Secondary motion in three-dimensional branching networks

A major aim of the present work is to understand and thoroughly document the generation, the three-dimensional distribution, and the evolution of the secondary motion as the fluid progresses downstream through a branched network. Six generations (G0-G5) of branches (involving 63 straight portions and 31 bifurcation modules) are computed in one go; such computational challenges are rarely taken in the literature. More than 30 × 10⁶ computational elements are employed for high precision of computed results and fine quality of the flow visualization diagrams. The study of co-planar vis-à-vis non-planar space-filling configurations establishes a quantitative evaluation of the dependence of the fluid dynamics on the three-dimensional arrangement of the same individual branches. As compared to the secondary motion in a simple curved pipe, three distinctive features, viz., the change of shape and size of the flow-cross-section, the division of non-uniform primary flow in a bifurcation module, and repeated switchover from clockwise to anticlockwise curvature and vice versa in the flow path, make the present situation more complex. It is shown that the straight portions in the network, in general, attenuate the secondary motion, while the three-dimensionally complex bifurcation modules generate secondary motion and may alter the number, arrangement, and structure of vortices. A comprehensive picture of the evolution of quantitative flow visualizations of the secondary motion is achieved by constructing contours of secondary velocity [Formula: see text], streamwise vorticity [Formula: see text], and [Formula: see text] iso-surfaces. It is demonstrated, for example, that for in-plane configuration, the vortices on any plane appear in pair (i.e., for each clockwise rotating vortex, there is an otherwise identical anticlockwise vortex), whereas the vortices on a plane for the out-of-plane configuration may be dissimilar, and there may even be an odd number of vortices. We have formulated three new parameters (E , [Formula: see text], and [Formula: see text]) for a quantitative description of the overall features of the secondary flow field. [Formula: see text] represents a non-uniformity index of the secondary flow in an individual branch, E represents the mass-flow-averaged relative kinetic energy of the secondary motion in an individual branch, and [Formula: see text] provides a measure of the non-uniformity of the secondary flow between various branches of the same generation Gn. The repeated enhancement of the secondary kinetic energy in the bifurcation modules is responsible for the occurrence of significant values of E even in generation G5. For both configurations, it is found that for any bifurcation module, the value of E is greater in that daughter branch in which the mass-flow rate is greater. Even though the various contour plots of the complex secondary flow structure appear visually very different from one another, the values of [Formula: see text] are found to lie within a small range ([Formula: see text]) for the six-generation networks studied. It is shown that [Formula: see text] grows as the generation number Gn increases. It is established that the out-of-plane configuration, in general, creates more secondary kinetic energy (higher E ), a similar level of non-uniformity in the secondary flow in an individual branch (similar [Formula: see text]), and a significantly lower level of non-uniformity in the distribution of secondary motion among various branches of the same generation (much lower [Formula: see text]), as compared to the in-plane arrangement of the same branches.