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water equations for natural river hydraulics.
Astrid Decoene, Luca Bonaventura, Edie Miglio, Fausto Saleri
To cite this version:
Astrid Decoene, Luca Bonaventura, Edie Miglio, Fausto Saleri. Asymptotic derivation of the section- averaged shallow water equations for natural river hydraulics.. 2008. �hal-00275460v2�
water equations for natural river hydraulis
Astrid Deoene
∗
, Lua Bonaventura
†
, Edie Miglio
‡
,Fausto Saleri
MOX Modelling and Sienti Computing
Dipartimentodi Matematia F. Brioshi,Politenio di Milano
Via Bonardi 9, 20133 Milano, Italy
Abstrat
The setion-averagedshallow water model usually applied in river and open hannel
hydraulisisderivedasymptotiallyuptoseondorder inthevertial/longitudinallength
ratio,starting from the three-dimensionalReynolds-averagedNavier-Stokesequations for
inompressible free surfae ows. The derivation is arried out under quite general as-
sumptionsonthegeometry ofthehannel, thusallowingfortheappliationof theresult-
ingequations to naturalriverswith arbitrarily shaped ross setions. As a resultof the
derivation, ageneralizedfrition termis obtained, that does notrelyon loal uniformity
assumptionsandthatanbeomputeddiretlyfromthree-dimensionalturbulenemodels,
withoutneedforloaluniformityassumptions. Themodiedequationsinludingthenovel
frition term are ompared to the lassialSaint Venant equations in the ase of steady
stateopenhannelows,whereanalytisolutionsareavailable,showingthatthesolutions
resultingfromthemodiedequationsetaremuhlosertothethree-dimensionalsolutions
thanthoseofthelassialequationset. Furthermore,itisshownthattheproposedformu-
lationyieldsresultsthatareverysimilartothoseobtainedwithempirialfritionlosures
widelyappliedinomputationalhydraulis. Thegeneralizedfritiontermderivedtherefore
justiesaposteriori these empiriallosures, while allowingto avoidthe assumptions on
loalowuniformityonwhihtheselosuresrely.
Keywords: Computational hydraulis; shallow water equations;Saint Venant equations.
1 Introdution
In environmental modelling of freesurfae ows, wheneverthe ratio between the vertial and
longitudinalsalesissmallenough,theso-alled Shallow Water approximationisusuallyintro-
dued, in order to redue the omputational ost implied by the numerial solution of three-
dimensional free surfae ow equations. Models based on this approximation are extensively
used to simulate various geophysial phenomena,suh as rivers and oastal ows,[6 ,8℄ oeans
and even avalanhes,[1℄ and they have been used in hydraulis for a very long time. When
the visosity is negleted and a retangular hannel setion is assumed, the derivation of the
∗
astrid.deoenemate.polimi.it
†
lua.bonaventurapolimi.it
‡
edie.migliomate.polimi.it
one-dimensionalShallow Watersystemis lassial,seee.g.[13 ℄. However, thisderivation isun-
satisfatory, sine visosity eets areadded a posteriori and thethree-dimensional geometry
isnot arbitrary.
In Ref.[7℄, Gerbeau and Perthame derive rigourously, by asymptoti analysis, a one -
dimensional visous Saint-Venant system from the two-dimensional Navier-Stokes equations
withmoleularvisosity,linearfritionboundary onditionsand atbathymetry. Theeet of
thevisosityisreovered inaone-dimensionalfrition termand inaone-dimensional diusion
term, both resulting from the derivation. The nal system is a seond order approximation
with respet to the ratio between the vertial and longitudinal sales of the original two-
dimensionalmodel. Othersystemshavebeenderivedinthesamespirit. InRef.[12℄,theasymp-
totianalysisismadethroughavariablehangeinareferenedomain,independentoftheratio
parameter and time. Marhe proposes inRef.[10℄ the derivation of a two-dimensional visous
shallowwatersystemtakingintoaountapillaryeets, varyingbathymetryandamoleular
visosity. However, inorder to simulate realisti river ows,three-dimensional geometries and
turbulenephenomenamustbetakenintoaount. Thus,theReynolds-averagedNavier-Stokes
equations (RANS) on an arbitrary three-dimensional domain are a more appropriate starting
pointforthe derivation ofsimpliedsystems. InRef.[5℄,Salerietal.derivedatwo-dimensional
visousshallowwatersystemfromthethree-dimensional RANSequations,takinginto aount
a non-at bathymetry, atmospheri pressure eets and onsidering a onstant vertial eddy
visosityandlinear fritionboundaryonditions.
In this paper, we have hosento proeedas inRef.[7℄, extendingthe analysisto thethree-
dimensionalRANSequationswithanisotropiReynoldstensorforfreesurfaeowsinarbitrary
geometry,withnonlinearfritionboundaryonditionsentirely analogoustothoseatuallyused
inpratieinhydraulisappliations. Wepresentarigourousderivationofthesetion-averaged
system, inluding the eets of eddy visosity and frition. This derivation is also aimed at
providing an adequate framework for the rigorous derivation of oupling between three- and
one-dimensionalfreesurfae models. Theequationsystemobtained allowsto omputethefree
surfae level of the ow as well as a setion-averaged veloity. If applied to hannels with
retangular ross-setion, this systemis similar to thelassial setion-averaged shallow water
equations [11℄,exeptforthefrition term. Indeed,our derivation showsthat, inorderto take
into aount eets up to theseond order inthe asymptoti parameter, the lassial frition
term should be orreted by a term whih depends on the turbulent vertial visosity. This
onlusion is in agreement withthe one of Gerbeau etal. in Ref.[7℄ for two-dimensional ows
with onstant visosity over a at bathymetry. Indeed, if the vertial visosity and frition
oeients aretaken to beonstant and the ow is homogeneousin thetransversal diretion,
we retrieve the same frition orretion as in Ref.[7℄. However, our derivation provides the
expression of the frition orretion term in a more general ase, whih inludes turbulent
ows,nonlinearfrition boundary onditions and three-dimensional arbitrarygeometry.
In partiular, we ompute the orretion term assoiated to a spei model for the verti-
al prole of turbulent veloity. Furthermore, for steady state open hannel ows admitting
analyti solutions of the three-dimensional aswell asthe simplied models, we show that the
solutions omputed inludingour orretion term aremuhloser tothose of thethree dimen-
sional model than those of the standard shallow water model. If empirial frition losures
areintrodued,asommonlydone inomputationalhydraulis(seee.g.[2 ℄), oneobtains results
very similar to those of our generalized frition term for steady state open hannel solutions.
Thus, the generalized frition term resulting from the present derivation justies a posteriori
these empirial losures, while allowing to avoid the assumptions on loal ow uniformity on
whih these losures rely. The frition orretion term an be easily inluded in setion aver-
aged modelssuh asthe one proposed byDeponti etal. inRef.[3℄, [4℄. Its useisalso expeted
to easethe oupling of three- and one-dimensional freesurfae models in theframework of an
integrated hydrologial basin model.
Insetion2ofthispaperwereviewthethree-dimensionalRANSequationsandtheirbound-
ary onditions. Then, we derive the setion-averaged shallow water modelin setion 3 and in
setion 4 we give the expression of the frition orretion term in the laminar and turbulent
ases. Finally, in setion 5, we ompare the analytial solutions of thethree-dimensional and
the setion-averaged models in the partiular ase of steady state open hannel ows with
retangular ross-setion, in order to show the auray gain ahieved by adding the frition
orretion.
2 The Reynolds-averaged Navier-Stokes equations
2.1 The three-dimensional equations with boundary onditions
We onsider the motion of an inompressible uid with onstant density ρ >0, in a three-
dimensional domain Ωt = Ω(t) whih is normal with respet to the vertial diretion z. We
denote by ω the projetion of Ωt on thexy-plane,dened asfollows:
ω(t) =
(x, y)∈R2/0≤x≤L, l1(x, t)≤y≤l2(x, t) ,
where l1 and l2 are the time and spae dependent transversal limits of the ow, and L its
length. We assumethebottom of thedomain to be xedand impervious. We allη andb the
funtionsdesribingthefreesurfaeandthebottom,whihisassumedtobexedintime. The
three-dimensional domainis dened by
Ωt=
(x, y, z)∈R3/(x, y)∈ω(t), b(x, y)≤z≤η(x, y, t) ,
as illustrated in Figure 1. The boundary of the domain Ωt is denoted by ∂Ωt and an be
deomposed into four separate parts: the freesurfae Γs(t),the bottom surfae Γb,the inow
boundary Γin(t) andtheoutow boundary Γout(t).
Figure 1: Three-dimensional domain
The governing equations for the motion of the uid are the inompressible Reynolds-
AveragedNavier-Stokes(RANS)equations inΩt,valid foranyt∈(0, T],whihan be written
asfollows:
dU
dt −∇· 1
ρσT
= f +g,
∇·U = 0,
(1)
where U = (u, v, w)T is the veloity of the uid, σT is the stress tensor, f = (fx, fy, fz)T
the sum of the external fores applied on the uid,
d
dt denotes the total time derivative and g = (0,0,−g)T the gravity aeleration. We only onsider Newtonian uids, for whih the tensor σT is written inthe following way:
σT = −pI + σ, (2)
where p is the pressure and σ the visous stress tensor. Sine we are onsidering ows in
preseneof gravity, thatis aligned withthevertial diretion, we onsidera turbulene model
giventhroughananisotropirelationshipbetweenthestresstensorσ andthestrain-ratetensor
D=∇U + (∇U)T.
Following Levermore andSammartino inRef. [9℄,we take:
σ =
σ11 µhD12 µvD13 µhD21 σ22 µvD23 µvD31 µvD32 µeD33
, (3)
where
σ11=µh(D11−1
2(D11+D22)) +µe1
2(D11+D22),
and
σ22=µh(D22−1
2(D11+D22)) +µe1
2(D11+D22).
Thepositiveoeientsµh,µv andµe aretheeddyvisosities. Theyan beinterpreted asthe eddyvisosityrelativetothe horizontalshearmotion,theeddyvisosityrelativetothevertial
shear motion, andthe bulkvisosityrelative to the expansionrate inthehorizontal diretion,
respetively.
The system is losed by suitable initial and boundary onditions. We denote by ns the
outward normalto the freesurfae, whihdependson time:
ns= 1
p1 +|∇η|2 (−∂η
∂x,−∂η
∂y,1)T,
and bynb theoutward normalto thebottom:
nb = 1
p1 +|∇b|2(∂b
∂x,∂b
∂y,−1)T,
whilevetors
tb,1
and
tb,2
forma basisfor the tangent planeto thebottom surfae:
tb,1= 1 s
1 +
∂b
∂x
2 (1,0,∂b
∂x)T,
and
tb,2= 1 s
1 +
∂b
∂y
2(0,1,∂b
∂y)T.
In a visous ow, the veloity is zero on a solidwall, sothat the so-alled no-slip ondition
should beapplied on thebottom:
U = 0 on Γb. (4)
However, the boundary layer at the bottom is hardly ever resolved at typial resolutions of
environmentalmodels. Furthermore,itisneessarytodesribeinsomeapproximatefashionthe
subgrid sale surfae roughness. Thus, ondition (4)is generallysubstituted by two boundary
onditions assignedat a small distane∆zr from thewall, whihrepresents thetypial length
sale of the bottom boundary layer. In addition, the veloity is onsidered zero at a distane
∆z0 ofthe wall,whihrepresentsthe typiallength saleofthebottomsurfae roughness,and
should bemuh smallerthan ∆zr:
∆z0 <<∆zr. (5)
The rstboundary onditionis akinemati ondition:
U ·nb = 0 at z=b(x, y) + ∆zr, (6)
and theseond oneis adynami onditionwhih aounts forfrition eets:
(1
ρσT ·nb)T ·tb = −α||U||U·tb at z=b(x, y) + ∆zr, (7)
where α > 0 is a dimensionless frition oeient. Note that nb and tb are, respetively, the outward normalanda tangent vetorto thebottom surfae,sothatondition (7)isindeedan
assumption on the prole of the tangential veloity omponent along the diretion normal to
the bottom surfae. A logarithmi wall lawis usually assumedfor tangential veloity nearthe
bottom,seee.g. [11 ℄,sothataparaboli modelishosenfor thevertial eddyvisosity,aswell
as a partiular value of the frition oeient α, depending on the value of ∆zr,that will be
desribedingreater detaillater. Inthefollowing we will denotezr(x, y) =b(x, y) + ∆zr.
Atthefreesurfae,theveloityoftheuidisequal totheveloityof thefreesurfae itself.
This isexpressedbythe following kinemati ondition:
∂η
∂t − U ·ns = 0 on Γs(t). (8)
The dynamialondition atthefree surfaetakesinto aount theatmospheri stress,
1
ρσT·ns = −1
ρpans on Γs(t), (9)
where pa isthe atmospheripressure.
2.2 Adimensionalization of the system
Let us onsider the following absolute sales: L for the total length, H for the depth and U
for the x-omponent of the veloity. We denote by ǫ the ratio between the vertial and the
longitudinalsales:
ǫ= H L.
Inaddition we introdue the following dimensionlessquantities:
νh = µh
ρ U L, νv = µv
ρ U L, νe= µe
ρ U L, G= H
U2g, pa= pa ρ U2.
ThesalefortimeisL/U,forthevertialveloityitisW =ǫ U,andforthepressureP =ρ U2.
For the sake of simpliity we indiate again by u, v, w, p, η and b, respetively, veloity omponents, pressure,freesurfae andbottomelevation, afterresaling. Usingthese notations
in(1) we obtain the following adimensionalized system, written asa funtion of the primitive
unknowns u,v,wand p:
∂u
∂t +∂u2
∂x +∂uv
∂y + ∂uw
∂z + ∂p
∂x = ∂
∂x
(νh+νe)∂u
∂x −(νh−νe)∂v
∂y
+ ∂
∂y
νh(∂u
∂y +∂v
∂x)
+ 1 ǫ2
∂
∂z
νv∂u
∂z
+ ∂
∂z
νv∂w
∂x
,
∂v
∂t +∂uv
∂x +∂v2
∂y +∂vw
∂z +∂p
∂y = ∂
∂x
νh(∂u
∂y+ ∂v
∂x)
+ ∂
∂y
(νh+νe)∂v
∂y −(νh−νe)∂u
∂x
+ 1 ǫ2
∂
∂z
νv∂v
∂z
+ ∂
∂z
νv∂w
∂y
,
ǫ2 ∂w
∂t +∂uw
∂x +∂vw
∂y + ∂w2
∂z
+∂p
∂z = −G+ ∂
∂x
νv∂u
∂z +ǫ2νv∂w
∂x
+ ∂
∂y
νv∂v
∂z
+ǫ2 ∂
∂y
νv∂w
∂y
+ ∂
∂z
2νe∂w
∂z
,
∂u
∂x + ∂v
∂y + ∂w
∂z = 0.
(10)
Coherently,the resaled boundary onditions are, onthefreesurfae Γs(t),
∂η
∂t +u∂η
∂x+v∂η
∂y = w,
∂η
∂x
p−(νh+νe)∂u
∂x + (νh−νe)∂v
∂y
−∂η
∂y
νh(∂v
∂x+∂u
∂y
+νv 1
ǫ2
∂u
∂z +∂w
∂x
= pa∂η
∂x,
−∂η
∂x
νh ∂v
∂x +∂u
∂y
+∂η
∂y
p−(νh+νe)∂v
∂y+ (νh−νe)∂u
∂x
+νv 1
ǫ2
∂v
∂z +∂w
∂y
= pa∂η
∂y,
−∂η
∂x
νv ∂u
∂z +ǫ2∂w
∂x
−∂η
∂y
νv ∂v
∂z +ǫ2∂w
∂y
−p + 2νe∂w
∂z = −pa,
(11)
and nearthebottom at z=zr,
u∂b
∂x+v∂b
∂y =w,
∂b
∂x
(νh+νe)∂u
∂x−(νh−νe)∂v
∂y−2νe∂w
∂z
+ ∂b
∂y
νh ∂v
∂x +∂u
∂y
+
∂b
∂x 2
− 1 ǫ2
! νv
∂u
∂z +ǫ2∂w
∂x
+ ∂b
∂x
∂b
∂y
νv ∂v
∂z +ǫ2∂w
∂y
= −α√
u2+v2+ǫ2w2 1
ǫu+ǫ∂b
∂xw
N(b, ǫ),
∂b
∂x
νh ∂v
∂x+∂u
∂y
+ ∂b
∂y
(νh+νe)∂v
∂y −(νh−νe)∂u
∂x−2νe∂w
∂z
+
∂b
∂y 2
− 1 ǫ2
! νv
∂v
∂z +ǫ2∂w
∂y
+ ∂b
∂x
∂b
∂y
νv ∂u
∂z +ǫ2∂w
∂x
= −α√
u2+v2+ǫ2w2 1
ǫv+ǫ∂b
∂yw
N(b, ǫ).
(12)
where
N(b, ǫ) = s
1 +ǫ2(∂b
∂x)2+ǫ2(∂b
∂y)2.
3 Derivation of the setion-averaged shallow water model
3.1 Seond order approximation in ǫ
Inorderto derive oursetion-averaged shallowwater model,a numberofapproximations have
to beperformed. Firstly,we assumethat the vertial eddyvisosityis rst orderwith respet
to theratio between thevertialand longitudinalsales, thatis,
νv = ǫ νv,0, (13)
whereνv,0isagivenpositivequantity. Thisassumptionanbejustiedbyasimpledimensional analysis. Indeed,followingthePrandtlhypothesis,theeddyvisosityishomogeneoustoalength
times aveloity,and more preisely
µ
ρ ∼ l2m||D||, (14)
where lm is the mixing length of the turbulent ow and ||D|| is the norm of the strain-rate tensor. When onsidering the vertial eddy visosity, lm is homogeneous to a depth and the strain-rate tensor reduesto the vertial aeleration, therefore weonlude that
µ ρ ∼ l2m
∂U
∂z
. (15)
NotethatPrandtl'smixinglength modelseefor instaneRef. [11 ℄ isbasedonthis assump-
tion. Adimensionalizing thisexpression ofµv gives:
U Lµˆv ∼ H2lˆm2 s
U2 H2
(∂uˆ
∂z)2+ (∂ˆv
∂z)2+ (ǫ∂wˆ
∂z)2
∼ U H lˆm2
∂uˆ
∂z
, (16)
where the hat denotes herethe adimensional variables. Thus
νv = µˆv
ρU L ∼ ǫ lˆm2 ρ
∂ˆu
∂z
= O(ǫ). (17)
Moreover, the horizontal and bulk visosities are of same order as the vertial eddy visosity,
and thereforewe an write:
νh = ǫ νh,0, νe = ǫ νe,0, (18)
where νh,0 and νe,0 are two given positive quantities. Finally, we assume a slow varying bathymetry in the longitudinal diretion, as it has been done often inthese derivations see
for instaneRef. [5℄ ,and we onsideraonstant atmospheripressure, that is
∂b
∂x =O(ǫ) and ∇pa= 0. (19)
Sine our aim is to obtain a seond order approximation with respet to ǫ of the three-
dimensional system, we neglet quantities of order O(ǫ2). In this way, under the previous
assumptions, (10) beomes:
∂u
∂t +∂u2
∂x +∂uv
∂y +∂uw
∂z + ∂p
∂x = ǫ ∂
∂x
(νh,0+νe,0)∂u
∂x−(νh,0−νe,0)∂v
∂y
+ ǫ ∂
∂y
νh,0(∂u
∂y + ∂v
∂x)
+1 ǫ
∂
∂z
νv,0
∂u
∂z
+ǫ ∂
∂z
νv,0
∂w
∂x
,
∂v
∂t +∂uv
∂x + ∂v2
∂y +∂vw
∂z +∂p
∂y = ǫ ∂
∂x
νh,0(∂u
∂y + ∂v
∂x)
+ ǫ ∂
∂y
(νh,0+νe,0)∂v
∂y−(νh,0−νe,0)∂u
∂x
+1 ǫ
∂
∂z
νv,0
∂v
∂z
+ ǫ ∂
∂z
νv,0
∂w
∂y
,
∂p
∂z = −G+ǫ ∂
∂x
νv,0∂u
∂z
+ǫ ∂
∂y
νv,0∂v
∂z
+ǫ ∂
∂z
2νe,0∂w
∂z
,
∂u
∂x + ∂v
∂y + ∂w
∂z = 0.
(20)
together withboundary onditions onthefree surfaeΓs(t),
∂η
∂t +u∂η
∂x +v∂η
∂y = w,
∂η
∂x
p−ǫ(νh,0+νe,0)∂u
∂x +ǫ(νh,0−νe,0)∂v
∂y
−∂η
∂y
ǫ νh,0(∂v
∂x+∂u
∂y)
+1 ǫνv,0
∂u
∂z +ǫ νv,0
∂w
∂x = pa∂η
∂x,
−∂η
∂x
ǫ νh,0
∂v
∂x +∂u
∂y
+∂η
∂y
p−ǫ(νh,0+νe,0)∂v
∂y +ǫ(νh,0−νe,0)∂u
∂x
+1 ǫνv,0
∂v
∂z +ǫ νv,0
∂w
∂y = pa∂η
∂y,
−∂η
∂x(ǫ νv,0
∂u
∂z)−∂η
∂y(ǫ νv,0
∂v
∂z)−p+ 2ǫνe,0
∂w
∂z = −pa,
(21)
and nearthe bottom at z=zr,
u∂b
∂x+v∂b
∂y =w,
∂b
∂x
ǫ(νh,0+νe,0)∂u
∂x−ǫ(νh,0−νe,0)∂v
∂y −2ǫ νe,0
∂w
∂z
+ ∂b
∂y
ǫ νh,0
∂v
∂x+∂u
∂y
− 1 ǫνv,0
∂u
∂z +ǫ νv,0
∂w
∂x
= −α
ǫ ||u||u,
∂b
∂x
ǫνh,0
∂v
∂x +∂u
∂y
+ ∂b
∂y
ǫ(νh,0+νe,0)∂v
∂y−ǫ(νh,0−νe,0)∂u
∂x
− 2ǫ νe,0
∂w
∂z
+
(∂b
∂y)2− 1
ǫ2 ǫ νv,0
∂v
∂z
= −α||u||
1
ǫv+ǫ∂b
∂yw
,
(22)
where u = (u, v) is thehorizontal veloity. Notiethat, negleting termsinO(ǫ) in (22)2 and
(22)3,we obtain thelassialboundaryondition onthebottom (seee.g. Ref. [2 ℄):
νv,0
∂u
∂z = −α||u||u .
3.2 Vertial integration of the equations
We willnowvertially-integrate system(20)between thefreesurfae andthereferenelevelzr
atwhih thebottomonditions aregiven. We willdenotebyh theorretedwaterdepth,that
isthereal water depthorretedbythe distane∆zr: h(x, y, t) = η(x, y, t)−zr(x, y)
= η(x, y, t)−b(x, y)−∆zr. (23)
Then, for any three-dimensional variable f, we denote with f¯the average along the vertial
diretion,
f¯(x, y, t) = 1 h(x, y, t)
Z η zr
f(x, y, z, t)dz.
Letus rst vertially-integrate the momentum equation(20)1 for ubetween thereferenebot-
tomlevel zr and thefreesurfae. Making useof the Leibnitzrule yields:
