Taylor–Culick flow
# Taylor–Culick flow

In fluid dynamics, Taylor–Culick flow describes the axisymmetric flow inside a long slender cylinder with one end closed, supplied by a constant flow injection through the sidewall. The flow is named after Geoffrey Ingram Taylor and F. E. C. Culick, since Taylor showed first in 1956 that the flow inside such a configuration is inviscid and rotational and later in 1966, Culick found a self-similar solution to the problem applied to solid-propellant rocket combustion. Although the solution is derived for the inviscid equation, it satisfies the non-slip condition at the wall since, as Taylor argued, any boundary layer at the sidewall will be blown off by flow injection. Hence, the flow is referred to as quasi-viscous.

## Flow description

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

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

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

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

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

