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-section with arbitrary pitch and arbitrary radius of coil is studied. A loose-coiling analysis leads to two dominant parameters, namely Dean number, Dn = Re lambda1/2, and Germano number, Gn = Re eta, where Re is the Reynolds number, lambda is the normalized curvature ratio and eta is the normalized torsion. The Germano number is embedded in the body-centred azimuthal velocity which appears as a group in the governing equations. When studying Gn effects on the helical flow in terms of the secondary flow pattern or the secondary flow structure viewed in the generic (non-orthogonal) coordinate system of large Dn, a third dimensionless group emerges, gamma = eta/(lambdaDn)1/2. For Dn < 20, the group gamma* = Gn Dn-2 = eta/(lambda Re) takes the place of gamma. Numerical simulations with the full Navier-Stokes equations confirmed the theoretical findings. It is revealed that the effect of torsion on the helical 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 changes from two vortices to one vortex is gamma* > 0.039 for Dn < 20 and gamma > 0.2 for Dn greater-than-or-equal-to 20. For flows with fixed high Dean number and lambda, increasing the torsion has the effect of changing the relative position of the secondary flow vortices and the eventual formation of a flow having a Poiseuille-type axial velocity with a superimposed swirling flow. In the orthogonal coordinate system, however, the secondary flow generally has two vortices with sources and sinks. In the small-gamma limit or when Dn is very small, the secondary flow is of the usual two-vortex type when viewed in the orthogonal 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 effects for Dn less-than-or-equal-to 5000 and gamma < 0.1.