Taylor–Culick flow

The axisymmetric inviscid equation is governed by the Hicks equation, that reduces when no swirl is present (i.e., zero circulation) to

${\displaystyle {\frac {\partial ^{2}\psi }{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}=-r^{2}f(\psi ),}$

where ${\displaystyle \psi }$ is the stream function, ${\displaystyle r}$ is the radial distance from the axis, and ${\displaystyle z}$ is the axial distance measured from the closed end of the cylinder. The function ${\displaystyle f(\psi )=\pi ^{2}\psi }$ is found to predict the correct solution. The solution satisfying the required boundary conditions is given by

${\displaystyle \psi =aUz\sin \left({\frac {\pi r^{2}}{2a^{2}}}\right),}$

where ${\displaystyle a}$ is the radius of the cylinder and ${\displaystyle U}$ is the injection velocity at the wall. Despite the simple-looking formula, the solution has been experimentally verified to be accurate.[3] The solution is wrong for distances of order ${\displaystyle z\sim a}$ since boundary layer separation at ${\displaystyle z=0}$ is inevitable; that is, the Taylor–Culick profile is correct for ${\displaystyle z\gg 1}$. The Taylor–Culick profile with injection at the closed end of the cylinder can also be solved analytically.[4]

• Berman flow