Past exam of the options I course of the University of Oxford / 2012 / K / K1

For a steady flow the local time derivative in the Euler equations vanishes. The vector identityreduces to a gradient when the vorticity $\nabla \times u$ is zero. With constant density and gravity $-gk$, the momentum equation becomesThus on each connected flow region,This is Bernoulli's theorem for a steady irrotational flow: irrotationality makes the same constant apply throughout a connected region, rather than only along an individual streamline.

This is a Rankine vortex. Write $r=\sqrt{x^2+y^2}$. Inside the core the velocity is $u=(\Gamma y/a^2,-\Gamma x/a^2,0)$, soThis is the core value obtained from the inner formula; outside, the given irrotationality makes the vorticity zero. For nonzero $\Gamma$, the sharp boundary $r=a$ has different one-sided velocity gradients, so a single classical vorticity is not defined exactly on that edge; the inner trace is the boxed value and the outer trace is zero.

Inside the core, direct differentiation gives $(u\cdot \nabla )u=-{\Gamma }^2(x,y,0)/a^4$. The steady Euler equations therefore give $p_r=\rho {\Gamma }^{2}r/a^4$ and $p_z=-\rho g$. Integrating,Outside, $|u{|}^2={\Gamma }^2/r^2$, so part (a) givesMatching pressure at the core edge $r=a$ yields $A=B-\rho {\Gamma }^2/a^2$. Hence the complete pressure field, up to one additive constant $B$, isBoth the velocity and pressure are continuous at $r=a$. The inside expression has been derived from the momentum equation because the core is rotational.

The Free surface of a Rankine vortex follows from setting its pressure constant. At a free surface with constant pressure $p_s$, solve the two pressure expressions from part (b) for $z$. With $C=(B-p_s)/(\rho g)$, this givesSubstituting $r^2=x^2+y^2$ recovers the required Cartesian form. The two values coincide at $r=a$, and both radial slopes there equal ${\Gamma }^2/(g{a}^3)$. The surface is parabolic in the core and approaches height $C$ as $r\to \infty$; at the centre its depression below $C$ is ${\Gamma }^2/(g{a}^2)$.