Training Aerospace Engineering Aerodynamic Forces — Lift & Drag
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Aerodynamic Forces — Lift & Drag

24 min Aerospace Engineering

Aerodynamic Forces — Lift & Drag

Flight depends on balancing gravitational, aerodynamic, and thrust forces. Lift and drag are the two primary aerodynamic forces.

Lift and Drag Equations

$$L = \frac{1}{2} \rho v^2 S \, C_L \qquad D = \frac{1}{2} \rho v^2 S \, C_D$$

where $\rho$ = air density, $v$ = airspeed, $S$ = wing planform area, $C_L$ = lift coefficient, $C_D$ = drag coefficient.

Bernoulli's Equation

$$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$$

Along a streamline in inviscid, incompressible flow. The pressure difference between the upper and lower surfaces creates lift.

Example 1

An aircraft wing has $S = 25$ m², $C_L = 1.2$, flying at $v = 70$ m/s at sea level ($\rho = 1.225$ kg/m³). Find the lift force.

$$L = \frac{1}{2}(1.225)(70^2)(25)(1.2) = \frac{1}{2}(1.225)(4{,}900)(25)(1.2) = 90{,}038 \text{ N} \approx 90 \text{ kN}$$

Example 2

The aircraft in Example 1 has a mass of 8{,}000 kg. Find the minimum speed for level flight at the same $C_L$.

For level flight: $L = W = mg = 8{,}000 \times 9.81 = 78{,}480$ N.

$$v_{\min} = \sqrt{\frac{2W}{\rho S C_L}} = \sqrt{\frac{2 \times 78{,}480}{1.225 \times 25 \times 1.2}} = \sqrt{4{,}281} = 65.4 \text{ m/s}$$

Example 3

If $C_D = 0.03$ for the same wing, find the drag force at 70 m/s.

$$D = \frac{1}{2}(1.225)(4{,}900)(25)(0.03) = 2{,}251 \text{ N} \approx 2.25 \text{ kN}$$

Lift-to-drag ratio: $L/D = 90/2.25 = 40$ (very efficient wing).

Practice Problems

1. Find the lift on a wing with $S = 16$ m², $C_L = 0.8$, $v = 60$ m/s, sea level.
2. An aircraft weighs 50 kN. At sea level with $C_L = 1.5$ and $S = 30$ m², find the stall speed.
3. At 10,000 m altitude, $\rho \approx 0.414$ kg/m³. How does the lift compare to sea level at the same speed?
4. Find the drag on a car with frontal area 2.2 m², $C_D = 0.32$, at 30 m/s (sea level).
5. A parachute has $C_D = 1.5$, $S = 30$ m². A 90 kg skydiver descends at terminal velocity. Find $v$.
6. If $L/D = 15$ and $W = 200$ kN, what is the drag in level flight?
7. Bernoulli: air flows at 50 m/s over the top of a wing and 40 m/s underneath. Find the pressure difference.
8. Dynamic pressure $q = \frac{1}{2}\rho v^2$. Find $q$ at Mach 0.8 at sea level ($v \approx 272$ m/s).
9. Induced drag: $C_{D_i} = C_L^2/(\pi e AR)$ where $AR = b^2/S$ and $e = 0.85$. Wing: $b = 10$ m, $S = 15$ m², $C_L = 1.0$. Find $C_{D_i}$.
10. Total drag $= C_{D_0} + C_{D_i}$. If $C_{D_0} = 0.02$, find total $C_D$ from #9.
11. A wing has $C_L = 0.4$ at $\alpha = 4°$ and $C_L = 1.2$ at $\alpha = 12°$. Find the lift slope $a = dC_L/d\alpha$.
12. At what angle of attack does the wing in #11 produce zero lift (assume linear)? $C_L = 0$ at $\alpha_0$.
Show Answer Key

1. $L = 0.5(1.225)(3600)(16)(0.8) = 28{,}224$ N $\approx 28.2$ kN

2. $v_s = \sqrt{2 \times 50{,}000/(1.225 \times 30 \times 1.5)} = \sqrt{1{,}360} = 36.9$ m/s

3. $L \propto \rho$; ratio $= 0.414/1.225 = 0.338$ — only 33.8% of sea-level lift

4. $D = 0.5(1.225)(900)(2.2)(0.32) = 388$ N

5. $D = W$; $v = \sqrt{2mg/(\rho S C_D)} = \sqrt{2(882.9)/(1.225 \times 30 \times 1.5)} = 5.67$ m/s

6. $D = W/(L/D) = 200/15 = 13.3$ kN

7. $\Delta P = \frac{1}{2}\rho(v_1^2 - v_2^2) = 0.5(1.225)(2500-1600) = 551$ Pa

8. $q = 0.5(1.225)(272^2) = 45{,}325$ Pa $\approx 45.3$ kPa

9. $AR = 100/15 = 6.67$; $C_{D_i} = 1/(\pi(0.85)(6.67)) = 0.0563$

10. $C_D = 0.02 + 0.0563 = 0.0763$

11. $a = (1.2-0.4)/(12-4) = 0.1$ per degree

12. $\alpha_0 = 4 - 0.4/0.1 = 0°$ (zero lift at $0°$) — or more precisely from $C_L = a(\alpha - \alpha_0)$: $\alpha_0 = 4 - 4 = 0°$

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