## Abstract

A major aim of the present study is to understand and thoroughly document the fluid dynamics in three-dimensional branching networks when an intermediate branch is partially or completely obstructed. Altogether, 26 different three-dimensional networks each comprising six generations of branches (involving 63 straight portions and 31 bifurcation modules) are constructed and appropriately meshed to conduct a systematic study of the effects of varying the locations of a blockage of a given relative extent and varying the extent of a blockage at a fixed location. The side-by-side consideration of two branching configurations (in-plane and \(90^{\circ }\) out-of-plane) gives a quantitative assessment of the dependence of flow alteration due to blockage on the three-dimensional arrangement of the same individual branches. A blockage in any branch affects the flow in both downstream and upstream branches. The presence of a blockage can make three-dimensional asymmetric alteration to the flow field, even when the blockage itself is geometrically symmetric. The overall mass flow rate entering the network is found to remain nearly unaltered if a blockage is shifted within the same generation but is progressively reduced if the blockage is shifted to upstream generations. A blockage anywhere in the network increases the degree of mass flow asymmetry \(\delta _{\mathrm{G}n} \) in any generation. The order of magnitude disparity in \(\delta _{\mathrm{G}n} \) between the in-plane and out-of-plane configurations, characteristic of unobstructed networks, can be significantly reduced in the presence of a single blockage. The present three-dimensional computations show that the effects of blockage on the mass flow distribution in a large network are complex, often non-intuitive and sometimes dramatic, and cannot be captured by any simple one-dimensional model.

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## Appendix: Mathematical formulation for the 1D model of the branching network

### Appendix: Mathematical formulation for the 1D model of the branching network

A realistic physical condition is implemented in the CFD solutions presented in the main body of the paper where it is assumed that the driving potential (\(\Delta p_{\mathrm{total}} \) across the network) remains unaltered; then the mass flow rates at the inlet and in various branches need to be determined as part of the flow solution in the 1D model.

### Unobstructed network

For a branching network comprising generations G0 to G*n*, there are \(2^{n}\) flow paths from the main inlet to the \(2^{n}\) branches of generation G*n*. The equation for the loss of total pressure along any of those flow paths may be written by applying the Hagen–Poiseuille equation sequentially across each branch that falls in that flow path. As an example, the equation for the flow path from G0B1 to G*n*B1 may be written as follows:

Here, \(Q_{ij} \) is the flow rate in the *j*th branch of generation G*i* which falls along the path for which the equation is written, \(\Delta p\) is the loss along that path in the 1D model, and \(R_{i} \) is the flow resistance in generation G*i* and is given by the following expression:

where \(l_{Gi}\) and \(d_{Gi} \) denote the length and diameter of a branch of generation G*i*. Similarly, equations can be constructed for the loss of total pressure along the remaining \(2^{n}-1\) flow paths from G0B1 to G*n*B*k* (where \(k=2,3,4,\ldots 2^{n})\) in the unobstructed network. The system of \(2^{n}\) simultaneous algebraic equations is solved to obtain the flow rates in the \(2^{n}\) branches of generation G*n*. The flow rates in all the other branches can then be determined by applying the flow conservation principle at each bifurcation. In the 1D model, since the viscous loss incurred along a path is obtained by sequentially applying the Hagen–Poiseuille formula to individual branches (where the static and total pressure drops are equal), the values of \(\Delta p\) and \(\Delta p_{\mathrm{total}} \) along a path are equal. Therefore, the value of \(\Delta p\) in the \(2^{n}\) equations is taken equal to the static pressure drop along each path obtained from the 3D computations.

Figure 23 shows that, for a fixed static pressure drop across the network, the mass flow rates obtained by the 1D model are significantly different (greater) from the results of 3D computations. Moreover, the 3D computations show that the mass flow rates in the various branches of the same generation can be significantly different. As an example, the 3D computations show that the difference between the minimum and maximum mass flow rates in generation G5 is as high as 85% of the average mass flow rate in G5. The 1D model predicts that the mass flow rates in all branches of the same generation of a geometrically symmetric network are equal. This shortcoming of the 1D model cannot be removed by improving the methodology of loss calculation (for example, by including, in future, a loss model for the flow in the bifurcation module [11]).

### 1D model in the presence of a blockage

When an obstruction occurs in any branch, additional terms accounting for the loss due to change of cross-sectional area in the obstructed branch needs to be included. For example, when an obstruction occurs in branch G2B1, the equation for the loss of total pressure along the flow path from G0B1 to G*n*B1 may be written as follows:

Here, \({R}'_{i} \) is the minor loss coefficient accounting for the change of cross-sectional area in the obstructed branch. The value of \({R}'_{i} \) is a function of the extent of blockage \(\varphi \) and branch length \(l_{\mathrm{G}i} \) [49]. Equation (A3) holds true for all flow paths containing the obstructed branch G2B1.

Since there are \(2^{n}\) possible flow paths from G0 to G*n*, we again obtain a system of \(2^{n}\) simultaneous equations which need to be solved to obtain the flow rates in the \(2^{n}\) branches of generation G*n*, but the equations are nonlinear this time. In general, if a blockage occurs in generation G*m* (where \(0<m\le n)\) of a G0–G*n* network, then the nonlinear equation (A3) holds along \(2^{n-m}\) flow paths, while equation (A1) holds for the remaining (\(2^{n}-2^{n-m})\) flow paths from G0B1 to G*n*B*k*(where \(k=1,2,3,\ldots 2^{n})\). An appropriate solution strategy is to be applied. In order to solve the system of equations, the value of \(\Delta p\) in the right-hand side of the equations is taken equal to the static pressure drop across the network obtained from the corresponding (same position and extent of blockage) 3D computations. A comparison of 3D and 1D computations in the presence of blockage is shown in Fig. 11 in the main body of the paper.

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Pradhan, K., Guha, A. A systematic study of blockage in three-dimensional branching networks with an application to model human bronchial tree.
*Theor. Comput. Fluid Dyn.* **34, **301–332 (2020). https://doi.org/10.1007/s00162-020-00523-1

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### Keywords

- Fluid dynamics
- CFD
- Branching network
- Blockage