finite volume simulation of unsteady water pipe ﬂow · oped simulations applying the shallow...

Drink. Water Eng. Sci., 7, 83–92, 2014www.drink-water-eng-sci.net/7/83/2014/doi:10.5194/dwes-7-83-2014© Author(s) 2014. CC Attribution 3.0 License.

Finite volume simulation of unsteady water pipe flow

J. Fernández-Pato and P. García-Navarro

LIFTEC (CSIC) – University of Zaragoza, C/María de Luna 5, 50018 Zaragoza, Spain

Correspondence to:J. Fernández-Pato ([email protected])

Received: 20 December 2013 – Published in Drink. Water Eng. Sci. Discuss.: 27 January 2014Revised: 11 July 2014 – Accepted: 27 July 2014 – Published: 21 August 2014

Abstract. The most commonly used hydraulic network models used in the drinking water community exclu-sively consider fully filled pipes. However, water flow numerical simulation in urban pipe systems may requireto model transitions between surface flow and pressurized flow in steady and transient situations. The governingequations for both flow types are different and this must be taken into account in order to get a complete numer-ical model for solving dynamically transients. In this work, a numerical simulation tool is developed, capableof simulating pipe networks mainly unpressurized, with isolated points of pressurization. For this purpose, themathematical model is reformulated by means of the Preissmann slot method. This technique provides a reason-able estimation of the water pressure in cases of pressurization. The numerical model is based on the first orderRoe’s scheme, in the frame of finite volume methods. The novelty of the method is that it is adapted to abrupttransient situations, with subcritical and supercritical flows. The validation has been done by means of severalcases with analytic solutions or empirical laboratory data. It has also been applied to some more complex andrealistic cases, like junctions or pipe networks.

1 Introduction

The numerical simulation of water pipe networks is char-acterized by the difficulty of simulating both transient andsteady states. Most of the time, the water flow is unpressur-ized but, under exceptional conditions, the limited storage ca-pacity of the pipes could cause a momentary pressure peak ifthe water level raises quickly. Under these conditions, a pres-surized flow should be taken into account, implying a changein the mathematical model in order to solve the problem ac-curately.

The most commonly used hydraulic network models usedin the drinking water community exclusively consider fullyfilled pipes. However, a complete pipe system model shouldbe able to solve steady and transient flows under pressur-ized and unpressurized situations and the transition betweenboth flows (mixed flows). Many of the models developed tostudy the propagation of hydraulic transients solve both sys-tem of equations, e.g.Gray (1954), Wiggert and Sundquist(1977). For this purpose, numerical schemes derived fromthe Method of Characteristics (MOC) are employed. Thismethod transforms the continuity and momentum partial dif-ferential equations into a set of ordinary differential equa-

tions, much easier to solve. MOC schemes can handle com-plex boundary conditions. However, the main issue withthese kind of methods is the need for interpolation in somecases, so that they are not strictly conservative. This resultsin a diffusion of the wave front that implies a wrong arrivaltime of the waves to the boundaries. Hence, a new efficientnumerical scheme is justified in order to solve accurately thewave front propagation.

Two main points of view have been considered by theresearchers to study mixed flow problems along the lastdecades. The first one consists of solving separately pressur-ized and shallow flows, treating the transitions between bothflow types as internal boundary conditions. These models arecapable of simulating sub-atmospheric pressures in the pipe,but the need for solving both mathematical models implies ahigh level of complexity. Other authors, e.g.García-Navarroet al. (1994), León et al.(2009), Kerger et al.(2010) devel-oped simulations applying the shallow water equations in aslim slot over the pipe (Preissmann slot method). An esti-mation of the pipe pressure can be obtained from the waterlevel in the slot. The great advantage of this model is the useof a single equation system for solving the complete prob-

Published by Copernicus Publications on behalf of the Delft University of Technology.

84 J. Fernández-Pato and P. García-Navarro: Finite volume simulation of unsteady water pipe flow

lem. This technique has been widely used to simulate localtransitions between pressurized and shallow flows but somestability problems have been reported (Trajkovic et al., 1999)when simulating cases with abrupt transitions. The reason isa big difference between the wave speeds (from∼ 10 m s−1

in shallow flow to∼ 1000 m s−1 in pressurized flow). In or-der to solve this issue, the slot width could be increased, butthis implies a loss of accuracy in the results, due to the no-mass/momentum conservation.

This work is focused on the development of a simulationmodel able to solve pipe system networks working mainlyunder free surface flow situations but exceptionally pressur-ized. Hence, it is based on the second option previously men-tioned and it is aimed to generalize the shallow water equa-tions by means of the Preissmann slot method. The explicitfirst order Roe scheme has been chosen as numerical methodfor all the simulations.

Implicit methods or parallel computation have been dis-missed, as the calculation times are not long. Fluxes andsource terms have been treated by means of an upwindscheme (García-Navarro and Vázquez-Cendón, 2000). Allthe symbols and units used are described in Table1.

This paper is an extension of the one presented at the 12thedition of the International Conference on “Computing andControl for the Water Industry – CCWI2013” (Fernández-Pato and García-Navarro, 2014).

2 Mathematical model

2.1 Shallow water equations

The unsteady open channel water flow are usually modelledby means of the 1D shallow water or St. Venant equations.These equations represent mass and momentum conservationalong the main direction of the flow and become a good de-scription for the behaviour of the most of the pipe-flow-kindproblems (Kundu et al., 2012; Abbott, 1979). The conserva-tive form of this systems of equations is presented in Eqs. (1)and (2):

∂A

∂t+

∂Q

∂x= 0 (1)

∂Q

∂t+

∂

∂x

(Q2

A+ gI1

)= gI2 + gA

(S0 − Sf

)(2)

whereA is the wetted cross section,Q is the discharge,g isthe acceleration due to gravity,I1 represents the hydrostaticpressure force term andI2 accounts for the pressure forcesdue to channel/pipe width changes:

I1 =

h(x,t)∫0

(h − η)b(x,η)dη (3)

Table 1. List of symbols and units.

Symbol Meaning Units

A Wetted cross-section m2

Af Full pipe cross-section m2

bs Preissmann slot width mb Pipe/channel width mc Wave speed m s−1

cWH Water hammer speed m s−1

D Diameter mE Pipe elastic modulus Pae Pipe thickness mFr Froude number –g Accelaration due to gravity m s−2

H Elevation of the hydraulic grade line mh Free surface depth mI1 Hydrostatic pressure force term m3

I2 Pressure force term due to channel/pipe width changes m2

K Fluid elastic modulus Pan Manning’s roughness coefficient s m−1/3

P Wetted perimeter mQ Water discharge m3 s−1

R Hydraulic radius mS0 Bed slope –Sf Energy grade line slope –t Time su Flow speed m s−1

λ Eigenvalue m s−1

e Eigenvector m s−1

v Cross-section averaged flow velocity m s−1

θ Angle between pipe and longitudinal coordinate radx Longitudinal coordinate mz Elevation mη Vertical coordinate mρ Fluid density kg m3

τ0 Boundary shear Pa

I2 =

h(x,t)∫0

(h − η)∂b(x,η)

∂xdη (4)

b =∂A(x,η)

∂η(5)

in which η andh(x, t) represent vertical coordinate and thefree surface depth, respectively. The remaining terms,S0 andSf , represent the bed slope and the energy grade line (definedin terms of the Manning roughness coefficient), respectively:

S0 = −∂z

∂x, Sf =

Q |Q|n2

A2R4/3(6)

whereR = A/P , P being the wetted perimeter. The coor-dinate system used for the formulation of the shallow waterequations is shown in Fig.1. A vectorial form of the Eqs. (1)and (2) is widely used:

∂U∂t

+∂F∂x

= R (7)

Drink. Water Eng. Sci., 7, 83–92, 2014 www.drink-water-eng-sci.net/7/83/2014/

J. Fernández-Pato and P. García-Navarro: Finite volume simulation of unsteady water pipe flow 852 J. Fernandez-Pato and P. Garcıa-Navarro: Finite volume simulation of unsteady water pipe flow

difference between the wave speeds (from∼ 10m/s in shal-low flow to∼ 1000m/s in pressurized flow). In order to solvethis issue, the slot width could be increased, but this implies aloss of accuracy in the results, due to the no-mass/momentumconservation.

This work is focused on the development of a simulationmodel able to solve pipe system networks working mainlyunder free surface flow situations but exceptionally pressur-ized. Hence, it is based on the second option previously men-tioned and it is aimed to generalize the shallow water equa-tions by means of the Preissmann slot method. The explicitfirst order Roe scheme has been chosen as numerical methodfor all the simulations.

Implicit methods or parallel computation have been dis-missed, as the calculation times are not long. Fluxes andsource terms have been treated by means of an upwindscheme (Garcıa-Navarro et al. (2000)). All the symbols andunits used are described in Table 1.

This paper is an extension of the one presented at the 12thedition of the International Conference on ”Computing andControl for the Water Industry - CCWI2013” (Fernandez-Pato et al. (2014)).

2 Mathematical model

2.1 Shallow water equations

The unsteady open channel water flow are usually modelledby means of the 1D shallow water or St. Venant equations.These equations represent mass and momentum conservationalong the main direction of the flow and become a good de-scription for the behaviour of the most of the pipe-flow-kindproblems (Kundu et al. (2012),Abbott et al. (1979)). The con-servative form of this systems of equations is presented in (1)and (2):

∂A∂t+∂Q∂x= 0 (1)

∂Q∂t+∂

∂x

(

Q2

A+ gI1

)

= gI2+ gA(

S 0− S f

)

(2)

whereA is the wetted cross section,Q is the discharge,g isthe acceleration due to gravity,I1 represents the hydrostaticpressure force term andI2 accounts for the pressure forcesdue to channel/pipe width changes:

I1 =

h(x,t)∫

0

(h− η)b(x,η)dη (3)

I2 =

h(x,t)∫

0

(h− η) ∂b(x,η)∂x

dη (4)

b =∂A(x,η)∂η

(5)

x

h(x,t)

z(x)

b(x,t)

A(x,t)

η

z(x)

h(x,t)

σ(x,η)

x

Figure 1. Coordinate system for shallow water equations.

in whichη andh(x, t) represent vertical coordinate and thefree surface depth, respectively. The remaining terms,S 0 andS f , represent the bed slope and the energy grade line (definedin terms of the Manning roughness coefficient), respectively:

S 0 = −∂z∂x, S f =

Q |Q|n2

A2R4/3(6)

whereR = A/P, P being the wetted perimeter. The coordinatesystem used for the formulation of the shallow water equa-tions is shown in Fig. 1. A vectorial form of the equations (1)and (2) is widely used:

∂U∂t+∂F∂x= R (7)

where

U = (A,Q)T (8)

F =(

Q,Q2/A+ gI1

)T(9)

R =(

0,gI2+ gA(

S 0− S f

))T(10)

In those cases in whichF = F(U), whenI2 = 0, it is possi-ble to rewrite the conservative system by means of the Jaco-bian matrix of the system 11:

∂U∂t+ J∂U∂x= R, J =

∂F∂U=

(

0 1c2− u2 2u

)

(11)

wherec is the wave speed (analogous to the speed of soundin gases), defined as follows:

c =

√

g∂I1

∂A(12)

The set of real eigenvalues (λ1,2) and eigenvectors (e1,2)of the system matrix can be used for diagonalizing it. Thesemeagnitudes represent the speed of propagation of the infor-mation:

λ1,2 = u± c, e1,2 = (1,u± c)T (13)

It is very common to characterize the flow type by meansof the Froude numberFr = u/c (analogous to the Mach num-ber in gases). It allows the classification of the flux into threemain regimes: subcriticalFr < 1, supercriticalFr > 1 andcritical Fr = 1.

Drink. Water Eng. Sci. www.drink-water-eng-sci.net

Figure 1. Coordinate system for shallow water equations.

where

U = (A,Q)T (8)

F =

(Q,Q2/A + gI1

)T

(9)

R =(0,gI2 + gA

(S0 − Sf

))T (10)

In those cases in whichF = F(U), whenI2 = 0, it is pos-sible to rewrite the conservative system by means of the Ja-cobian matrix of the system Eq. (11):

∂U∂t

+ J∂U∂x

= R, J =∂F∂U

=

(0 1

c2− u2 2u

)(11)

wherec is the wave speed (analogous to the speed of soundin gases), defined as follows:

c =

√g

∂I1

∂A(12)

The set of real eigenvalues (λ1,2) and eigenvectors (e1,2) ofthe system matrix can be used for diagonalizing it. Thesemagnitudes represent the speed of propagation of the infor-mation:

λ1,2= u ± c, e1,2

= (1,u ± c)T (13)

It is very common to characterize the flow type by means ofthe Froude numberFr = u/c (analogous to the Mach num-ber in gases). It allows the classification of the flux into threemain regimes: subcriticalFr < 1, supercriticalFr > 1 andcritical Fr = 1.

2.2 Water-hammer equations

The instant response for changes in a pipe flow is closely re-lated to the elastic compresibility of both the fluid and thepipe wall material. Unsteady flow in pipes is commonly de-scribed by the cross-section integrated mass and momentumequations (Chaudhry and Mays, 1994):

∂H

∂t+ v

∂H

∂x+ v sinθ +

c2WH

g

∂v

∂x= 0 (14)

4 J. Fernandez-Pato and P. Garcıa-Navarro: Finite volume simulation of unsteady water pipe flow

Figure 2. (a) Pure shallow flow (b) Pressurized pipe.

3 Preissmann slot model

The Preissmann slot approach consist in assuming that thetop of the pipe/closed channel is connected to a fictitious nar-row slot, which is open to the atmosphere. This fact allows toconsider a free surface flow situation, even when the pipe ispressurized, by including the slot in the shallow water equa-tions (see Fig. 2). The slot widthbs is ideally chosen equal-ing the speed of gravity waves in the slot to the water ham-mer wavespeed. Therefore, the water level in the slot corre-sponds to the pressure head level. This model is based on thepreviously remarked similarity between the wave equationswhich describe free surface and pressurized flows. The wa-ter hammer flow comes from the capacity of the pipe systemto change the area and fluid density, so forcing the equiv-alence between both models requires that the slot stores asmuch fluid as the pipe would by means of a change in areaand fluid density. The pressure term and the wavespeed forA ≤ bH are:

h =Ab

(19)

I1 =A2

2b(20)

c =

√

g∂I1

∂A=

√

gAb

(21)

and forA > bH:

h = H +A− bH

bs(22)

I1 = bH

(

A− bHbs

+H2

)

+(A− bH)2

2bs(23)

c =

√

g∂I1

∂A=

√

gAbs

(24)

The ideal choice for the slot width results in:

cWH = c⇒ bs = gA f

c2WH

(25)

in which A f stands for the full pipe cross-section.

4 Finite volume numerical model

4.1 Explicit first order Roe scheme

Roe scheme is based on a local linearization of the conservedvariables and fluxes:

δF = JδU (26)

It is necessary to build an approximate Jacobian matrixJwhose eigenvalues and eigenvectors satisfy:

δUi+1/2 = Ui+1−Ui =

2∑

k=1

(αkek)i+1/2 (27)

δFi+1/2 = Fi+1−Fi = Ji+1/2δUi+1/2 =

2∑

k=1

(

λkαkek

)

i+1/2(28)

The wave strenght coefficients αk represent the variablevariation coordinates in the Jacobian matrix basis. The eigen-values and eigenvectors are expressed in terms of the averageflux velocity and wave speed:

λk = (u± c) , ek = (u± c)T (29)

where

ui+1/2 =Qi+1√

Ai +Qi√

Ai+1√

AiAi+1

(√Ai+1+

√Ai

) (30)

ci+1/2 =

√

g2

[(Ab

)

i+

(Ab

)

i+1

]

(31)

Following (Garcıa-Navarro et al. (2000)), the source termsof the equation system (Eq. (7)) are also discretizated usingan upwind scheme:

(

Rδx)

i+1/2=

∑

k+

βkek

i+1/2

+

∑

k−βkek

i+1/2

(32)

in which the source streghtsβk take the role of ˜αk coefficientsfor the fluxes. Then, the complete discretization of the systembecomes:

Un+1i = Un

i −δtδx

∑

k+

(

λkαk − βk

)

ek

i+1/2

+

+

∑

k−

(

λkαk − βk

)

ek

i+1/2

(33)

4.2 Boundary conditions and stability conditions

In order to solve a numerical problem, it is necessary tocharacterize the domain limits by imposing some physical


Figure 2. (a) Pure shallow flow(b) Pressurized pipe.

J. Fernandez-Pato and P. Garcıa-Navarro: Finite volume simulation of unsteady water pipe flow 5

Table 2. Boundary conditions.

Flow regime and boundary Number of physical BC to impose

Upstream subcritical flow 1Upstream supercritical flow 2Downstream subcritical flow 1Downstream supercritical flow 0

2

3

1

1 IMAX

1

1

I MAX2

1

IMAX

3

Figure 3. Example of pipe junction.

boundary conditions (BC). The number of boundary condi-tions depends on the flow regime (subcritical or supercrit-ical). Hence, there are four posibilities for a 1D numericalproblem (see Table 2).

In the cases that consider pipe junctions, additional inter-nal boundary conditions are necessary in order to model thisfeature. Fig. 3 shows an example of a 3 pipe junction.The water level equality condition is imposed at the junctionfor all the pipes:

h1 = h2 = · · · = hN (34)

Discharge continuity condition is formulated depending onthe flow regime. In the cases presented, water flows in a sub-critical regime:

Q1

(

I1MAX

)

=

N∑

i=1

Qi (1)+Q3 (1) (35)

Storage wells are another common junction type. For thesecases, the boundary conditions are modified as follows:

h1 = h2 = · · · = hN = Hwell,

N∑

i=1

Qi = AwelldHwell

dt(36)

whereAwell andHwell are the well top area and depth, respec-tively.

The presented method use an explicit time discretizationand it requires a control on the time step in order to avoid

instabilities. Hence the Courant-Friedrichs-Lewy (CFL) con-dition is applied on every branch (j) of the network:

∆t jmax =

∆x j

max(∣

∣

∣u j∣

∣

∣+ c j) (37)

Then, the minimum of all the computed time steps is chosen:

∆tmax =min{

∆t jmax

}

, CFL =∆t∆tmax

≤ 1 (38)

5 Test cases

The model is applied to several analytic or experimental testcases in order to evaluate its accuracy.

5.1 Steady state over a bump

Following Murillo et al. (2012) a frictionless rectangular25mx 1m prismatic channel is considered. The variable bed levelis given by:

z (8≤ x ≤ 12) = 0.2−0.05(x−10)2 (39)

and the initial conditions for the water depth and discharge:

h(x,0)= 0.5− z(x), u(x,0)= 0 (40)

The different boundary conditions for the test cases simulatedare summarized in Table 3.

Fig. 4 (a) shows the generation of hydraulic jump that con-nects both subcritical and supercritical regimes. In the secondtest case, shown in Fig. 4 (b) the connection is made withoutany shock wave and it can be mathematically proved (andnumerically checked) that this transition takes place in thehighest part of the bump.

5.2 Dam-break

Dam-break is a classical example of non-linear flow withshocks. It has been widely used to test the accuracy and con-servation of the numerical scheme, by comparing with its an-alytical solution. In the case presented, an initial discontinu-ity of 1m : 0.5m ratio with no friction was considered. Fig. 5shows the good agreement between numerical and analyticalsolution at the given time.

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Figure 3. Example of pipe junction.

∂v

∂t+ v

∂v

∂x+ g

∂H

∂x+

4τ0

ρD= 0 (15)

in which H(x, t) = elevation of the hydraulic gradeline, v(x, t) = local cross-section averaged flow velocity,θ(x, t) = angle between the pipe and the horizontal level,D = diameter,ρ = fluid density,τ0 is the boundary shear, typ-ically estimated by means of a Manning or Darcy-Weissbachfriction model. The magnitudecWH accounts for the elasticwaves speed in the pipe:

cWH =

√K/ρ

1+DKeE

(16)

beinge the pipe thickness andE, K the elastic modulus of thepipe material and fluid, respectively. By neglecting convec-tive terms, it is possible to reach a linear hyperbolic equationsystem:

∂H

∂t+

c2WH

g

∂v

∂x= 0 (17)

∂v

∂t+ g

∂H

∂x+

4τ0

ρD= 0 (18)

These equations conform a hyperbolic system, analogous tothe shallow water equations, considering the pressure and thevelocity as the conserved variables.

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Table 3. Steady state over a bump.

Test UpstreamQ(

m3/s)

Downstreamh (m)

#1 0.18 0.33#2 1.53 0.66 (sub)

Figure 4. Steady state over a bump. Initial state (dashed line). Bed level (grey). Numerical solution (blue): (a) Test #1; (b) Test #2.

5.3 Wiggert test case

The experimental case designed by Wiggert (Wiggert(1972)) and widely numerically reproduced (e.g. Kerger etal. (2010), Bourdarias et al. (2007)) consists in a horizontal30 m long and 0.51 m wide flume. A 10 m roof is placed inthe middle section, setting up a closed rectangular pipe 0.148m in height (see Fig. 6). A 0.01m−1/3s Manning roughnesscoefficient is assumed. As initial conditions, a water levelof 0.128 m and zero discharge are considered. Then a wavecoming from the left causes the pressurization of the pipe.The imposed downstream boundary conditions are the samevalues measured by Wiggert (Fig. 7).

Fig. 8 shows the numerical results for the four gauges. Theexperimental comparison is done only for the second one,due to data availability issues. An overall good agreement isobserved with only small numerical oscillations presented.

5.4 Transient mixed flow

The aim of the next test case is to prove the model in the sim-ulation of large-scale strong transients. It considers a uniformslope (0.1%) rectangular pipe connected to a downstreamvalve (Leon (2007) and Leon et al. (2009)). The lenght,width and height are 10 km, 10 m and 9.5 m, respectively,and the chosen Manning roughness is 0.015.

As initial condition, a uniform water depth (8.57 m) anddischarge (Q = 240m3/s) are assumed. From this state, thedownstream valve is closed, generating a shock wave movingupstream and pressurizing the pipe. Fig. 9 shows the numer-ical results obtained for the pressure head at three differenttimes. For this simulation, a 500 cells mesh and a timestepgiven by CFL=0.5 have beed used. It is clearly observed that

the pipe pressure raises gradually with the upwards shock-wave movement.

5.5 Pressurized flow transients

A full-pressurized (Leon (2007) and Leon et al. (2009)) 10km flat, frictionless pipe of rectangular section (10 m x 7.853m) is connected upstream to a water reservoir that guaranteesa constant pressure head of 200 m. A closure valve is placeddownstream. When the valve is suddenly closed, a waterham-mer wave is generated moving upwards. Fig. 9 shows the nu-merical results with CFL=0.8 and a 500 cells mesh.

This case simulates a completely pressurized pipe, so it isnecessary to be very precise in the slot width selection in or-der to ensure the numerical solution accuracy. A waterham-mer speed of 1000 m/s and a initial flow speed of 2.0 m/s areassumed. Following (Eq. (25)) condition:

cWH = c⇒ bs = gA f

c2WH

= 0.77mm (41)

6 Pipe networks

6.1 Transient flow in a 3-pipe junction

A case proposed in Wixcey (1990) is presented first. A 5km long prismatic main channel (1) dividing into two sec-ondary branches (2 and 3) also 5 km long with the same 1m wide rectangular geometry have been considered, as pre-sented in fig. 3. Manning roughness coefficient for all thepipes is 0.01m−1/3s. The main branch has a slope of 0.002and the secondary ones 0.001.


Figure 4. Steady state over a bump. Initial state (dashed line). Bed level (grey). Numerical solution (blue):(a) Test #1;(b) Test #2.

Table 2. Boundary conditions.

Flow regime and boundary Number of physicalBC to impose

Upstream subcritical flow 1Upstream supercritical flow 2Downstream subcritical flow 1Downstream supercritical flow 0

Table 3. Steady state over a bump.

Test Upstream Downstream

Q(m3 s−1

)h(m)

#1 0.18 0.33#2 1.53 0.66 (sub)

3 Preissmann slot model

The Preissmann slot approach consist in assuming that thetop of the pipe/closed channel is connected to a fictitious nar-row slot, which is open to the atmosphere. This fact allows toconsider a free surface flow situation, even when the pipe ispressurized, by including the slot in the shallow water equa-tions (see Fig.2). The slot widthbs is ideally chosen equal-ing the speed of gravity waves in the slot to the water ham-mer wavespeed. Therefore, the water level in the slot corre-sponds to the pressure head level. This model is based on thepreviously remarked similarity between the wave equationswhich describe free surface and pressurized flows. The wa-ter hammer flow comes from the capacity of the pipe systemto change the area and fluid density, so forcing the equiv-alence between both models requires that the slot stores asmuch fluid as the pipe would by means of a change in areaand fluid density. The pressure term and the wavespeed forA ≤ bH are:

h =A

b(19)


0.4

0.5

0.6

0.7

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

h (

m)

x (m)

0.9

0.9 1.0

Figure 5. Dam-break test case. Initial state (dashed line). Numerical results (blue dots). Analytical solution (cyan line).

10 m

Wave

2.0 m 3.5 m 0.5 m

0.51 m

0.148 m

1 2 3 4

3.5 m0.5 m

Figure 6. Wiggert experimental setup.

0.12

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.13

a b

Figure 7. Upstream (a) and downstream (b) pressured head level.


Figure 5. Dam-break test case. Initial state (dashed line). Numeri-cal results (blue dots). Analytical solution (cyan line).

I1 =A2

2b(20)

c =

√g

∂I1

∂A=

√g

A

b(21)

and forA > bH :

h = H +A − bH

bs

(22)

I1 = bH

(A − bH

bs

+H

2

)+

(A − bH)2

2bs

(23)

c =

√g

∂I1

∂A=

√g

A

bs

(24)

The ideal choice for the slot width results in:

cWH = c ⇒ bs = gAf

c2WH

(25)

in whichAf stands for the full pipe cross-section.


J. Fernández-Pato and P. García-Navarro: Finite volume simulation of unsteady water pipe flow 87


0.4

0.5

0.6

0.7

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

h (

m)

x (m)

0.9

0.9 1.0


10 m

Wave

2.0 m 3.5 m 0.5 m

0.51 m

0.148 m

1 2 3 4

3.5 m0.5 m


0.12

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.13

a b





0.4

0.5

0.6

0.7

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

h (

m)

x (m)

0.9

0.9 1.0


10 m

Wave

2.0 m 3.5 m 0.5 m

0.51 m

0.148 m

1 2 3 4

3.5 m0.5 m


0.12

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.13

a b



Figure 7. Upstream(a) and downstream(b) pressured head level.

4 Finite volume numerical model

4.1 Explicit first order Roe scheme

Roe scheme is based on a local linearization of the conservedvariables and fluxes:

δF = JδU (26)

It is necessary to build an approximate Jacobian matrixJwhose eigenvalues and eigenvectors satisfy:

δUi−1/2 = Ui+1 − Ui =

2∑k=1

(αkek

)i+1/2 (27)

δFi+1/2 = Fi+1 − Fi = Ji+1/2δUi+1/2 (28)

=

2∑k=1

(λkαkek

)i+1/2

The wave strenght coefficientsαk represent the variablevariation coordinates in the Jacobian matrix basis. The eigen-values and eigenvectors are expressed in terms of the averageflux velocity and wave speed:

λk = (u ± c) , ek = (u ± c)T (29)

where

ui+1/2 =Qi+1

√Ai + Qi

√Ai+1

√AiAi+1

(√Ai+1 +

√Ai

) (30)

ci+1/2 =

√g

2

[(A

b

)i

+

(A

b

)i+1

](31)

Following (García-Navarro and Vázquez-Cendón(2000)),the source terms of the equation system (Eq.7) are also dis-cretizated using an upwind scheme:

(Rδx

)i+1/2

=

(∑k+

βkek

)i+1/2

+

(∑k−

βkek

)i+1/2

(32)

in which the source streghtsβk take the role ofαk coefficientsfor the fluxes. Then, the complete discretization of the systembecomes:

Un+1i = Un

i −δt

δx

(∑k+

(λkαk − βk

)ek

)i−1/2

(33)

+

(∑k−

(λkαk − βk

)ek

)i+1/2

4.2 Boundary conditions and stability conditions

In order to solve a numerical problem, it is necessary tocharacterize the domain limits by imposing some physicalboundary conditions (BC). The number of boundary condi-tions depends on the flow regime (subcritical or supercriti-cal). Hence, there are four posibilities for a 1-D numericalproblem (see Table2).


88 J. Fernández-Pato and P. García-Navarro: Finite volume simulation of unsteady water pipe flow8 J. Fernandez-Pato and P. Garcıa-Navarro: Finite volume simulation of unsteady water pipe flow

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16h (

m)

t (s)

a b

c d

Figure 8. Comparision between numerical results (blue), experimental data (green dots) and numerical results from Kerger et al. (2010) (reddots) for the four gauges of the Wiggert test case setup.

190

195

200

205

210

215

220

0 2000 4000 6000 8000 10000

Pre

ssure

head (

m)

x (m)

Figure 9. Shockwave propagation in transient mixed flow att=100 s (red),t=200 s (green) andt=300 s (blue).

0

50

100

150

200

250

300

350

400

450

0 2000 4000 6000 8000 10000

Pre

ssure

head (

m)

x (m)

Figure 10. Waterhammer simulation. Pressured head att=3 s (red),t=6 s (green) andt=9 s (blue).


Figure 8. Comparision between numerical results (blue), experimental data (green dots) and numerical results fromKerger et al.(2010) (reddots) for the four gauges of the Wiggert test case setup.

In the cases that consider pipe junctions, additional inter-nal boundary conditions are necessary in order to model thisfeature. Figure3 shows an example of a 3 pipe junction.

The water level equality condition is imposed at the junc-tion for all the pipes:

h1 = h2 = . . . = hN (34)

Discharge continuity condition is formulated depending onthe flow regime. In the cases presented, water flows in a sub-critical regime:

N∑i=1

Qi = 0 (35)

Storage wells are another common junction type. For thesecases, the boundary conditions are modified as follows:

h1 = h2 = ·· · = hN = Hwell, (36)N∑

i=1

Qi = AwelldHwell

dt

whereAwell andHwell are the well top area and depth, re-spectively.

The presented method uses an explicit time discretizationand it requires a control on the time step in order to avoidinstabilities. Hence the Courant-Friedrichs-Lewy (CFL) con-dition is applied on every branch (j ) of the network:

1tjmax =

1xj

max(∣∣uj

∣∣+ cj) (37)

Then, the minimum of all the computed time steps is chosen:

1tmax = min{1t

jmax

}, CFL =

1t

1tmax≤ 1 (38)

5 Test cases

The model is applied to several analytic or experimental testcases in order to evaluate its accuracy.

5.1 Steady state over a bump

Following Murillo and García-Navarro(2012) a frictionlessrectangular 25 m× 1 m prismatic channel is considered. Thevariable bed level is given by:

z(8 ≤ x ≤ 12) = 0.2− 0.05(x − 10)2 (39)

and the initial conditions for the water depth and discharge:

h(x,0) = 0.5− z(x), u(x,0) = 0 (40)

The different boundary conditions for the test cases simu-lated are summarized in Table3.

Figure4a shows the generation of hydraulic jump that con-nects both subcritical and supercritical regimes. In the secondtest case, shown in Fig.4b the connection is made withoutany shock wave and it can be mathematically proved (andnumerically checked) that this transition takes place in thehighest part of the bump.


J. Fernández-Pato and P. García-Navarro: Finite volume simulation of unsteady water pipe flow 89


0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

a b

c d


190

195

200

205

210

215

220

0 2000 4000 6000 8000 10000

Pre

ssure

head (

m)

x (m)


0

50

100

150

200

250

300

350

400

450

0 2000 4000 6000 8000 10000

Pre

ssure

head (

m)

x (m)



Figure 9. Shockwave propagation in transient mixed flow att = 100 s (red),t = 200 s (green) andt = 300 s (blue).


0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 2 4 6 8 10 12 14 16

h (

m)

t (s)

a b

c d


190

195

200

205

210

215

220

0 2000 4000 6000 8000 10000

Pre

ssure

head (

m)

x (m)


0

50

100

150

200

250

300

350

400

450

0 2000 4000 6000 8000 10000

Pre

ssure

head (

m)

x (m)



Figure 10. Waterhammer simulation. Pressured head att = 3 s(red),t = 6 s (green) andt = 9 s (blue).

5.2 Dam-break

Dam-break is a classical example of non-linear flow withshocks. It has been widely used to test the accuracy and con-servation of the numerical scheme, by comparing with itsanalytical solution. In the case presented, an initial discon-tinuity of 1 m: 0.5 m ratio with no friction was considered.Figure5 shows the good agreement between numerical andanalytical solution at the given time.

5.3 Wiggert test case

The experimental case designed byWiggert (1972) andwidely numerically reproduced (e.g.Kerger et al., 2010;Bourdarias and Gerbi, 2007) consists in a horizontal 30 mlong and 0.51 m wide flume. A 10 m roof is placed in themiddle section, setting up a closed rectangular pipe 0.148 min height (see Fig.6). A 0.01 s m−1/3 Manning roughnesscoefficient is assumed. As initial conditions, a water levelof 0.128 m and zero discharge are considered. Then a wavecoming from the left causes the pressurization of the pipe.The imposed downstream boundary conditions are the samevalues measured byWiggert(1972) (Fig. 7).

Figure8 shows the numerical results for the four gauges.The experimental comparison is done only for the secondone, due to data availability issues. An overall good agree-ment is observed with only small numerical oscillations pre-sented.


Q

QMAX

QMIN

t(s)0 1800900600300 1200 1500

Figure 11. Triangular inlet hydrograph in the pipe 1.

0

0.2

0.4

0.6

0.8

0 1000 2000 3000 4000 5000 6000

h(m

) (T

3)

t (s)

0

0.2

0.4

0.6

0.8

0 1000 2000 3000 4000 5000 6000

h(m

) (T

2)

t (s)

0 0.2 0.4 0.6 0.8

1 1.2

0 1000 2000 3000 4000 5000 6000

h(m

) (T

1)

t (s)

Figure 12. Pressurized transient in a pipe junction. Pressured head vs timex=500 m (red),x=1000 m (green) andx=5000 m (blue).

2

3

1

J1

J2

W1

W2

1 IMAX

1

1

I MAX2

1

IMAX

3

5

6

4

7

1

IMAX

5

1

I MAX6

1 IMAX

7

1

IMAX

4

Figure 13. (Scheme of the looped pipe network.



5.4 Transient mixed flow

The aim of the next test case is to prove the model in the sim-ulation of large-scale strong transients. It considers a uniformslope (0.1 %) rectangular pipe connected to a downstreamvalve (León, 2006; León et al., 2009). The lenght, width andheight are 10 km, 10 m and 9.5 m, respectively, and the cho-sen Manning roughness is 0.015.

As initial condition, a uniform water depth (8.57 m) anddischarge (Q = 240 m3 s−1) are assumed. From this state,the downstream valve is closed, generating a shock wavemoving upstream and pressurizing the pipe. Figure9 showsthe numerical results obtained for the pressure head at threedifferent times. For this simulation, a 500 cells mesh and atimestep given by CFL= 0.5 have beed used. It is clearlyobserved that the pipe pressure raises gradually with the up-wards shockwave movement.

5.5 Pressurized flow transients

A full-pressurized (León, 2006; León et al., 2009) 10 km flat,frictionless pipe of rectangular section (10 m× 7.853 m) isconnected upstream to a water reservoir that guarantees aconstant pressure head of 200 m. A closure valve is placeddownstream. When the valve is suddenly closed, a waterham-mer wave is generated moving upwards. Figure9 shows thenumerical results with CFL= 0.8 and a 500 cells mesh.

This case simulates a completely pressurized pipe, so it isnecessary to be very precise in the slot width selection in or-der to ensure the numerical solution accuracy. A waterham-mer speed of 1000 m s−1 and a initial flow speed of 2.0 m s−1

are assumed. Following (Eq.25) condition:

cWH = c ⇒ bs = gAf

c2WH

= 0.77mm (41)

6 Pipe networks

6.1 Transient flow in a 3-pipe junction

A case proposed inWixcey (1990) is presented first. A 5 kmlong prismatic main channel (1) dividing into two secondarybranches (2 and 3) also 5 km long with the same 1 m widerectangular geometry have been considered, as presented in




Q

QMAX

QMIN

t(s)0 1800900600300 1200 1500


0

0.2

0.4

0.6

0.8

0 1000 2000 3000 4000 5000 6000

h(m

) (T

3)

t (s)

0

0.2

0.4

0.6

0.8

0 1000 2000 3000 4000 5000 6000

h(m

) (T

2)

t (s)

0 0.2 0.4 0.6 0.8

1 1.2

0 1000 2000 3000 4000 5000 6000

h(m

) (T

1)

t (s)


2

3

1

J1

J2

W1

W2

1 IMAX

1

1

I MAX2

1

IMAX

3

5

6

4

7

1

IMAX

5

1

I MAX6

1 IMAX

7

1

IMAX

4



Figure 12. Pressurized transient in a pipe junction. Pressured head vs timex = 500 m (red),x = 1000 m (green) andx = 5000 m (blue).


Q

QMAX

QMIN

t(s)0 1800900600300 1200 1500


0

0.2

0.4

0.6

0.8

0 1000 2000 3000 4000 5000 6000

h(m

) (T

3)

t (s)

0

0.2

0.4

0.6

0.8

0 1000 2000 3000 4000 5000 6000

h(m

) (T

2)

t (s)

0 0.2 0.4 0.6 0.8

1 1.2

0 1000 2000 3000 4000 5000 6000

h(m

) (T

1)

t (s)


2

3

1

J1

J2

W1

W2

1 IMAX

1

1

I MAX2

1

IMAX

3

5

6

4

7

1

IMAX

5

1

I MAX6

1 IMAX

7

1

IMAX

4




Fig. 3. Manning roughness coefficient for all the pipes is0.01 s m−1/3. The main branch has a slope of 0.002 and thesecondary ones 0.001.

First, a steady state has been calculated from the initialconditions:

Q1(i,0) = 0.1m3 s−1, (42)

Q2(i,0) = Q3(i,0) = 0.05m3 s−1

h1(i,0) = h2(i,0) = h3(i,0) = 0.2m (43)

using as boundary condition the inlet discharge:

Q1(1, t) = 0.1m3 s−1 (44)

and free outflow boundary condition at the outlet of pipes 2and 3.

Using this steady state as initial condition for the tran-sient calculation, a triangular function of peak dischargeQMAX = 3.2 m3 s−1 and a period of 600 s (Fig.11) is im-posed at the beginning of the pipe 1. Figure12 shows thenumerical results using a 100 cells mesh and a CFL= 0.9condition.

6.2 Transient flow in a 7-pipe looped network

In this section a simple looped pipe network (see Fig.13)is presentedWixcey (1990). Each pipe is closed, squared,1 m wide and 100 m long. The bed slopes areS0,1 = 0.002,S0,2 = S0,3 = 0.001,S0,4 = 0.0, S0,5 = S0,6 = 0.001,S0,7 =

0.002 and a 0.01 s m−1/3 Manning friction coefficient is con-sidered. A first calculation provided the steady state from thefollowing initial conditions:

Q1(i,0) = Q7(i,0) = 0.1m3 s−1 (45)


J. Fernández-Pato and P. García-Navarro: Finite volume simulation of unsteady water pipe flow 91J. Fernandez-Pato and P. Garcıa-Navarro: Finite volume simulation of unsteady water pipe flow 11

0 500 1000 1500 2000

h7(m

)

t (s)

h5(m

)h4(m

)

0

0.2

0.4

0.6

0.8

1

h2(m

)

0

0.4

0.8

1.2

h1(m

)

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Q1(m

3/s

)

0

3

Q2(m

3/s

)Q

4(m

3/s

)Q

5(m

3/s

)Q

7(m

3/s

)

2

1

0

3

2

1

0

3

2

1

0

3

2

1

0 500 1000 1500 2000t (s)

0.2

0

-0.2

a b

Figure 14. Time histories at grid pointsi=N/2 (red) andi=N (blue) for punctually pressurized flow: (a) Water depth; (b) Discharge. Thedischarge for pipe 4 is recorded at locationsi=1 (red),i=N/2 (grey) andi=N (blue).

ulating highly transient mixed flows. Journal of Computationaland Applied Mathematics 235, 2030-2040.

Kundu, P.K., Cohen, I.M., Dowling, D.R., 2012. Fluid Mechanics.Academic Press, pp. 920.

Leon, A.S., 2007. Improved Modeling of Unsteady Free Surface,Pressurized and Mixed Flows in Storm-sewer Systems (Disserta-tion).

Leon, A.S., Ghidaoui, M.S., Schmidt, A.R., Garcıa, M.H., 2009.Application of Godunov-type schemes to transient mixed flows.Journal of Hydraulic Research 47 (2), 147-156

Murillo, J., Garcıa-Navarro, P., 2012. Augmented versions of theHLL and HLLC Riemann Solvers including source terms in oneand two dimensions for shallow flow applications. Journal ofComputational Physics 231 (20) 6861-6906.

Trajkovic, B., Ivetic, M., Calomino, F., D’Ippolito, A, 1999. Inves-tigation of transition from free surface to pressurized flow in acircular pipe. Water Science and Technology 39 (9), 105-112.

Wiggert, D.C., 1972. Transient flow in free-surface, pressurizedsystems. Journal of the Hydraulics Division, Proceedings of theAmerican Society of Civil Engineers 98 (1), 11-26.

Wiggert, D.C., Sundquist, M.J., 1977. Fixed-grid Characteristics forPipeline Transients. Journal of the Hydraulics Division, Proceed-ings of the American Society of Civil Engineers 103, 1403-1416.

Wixcey, J.R., 1990. An investigation of algorithms for open chan-nel flow calculations. Numerical Analysis Internal Report, 21,Departament of Mathematics, University of Reading.


Figure 14. Time histories at grid pointsi = N/2 (red) andi = N (blue) for punctually pressurized flow:(a) Water depth;(b) Discharge. Thedischarge for pipe 4 is recorded at locationsi = 1 (red),i = N/2 (grey) andi = N (blue).

Q2(i,0) = Q3(i,0) = Q5(i,0) (46)

= Q6(i,0) = 0.05m3 s−1

Q4(i,0) = 0 (47)

hj (i,0) = 0.2m, (j = 1, . . . ,7) (48)

and the upstream boundary condition:

Q1(1, t) = 0.1m3 s−1 (49)

Free outflow downwards boundary condition has been im-posed.

JunctionsJ1 and J2 are treated as normal confluences(Eq. 34) and a storage well of 5 m2 top surface is assumedin junctionsW1 andW2 (Eq. 36). The previously computedsteady state is now used as an initial condition for a secondcalculation. A triangular function of peak dischargeQMAXand a period of 600 s (Fig.11) is imposed at the beginning ofthe pipe 1. The minimum discharge is 0.1 m3 s−1.

Two different situations were studied. In the first one thepeak discharge is fixed toQMAX = 2.0 m3 s−1 and the flowremains unpressurized all over the pipe system. In the sec-ond case, the maximum discharge is increased toQMAX =

3.0 m3 s−1, so pressurization occurs in some points of pipe1. Figure14 shows the results for the pressurized case. Theresults in pipes 3 and 6 have been omitted because of sym-metry reasons. The discharge at the center of the pipe 4 isconstantly zero and equal in magnitude but opposite in signin the symmetric points, which verify the simmetric characterof the pipe network.

7 Conclusions

The present work leads to the conclusion of the reasonablygood applicability of the Preissmann slot model for an es-timation of the pressure values in unsteady situations be-tween both shallow and pressurized flows. In this secondcase, a small deviation of the ideal slot width induces a massand momentum conservation error. On the other hand, it isclear that wider slots increases the stability of the numericalmodel. Hence, in benefit of the results accuracy, it is remark-ably important to find an agreement between both facts.

The model takes advantages of the similarity of both equa-tion systems (shallow water and pressurized flow) and pro-vides a simple way to simulate occasionally pressurizedpipes, treating the system as an open channel in the Preiss-mann slot. This results in an easier implementation of themodel because it avoids the managing of two separate sys-tems of equations (shallow water and water hammer) in or-der to model separately the pressurized and the free surfaceflows.

The use of an explicit scheme implies a limitation onthe computational time step, which increases the calculationtime, specially in the in the pressurized cases, due to thesmall width of the slot.

Finally, the authors want to remark the possibility ofadapting the method to more realistic systems, like pipe net-works which can be punctually pressurized. Some additionalinternal boundary conditions are necessary in these cases, inorder to represent pipe junctions and storage wells.

Edited by: L. Berardi



References

Abbott, M. B.: Computational Hydraulics. Elements of the The-ory of the Free Surface flows, Pitman Publishing Ltd., London,324pp., 1979.

Bourdarias, C. and Gerbi, S.: A finite volume scheme for a modelcoupling free surface and pressurised flows in pipes, J. Computat.Appl. Mathemat., 209, 109–131, 2007.

Chaudhry, M. H. and Mays, L. W. (Eds.): Computer modeling offree-surfaces and pressurized flows, Kluwer Academic Publish-ers, Boston, 741 pp., 1994.

Fernández-Pato, J. and García-Navarro, P.: A Pipe Network Sim-ulation Model with Dynamic Transition between Free Surfaceand Pressurized Flow, Proc., 12th Int. Conf. on “Computing andControl for the Water Industry – CCWI2013”, Perugia, ProcediaEngineering, Elsevier, 70, 641–650, 2014.

García-Navarro, P. and Vázquez-Cendón, M. E.: On numericaltreatment of the source terms in the shallow water equations,Comput. Fluids, 29, 951–979, 2000.

García-Navarro, P., Alcrudo, F., and Priestley, A.: An implicitmethod for water flow modelling in channels and pipes, J. Hy-draul. Res., 32, 721–742, 1994.

Gray, C. A. M.: Analysis of Water Hammer by Characteristics,Trans. Am. Soc. Civ. Engin., 119, 1176–1189, 1954.

Kerger, F., Archambeau, P., Erpicum, S., Dewals, B. J., and Pirotton,M.: An exact Riemann solver and a Godunov scheme for simulat-ing highly transient mixed flows, J. Computat. Appl. Mathemat.,235, 2030–2040, 2010.

Kundu, P. K., Cohen, I. M., and Dowling, D. R.: Fluid Mechanics,Academic Press, 891 pp., 2012.

León, A. S.: Improved Modeling of Unsteady Free Surface, Pres-surized and Mixed Flows in Storm-sewer Systems (Dissertation),2006.

León, A. S., Ghidaoui, M. S., Schmidt, A. R., and García, M. H.:Application of Godunov-type schemes to transient mixed flows,J. Hydraul. Res., 47, 147–156, 2009.

Murillo, J. and García-Navarro, P.: Augmented versions of theHLL and HLLC Riemann solvers including source terms in oneand two dimensions for shallow flow applications, J. Computat.Phys., 231, 6861–6906, 2012.

Trajkovic, B., Ivetic, M., Calomino, F., and D’Ippolito, A.: Inves-tigation of transition from free surface to pressurized flow in acircular pipe, Water Sci. Technol., 39, 105–112, 1999.

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finite volume simulation of unsteady water pipe ﬂow · oped simulations applying the shallow...

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