Example

Question:

A fully loaded Boeing aircraft has a mass of \(3.3 \times 10^5\,\mathrm{kg}\). Its total wing area is \(500\,\mathrm{m}^2\). It is in level flight with a speed of \(960\,\mathrm{km/h}\).
(a) Estimate the pressure difference between the lower and upper surfaces of the wings.
(b) Estimate the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface. [The density of air is \(\rho = 1.2\,\mathrm{kg\,m}^{-3}\)]

Solution:

(a) The weight of the Boeing aircraft is balanced by the upward force due to the pressure difference: \[ \Delta P \times A = 3.3 \times 10^5\,\mathrm{kg} \times 9.8 \] \[ \Delta P = \frac{3.3 \times 10^5\,\mathrm{kg} \times 9.8\,\mathrm{m\,s}^{-2}}{500\,\mathrm{m}^2} = 6.5 \times 10^3\,\mathrm{N\,m}^{-2} \]
(b) The pressure difference is related to the speeds above and below the wing: \[ \Delta P = \frac{\rho}{2}(v_2^2 - v_1^2) \] where \(v_2\) is the speed of air over the upper surface and \(v_1\) under the bottom surface.
Solving for the fractional increase:
Take the average speed \(v_{av} = \frac{v_2 + v_1}{2} = 960\,\mathrm{km/h} = 267\,\mathrm{m\,s}^{-1}\), \[ \frac{v_2 - v_1}{v_{av}} = \frac{\Delta P}{\rho v_{av}^2} \approx 0.08 \]
Thus, the speed above the wing needs to be only 8% higher than that below.