📐 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:
$$(L/D)_{\text{max}} = \frac{1}{2 \sqrt{C_{D,0} / (\pi e AR)}}, \quad C_{L,\text{range}} = \sqrt{\pi e AR C_{D,0}}$$
$$V_{\text{range}} = \sqrt{\frac{2 W}{\rho S C_{L,\text{range}}}}, \quad V_{\text{endurance}} = \frac{V_{\text{range}}}{3^{1/4}} \approx 0.76\, V_{\text{range}}$$
14.2 : 1
Maximum $(L/D)_{\text{max}}$
136.6 km/h
Best Glide Speed ($V_{\text{range}}$)
103.8 km/h
Max Endurance ($V_{\text{end}}$)
93.9 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 | 0.709 | 14.2 | 37.9 m/s (137 km/h) | 705.2 N |
| Max Loiter Endurance | 1.228 | 12.3 | 28.8 m/s (104 km/h) | 814.3 N |
| Clean Stall Boundary | 1.50 | - | 26.1 m/s (94 km/h) | - |
📈 Thrust Required & Total Drag Curve vs Airspeed: $D(V)$
Min Drag at $V = 137\ \text{km/h}$🔍 View Raw GNU Fortran Double-Precision Solver Output
CD0= 2.50000000E-02 OSWALD_E= 0.8000 AR= 8.0000 RHO= 1.22500000E+00 S= 1.60000000E+01 W= 1.00000000E+04 CL_MAX= 1.5000 K= 4.97359197E-02 CL_RANGE= 0.708982 CD_RANGE= 0.050000 LD_RANGE= 14.1796 V_RANGE= 37.938 CL_ENDUR= 1.227992 CD_ENDUR= 0.100000 LD_ENDUR= 12.2799 V_ENDUR= 28.826 CL_CLIMB= 0.409331 CD_CLIMB= 0.033333 LD_CLIMB= 12.2799 V_CLIMB= 49.929 V_MIN= 26.082 CD_VMIN= 0.136906 LD_VMIN= 10.9564 SENSITIVITY_START 1.25000000E-02, 20.0530 1.28787879E-02, 19.7559 1.32575758E-02, 19.4717 1.36363636E-02, 19.1993 1.40151515E-02, 18.9381 1.43939394E-02, 18.6872 1.47727273E-02, 18.4461 1.51515152E-02, 18.2141 1.55303030E-02, 17.9906 1.59090909E-02, 17.7751 1.62878788E-02, 17.5672 1.66666667E-02, 17.3664 1.70454545E-02, 17.1724 1.74242424E-02, 16.9847 1.78030303E-02, 16.8030 1.81818182E-02, 16.6271 1.85606061E-02, 16.4566 1.89393939E-02, 16.2912 1.93181818E-02, 16.1306 1.96969697E-02, 15.9748 2.00757576E-02, 15.8234 2.04545455E-02, 15.6762 2.08333333E-02, 15.5330 2.12121212E-02, 15.3937 2.15909091E-02, 15.2581 2.19696970E-02, 15.1260 2.23484848E-02, 14.9972 2.27272727E-02, 14.8717 2.31060606E-02, 14.7493 2.34848485E-02, 14.6299 2.38636364E-02, 14.5133 2.42424242E-02, 14.3995 2.46212121E-02, 14.2883 2.50000000E-02, 14.1796 2.53787879E-02, 14.0734 2.57575758E-02, 13.9696 2.61363636E-02, 13.8680 2.65151515E-02, 13.7685 2.68939394E-02, 13.6712 2.72727273E-02, 13.5760 2.76515152E-02, 13.4827 2.80303030E-02, 13.3912 2.84090909E-02, 13.3017 2.87878788E-02, 13.2139 2.91666667E-02, 13.1278 2.95454545E-02, 13.0434 2.99242424E-02, 12.9605 3.03030303E-02, 12.8793 3.06818182E-02, 12.7995 3.10606061E-02, 12.7213 3.14393939E-02, 12.6444 3.18181818E-02, 12.5689 3.21969697E-02, 12.4947 3.25757576E-02, 12.4219 3.29545455E-02, 12.3503 3.33333333E-02, 12.2799 3.37121212E-02, 12.2107 3.40909091E-02, 12.1427 3.44696970E-02, 12.0758 3.48484848E-02, 12.0100 3.52272727E-02, 11.9453 3.56060606E-02, 11.8815 3.59848485E-02, 11.8188 3.63636364E-02, 11.7571 3.67424242E-02, 11.6964 3.71212121E-02, 11.6365 3.75000000E-02, 11.5776 3.78787879E-02, 11.5196 3.82575758E-02, 11.4624 3.86363636E-02, 11.4061 3.90151515E-02, 11.3506 3.93939394E-02, 11.2959 3.97727273E-02, 11.2420 4.01515152E-02, 11.1888 4.05303030E-02, 11.1364 4.09090909E-02, 11.0847 4.12878788E-02, 11.0338 4.16666667E-02, 10.9835 4.20454545E-02, 10.9339 4.24242424E-02, 10.8850 4.28030303E-02, 10.8367 4.31818182E-02, 10.7891 4.35606061E-02, 10.7421 4.39393939E-02, 10.6957 4.43181818E-02, 10.6499 4.46969697E-02, 10.6046 4.50757576E-02, 10.5600 4.54545455E-02, 10.5159 4.58333333E-02, 10.4724 4.62121212E-02, 10.4293 4.65909091E-02, 10.3869 4.69696970E-02, 10.3449 4.73484848E-02, 10.3034 4.77272727E-02, 10.2625 4.81060606E-02, 10.2220 4.84848485E-02, 10.1820 4.88636364E-02, 10.1424 4.92424242E-02, 10.1033 4.96212121E-02, 10.0647 5.00000000E-02, 10.0265 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 $C_L = \mathbf{0.853}$.