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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: 459 kg
📖 Optimal Performance Formulae: $$(L/D)_{\text{max}} = \frac{1}{2 \sqrt{C_{D,0} / (\pi e AR)}}, \quad C_{L,\text{range}} = \sqrt{\pi e AR C_{D,0}}$$ $$V_{\text{range}} = \sqrt{\frac{2 W}{\rho S C_{L,\text{range}}}}, \quad V_{\text{endurance}} = \frac{V_{\text{range}}}{3^{1/4}} \approx 0.76\, V_{\text{range}}$$
39.7 : 1
Maximum $(L/D)_{\text{max}}$
97.4 km/h
Best Glide Speed ($V_{\text{range}}$)
74.0 km/h
Max Endurance ($V_{\text{end}}$)
71.9 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 0.874 39.7 27.0 m/s (97 km/h) 113.3 N
Max Loiter Endurance 1.513 34.4 20.5 m/s (74 km/h) 130.9 N
Clean Stall Boundary 1.60 - 20.0 m/s (72 km/h) -

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

Min Drag at $V = 97\ \text{km/h}$
🔍 View Raw GNU Fortran Double-Precision Solver Output 
CD0= 1.10000000E-02
OSWALD_E=  0.9200
AR=   24.0000
RHO= 1.22500000E+00
S= 1.15000000E+01
W= 4.50000000E+03
CL_MAX=  1.6000
K= 1.44162086E-02
CL_RANGE=  0.873516
CD_RANGE=  0.022000
LD_RANGE=   39.7053
V_RANGE=    27.044
CL_ENDUR=  1.512974
CD_ENDUR=  0.044000
LD_ENDUR=   34.3858
V_ENDUR=    20.549
CL_CLIMB=  0.504325
CD_CLIMB=  0.014667
LD_CLIMB=   34.3858
V_CLIMB=    35.592
V_MIN=    19.982
CD_VMIN=  0.047905
LD_VMIN=   33.3991
SENSITIVITY_START
 5.50000000E-03,     56.1517
 5.66666667E-03,     55.3198
 5.83333333E-03,     54.5238
 6.00000000E-03,     53.7612
 6.16666667E-03,     53.0297
 6.33333333E-03,     52.3273
 6.50000000E-03,     51.6521
 6.66666667E-03,     51.0023
 6.83333333E-03,     50.3765
 7.00000000E-03,     49.7732
 7.16666667E-03,     49.1910
 7.33333333E-03,     48.6288
 7.50000000E-03,     48.0855
 7.66666667E-03,     47.5599
 7.83333333E-03,     47.0513
 8.00000000E-03,     46.5586
 8.16666667E-03,     46.0810
 8.33333333E-03,     45.6179
 8.50000000E-03,     45.1684
 8.66666667E-03,     44.7320
 8.83333333E-03,     44.3080
 9.00000000E-03,     43.8958
 9.16666667E-03,     43.4949
 9.33333333E-03,     43.1048
 9.50000000E-03,     42.7251
 9.66666667E-03,     42.3551
 9.83333333E-03,     41.9947
 1.00000000E-02,     41.6432
 1.01666667E-02,     41.3005
 1.03333333E-02,     40.9661
 1.05000000E-02,     40.6396
 1.06666667E-02,     40.3209
 1.08333333E-02,     40.0095
 1.10000000E-02,     39.7053
 1.11666667E-02,     39.4078
 1.13333333E-02,     39.1170
 1.15000000E-02,     38.8325
 1.16666667E-02,     38.5541
 1.18333333E-02,     38.2817
 1.20000000E-02,     38.0149
 1.21666667E-02,     37.7536
 1.23333333E-02,     37.4977
 1.25000000E-02,     37.2468
 1.26666667E-02,     37.0010
 1.28333333E-02,     36.7599
 1.30000000E-02,     36.5235
 1.31666667E-02,     36.2916
 1.33333333E-02,     36.0641
 1.35000000E-02,     35.8408
 1.36666667E-02,     35.6216
 1.38333333E-02,     35.4063
 1.40000000E-02,     35.1950
 1.41666667E-02,     34.9873
 1.43333333E-02,     34.7833
 1.45000000E-02,     34.5828
 1.46666667E-02,     34.3858
 1.48333333E-02,     34.1920
 1.50000000E-02,     34.0016
 1.51666667E-02,     33.8142
 1.53333333E-02,     33.6299
 1.55000000E-02,     33.4487
 1.56666667E-02,     33.2703
 1.58333333E-02,     33.0947
 1.60000000E-02,     32.9219
 1.61666667E-02,     32.7517
 1.63333333E-02,     32.5842
 1.65000000E-02,     32.4192
 1.66666667E-02,     32.2567
 1.68333333E-02,     32.0966
 1.70000000E-02,     31.9389
 1.71666667E-02,     31.7835
 1.73333333E-02,     31.6303
 1.75000000E-02,     31.4793
 1.76666667E-02,     31.3305
 1.78333333E-02,     31.1837
 1.80000000E-02,     31.0390
 1.81666667E-02,     30.8963
 1.83333333E-02,     30.7556
 1.85000000E-02,     30.6167
 1.86666667E-02,     30.4797
 1.88333333E-02,     30.3446
 1.90000000E-02,     30.2112
 1.91666667E-02,     30.0795
 1.93333333E-02,     29.9496
 1.95000000E-02,     29.8213
 1.96666667E-02,     29.6947
 1.98333333E-02,     29.5697
 2.00000000E-02,     29.4462
 2.01666667E-02,     29.3243
 2.03333333E-02,     29.2039
 2.05000000E-02,     29.0849
 2.06666667E-02,     28.9674
 2.08333333E-02,     28.8513
 2.10000000E-02,     28.7366
 2.11666667E-02,     28.6232
 2.13333333E-02,     28.5112
 2.15000000E-02,     28.4005
 2.16666667E-02,     28.2910
 2.18333333E-02,     28.1828
 2.20000000E-02,     28.0759
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 $C_L = \mathbf{0.853}$.