Direct sensitivity computation for the Saint-Venant equations with hydraulic jumps

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Direct sensitivity computation for the Saint-Venant

equations with hydraulic jumps

Carole Delenne, Vincent Guinot, Bernard Cappelaere

To cite this version:

Carole Delenne, Vincent Guinot, Bernard Cappelaere. Direct sensitivity computation for the

Saint-Venant equations with hydraulic jumps. Comptes Rendus Mécanique, Elsevier Masson, 2008, 336

(10), pp.766-771. �10.1016/j.crme.2008.09.006�. �hal-01196906�

with hydraulic jumps. CaroleDELENNE

a

,Vincent GUINOT

a

,Bernard CAPPELAERE

a

a

HydroSciencesUMR5569(CNRS,IRD,UM1,UM2),AvenueJeanbrau 34090Montpellier

Abstract

ThispaperpresentsanewRiemannsolvertosolvetheSaint-Venantequationsinconjunctionwiththesensitivity problemwhenthesolutions arediscontinuous.Thesolveris basedontheaprioriassumptionoftworarefaction waves. The presence of shocks is detected a posteriori and an extra sensitivity term in the form of a Dirac sourcetermis accountedfor inthe sensitivity balanceequations.Tocite this article: C.Delenne,V.Guinot,B. Cappelaere,C.R. Mecanique??(2008).

Résumé

CalculdirectdesensibilitépourleséquationsdeSaint-Venantavecressautshydrauliques.Onpropose iciunsolveurdeRiemannpourrésoudreleséquationsdesensibilitéconjointementàlaprojectionsurunedimension deséquationsdeSaint-Venantdanslecasdesolutionsdiscontinues.Lesolveurestbasésurlasuppositionapriori dedeuxondesderaréfaction.La présencedechocsestdétectéeaposteriorietuntermesupplémentaire,sous la formed'untermesourcedeDirac,estintroduitdansl'équilibredeséquationsdesensibilité.Pourcitercetarticle: C.Delenne,V.Guinot,B.Cappelaere,C.R.Mecanique??(2008).

Keywords: Computationaluidmechanics;Sensitivity;Hyperbolicconservationlaws;shocks Mots-clés:Mécaniquedesuidesnumérique;Sensibilités;Loisdeconservationhyperboliques;Chocs

Versionfrançaiseabrégée

Onchercheàrésoudreleséquationsdesensibilitéconjointementàlaprojectionsurunedimensiondes équations deSaint Venantdans le cas desolutionsdiscontinues.Dans ce cas, en eet,les équations en sensibilité ne peuvent pas être obtenues par une simple dérivation des équations hydrodynamiques de base etunterme sourcedeDiracapparaîtauniveaudeschocs.On proposeiciune méthodenumérique

Emailaddresses:delenne@msem.univ-montp2. fr (CaroleDELENNE),guinot@msem.univ-montp2.f r(Vincent GUINOT),cappelaere@msem.univ-mont p2. fr (BernardCAPPELAERE).

deux cellules

i

et

j

(issues d'une discrétisation de l'espace) avec des états gauche et droit dénis par ces deux cellules respectivement; (ii) déterminerles valeurs dela variable d'écoulementet/ou des ux dansles régionsd'étatsconstantsen utilisant lesinvariantsdeRiemann, (iii) déterminerlesvitesses de propagationdesdiérentesondeset lalocalisationdediscontinuitéparrapportàcesondes;(iv)calculer les ux nécessaires pour équilibrer l'équation entre les cellules

i

et

j

grâce à la valeur de la variable d'écoulement auniveau de ladiscontinuité initiale.La solutionest alors calculéepour diérentspas de tempsenutilisantunschémadediscrétisationexplicite.

The Riemann problem of the Saint-Venant and sensitivity equations is rst recalled and then the applicationoftheproposed solverisdetailed.

1. The Riemannproblem

Theone-dimensionalprojectionoftheSaint-Venantequationsisa

3 × 3

HyperbolicSystemof Conser-vationLaws(HSCL), whichcanbewritten invectorformas:

∂U

∂t

+

∂F(U, φ)

∂x

=

0

U

_{(x, 0)}

_{= U}

₀

_{(x, φ)}

U

_(x

_b

_{, t)}

_{= U}

_b

_{(t, φ)}



















with

U

=











h

q

r











, F =











q

q

2

/h + gh

2

/2

qr/h











(1)

with

U

: theconservedvariable;

F

:theux function;

φ

:aparameteron which theux depends;

g

: the gravitationalacceleration;

h

: the water depth;

q

(resp.

r

) : the unit discharge in the

x

(resp.

y

) direction;

u = q/h

(resp.

v = r/h

):theowvelocityinthe

x

(resp.

y

)direction.Thesuperscripts

b

and

0

indicate thedomainboundaryabscissaandtheinitialconditionrespectively.

Dierentiatingthegoverningequation1withrespecttotheparameter

φ

leadstotheequationforthe sensitivity

s

= (η, θ, ρ)

of

U

to

φ

:

∂s

∂t

+

∂

∂x

(As) = −

∂

∂x

 ∂F

∂φ



(2) with

A

= ∂F/∂U

.

However,thisderivationiscorrectonlyundertheassumptionofacontinuousanddierentiablesolution

U

[7].Inthepresenceofadiscontinuity,theso-calledRankin-Hugoniotconditions(orjumprelationships) mustbeused:

F

L

_{− F}

R

= (U

L

_{− U}

R

)c

s

where

c

s

isthediscontinuityspeed,andwhere theindex LandRdenotetheleftandrightstatesacross thediscontinuity.Moreover,since

U

and

s

areindependentvariables, thesensitivities

s

L

and

s

R

onthe leftandright-handsideofthediscontinuityareindependentfrom

U

L

and

U

R

.Consequently,thejump relationshipforthesensitivityismuchmorecomplexthanthatfortheowvariable([1],[6])andimplies aspecicsourceterm

R

,inaformofaDiracfunction,which takeseectonlyatthediscontinuity:

H

L

_{− H}

R

+ R = (s

L

_{− s}

R

)c

_s

with

s

theconservedvariableand

H

= As

theuxfunction forsensitivity.Then,

R

_{= ∆ [(A − c}

_s

I

_{) s]}

(3) Thesource term

R

is non zeroonlywhen there isashock.Indeed, whenthe discontinuityis acontact one,

c

s

isaneigenvalueofthematrix

A

and

(A

R

,L

− c

s

I

) = 0

.

Because this Jacobianmatrix

A

depends onthe solution

U

, the Riemann problem of thesensitivity cannot be considered independently from the one of the ow variable. We thus consider the following initial-valueproblem:

∂U

∂t

+

∂F

∂x

=

0

∂s

∂t

+

∂H

∂x

=

∆ [(A − c

s

I

) s]

(U, s)(x, 0) =

(

(U

L

, s

L

)

for

x < 0

(U

R

, s

R

)

for

x > 0































(4)

whichisa

3 × 3

HSCLwiththefollowingeigenvalues:



λ

(1)

_{, λ}

(2)

_{, λ}

(3)



_{= (u − c, u, u + c)}

(5) ThegeneralsolutionoftheRiemannproblemismadeofthreewavesseparatingtwointernalregionsof constantstate.IntheexactsolutionoftheRiemannproblem,thesecondwave(withcelerity

λ

(2)

_{= u}

)is acontactdiscontinuity,while therstand thirdwavesmaybeofanytype,dependingon

U

L

and

U

R

. Althoughnotstrictlyvalidacrossashock,theRiemanninvariantsmaybeusedtoapproximatethejump relationships(e.g. [8],[10],[4], [3], [9]).Then, thenature ofthe wavesin theRiemann problemmaybe guessed a priori without a posteriori verication and the resulting systemof algebraicequations may besolvedtodeterminedirectlythesolutionintheintermediateregionsofconstantstate.Theproposed approximate-stateRiemannsolverusestheassumptionthattheeigenvalues(5)aretheceleritiesofthree rarefactionwaves.Inwhat follows,weassessthesensitivityofthesolutiontotheinitialvalueoftheleft orrightstateof

U

.

2. The Approximate-state Riemannsolver

2.1. Discretization

Thesolutionwillbeadvancedintimeusingthefollowingdiscretizationof (4):

U

n+1

i

= U

n

i

+

∆t

∆x

i



F

n+

1

/

2

i−

1

/

2

− F

n+

1

/

2

i+

1

/

2



(6)

s

n+1

i

= s

n

i

+

∆t

∆x

i



H

n+

1

/

2

i−

1

/

2

− H

n+

1

/

2

i+

1

/

2

+ R

n+

1

/

2

i−

1

/

2

,i

+ R

n+

1

/

2

i+

1

/

2

,i



(7) where

U

n

i

and

s

n

i

are the average values of

U

and

s

overthe cell

i

at the time

n

;

F

n+

1

/

2

i−

1

/

2

and

H

n+

1

/

2

i−

1

/

2

are theaverage valuesofthe uxes

F

and

H

at the interface

i −

1

_/

₂

betweenthe cells

i − 1

and

i

over thetimeinterval

t

n

_{, t}

n+1



;

R

n+

1

/

2

i−

1

/

2

,i

and

R

n+

1

/

2

i+

1

/

2

,i

representthecontributionsofthesourcetermspossibly generatedbyshockspropagatingrespectivelyfromtheinterfaces

i +

1

_/

₂

and

i −

1

_/

₂

intothecell

i

;

∆t

is thecomputationaltimestepand

∆x

i

isthewidthofthecell

i

.

Foreachinterfacebetweentwocells

i

and

j

(issuedfromaspatial discretization),aRiemannproblem is dened with leftand rightstatestakenfrom these cells. Then,the proposed solveruses theclassical HLLapproachto computethesolutionin thecontinuouscaseandaspecictreatmentoftheshocks.

TheRiemanninvariantsfor(4)aredened as[5]:























d(u − 2c) = 0

for

dx

dt

= u − c

dv = 0

for

dx

dt

= u

d(u + 2c) = 0

for

dx

dt

= u + c

(8)

andtherefore,noting

χ

,

ν

,

ω

thesensitivitiesof

c

,

u

,

v

withrespectto theparameter

φ

:























d(ν − 2χ) = 0

for

dx

dt

= u − c

dω = 0

for

dx

dt

= u

d(ν + 2χ) = 0

for

dx

dt

= u + c

(9)

Intheregionsofconstantstate(seegure1),usingtheapproximateexpressions(8)and(9)yields:



























c

∗,1

= c

∗,2

=

1

2

(c

L

+ c

R

) +

1

4

(u

L

− u

R

)

u

∗,1

= u

∗,2

=

1

2

(u

L

+ u

R

) + c

L

− c

R

v

∗,1

= v

L

v

∗,2

= v

R

(10)



























χ

∗,1

= χ

∗,2

=

1

2

(χ

L

+ χ

R

) +

1

4

(ν

L

− ν

R

)

ν

∗,1

= ν

∗,2

=

1

2

(ν

L

+ ν

R

) + χ

L

− χ

R

ω

∗,1

= ω

L

ω

∗,2

= ω

R

(11)

wherethesubscript*,1and*,2denotethevaluesofthevariablesintheintermediateregionsofconstant stateontheleftandright-handsidesofthecontactdiscontinuity,respectively.Theux

H

intheregions ofconstantstateisthendetermineduniquelyfrom(11):

H

_∗,p

₌











θ

∗,p

c

2

∗,p

− u

2

∗,p

η

∗,p

+ 2u

∗,p

θ

∗,p

−u

∗,p

ν

∗,p

η

∗,p

+ ν

∗,p

θ

∗,p

+ u

∗,p

ρ

∗,p











p = 1, 2

Usingthefactthat

h = c

2

_/g

,

q = hu

and

r = hv

,yieldsthesensitivity

s

∗,p

:











η

θ

ρ











∗,p

=











2cχ/g

ηu + hν

ηv + hω











∗,p

t

x

U

L

U

R

U

*, 1

U

*, 2

dx

dt

=

u c

-dx

dt

=

u c

+

dx

dt

=

u

Fig.1.SchemeofthedierentregionsfortheSaint-Venantequationssolution

2.3. Shocktreatment

Ashockappearsontherstorthirdwave(andconsequentlyasourceterm

R

)wheneither

u

L

− c

L

>

u

R

− c

R

or

u

L

+ c

L

> u

R

+ c

R

. The shock speedis estimated using oneof the followingrelationships (withthesuperscript

L

and

1

or

R

and

2

iftheshockisontherstorthirdwaverespectively):

c

s

=

q

{L,R}

− q

∗,{1,2}

h

{L,R}

− h

∗,{1,2}

c

s

=

q

2

_{/h + gh}

2

_/2

{L,R}

− q

2

_{/h + gh}

2

_/2

∗,{1,2}

q

{L,R}

− q

∗,{1,2}

or

u

{L,R}

+ u

∗,{1,2}

/2

iftheprevioustwodenominatorsarenull.

If theshockspeed in negative, thecorresponding wavetravels from theinterface

i −

1

_/

₂

intothe cell

i − 1

anddoesnotyieldanysourceterminthecell

i

.Incontrast,ifthespeedispositive,thewavetravels intothecell

i

andcontributesto thesourceterm

R

n+

1

/

2

i−

1

/

2

,i

byaquantitygivenbyequation(3).

2.4. Fluxesattheinterface

Attheend oftheprocess, theuxescan becomputedattheinterfaceusing:

F

n+

1

/

2

i−

1

/

2

= F



U

n+

1

/

2

i−

1

/

2



A

n+

1

/

2

i−

1

/

2

= A



U

n+

1

/

2

i−

1

/

2



H

n+

1

/

2

i−

1

/

2

= A

n+

1

/

2

i−

1

/

2

s

n+

1

/

2

i−

1

/

2

Thewholeprocessisthenrepeatedin timeusing6and7.

2.5. Example

Thegure2showsthecalculationofthewaterheightandthesensitivitytotheinitialleftwaterheight value

h

L

,in thecaseofthedam-breakproblem.

h (m)

0

2

-L /2

x

+L /2

h (-)

0

1

analytical

numerical

-L /2

x

+L /2

Fig.2.Waterheight

h

andsensitivity

η

totheinitialleftwaterheightvalue

h

L

Conclusion

TheproposedapproximatestateRiemannsolverhasbeenappliedhereto solvetheshallowwaterand sensitivity equationswhen shocksare presentin thesolution. Itwill begeneralizedto otherHSCLand othercasestudyinforthcomingpublications.

Références

[1] Bardos,C.,Pironneau,O.,2002,Aformalismforthedierentiationofconservationlaws.C.RAcad.Sci.Paris,Ser.I 335,p.839845.

[2] Dukowicz,J.K.,1985,Ageneral,non-iterativeRiemannsolverforGodunov'smethod.JournalofComputationalPhysics 61,119-137.

[3] Guinot, V., 2000, Riemann solvers for water hammer simulationsby Godunov method. International Journal for NumericalMethodsinEngineering49,851-870.

[4] Guinot,V.,Godunov-typeschemes.Anintroductionforengineers.Elsevier.

[5] Guinot, V., 2006, Ondes en mécanique des uides. Modélisation et simulation numérique. Hermes Publishing (in French).

[6] Guinot, V.,Leménager, M.,Cappelaere, B., 2007, Sensitivity equations for hyperbolic conservation law-based ow models.AdvancesinWaterResources,30:1943-1961.

[7] Gunzburger,M.D.,1999,Sensitivities,adjointsandowoptimization,International JournalforNumericalMethods inFluids,31:53-78.

[8] Lax, PD.,1957, Hyperbolic systemsof conservation laws. Communications in Pure and Applied Mathematics,10, 537-566.

[9] Lhomme,J.,Guinot,V.,inpress,Ageneral,approximate-stateRiemannsolverforhyperbolicsystemsofconservation lawswithsourceterms.InternationalJournalforNumericalMethodsinFluids.