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 / 2 / Solution

For planar incompressible flow, $u_x+w_z=0$. Locally this permits a streamfunction with $u={\psi }_z$, $w=-{\psi }_x$, because equality of the mixed derivatives enforces incompressibility. The remaining component of the curl is $u_z-w_x={\psi }_{zz}+{\psi }_{xx}$, so irrotational flow requires Laplace's equation for $\psi$.

Put $K=\pi /b$. The sinusoidal contribution to the supplied streamfunction has ${\psi }_{xx}=U_{0}a{K}^2\cos (Kx)e^{-Kz}$ and ${\psi }_{zz}=-U_{0}a{K}^2\cos (Kx)e^{-Kz}$; the linear term is also harmonic. HenceAs $z\to \infty$, these approach the specified uniform flow, $(u,w)=(U_0,0)$.

The impermeability kinematic boundary condition is $w=u{z}_s^{\prime }(x)$ on $z=z_s(x)$, equivalently $\psi$ is constant on the wall. For $aK\ll 1$, evaluating the perturbed velocities at $z=0$ gives $w=-U_{0}aK\sin (Kx)=U_0{z}_s^{\prime }$ to first order. The residual is of order $U_0(aK{)}^2$. Equivalently $\psi (x,z_s)=U_{0}a\cos (Kx)[1-e^{-Ka\cos (Kx)}]=O(U_{0}K{a}^2)$. The surface velocity isThe first expression is the supplied approximate field evaluated exactly at the wall; its accuracy as a wall-flow solution is still only first order in $a/b$.