Past exam of the flows, fluctuations and complexity and symmetry and relativity paper of Part B of the physics course of the University of Oxford / 2011 / 1 / 3 / Solution

Use the rest frame of the hydrofoil, with undisturbed velocity $U_0{\mathbf{e}}_x$. Let $p_{\ast }$ be the undisturbed pressure at the lower surface level $z=0$, so the undisturbed hydrostatic pressure is $p_{\ast }-\rho gz$. The Bernoulli equation gives, with $h=a\cos (\pi x/b)$ and $ϵ=\pi a/b$,These formulas implement the prescribed neglect of the other velocity components. To the controlled first order in $a/b$, $p_{\mathrm{lower}}-p_{\mathrm{upper}}=\rho gh+\rho U_0^2ϵ\cos (\pi x/b)$. The vertical component of pressure acting on an inclined surface is integrated over its horizontal projection, soThe total upward fluid force per unit span is $L_{\mathrm{dynamic}}^{\prime }+B^{\prime }$. The lift due to motion is $L_{\mathrm{dynamic}}^{\prime }$; $B^{\prime }$ is the separate buoyancy. If the given approximate speed is squared without truncation, the formal additional dynamic term is $\rho U_0^2{\pi }^2{a}^2/(4b)$, but a controlled second-order prediction would require the omitted second-order flow and transverse-velocity contributions too.