First- and higher-order effects of curvature and torsion on the flow in a helical rectangular duct

A series expansion method is employed to determine the first-order terms in curvature epsilon and torsion eta of fully developed laminar flow in helical square ducts and in helical rectangular ducts of aspect ratio two. The first-order solutions are compared to solutions of the full governing equations. For toroidal square ducts with zero pitch, the first-order solution is fairly accurate for Dean numbers, De = Re epsilon(1/2), up to about 20, and for straight twisted square ducts the first-order solution is accurate for Germane numbers, Gn = eta Re, up to at least 50 where Re is the Reynolds number. Important conclusions are that the flow in a helical duct with a finite pitch or torsion to the first order (i.e. with higher-order terms in epsilon and eta neglected) is obtained as a superposition of the flow in a toroidal duct with zero pitch and a straight twisted duct; that the secondary flow in helical non-circular ducts for sufficiently small Re is dominated by torsion effects; and that for increasing Re, the secondary flow eventually is dominated by effects due to curvature. Torsion has a stronger impact on the flow for aspect ratios greater than one. A characteristic combined higher-order effect of curvature and torsion is an enlargement of the lower vortex of the secondary flow at the expense of the upper vortex, and also a shift of the maximum axial flow towards the upper wall, For higher Reynolds numbers, bifurcation phenomena appear. The extent of a few solution branches for helical ducts with finite pitch or torsion is determined. For ducts with small torsion it is found that the extent of the stable solution branches is affected little by torsion. Physical velocity components are employed to describe the flow. The contravariant components are found useful when describing the convective transport in the duct.