Flow in curved ducts: bifurcation structure for stationary ducts

In developed laminar flow in stationary curved ducts there is, in addition to the two-vortex secondary flow structure first analysed by Dean (1927), another solution branch with a four-vortex secondary flow. These two branches have recently been shown to be joined by a third branch, but stability characteristics and the possible presence of additional branches have yet to be described. In this paper orthogonal collocation is used in conjunction with continuation techniques to characterize the bifurcation structure for symmetric flows. The two- and four-vortex solutions are stable to symmetric disturbances, while the recently reported branch joining them is unstable. A more systematic exploration of the parameter space than has hitherto been reported is performed by examining the morphogenesis of the bifurcation structure within the general framework of properties described by Benjamin (1978). The starting point is the ‘perfect’ problem of flow in an infinite curved slit, which bifurcates to give rise to a cellular structure. Addition of ‘stickiness’ at the cell boundaries turns each pair of cells into a curved duct of rectangular cross-section which, by a geometry change, leads to the curved circular tube. For the perfect problem a large number of solution branches are present, but the addition of stickiness turns most of them into isolae which vanish before the no-slip limit is reached. The solution branches that remain include, in addition to the three described previously, another solution family not connected to the other one. This family comprises two branches, both four-vortex in character and unstable to symmetric disturbances.