Axially invariant laminar flow in helical pipes with a finite pitch

Abstract

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 = Reλ½, and Germano number, Gn = Reη, where Re is the Reynolds number, λ is the normalized curvature ratio and η 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, γ = η/(λDn)½. For Dn < 20, the group γ* = Gn Dn-2 = η/(λRe) takes the place of γ.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 γ ≤ 0.01 for moderate Dn. The critical value for which the secondary flow pattern changes from two vortices to one vortex is γ* > 0.039 for Dn < 20 and γ > 0.2 for Dn ≥ 20. For flows with fixed high Dean number and A, 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-γ 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-γ 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, A and γ effects for Dn ≤ 5000 and γ < 0.1.

Helically Coiled Heat Exchangers

“…5, influences heat transfer around the periphery. The fluctuational behaviour of the Nusselt number has been reported by other investigators as well (Lin and Ebadian, 1999;Liu, 1992;Patankar et al, 1974). In the later regions of the coil, the Nusselt number at the top and bottom differ only marginally.…”

Some Compositional Viscosity Correlations for Crude Oils from Russia and Norway

“…The value is somewhat higher than the value typically found for the random packing spheres (0.64) 26 . ⋅ .…”

Flow in pipes with non-uniform curvature and torsion

“…Tuttle (1990) and Kao (1987) showed that, for pipes with circular cross-section, torsion causes a skewing of Dean's two-vortex structure. The numerical work by Liu & Masliyah (1993) showed that a single-vortex solution exists provided that the torsion is much greater than the curvature. Yamamoto, Yanase & Yoshida (1994) used numerical techniques to solve the full Navier-Stokes equations for a wide range of parameters; they found no evidence that helical pipes admit a stable four-vortex solution.…”