Finding order in complexity: A study of the fluid dynamics in a three-dimensional branching network

Abstract

The complex fluid dynamics associated with the flow in three-dimensional dichotomously branching networks is investigated. The flow physics described here is generic, though the particular flow geometry employed represents a model human bronchial tree. Up to six generations of branches (involving 63 straight portions and 31 bifurcation modules) are computed in one go; such computational challenges are rarely taken in the literature. In the present study, two branching configurations are considered side by side: the most widely studied in-plane configuration in which the centrelines of all generations lie on the same plane, and the 90∘ out-of-plane configuration in which the centreline of each generation is rotated with respect to its grandmother generation following a systematic methodology to form a space-filling three-dimensional structure. The paper develops a physical understanding of the fluid dynamics of branching networks and its dependence on the configuration (in-plane versus out-of-plane) and the extent (four, five, or six generations) of the network under consideration. The study of co-planar vis-à-vis non-planar configurations establishes a quantitative evaluation of the dependence of the fluid dynamics on the three-dimensional arrangement of the same individual branches. It is shown that apparent symmetry in the geometry of any two branches does not automatically imply symmetry in the flow field in those two branches. With the help of velocity contours, pressure contours, and distribution of mass flow in each branch, a qualitative and quantitative study is performed on the nature and evolution of flow asymmetry. The computations show that the degree of mass-flow asymmetry is smaller for the out-of-plane configuration (which is a more realistic model of a human bronchial tree) as compared to that for the in-plane configuration. The mass-flow asymmetry grows in each successive generation (starting from generation G2 for in-plane and G3 for out-of-plane configurations). In addition to mass-flow distribution, other types of asymmetries in the flow field are also analysed. It is established that, in spite of the complexity of the flow solutions, there also exists a systematic order such that it is possible to ascertain the flow field in all branches of a particular generation by determining the flow field in some systematically selected branches of that generation, indicating a possible route to the saving of computational resource and time.