## Abstract

We consider flow in large, regular networks configured in the form of a self-similar branching tree, in which each branch of the tree splits into two with a constant diameter and length ratio. Where appropriate, the limit of an infinite network is also considered. The laminar or turbulent nature of the flow at the inlet may be different from that at the outlet as a consequence of the change in Reynolds number from one generation to the next. A steady flow is shown to be globally stable for a friction law that is non-decreasing with respect to the flow rate. A difference between critical Reynolds numbers for transition and relaminarization may cause the flow in the entire network to oscillate. Control of branch flow rates and junction pressures is investigated using valves and pumps as the input variables. Fault detection due to blockage is discussed. The network hydrodynamics are only partially controllable if individual pumps are used to control the pressure at the junctions.

## 1. Introduction

Tree-like configurations are a very common form of flow networks in both natural and artificial systems. A common engineering example is the piping that distributes water in a city. The water is pumped from the source to the end user through a series of pipes of usually decreasing cross-sectional areas and lengths. A similar network provides heating and cooling fluids in district and building climate-control systems. In nature, the collection of water in a river basin is through streams and tributaries of different flow rates. The mammalian vascular systems and bronchial trees also have similar configurations. Many of these trees not only have flow, but also provide transport of a scalar, such as heat or species concentration.

Though there have been many studies of flows in branching trees, the geometrical complexity of an irregular tree with a large number of generations has precluded the search for general results. The magnitude of the problem can be reduced, however, by making simplifications in the model of flow dynamics and assuming regularity in the geometry. Most studies of blood flow assume that the flow rate is proportional to the longitudinal pressure gradient in a blood vessel (Cohn 1954; Berne & Levy 1992; Schmid-Schönbein 1999; Gafiychuk & Lubashevsky 2001). While this assumption does not take into account entry effects, flow separation near bifurcations, elasticity of the walls and other observed phenomena, it provides a starting point for the analysis of complex networks. A comprehensive overview of the mechanics of vascular networks is given by Pedley (2000). Bejan and co-workers have done extensive work on trees and networks, mostly from a *constructal* and optimization point of view. As examples of this, heat transfer from a fractal piping network within a porous environment (Bejan & Errera 1997), tree architecture in hot-water distribution piping to minimize flow resistance (Bejan 2000) and trees combined with closed-loop structures for cooling purposes (Wechsatol *et al*. 2005) can be cited. Following a constructal approach, it has been shown that the flow architecture of the lungs for which oxygen and carbon dioxide are best delivered and removed, respectively, is a bronchial tree with 23 levels of bifurcation (Reis *et al*. 2004). Among other applications, fault detection and identification in piping networks (Caputo & Pelagagge 2002), fractal-like microchannel networks for cooling purposes (Chen & Cheng 2002; Pence 2002; Alharbi *et al*. 2004), mine ventilation networks (Hu *et al*. 2003; Koroleva & Krstic 2005) and fuel cells (Senn & Poulikakos 2004) have been studied.

Control is an inherent part of many networks. In the cardiovascular system, for instance, control of the blood pressure through baroreflexive feedback prevents neurogenic syncope in normal humans (Morillo & Villar 1997), while control in water distribution systems is obtained by individual pumps at all levels of the tree (Brion & Mays 1991). Other control inputs are possible; for example, the effect of heart rate on heating and cooling in the microvascular branching network of reptiles has been investigated (Seebacher 2000). Other applications of the control of networks have included heat exchanger networks and a tree-shaped network of vibrating strings (Dager & Zuazua 2001).

In this study, large, regular tree networks are considered. Simple but realistic assumptions are made regarding their geometrical self-similarity and hydrodynamics. The steady-state and dynamic behaviour of the tree both without and with control is analysed.

## 2. Network topology

The tree to be studied is schematically shown in figure 1. Each branch in a given generation bifurcates into two in the following generation. There are a total of *N* generations, where *N* is assumed to be large. If *i* indicates the generation number of a branch and *j* the branch number within that generation, then each branch can be represented uniquely as a pair of integers , where and ; is the number of branches in generation *i*. An alternative notation where branch is referred to as , where , will sometimes be used for purposes of compactness in analytical and computational representations. Similarly, the junctions are uniquely identified either by the indices or the index corresponding to the branch that leads *toward* it. There are a total of branches with junctions in the tree.

The inlet to the tree is a pipe of diameter , length and volumetric flow rate ; is time and ‘*’ indicates a dimensional quantity. Each branch is a circular pipe of diameter and length that carries a volume flow rate . It bifurcates into two branches, each of diameter and length . It has been shown that for laminar and turbulent flows, respectively, diameter ratios of and and length ratios of and minimize the resistance to flow for a given flow volume (Murray 1926; Bejan *et al*. 2000; Lorente *et al*. 2002); in physiology this is known as Murray's law. In this study, the diameter and length reduction is taken to be the same, and though, in principle, there is no geometrical restriction on *β*, it is assumed here to be a constant for the entire tree and greater than unity in value. Thus, and , where and . The pressure difference in branch between its own inlet and outlet is , where is the floor function. The overall pressure difference between the inlet and the outlet of the tree is .

Infinite networks are an analytical approximation for large, finite networks; however, physical quantities associated with the infinite limit must be finite. The total length of an *N*-generation tree is(2.1)For an infinite tree, we get(2.2)The total lengths of trees of various generations are shown in figure 2 as a function of *β*. It is seen that relatively few generations (of the order of 10 or so) are needed for an infinite tree approximation to be valid. Also, as .

In a similar manner, other geometrical parameters of the tree network can be computed. The lateral area of every generation is times the previous one, and for the internal fluid volume the scaling is , so that for convergence the value of *β* should be larger than and , respectively, in an infinite tree.

## 3. Mathematical model

A one-dimensional flow model is used to study the hydrodynamics of the network. We make the assumption that the flow is fully developed, and inlet and exit effects are neglected. For a branch , the momentum equation is(3.1)where is fluid density. The three terms from the left are the fluid acceleration, viscous resistance and pressure force, respectively. The resistance term is different for laminar and turbulent flow and is given by(3.2)where is the dimensional kinematic viscosity. The Reynolds number of the flow in branch is and is the critical Reynolds number for transition. The laminar flow formula corresponds to Poiseuille flow, and the turbulent value is modelled using the Blasius formula for flow in a smooth pipe (Fox & McDonald 1998).

The fluid is assumed to be incompressible, so that continuity at the junction can be expressed as(3.3)The flow rates, pressures and time are non-dimensionalized, using , and , respectively. Equations (3.1) and (3.3) become(3.4a)(3.4b)where(3.5)and .

## 4. Steady-state analysis

Since all branches of a generation are identical, there is at least one solution to the steady-state problem, in which there is no variation with *j* for a given *i*. The flow rate in each branch is then , where is the non-dimensional flow rate in the first branch. From equation (3.4*a*), the steady-state pressure drop in generation *i* is(4.1)These pressure drops can be summed over all the generations from *i*=1 to *N* to give the total pressure drop (Cohn 1954).

The local Reynolds number in a branch is . increases monotonically if *β*>2 or decreases if *β*<2 from the first to the last generation from a value of to . The flow in each branch of the network is laminar or turbulent depending on whether is less than or greater than , respectively. The last generation before the flow changes to a different regime is(4.2)where is the ceiling function and is the critical Reynolds number. Thus, there are four possible cases.

*Laminar in-laminar out*. The flow is laminar in the entire network. The total pressure drop is(4.3)There is a singularity in at . For , the pressure drop tends to a finite value , only if . Note that Murray's law also indicates as an optimum geometrical scale (Murray 1926; Bejan*et al*. 2000; Lorente*et al*. 2002), so that an infinite network of this form is not feasible with finite total length.*Turbulent in-turbulent out*. The flow is turbulent everywhere and(4.4)There is a singularity at . This tree can be infinite if , in which case .*Laminar in-turbulent out*. The entrance is laminar, but there is transition to turbulent flow in some intermediate generation . We have then(4.5)There are singularities at and . An infinite tree is possible if , so that , with a singularity at .*Turbulent in-laminar out*. The entrance is turbulent, but there is relaminarization at some intermediate generation . We have(4.6)Again there are singularities at and . For , an infinite tree with pressure drop of is possible.

## 5. Dynamics of flow

The general problem of the dynamic behaviour of an infinitely large, nonlinear network is intractable at this time. We will confine ourselves to the linear case with laminar flow. Using equations (3.4*a*) and (3.5*a*), we get(5.1)Since , where , we have(5.2)Adding all the individual equations, we get(5.3)where the pressure drops in each generation add up to give the overall pressure difference . This equation can be solved for a given . The time constant of the network is(5.4)For an infinite network,(5.5)Figure 3 shows the variation of the time constant with *β* for several trees.

## 6. Global stability of steady flow

It will be shown that for certain frictional behaviour the steady flow in a regular tree network is stable to all perturbations, large or small. The junction index notation *k* is used for the pressure. Equation (3.4*b*) for and represents the continuity equations in a network of *N* generations. The momentum equation of the inlet branch is(6.1)The momentum equations for the inner branches are(6.2)where and . The momentum equations of the outlet branches are(6.3)where . is assumed to be a non-decreasing function of the volumetric flow rate.

Let and , the bar denoting a steady-state value and the prime a small perturbation. Substituting in equations (3.4*b*) and (6.1)–(6.3) and subtracting the steady-state equations, we get(6.4a)(6.4b)(6.4c)(6.4d)Consider the function , which is a weighted sum of the kinetic energy of each branch. Differentiating *V* with respect to time and using equations (6.4*b*–*d*), we have(6.5)The first term on the right-hand side is non-positive for any volumetric flow if , which is true for a non-decreasing . Looking at the rest of the terms in equation (6.5), we see that they are(6.6a)(6.6b)(6.6c)(6.6d)Since and with equality only at the origin, *V* is a Lyapunov function: the hydrodynamic steady state is globally stable for a non-decreasing friction law , and consequently also unique.

## 7. Oscillatory flow

Since the nature of the flow may change as the conduit diameter decreases, transition from laminar to turbulent flow or vice versa becomes relevant for tree networks, and flow phenomena like relaminarization may occur (Narasimha & Sreenivasan 1979; Iida & Nagano 1998; Greenblatt & Moss 1999). For a single pipe with fixed geometry, the pressure difference can be varied to drive the flow to either laminar or turbulent flow and the transition zone has little relevance for the steady state. However, in a tree network with flow driven by the pressure difference between the network inlet and outlet, it may happen that the flow in some generation be trapped in the transition regime and stay oscillating between the two regimes, affecting, thus, the entire network both upstream and downstream of the transition generation. This is due to differences between the critical Reynolds numbers for the accelerating and decelerating stages in a pulsating flow that have been observed; relaminarization of the flow was seen during acceleration and turbulent bursts on deceleration (Iguchi & Ohmi 1982; Ohmi *et al*. 1982). To analyse this, we will assume two critical Reynolds numbers: for laminar-to-turbulent transition for a flow under acceleration and for relaminarization of a flow under deceleration, where is a constant.

### (a) Single pipe

Consider first a single pipe where the flow is driven by a constant pressure difference between the inlet and the outlet. For some values of , there exists a region where the flow is neither fully laminar nor fully turbulent, but oscillates between them. Numerical solutions of the momentum equation for a single pipe were obtained for confirmation of these oscillations. Equation (3.4*a*) was numerically integrated using an implicit Euler scheme, the Matlab subroutine *fsolve* and the following parameters: time-step of 0.01, , and . The time-step was much smaller than the period of the oscillation, and was also small enough for the solution to become independent of it. Oscillations between flow rates of and can be observed in figure 4, which shows the flow rate dynamics for .

A local, linearized analysis of equation (3.4*a*) also enables an analytical estimation of the dimensionless frequency of the oscillation to be(7.1)where , , and . The laminar and turbulent steady-state flows are and , respectively. Figure 5 shows a comparison between the numerically and analytically computed *f* as a function of *ϵ*.

It was shown in §6 that a non-decreasing friction law leads to global stability of the steady state, a condition which is violated here because of hysteresis due to different critical Reynolds numbers for accelerating and decelerating flows. In the transition region between and , which correspond, respectively, to the laminar and turbulent boundaries, there are two possible resistance values for a single *q*. Figure 6 shows the laminar and turbulence resistance curves as a function of the volumetric flow rate; the solid lines represent the region in which each is valid. It is thus possible that for turbulent flow, the higher friction reduces the flow rate and makes it laminar. The laminar friction is now lower so that the flow rate increases, eventually making the flow turbulent, and the cycle repeats itself.

### (b) Network of pipes

If the inlet flow is laminar, the entire flow downstream will also be laminar. If the inlet flow is turbulent, the downstream flow may be turbulent all the way to the outlet, depending on the magnitude of the inlet flow, the number of generations and the geometric scale factor. However, if the network is large enough, the flow will become laminar at some point. Figures 7 and 8 show the oscillations in volumetric flows and junction pressures, respectively, of a four-generation network when the second generation is in transition. For this, equations (3.4*a*) and (3.4*b*) were integrated using a time-step of 0.01, , , and .

Figure 9 shows calculations for a tree of 10 generations where the flow at the inlet branch is turbulent. In the shaded region, the parameters are such that the flow oscillates. In this case at least one generation of the tree is in the transition regime, so that the flow in the entire network is affected. Outside the shaded areas, the network is a turbulent in-laminar out tree without oscillation; there is no branch of the tree within the hysteresis region of the friction law, resulting in one generation being fully turbulent and the next fully laminar. For a small enough *ϵ*, there are no oscillations for any *β*.

## 8. Flow control

Pumps and valves are the most common devices used to drive flow within large networks and to control the flow rates in the branches and the pressures at the junctions. The flow will be assumed to be laminar in following analysis, so that controllability theory for linear systems can be used. Thus, the momentum equation is(8.1)where the additional term represents the pressure rise due to the pump. It is assumed that in generation *i*, the pressure , where *γ* is a scaling parameter and *P* is the pressure rise of the pump at the inlet branch.

### (a) State and output control

Controllability refers to the property of being able to take the system or its output from a given state to any other. According to linear theory (Paraskevopoulos 2002), the state of a linear system described by , where is the control input, is controllable if and only if the matrix is full rank. On the other hand, if the output is governed by , it is controllable if and only if is full rank.

For a pump on a single pipe, the pressure drop across it can be varied and thus the flow rate. For a network with a single pump at the entrance of the network, if there is a set of resistive loads, such that the pressure at the exit reaches specific values, then the outlet pressure is controllable.

### (b) Network control by means of valves

Consider flow in a large network driven by a constant pressure difference between the inlet and outlet. Valves are used as the control input to attempt to control the hydrodynamics. The variables of state are the flows and pressures at every branch and junction, respectively, and by placing one valve in any branch of the tree the network is not controllable. However, the tree is output controllable if the output of interest is just one flow or a pressure at a junction. If a second valve is added, the network is still not state controllable but is output controllable in two variables. The output controllability range is affected by the valves themselves, since each valve has an effect on the flow. Even if a valve is used in every branch, the network state space would still not be completely controllable. Consider a two-generation network, as shown in figure 10, with the state variables being the three flows and the pressure at the junction. Two flows may be specified, but the third one cannot be chosen. This means that flows at the junctions in any tree network are not totally independent because of continuity. Mathematically, the algebraic relations impose a constraint on the differential variables.

### (c) Fault detection in networks

A tree network with valves in every branch allows analysis of the problem of fault detection, since valve changes in any branch have an effect on other branches. Consider a four-generation tree with valves in every branch and flow driven by an overall pressure difference . One branch at a time is completely blocked and the flow rates are computed for and . A four-generation tree has 15 branches, and only the seven branches within the first three generations are blocked. A matrix can be defined representing the outlet flow rate obtained upon blocking completely a single branch, which in this case can be calculated to be

Measuring all the outlet flows can indicate where the blockage occurred. In each column of , the blockage is at the last common branch of those branches with no flow. The first common branch is (1,1). For instance, in column six the last common branch to branches (4,5) and (4,6) is (3,3); thus, this branch is completely blocked. Furthermore, for *n* pairs of zero flow measurements, the fault is going to be *n* generations upstream.

The problem of partial blockage is more difficult. If one branch is partially blocked, the downstream flows at the outlet are lower than those without a blockage. A single partial blockage can be located as before, but if there is more than one it is not possible to locate the blockage by only looking at the outlet flows.

### (d) Control by means of individual pumps

The use of individual pumps as inputs to control the junction pressures is also common. Independent pumps that produce a pressure rise may be placed in every branch of the tree. The hydrodynamic network model, equations (8.1) and (3.4*b*), constitutes a differential–algebraic system of equations (DAE). They must be recast into a purely differential system. DAE systems have been extensively studied in this regard, and, in general, this is not a trivial task (Dai 1989; Brenan *et al*. 1996). One option here is to exclude the pipes whose flow rates are dependent on those of the others. An equivalent approach that we will use, which is possible only because of linearity, is to write differential equations with the pressures at the junctions as the only unknowns. The procedure is first presented for a two-generation network, and then extended to larger networks.

For the two-generation network shown in figure 10, the momentum equations are(8.2a)(8.2b)(8.2c)In addition, we have mass conservation at the junction that can be written as(8.3)Equations (8.2*a*)–(8.3) can be reduced to (see appendix A for details)(8.4)in which the flow rates do not appear.

For a large network, equation (8.4) can be written for each one of the junctions in the tree. The variables to be controlled are the pressures at the junctions that can be written as . The control variables are the pump pressure rise terms that can also be written as and . Thus,(8.5)where , , and ; contains the known inlet and outlet pressures of the tree.

For example, for the four-generation, 15-pipe, seven-junction tree in figure 1, it can be shown thatwhere and . and are known matrices that are too large to be shown here. The matrix is symmetric and non-singular.

Multiplying by and writing , equation (8.5) takes the form(8.6)where , and . contains the junction pressure terms ; and contain the pressure rise terms and produced by the control branch pumps, respectively. The output of the system is taken to be the branch volumetric flow rates .

### (e) State *R*-controllability and output controllability

Defining(8.7)where , equation (8.6) becomeswhere is temporarily the input variable. The system has the form given in §8*a*. * x* is controllable if , defined by and the identity matrix, is of rank and exists.

Since the DAE model has been reduced to a differential one, controllability of the reduced differential system would indicate that the network is only partially controllable or *R-controllable* (Yip & Sincovec 1981). The original state vector contains flow rates in addition to pressure terms, all of which are not controllable. If contains only volumetric flows, the flows can be determined from the junction pressures. The flows are output controllable if , defined as in §8*a*, is of rank .

For a maximum number of control pumps, , i.e. there is one pump per branch in the network. If every single available pump is used , then is not unique and the controllability depends on * A*. From equation (8.7), can be determined for any given , indicating that the system is also controllable for as the input variable.

If a pump is placed before every junction, then . The size of matrices and will be and the existence of depends on , where and . If and are of rank , the system is controllable.

If , there are more junctions than pumps and the junction pressures are not controllable; there are less control inputs than control variables. For example, for a network of three generations with a single pump located in any branch, the three junction pressures cannot be regulated by the pump. The input vector has zero elements.

### (f) Numerical computations

Calculations were performed for networks of two, three, four, five and six generations with pumps in every branch. The results are summarized in table 1, where the number of generations, branches, junctions and output flows determine the rank of the state and output controllability matrices. A two-generation tree is the smallest possible network; using pumps in every branch, the two-generation tree is state controllable and only two flows are output controllable. The third flow cannot be specified because of its dependence on the other two. In a three-generation network, pumps in every branch make it state controllable; this is also true for four, five and six generation networks. This means that by using individual control pumps in an *N* network tree, the pressure junctions are state controllable and flows are output controllable.

## 9. Conclusions

Flows in large, regular tree networks have been analysed here using simple but realistic assumptions regarding their geometrical self-similarity and hydrodynamics. The effect of the geometric scale factor on the steady, unsteady and stability characteristics of flow in the tree is determined. In some cases, infinite trees can be shown to be viable as an appropriate limit. It is found that if the critical Reynolds number for laminar-to-turbulent flow transition is different from turbulent-to-laminar relaminarization, the flow in the tree may oscillate in time.

It is possible to detect the location of faults when there is partial or complete blockage of a single branch within the tree. The flows in the network are not completely state controllable by using valves as the input control elements. Using individual pumps as the control elements, it is found that the pressure at the junctions is controllable and the flows are *R*-controllable.

## Acknowledgments

We thank the Universidad Nacional Autónoma de México for the support given to W.F. during this research.

## Footnotes

↵† Present address: Beckman Laser Institute and Medical Clinic, University of California, Irvine, CA 92612, USA.

- Received December 19, 2005.
- Accepted February 27, 2006.

- © 2006 The Royal Society