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Aerospace
Lift coefficient at maximum lift-to-drag ratio
Lift coefficient that maximises CL/CD for the parabolic polar CD = CD0 + kCL², found by differentiating at fixed configuration.
- Ref
- AER-23
- Rev
- v1
- Field
- Aerospace
- Result
- CL,opt
For the parabolic polar CD = CD0 + kCL², the ratio CL/CD is greatest where its derivative vanishes: CL = √(CD0/k). At that point the zero-lift drag equals the induced drag, so the total drag coefficient is 2CD0. This is the lift coefficient for best glide, and for best range of a jet. It must lie below the stall lift coefficient to be usable.
Lift coefficient that maximises CL/CD for the parabolic polar CD = CD0 + kCL², found by differentiating at fixed configuration. Here the zero-lift drag equals the induced drag. It must lie within the usable range of the polar, below stall.
Calculator
- Maximum lift-to-drag ratio, 1/(2√(CD0 k))15.81
- Drag coefficient at that point, 2CD00.05
- Best glide angle, atan(1/(L/D)max)3.619 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The best lift-to-drag ratio is 15.81, reached at CL = 0.7906, where the drag coefficient is 0.05.
- Flying at a lift coefficient away from 0.791 lowers L/D; at CL = 0.395 or 1.58 it drops to 12.65.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| CD0 Zero-lift drag coefficient | 0.01 to 0.1 | 0.5 | 1.581 | 1.081 | +10% | +4.88 % |
| k Induced drag factor | 0.02 to 0.1 | 0.5 | 1.118 | 0.618 | +10% | −4.65 % |
Sweep of Induced drag factor
| Induced drag factor | Best-L/D lift coefficient |
|---|---|
| 0.02 | 1.118 |
| 0.03 | 0.9129 |
| 0.04 | 0.7906 |
| 0.05 | 0.7071 |
| 0.06 | 0.6455 |
| 0.07 | 0.5976 |
| 0.08 | 0.559 |
| 0.09 | 0.527 |
| 0.1 | 0.5 |
Sweep one variable
Variables and how each one contributes
Best-L/D lift coefficient
What the equation gives you. Everything else feeds it.
Zero-lift drag coefficient
The zero-lift drag coefficient. Raises the optimum lift coefficient with the square root. Raising it by 10 % (to 0.0275) moves the result by +4.88 %, from 0.7906 to 0.8292.
Induced drag factor
The induced drag factor, 1/(πeAR). Lowers the optimum lift coefficient with the square root of its inverse. Raising it by 10 % (to 0.044) moves the result by −4.65 %, from 0.7906 to 0.7538.
Worked example with the values above
- Ratio: CD0/k = 0.025 / 0.04 = 0.625.
- Square root: √0.625.
- CL,opt = 0.790569.
Assumptions and limits
- A parabolic polar CD = CD0 + kCL², constant over the speed range and with no compressibility rise.
- Steady level, unaccelerated flight with lift equal to weight, at constant density.
Common mistakes
- Using a polar from a different configuration, such as flaps or gear down.
- Forgetting stall and Mach limits on the mathematical solution.
- Mixing up the minimum-drag speed and the minimum-power speed.
Statistics and uncertainty
The parabolic drag polar fits real aircraft well only over the middle of the lift range. Speeds from it are usually accurate to 5 % when CD0 and k come from flight tests; with handbook estimates they can be off by 10 % to 20 %, and the result moves with the one-quarter power of the drag ratio.
Where it is used
- Plotting thrust and power curves and finding best speeds for range and endurance.
- Estimating how stall speed changes in turns.
Maths you need
- Drag polar: CD = CD0 + kCL²
- Level flight: L = W and T = D
- Load factor: n = L/W