Drink. Water Eng. Sci., 7, 83–92, 2014 www.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 2014 Revised: 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 require to model transitions between surface flow and pressurized flow in steady and transient situations. The governing equations for both flow types are different and this must be taken into account in order to get a complete numerical model for solving dynamically transients. In this work, a numerical simulation tool is developed, capable of simulating pipe networks mainly unpressurized, with isolated points of pressurization. For this purpose, the mathematical model is reformulated by means of the Preissmann slot method. This technique provides a reasonable estimation of the water pressure in cases of pressurization. The numerical model is based on the first order Roe’s scheme, in the frame of finite volume methods. The novelty of the method is that it is adapted to abrupt transient situations, with subcritical and supercritical flows. The validation has been done by means of several cases with analytic solutions or empirical laboratory data. It has also been applied to some more complex and realistic cases, like junctions or pipe networks.
1
Introduction
The numerical simulation of water pipe networks is characterized by the difficulty of simulating both transient and steady states. Most of the time, the water flow is unpressurized but, under exceptional conditions, the limited storage capacity of the pipes could cause a momentary pressure peak if the water level raises quickly. Under these conditions, a pressurized flow should be taken into account, implying a change in the mathematical model in order to solve the problem accurately. The most commonly used hydraulic network models used in the drinking water community exclusively consider fully filled pipes. However, a complete pipe system model should be able to solve steady and transient flows under pressurized and unpressurized situations and the transition between both flows (mixed flows). Many of the models developed to study the propagation of hydraulic transients solve both system of equations, e.g. Gray (1954), Wiggert and Sundquist (1977). For this purpose, numerical schemes derived from the Method of Characteristics (MOC) are employed. This method transforms the continuity and momentum partial differential equations into a set of ordinary differential equa-
tions, much easier to solve. MOC schemes can handle complex boundary conditions. However, the main issue with these kind of methods is the need for interpolation in some cases, so that they are not strictly conservative. This results in a diffusion of the wave front that implies a wrong arrival time of the waves to the boundaries. Hence, a new efficient numerical scheme is justified in order to solve accurately the wave front propagation. Two main points of view have been considered by the researchers to study mixed flow problems along the last decades. The first one consists of solving separately pressurized and shallow flows, treating the transitions between both flow types as internal boundary conditions. These models are capable of simulating sub-atmospheric pressures in the pipe, but the need for solving both mathematical models implies a high level of complexity. Other authors, e.g. García-Navarro et al. (1994), León et al. (2009), Kerger et al. (2010) developed simulations applying the shallow water equations in a slim slot over the pipe (Preissmann slot method). An estimation of the pipe pressure can be obtained from the water level in the slot. The great advantage of this model is the use of 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 local transitions between pressurized and shallow flows but some stability problems have been reported (Trajkovic et al., 1999) when simulating cases with abrupt transitions. The reason is a big difference between the wave speeds (from ∼ 10 m s−1 in shallow flow to ∼ 1000 m s−1 in pressurized flow). In order to solve this issue, the slot width could be increased, but this implies a loss of accuracy in the results, due to the nomass/momentum conservation. This work is focused on the development of a simulation model able to solve pipe system networks working mainly under free surface flow situations but exceptionally pressurized. Hence, it is based on the second option previously mentioned and it is aimed to generalize the shallow water equations by means of the Preissmann slot method. The explicit first order Roe scheme has been chosen as numerical method for all the simulations. Implicit methods or parallel computation have been dismissed, as the calculation times are not long. Fluxes and source terms have been treated by means of an upwind scheme (García-Navarro and Vázquez-Cendón, 2000). All the symbols and units used are described in Table 1. This paper is an extension of the one presented at the 12th edition of the International Conference on “Computing and Control for the Water Industry – CCWI2013” (FernándezPato and García-Navarro, 2014). 2
Mathematical model
2.1
Table 1. List of symbols and units. Symbol
Meaning
Units
A Af bs b c cWH D E e Fr g H h I1 I2 K n P Q R S0 Sf t u λ e v θ x z η ρ τ0
Wetted cross-section Full pipe cross-section Preissmann slot width Pipe/channel width Wave speed Water hammer speed Diameter Pipe elastic modulus Pipe thickness Froude number Accelaration due to gravity Elevation of the hydraulic grade line Free surface depth Hydrostatic pressure force term Pressure force term due to channel/pipe width changes Fluid elastic modulus Manning’s roughness coefficient Wetted perimeter Water discharge Hydraulic radius Bed slope Energy grade line slope Time Flow speed Eigenvalue Eigenvector Cross-section averaged flow velocity Angle between pipe and longitudinal coordinate Longitudinal coordinate Elevation Vertical coordinate Fluid density Boundary shear
m2 m2 m m m s−1 m s−1 m Pa m – m s−2 m m m3 m2 Pa s m−1/3 m m3 s−1 m – – s m s−1 m s−1 m s−1 m s−1 rad m m m kg m3 Pa
Shallow water equations
The unsteady open channel water flow are usually modelled by means of the 1D shallow water or St. Venant equations. These equations represent mass and momentum conservation along the main direction of the flow and become a good description for the behaviour of the most of the pipe-flow-kind problems (Kundu et al., 2012; Abbott, 1979). The conservative form of this systems of equations is presented in Eqs. (1) and (2): ∂A ∂Q + =0 ∂t ∂x
(1)
∂Q ∂ Q2 + + gI1 = gI2 + gA S0 − Sf ∂t ∂x A
(2)
where A is the wetted cross section, Q is the discharge, g is the acceleration due to gravity, I1 represents the hydrostatic pressure force term and I2 accounts for the pressure forces due to channel/pipe width changes: h(x,t) Z
I1 =
(h − η) b(x, η)dη 0
Drink. Water Eng. Sci., 7, 83–92, 2014
(3)
h(x,t) Z
(h − η)
I2 =
∂b(x, η) dη ∂x
(4)
0
b=
∂A(x, η) ∂η
(5)
in which η and h(x, t) represent vertical coordinate and the free surface depth, respectively. The remaining terms, S0 and Sf , represent the bed slope and the energy grade line (defined in terms of the Manning roughness coefficient), respectively:
S0 = −
∂z , ∂x
Sf =
Q |Q| n2 A2 R 4/3
(6)
where R = A/P , P being the wetted perimeter. The coordinate system used for the formulation of the shallow water equations is shown in Fig. 1. A vectorial form of the Eqs. (1) and (2) is widely used: ∂U ∂F + =R ∂t ∂x
(7) www.drink-water-eng-sci.net/7/83/2014/
J. Fern´andez-Pato and P. Garc´ıa-Navarro: Finite volume simulation of uns J. Fernández-Pato P. García-Navarro: volume of unsteady water pipe flow 85ıa-Navarro: Fin d P. Garc´ıa-Navarro: Finite volumeand simulation of unsteady Finite water pipe flowsimulation 4 J. Fern´andez-Pato and P. Garc´ Table 2. Boundary conditions.
s in shalr to solve implies a omentum
imulation ng mainly y pressurusly menater equae explicit al method
been disuxes and n upwind mbols and
t the 12th uting and ern´andez-
modelled equations. servation good deflow-kind The connted in (1)
(1) (2)
arge, g is ydrostatic ure forces
(3)
(4)
(5)
b(x,t)
4 Finite Flow regime and boundary
σ(x,η) A(x,t)
h(x,t) η
h(x,t) z(x)
^ x
z(x)
^ x
Figure 1. Coordinate system for shallow water equations.
in which η and h(x, t) represent vertical coordinate and the where free surface depth, respectively. The remaining terms, S 0 and U (8) (A, Q)T the bed slope and the energy grade line (defined S f= , represent in terms of the Manning roughness coefficient), respectively: 2 ∂z Q |Q| n S f = T2 4/3 S0 = − , 2 F = Q, ∂xQ /A + gI1 A R
(6) (9)
where R = A/P, P being the wetted perimeter. The coordinate system used for the formulation of the shallow water equaT tions in Fig. (1) R = is 0,shown gI2 + gA S0 1. −A Sfvectorial form of the equations(10) and (2) is widely used: In those cases in which F = F(U), when I2 = 0, it is possible to rewrite the conservative system by means of the Ja∂U ∂F cobian + matrix = R of the system Eq. (11): (7) ∂t ∂x ∂U ∂U ∂F 0 1 where +J = R, J= = 2 (11) c − u2 2u ∂t ∂x ∂U T (8) U = (A, Q) where c is the waveTspeed (analogous to the speed of sound in gases), F= Q, Q2defined /A + gI1as follows: (9) r T ∂I21+ gA S 0 − S f R = 0, gI (10) c= g (12) ∂A In those cases in which F = F(U), when I2 = 0, it is possible tosetrewrite conservative system means of the JacoThe of realthe eigenvalues (λ1,2 ) and by eigenvectors (e1,2 ) of bian matrix of the system 11: the system matrix can be used for diagonalizing it. These ! of the informagnitudes speed of propagation ∂U ∂U represent the ∂F 0 1 (11) +J = R, J= = 2 mation: c − u2 2u ∂t ∂x ∂U
1,2 λwhere = cu is ± the c, wave e1,2 = (1,(analogous u ± c)T to the speed of sound (13) speed in gases), defined as follows: It is very common to characterize the flow type by means of r the Froude ∂I1 number F r = u/c (analogous to the Mach num(12) c = in gases). g ber ∂A It allows the classification of the flux into three main regimes: subcritical F r < 1, supercritical F r > 1 and The set real eigenvalues (λ1,2 ) and eigenvectors (e1,2 ) critical F r of = 1. of the system matrix can be used for diagonalizing it. These meagnitudes representequations the speed of propagation of the infor2.2 Water-hammer mation: The instant response for changes in a pipe flow is closely reλ1,2 =tou ±the c, elastic e1,2compresibility = (1, u ± c)T of both the fluid and(13) lated the pipe wall material. Unsteady flow in pipes is commonly deIt is very common to characterize the flow type by means scribed by the cross-section integrated mass and momentum of the Froude number Fr = u/c (analogous to the Mach numequations (Chaudhry and Mays, 1994): ber in gases). It allows the classification of the flux into three main subcritical 2 Fr < 1, supercritical Fr > 1 and cWH ∂H regimes: ∂H ∂v + vFr = + =0 (14) critical 1. v sin θ + ∂t ∂x g ∂x
www.drink-water-eng-sci.net www.drink-water-eng-sci.net/7/83/2014/
Upstream subcritical flow Upstream supercritical flow Downstream subcritical flow Downstream supercritical flow Figure 2. (a) Pure shallow flow (b) Figure (b) Pressurized Pressurized pipe. pipe.
3
Preissmann slot model
2
2
I MAX
The Preissmann slot approach consist in assuming that the top of the pipe/closed channel is connected to a fictitious nar1 1 row slot, which is open to 1the atmosphere. This fact allows to I MAX 1 consider a free surface flow situation, even when the pipe is 1 pressurized, by including the slot in the shallow water equations (see Fig. 2). The slot width b s is ideally chosen equaling the speed of gravity waves in the slot to3 the water ham3 mer wavespeed. Therefore, the water level in the slotIMcorreAX sponds to the pressure head level. This model is based on the previously remarked similarity between the wave equations Figure 3. 3. Example of pipe junction. Figure which describe free surface and pressurized flows. The water hammer flow comes from the capacity of the pipe system to change conditions the area and fluid density, soofforcing the equivboundary (BC). The number boundary condialence between both models requires that the slot stores as tions depends on the flow regime (subcritical or supercritmuch fluid the by means of aachange in area ∂v Hence, ∂v as there ∂Hpipe 0 ical). are4τwould four 1D numerical +fluid v density. + g The + = posibilities 0 term andfor (15) and pressure the wavespeed for ∂t ∂x(see Table ∂x 2).ρD problem A ≤InbH are: the cases that consider pipe junctions, additional interin which Hconditions (x, t) = elevation of the hydraulic nal boundary are necessary in order to modelgrade this A line, t) = local cross-section averaged flow velocity, (19) hfeature. = v(x, Fig. 3 shows an example of a 3 pipe junction. θThe (x, bwater t) = angle the pipe and the horizontal level, levelbetween equality condition is imposed at the junction D = diameter, ρ = fluid density, τ is the boundary shear, typ0 for all 2the pipes: Aestimated by means of a Manning or Darcy-Weissbach ically (20) I1 = friction model. magnitude cWH accounts for the elastic h1 = 2b h2 = · · · = hThe (34) N waves the pipe: rspeed inr ∂I1 A on cDischarge = gs continuity = g condition is formulated depending(21) K/ρ Inbthe cases presented, water flows in a subthe flow ∂A regime. cWH = (16) DK critical and for regime: A 1>+bH: eE
N A bH e the −X being and E, K the elastic modulus of(22) the hQ= H1 + pipe thickness Q (1) + Q3 (1) (35) 1 I MAX =b pipe materials and ifluid, respectively. By neglecting conveci=1 tive terms, it is possible to reach a linear hyperbolic equation ! system: H (A − bH)2 A − bH Storage wells are another + (23) + common junction type. For these I1 = bH bs 2conditions 2bare cases, the boundary modified as follows: s 2 ∂H cWH ∂v + =0 (17) r r N X ∂t g 1 ∂x A ∂I dHwell (24) ch1== h2g= · · ·== hNg= Hwell , (36) Qi = Awell ∂A bs dt i=1
The ∂v ideal ∂Hchoice 4τ for the slot width results in: where and H0well + gAwell + = 0are the well top area and depth, respec(18) ∂t ∂x ρD A tively. f cWH = cpresented ⇒ b s = gmethod (25) The use an explicit time discretization c2WH These equations conform a hyperbolic system, analogous to and it requires a control on the time step in order to avoid the shallow water equations, considering the pressure and the in whichasA fthe stands for thevariables. full pipe cross-section. velocity conserved www.drink-water-eng-sci.net Drink. Water Eng. Sci. Drink. Water Eng. Sci., 7, 83–92, 2014
Number ph 4.1 of Exp
1 Roe schem 2 variables 1 0
˜ δF = JδU
It is ne instabiliti whose eig dition is a
δUji+1/2 = ∆tmax = m
Then, the δF i+1/2 =
∆tmax = m The wa variation values and 5 Test c flux veloc The mode ˜ k = (˜uin±o λcases
where 5.1 Ste
uFollowing ˜ i+1/2 = √ x 1m pris is given r b
cz˜ i+1/2 (8 ≤ =x ≤
andFollow the in of the equ h(x,upwind 0) = 0 an
The differ ˜ Rδx are summ i+1/2 Fig. 4 ( nects both in which t test case, for the flu any shock becomes: numerica highest pa
n+1 Dam 5.2 U =U i
Dam-brea shocks. It servation alytical so ity of 1m 4.2 showsBou the solution In order a
characteri
Test #1
86
Upstream Q m3 /s
Downstream h (m)
0.18
0.33
#2and1.53 0.66 Finite (sub) volume simulation of unsteady water pipe flow J. Fernández-Pato P. García-Navarro:
J. Fern´andez-Pato and P. Garc´ıa-Navarro: Finite volume simulation of unsteady water pipe flow Figure 4. Steady state over a bump. Initial state (dashed line). Bed level (grey). Numerical solution (blue): (a) Test #1; (b) Test #2.
Figure 4. Steady state over a bump. Initial state (dashed line). Bed level (grey). Numerical solution (blue): (a) Test #1; (b) Test #2.
h (m)
5.3 Wiggert test case the pipe pressure raises gradually with the upwards shockTable 2. Boundary conditions. wave 1movement. The experimental case designed by Wiggert (Wiggert 0.9 (1972)) and widely numerically reproduced (e.g. Kerger et 5.5 Pressurized flow transients al. (2010), Bourdarias et al. (2007)) consists in a horizontal 0.8 Flow30regime Number m longand andboundary 0.51 m wide flume. A 10 of m physical roof is placed in A full-pressurized (Le´on (2007) and Le´on et al. (2009)) 10 BCrectangular to impose pipe 0.148 the middle section, setting up a closed 0.7 frictionless pipe of rectangular section (10 m x 7.853 km flat, m in height (see Fig. 6). A 0.01m−1/3 s Manning roughness m) is connected upstream to a water reservoir that guarantees Upstream subcritical flow 1 coefficient is assumed. As initial conditions, a water level 0.6 a constant pressure head of 200 m. A closure valve is placed Upstream supercritical 2 of 0.128 m and zeroflow discharge are considered. Then a wave downstream. When the valve is suddenly closed, a waterhamDownstream subcritical 1 coming from the leftflow causes the pressurization of the pipe. mer 0.5 wave is generated moving upwards. Fig. 9 shows the nuDownstream supercritical flow 0 The imposed downstream boundary conditions are the same merical results with CFL=0.8 and a 500 cells mesh. 0.4 values measured by Wiggert (Fig. 7). 0.9 1.0 0 0.1 0.2 a completely 0.3 0.4 pressurized 0.5 0.6 pipe, 0.7 so it 0.8is This case simulates x (m) Fig. 8 shows the numerical results for the four gauges. The necessary to be very precise in the slot width selection in orexperimental comparison the second one, Table 3. Steady state over a bump. is done only for derInitial to ensure (dashed the numerical solutionresults accuracy. A waterhamFigure 5. Dam-break test case. line).test Numerical (blue Analytical (cyan line). Figure state 5. Dam-break case. Initial statedots). (dashed line).solution Numeridue to data availability issues. An overall good agreement is mer speed of 1000 m/s and a initial flow speed of 2.0 m/s are cal results (blue dots). Analytical solution (cyan line). observed with only small numerical oscillations presented. assumed. Following (Eq. (25)) condition:
Test
Upstream 3 s−1 Q m 5.4 Transient mixed flow
Downstream
h (m)
Af (41) cWH = c ⇒ b s = g 2 = 0.77mm #1 of the 0.18 The aim next test case is0.33 to prove the model in the simcWH ulation It considers a uniform #2of large-scale 1.53 strong transients. 0.66 (sub) A2 I1 = 10 m slope (0.1%) rectangular pipe connected to a downstream 6 Pipe 2bnetworks valve (Le´on (2007) and Le´on et al. (2009)). The lenght, width and height are 10 km, 10 m and 9.5 m, respectively, 6.1 Transient flow in a 3-pipe junction and the chosen Manning roughness is 0.015. r r 3 Preissmann slot model Wave in Wixcey (1990) is presented first. A 5 As initial condition, a uniform water depth (8.57 m) and 2 ∂I1 1A case proposed 4 A 3 = main c =longgprismatic g channel (1) dividing into two seckm discharge (Q = 240m3 /s) are assumed. From this state, the ∂A b The Preissmann slotvalve approach consist in assuming that the 0.5 mondary branches downstream is closed, generating a shock wave moving 2.0 m 3.5(2 m and 0.5 m 3) also 5 km long with the same 1 3.5 m top of the upstream pipe/closed channel is connected fictitious narm wide rectangular geometry have been considered, as preand pressurizing the pipe. Fig.to9 ashows the numerand for A > bH : Figure 6. Wiggert experimental setup. sented in fig. 3. Manning roughness coefficient for all the ical results obtained foratmosphere. the pressure head atfact three different row slot, which is open to the This allows to −1/3 A− bH pipes is 0.01m s. The main branch has a slope of 0.002 times. For this simulation, a 500 cells mesh consider a free surface flow situation, even whenand thea timestep pipe is h =the Hsecondary + and ones 0.001. given by CFL=0.5 have beed used. It is clearly observed that b
h=
A b
I1 = bH
A − bH H + bs 2
(22)
www.drink-water-eng-sci.net (A − bH )2 + b 2b0.21s
(23)
0.2 0.19
∂I1 g = ∂A
c=
s
0.18
A g bs
h (m)
r
0.17
(24)
0.16 0.15
0.14 The ideal choice for the slot width results in: 0.13
6
8
10
12
cWH = t (s) c ⇒ bs = g
Af 14
0.12
16
0
2
4
2 cWH Figure 7. Upstream (a) and downstream (b) pressured head level.
(19)
(21)
s
h (m)
pressurized, by including the slot in the shallow water equations (see Fig. 2). The slot width bs is ideally chosen equalDrink.ofWater Eng.waves Sci. in the slot to the water haming the speed gravity mer wavespeed. Therefore, the water level in the slot correa sponds to the pressure head level. This model is0.21based on the 0.2 previously remarked similarity between the wave equations 0.19 which describe free surface and pressurized flows. The wa0.18 ter hammer flow comes from the capacity of the0.17pipe system to change the area and fluid density, so forcing 0.16 the equivalence between both models requires that the 0.15 slot stores as 0.14 much fluid as the pipe would by means of a change in area 0.13 and fluid density. The pressure term and the wavespeed for 0.12 0 2 4 A ≤ bH are:
0.51 m (20)
6
8
(25) t (s)
10
12
14
in which Af stands for the full pipe cross-section.
Drink. Water Eng. Sci., 7, 83–92, 2014
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Drink. Water E
Figure 5. Dam-break test case. Initial state (dashed line). Numerical results (blue dots). Analytical solution (cyan line).
10 m
0.51 m
J. Fernández-Pato and P. García-Navarro: Finite volume simulation of unsteady water pipe flow
87 0.148 m
10 m
Wave 2
1
0.5 m
2.0 m
3.5 m
0.51 m
4
3
3.5 m
0.5 m
0.148 m
Figure 6. Wiggert experimental setup. Wave 2
1
0.5 m
2.0 m
3.5 m
4
3
3.5 m
0.5 m
6. Wiggert experimental setup. FigureFigure 6. Wiggert experimental setup. a
b 0.21
0.21
0.2
0.2
0.19
0.19 0.18
0.18
b
h (m)
0.17 0.21 0.16 0.2 0.15 0.19 0.14 0.18 0.13 0.17 0.12 0.16 0
2
4
6
8
10
12
14
16
h (m)
h (m)
h (m)
a
0.17 0.21 0.16 0.2 0.15 0.19 0.14 0.18 0.13 0.17 0.12 0.16 0
2
4
6
0.13
12
14
16
10
12
14
16
0.14 0.13 0.12
0.12 0
2
4
6
8
10
12
14
0
16
2
4
Finite volume numerical model Figure 7. Upstream (a) and downstream (b) pressured head level.
www.drink-water-eng-sci.net Explicit first order Roe scheme
Roe scheme is based on a local linearization of the conserved variables and fluxes: ˜www.drink-water-eng-sci.net
δF = JδU
(26)
It is necessary to build an approximate Jacobian matrix J˜ whose eigenvalues and eigenvectors satisfy: δUi−1/2 = Ui+1 − Ui =
α˜ k e˜ k
i+1/2
(27)
k=1
δFi+1/2 = Fi+1 − Fi = J˜ i+1/2 δUi+1/2 2 X = λ˜ k α˜ k e˜ k k=1
(28)
i+1/2
The wave strenght coefficients α˜ k represent the variable variation coordinates in the Jacobian matrix basis. The eigenvalues and eigenvectors are expressed in terms of the average flux velocity and wave speed: λ˜ k = (u˜ ± c) ˜ ,
e˜ k = (u˜ ± c) ˜T
(29)
where √ √ Qi+1 Ai + Qi Ai+1 u˜ i+1/2 = √ √ √ Ai Ai+1 Ai+1 + Ai www.drink-water-eng-sci.net/7/83/2014/
8
s Drink. Water Eng. Sci. g A A + (31) c˜i+1/2 = 2 b i b i+1 Following (García-Navarro and Vázquez-Cendón (2000)), the source terms of the equation system (Eq. 7) areSci. also disDrink. Water Eng. cretizated using an upwind scheme: ! ! X X ˜ Rδx = β˜k e˜ k + β˜k e˜ k (32) i+1/2
2 X
6
t (s)
t (s)
4.1
10
0.15
7. Upstream (a) downstream and downstream pressuredhead headlevel. level. FigureFigure 7. 0.14 Upstream (a) and (b)(b) pressured
4
8 t (s)
t (s)
0.15
(30)
k+
i+1/2
k−
i+1/2
in which the source streghts β˜k take the role of α˜ k coefficients for the fluxes. Then, the complete discretization of the system becomes: ! X δt n+1 n Ui = Ui − λ˜ k α˜ k − β˜k e˜ k (33) δx k+ i−1/2 ! X + λ˜ k α˜ k − β˜k e˜ k k−
4.2
i+1/2
Boundary conditions and stability conditions
In order to solve a numerical problem, it is necessary to characterize the domain limits by imposing some physical boundary conditions (BC). The number of boundary conditions depends on the flow regime (subcritical or supercritical). Hence, there are four posibilities for a 1-D numerical problem (see Table 2). Drink. Water Eng. Sci., 7, 83–92, 2014
888
J. Fern´andez-Pato Garc´ıa-Navarro:Finite Finitevolume volume simulation simulation of pipe flow J. Fernández-Pato andand P. P. García-Navarro: of unsteady unsteadywater water pipe flow
a
b 0.21
0.2
0.2
0.19
0.19
0.18
0.18
0.17
0.17
h (m)
h (m)
0.21
0.16
0.15
0.14
0.14
0.13
0.13
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0.12
0
2
4
6
8 t (s)
c
10
12
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10
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14
16
t (s)
d 0.21
0.21
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0.19
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0.17
h (m)
h (m)
0.16
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0.13
0.13 0.12
0.12 0
2
4
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14
0
16
2
4
t (s)
6
8 t (s)
Figure 8. Comparision betweennumerical numericalresults results(blue), (blue),experimental experimental data (green Kerger et et al.al. (2010) (red(red Figure 8. Comparision between (greendots) dots)and andnumerical numericalresults resultsfrom from Kerger (2010) dots) four gauges theWiggert Wiggerttest testcase casesetup. setup. dots) forfor thethe four gauges ofofthe 220
h1 = h2 = . . . = hN
Pressure head (m)
In the cases that consider pipe junctions, additional inter215 nal boundary conditions are necessary in order to model this feature. Figure 3 shows an example of a 3 pipe junction. 210 The water level equality condition is imposed at the junc205 tion for all the pipes: 200
(34)
Discharge continuity condition195is formulated depending on the flow regime. In the cases presented, water flows in a sub190 0 2000 4000 critical regime: N X
Then, the minimum of all the computed time steps is chosen: n o j 1tmax = min 1tmax , 5
CFL =
1t ≤1 1tmax
(38)
Test cases
The model is applied to several analytic or experimental test cases6000 in order to 8000 evaluate its accuracy. 10000 x (m)
5.1 Steady state over a bump Figure Q = 9. 0 Shockwave propagation in transient mixed flow at t=100 (35) s (red), t=200 s (green) and t=300 s (blue). i
i=1 450 Storage wells are another common junction type. For these cases, the boundary conditions 400 are modified as follows: 350
N X i=1
Qi = Awell
dHwell dt
Pressure head (m)
h1 = h2 = · · · = hN = Hwell ,
300 250
(36)
Following Murillo and García-Navarro (2012) a frictionless rectangular 25 m × 1 m prismatic channel is considered. The variable 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:
200 150
h(x, 0) = 0.5 − z(x),
u(x, 0) = 0
(40)
where Awell and Hwell are the100well top area and depth, reThe different boundary conditions for the test cases simuspectively. 50 lated are summarized in Table 3. The presented method uses an0 explicit time discretization 0 2000 4000 6000 8000 10000 of hydraulic jump that conFigure 4a shows the generation and it requires a control on the time step in order to avoid x (m) nects both subcritical and supercritical regimes. In the second instabilities. Hence the Courant-Friedrichs-Lewy (CFL) concase, in Fig. 4b the connection is made without Figure 10. Waterhammer simulation. Pressured head at t=3 s (red), t=6 stest (green) andshown t=9 s (blue). dition is applied on every branch (j ) of the network: any shock wave and it can be mathematically proved (and 1x j numerically checked) that this transition takes place in the j (37) 1tDrink. max = Water Eng. Sci. www.drink-water-eng-sci.net j j highest part of the bump. max u + c Drink. Water Eng. Sci., 7, 83–92, 2014
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0.14 0.15 0.13 0.14 0.12
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J. Fern´andez-Pato and P. Garc´ıa-Navarro: Finite volume simulation of unstea 10
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Comparision between numerical results (blue), experimental data (green dots) and numerical resultst (s)from Kerger et al. (2010) (red t (s) J. Wiggert Fernández-Pato he four gauges of the test case setup. and P. García-Navarro: Finite volume simulation of unsteady Comparision between numerical results (blue), experimental data (green dots) and numerical results from Kerger et al. (2010) (red he four gauges of the Wiggert test case setup.
89
Q
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215 220 PressurePressure head (m)head (m)
water pipe flow
210 215 205 210 200 205
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x (m) 190
0
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Shockwave propagation in transient mixed flow at t=100 s (red), x (m) t=200 s (green) and t=300 s (blue).
Figure 9. Shockwave propagation in transient mixed flow at
t = 100 s (red),mixed t = 200 and tt=200 = 300 s (blue). Shockwave propagation in transient flow sat(green) t=100 s (red), s (green) and t=300 s (blue).
5.4
Transient mixed flow
450
5.2
h(m) (T1)
1.2 1 0.8 0.6 0.4 0.2 0
5000
h(m) (T2)
Dam-break
h(m) (T3)
PressurePressure head (m)head (m)
The aim of the next test case is to prove the model in the simulation of large-scale strong transients. It considers a uniform slope (0.1 %) rectangular pipe connected to a downstream valve (León, 2006; León et al., 2009). The lenght, width and height are 10 km, 10 m and 9.5 m, respectively, and the cho0 1000 2000 3000 4000 sen Manning roughness is 0.015. t (s) As initial condition, a uniform water depth (8.57 m) and 0.8 discharge (Q = 240 m3 s−1 ) are assumed. From this state, 0.6 2000 4000 6000 8000 10000 x (m) the downstream valve is closed, generating a shock wave 0.4 0 0 2000 4000 6000 8000 10000 moving upstream and pressurizing the pipe. Figure 9 shows 0. Waterhammer simulation. Pressured head at t=3 s (red), t=6 s (green) 0.2 and t=9 s (blue). x (m) the numerical results obtained for the pressure head at three 0 0. Waterhammer simulation. head at t=3 s (red), t=6 s (green) and (blue). at t = 1000 0 t=9 s head 2000 For this simulation, 3000 a 500 cells mesh 4000 and a Figure Pressured 10. Waterhammer simulation. Pressured 3s different times. Water Eng. Sci. (red), t = 6 s (green) and t = 9 s (blue). www.drink-water-eng-sci.net t (s) beed used. It is clearly timestep given by CFL = 0.5 have 0.8 observed that the pipe pressure raises gradually with the upWater Eng. Sci. www.drink-water-eng-sci.net 0.6 wards shockwave movement. 400 450 350 400 300 350 250 300 200 250 150 200 100 150 50 100 0 0 50
5000
0.4 0.2
5.5 Pressurized flow transients Dam-break is a classical example of0 non-linear flow with 0 accuracy and con1000 2000 3000 4000 5000 shocks. It has been widely used to test the A full-pressurized (León, 2006;t León et al., 2009) 10 km flat, (s) servation of the numerical scheme, by comparing with its frictionless pipe of rectangular section (10 m × 7.853 m) is analytical solution. In the case presented, an initial disconFigure 12. Pressurized transient in aconnected pipe junction. Pressured vs time x=500 mthat (red),guarantees x=1000 m (green) and x=500 upstream tohead a water reservoir a tinuity of 1 m : 0.5 m ratio with no friction was considered. constant pressure head of 200 m. A closure valve is placed Figure 5 shows the good agreement between numerical and downstream. When the valve is suddenly closed, a waterhamanalytical solution at the given time. mer wave is generated moving upwards. Figure 9 shows the numerical results with CFL = 0.8 and a 500 cells mesh. 5.3 Wiggert test case This case simulates a completely pressurized pipe, so it is necessary to be very precise inW1the slot width selection in or2 1 The experimental case designed by Wiggert (1972) and I MAX der to ensure2 the numerical solution accuracy. 5A waterham1 widely numerically reproduced (e.g. Kerger et al., 2010; mer speed of 1000 m s−1 and a initial flow speed of 2.0 m s−1 Bourdarias and Gerbi, 2007) consists in a horizontal 30 m are assumed. Following (Eq. 25) condition: 1 long and 0.51 m wide flume. A 10 m roof is placed in the IM5 1 AX middle section, setting up a closed rectangular pipe 0.148 m 1 Af 4 J1 I MAX 1 1 −1/3 = 0.77 mm (41)J2 c = c ⇒ b = g WH s in height (see Fig. 6). A 0.01 s m Manning roughness 2 6 1 X cWH I MA coefficient is assumed. As initial conditions, a water level of 0.128 m and zero discharge are considered. Then a wave 4 I MAX coming from the left causes the pressurization of the pipe. 3 6 6 Pipe networks IM3 1 AX The imposed downstream boundary conditions are the same W2 values measured by Wiggert (1972) (Fig. 7). 6.1 Transient flow in a 3-pipe junction Figure 8 shows the numerical results for the four gauges. the looped pipe network. The experimental comparison isFigure done 13. only(Scheme for theofsecond A case proposed in Wixcey (1990) is presented first. A 5 km one, due to data availability issues. An overall good agreelong prismatic main channel (1) dividing into two secondary ment is observed with only small numerical oscillations prebranches (2 and 3) also 5 km long with the same 1 m wide sented. rectangular geometry have been considered, as presented in Drink. Water Eng. Sci. www.drink www.drink-water-eng-sci.net/7/83/2014/
Drink. Water Eng. Sci., 7, 83–92, 2014
Figure 11. Triangular inlet hydrograph in the pipe 1. 0
300
600
900
1200
1500
t(s)
1800
h(m) (T2) h(m) (T2)
90
h(m) (T1) h(m) (T1)
Figure 11. Triangular inlet hydrograph in the pipe 1. 1.2 1 0.8 0.6 0.4 1.2 0.2 1 0.80 0.6 0 0.4 0.2 0 0.8 0 0.6
J. Fernández-Pato and P. García-Navarro: Finite volume simulation of unsteady water pipe flow
1000
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0 0.8 0.6 0
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t (s)
0 1000 in a pipe junction. 2000 3000 x=500 m (red), x=1000 4000 m (green) and x=5000 5000 Figure 012. Pressurized transient Pressured head vs time m (blue).
6000
t (s)
Figure Pressurized transientinina apipe pipejunction. junction.Pressured Pressured head head vs time x=500 mm (red), x=1000 m (green) and x=5000 (blue). Figure 12.12. Pressurized transient x = 500 (red), x = 1000 m (green) and x =m5000 m (blue).
W1 1
2
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I MAX
5 1 W1
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1
I
1
1 MAX
I MAX
5 1
J1
I
1 MAX
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J2 I
1 3
4 MAX
3
Figure (Scheme loopedpipe pipenetwork. network. Figure 13.13. (Scheme of of thethe looped
I MAX
6
IM3
1
AX
W2
Figure 13. (Scheme of the looped pipe network.
(42)
Q2 (i, 0) = Q3 (i, 0) = 0.05 m3 s−1
h1 (i, 0) = h2 (i, 0) = h3 (i, 0) = 0.2 m
7
I MAX
1
W2 4 I MAX
Fig. 3. Manning roughness coefficient for all the pipes is Drink. Water Eng. Sci.branch has a slope of 0.002 and the 0.01 s m−1/3 . The main secondary ones 0.001. First, steadyEng. state Drink.aWater Sci.has been calculated from the initial conditions: Q1 (i, 0) = 0.1 m3 s−1 ,
1
6
6
IM3
AX
Using this steady state as initial condition for the transient calculation, a triangularwww.drink-water-eng-sci.net function of peak discharge QMAX = 3.2 m3 s−1 and a period of 600 s (Fig. 11) is imposed at the beginning of thewww.drink-water-eng-sci.net pipe 1. Figure 12 shows the numerical results using a 100 cells mesh and a CFL = 0.9 condition. 6.2
(43)
using as boundary condition the inlet discharge: (44)
and free outflow boundary condition at the outlet of pipes 2 and 3. Drink. Water Eng. Sci., 7, 83–92, 2014
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II
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I MAX
1
6 5MAX MA X
J1
Q1 (1, t) = 0.1 m3 s−1
7
IM5
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1 1
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Transient flow in a 7-pipe looped network
In this section a simple looped pipe network (see Fig. 13) is presented Wixcey (1990). Each pipe is closed, squared, 1 m wide and 100 m long. The bed slopes are S0,1 = 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 considered. A first calculation provided the steady state from the following initial conditions: Q1 (i, 0) = Q7 (i, 0) = 0.1 m3 s−1
(45)
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J. Fernández-Pato Finitevolume volumesimulation simulation unsteady water J. Fern´andez-Patoand and P. P. García-Navarro: Garc´ıa-Navarro: Finite of of unsteady water pipepipe flow flow a
b 1.2 Q1(m3/s)
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0.8
2 1
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h1(m)
1191
2 1 0
0
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t (s)
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2000
0
500
1000
t (s)
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Figure Time histories at grid points pressurized flow: (a)(a) Water depth; (b)(b) Discharge. The Figure 14.14. Time histories at grid points i =i=N/2 N/2 (red) and ii=N = N(blue) (blue)forforpunctually punctually pressurized flow: Water depth; Discharge. The discharge pipe 4 is recordedatatlocations locationsi i=1 (grey) and i=N discharge forfor pipe 4 is recorded = 1(red), (red),i=N/2 i = N/2 (grey) and i(blue). = N (blue).
highly transient mixed flows. Journal of Computational Q2 (i,ulating 0) = Q (46) 3 (i, 0) = Q5 (i, 0)
and Applied Mathematics 235, 2030-2040. = Kundu, Q6 (i, 0) = 0.05 m3I.M., s−1 Dowling, D.R., 2012. Fluid Mechanics. P.K., Cohen, Academic Press, pp. 920. Le´on, A.S., 2007. Improved Modeling of Unsteady Free Surface, Q4 (i,Pressurized 0) = 0 and Mixed Flows in Storm-sewer Systems (Disserta(47) tion). Le´on, A.S., Ghidaoui, M.S., Schmidt, A.R., Garc´ıa, M.H., 2009. Application of Godunov-type schemes to transient mixed flows. hj (i,Journal 0) = 0.2 = 1, . . .47 , 7) (48) of m, Hydraulic(jResearch (2), 147-156 Murillo, J., Garc´ıa-Navarro, P., 2012. Augmented versions of the and the upstream boundary condition: HLL and HLLC Riemann Solvers including source terms in one two0.1 dimensions of Q1 (1,and t) = m3 s−1 for shallow flow applications. Journal (49) Computational Physics 231 (20) 6861-6906. Trajkovic, Ivetic, M., Calomino, F., condition D’Ippolito, has A, 1999. Free outflowB.,downwards boundary beenInvesimtigation of transition from free surface to pressurized flow in a posed. circular pipe. WaterJScience and Technology 39 (9), 105-112. Junctions J1 and 2 are treated as normal confluences Wiggert, D.C., 1972. Transient flow2 in free-surface, pressurized (Eq. 34) and a storage well of 5 m top surface is assumed systems. Journal of the Hydraulics Division, Proceedings of the in junctions W1 and W (Eq. 36). The previously computed American Society of 2Civil Engineers 98 (1), 11-26. steady state is now used an1977. initial condition for a second Wiggert, D.C., Sundquist, as M.J., Fixed-grid Characteristics for calculation. A triangular function of peak discharge QMAX Pipeline Transients. Journal of the Hydraulics Division, Proceedand aings period ofAmerican 600 s (Fig. 11) of is imposed at the103, beginning of of the Society Civil Engineers 1403-1416. An investigation theWixcey, pipe 1.J.R., The1990. minimum dischargeofis algorithms 0.1 m3 s−1for . open channeldifferent flow calculations. Numerical AnalysisInInternal Report, 21, Two situations were studied. the first one the Mathematics, peak Departament discharge isoffixed to Q University = 2.0 mof3 Reading. s−1 and the flow MAX
remains unpressurized all over the pipe system. In the second case, the maximum discharge is increased to QMAX = 3.0 m3 s−1 , so pressurization occurs in some points of pipe 1. Figure 14 shows the results for the pressurized case. The results in pipes 3 and 6 have been omitted because of symmetry reasons. The discharge at the center of the pipe 4 is constantly zero and equal in magnitude but opposite in sign in the symmetric points, which verify the simmetric character of the pipe network. www.drink-water-eng-sci.net www.drink-water-eng-sci.net/7/83/2014/
7
Conclusions
The present work leads to the conclusion of the reasonably good applicability of the Preissmann slot model for an estimation of the pressure values in unsteady situations between both shallow and pressurized flows. In this second case, a small deviation of the ideal slot width induces a mass and momentum conservation error. On the other hand, it is clear that wider slots increases the stability of the numerical model. Hence, in benefit of the results accuracy, it is remarkably important to find an agreement between both facts. The model takes advantages of the similarity of both equation systems (shallow water and pressurized flow) and provides a simple way to simulate occasionally pressurized pipes, treating the system as an open channel in the Preissmann slot. This results in an easier implementation of the model because it avoids the managing of two separate systems of equations (shallow water and water hammer) in order to model separately the pressurized and the free surface flows. The use of an explicit scheme implies a limitation on the computational time step, which increases the calculation time, specially in the in the pressurized cases, due to the small width of the slot. Finally, the authors want to remark the possibility of adapting the method to more realistic systems, like pipe networks which can be punctually pressurized. Some additional internal boundary conditions are necessary in these cases, in order to represent pipe junctions and storage wells. Edited by: L. Berardi
Drink. Water Eng. Sci. Drink. Water Eng. Sci., 7, 83–92, 2014
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References Abbott, M. B.: Computational Hydraulics. Elements of the Theory of the Free Surface flows, Pitman Publishing Ltd., London, 324pp., 1979. Bourdarias, C. and Gerbi, S.: A finite volume scheme for a model coupling 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 of free-surfaces and pressurized flows, Kluwer Academic Publishers, Boston, 741 pp., 1994. Fernández-Pato, J. and García-Navarro, P.: A Pipe Network Simulation Model with Dynamic Transition between Free Surface and Pressurized Flow, Proc., 12th Int. Conf. on “Computing and Control for the Water Industry – CCWI2013”, Perugia, Procedia Engineering, Elsevier, 70, 641–650, 2014. García-Navarro, P. and Vázquez-Cendón, M. E.: On numerical treatment of the source terms in the shallow water equations, Comput. Fluids, 29, 951–979, 2000. García-Navarro, P., Alcrudo, F., and Priestley, A.: An implicit method for water flow modelling in channels and pipes, J. Hydraul. 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 simulating highly transient mixed flows, J. Computat. Appl. Mathemat., 235, 2030–2040, 2010.
Drink. Water Eng. Sci., 7, 83–92, 2014
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, Pressurized 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 the HLL and HLLC Riemann solvers including source terms in one and two dimensions for shallow flow applications, J. Computat. Phys., 231, 6861–6906, 2012. Trajkovic, B., Ivetic, M., Calomino, F., and D’Ippolito, A.: Investigation of transition from free surface to pressurized flow in a circular pipe, Water Sci. Technol., 39, 105–112, 1999. Wiggert, D. C.: Transient flow in free-surface, pressurized systems, Journal of the Hydraulics Division, Proc. Am. Soc. Civ. Engin., 98, 11–27, 1972. Wiggert, D. C. and Sundquist, M. J.: Fixed-grid Characteristics for Pipeline Transients, Journal of the Hydraulics Division, Proc. Am. Soc. Civ. Engin., 103, 1403–1416, 1977. Wixcey, J. R.: An investigation of algorithms for open channel flow calculations, Numerical Analysis Internal Report, 21, Departament of Mathematics, University of Reading, 1990.
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