Elemental T and Y Shapes of Tree Networks of Ducts with Various Cross-Sectional Shapes

This paper reports optimal bifurcation shapes (T and Y) in turbulent regime of tree-shaped flows. Unlike earlier studies of T and Y constructs, here the effect of pressure losses at the junction is taken into account, and the wall roughness and duct cross-sectional shapes are free to vary. The optimal ratio of duct cross-sectional areas (as a generalization of Murray's law), the optimal ratio of duct lengths, and the optimal angle between the branches of the Y are presented. These optimal geometrical features are reported as functions of the flow direction (splitting flow versus merging flow), wall roughness, duct cross-sectional shape, and svelteness. The svelteness, Sv, is a global property defined as the external length scale of the flow construct divided by the internal length scale. It is shown that the effect of junction pressure losses on the optimized architecture can be neglected when Sv3∕2 is greater than approximately 104. Two dimensionless terms are introduced and shown to be useful for the optimization of flow networks.