the numerical treatment of wet/dry fronts in shallow flows...

ELSEVIER

Available online at www.sciencedirect.com MATHEMATICAL AND

SO,SNOS ~ _ _ D , ~ O T " COMPUTER MODELLING

Mathematical and Computer Modelling 42 (2005) 419-439 www.elsevier.com/locate/mcm

The Numerical Treatment of W e t / D r y Fronts in Shallow Flows:

Application to One-Layer and Two-Layer Systems

M . J . C A S T R O D e p a r t a m e n t o de An£1isis M a t e m £ t i c o Unive r s idad de M£laga, 29080, M£laga

A. M. FERREIRO FERREIRO AND J. A. GARCfA-RODRfGUEZ D e p a r t a m e n t o de M a t e m £ t i c a Apl icada , Un ive r s idad de San t i ago de C o m p o s t e l a

15706, San t i ago de C o m p o s t e l a

J. M. GONZJ~LEZ-VIDA, J. MACiAS AND C. PARt~S D e p a r t a m e n t o de An£1isis M a t e m £ t i c o Un ive r s idad de M£laga, 29080, M£laga

M . E L E N A V J ~ Z Q U E Z - C E N D S N D e p a r t a m e n t o de M a t e m £ t i c a Apl icada , Un ive r s idad de San t i ago de C o m p o s t e l a

15706, San t iago de C o m p o s t e l a

(Received and accepted January 2004)

Abstract--This paper deals with the numerical difficulties related to the appearance of the so- called wet /dry fronts tha t may occur during the simulation of free-surface waves in shallow fluids and internal waves in stratified fluids composed by two shallow layers of immiscible liquids. In the one-layer case, the fluid is supposed to be governed by the shallow water equations. In the case of two layers, the system to solve is formulated under the form of two-coupled system of shallow water equations. In bo th cases, the discretization of the equations are performed by means of the Q-schemes of Roe upwinding the source term developed in [1,2] for the one-layer system, and [3] for the two-layer system. These schemes satisfy the so-called C-property: they solve exactly steady solutions corresponding to water at rest.

In order to handle properly with wet /dry fronts, a numerical scheme has to verify also an extended C-property: it has to solve exactly steady solutions corresponding to water at rest including wet /dry transitions. In [4-6], some numerical schemes satisfying this property has been obtained. In this paper, we propose an improvement of these schemes and its extension to two-layer systems.

We present some numerical results: for one-layer fluids, we compare the numerical results with some measurements corresponding to a physical model. For two-layer systems, we use the numerical scheme to perform a lock-exchange experiment, and we verify its validity by comparing the steady state reached with an approximate solution obtained by Armi and Farmer in [7,8] by using a simplified model. @ 2005 Elsevier Ltd. All rights reserved.

K e y w o r d s - - Q - s c h e m e s , Coupled conservation laws, Source terms, 1D shallow water equations, Two-layer flows, Hyperbolic systems, Wet /dry front, Lock-exchange experiment.

This research has been partially supported by the C.I.C.Y.T. (Project REN2003-07530-C02-02).

0895-7177/05/$ - see front mat ter @ 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2004.01.016

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420 M.J. CASTRO et al.

1. I N T R O D U C T I O N

This paper is concerned with the simulation of free-surface waves in shallow fluids and internal waves in stratified fluids composed by two shallow layers of immiscible liquids. More precisely, we deal with the numerical difficulties related to the appearance of the so-called wet/dry fronts.

In the case of a single layer, these fronts develop when, due to the initial conditions or as a

consequence of the fluid motion, the thickness of the layer vanishes in a zone of the domain. These situations arise very frequently in practical applications as flood waves, dam-breaks, breaking of waves on beaches, ere . . . .

A similar difficulty appears in two-layer fluids when the thickness of at least one of the layers vanishes. In this case, a front develop separating two-layer and one-layer flows or two-layer flows and dry bed. By analogy, these situations will be also called we t /d ry fronts. These fronts appear also in practical applications in coastal simulations or in lock-exchange experiments, in which the two layers are initially separated by a dam that is suddenly broken.

In this article, the motions of the fluids are supposed to be governed by the shallow water system of partial differential equations. Moreover, viscous stresses are neglected, so that the systems to solve are of hyperbolic nature. For the sake of simplicity, we only consider the case

of one-dimensional flows along open channels with rectangular cross-sections of constant breadth and variable bottom. Nevertheless, the numerical schemes proposed here can be easily extended

to the more general situation of ld flows in channels with nonrectangular cross sections or 2d flows (see [1,9,10]).

In the case of a single layer, the system is formulated under the form of a conservation law with

source terms and its discretization is performed by means of the numerical scheme presented by Bermtldez and Vgzquez-Cend6n [1,11,12], consisting in a Q-scheme upwinding the source terms. In [4-6], the simulation of wet /dry fronts with this scheme was considered. The authors of these works proposed some modifications of this scheme that preserves the so-called C-property of conservation, that is, its ability to exactly solving steady-state solutions corresponding to water

at rest. In this article, we propose an improvement of this modified scheme.

In the case of two layers, the system is formulated under the form of two-coupled systems of

conservations laws with source term that is diseretized by using the numerical scheme presented in [3], which is in fact an extension of the scheme used in the one-layer case. We extend to this

case, the improved wet /d ry treatment proposed for one-layer fluids.

The outline of this paper is as follows: the first and second sections are devoted, respectively,

to one-layer and two-layer flows. In both cases, we recall the equations, the numerical scheme in the wet bed case, and we introduce the wet /dry fronts treatment. We verify that the modified schemes satisfies an extended C-property.

In the third section, we present first some numerical experiments corresponding to one-layer flows: we compare the numerical results with some measurements corresponding to a physical model in order to test the wet /dry treatment. Next, we apply the two-layer model to perform a lock-exchange experiment: the layers are initially separated by a dam that is suddenly broken. The model is run until a steady state is reached representing a secular exchange flow trough the channel. This steady state is compared with an approximate solution obtained by Armi and Farmer in [7,8] by using a simplified model.

Conclusions are drawn in Section 4.

2. O N E - L A Y E R F L O W S

2.1. E q u a t i o n s

The one-dimensional shallow water system considered here represents mass and momentum conservation for flows through a rectangular channel with constant breadth. It is obtained by

The Numerical Treatment 421

reference level

Figure 1. One-layer sketch.

taking cross-sectional averages in the incompressible Euler equations. We formulate the system

under the form of a conservation law with source terms as follows:

in which,

OW x t OF )+ s(.,w), (2.1)

W ( x ' t ) = [h(x ' t ) ' (2.2)

where the coordinates x,t refer, respectively, to the axis of the channel and the time. We suppose that x C [0, L] and t E [0, T]. h(x,t) and q(x,t) represent, respectively, the water depth (see Figure 1) and the discharge, that are related to the velocity u(x, t) by the formula

q(x, t) = u(x, t)h(x, t).

The flux is given by

F ( W ) = gh2 ,

where g is the gravity.

The source term due to bed slopes (see Figure 1) is written as

(2.3)

E°l S ( x , W ) = db , (2.4) -gh

where b(x) is the bot tom function, i.e., channel bot tom is defined by the equation z = b(x), z being the vertical coordinate.

Vertical viscous effects can be easily included into the model by adding new source terms that parameterize the friction forces due to the wind, wails, bot tom . . . . Nevertheless, these terms do not play any role in the treatment of wet /dry fronts. Therefore, in order to avoid an excess of

notation, we will not include these terms in the presentation of the numerical schemes.

2.2. T h e N u m e r i c a l S c h e m e

In the case where h(x, t) > 0, i.e., when wet /dry fronts do not appear, the system is discretized by using a Q-scheme upwinding the source term as proposed in [1,11,12].

To begin with, we decompose the space domain in computing cells Ii = [x~-1/2,x~+l/2]. Although the numerical scheme is effectively thought and used for irregular meshes, for the

422 M.J . CASTRO et al.

sake of simplicity, we assume that these cells have a constant size Ax and that Xi+l/2 = i A x . xi = (i - 1/2)Ax is thus, the center of the cell Ii. Let At be the time step and t n = nat.

We consider approximations of solution W, which are constant on each finite volume Is. The value of the approximated solution at I~ at the time tn will be represented by

w : = [h?] q? J •

We will also use the following notation for the approximations of the velocity:

q7 U i ~----£.

hi

Xi-11 Xi-1/i X i[ [ X i+1/2 [Xi +1

I I I i

Figure 2. Finite volume.

In the description of the numerical schemes, Jacobian matrix of the system

- u 2 + c 2 2u '

where

C ~- V / ~ .

The eigenvalues are

"~i =U--C,

A2 = u + c .

If h > O, the system is strictly hyperbolic and A(W) diagonalizes as foltows:

0 l I ~ ( W ) - I A ( w ) K ( W ) = A(W) = A2 '

where K(W) is a matrix whose columns are eigenvectors of A(W)

[11] K ( W ) = ~1 ~2 '

We will use also the matrices

0 ] 0 ] sgn(~2) ' (~2) ~ '

A(w)- = K(W)A(W)±K(W) -1, IA(w)I = A ( w ) + - A ( w ) -

Using these notations, the numerical scheme reads as follows:

W/n@ A~ A t ( p / t l / 2 S i _ l / 2 @ pi;1/2Si+l/2" w n + l = " ~XX (Fi-1/2 -- Fi+1/2) + "~X

we will use the following matrices: A is the

(2.5)

(2.6)

(2.7)

(z8)

(2.9)

(2.10)


The numerical fluxes are defined by

, W ?Z _ _ , Fi+Wg~ = [I(F(W~D-FF(W~+I)) -- 7lAi+v~l ( ,+1 w:) (2.11)

Here,

[ n 2 1

is the Jacobian matrix evaluated at an 'intermediate state' between W/n and W ~ 1. of the Q-scheme of Van Leer is

(2.12)

In the case

qn +q~ / h~ +h~+l i+1 ~ V (2.13) ui+l/2 - h~+ 1 + h~ ' ci+1/2 = g 2 '

while the choice corresponding to the Q-scheme of Roe is

n V/~+l'lt/;1 + x/~u~ n i h~ + h~+ i (2.14) Ui+l/2 = ~ i "-~ ~ ' Ci+1/2 = °0 2

As it is well known, this last choice has the following property (see [13]):

.,4.i+1/2 • (W~_ 1 -- W ? ) = F (W~_I) -- F ( W ~ ) . (2.15)

Coming back to the description of scheme (2.10), the numerical source term is given by

[ o ] n 2

Si+1/2 : - - ( C i + l / 2 ) ( b ( x i + l ) - b ( x { ) ) '

and it is upwinded by using the projection matrices

(2.16)

1 Pgl/2 = -~K,+1/2 (Id 4- sgn(A,+l/2)) K -1 i+1/2 ' (2.17)

where Ai+i/2 is the diagonal matrix whose coefficients are the eigenvalues Ai+l/2,j, j = 1, 2 of Ai+z/2, and Ki+l/2 is a matrix whose columns are eigenvectors corresponding to these eigenvalues.

Concerning the stability requirements, we use the following CFL-condition:

max {1~,+1/2,~[ , 1 < l < 2, 1 < i < M} ~-~x - < 7 ' A t

where 0 < 3' < 1. Finally, in order to prevent the numerical viscosity of the Q-schemes from vanishing when any

of the eigenvalues of the matrices IA~+1/21 are zero, Harten regularization is applied [14].

REMAaK 1. If the intermediate state of Roe (2.14) is chosen, the numerical Q-scheme (2.10) could be rewritten as follows

) W ? +1 = W ? -[- ~ 1/2 - F i ; 1 / 2 ,

where ~ 1 / ~ = Y (W?) + ~ 1 / ~ (A~+1/2 (w~;1 - win) - S,+1/2),

(2.19) F+1/2 = F (W~+I) - P+l/2 (Ai+l/2 ( W ~ r l - W/~) - Si+1/2) ,

The equivalence of both schemes is easily proved using the Roe property (2.15) and the matrix identities P~l/2.di+l/2 + = Ai+l/2.


2.3. C o n s e r v a t i o n P r o p e r t y

In [11], a well-balanced condition called conservation property or C-property which prevent the appearance of nonphysical oscillations due to the discretization of the source terms was introduced.

DEFINITION 1. A numerical scheme is said to verify the C-property if it solves exactly the steady- state solutions corresponding to water at rest, i.e.,

q ~ 0, h _= ~ - b, (2.20)

where ~ is a constant such that 77 > max{b(x), x e [0, L]}.

In [9], a theorem providing sufficient conditions to ensure this property was shown for the case of two-layer fluids. This theorem is easily adapted to the one-layer case as follows.

THEOREM 1. Let us consider a numerical scheme of form (2.10) for solving (2.1). Given a steady-state solution of form (2.20), we define

[ h ( x J ] (2.21) W~ = Lq(z,) 3"

If, for each i, the following equalities hold

Ai+ l12 " ([/V/+I- ~2i) = Si+112, (2.23) then the numerical scheme satisfies the C-property.

PROOF. To prove the theorem, we apply the numerical scheme to the initial condition

w ° = w~, v ~

Using (2.22) and the definition of the numerical fluxes (2.11), the first stage of the scheme can be written as follows:

. (wo _ wo_l) + A;+1/2. (W°+l - wo) ) wT=w°- A--7

+A-zx Now, using the matrix identities

and (2.23), we deduce

± P~l/2Ai+1/2 = A~+I/2

W~ z = W°~, Vi.

COROLLARY 1. ~'fa numerical scheme of form (2.10) with numerical sources given by (2.16), is such that the matrices ¢4i+1/2 coincide with those of the Q-scheme of Roe when the scheme is applied to the initial condition (2.21) with ( h, q) given by (2.20), then it satisfies the C-property.

PROOF. First, it is trivial to prove that (2.22) is satisfied if Roe's intermediate states are chosen, as this condition is a particular case of the more general property (2.15).

On the other hand, an easy computation shows that, in the particular case of the states (2.21), one has

[, ,° ][ 0 ] A i + l / 2 ( w i n + l - W i n ) = / i+l/2X~ 2 n 2 co


V

Figure 3. Water at rest steady state with wet/dry front.

As a consequence, both the Q-schemes of Roe and Van Leer satisfy the conservation property.

2.4. T h e W e t / D r y T r e a t m e n t

In order to verify if a numerical scheme can handle correctly with wet /dry fronts, it is reasonable to test, firstly, its ability in solving steady-state solutions including wet /dry transitions. It is thus, quite natural to extend the definition of the C-property as follows.

DEFINITION 2. A numerical scheme is said to verify the extended C-property if it solves exactly the steady-state solutions of the form

; ~ - b(x), i f b(x) < 7, q O, h(x) (2.24) \ 0, in other case,

where r/is a constant (see Figure 3).

REMARK 2. It can be easily verify that, if function b is regular enough, the pairs (h, q) given by (2.24) are in effect solutions of (2.1) in the sense of the distributions. These solutions represent water at rest where b(x) < z/and no water elsewhere.

Clearly, Theorem 1 still holds for this extended condition. Nevertheless, the Q-schemes of Roe and VanLeer introduced above does not satisfy property (2.23) in this case. In effect, let us consider a steady-state solution (h, q) given by (2.24). If we consider an intercell xi+1/2 placed between a wet cell I~ and a dry cell Ii+l, we have now

.a,+l/~. (w?+~ - w ? ) = [ n 0 0

The modification of the numerical source terms proposed in [5] allows to override this problem

[ ° [ rcn ,2hn? , if h ~ < b ( x i + l ) - b ( x ~ ) , L--k i+1/21 i J

= 0 if h~'+l < b(x~) - b(xi+l), (2.25) n 2 n ' Si+1/2 (c i+ l /2) hi+l

[ 0 1 -(c~+1/2) 2 (b(z~+l) - b(xi)) ' in other cases.

It is easy now to verify that, with this redefinition of the source terms, the Q-scheme of Roe satisfies property (2.23). Therefore, we can state the following result.

426 M . J . CASTRO et aL

A b

I i I i + l

(a) Wet/Dry front. (b) Redefinition of the bot tom function.

Figure 4. The wet/dry front treatment as a redefinition of the bot tom function.

COROLLARY 2. I r a numerical scheme of form (2.10), with numerical sources given by (2.25), is such that the matrices A~+1/2 coincide with those of the Q-scheme of Roe when the scheme is applied to the initial condition (2.21) with (h, q) given by (2.24), then it satisfies the extended C-property.

The numerical source terms (2.25) can be understood as a redefinition of the discretized bot tom function: when the bot tom emerges at an intercell, it is redefined in the dry cell in order to avoid the appearance of spurious pressure forces, as it is illustrated in Figures 4a and 4b.

While this redefinition of the discretized bot tom function allows to t reat adequately emerging bottom situations for fluids at rest, further corrections have to be done to the numerical scheme in the ease of fluids in motion. Otherwise, the fluid could climb bot tom steps of arbitrary size.

In [5], emerging bot tom situations were treated by imposing the condition

qn+l (2.26) i ----0

in the wet cell, in order to simulate the fact that the discharge across a wet /d ry front is zero. In [6], it was remarked that, with this treatment, the advance of wet-dry fronts are not correctly

simulated: the computed velocities of the fronts are lower than the observed in experiments, the error being specially important in the case of water advancing on a dry bed on increasing depth. In this latter work, an improvement was proposed: the above correction is applied only if the water layer flows in the decreasing depth direction, tha t is, if

sgn (q~) = sgn (bi+l - bi).

This correction improved the computed velocities of wet-dry fronts, but the corresponding numerical scheme presents a drawback: negative values of h appear at the dry cells.

We propose here a modification of Roe scheme based on the remark that condition (2.26) is natural only at the wet /d ry front and not on the whole wet cell. Therefore, it has to be treated like an internal boundary condition of the problem: in fact, the wet-dry front is a free boundary of the problem. To do this, we propose the following procedure: let us consider a situation like that shown in Figure 4a, where the bot tom emerges at xi+1/2 at time tn, Ii is a wet cell, and l i+i is a dry cell, i.e.,

W +l = [0, 0IT;

then we have the following.

• The modified numerical source term given by (2.25) is computed.


• A new state I/V/~ defined by

is considered. • W~ and W ~ I are computed by using (2.18) with the fluxes F~+1/2 (see (2.19)) associated

to the states 17~, W ~ I and the modified source term.

The numerical scheme is similar if It is the dry cell and Ii+l is the wet one, and Roe scheme without any modification is applied when no bottom emergence occur.

REMARK 3. Observe that the states @ and W ~ I correspond to an emerging bottom situation

with water at rest. Therefore, Ai+t/2(W~+ 1 - I~V n) = S~+1/2 which implies that F/;1/2 =/?(~n) and _F~;1/2 = F(W~+a) = [0, 0] n-. As a consequence, the modification proposed can be interpreted as follows: first, the solution at xi+1/2 is approached by that corresponding to an emerging bottom

situation for a fluid at rest, ITaly; then, in order to compute W~ +1 the numerical flux corresponding to this approximate solution at the intercell is used, while there is no flux through xi+1/2 for Ii+1. Clearly, the smaller is ]q~t the better this approximation should be.

REMARK 4. Observe also that, with the previous numerical procedure, _n+l ui is not necessarily equal to zero, and that no negative values of h at the dry cell are produced.

3. T W O - L A Y E R F L U I D S

3.1. E q u a t i o n s

The one-dimensional two-layer shallow water system considered here represents mass and momentum conservation for the flow of two superposed layer of immiscible fluids of constant densities through a rectangular channel. It is obtained by taking cross-sectional averages in the incompressible Euler equations for each layer. We formulate the system under the form of two-coupled systems of conservation laws with source terms, in the sense introduced in [3]

0 w ( x , t ) + 0-~-F (W(x, t)) = B ( W ( x , t ) ) O - ~ W ( x , t ) + S ( x , W ) (3.28)

Ot o x ox

where

being

w ( x , t ) = [ LW2(x,t) ,

F ( w ) = rF(W1)] k F(W2) ,

B ( W ) = [ 0

B2(W2) [

r ( w j ) = 9 h2 ,

[0 00] , B I ( W 1 ) = _ g h l ,

s(x,w)_- is1 l(X, Wl)]

j = 1, 2; (3.29)

j = 1, 2;

[0 001 B2(W2) = - g r h 2 '

(3.30)

(3.31)

(3.32)

[0] db , j = 1, 2. (3.33) Sl,j(x, wj) = -9hj

In these formulae, index 1 makes reference to the upper layer and index 2 to the lower one (see Figure 5 for details), x, t, g, and b are as in the previous section. L)j is the density of the jth layer (Lh < ~)2) and r ---- Pl/~2. h j ( x , t ) , q j (x , t ) , and uj(x, t) represent, respectively, the water depth, the discharge, and the velocity of the jth layer.


ref. level 1

ref. level_ 2

QI

j ~

hl

92

Layer 1

- ~ h 2 - - L a y e 2

Figure 5. Two-layer sketch.

3.2. The Numer ica l Scheme

In the case, where hi(x, t) > 0, j = 1, 2, we use the generalized Q-schemes introduced in [3]. As in the previous section, some matrices have to be introduced in order to describe the scheme. In this case, the matrix of the system is

A(W) = [ A(W1) -BI(Wt)I -B2(W~) A(W~) '

where A(W) is the Jacobian matrix of the one-layer system given by (2.5). The eigenvalues of this matrix may become complex corresponding to the development of the so-called Kelvin- Helmholtz instabilities at the interface. Nevertheless, in this work we only consider the case when the matrix A has four different real eigenvalues is considered, that is, the flow is supposed to be stable and the system is hyperbolic. Therefore,

0

K ( W ) - I A ( W ) K ( W ) = A(W) -- "-. , (3.34)

,~4

where K ( W ) is a matrix whose oolumns are eigenveotors of A ( W ) .

We will also use the matrices

s g n ( A ( W ) ) = "sgn(A1) 0 1

0 sgn(A4)

A(W) i = K ( W ) A ( W ) ± K ( W ) -1,

[ (~10)~ 0 A ( W ) ± = ... (/~4) 4-

IA(W)[ = A(W) + - A ( W ) - .

The approximation of W(xi, nat) given by the numerical scheme will be represented by

W~ [ w n 1 [hn" 1 ~-~ /n 1 , n : n~'3 , j = 1, 2. wi,2J w~,j q~,J

We will also use the following notation for the approximations of the velocities:

ft u n . q i , j

~,3 -- h n. ' j = 1,2. q-~2

The Numerical Trea tment 429

Using these notations, the scheme writes as follows: At w V 1 = w ~ + ~ (F~-v~ - F~+~/~)

At + ~ (Bi-1/2. (W~ - Wig_l) + Bi+1/2. ( W ~ + I - W ~ ) )

At (p+i_l/2Si_l/2 4- P~+1/2Si+l/2), +-X-£~ where

(3.3s)

with

[ 0 1 1 _ n 2 n 2

.¢~i+1/2,j : (~ti+l/2,j) 7-(Ci+l/2,j) 2Un÷l /2 , j '

[ 0 00] [ 0 n 2 Bi÷,/2,1 (ci÷1/2,1) hi ~ i ÷ 1 / 2 , 2 : ( i ÷ 1 / 2 , 1 ) 2 -- ' -- e n r h2 '

where

"Lti+l/%J : hi+l,J + ~,3 n h n. .' Ci+l/2'J : -2-

for the Q-scheme of Van Leer and

n

n + ,J

for the Q-scheme of Roe. Matrix Bi+u2 is defined by

[ 0 B i + l / 2 = Bi+1/2,2

hn " ci%1/2,j g ~'J + h i + l d

= -2- '

Bi+l/2'11 0

The discretization of the source terms is as follows:

n 2 ~c~+1/2,1J ( b ( x ~ + l ) - b(x~))

S i + 1 / 2 ~- 0 (3.38) n 2

(Ci+1/2,2) (b (X i+l ) -b (x i ) ) and the upwinding is performed now by means of the projection matrices

1K (Id ± sgn (Ai+1/2))~+1/2' = g - 1 P~+l/2 ~ i+1/2

As in the case of the one-layer system, we use a C F L - c o n d i t i o n

max {[Ai+l/2,11 , 1 < l < 4, 1 < i < M} ~xx < ~ / ' A t

where 0 < ~/< 1 and we also apply Harten regularization.

REMARK 5. Again, if the intermediate state of Roe (3.37) is chosen, the numerical @scheme (3.35) can be rewritten as follows

w V 1 = w r + ~ F~+_1/2 - FL~/~ , (a.ag)

w h e r e

F++1/2 = F ( W i + l ) - P + + l / 2 ( A i + l / 2 ( W i + I - W i ) - S i + 1 / 2 ) . (3 .40)

(3.37)

1 1 (F (W~) + F (W7+1)) ~ [A,+I/~[ • (W,"+I - W~) (3.36) F i + l / z = "~ - •

Here, Ai+l/2 represents the matrix of the system evaluated at a chosen 'intermediate state', In the case of the generalized @schemes of Roe and Van Leer presented in [3], this matrix is as follows:

[ A i + l / 2 , 1 - B i + l / 2 , 1 ] Ai+l/2 = _Bi+1/2, 2 Ai+1/2,2 j '


3.3. C o n s e r v a t i o n P r o p e r t y

In [9], the C-property was generalized to bilayer system in the more general context of a channel with variable depth and width and nonrectangular cross-section. In the context of the present work, the formulation of this condition is as follows.

DEFINITION 3. It is said that a numerical scheme for solving (3.28) satisfies the C-property if any steady-state stationary solution given by

ql = O, q2 --= O, hl = hl, h2 -= ~72 - b, (3.41)

where hi and ~2 are constants such that hi > 0 and ~2 > max{b(x ) , x E [0, L]), is computed exactly.

In the cited work, the following results were shown.

THEOREM 2. Let us consider a numerical scheme of form (3.35) for solving (3.28). Given any steady solution (hi, ql, h2, q2) of form (3.41), we define

[ ~i,1 1 ~Vij : [hj(x~) 1 (3.42) W ~ = vC,2J' ' L q j ( x j "

ff~ for each i, the following equalities hold

A,+1/2" (~v~V',+, - VV" 0 : Si+,/2, (3.44)

then the scheme satisfies the C-property.

COROLLARY 3. I f a numerical scheme of form (3.35),(3.36) with source terms given by (3.38) is such that the matrices Ai+I/2 coincide with those of the Q-scheme of Roe or Van Leer when the scheme is applied to the initial condition (3.42) with (hi, ql, h2, q2) given by (3.41), then it satisfies the C-property.

The proofs are similar to those corresponding to one-layer systems.

V

| i I i+l I i I i+l

(a) Wet/dry front: disappearance of both layers. (b) Wet/dry front: disappearance of the lower layer.

Figure 6. Wet/dry fronts at the two-layer system.

The Numerica l Treatment 431

3.4. T h e W e t / D r y T r e a t m e n t

As in the case of one-layer systems, first we extend the C-property in order to include steady- state solutions corresponding to water at rest with wet/dry transitions. Notice that, in two-layer system, two types of wet/dry transitions may occur in water at rest when the bottom emerges: disappearance of both layers (see Figure 6a) or disappearance of the lower layer (see Figure 6b). These two possibilities are taking into account in the following family of steady-state solutions:

ql ~ 0, q2 ~ 0,

h i ( x ) = r h - b(x), i f r12 _< b(x) < ~1, h 2 ( x ) = r12 - b(x), i f b(x) < r12 , 0, if ~1 ~ b(x), 0, if r/2 _< b(x),

where r h and 7]2 are two constant such that 7]2 < rll (see Figure 7).

DEFINITION 4. It is said that a numerical scheme for solving (3.28) satis/~es the extended C- property if any steady-state solution given by (3.45) is computed exactly.

Again, as in the one-layer system, if function b is regular enough, (hi, ql, h2, q2) given by (3.45) are solutions of (3.28) in the sense of the distributions. Theorem 2 is still valid for this extended condition, but again the generalized Q-schemes of Roe and Van Leer introduced above do not satisfy property (3.44) at the intercells placed at a wet/dry front.

In order to override this problem, a modification of the numerical source terms similar to that proposed for the one-layer system can be performed

8 1 + 1 / 2 =

I 0 n 2

- - ( C i+1 /2 ,1 ) (hinl "~-h~,2 )

0 n 2

n hr~ r~ ci+1/2,1 +1,1 ~- hi+1,2

n 2 n n L(ci+1/2,2) (rhi+l,l + hi+l,2) J o ] -(C~+l/2A)2(b(Xi+l)-b(xi))

n 2 0 n ' - - (Ci+1/2,2 ) (r(b(xi+l)-b(xl)--hi ,2) +h~',2)

0 n 2

- - ( c i + l / 2 j ) (b(xi+l)-b(xi)) 0

n 2

0 n 2

- ( b ( x , + l ) -

0 n 2 --(ci+1/2,2) (b(~i+l) --b(xi))

if h~i,1 -t- h~, 2 < b(xi+l) - b(xi),

if hn+l,1 -t- h~'+1,2 < b(xi) - b(xi+l),

if h '~ i,2 < b(Xi+l) - b(x~) < hni,1 + h~.2,, (3.46)

if h n i+1,2 < b(xi) -b(xi+l) < h~+l, 1 + h~+ 1, 2 ,

in o ther cases.

Some straightforward calculations allow to prove that, with this redefinition of the source term, the generalized Q-schemes of Roe and Van Leer satisfies property (3.44). Therefore, the following result can be easily shown.

432 M . J . CASTRO et al.

V

hi

I h2

Figure 7. W e t / d r y fronts in a two-layer sys tem: water at rest .

COROLLARY 4. /Y a numerical scheme of form (3.35), with numerical sources given by (3.46), is such that the mat r i ce s A i + l / 2 coincide with those of the Q-scheme of Roe when the scheme is applied to the initial condition (3.42) with (hi, ql, h2, q2) given by (3.45), then it satisfies the extended C-property for the two-layer system.

As in the one-layer system, the numerical source terms (3.46) can be understood as a redefinition of the diseretized bot tom function in two-layer ceils with a neighbor that is a dry or a one-layer cell. Notice that , in this case, this redefinition is different for each layer.

Again, in the case of fluids in motion, further modifications have to performed in emerging bot tom situations: the conditions

ql = 0, q2 = O, (3.47)

o r

q2 = 0, (3.48)

have to be taken into account at the wet-dry front in situations that are, respectively, like those

represented in Figure 6a or in Figure 6b. These modifications are performed by generalizing those presented in the one-layer case: let us suppose that the bot tom emerges at the intercell xi+1/2

at time t~ and that Ii is a two-layer cell and Ii+l a one-layer or a dry cell. Then, we have the following.

• The source terms given by (3.46) are computed. • The solution at the intercell is approached by

= h n n 0] T ~ n [ i,l' O, hi,2,

if Ii+l is a dry cell, or by

- ,~= h ~ 7~ h ~ 0IT W i I i,1, qi,l~ i ,2,

if Ii+l is a one-layer cell. • The numerical fluxes F~+I/2 given by (3.40) are calculated using the states ~r~, W$+l"

• Scheme (3.39) is used to calculate W '~+1 w ~ + l i , " ' i + 1 "


5 10 15 2O 25 30

(a) Test 1. Initial conditions.

35 40

0.;

0.1

02

0.=

0.~

h comput, by ~61

i , i 21 r 5 10 15 0 25 30

(b) Test 1. Steady-state solution.

Figure 8. Test 1. Mass conservation.

4. N U M E R I C A L R E S U L T S

I 35 40

4.1. O n e - L a y e r S y s t e m

4.1.1. Te s t 1. M a s s c o n s e r v a t i o n

The objective of this first numerical experiment is to verify if mass conservation is preserved when the numerical scheme presented in Section 2 is used. We consider a straight channel with rectangular cross-section and constant width. The axis of the channel is identified with the interval [0, 40]. The bot tom is given by

f O, 0 _< z' < 30, b(x)

0.6, 3 0 < x < 4 0 .

In this channel, we perform a dam break experiment. The dam, placed at x = 15~5, is simulated

through the initial conditions (see Figure 8a)

h(x,O) = { 0.4, 0 _ < x < 15.5, 0, 15.5 < x < 40, q(x, O) = O.

At x = 0, the boundary condition q(0, t) = 0 is imposed. Due to this boundary condition and to the presence of a step at x = 30, there is no mass loses through the boundaries during the

simulation and, thus, the total mass has to be preserved. The numerical scheme is run with Ax = 0.2 and C FL = 0.9 until a steady-state solution

corresponding to water at rest is reached. The difference between the initial and the final mass given by the numerical solution is of 1.6.10 -14 . If the numerical scheme proposed by [6] is used, there is an increment of water mass of 0.176634 due to the appearance of negative values of h at dry cell. In Figure 8b, we compare the steady-state solution computed with the numerical scheme proposed in [6] and with the improvement proposed here.

4.1.2. Te s t 2. S h o c k ove r a sol id wall

In this numerical experiment we consider a column of water of 1 m depth, moving towards a vertical wall with a constant velocity of 10 m/s. The axis of the channel is now the interval [0, 12],

and the bot tom is given by the function

j" 0, 0 _ < x < 1 0 , b(x)

l 10, 1 0 < x < 12.


[ .~- Water Depth {Numer.) - - W m e r Depth (Exsct) ..... t B t tom

2 4 8 8 10

(a) t = 0.

2 4 s

(b) t = 0.

I

10 12

BoSom i + t= l .gO373aea

- - - - - - . - - - - - - . - - . - - . - - . - - - . - - - - . - - - -

. . . . . . + . . + . . + . . . . + . . . . ~ . . . . + . . . . . ~ . . . . . + . . . . . . + . . . . . + + + + + + + + + +

9.6 9.7 9.8 9.9 10 10.1 10.2 10,3 1(1.4 10,5

(c) t = 1.903733e - 3.

===========================

i

i i i + i i i

e s lo 12

(e) t = i .

Figure 9. Water depth and

Discharge (numer.)

~ 1 ,g03733e3 I -- - DiScharge (exact)

i i i i 4 6 s

(d) t = 1.903733e - 3.

10 12

~F

e ~

7~

6~ ,t

2~

+ Dischargo (num~r.)

OF i i i 0 2 4 e

(f) t = 1.

discharge of Test 2.

The in i t ia l condi t ion is given by (see F igure 9)

h(z,0)={ 1, 0_<z<10, {10, 0_<z<10, 0, 10<z<12, q(z,O)= 0, 10_<z<12.

Concern ing the b o u n d a r y condi t ions , b o t h the wa te r d e p t h and the d ischarge are imposed at

x = 0 h(0, t) = 1, q(0, t) = 10,

while no b o u n d a r y condi t ion is cons idered at z = 12. In th is numer ica l expe r imen t , A z = 0.05

and O F L = 0.9. Accord ing to R e m a r k 3, as the discharge in the wet cell of the emerg ing b o t t o m s i t ua t ion arising

at t ime t = 0 is far to be near of zero~ the numer ica l so lu t ion presen ts an i m p o r t a n t er ror a t the

first i t e r a t i on (t = 1 . 9 0 3 7 3 . 1 0 - a ) : observe F igures 9c and 9d, where the c o m p u t e d discharge is

The Numerical Trea tment 435

compared with that corresponding to the the exact solution. Nevertheless, this initial error is rapidly dumped and after some iterations only a small disturbance on the discharges near the traveling shock can be observed (see Figure 9).

4.1.3. Tes t 3. C o m p a r i s o n w i t h e x p e r i m e n t a l d a t a

In order to test the performance of the numerical treatment of wet /dry fronts proposed here, we compare the numerical results with some measurements corresponding to a physical model developed by L. Cea and J. Puertas at hydraulic laboratories in the C.I.T.E.E.C. (University of A Corufia). The experiment was performed in a channel of 17 meters long, in which the experimental devices of measure were located at the ceiling (see Figure 10). The flow was generated by means of a piston which pushed a plate with an slope of 60 °. The movement of this plate was constant and the imposed velocity was of 580 mm/s (quick movement).

The numerical domain corresponds to the five meter long portion of the channel where the measure devices and the obstacle were placed (see Figure 11). The obstacle is a cube of 0.36 meters high and 0.51 meters long (see Figure 11). As boundary conditions at the left side of the channel, the measured free surface elevation at point P1 is imposed. At the right side, we use a duplication of the state condition. The initial condition is depicted in Figure 11: water at rest with a depth of 0.30 to the left of the obstacle, and no water to the right. An uniform mesh with 452 nodes is used. In order to simulate the drag force of the bottom, we use the following empirical expression (see [4]):

f~=gM2ql ~ h -4/3,

where M is the Manning number, and it is set equals to M = 0.0175. In [4], a suitable numerical treatment of this term is also proposed.

In Figure 12, we compare the free surface obtained with the numerical scheme with CFL = 0.5 (solid lines) with the measurement data at points P2, P3, P4, and P5 (circles) at times t = 2, 2.4, 5, 9, and 14.5. As it can be seen, the water jumps over the obstacle producing wave breaking and fronts going downstream and upstream.

P1 : P2 P3 P4 P5

(3' O.S 1.5 2.5 3.5 Generating ~ > < >

plaque a.51 o.51

Figure 10. Piston which generates the wave and si tuat ion of the measure devices.

V

0.30 cm

P2 P3 P4 P5

P1 = 0 0.5 1.5 2.5 3.5

0.36 m

3,51 rn 0.51 m

I

5

Figure 11. Numerical domain and si tuat ion of the measure points.

436 M . J . CASTRO et at.

o7 I o.e

o s

0.4

0,~

j ~ H

0.5 ~ ~,5 2 ~ ~ 3S 4 4S

(a) T ime = 2

0.5 1 1.5 2 2.5 ~ 3.5 4 4 5

(c) Time = 3 5

0.5 I 1.5 2 2.5 3 3.S 4 4.S

(e) Time = 9.

o,.'

o.,

0.2

I

o7!

o.~

o.~

o.~

o,:

o.~

o.I

S

i

o,s 1 1.s 2 ~.s 3 3.s ~ 4.s

(b) T i m e = 2 4

, , i , , , , J , 0s ~ 1.z 2 2.S 3 3.S 4 4.S

(d) Time ---- 5,

f - j ~ .....

F i

, , , ) l , J l J

0.~ I 1.5 2 2.~ 3 3.5 4 4 3

(f) Time = 20.

Figure 12. Wave evolution at different t imes.

0 7

O~

OS

O.4

o.~

0.5 1 1,5 2 2.5 3 3.5 4 4.5

(a) Water begins flowing along the wall.

0.s 1 l.s 2 2.s 3 3.~ 4 ¢.s

(b) Unstable simulation produced by the scheme given in [5J

Figure 13.


If the we t /d ry condition proposed in [5] is used, that is, if q = 0 is imposed always at the wet

cells in emerging bot tom situations, the numerical scheme fails when the water climbs over the

obstacle, as it can be observed in Figure 13 while the numerical scheme proposed here produces good simulations (see Figure 12).

Finally, in Figure 14, we present a comparison between experimental depths (dots) and numerical simulations (solid) at each measure device placement. We can observe in the picture corresponding to P5 (measure device next to the wall) a very good agreement.

0.4

0.2

0.4

0.2

P1

0.4

0.2

0.4

0.2

P2

o t o t 0 5 10 15 20 0 5 10 15 20

P3 o.4 0.2

0

P4

0 t 4 t 0 5 10 15 20 0 5 10 15 20

0 0

P5

" ~ ~ Aprox

- - Exp

i i i t

5 10 15 20

Figure 14. Comparison between experimental water depth (points) and the simulated depth (continuous line).

4.2. T w o - L a y e r S y s t e m

In [15], some steady-state solutions corresponding to flow exchange through channels with rectangular cross-sections and a single bump were approximated. These approximations are based on a simplified two-layer hydraulic model where the density ratio is supposed to be close to one and the rigid lid approximation is used. Moreover, these authors characterize the so-called maximal solution, tha t is, the solution corresponding to a maximal exchange through the channel.

The aim of this section is to obtain a steady-state solution by performing a lock-exchange experiment with the two-layer model and compare it with the approximation of the maximal solution provided in [15].

As initial condition, we consider the two layers separated by a vertical "artificial barrier" in this case located at x -- 2 (see Figure 15a). This barrier is removed at time t = 0 and the

scheme is run until a steady state is reached (see Figure 15f). The time evolution from the initial state to the steady-state solution are shown in Figures 15b-15e: a satisfactory treatment of the two-layer/one-layer front can be observed. In the numerical experiment presented here, the rectangular channel is one meter width and six meters long. Its bot tom topography is defined

438 M . J . CASTRO et al.

0

-O.2

-0.4

-oa

-1

.t.4

-1.e

Fluid A

-2

/ /

r t

t

t / ~

i i -1 0

(a) Time = 2.

o

~.2

-0.4

-o.s

-0.8

-1

-1.2

-1.4

-t.e

-1 .s

%

Free Surface t=2 Inte~ac~ t=2 Bottom topogrsphy /

~ ~ i *~ ; ""'--~-_

(b) T i m e = 2 . 4 .

O

-0*2

-0A

-O.B

"0.6

"1

-1.2

-I ,4

"1.8

-I,8

-2"

o

-02

"0.4

.o.e

..o.s

-1

,I.2

-1A

4.6

-1 .a

~ Fre~ Sudace t=5 Interface t=5

- Bottom t~oo~lr~h:/

(c) T ime = 3.5.

~ Free Surface t=10 Irtterface t=l 0

- Botlom topograph~ + ~ : ; : : ; :~: : ; ; ; ;~k~

. + t x

-~ -1 o 2

0

-o.2 ~_ Free Surface t=8 - Interface

~o.4 Bottom topoEIraphy

-0.8

"1.2 *++

-I.4 +1/ \

(d) T ime = 5.

~.~ . Free aufface + Interface

-0.4 - - Bottom topography F&A so lu t ion

-o.e

-1.2 t / ~\

/ -1.4 /

-1.! / ~ \

-1.a

~ '-2"" , ~ ~ "--~..

(e) Time = 9. (f) T ime = 20.

Figure 15. Maximal exchange flow th rough a rectangular channel wi th a single bump: initial condition, t ime evolution until s ta t ionary solution and comparison wi th F&A s ta t ionary solution.

by the function b(x) = e x p ( - x 2) - 2, where x E [-3, 3]. As boundary conditions only the ratio between fluxes is imposed and it is set to 1, i.e., ql = -q2 at each channel end. The ratio between densities r = L)I/Q2 has been taken equal to 0.98, Ax = 6/150, and CFL = 0.9.

In Figure 15f, we compare the interface corresponding to the steady state reached with that corresponding to the maximal solution provided in [15]. Notice that only some small differences between both solutions appear due to the rigid lid approximation use in [15]. At the steady state, the discharges obtained are ql = -q2 = 9.37710 -2 ma/s while the maximal flow exchange is approximated in [15] by ql = -q2 = 9.213 10 .2 ma/s.

5. C O N C L U S I O N S

In this paper, we have extended the definition of the so-called C-property to numerical schemes for one-layer and two-layer shallow water equations with we t /d ry fronts. We have presented an


improvement of the numer ica l schemes proposed in [4-6] sat isfying this ex tended condi t ion and

we have also cons t ruc ted its na tu ra l extension to two-layer shal low-water flows.

Physical exper iments have been developed in l abora to ry to prove the performance of the nu-

merical model for one-layer flows.

The general iza t ion of w e t / d r y techniques for two-layer model has been tes ted by means of a

lock-exchange exper iment . In this exper iment , we have compared the the numer ica l s teady-s ta te

solut ion reached wi th the approximate solut ion provided by Armi and Farmer for channels wi th

rec tangular cross-sections and simple geometries.

Fu tu re lines of research focus on the development of two-layer shallow water experiments ,

s imilar to those presented here for the one-layer case, in order to test also the accuracy of the

t r ans ien t approximat ions .

R E F E R E N C E S

1. M.E. V~zquez-Cenddn, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. Comp. Physics 148, 497-526, (1999).

2. P. Garcfa-Navarro and M.E. V£zquez-Cenddn, On numerical treatment of the source terms in the shallow water equations, Computers and Fluids 29 (8), 17-45, (2000).

3. M.J. Castro, J. Macfas and C. Par6s, A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system, Math. Model. and Numer. An. 35 (1), 107-127, (2001).

4. P. Brufau, Simulaci6n bidimensional de fiujos hidrodin£micos transitorios en geometrfas irregulares, Tesis Doctoral, Univ. de Zaragoza, (2000).

5. P. Brufau, M.E. V£zquez-Cenddn, Garcfa-Navarro, A numerical model for the flooding and drying of irregular domain, J. Numer. Meth. Fluids 39, 247-275, (2002).

6. A. Perreiro, Resolucidn y validaeidn experimental del modelo de aguas poco profundas unidimensional in- ciuyendo £reas secas. Trabajo de DEA, Universidad de Santiago de Compostela, (2002).

7. L. Armi, The hydraulics of two flowing layers with different densities, J. Fluid Mech. 163, 27-58, (1986). 8. L. Armi and D. Farmer, Maximal two-layer exchange through a contraction with barotropic net flow, J. Fluid

Mech. 164, 27-51, (1986). 9. M.J. Castro, J.A. Garcfa-Rodrfguez, J. Macfas, C. Par6s and M.E. V£zquez-Cend6n, Two-layer numerical

model for solving exchange flows in channels with arbitrary section, J. Comput. Physics. 195, 202-235, (2004).

10. J.M. Gonz£1ez-Vida, Desarrollo de esquemas num6ricos para el tratamiento de frentes seco-mojado en sistemas de aguas someras, Ph.D. Thesis, Univesidad de M£1aga, (2003).

11. A. Berm6dez and M.E. V£zquez, Upwind methods for hyperbolic conservation laws with source terms, Com- puters and Fluids 2a (8), 1049-1071, (1994).

12. M.E. Vg.zquez-Cenddn, Estudio de Esquemas Descentrados para su Aplicacidn alas leyes de Conservaci6n Hiperbdlicas con T6rminos Fuente, Ph.D. Thesis, Universidad de Santiago de Compostela, (1994).

13. P.L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes, d. Comput. Phys. 43, 357-371, (1981).

14. A. Harten, On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. of Numer. Anal. 21 (1), 1-23, (1984).

15. D. Farmer and L. Armi, Maximal two-layer exchange over a sill and through a combination of a sill and contraction with barotropic flow, d. Fluid Mech. 164, 53-76, (1986).

16. M.J. Castro, J. Macfas, C. Par6s, J.A. Rubal and M.E. V£zquez Cenddn, Two-layer numerical model for solving exchange flows through channels with irregular geometry, In Proceedings of "ECCOMAS 2001": Swansea, (2001).

17. J. Garcfa Lafuente, J.L. Almaz£n, F. Castillejo, A. Khribeche and A. Hakimi, Sea level in the Strait of Gibraltar: Tides, International Hydrographic Review LXVII (1), 111-130, (1990).

18. P.L. Roe, Upwinding differenced schemes for hyperbolic conservation laws with source terms, In Proceedings of the Conference on Hyperbolic Problems, (Edited by Carasso, Raviart, and Serre), pp. 41-51, Springer, (1986).

19. E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, Springer, (1997).

20. E.F. Toro and M.E. V£zquez-Cenddn, Model hyperbolic systems with source terms: Exact and numerical solutions, In Proceedings of Codunov Methods: Theory and Applications, (2000).

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