Physics: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 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
- [math]\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), }[/math]
where [math]\displaystyle{ \psi }[/math] is the stream function, [math]\displaystyle{ r }[/math] is the radial distance from the axis, and [math]\displaystyle{ z }[/math] is the axial distance measured from the closed end of the cylinder. The function [math]\displaystyle{ f(\psi) = \pi^2\psi }[/math] is found to predict the correct solution. The solution satisfying the required boundary conditions is given by
- [math]\displaystyle{ \psi= aU z \sin \left(\frac{\pi r^2}{2 a^2}\right), }[/math]
where [math]\displaystyle{ a }[/math] is the radius of the cylinder and [math]\displaystyle{ U }[/math] 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 [math]\displaystyle{ z\sim a }[/math] since boundary layer separation at [math]\displaystyle{ z=0 }[/math] is inevitable; that is, the Taylor–Culick profile is correct for [math]\displaystyle{ z\gg 1 }[/math]. The Taylor–Culick profile with injection at the closed end of the cylinder can also be solved analytically.[4]
See also
References
- ↑ 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.
- ↑ Culick, F. E. C. (1966). Rotational axisymmetric mean flow and damping of acoustic waves in asolid propellant rocket. AIAA Journal, 4(8), 1462–1464.
- ↑ 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.
- ↑ Majdalani, J., & Saad, T. (2007). The Taylor–Culick profile with arbitrary headwall injection. Physics of Fluids, 19(9), 093601.
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