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📐 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
Optimal Lift-to-Drag Ratio Cfd
📊 Solver Telemetry ● ACTIVE
👁️ Views 22
⚡ Solves 18
💾 Downloads 654 📦 Fortran Code 4.4 KB
📅 Released Jun 2026
⏱️ Latency < 1 ms
⚡ TOOLS & REPORTS:
💾 Download Fortran 90
Aircraft Configurations: High-Performance Competition Sailplane Cessna 172 Skyhawk (Light Aircraft) Commercial Wide-Body Jet Cruise Solar High-Altitude Endurance UAV

📥 Aircraft Weight, Polar & Wing Geometry

Mass: 357 kg
📖 Optimal Performance Formulae: $$(L/D)_{\text{max}} = \frac{1}{2 \sqrt{C_{D,0} / (\pi e AR)}}, \quad C_{L,\text{range}} = \sqrt{\pi e AR C_{D,0}}$$ $$V_{\text{range}} = \sqrt{\frac{2 W}{\rho S C_{L,\text{range}}}}, \quad V_{\text{endurance}} = \frac{V_{\text{range}}}{3^{1/4}} \approx 0.76\, V_{\text{range}}$$
40.2 : 1
Maximum $(L/D)_{\text{max}}$
109.3 km/h
Best Glide Speed ($V_{\text{range}}$)
83.0 km/h
Max Endurance ($V_{\text{end}}$)
101.7 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 1.125 40.2 30.4 m/s (109 km/h) 87.1 N
Max Loiter Endurance 1.949 34.8 23.1 m/s (83 km/h) 100.5 N
Clean Stall Boundary 1.30 - 28.2 m/s (102 km/h) -

📈 Thrust Required & Total Drag Curve vs Airspeed: $D(V)$

Min Drag at $V = 109\ \text{km/h}$
🔍 View Raw GNU Fortran Double-Precision Solver Output 
CD0= 1.40000000E-02
OSWALD_E=  0.9000
AR=   32.0000
RHO= 1.50000000E-01
S= 4.50000000E+01
W= 3.50000000E+03
CL_MAX=  1.3000
K= 1.10524266E-02
CL_RANGE=  1.125473
CD_RANGE=  0.028000
LD_RANGE=   40.1955
V_RANGE=    30.355
CL_ENDUR=  1.949377
CD_ENDUR=  0.056000
LD_ENDUR=   34.8103
V_ENDUR=    23.065
CL_CLIMB=  0.649792
CD_CLIMB=  0.018667
LD_CLIMB=   34.8103
V_CLIMB=    39.949
V_MIN=    28.244
CD_VMIN=  0.032679
LD_VMIN=   39.7814
SENSITIVITY_START
 7.00000000E-03,     56.8450
 7.21212121E-03,     56.0028
 7.42424242E-03,     55.1970
 7.63636364E-03,     54.4249
 7.84848485E-03,     53.6844
 8.06060606E-03,     52.9733
 8.27272727E-03,     52.2898
 8.48484848E-03,     51.6320
 8.69696970E-03,     50.9985
 8.90909091E-03,     50.3877
 9.12121212E-03,     49.7983
 9.33333333E-03,     49.2292
 9.54545455E-03,     48.6791
 9.75757576E-03,     48.1471
 9.96969697E-03,     47.6322
 1.01818182E-02,     47.1334
 1.03939394E-02,     46.6499
 1.06060606E-02,     46.1811
 1.08181818E-02,     45.7261
 1.10303030E-02,     45.2843
 1.12424242E-02,     44.8550
 1.14545455E-02,     44.4378
 1.16666667E-02,     44.0319
 1.18787879E-02,     43.6370
 1.20909091E-02,     43.2525
 1.23030303E-02,     42.8781
 1.25151515E-02,     42.5131
 1.27272727E-02,     42.1574
 1.29393939E-02,     41.8104
 1.31515152E-02,     41.4718
 1.33636364E-02,     41.1414
 1.35757576E-02,     40.8187
 1.37878788E-02,     40.5035
 1.40000000E-02,     40.1955
 1.42121212E-02,     39.8944
 1.44242424E-02,     39.6000
 1.46363636E-02,     39.3120
 1.48484848E-02,     39.0301
 1.50606061E-02,     38.7543
 1.52727273E-02,     38.4842
 1.54848485E-02,     38.2197
 1.56969697E-02,     37.9606
 1.59090909E-02,     37.7067
 1.61212121E-02,     37.4578
 1.63333333E-02,     37.2138
 1.65454545E-02,     36.9745
 1.67575758E-02,     36.7397
 1.69696970E-02,     36.5094
 1.71818182E-02,     36.2833
 1.73939394E-02,     36.0614
 1.76060606E-02,     35.8435
 1.78181818E-02,     35.6295
 1.80303030E-02,     35.4193
 1.82424242E-02,     35.2127
 1.84545455E-02,     35.0098
 1.86666667E-02,     34.8103
 1.88787879E-02,     34.6142
 1.90909091E-02,     34.4213
 1.93030303E-02,     34.2317
 1.95151515E-02,     34.0451
 1.97272727E-02,     33.8616
 1.99393939E-02,     33.6810
 2.01515152E-02,     33.5033
 2.03636364E-02,     33.3283
 2.05757576E-02,     33.1561
 2.07878788E-02,     32.9865
 2.10000000E-02,     32.8195
 2.12121212E-02,     32.6550
 2.14242424E-02,     32.4929
 2.16363636E-02,     32.3332
 2.18484848E-02,     32.1759
 2.20606061E-02,     32.0208
 2.22727273E-02,     31.8680
 2.24848485E-02,     31.7173
 2.26969697E-02,     31.5687
 2.29090909E-02,     31.4222
 2.31212121E-02,     31.2778
 2.33333333E-02,     31.1353
 2.35454545E-02,     30.9947
 2.37575758E-02,     30.8560
 2.39696970E-02,     30.7192
 2.41818182E-02,     30.5842
 2.43939394E-02,     30.4509
 2.46060606E-02,     30.3194
 2.48181818E-02,     30.1895
 2.50303030E-02,     30.0613
 2.52424242E-02,     29.9348
 2.54545455E-02,     29.8098
 2.56666667E-02,     29.6863
 2.58787879E-02,     29.5644
 2.60909091E-02,     29.4440
 2.63030303E-02,     29.3250
 2.65151515E-02,     29.2075
 2.67272727E-02,     29.0913
 2.69393939E-02,     28.9766
 2.71515152E-02,     28.8632
 2.73636364E-02,     28.7511
 2.75757576E-02,     28.6403
 2.77878788E-02,     28.5308
 2.80000000E-02,     28.4225
SENSITIVITY_END
💾 Download .f90 Code

📘 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

  1. McCormick, B. W.: Aerodynamics, Aeronautics & Flight Mechanics, Wiley.
  2. 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 $C_L = \mathbf{0.853}$.