conservative model unsteady flows deformable closed pipes

8/10/2019 Conservative Model Unsteady Flows Deformable Closed Pipes

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A conservative model for unsteady flows in deformable closed pipesand its implicit second-order finite volume discretisation

C. Bourdarias, S. Gerbi *

Universite de Savoie, LAMA, GM3, 73376 Le Bourget-du-Lac Cedex, France

Received 30 October 2006; received in revised form 20 April 2007; accepted 30 September 2007Available online 16 January 2008

Abstract

We present a one-dimensional model for compressible flows in a deformable pipe which is an alternative to the Allievi equations and isintended to be coupled in a natural waywith the shallow water equations to simulate mixed flows. The numerical simulation is per-formed using a second-order linearly implicit scheme adapted from the Roe scheme. The validation is performed in the case of waterhammer in a rigid pipe: we compare the numerical results provided by an industrial code with those of our spatial second-order implicitscheme. It appears that the maximum value of the pressure within the pipe for large CFL numbers and a coarse discretisation is accu-rately computed.2008 Elsevier Ltd. All rights reserved.

1. Introduction

The work presented in this article is the first step in amore general project: the modelisation of unsteady mixedflows in open channels and in pipes and its finite volumediscretisation. Since we are interested in flows occurringin closed pipes, it may happen that some parts of the floware free surface (this means that only a part of the cross sec-tion of the pipe is filled) and other parts are pressurised(this means that all the cross section of the pipe is filled).The Saint Venant equations, which are written in a conser-vative form, are usually used to describe free-surface flowsof water in open channels. They are also used in the context

of mixed flows using the artifice of the Preissman slot[18,4]: Cunge and Wegner [5] studied the pressurised flowin the pipe as if it were a free-surface flow by assuming anarrow slot to exist in the upper part of the pipe, the widthof the slot being calculated to provide the correct sonicspeed. This approach has been credited to Preissmann.

Implementing the Preissmann slot technique has the

advantage of using only one flow type (free-surface flow)throughout the whole pipe and of being able to easilyquantify the pressure head when the pipe pressurises. Nev-ertheless, as pointed out by several authors (see [17] forinstance) the pressurising phenomenon is a dynamic shockrequiring a full dynamic treatment even if inflows and otherboundary conditions change very slowly. In addition, thePreissmann slot technique is unable to take into accountthe depressurisation phenomenon which occurs during awaterhammer.

The commonly used model to describe pressurised flowsin pipe-lines is the system of Allievi equations. These

equations are usually solved with the characteristicsmethod (see [19] for instance). The resulting system offirst-order partial differential equations is not written undera conservative form since this model is derived by neglect-ing some acceleration terms. This non-conservative formu-lation is not appropriate for a finite volume discretization.Above all, it appears that a unifiedmodelisation with acommon set of conservative variables (see below) couldbe of a great interest for the coupling between free-surfaceand pressurised flows and its numerical simulation could bemore effective.

0045-7930/$ - see front matter 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compfluid.2007.09.007

* Corresponding author.E-mail addresses: [email protected](C. Bourdarias),

[email protected](S. Gerbi).

www.elsevier.com/locate/compfluid

Available online at www.sciencedirect.com

Computers & Fluids 37 (2008) 12251237
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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In this paper, we derive a 1D model from 3D compress-ible Euler equations by integration over sections orthogo-nal to the flow direction and by using a linearizedpressure law. We will consider entirely rigid as well asdeformable pipes (through the use of a Hookes law forthe deformation of the pipe due to the change of pressure).This model is a 1D first-order hyperbolic system of partialdifferential equations written in conservative form which isformally very close to the SaintVenant equations for shal-

low water. This particular fact is used to derive a modeland a numerical scheme for mixed flows which are pre-sented in [2]. The goal of the present paper is essentiallyto build the new model for pressurised flows, to describea finite volume discretisation and to carry out a code tocode validation by the resolution of standard critical testsof waterhammer.

The paper is organised as follows. In Section 2, wepresent a formal derivation of the model from 3D com-pressible Euler equations. Section3 is devoted to the finitevolume discretisation of the proposed model. Since theCFL condition for explicit scheme is very restrictive, wechoose, following [10,7], to discretise the system of first-order conservative partial differential equations by a linearimplicit finite volume scheme to avoid the usual CFL con-dition for an explicit spatial discretization. For the bound-ary condition treatment, we propose an adaptation of aclassical method ([6,13] for instance) to the implicit case.We recall the principle of the first-order explicit Roescheme and then we adapt it to build a linearly implicitmethod. Then the method to build the missing bound-ary conditions is described, and the second-order MUSCLscheme is formulated. Finally in Section4, we present thenumerical validation of this study by the comparisonbetween the resolution of this model and the resolution

of the Allievi equation solved by the research code

belier used at Center in Hydraulics Engineering ofElectricite De France (EDF) [20] for the case of criticalwaterhammer tests.

2. Formal derivation of the model

This model is derived from 3D system of compressibleEuler equations by integrating over sections orthogonalto the flow axis. Hence, we neglect the second and third

equation for the conservation of the momentum. In thecartesian coordinate system (shown in the Fig. 1), wherethe vector~iis along the pipe axis, the equation for conser-vation of mass and the first equation for the conservationof momentum are

otqdivq~U 0; 1otqu divqu~U FxoxP 2

with the speed vector ~U u~iv~jw~ku~i~V . We usethe linearized pressure law:

P Paq

q0

bq0 3in which q represents the density of the liquid, q0, the den-sity at atmospheric pressurePa and bthe water compress-ibility coefficient (which is the inverse of the bulk modulusof elasticity). The sonic speed is then given by

c 1ffiffiffiffiffiffiffiffibq0

p : 4Practically, b5:0 1010 m2=N and thusc1400 m=s.

Exterior strengths, withx-componentFx, are the gravity~gand the friction

qgSf~i given by the ManningStrickler

law (see[18]),

Nomenclature

c pressure wave speed in a rigid pipeb inverse of the bulk modulus elasticity

Pa atmospheric pressureq

0 liquid mass density at pressure PaX cross section, with boundary oXx center ofXx longitudinal coordinateS function ofx and P s.t. ASx;P~U velocity

p elev. of hydr. grade line Pq0g

L pipe lengthE pipe wall modulus of elasticity~N outward unit normal vector/ angle between~n and ~N

Rh hydraulic radius APmKs Strickler coefficient

~n outward unit normal vector in theX-plane

a pressure wave speedg gravitational accelerationP pressure of liquidq

mass density at press.PA pipe cross-sectional area~m ~xm, where m2oXt timeu x-liquid mean velocity~V velocity in theX-planeQ dischargeAuz elevation of pipe bottome pipe wall thicknessd pipe inner diameter

Pm pipe inner perimeterSf frictionh slope of pipe axis

1226 C. Bourdarias, S. Gerbi / Computers & Fluids 37 (2008) 12251237


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Sf KAujuj with KA 1K2

s

RhA

4=3;

where AAx; t is the given surface area of a sectionXx; t normal to the pipe axis at position x and at timet, (seeFig. 1), Ks > 0 is the Strickler coefficient of rough-ness and RhA is the so-called hydraulic radius given by

RhA APm ;Pm being the perimeter ofX. For instance, inthe case of a fully-filled circular pipe with diameter d weget Rhd=4.

Then Eqs.(1) and (2) become

otqoxqu divy;zq~V 0; 5otqu oxqu2 divy;zqu~V qgsin hSf c2oxq:

6Eqs.(5) and (6)are integrated over a cross section Xx; t(Fig. 1, left).

In the following, overlined letters represent averagedquantities over Xx; t. For Eq. (5) we have successively,with the approximation ququ:ZXx;t

otqo tZXx;t

qZ

oXx;tqot~m~n;

where m2oX; ~m stands for Om!

and~n ~mj~mj is the outwardunit vector at the point min the X-plane (seeFig. 1),

ZXx;t

oxqu oxquA ZoXx;t

quox~m~n;ZXx;t

divy;zq~V Z

oXx;tq~V~n:

Therefore we get the following equation for the conserva-tion of the mass:

otqA oxqQ Z

oXx;tqot~muox~m~V ~n; 7

where QAu is the discharge of the flow.Next, with the approximations qu

qu and qu2

qu2,

we get for Eq.(6):

ZXx;totqu otZXx;t

quZoXx;t

quot~m~n

o tqQ Z

oXx;tquot~m~n;Z

Xx;toxqu2 ox

ZXx;t

qu2 Z

oXx;tqu2ox~m~n

oxqu2A Z

oXx;tqu2ox~m~n;Z

Xx;tdivy;zqu~V

ZoXx;t

qu~V~n;ZXx;t

c2oxqc2oxZXx;t

qc2Z

oXx;tqox~m~n

c2

oxqA c2

qoxA:

Thus Eq.(6) becomes the equation for the conservation ofmomentum:

otqQ ox qQ2

A c2qA

gqAsinh Sf c2qoxA

Z

oXx;tquot~m uox~m ~V ~n:

8To treat the integral terms appearing in Eqs. (7) and (8), weuse the no-leak condition at the tube wall oXwhich writes:

~U~N o t~m~N;where~Nis the outward unit vector at the point m. It meansthat the normal velocity of the water at the point m2oXand the normal velocity of m (due to the deformation ofthe pipe under pressure rise or decrease) are the same.Thus, in the present case of axial symmetry, an easy com-putation gives

ot~muox~m~V ~n oxrot~m~i tan /ot~m~i;where rxmrx; t is the radius of X and /

~n; ~N

2 p=2;p=2

, with the orientation given by

~i;~n

(see Fig. 1, right). For instance we have tan/< 0 in

Nn

i

i

y

z

jk

g

x

(x,t)

n

m

r

R

mO

pipe axis

inner wall

inner wall

Fig. 1. Geometric characteristics of the pipe.

C. Bourdarias, S. Gerbi / Computers & Fluids 37 (2008) 12251237 1227
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contracting sections. On the other hand, neglecting the tan-gential deformations in the longitudinal sections of thepipe, we set ot~m otR~N where RRx; t is the radiusof normal curvature at the point m along oX (see Fig. 1,right) satisfying Rr= cos/. Neglecting the time varia-tions of/, this leads to ot~m~i tan/pd otA and we getZ

oXx;tq

qu

ot~m uox~m ~V ~n1A

qA

qQ

tan2 /otA:Using tan/ 1

2oxd0, whered0is the diameter at rest, omit-

ting the overlined notations and setting the conservativevariables: M qA and DqQ, Eq. (7) for the conserva-tion of the mass becomes:

otM oxD 14

otA

Aoxd02M: 9

To deal with the deformation of the section due to thechange of pressure, we use a linear elastic law (derived from

the Hookes law for an elastic material): dR R2

eEdP. SettingAAx; t Sx;Px; t 10for a pipe with a circular cross section, this law writes (see[18]for instance, in the case of a constant radius):

oS

oP Sd

eEcos/; 11

where ddx; t is the diameter of the pipe, e is the wallthickness (assumed to be a constant for the sake of simplic-ity), E is Youngs modulus of elasticity for the wall mate-rial. //x is the angle ~n; ~N between the outwardunit vector of the cross section X at a point m of the inner

wall of the pipe and the outward unit vector ~Nat the samepoint (seeFig. 1, right side). It is assumed to be constant intime for the sake of simplicity.

From(10) and (11)we get

oxAoxS AdeEcos/

oxP

and then, using(3), the term c2qoxAin(8)writes

c2qoxAc2q deEcos/

Aoxqc2qoxS

c4q deEcos/

oxqA c4q2 deEcos/

oxAc2qoxS

This leads easily to

c2qoxAoxc2 a2qA a2qoxSqAoxa2

with a given by cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1qq0

dbeEcos/

p (for a circular pipe) and approx-imated by

aax; t cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 dx;t

beEcos/x

q ; 12a represents the sonic speed for the considered pipe andtakes into account the deformation of the pipe (by thedependancy in time) and also the non-uniformness of the

pipe by the dependancy in the diameter at the position x.

In the infinitely rigid caseE 1, we recover the sonicspeedc given by Eq.(4)whereas in the rigid case for a non-uniform pipe, we must take into account the variation ofthe section with respect to the position x and thus the var-iation of the sonic speed oxa.

The term oxS is approximated by oxS0 where S0 is the

area of the cross section at restS0x Sx;Pa. ThusEq.(8) for the conservation of the momentum becomes

otD ox D2

Ma2M

gMsin hx gKADjDj

M

Moxa2 a2MA

oxS0

14

otA

Aoxd02D: 13

Lastly, proceeding for otA in the same way as for oxA, weget

otAkotM 14with kkx; t dx;t

eEcos/x ax; t2.We can now formulate the system of coupled first-order

hyperbolic partial differential equation obtained for thetwo following cases:

For a rigid pipe

otUoxFU Grx;U; 15where

U qA;qQt M;Dt; FU D;D2

Ma2M

tand the source term Gr writes

Grx;U 0;gMsinhxgKADjDjM

Moxa2 a2MA

oxS0 t

:

16For a deformable pipe

otUoxFU Gdx;U; 17otAkotM; 18where

U qA;qQt M;Dt; FU D;D2

Ma2M

tand

Gdx;U Gr14

otA

Aoxd02U: 19

Let us mention that the above model is formally close tothe system of SaintVenant for shallow water, where theunknown state vector Uand the flux FU are given by

U A;Qt and FU Q;Q2

A gI1A

t;

AAx; tbeing the so-called wetted area andgI1Aarisingfrom thehydrostatic pressure law. Then a common set of con-

servative variables is for instance U Aeq qAq0 ;Qeq qQq0 t



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(free-surface-equivalent areaand free-surface-equivalentdischarge). The two models mainly differ by the pressurelaw. It is the reason why it seems natural to formulate themixed flow problemsas a first-order hyperbolic systemof par-tial differential equations with discontinuities (in the gradientof the pressure) located at the interface between the two types

of flows (free-surface/pressurised). The modelisation, thetreatment of this type of discontinuity, the numerical resultsand the experimental validation are presented in an article(in press at the present time)[2].

3. The finite volume discretisation and its implicit

formulation

For the numerical approximation of(9), (13) and (14)with suitable initial and boundary conditions, we use afinite volume method with a Roe-like numerical flux in apartial linear implicit version. We present here a second-

order extension of the implicit first-order scheme describedin[3] and adapted to a pipe with contracting or expandingsections.

The main axis of the pipe, with length L, is divided in Nmeshes mi xi1=2;xi1=2; 1 6 i 6 N such that

x120 and xN1

2L. We denote by x i the center ofm i and

by hi its length. Dt denotes the time step. We set t00and for nP 0, tn1tnDt. The discrete unknowns areUni

MniDni

; 1 6 i 6 N; 0 6 n 6 nmax that approximate

the mean value of Ux; t on mi tn; tn1; and Ani thatapproximates Axi; tn. The upstream and downstreamboundary states Un0;U

nN

1 are associated to fictive meshes

denoted 0 and N1.Let R be a diagonalizable dd real matrix with real

eigenvalues k1; . . . ; kd associated to eigenvectorsr1; . . . ; rd :diagki P1MP. We setjRj PdiagjkijP1(absolute valueof the matrix R).

3.1. Principle of explicit first-order Roe scheme. Well

balanced VF-Roe scheme

In this section we recall the principle of the explicit first-order Roe scheme and explain how to bypass the difficul-ties which arise in the case of a deformable pipe and/or a

non-uniform geometry.Roes scheme, derived from Godunovs method, is

based on the use of an approximate Riemann solver (see[7]for instance). It takes the conservative form:

Un1i UniDt

Fni1=2Fni1=2

hiGni ; 1 6 i 6 N; nP 0:

20For all 1 6 i 6 N;U0i is computed as the averaged value ofthe initial dataU0on the meshmi. The right hand sideG

ni is

a centered approximation of the contribution of the sourceterms in the mesh mi, the numerical flux F

ni

1=2 is given by

Fni1=2FUi1=20;Uni ;Uni1 where Ui1=2n;Ul;Ur is

the exact solution, along the line nx=t, of the linearizedRiemann problem:

PRLoUotRUl;Ur oUox0;

Ux; 0 Ul if x< 0;Ur if x> 0;

8>:the function R with values in the set of 22 real matrices

being subject to the conditions

(1) R is continuous,(2) RUl;Ur UlUr FUl FUr(Roes condi-

tion, ensuring conservativity)(3) RUl;Ur is diagonalizable with real eigenvalues.

The two first conditions imply RU;U JFU, Jaco-bian matrix of the function F(consistency). In its centeredform, the numerical flux takes the form

Fni1=2

F

Uni

F

Uni

1

2 1

2 jRUni ;U

ni1j U

niU

ni1:

The time step (at time tn) in the resulting scheme isclassically subject to a CourantFriedrichLevy typecondition:

CFL Dt6 Cinf

16i6Nhi

maxfj~knk;i1=2j; 1 6 k6 2; 1 6 i 6 Ng

;

C20; 1;where ~knk;i1=2 is the kth eigenvalue of the matrix

RUni ;Uni1:Cis called the CFL coefficient.Following Roes method ([15]in the case of one-dimen-sional Euler equations) we get, in the case of an entirely

rigid pipe, the following Roe matrix:

RUl;Ur 0 1

c2 ~u2 2~u

with ~uul

ffiffiffiffiffiffiMl

p urffiffiffiffiffiffi

Mrpffiffiffiffiffiffi

Mlp ffiffiffiffiffiffiMrp :

The eigenvalues of RUl;Ur are ~k1~uc< 0 0

(8>: 24witheJ eJWni ;Wni1 Bxi1=2; WniWni12 .eJ has the eigenvalues ~k1~k20; ~k3~uni1=2~ai1=2and ~k4~uni1=2~ai1=2, with

~uni1=2 DiDi1MiMi1 and ~a

2i1=2

a2i a2i12

; a2i c2Wxi:

The eigenvectors are

~r1~a2i1=2 ~uni1=22

0

geMni1=20

0BBB@1CCCA; ~r2

c2 lnS0i1=2g0

0

0BBB@1CCCA;

~r3

0

0

1

~uni1=2~ai1=2

0BBB@1CCCA and ~r4

0

0

1

~uni1=2~ai1=2

0BBB@1CCCA

with eMni1=2 Mn

iMn

i12

and

S0

i

1=2

S0

xi

1=2

. The

solution of the Riemann problem(24)consists in four con-stant states connected by shocks propagating along thelines nx=tki. Since the velocity is practically alwaysless (in magnitude) that the sonic speed, we have~a< ~u< ~a and then ~k2< 0~k1 < ~k3. Since the valuesof the corrected elevation Z and ofW are known, weare looking for the states on each sides of the linen0 de-noted byMn;i1=2;Dn;i1=2for the left side andMn;i1=2;Dn;i1=2for the right side (seeFig. 2above).Fig. 3.

Moreover, as the third component of~r1 is null, the dis-charge D is continuous through the line n0. Thus

Dn;i

1=2

D

n;i

1=2. In the sequel, we denote D

ni

1=2 this value.

A classical computation gives



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Mn;i1=2Mni

geMni1=22~ai1=2~ai1=2~uni1=2

Zi1Zi

eMni1=2lnS0i1=22~ai1=2~ai1=2~uni1=2 a2i1a2i ~u

ni1=2~ai1=2

2~ai1=2Mni1Mni

1

2~ai1=2Dni1Dni ;

Dni1=2DnigeMni1=2

2~ai1=2Zi1Zi

eMni1=2lnS0i1=2

2~ai1=2a2i1a2i

~uni1=22 ~a2i1=2

2~ai1=2Mni1Mni

~un

i1=2~ai1=22~ai1=2

Dni1Dni ;

Mn;i1=2Mn;i1=2

geMni1=2~uni1=22 ~a2i1=2

Zi1Zi

eMni1=2lnS0i1=2~un

i1=22 ~a2i1=2a2i1a2i

and then, for instance, Wi1=20;Wni ; Wni1 Zi;Wi;M

n;i1=2;D

ni1=2t.

Adding the variables Z and W in the system (22) pro-

duced upwinding terms:

geMni1=2Zi1Zi and eMni1=2lnS0i1=2a2i1a2i :Considering the two last components of(23)we get our ex-plicit first-order scheme under the form:

Un1i UniDt

Fn;i1=2Fn;i1=2

hiHni ; 1 6 i 6 N; nP 0;

25where

Fn;i1=2FUi1=20;Uni ;Uni1

with

Ui1=20;Uni ;Uni1 Mn;i1=2;Dni1=2t

and

Hni gKAiDnij Dnij

MniHUni :

Practically, with b5:01010 m2=N and in the unde-formable case, the (CFL) condition gives Dt6 0:7103Dx.This severe restriction is a motivation for the constructionof an implicit scheme.

3.2. First-order implicit scheme

A fully implicit scheme based on the previous approachwould write

Un1i UniD

t

Fn1;i1=2 Fn1;i1=2

hi Hn1i

26

with

Fn1;i1=2FUi1=20;Un1i ;Un1i1 27

Hn1i gKAiDn1i ; jDn1i j

Mn1iHUn1i : 28

It would obviously be of high cost and not easily extendedto second order. Another approach (see [10,8]) consists inlinearizing F and the source term around their values attime tn. Then we get, instead of(26)(28)

Un1i

Uni

Dt eFn1;i1=2 eFn1;i1=2hi eHn1i 29

witheFn1;i1=2 Fn;i1=2JFUi1=20;Uni ;Uni1

Ui1=20;Un1i ;Un1i1 Ui1=20;Uni ;Uni1

;eHn1i HniJHUni Un1i Uni :Thanks to the homogeneity property of the flux function F,

JFU U FU;

the numerical flux is given by

(3)

(4)

Wi+1

Wi

n n

Mi+1/2

D

i+1/2

n,

Mi+1/2

Dn,+

i+1/2

n,+n,

(1) (2)

x

t

Fig. 2. Solution of the Riemann problem (24). The number of the linescorresponds to those of the eigenvalues.

x x x x x xi ii+1 i+1i1 i1monotonic case extremum

Fig. 3. Slopes reconstruction.



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eFn1i1=2JFUi1=20;Uni ;Uni1 Ui1=20;Un1i ; Un1i1 : 30

At this stage (29) is a linear system of 2Nequations and2N4 scalar unknowns because the upstream and down-stream states Un10 and U

n1N

1 have to be determined. In

the next subsection we specify our procedure to close thissystem.

Lastly, let us explain briefly how to deal with a deform-able pipe. The cross area Ani being known at time tnin meshmi, the source term

14

otA

Aoxd02U

is discretized by

kni oxd02i4Ani Dt

Mn1i Mni Mni

Dni

;

according to the following time discrete version of Eq.(14):

An1i Anikni Mn1i Mni ; 31where we set:

kni dni

eEcos/xi ani 2,

dni diameter asociated to Ani , and ani cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 dni

beEcos/xi

q :This leads to a minor modification of the system arising

from(29).Notice that it is convenient, in the source term HUni , to

replaceKAiby KAn

i and in the matrix eJto replace ~a2i1=2by~an2i1=2

ani2an

i12

2 .

The cross area A is then updated through (31). Noticethat the computation of A allows us to get the pressurevia qM=A and the discharge QD=q.

3.3. Boundary conditions

To achieve the description of the implicit scheme, wepresent now a way to take the boundary conditions intoaccount. We recall that the upstream and downstream statevectors (corresponding to x1=2 and xN1=2) at time tn are

respectively denoted Un0 Mn

0Dn0

and UnN1 Mn

N1DnN1 .

From a mathematical point of view, we must give as manyscalar boundary conditions as incoming characteristiccurves, that is one at each end of the pipe. For example

Mn0M0; tn and DnN1DL; tn are given quantities or,more generally, we impose some condition fuM;D; t 0at the upstream end and fdM;D; t 0 at the downstreamend.

Numerically, the computation of the boundary fluxes(via the resolution of the Riemann problem (24)) requirescomplete state vectors that one can consider as exteriorvalues on fictive meshes. Un0 and U

nN

1 play this role and

are supposed to be known at time tn. Thus the problem is

to determine or estimate the boundary statesUn10 and U

n1N1 in(29).

Using an explicit characteristic method (for instance) isnot convenient because it is subject to a CFL type condi-tion. The method described below is an adaptation to theimplicit scheme of those studied by Dubois[6] and Kum-

baro [13] (see also [7]). It allows to update the boundarystates using known values at the same level time, so it isnaturally implicit. Let us recall the original procedure inthe case of the upstream state, for instance.

We start with given interior vector statesUn1i 1 6 i 6 N and any given relationship

fuMn10 ;Dn10 ; tn1 0:We have to build a complete boundary state using thesedata at the same level time and not at the previous oneas in the characteristic method. The vector statesWn10 and W

n11 are expressed in the basis of eigenvectors

of the matrixeJ in (24) where we assume that Z0Z1 and W0W1 (notice that these eigenvectors dependon the unknown Un10 ):

Wn10 an10 ~r1bn10 ~r2cn10 ~r3 dn10 ~r4 andWn11 an11 ~r1bn11 ~r2cn11 ~r3 dn11 ~r4:The classical BC method consists of connecting Un10 toUn11 by an unique jump through the incoming characteris-tic x~k4t, thus settingan10 an11 ; bn10 bn11 and cn10 cn11or equivalently Dn11

Dn10

~un11=2

~a1=2

Mn11

Mn10

.

Then we get Un

10 as the solution of the nonlinear system:

Dn11 Dn10 ~un11=2 ~a1=2Mn11 Mn10 ;fuMn10 ;Dn10 ; tn1 0:

( 32

Applying this method for nonlinear boundary conditions(BC in brief) at time tn1 is not compatible with the frame-work of linearised implicit method as described above asabove. We propose to apply it at time tn1 after a first stepwhich consists in completing (29) with four linear equa-tions thanks to a modified procedure that we present fur-ther. Solving the resulting system supplies a first estimateof interior and unknown boundary states, then we make

use of the standard BC method. Our adaptation to the im-plicit scheme consists of two steps.

First step: we apply the previous procedure using theeigenvectors of the matrixeJ at time tn instead of timetn1. From(32)we get the linear equation (with unknowns

Mn10 and Dn10 ):

Dn11 Dn10 ~un1=2~a1=2Mn11 Mn10 33and we use a linearized version flinu Mn10 ;Dn10 ; tn1 0 ofthe upstream boundary condition (around the vector stateUn0). Similarly for the downstrean state we get

D

n

1

N1Dn

1

N ~aN1=2~un

N1=2Mn

1

N1Mn

1

N 34



9/13

and flind Mn1N1;Dn1N1; tn1 0.The linear system arising from (29) is now completely

determined.Second step: equipped with interior vector states issued

from the first step, we apply the standard BC method(32)at time tn1.

Remark 1. Commonly used boundary conditions are forinstance an upstream constant total head and the down-stream flow Qt (as in the presented numerical results).The total load in the pipe is defined by H z u22gp,where z is the altitude (m), u the flow speed (m/s), p therelative pressure (m) expressed in equivalent water heightdefined by pc 2qq0

q0g . We impose at the entrance of the

pipe with altitude z0 a constant load H0. Thus we get

Dn10Mn1

0

2 2Mn10bq2

0A0

2gH0z0 2bq0 as boundary datum in(32). The downstream boundary condition is Qn1N1Qtn1 which writes D

n1N1A

n1N1

Mn1N1

Qtn1 with An1N1A

n

N1kn

N1Mn

1

i Mni and a suitable definition of

knN1. A similar technique as above is then applied.

3.4. Space second-order implicit scheme

The main inconvenience with usual Roe-type schemes(as the Godunov one) as well as the implicit scheme is thatthis type of discretisation is very diffusive. One way toimprove the scheme is to perform a second-order space dis-cretisation (which can be coupled with a second-orderRungeKutta time discretisation). This type of improve-ment (for explicit schemes) seems to be practically a relax-

ation of the CFL condition more than a better spaceapproximation.

We recall in the following section the principle of theMUSCL method for a second-order explicit space discret-isation and we propose a linearly implicit version of thiscommonly used method to improve spatial approximation.

3.4.1. MUSCL method

The MUSCL method (monotonic upstream scheme forconservation laws), due to Van Leer [14], is based on areconstruction step which consists in constructing apiecewise linear function from cell-averages. Then we cansolve new local Riemann problems extrapolating the aver-aged states in each cell: this method, quite simple to imple-ment, is a natural extension of the previous scheme.

The steps are as follows (with the same notations as inthe explicit scheme):

(1) From given cell-averages Uni ; 1 6 i 6 N at time tn,

define slope vectors Rni Rni;1Rni;2

.

(2) On each cellxi1=2;xi1=2 define a piecewise linearapproximate solution

Unx Uni xxiRni :

(3) Compute fluxes:

Fn;i1=2FUi1=20;Un;i ;Un;i1;

where

Un;i Uni h i2Rni ;

Un;i1Uni1 h i12 Rni1:

(

(4) Compute updated cell-averages Un1i applying(25).

The first step (slopes reconstruction) is crucial. A natural

way would consist in using the central difference Un

i1Uni1xi1xi1

which is a second-order approximation ofU0xi. To satisfysome monotonicity and TVD properties, we apply a limita-tion procedure, for instance the following classical one (see[11]for instance). For k1; 2 we set

Rni;kminmodUni1;kUni;k

xi1xi ;Uni;kUni1;k

xixi1

35

where the minmod function is defined over R2 by

minmoda; b minjaj; jbjsgna if sgna sgnb;0 else:

3.4.2. An adaptation to the implicit scheme

A direct application of this method to the previous first-order scheme would introduce nonlinearities due to slopescomputations. Instead we propose the following procedurein three steps:

First step:

From the averaged statesUni ; i

1; . . . ;N, at timetn, wedefine the slope vectors Rni following(35). Then we com-pute at each interface xi1=2 the corresponding statesU

n;i ;U

n;i1 and finally U

i1=20;Un;i ;Un;i1.

Second step:

A first estimateeUn1i ofUn1i is computed applying thefirst-order scheme (29) together with the treatment of theboundary conditions as in Section 3.3. We deduce a firstestimate ~Rn1i of the slopesR

n1i .

Third step:

The second-order fully implicit scheme, would writeunder the form(26)with

F

n1;i1=2FUi1=20

;U

n1;i ;U

n1;i1 ;

Hn1i HUn1i :

The linear implicit second-order scheme is obtained by lin-earizing Fn1;i1=2 around U

n;i and U

n;i1, by using in (24) a

matrixeJ associated to statesUn;i ;Un;i1at time tn insteadofUn1;i ;Un1;i1 and finally by using the estimated slope~Rn1i to compute U

n1;i and U

n1;i1 . It leads to a scheme

under the form:

Un1i UniDt

eFn1;i1=2 eFn1;i1=2

hi

eHn1i 36

with



10/13

eFn1;i1=2 JFUi1=20;Un;i ;Un;i1 Ui1=20;Un1;i ;Un1;i1 ;eHn1i Hni JHUni Un1i Uni ;where

Un1;i

Un1i

h i

2~Rn1i

Un1;i1 Un1i1 h i12 ~Rn1i1(Finally the updated states Un1i are solutions of the linearsystem arising from (36) combined with Eqs. (33) and(34)where Mn11 ;D

n11 ;M

n1N and D

n1N are, respectively, re-

placed with Mn1;1 ;Dn1;1 ;M

n1;N and D

n1;N . The treat-

ment of the boundary conditions is performed using (32)modified in a similar way. Finally, in the case of a deform-able pipe, the procedure is the same as in the first-order im-plicit scheme.

4. Numerical validation in the case of a constant radius

circular pipe

We present now numerical results of a water hammertest. The pipe of circular cross section of 2 m2 and thickness20 cm is 2000 m long. The altitude of the upstream end ofthe pipe is 250 m and the angle is 5. The Young modulus is23109 Pa since the pipe is supposed to be built in con-crete. The total upstream head is 300 m. The initial down-stream discharge is 10 m3=s and we cut the flow in 5 s. Thenumber of mesh points is either 1000 or 150 depending onthe tests and is presented for each of them. Let us define thepiezometric line by

piezozdp with pc2qq0q0g

:

4.1. Comparison with the solution of Allievi equations

To ensure that the model that we propose describes pre-cisely flows in closed pipe-lines, we present a validation ofit by comparing numerical results of the proposed modelwith the ones obtained by solving Allievi equations bythe method of characteristics with the so-called beliercode used by the Center in Hydraulics Engineering ofElectricite De France (EDF) [20]. Our code is written inFortran 77.

A first simulation of the water hammer test was done fora CFL coefficient equal to 1 (i.e.CFL1) and a spatial dis-cretisation of 1000 mesh points (the mesh size is equal to2 m). In the Fig. 4, we present a comparison between theresults obtained by our first-order scheme and the onesobtained by the beliercode at the middle of the pipe:the behavior of the piezometric line and the discharge atthe middle of the pipe. One can observe that the resultsfor the proposed model are in very good agreement withthe solution of Allievi equations. A little smoothing effectand absorption may be probably due to the first-order dis-

cretisation type. It is the reason why we perform the same

test, in the same condition (CFL1 and 1000 mesh points)with the second-order scheme. The Fig. 5 shows that thissmoothing effect is much less important with this second-order scheme. It is the reason why in the following numer-ical results, we consider the second-order scheme with aCFL condition equal to 1 as the reference solution.

4.2. Influence of the CFL coefficient

The purpose of our scheme is to perform a solution witha not accurate spatial detail but with very low computa-tional costs. As argued before, we choose to perform a lin-ear implicit spatial discretisation (first order and secondorder) to get rid of the CFL condition imposed for the sta-bility of explicit discretisation of hyperbolic systems of par-tial differential equations. Since the proposed spacediscretisation is only linear implicit (to keep a low CPUtime), we can expect a deterioration of the results forCFL> 1. Therefore, we performed the same test as abovefor different CFL coefficients and 1000 mesh points. Wepresent in theFig. 6, the same quantities (piezometric lineand discharge) at the middle of the pipe for the second-order space discretisation scheme. One can observe that

even for CFL10, the results are good from the point of

150

200

250

300

350

400

450

500

550

0 5 10 15 20 25 30 35 40

Piezometricline(m)

Time (second)

1st orderbelier

-2

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35 40

Discharge(m3/s)

Time (second)

1st orderbelier

Fig. 4. Validation of the model: first-order scheme. Piezometric line (top)and discharge (bottom) at the middle of the pipe.



11/13

view of the maximum pressure calculation and that the sec-ond-order implicit discretisation is very stable in regard ofthe time discretisation. The very good behavior of thisscheme from the point of view of the time discretisationis a very good argument for using it, even though we mustsolve at each time step a linear system (which contains only6 diagonals). Moreover, we wanted to study the behaviorof the scheme from the point of view of the spatial discret-isation. Thus we perform the same test for different CFLcoefficients and for only 150 mesh points (the mesh size isthen 13.33 m). Before doing this test, we suspected a bad

behavior for a coarse space mesh and great CFL coeffi-cients since the scale of the physical effects of a water ham-mer is very small against this space mesh size and this timestep.Fig. 7shows the behavior of the second-order schemefor a coarse grid discretisation and for different CFL num-bers. A great smoothing effect is present but it seems thatthe highest pressure value is well computed.

Nevertheless, the hydraulics engineers are interested inthe value of the highest pressure in the pipe to make a goodevaluation of the size and the strength of the pipe. There-fore we present inTable 1the highest piezometric line com-puted in the pipe and the relative error with the highest

piezometric line computed by the code belier for the pre-

ceding water hammer test, Pmax688:442. The relativeerror is very small and therefore even for a coarse meshand a great CFL coefficient, the highest piezometric levelis wellcomputed in a very small CPU time.

4.3. Case of contracting or expanding sections

To illustrate the effect of the variations of the sectionsand the important contribution of the term of geometry(terms a2 M

A

oxS0 and Moxa2 in Eq. (13)), we consider the

case of an immediate flow shut down in a frictionlesscone-shaped pipe. We thus computed the waterhammerpressure rise at the middle of the pipe DH and relate it tothe pressure rises calculated for an equivalent pipe segmentDHeq as presented in[1,18].

Let us hereby recall how do we transform a real pipewith expanding or contracting sections to an equivalentpipe of constant dimension. The similarity conditionsshould be preserved to transform a pipe with expandingor contracting sections to an equivalent uniform pipe seg-ment. These conditions imply that the following propertiesshould not change in the equivalent pipe segment, com-

pared to the real pipe of the same length L:

150

200

250

300

350

400

450

500

550

0 5 10 15 20 25 30 35 40

Piezometr

icline(m)

Time (second)

2nd orderbelier

-2

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35 40

Discharge(m3/s)

Time (second)

2nd orderbelier

Fig. 5. Validation of the model: second-order scheme Piezometric line(top) and discharge (bottom) at the middle of the pipe.

150

200

250

300

350

400

450

500

550

0 5 10 15 20 25 30 35 40

Piezometricline(m)

Time (second)

CFL = 1CFL = 2CFL = 5CFL = 10

-2

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35 40

Discharge(m3/s)

Time (second)

CFL = 1CFL = 2CFL = 5CFL = 10

Fig. 6. Numerical behavior for a fine space mesh and different CFLnumbers. Piezometric line (top) and discharge (bottom) at the middle ofthe pipe.



12/13

(1) inertia forces due to the same way of flow shut down,(2) pressure wave propagation time.

Condition (1) leads to the following formula for theequivalent cross-sectional area Seq of the pipe segment:

Seq LRL0

dxSx

:

150

200

250

300

350

400

450

500

550

0 5 10 15 20 25 30 35 40

Piezometricline(m)

Time (second)

CFL = 1CFL = 2CFL = 5

-2

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35 40

Discharge(m3/s)

Time (second)

CFL = 1CFL = 2CFL = 5

Fig. 7. Numerical behavior for a coarse space mesh and different CFLnumbers. Piezometric line (top) and discharge (bottom) at the middle of

the pipe.

Table 1Highest piezometric level and relative error (in bold)

CFL 1 2 5 10

Nx1000 688 685 680 6730.04% 0.4% 1.2% 2.2%

Nx150 677 669 655 6381.7% 2.8% 4.9% 7.3%

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3 3.5 4

H/Heq

D1/D2

Finite volume methodEquivalent pipe

Fig. 8. Comparison in the prediction of pressure rises in cone-shaped

pipes between the present method and the equivalent pipe method.

0

100

200

300

400

500

600

700

800

0 10 20 30 40 50 60

Piezometricline(m)

Time (second)

RigidDeformable

-6

-4

-2

0

2

4

6

8

10

12

0 10 20 30 40 50 60

Discharge

(m3/s)

Time (second)

RigidDeformable

1.5948

1.595

1.5952

1.5954

1.5956

1.5958

1.596

1.5962

1.5964

1.5966

1.5968

0 10 20 30 40 50 60

Diameter(m)

Time (second)

RigidDeformable

Fig. 9. Influence of the pipe deformation. Piezometric line (top), discharge

(middle) and diameter (bottom) at the middle of the pipe.



13/13

The equivalent pressure wave speedaeq can be found fromthe condition (2):

aeq LRL0

dxax

:

We thus perform tests of flows shut down at downstream

end of a cone-shaped pipe of length L= 1000 m, the up-stream diameter D1 being always equal to 1 m whereasthe downstream diameter D2 varies. We use 100 meshpoints and the CFL number is chosen equal to 0.8. Beforethe shut down, the downstream discharge is 1 m3/s.Fig. 8shows the ratio DH=DHeqversus the ratioD1=D2. The mod-el and the proposed finite volume discretisation shows avery good agreement with the equivalent pipe theory.

4.4. Effect of the pipe deformation

To see the effect of the deformation of the pipe on the

values of the piezometric line and the discharge, we per-form the same water hammer test where we take intoaccount the deformation of the pipe. Let us recall that fromEq. (12) the sonic speed is in this case non-constant anddepends on Youngs modulus, the thickness of the walland the diameter of the pipe. We present in theFig. 9thepiezometric line, the discharge and the diameter of the pipeat the middle of the pipe computed by the second-orderscheme with CFL1 and 1000 mesh points.

One can remark that in the deformable case, the pres-sure is less high since the pipe absorbs a great part of thestress and the frequency of the smaller (since the sonicspeed is smaller in the deformable case). These results must

be viewed as an illustration of our model.

Acknowledgements

The authors thank EDF-CIH (France) for helpfulldiscussions and supplying us the numerical results of thebeliercode. They also thank the referees for their appro-priate advises in the presentation of this article.

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conservative model unsteady flows deformable closed pipes

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