Fluid Flow into a Curved Pipe
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 351 (1664):
71--87 (Oct 8, 1976)DOI: 10.1098/rspa.1976.0130
Abstract
The influence of curvature on a pipeflow is discussed for a pipe that starts bending uniformly after an initial straight section. The Reynolds number and curvature are assumed large and small respectively, and the motion is examined first for distances from the starting of the bend that are comparable with the tubewidth. When the Dean number is finite, the coreflow remains practically undisturbed, i.e. unidirectional, until the bend and thereafter streams uniformly towards the outside of the curve, inducing a three dimensional boundary layer. This layer, however, has to react before the bending in order to adjust to the downstream conditions. It does so by means of a novel kind of upstream response. The azimuthal pressure variation generated by the bend is felt upstream and therefore both drives an inwards azimuthal motion in the boundary layer and produces an axial shear maximum at the inside wall. In the curved section the centrifuging then causes the maximum to shift to the outer bend at 1.51 pipe-radii beyond the start of bending. Finally, the theory is extended to longer lengthscales, to large Dean numbers and to general initial profiles.