AXIALLY INVARIANT LAMINAR-FLOW IN HELICAL PIPES WITH A FINITE PITCH

Steady axially invariant (fully developed) incompressible laminar flow of a Newtonian fluid in helical pipes of constant circular cross-sect ion with arbitrary pitch and arbitrary radius of coil is studied. A lo ose-coiling analysis leads to two dominant parameters, namely Dean num ber, Dn = Re lambda1/2, and Germano number, Gn = Re eta, where Re is t he Reynolds number, lambda is the normalized curvature ratio and eta i s the normalized torsion. The Germano number is embedded in the body-c entred azimuthal velocity which appears as a group in the governing eq uations. When studying Gn effects on the helical flow in terms of the secondary flow pattern or the secondary flow structure viewed in the g eneric (non-orthogonal) coordinate system of large Dn, a third dimensi onless group emerges, gamma = eta/(lambdaDn)1/2. For Dn < 20, the grou p gamma = Gn Dn-2 = eta/(lambda Re) takes the place of gamma. Numeric al simulations with the full Navier-Stokes equations confirmed the the oretical findings. It is revealed that the effect of torsion on the he lical flow can be neglected when gamma less-than-or-equal-to 0.01 for moderate Dn. The critical value for which the secondary flow pattern c hanges from two vortices to one vortex is gamma > 0.039 for Dn < 20 a nd gamma > 0.2 for Dn greater-than-or-equal-to 20. For flows with fixe d high Dean number and lambda, increasing the torsion has the effect o f changing the relative position of the secondary flow vortices and th e eventual formation of a flow having a Poiseuille-type axial velocity with a superimposed swirling flow. In the orthogonal coordinate syste m, however, the secondary flow generally has two vortices with sources and sinks. In the small-gamma limit or when Dn is very small, the sec ondary flow is of the usual two-vortex type when viewed in the orthogo nal coordinate system. In the large-gamma limit, the appearance of the secondary flow in the orthogonal coordinate system is also two-vortex like but its orientation is inclined towards the upper wall. The flow friction factor is correlated to account for Dn, lambda and gamma eff ects for Dn less-than-or-equal-to 5000 and gamma < 0.1.