Serway & Jewett — Fluid Mechanics: Problem 48

The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in Figure P14.48. The hydrofoil has a shape like that of an airplane wing. Its area projected onto a horizontal surface is \(A\). When the boat is towed at sufficiently high speed, water of density \(\rho\) moves in streamline flow so that its average speed at the top of the hydrofoil is \(n\) times larger than its speed \(v_b\) below the hydrofoil. (a) Ignoring the buoyant force, show that the upward lift force exerted by the water on the hydrofoil has a magnitude \(F \approx \frac{1}{2}(n^2 - 1)\rho v_b^2 A\). (b) The boat has mass \(M\). Show that the liftoff speed is given by \[v \approx \sqrt{\frac{2Mg}{(n^2 - 1)A\rho}}\]