Entry flow in a heated straight tube

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Abstract
The developing flow in the entry region of a heated horizontal pipe is analysed. The asymptotic solution of the developing flow near the entrance of the heated straight pipe, distance O(a), is obtained by perturbing the solution of the developing flow in an unheated straight pipe. Two vortices result from the combination of the radial-directional and the downward motions of the fluid particles which are induced by the displacement of the boundary layer and develop along the pipe. The axial velocity has a concave profile in the inviscid core with its maximum off the centre-line near the entrance and it grows toward a uniformly distributed profile downstream. The downward stream caused by the displacement of the secondary boundary layer forces the axial velocity profile to turn anticlockwise continuously along the pipe if the flow is from left to right. The core flow induced by the axial boundary-layer displacement generates a favourable pressure gradient. Simultaneously, the secondary boundary-layer displacement affects the core flow to induce a favourable pressure gradient on the bottom of the pipe and an unfavourable pressure gradient on the top wall. The effect of the axial boundary-layer displacement is stronger than that of the secondary boundary layer near to the entrance. Downstream the growth of the boundary-layer thickness is suppressed by the inviscid secondary flow. It is expected that the displacement effect of the secondary boundary layer becomes dominant downstream from the region of O(d) when Gr is large.

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