The three-dimensional boundary layer in the entry region of curved pipes with finite curvature ratio

The major flow development in the region within a distance O((aR)1/2) from the entrance of a curved pipe occurs near the pipe wall, where a is the radius of the pipe cross section, assumed circular, and R is the radius of curvature of the central axis of the pipe. A three-dimensional boundary-layer solution is obtained for elucidating the physics of this developing flow; in particular, the effect of nonzero curvature ratio α=a/R on the geometric similarity of the flow. The numerical results show that the series solution in terms of α is valid only when α≤0.1 and s≤0.1 (aR)1/2, where s is the distance from the inlet along the pipe axis. The crossover of the axial wall shear is purely a geometric property and its location is a strong function of α. It is also demonstrated that (aR)1/2 is the proper length scale by showing that the solution of the first region, s∼O(a), is included in that of the second, s∼O(aR)1/2.