A bifurcation study of laminar flow in a curved tube of rectangular cross-section

Abstract

The spiralling flow within a curved tube of rectangular cross-section normally forms two distinct cells in the plane of the cross-section, but a transition to a different, four-cell flow is known to occur at a critical value of the axial pressure gradient. This paper shows that, for a square cross-section, the transition is a result of a complex structure of multiple, symmetric and asymmetric solutions. The structure is revealed by solving extended systems of equations for steady-state, fully developed, laminar flow, in a finite-element approximation, to locate exactly the positions of singular points. Continuation methods are used to trace the paths of these points as the aspect ratio and radius of curvature vary. For a square cross-section, multiple, symmetric flows of both two and four cells are found in two distinct ranges of axial pressure gradient q. In addition, a pair of asymmetric solutions arise from a symmetry-breaking bifurcation point on the primary flow branch. The paths of limit points and symmetry-breaking bifurcation points are obtained as the aspect ratio γ varies, and a transcritical bifurcation point is found at γ = 1.43. For larger aspect ratios the secondary four-cell branch is disconnected from that of the primary two-cell flow. The bifurcation set in the two parameters q and γ has a complex structure, with a number of higher-order singularities; in particular, the path of limit points is found to cross in the manner of an unfolded swallowtail catastrophe. A stability analysis shows that all multiple solutions except the two-cell type are unstable with respect to either symmetric or antisymmetric perturbations. For values of the aspect ratio less than 1.43 there is a range of axial pressure gradients for which there is no stable flow. The critical Dean numbers of the singular points are found to vary strongly at small values of radius of curvature β, but the bifurcation set remains qualitatively the same. Previous work is interpreted in the light of the present results.