Steady flow in a dividing pipe

Abstract

The high Reynolds number flow through a circular pipe divided along a diameter by a semi-infinite splitter plate is considered. Matched asymptotic expansions are used to analyse the developing flow, which is decomposed into four regions: a boundary layer of Blasius type growing along the plate, an inviscid core, a viscous layer close to the curved wall and a nonlinear corner region. The core solution is found numerically, initially in the long-distance down-pipe limit and thereafter the full problem is solved using down-pipe Fourier transforms. The accuracy in the corners of the semicircular cross-section is improved by subtracting out the singularity in the velocity perturbation. The linear viscous wall layer is solved analytically in terms of a displacement function determined from the core. A plausible structure for the corner region and equations governing the motion therein are presented although no solution is attempted. The presence of the plate has little effect ahead of the bifurcation, but wall shear on the curved wall is found to increase from its undisturbed value downstream.