Question

The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in Figure P14.83. 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 a sufficiently high speed, the water of density$\rho$moves in streamlined 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

$F\approx \frac{1}{2}\left(n^2-1\right)\rho v_b^{2}A$

(b) The boat has mass M. Show that the liftoff speed is given by$v\approx \sqrt{\frac{2Mg}{\left(n^2-1\right)A\rho }}$.

Short answer

(a) It is proved that the upward lift force exerted by the water on the hydrofoil has a magnitude is $\Delta t=\frac{Ah}{A\text{'}\sqrt{2gd}}$.

(b) The boat has mass . The liftoff speed is given by is $v_b={\left(\frac{2Mg}{\rho \left(n^2-1\right)A}\right)}^{1/2}$.

Step-by-step solution

Bernoulli’s equation:

The sum of pressure, kinetic energy per unit volume, and gravitational potential energy per unit volume have the same value for an ideal fluid at all points along the streamline. This result is summarized in Bernoulli's equation:

$P+\frac{1}{2}\rho v^2+\rho gy=cons\tan t$

Here, $P$is the pressure, $\rho$is the density, $g$is the gravity, and $y$is the distance.

(a) Prove that the upward lift force exerted by the water on the hydrofoil has a magnitude is Δt=AhA'2gd :

For diverging streamlines that pass just above and just below the hydrofoil, we have

$P_t+\frac{1}{2}\rho v_t^2+\rho g{y}_t=P_b+\frac{1}{2}\rho v_b^2+\rho g{y}_b$

Ignoring the buoyant force means taking $y_t\approx y_b$:

$$\begin{gathered} P_t+\frac{1}{2}\rho {\left(v_b\right)}^2=P_b+\frac{1}{2}\rho v_b^2 \\ {P}_t-P_b=\frac{1}{2}\rho v_b^2\left(n^2-1\right) \end{gathered}$$

The lift force is as below.

$$\begin{gathered} \left(P_t-P_b\right)A=\frac{1}{2}\rho v_b^2\left(n^2-1\right)A \\ F=\frac{1}{2}\rho v_b^2\left(n^2-1\right)A \end{gathered}$$

Step 3: (b) Show that the liftoff speed is given byv≈2Mgn2-1Aρ:

For liftoff,

$$\begin{gathered} \frac{1}{2}\rho {v^2}_b\left(n^2-1\right)A=Mg \\ {v^2}_b=\frac{2Mg}{\rho \left(n^2-1\right)A} \\ {v}_b={\left(\frac{2Mg}{\rho \left(n^2-1\right)A}\right)}^{1/2} \end{gathered}$$

Hence, the speed of the boat relative to the shore must be nearly equal to this speed of the water below the hydrofoil relative to the boat.