📐 Optimal Lift-to-Drag Ratio
Determine the maximum lift-to-drag ratio (L/D) and optimal cruise angle of attack for aircraft wings.
⚡ Fortran 90 Engine
Double Precision (IEEE 754)
✓ ISO / ASME Validated
Cfd
📊 Solver Telemetry
● ACTIVE
📥 Aircraft Weight, Polar & Wing Geometry
📖 Optimal Performance Formulae:
$$(L/D)_{\text{max}} = \frac{1}{2 \sqrt{C_{D,0} / (\pi e AR)}}, \quad C_{L,\text{range}} = \sqrt{\pi e AR C_{D,0}}$$
$$V_{\text{range}} = \sqrt{\frac{2 W}{\rho S C_{L,\text{range}}}}, \quad V_{\text{endurance}} = \frac{V_{\text{range}}}{3^{1/4}} \approx 0.76\, V_{\text{range}}$$
13.0 : 1
Maximum $(L/D)_{\text{max}}$
142.8 km/h
Best Glide Speed ($V_{\text{range}}$)
108.5 km/h
Max Endurance ($V_{\text{end}}$)
97.9 km/h
Stall Speed ($V_{\text{stall}}$)
⚡ Flight Optimization Regimes Comparison
| Operating Regime | Optimal $C_L$ | $L/D$ Efficiency | True Airspeed (m/s) | Thrust Req. (N) |
|---|---|---|---|---|
| Max Range / Best Glide | 0.704 | 13.0 | 39.7 m/s (143 km/h) | 843.2 N |
| Max Loiter Endurance | 1.220 | 11.3 | 30.1 m/s (109 km/h) | 973.7 N |
| Clean Stall Boundary | 1.50 | - | 27.2 m/s (98 km/h) | - |
📈 Thrust Required & Total Drag Curve vs Airspeed: $D(V)$
Min Drag at $V = 143\ \text{km/h}$🔍 View Raw GNU Fortran Double-Precision Solver Output
CD0= 2.70000000E-02 OSWALD_E= 0.7800 AR= 7.5000 RHO= 1.22500000E+00 S= 1.62000000E+01 W= 1.10000000E+04 CL_MAX= 1.5000 K= 5.44119464E-02 CL_RANGE= 0.704425 CD_RANGE= 0.054000 LD_RANGE= 13.0449 V_RANGE= 39.671 CL_ENDUR= 1.220100 CD_ENDUR= 0.108000 LD_ENDUR= 11.2972 V_ENDUR= 30.143 CL_CLIMB= 0.406700 CD_CLIMB= 0.036000 LD_CLIMB= 11.2972 V_CLIMB= 52.209 V_MIN= 27.186 CD_VMIN= 0.149427 LD_VMIN= 10.0384 SENSITIVITY_START 1.35000000E-02, 18.4483 1.39090909E-02, 18.1750 1.43181818E-02, 17.9134 1.47272727E-02, 17.6629 1.51363636E-02, 17.4226 1.55454545E-02, 17.1918 1.59545455E-02, 16.9700 1.63636364E-02, 16.7565 1.67727273E-02, 16.5509 1.71818182E-02, 16.3527 1.75909091E-02, 16.1614 1.80000000E-02, 15.9767 1.84090909E-02, 15.7982 1.88181818E-02, 15.6255 1.92272727E-02, 15.4584 1.96363636E-02, 15.2965 2.00454545E-02, 15.1396 2.04545455E-02, 14.9875 2.08636364E-02, 14.8398 2.12727273E-02, 14.6964 2.16818182E-02, 14.5571 2.20909091E-02, 14.4217 2.25000000E-02, 14.2900 2.29090909E-02, 14.1618 2.33181818E-02, 14.0370 2.37272727E-02, 13.9155 2.41363636E-02, 13.7971 2.45454545E-02, 13.6816 2.49545455E-02, 13.5690 2.53636364E-02, 13.4591 2.57727273E-02, 13.3519 2.61818182E-02, 13.2472 2.65909091E-02, 13.1449 2.70000000E-02, 13.0449 2.74090909E-02, 12.9472 2.78181818E-02, 12.8516 2.82272727E-02, 12.7582 2.86363636E-02, 12.6667 2.90454545E-02, 12.5772 2.94545455E-02, 12.4895 2.98636364E-02, 12.4037 3.02727273E-02, 12.3196 3.06818182E-02, 12.2372 3.10909091E-02, 12.1564 3.15000000E-02, 12.0772 3.19090909E-02, 11.9996 3.23181818E-02, 11.9234 3.27272727E-02, 11.8486 3.31363636E-02, 11.7753 3.35454545E-02, 11.7032 3.39545455E-02, 11.6325 3.43636364E-02, 11.5631 3.47727273E-02, 11.4949 3.51818182E-02, 11.4278 3.55909091E-02, 11.3620 3.60000000E-02, 11.2972 3.64090909E-02, 11.2336 3.68181818E-02, 11.1710 3.72272727E-02, 11.1094 3.76363636E-02, 11.0489 3.80454545E-02, 10.9893 3.84545455E-02, 10.9307 3.88636364E-02, 10.8730 3.92727273E-02, 10.8163 3.96818182E-02, 10.7604 4.00909091E-02, 10.7053 4.05000000E-02, 10.6511 4.09090909E-02, 10.5977 4.13181818E-02, 10.5451 4.17272727E-02, 10.4933 4.21363636E-02, 10.4423 4.25454545E-02, 10.3919 4.29545455E-02, 10.3423 4.33636364E-02, 10.2934 4.37727273E-02, 10.2452 4.41818182E-02, 10.1977 4.45909091E-02, 10.1508 4.50000000E-02, 10.1045 4.54090909E-02, 10.0589 4.58181818E-02, 10.0139 4.62272727E-02, 9.9695 4.66363636E-02, 9.9257 4.70454545E-02, 9.8824 4.74545455E-02, 9.8397 4.78636364E-02, 9.7976 4.82727273E-02, 9.7560 4.86818182E-02, 9.7149 4.90909091E-02, 9.6744 4.95000000E-02, 9.6343 4.99090909E-02, 9.5947 5.03181818E-02, 9.5557 5.07272727E-02, 9.5170 5.11363636E-02, 9.4789 5.15454545E-02, 9.4412 5.19545455E-02, 9.4040 5.23636364E-02, 9.3672 5.27727273E-02, 9.3308 5.31818182E-02, 9.2948 5.35909091E-02, 9.2593 5.40000000E-02, 9.2241 SENSITIVITY_END
📘 Calculation Methodology: Maximum Aerodynamic Efficiency (L/D)max
Mathematical Model & Theory
The maximum lift-to-drag ratio $(L/D)_{max}$ occurs where parasite drag equals induced drag ($C_{D,0} = C_{D,i}$), defining the optimum cruise condition and minimum glide angle:
$$C_{L,opt} = \sqrt{\pi e AR C_{D,0}}, \quad \left(\frac{L}{D}\right)_{max} = \frac{1}{2}\sqrt{\frac{\pi e AR}{C_{D,0}}}$$
$$\gamma_{min} = \arctan\left(\frac{1}{(L/D)_{max}}\right)$$
Assumptions
- Parabolic drag polar without shock wave separation.
- Subsonic attached flow.
Academic References
- McCormick, B. W.: Aerodynamics, Aeronautics & Flight Mechanics, Wiley.
- Hoerner, S. F.: Fluid-Dynamic Drag.
Worked Engineering Example
Problem Statement:
A glider has $AR = 18.0$, $e = 0.92$, $C_{D,0} = 0.014$. Determine $(L/D)_{max}$ and optimal $C_L$.
Step-by-step Solution:
1. $C_{L,opt} = \sqrt{\pi \times 0.92 \times 18.0 \times 0.014} \approx 0.853$.
2. $(L/D)_{max} = 0.5 \times \sqrt{\pi \times 0.92 \times 18.0 / 0.014} \approx 30.48$.
Final Result:
Maximum efficiency is $(L/D)_{max} = \mathbf{30.48}$ at $C_L = \mathbf{0.853}$.
A glider has $AR = 18.0$, $e = 0.92$, $C_{D,0} = 0.014$. Determine $(L/D)_{max}$ and optimal $C_L$.
Step-by-step Solution:
1. $C_{L,opt} = \sqrt{\pi \times 0.92 \times 18.0 \times 0.014} \approx 0.853$.
2. $(L/D)_{max} = 0.5 \times \sqrt{\pi \times 0.92 \times 18.0 / 0.014} \approx 30.48$.
Final Result:
Maximum efficiency is $(L/D)_{max} = \mathbf{30.48}$ at $C_L = \mathbf{0.853}$.