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[1] and later in 1966, Culick found a self-similar solution to the problem applied to solid-propellant rocket combustion.[2] Although the solution is derived for inviscid equation, it satisfies the non-slip condition at the wall since as Taylor argued that the boundary layer that be supposed to exist if any at the sidewall will be blown off by flow injection. Hence, the flow is referred to as quasi-viscous.

Flow description

The axisymmetric invscid equation is governed by 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 solution, the solution is verified to be accurate experimentally.[3] The solution is wrong for distances of order ${\displaystyle z\sim a}$ since boundary layer separation at ${\displaystyle z=0}$ is inevitable, i.e., Taylor–Culick profile is correct for ${\displaystyle z\gg 1}$. Taylor–Culick profile with injection at the closed end of the cylinder can be solved analytically.[4]

See also

• Berman flow

References

1. Taylor, G. I. (1956). Fluid flow in regions bounded by porous surfaces. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 234(1199), 456–475.
2. Culick, F. E. C. (1966). Rotational axisymmetric mean flow and damping of acoustic waves in asolid propellant rocket. AIAA Journal, 4(8), 1462–1464.
3. Dunlap, R., Willouchby, P. G., & Hermsen, R. W. (1974). Flowfield in the combustion chamber of a solid propellant rocket motor. AIAA journal, 12(10), 1440–1442.
4. Majdalani, J., & Saad, T. (2007). The Taylor–Culick profile with arbitrary headwall injection. Physics of Fluids, 19(9), 093601.