for Alpha effective. Alpha effective is half of the downwash angle. If you remember from the previous paragraph, the effective AoA is caused by the airflow already starting to turn and is present even in a wind tunnel. The effective angle of attack is based from the effective airflow angle. The top of the induced angle of attack is the base of the effective angle of attack, and matches the effective airflow angle (which is averaged). The is shown in red by the image below.
Induced drag is created by the difference between the effective airflow and the relative airflow, and is perpendicular to the induced AoA (which is equal to the effective airflow angle). Mathematically this looks like:
Actual alpha = Alpha induced + Alpha effective
Because of the lack of detailed information available, and far too many so called 'internet experts', there are several popular misconceptions about induced drag. One of them is that the wingtip vortex creates an actual drag force. This is not true. The force the vortex applies to the wing is dissipated by the counter balance of the wing. In other words, as the vortex tries to rotate the wing, nose up... the airflow is countering it and trying to rotate the airplane nose down. The wing wants to weathervane into the wind. The two forces find a balance point which is the new induced angle. At this point the vortex force is dissipated, but not without a consequence. The wing angle and lift angle are now tilted rearward. Induced drag is solely the product of the rearward lift that is produced when the lift angle is tilted slightly rearward. This also produces a higher Lift coefficient when measured against the wing AoA, but an equal or lesser Lift coefficient when measured against the fuselage AoA. This is because the higher downwash angle will produce more lift, but the rear tilt of the lift transmits less lift upwards.
note: Contrary to popular belief the vortex force actually increases parasite drag by increasing the frontal area of the wings and entire fuselage. This is accounted for in CDparasite so we can forget about it for this discussion.
Finding Oswald e:
Now that we understand ae and ai we can find Oswald e, but in order to find Oswald e we need to find Alpha induced or CDi first; or 2D Cl_alpha. This is why CL^2 / pi AR e is not useable in the real world (until after someone charts e the first time or finds CDi or ai.)
To make this easier to understand let's use a real airplane, and real specs. I am going to show you how to do this with the absolute minimal of data. But first, a simple chart...
For the lazy ones out there, use this:
CDp can be assumed to be = total drag at minimum drag speed.
For the rest of you, you will need:
Aircraft weight in level flight: 100,000 pounds
Aircraft pitch in degrees: 3.28 degrees
Zero Lift alpha to body, aZL_body: -2.05
250KIAS speed in Mach at sea level: 0.3779
wing span b = 93.333
wing area S = 1000.7
Aspect ratio AR = b^2/S = 8.705
These numbers are based on a Douglas DC9-30.
Finding e:
Step 1
Find pressure Q for your speed.
1481.4 * Mach^2 = 1481.4 * 0.3779^2 = 211.55 Q
Step 2
Find CL
weight / (Q * S) = 100,000 / (211.55 * 1000.7) = 0.47237 CL measured from weight
Step 3
Find Cla
2pi * { AR / [2 + sqrt(4 + AR^2)] } = 6.28318531 * { 8.705 / [2 + sqrt( 4 + 8.705^2)] } = 5.0033 Cla
Step 4
Find CLa
Cla * [ AR / (AR + 2) ] = 5.0033 * [8.705 / (8.705 + 2)] = 4.0685 CLa
CLa / Cla = very close to e. There are a few things that will cause small errors.
Step 5
Find Cl and CL for Cla method
(Aircraft pitch in degrees - aZL_body) * Cla = (3.28 - -2.05) * 5.0033 = 26.667589 Cl
(Aircraft pitch in degrees - aZL_body) * CLa = (3.28 - -2.05) * 4.0685 = 21.685105 CL
(if those Cl and CL look weird, don't worry...they are in radians...they relate directly to pi*AR, which for this airplane is 27.3476.)
-OR-
Find alphas for alpha_induced method (much more accurate)
(CL measured from weight / Cla) * 57.2957 = (0.47237 / 5.0033) * 57.2957 = 5.4094 degrees alpha2D. (We can compare this to our pitch - zero lift value of 5.33 and see our zero lift alpha has changed a bit.)
(CL measured from weight / CLa) * 57.2957 = (0.47237 / 4.0685) * 57.2957 = 6.6523 degrees alpha3D. (We can compare this to our pitch - zero lift value of 5.33 and see the effect of alpha_induced.)
Step 6
Find e using Cla method...
CL / Cl = 21.685105 / 26.667589 = 0.81316 e
This method is much more prone to errors. See step 7 for more accuracy.
-OR-
Find ai_radians for ai method...
[alpha3D - alpha2D] / 57.2957 = (6.6523 - 5.4094) / 57.2957 = 0.021693 ai_radians
Step 7
Final calculations:
Method #1
Here's a good way to tell how far off you are.
Compare:
CL / Cl against CL / (pi*AR)
CL / Cl = 21.685105 / 26.667589 = 0.81316 e
CL / (pi*AR) = 21.685105 / (pi * 8.705) = 0.79294 e
0.81316 e against 0.79295 e. There is no way that I know of to confirm if the error is valid or not. It could be depending on the environment.
Method #2
e = CL / (pi AR * ai_radians) = 0.47237 / (pi * 8.7049 * 0.021693 ai_radians) = 0.79625 e
This is the most accurate one.
0.81316 e
0.79294 e
0.79625 e
The others will vary in error from 100% to +/- 5% drag error.
Pretty easy. If you have questions or need clarifications, feel free to ask.
Note: In applying this in FS, CDi will be low at high altitude and high mach due to the CDi bug. I would suggest repeating these steps using different scenarios to find the e for the phase of flight you use most often. For airliners and biz jets, it would be cruise altitude at cruise mach. For fighters, find their combat envelope and apply the accuracy there. Most everything else is low and slow and won't vary much.
================================
For those who are interested in the process and extended formulas, the lift force produced by the wing is found by:
F = density * (pi*b^2/4) * Velocity * (Velocity * sin(downwash angle))
Of course, starting from here we already have the downwash angle so Alpha induced = downwash angle / 2.
So in the real world we only need b=wingspan, an airspeed indicator, and a sensor (or string) to measure the wash angle. From these three things we easily find ai; and all the aerodynamic coefficients of the wing.
Here's a quick diagram for reference.
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