Entry flow in a curved pipe

Abstract

A secondary flow is set up when a fluid flows through a stationary curved pipe. The fluid in the middle of the pipe moves outwards and that near the wall inwards. Dean showed that the dynamical similarity of this fully developed flow depends on a non-dimensional parameter $D = 2(a/R)^{\frac{1}{2}}(a\overline{W}/\nu) $, where $\overline{W} $ is the mean velocity along the pipe, v is the coefficient of kinematic viscosity and a is the radius of the pipe, which is bent into a coil of radius R. Dean's analysis was limited to small values of D. Later, Barua developed an asymptotic boundary-layer theory for large values of D and showed for these values of D that the resistance coefficient γ_c is much larger than that for the corresponding straight pipe. The present work deals with the flow in a curved pipe as it develops from a uniformly distributed velocity at the entrance to a fully developed profile. Barua's results for the fully developed flow are adopted as downstream conditions in the present work. The ratio of the entry lengths of the curved ipe and the corresponding straight one is shown to be proportional to D^−1/2 when D is large. Thus, the entry length for a curved pipe is much shorter than that for the corresponding straight pipe.