Exact solutions to the Riemann problem of the shallow water equations with a bottom step

Computers & Fluids 30 (2001) 643±671

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Exact solutions to the Riemann problem of the shallow water equations with a bottom step Francisco Alcrudo a,*,1, Fayssal Benkhaldoun b a

LITEC(CSIC) ± Area de Mec anica de Fluidos, CPS-Universidad de Zaragoza, c./Maria de Luna, 3, 50.015 Zaragoza, Spain b D epartement G enie Math ematiques, Laboratoire LMI, INSA de ROUEN, B.P. 08 ± Place Emile Blondel, 76.131 Mont Saint Aignan Cedex, France Received 22 July 2000; received in revised form 22 October 2000; accepted 10 November 2000

Abstract The similarity solution to the Riemann problem of the one dimensional shallow water equations (SWE) with a bottom step discontinuity is presented. The step is placed at the same location where the ¯ow variables are initially discontinuous. While the solutions found are still a superposition of travelling waves belonging to the two well-known families of the shallow water system, namely hydraulic jumps and rarefactions, the appearance of a standing discontinuity at the step position produces a very interesting solution pattern. This is mainly due to the asymmetry introduced by the step. The adopted solution procedure combines the basic theory of hyperbolic systems of conservation laws together with a sound interpretation of the physical concepts embedded in the shallow water system. This ®nally leads to a set of algebraic equations that must be iteratively solved. The ideas contained in this paper may be of valuable help to the understanding of the behaviour of the SWE with source terms, that constitute the core of many mathematical models for free surface ¯ow simulation. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Shallow water equations; Source terms; Riemann problems; Systems of conservation laws

1. Introduction Although the shallow water equations (SWE) represent a very simple model of the movement of layers of incompressible ¯uids with a free surface, considerable attention has been paid to its study for di€erent reasons. Firstly, they form the basis of many mathematical models in use for *

Corresponding author. Tel.: +34-976-76-18-81; fax: +34-976-76-18-82. E-mail addresses: [email protected] (F. Alcrudo), [email protected] (F. Benkhaldoun). 1 Supported by CONSID (CAI-DGA) and INSA de Rouen. 0045-7930/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 3 0 ( 0 1 ) 0 0 0 1 3 - 5

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the simulation of certain ¯uid ¯ow phenomena. Currents in estuaries, lakes and sloping beaches, tidal waves, bore propagation, ¯ood routing in natural and man-made streams, submersion waves and atmospheric ¯ows, among others, can be reasonably well described by the SWE (see Ref. [15]). Secondly, their structure is very close to that of the gas dynamics equations while being simpler to interpret and manipulate. Indeed they represent a nonlinear system of conservation laws (SCL) that mimic some of the features (like shock waves) of more complicated systems such as those describing reactive gas dynamics, combustion, ¯ow in aquifers etc. [12]. Most problems cited above have no analytical solution and must therefore be solved with the aid of numerical algorithms. Hence the interest in developing smart and robust schemes that are well suited to represent such phenomena. Recently much attention has been paid to the numerical study of SCL with source terms [1,5,8,9,14]. The acronym SCL with source terms is somewhat contradictory, for the presence of a source term destroys conservation of at least some of the variables, but it is often found in the literature meaning a SCL when only the homogeneous part is considered. While the numerical treatment of homogeneous SCL (at least in one dimension) can be regarded as a solved problem, this is no longer the case when a source term is present. In many situations it becomes dominant, even sti€ and robust methods known to work very well in the homogeneous case fail. In particular, regarding the one dimensional (1-D) SWE, source terms are present to account for external forces acting on the ¯uid mass, like Coriolis acceleration, wind stresses or bottom and lateral reactions. The last two can become very important and completely dominate the ¯ow in many situations of interest (natural streams, ¯ow in narrow mountain valleys, sloping channels, etc.). Moreover, if they are of very irregular nature, many numerical methods that behave nicely in absence of source terms can produce completely wrong solutions (see for instance Refs. [4,14]). While steady solutions to the 1-D SWE either with or without source terms can be easily computed to arbitrary accuracy for determining open channel water pro®les [2,10], there are very few time dependent solutions known, more even so if source terms are present. The 1-D Riemann problem (also referred to as the discontinuity in the initial conditions problem), in absence of source terms (i.e. on a frictionless ¯at bed) has been widely described, specially regarding dam break waves [11]. Glaister [3] solved it in an approximate way in cylindrical coordinates, that introduce a source term generated by the metrics. Watson et al. [16] obtained exact solutions to the same problem on a constant slope beach (i.e. with a linear source term) by means of an ingenious transformation of variables that led back to a homogeneous system. It must be said that the cited examples belong in fact to a restricted set of the SWE solution space: Firstly, they are of the similarity type in which the ¯ow variables do not depend separately on the space, x, and time, t, coordinates, but on the ratio x=t. Also they can be considered singular since the ¯ow variables are discontinuous at the initial instant and this character is preserved at subsequent times in the form of travelling shocks and weak discontinuities (jumps in the ®rst derivatives). This turns out in fact to be an advantage when these solutions are used to construct and test numerical methods, that, if well suited to reproduce such diculties, are presumed to behave at least as well when the solution is smooth. In any case, no other exact time dependent solutions to the SWE are known to the authors so far. In this work the exact solution to the Riemann problem of the 1-D SWE with a bottom step is presented. The situation considered is identical to the standard Riemann problem except for the

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presence of a bottom discontinuity. This causes the source term arising from the bed slope to become singular at that point, more precisely a Dirac delta function. Greenberg and LeRoux [5] attacked this problem for a model scalar equation with the aim of building a well-balanced scheme for the numerical processing of source terms. Their quite successful scheme is based on exact solutions to the scalar equation, that they achieved by solving ®rst the regularised Riemann problem obtained when substituting the step for a ramp: Once the exact solution for the ramp is known (for which all the interactions between left and right running waves had to be considered), a limiting process increasing the ramp angle to 90° produces the sought solution. The approach adopted here is not, unfortunately, that much elegant, but nevertheless leads to the desired result. This paper is structured as follows: Section 2 recalls some basic concepts about the 1-D SWE including those regarding their physical meaning. Section 3 describes steady state solutions together with a method for computation. This leads to a brief study on stationary ¯ow over a step, in Section 4. Section 5 merges the ideas of previous paragraphs with the basic theory of SCL in order to construct the solution to the Riemann problem with a step. Section 6 describes the possible di€erent con®gurations, and ®nally several conclusions are drawn.

2. Physical model We consider here 1-D shallow water ¯ow in rectangular channels or streams of unit width, and understand by such the movement of a layer of ¯uid as a whole over a variable bed as depicted in Fig. 1. The ¯ow can be described by specifying at any point and instant of time the thickness of the water layer, or depth, h, and the average horizontal velocity of the ¯uid, u. The elevation of the free surface of the ¯uid with respect to an arbitrary datum is marked by f…x; t†, while the position of the bottom itself is denoted by function zB …x† (no modi®cation of the channel bed with time is considered, hence disregarding any erosion or sedimentation process). Therefore f…x; t† ˆ h…x; t† ‡ zB …x†

…1†

This simple description implicitly assumes that no other than horizontal movement takes place, thus neglecting vertical accelerations and velocities what in turn leads to a hydrostatic pressure distribution. The approximation holds provided the slope of the bed function is small. If this is

Fig. 1. Typical con®guration in 1-D shallow water ¯ow.

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not the case, the SWE simply do not describe a physically meaningful ¯ow, for vertical movement becomes important, the horizontal velocity pro®le may be nonuniform and pressure distribution deviates from hydrostatic. The motion equations can be easily deduced by simply writing conservation of mass and momentum along the direction of motion, x, (see for instance Refs. [11,15]) and read oU oF ‡ ˆ S: ot ox

…2†

Here t represents time and U, F and S the conserved variables, ¯ux function and source vectors respectively       0 h hu Uˆ ; Fˆ ; Sˆ …3† gh dzdxB hu hu2 ‡ gh2 =2 where g is the acceleration of gravity. Only momentum sources arising from bed slope (due to the e€ect of gravity and the bottom reaction) are considered here. Frictional forces (with the channel bed and lateral walls plus wind stresses) are usually accounted for with empirical formulae, such as Manning's or Chezy's, but they are not included here for brevity, since the Riemann solutions sought correspond to frictionless situations. The description can be easily extended to two dimensions by considering also horizontal movement in the transverse direction to x [15]. We will call Eq. (2) with de®nitions (3) the mass±momentum representation of shallow water ¯ows. The homogeneous system obtained if source terms are neglected is of the hyperbolic type and of very similar structure to the 1-D isentropic gas dynamic equations. Calling A the Jacobian matrix of the ¯ux vector, F, one has   oF 0 1 ˆ 2 …4† Aˆ c u2 2u oU where c ˆ …gh†1=2 is the velocity of propagation of small amplitude disturbances and plays the role of the speed of sound of the system. The two eigenvalues of A, a1;2 , are always real and distinct provided the water depth is nonzero a1;2 ˆ u  c

…5†

and the system can be easily diagonalised and written in characteristic form o…u  2c† o…u  2c† ‡ …u  c† ˆ ot ox

g

dzB dx

…6†

An important parameter to shallow water ¯ows is the Froude number, de®ned here as F ˆ u=c, which represents (the square root of) the ratio of inertial to gravitational forces. When F is below one the ¯ow is said subcritical while if over one it is called supercritical, in complete analogy with sub and supersonic ¯ow in gas dynamics. It is worth noting that Eq. (2) represents the strong conservation or divergent form of the SWE and all the corresponding theory applies. In particular it admits discontinuous solutions or shocks, that in the present context are called hydraulic jumps, either travelling or stationary. Such discontinuities can be seen in man-made channels and natural streams in the form of abrupt variations in water depth with intense turbulent mixing (the actual ¯ow becomes fully 3-D there).

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Although not strictly discontinuous to the human eye (up to a few metres long) as it happens with shock waves in gas dynamics, they e€ectively represent discontinuous variations given the longitudinal dimensions of channel ¯ow (kilometres). It can be proved that there is a loss of mechanical energy of the shallow water ¯ow in a hydraulic jump solely determined by the mass and momentum balance across it, regardless of the physical mechanisms (turbulent mixing) that actually dissipate the energy (see Refs. [2,10,15]). Hydraulic jumps and their characteristics are dealt with in more detail in Section 5. If the functions involved are smooth, the above equations can be easily manipulated to yield di€erent formulations of shallow water ¯ow. Among them one ®nds what will be called the mass± energy representation for the reasons given below:     o o h hu ˆ0 …7† ‡ ot u=g ox …h ‡ u2 =2g ‡ zB † The amount on the second row of the ¯ux is the total available head, Htot , and represents the total mechanical energy of the ¯ow expressed in terms of potential energy divided by g. Htot ˆ h ‡ u2 =2g ‡ zB ˆ f ‡ u2 =2g

…8†

Note that Eq. (7) is a homogeneous system, i.e. the source term present in the mass±momentum representation (Eqs. (2) and (3)) has been embedded in the ¯ux term. However it must be borne in mind that Eqs. (2) and (7) are not equivalent if any of the functions involved is not smooth: In particular the mass±momentum representation is the only valid in case the solution contains hydraulic jumps, for it has been said that mechanical energy is dissipated across a hydraulic jump while mass and momentum are conserved. On the other hand, in case the bottom surface function, zB …x†, is not continuous, the mass±energy formulation may still be valid, but the mass±momentum one must be discarded because the slope becomes in®nite in such situation.

3. Steady state solutions in a reach This section is devoted to the study of steady state solutions of the 1-D SWE over a variable bed with no friction. We consider here the solution procedure to obtain the ¯ow variables at any section of a stream, when they are known in a certain position within the reach. If the time derivative in Eq. (2) is identically zero, and for a smooth bed function, zB …x†, the situation is reduced to a boundary value problem of the form q ˆ hu ˆ constant

…9†

which constitutes a ®rst integral of the SWE, plus d…hu2 ‡ gh2 =2† dzB ‡ gh ˆ0 dx dx

…10†

that can be integrated with appropriate boundary conditions provided no hydraulic jumps are present. The computation starts at a control section where the ¯ow variables, h, u, are known and proceeds upstream in case the ¯ow is subcritical or downstream for supercritical ¯ow. This puts

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certain restrictions on the boundary conditions allowed, and the procedure can be easily extended to include friction losses. However, it is more convenient in this case to make use of Eq. (7) that provides a second integral of the motion equations, i.e. Htot ˆ h ‡ u2 =2g ‡ zB ˆ constant

…11†

Htot ˆ h ‡ q2 =2gh2 ‡ zB ˆ constant

…12†

or what reduces the problem to solving a cubic equation for h. De®ning the speci®c head, Hs (i.e. total head referred to the channel bed at the considered section) Hs ˆ Htot

zB ˆ h ‡ u2 =2g

Eq. (12) can be rewritten in dimensionless form as  3  2 h h q2 ‡ ˆ0 Hs Hs 2gHs3

…13†

…14†

which is a simple cubic equation of the form x3

x2 ‡ a ˆ 0

…15†

aˆ

q2 P0 2gHs3

…16†

The ratio x ˆ h=Hs can be easily related to the Froude number of the stream since h h 1 ˆ ˆ 2 Hs h ‡ u =2g 1 ‡ F 2 =2

…17†

thus x ! 0 corresponds to F ! 1 and x ! 1 to F ! 0. Critical ¯ow …F ˆ 1† is obtained when h=Hs ˆ 2=3. Referring to Fig. 2, and assuming the ¯ow state is known at Section 1, the procedure to compute the ¯ow at another position in the stream close to Section 1, like Section 2 in the ®gure, would be as follows:

Fig. 2. Two consecutive sections in steady ¯ow.

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1. compute the value of Htot from known variables at Section 1. Htot ˆ h1 ‡ u21 =2g ‡ zB1

…18†

2. obtain the speci®c head at Section 2, Hs2 , with Hs2 ˆ Htot

zB2 ˆ Hs1 ‡ zB1

zB2

…19†

3. compute the value of a in Section 2, a2 , from Eq. (16). 4. solve Eq. (15) and obtain h2 as h2 ˆ Hs2 x

…20†

5. then compute the value of u2 from u2 ˆ q=h2 ˆ u1 h1 =h2

…21†

There are however certain subtleties that make the above procedure slightly more complicated. Firstly, we are only interested in real solutions to Eq. (15) for which 0 6 x 6 1, what implies h2 P 0 and u2 real. Under these conditions the number of useful solutions to Eq. (15) is quite restricted: · If a < 0 there are two complex conjugate solutions, x1 , x2 and a real one, x3 > 1 what implies u22 < 0 and thus must be discarded. · If a ˆ 0 there is a double real solution x1 ˆ x2 ˆ 0, while the third one is x3 ˆ 1. The ®rst two 1=2 correspond to the limiting case h2 ˆ 0, u2 ˆ …2gHs2 † . The third one to ¯uid at rest, h2 ˆ Hs2 , u2 ˆ 0. · For 0 < a < 4=27 there are three real solutions. The ®rst one always lies in the interval … 1=3; 0† and is thus useless. The second, x2 , lies in …0; 2=3† and the third one, x3 , in …2=3; 1†. The latter solutions are of interest to our problem since they correspond to positive water depths and real velocities. In particular x2 represents supercritical ¯ow and x3 subcritical ¯ow at Section 2. This can be easily seen by writing   2 2 2 u22 …22† h2 ‡ x < ! h2 < Hs2 ˆ ! u22 > gh2 ˆ c22 3 3 3 2g and the opposite is true for x3 . Two valid solutions are available in this case. Normally the choice of one or the other depends on the ¯ow conditions at Section 1 for reasons that will be explained below. · When a ˆ 4=27 exactly, x1 ˆ 1=3, x2 ˆ x3 ˆ 2=3. The ®rst one has no meaning and the double second root corresponds to critical ¯ow at Section 2. · If a > 4=27, x1 < 1=3 and x2 and x3 are complex conjugate, so no useful solutions are available in this case. It is thus clear that only for 0 6 a 6 4=27 the problem has physically meaningful solutions. As regards the choice of the subcritical or supercritical solution, attention must be paid to the conditions in Section 1 and the form of the bed, zB …x†, between points 1 and 2. It is well known that transitions from subcritical to supercritical channel ¯ow can only occur at points of maximum of the bottom function [2,6,10].

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Fig. 3. Transcritical, steady ¯ow transition.

In particular, subcritical ¯ow accelerates when approaching a bottom elevation (see Fig. 3) eventually reaching critical conditions …F ˆ 1† at its top and then expanding to supercritical on the downslope side if downstream ¯ow conditions allow for it (i.e. if the ¯ow is not forced subcritical by a control structure such as a weir etc.) or slowing down back to subcritical ¯ow if not [2,6,10]. The analogy with gas ¯ow in a convergent divergent nozzle is also evident. In case water arrives supercritical to a bottom elevation, it decelerates on the uphill side, eventually reaching critical conditions at its top, then decelerating to subcritical or expanding again to supercritical ¯ow on the downhill side, depending on the downstream ¯ow conditions. From this reasoning it is clear that if there is no maximum in the bottom function between Sections 1 and 2 the ¯ow state (either subcritical or supercritical) in Section 2 will be the same as in Section 1. In case there is a maximum, ¯ow conditions would have to be computed at the crest ®rst, in order to check if critical ¯ow is reached there and then proceed downstream. Supercritical ¯ow can also be decelerated to subcritical conditions when passing across a hydraulic jump, but in that case energy is dissipated, total available head decreases and the calculation must be based on di€erent arguments. Transitions produced by a hydraulic jump are considered in Section 5. With some di€erences, the method described above is known in the literature as the standard step method [2,6] for computing water surface pro®les in channels, and usually includes the e€ects of friction but they will not be considered here because we are concerned with a frictionless Riemann problem. 4. Stationary step transitions Let us consider here frictionless shallow water ¯ow over a step of height Dz that, without loss of generality, is facing to the left (see Figs. 4 and 5). The ¯ow direction is arbitrary. If ¯ow conditions are known at one side of the step, then it is possible to apply the procedure described in previous section to obtain the conditions to the other side. A word of caution must be expressed here since actual ¯ow in the very vicinity of a step is neither frictionless nor can be assumed to ful®ll the shallow water approximations (Fig. 4). In fact a recirculation bubble appears at the inner corner that dissipates mechanical energy. One can, however, consider the ¯ow far enough upstream and downstream of the step where it has ac-

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Fig. 4. Actual steady ¯ow over a step.

Fig. 5. Idealised steady ¯ow over a step (SST).

commodated to shallow water assumptions, and water depth and velocity are again well-de®ned quantities (Fig. 5). One might also consider an idealised step ¯ow for which energy losses are neglected (nevertheless this condition can be easily relaxed, see below). The problem considered here is that of relating the ¯ow state to the left of the step …hl ; ul † to that on the right …hr ; ur †. Since a step can be thought of as an (abrupt) bottom elevation, the considerations given in previous section apply. In particular, transitions between subcritical and supercritical states are forbidden, for the step has only an up or downslope (depending on the ¯ow direction) and thus the limiting state that can be reached is critical ¯ow …F ˆ 1†. It must be pointed out that the transition ¯ow pattern is determined by the step height, Dz, plus two more variables, either the left …hl ; ul † or right …hr ; ur † state or a combination of both. Conservation of mass and energy (total head) across the step are the required relationships needed to compute the other two unknown quantities by the procedure discussed in Section 3, and thus completely determine the problem. If energy losses were to be considered, recalling the usually accepted scaling with the kinetic energy of the ¯ow [2,10], they can be computed with formulae like DHtot ˆ K

u2l 2g

…23†

where DHtot is the total head loss and K an empirical dimensionless value depending on the Reynolds number and geometry of the step. The inclusion of energy losses would complicate the

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computational procedure but would not signi®cantly modify the structure of the solution and, as it has been said before, they will not be taken into account in this work. Since in steady shallow water ¯ow across a step there can be no transitions between sub and supercritical conditions, each case will be considered separately. In subcritical ¯ow facing the step (i.e. from left to right in the con®guration chosen in this paper) water accelerates while passing over the step as pointed out in Section 3. Since speci®c discharge, q, is constant ql ˆ hl ul ˆ qr ˆ hr ur

…24†

ul < ur ! hl > hr ! Fl < Fr 6 1

…25†

then There is, however, a limit on the incoming Froude number …Fl † such that the ¯ow becomes critical to the right of the step. In such situation the step works as a control section and the ¯ow is choked as it happens in a convergent nozzle. The limiting left Froude number depends on the ratio of the step height to the incoming water depth (or to the total head depending on the choice of variables). It can be calculated by noting that ar in Eqs. (15) and (16) can be rewritten as a function of the Froude number ar ˆ

q2r 4Fr2 ˆ
3 2gHs3r 2 ‡ Fr2

…26†

but also Hsr ˆ Hsl

Dz

…27†

thus ar ˆ

q2l 2g Hsl


3 ˆ …2 ‡ Fl2 Dz

4Fl2 2Dz=hl †

3

…28†

and since Fr ˆ 1 implies ar ˆ 4=27, an algebraic equation is set that can be (iteratively) solved for Fl . Only care has to be taken to pick the subcritical …Fl 6 1† solution. Once Fl is known, ®nding the rest of the ¯ow variables is straightforward. As can be easily seen, the calculation procedure outlined in Section 3 makes no di€erence as to the direction of movement. Therefore subcritical ¯ow from right to left decelerates further when passing over the step, showing exactly the symmetric behaviour to that from left to right, and the same relationships apply, in particular inequalities (25) and the limit expressed in Eqs. (26)±(28). A supercritical stream ¯owing across a bottom elevation (i.e. from left to right in our con®guration) is decelerated to lower Froude numbers according to what was said in Section 3. The following inequalities hold ul > ur ! hl < hr ! Fl > Fr P 1

…29†

In case the ¯ow direction is reversed, a supercritical stream increases its Froude number when expanding down the step, and the same relations hold. Again, regardless of the ¯ow direction, there is a limit on the left conditions that correspond to critical ¯ow on the right. These can be

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calculated also with Eqs. (26)±(28). However, in this case the supercritical solution …Fl P 1† will be chosen. The concept of idealised, steady, 1-D ¯ow over a step as described in this section will be often recalled in the rest of this paper, and will be referred to as a stationary step transition (SST). The transition will be called subcritical if the absolute value of the ¯ow Froude number remains below 1 to both sides of the step, supercritical if jF j > 1 to both sides of it and critical in case jF j ˆ 1 to the right of the step regardless of its value to the left of it. 5. The Riemann problem with a bottom step We are seeking solutions to system (2) with the following bed function:  0 if x < 0 zB …x† ˆ Dz if x > 0 and initial conditions  Ul if x < 0 U…x; 0† ˆ Ur if x > 0

…30†

…31†

in x 2 … 1; ‡1† and for t > 0 as depicted in Fig. 6. It is worth noting that the problem is doubly singular, because not only the initial conditions are discontinuous, but also the bottom function is. Since there are no horizontal reference lengths we are led to think that its solution must be of the similarity type, therefore depending on the ratio x=t rather than on x and t separately, just as it happens for the standard Riemann problem (without a bottom step). 5.1. The standard Riemann problem solution (Dz ˆ 0) Let us recall here the ideas underlying the construction of the solution to the standard Riemann problem. It is well known that such solution is made up of simple waves (see Refs. [7,11]). There are only two types of simple waves for the 1-D SWE, namely shocks (hydraulic jumps) and

Fig. 6. Set-up of the Riemann problem with a step.

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Fig. 7. The two families of shocks.

expansions. Shocks and rarefactions, in turn, can belong to either of the two possible families. A shock, S i , belongs to the ith family if its path intersects the characteristics associated with the ith eigenvalue of the ¯ux Jacobian, ai . The equation for the characteristic lines of the ith family, C i , is dx …32† ˆ ai dt A simple sketch of the shock families and its relation to characteristics is shown in Fig. 7. The Hugoniot locus for ¯ow state U0 ˆ …h0 ; h0 u0 †T is the set of ¯ow states U ˆ …h; hu†T that can be connected to U0 by means of a shock of either family [7]. The Hugoniot locus for the SWE is composed of two one parameter families of states (one parameter family per every shock family). T We will denote by Si …U0 † the one parameter family of states U ˆ …h; hu† that can be connected i to U0 by an i-shock, S , i.e.  Si …U0 † ˆ U; F…U† F…U0 † ˆ VSi …U U0 † …33† Ci

where VSi is the shock speed. Note that Si …U0 † is a subset of the Hugoniot locus of U0 . Taking the water depth, h, of the connecting state as the parameter, one can easily ®nd for the water velocity of the S1;2 …U0 † states s   g 1 1 2 ‡ u1;2 ˆ u0  … h h0 † …34† 2 h h0 The corresponding speed of propagation of such shocks, VS , is s g h …h ‡ h0 † VS1;2 ˆ u0  2 h0

…35†

Only care has to be taken that the shock is a physically feasible hydraulic jump that complies with the proper entropy condition, expressed here in geometric form as follows: ail P VSi P air

…36†

where here subindices l and r refer to states immediately to the left and right of the hydraulic jump. It can be easily proved [10] that shocks ful®lling Eq. (36) are those that dissipate mechanical energy in contrast with nonphysical jumps for which total head is increased across them. Thus, for the SCL represented by 1-D SWE, the entropy condition can be easily understood as an energy condition.

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Fig. 8. The two families of rarefactions.

Rarefaction waves (Fig. 8), Ri , are named of the ith family if along them the ratio x=t equals the ith eigenvalue of the ¯ux Jacobian, ai x R1;2 …37† ˆ a1;2 t The set of states, U, that can be connected to a given one, U0 , by means of an i-rarefaction will be denoted by Ri …U0 †. Again taking the water depth of the connecting state as a parameter, one ®nds for the corresponding ¯uid velocity of the R1;2 …U0 † states p p u1;2 ˆ u0  2 gh0  gh …38† It can be easily seen that the leftmost and rightmost weak discontinuities (corners) of an Ri rarefaction follow the path of the corresponding characteristic lines, Cli and Cri , where here subindices l and r correspond now to the constant states to the left and right of the rarefaction wave respectively. Therefore the speeds, VRi l and VRi r , at which an Ri rarefaction spreads at either of its sides coincide with the corresponding eigenvalues VRi l;r ˆ ail;r

…39†

In general, we will denote by Wi …U0 † the set of states, U, that can be connected to U0 by means of an i-wave (either a shock or a rarefaction). It is worth noting that Wi …U0 † is a one parameter family of states. From these considerations and a little bit of algebra involving the above equations, it can be proved that a 2-wave, W 2 , be it a rarefaction or a shock, always moves faster than a 1-wave, W 1 , when they are separated by a constant state, Um . The solution to the standard Riemann problem for the 1-D SWE is thus made up of a 1-wave connecting Ul with an intermediate constant state, Um , followed by a 2-wave connecting the latter with Ur [7]. It can be represented by the following sketch: W1

W2

Ul !Um !Ur

…40†

The problem is solved once state Um is known. 5.2. Solution to the Riemann problem with a step (Dz 6ˆ 0) Let us now return to the Riemann problem with a bottom step. Without loss of generality and for all numerical applications in the rest of this paper we will consider the step of unit height (Dz ˆ 1 m).

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From similarity considerations it is clear that to either side but very close to the step …x ˆ 0 † the solution must be constant in time since if U… x; t† ˆ f … x=t†

…41†

immediately to the left and right of the step one has

U 0 ; t ˆ f 0 =t ˆ f…0 †

…42†

just like it happens without a step at x ˆ 0. In this case, however, U…0 ; t† 6ˆ U…0‡ ; t†, but they must be connected by a SST as those described in Section 3, instead. We will call these constant states U2 and U3 , i.e. U3 ˆ U…0‡ ; t† ˆ f …0‡ †

…43†

U2 ˆ U…0 ; t† ˆ f …0 †

…44†

understanding that U3 ˆ SST…U2 †

…45†

the above equation meaning that U3 is the conjugate of U2 across a SST (and vice versa). Away from the step, the bottom surface function is ¯at. Also, the initial constant states Ul and Ur stand far to the left and right of the step at any (®nite) time. It is therefore reasonable to think that between the step and the two initial constant states, the solution is made up of simple waves connecting Ul and U2 and then U3 and Ur , just like it happens in the standard Riemann solution. Since U2 uniquely determines U3 (and vice versa), only two parameters need to be found, for instance the two components of state U2 , (h2 and u2 ), what leaves room for only two simple waves. (Recall that the Wi …U0 †'s are one parameter families of states.) Hence, the solution could be represented by the following diagram: W1

W2

SST

Ul !U2 ! U3 !Ur

…46†

However there is no a priori reason why the waves do not travel or spread across the step, what would give rise to other possibilities than that shown above. 5.2.1. A simple case Before attacking more general cases, let us concentrate on the diagram represented by Eq. (46). A particular situation of such type can be obtained with initial conditions hl ˆ 4 m ul ˆ 0 m s

1

hr ˆ 1 m ur ˆ 0 m s

1

…47†

that produce a 1-rarefaction spreading to the left and a 2-shock travelling right as can be seen in Fig. 9 at t ˆ 1 s, where the water free surface, h ‡ zB , (left) and Froude number, F, (right), have been plotted versus x. The solution at t ˆ 0 s can also be seen as a dotted line, together with the step bottom function, zB …x† in solid, on the left graph (Dz ˆ 1 m). It has been deemed more adequate to plot Froude number, F, instead of water velocity, u, since the former gives direct information on the criticality of the ¯ow.

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657

Fig. 9. 1-Rarefaction followed by a subcritical transition and a 2-shock (t ˆ 1 s). (Left) Free surface elevation. (Right) Froude number.

The steps to obtain U2 and U3 for the above situation can be described as follows: compute the one parameter family of states R1 …Ul †,
compute the conjugate across the step transition of the above family, SST R1 …Ul † , compute the one parameter family of states S2 …Ur †,
compute the conjugate across the step transition of the above family, SST S2 …Ur † , obtain U2 from the intersection
U2 ˆ R1 …Ul † \ SST S2 …Ur † 6. obtain U3 from the intersection
U3 ˆ S2 …Ur † \ SST R1 …Ul † 1. 2. 3. 4. 5.

…48† …49†

The last step can be spared recalling Eq. (45) once U2 has been found from step 5. A sketch of the solution procedure on the phase plane …h; F † can be seen in Fig. 10 while the particular state-wave diagram reads R1

SST

S2

Ul !U2 ! U3 !Ur

…50†

Fig. 10. Solution in the phase plane …h; F †.

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5.2.2. The general case For the particular situation described in the former paragraph the simple waves were, from left to right, a R1 and a S 2 , but other initial conditions can give rise to di€erent combinations of rarefactions and shocks. Note however that the leftmost running wave must necessarily belong to the ®rst family, W 1 , while the rightmost running wave must be of the second family, W 2 . Otherwise the left wave would catch up with the right one, or rather, the initial discontinuity would never break up in running waves. Furthermore, depending on the velocity of propagation (shocks) or the spreading rate (rarefactions) the waves can be located to the left or right of the step, or even accommodate it in between. This last possibility can only be accomplished by rarefactions that have a ®nite (increasing) width, while shocks, which are actual discontinuities, must propagate either to the left or to the right of the step. In any case, the computation of the solution is based upon the same principle: Find the appropriate simple waves satisfying the entropy conditions that connect Ul and Ur via states U2 and U3 across the step. The possible di€erent con®gurations found in the course of this work are discussed in next section.

6. The set of solutions Among the diverse classi®cation criteria for the solutions, we have chosen one based on the nature of the simple waves involved. Hence, one set of cases consists of solutions containing 1rarefactions and 2-shocks, another set 1-shocks and 2-shocks and so forth. Within one set, different cases are considered regarding the location of the simple waves with respect to the step position (located at x ˆ 0). 6.1. Solutions containing R1 and S 2 Starting with the case when both R1 and S 2 propagate to the left of the step one ®nds the following ®ve situations. 6.1.1. 1-Rarefaction, 2-shock and supercritical SST In this case the ¯ow is supercritical from right to left in the whole domain. Therefore U3 ˆ Ur and another intermediate state, Um , separates R1 and S 2 . The state-wave diagram reads R1

S2

SST

Ul !Um !U2 ! U3 ˆ Ur

…51†

A numerical example corresponding to the initial conditions hl ˆ 4 m ul ˆ 10 m s

1

hr ˆ 1 m ur ˆ 6 m s

1

…52†

can be seen in Fig. 11. As in Fig. 10, left plot corresponds to the water free surface, f ˆ h ‡ zB , at t ˆ 0:5 s, where also the initial condition is visible as a dotted line, together with the bottom step in solid (Dz ˆ 1 m). Right plot shows Froude number distribution, with dotted initial data.

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659

Fig. 11. 1-Rarefaction, 2-shock and supercritical transition (t ˆ 0:5 s). (Left) Free surface elevation. (Right) Froude number.

A degenerate case of this con®guration arises when the initial conditions for x < 0 (Ul ) produce a retreating supercritical R1 , but there is no water to the right of the step. The solution then contains only R1 and if the null state is denoted by 0 ˆ …0; 0†, the state-wave diagram is R1

SST

Ul !0! 0 ˆ Ur

…53†

Such solution appears when ul < 2cl < 0 and hr ˆ 0 and is completely analogous to the wellknown dam break over a ¯at dry bed, save for a superimposed negative velocity overall. 6.1.2. 1-Rarefaction, 2-shock, critical SST and 2-rarefaction This three wave pattern can be seen in Fig. 12 and is caused by the restrictions regarding transcritical step transitions (see below). The state-wave diagram is here R1

S2

SST

R2

Ul !Um !U2 ! U3 !Ur

…54†

and U3 is necessarily a critical state, i.e. jF3 j ˆ 1. As explained in Sections 3 and 4 a subcritical ¯ow will remain subcritical when crossing a bottom step and the same will happen for a supercritical stream. Only if the ¯ow regime is exactly

Fig. 12. 1-Rarefaction, 2-shock and critical transition followed by a 2-rarefaction (t ˆ 0:5 s). (Left) Free surface elevation. (Right) Froude number.

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critical …jF j ˆ 1† at the highest side of the step, has the stream the choice of expanding to supercritical conditions or slow down to subcritical ones down the step. The actual choice depends of course on the ¯ow conditions down the step. For this case S 2 is a left propagating shock, connecting states Um and U2 (recall that superindices here denote the shock or eigenvalue family, and not the square of the quantity), i.e. VS2 < 0

…55†

and from the entropy condition of a 2-shock one has a2m > VS2 > a22 ˆ u2 ‡ c2

…56†

hence u2 <

c2

…57†

what means that the ¯ow must be from right to left, and supercritical to the left of the step. These conditions were also met in previous case, but the ¯ow there was supercritical everywhere. In such case there was no restriction to the ¯ow to accelerate from a supercritical state to a faster supercritical one while passing across the stationary step transition. In the present situation, however, the ¯ow is subcritical to the right of the step and the only way that supercritical conditions can be achieved to the left of it is by reaching ®rst critical ¯ow at its top. This is accomplished by means of a right running rarefaction, what explains the need for the third wave, R2 , connecting Ur with the ®xed critical conditions U3 . Note that only an R2 can bring the stream to F3 ˆ 1 at the step (an R1 expands towards the right, see Fig. 8). Also it is easily seen by manipulating the corresponding entropy conditions that either an S 1 or an S 2 to the right of the step cannot meet the F3 ˆ 1 requirement. The particular initial conditions for the ¯ow in Fig. 12 are hl ˆ 4 m ul ˆ 10 m s

1

hr ˆ 2 m ur ˆ 0 m s

1

…58†

6.1.3. 1-Rarefaction, subcritical SST and 2-shock This case has been thoroughly explained earlier in Section 5.2.1 when discussing the general solution structure. A practical example can be seen in Fig. 9. Note that ¯ow now is from left to right in contrast with the two previous cases. 6.1.4. 1-Rarefaction, critical SST, 1-rarefaction and 2-shock A natural sequel to previous case when the R1 becomes transcritical, thus extending across the time axis and therefore embracing the step. The following state-wave diagram applies: R1

SST

R1

S2

Ul !U2 ! U3 !Um !Ur

…59†

where again U3 is a critical state, F3 ˆ 1. Note that R1 extends only to a certain position left of the step connecting then with constant state U2 . The rightmost position of R1 is ®xed by the condition that the conjugate of U2 across the step transition leads to a critical U3 and can be computed using Eqs. (26)±(28). Fig. 13 shows a practical situation.

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661

Fig. 13. Transcritical 1-rarefaction across the step and 2-shock (t ˆ 0:5 s).

This con®guration also admits a dry bed solution when Ur is the vacuum. The state-wave diagram is then R1

SST

R1

Ul !U2 ! U3 !0 ˆ Ur

…60†

and U3 remains critical. 6.1.5. Supercritical SST, 1-rarefaction and 2-shock Flow is supercritical to the left of the step and decelerates to slower supercritical conditions across it. Then it expands along R1 and reaches S 2 . The state-wave diagram is SST

R1

S2

Ul ˆ U2 ! U3 !Um !Ur

…61†

An example is shown in Fig. 14, where it may seem surprising the fact that the ¯ow is supercritical to the left of S 2 and is even more supercritical to its right. It must be borne in mind that S 2 is a right running shock and that if brought to rest by an appropriate Galilean velocity transformation, the ¯ow hits the shock in supercritical conditions and leaves it in subcritical ones as expected. If Ur ˆ 0, S 2 disappears (in fact is a vanishing intensity shock at the foot of R1 ) and one has the dry bed pattern

Fig. 14. Supercritical transition, 1-rarefaction and 2-shock (t ˆ 0:5 s).

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F. Alcrudo, F. Benkhaldoun / Computers & Fluids 30 (2001) 643±671 SST

R1

Ul ˆ U2 ! U3 !0 ˆ Ur

…62†

6.2. Solutions containing S 1 and S 2 Since the waves have zero width in this set of cases, there is no possibility that they spread across the step what in some situations forces the appearance of an additional rarefaction wave to match with critical conditions at the crest of the step. Again, ®ve di€erent con®gurations have been found. 6.2.1. 1-Shock, 2-shock and supercritical SST This situation corresponds to supercritical ¯ow from right to left hitting two consecutive left moving shocks. Needless to say that S 1 moves faster to the left than S 2 (VS1 < VS2 < 0) as expected. Note that U3 ˆ Ur giving the following state-wave diagram: S1

S2

SST

Ul !Um !U2 ! U3 ˆ Ur

…63†

A typical situation is shown on Fig. 15. 6.2.2. 1-Shock, 2-shock, critical SST and 2-rarefaction This is the analogue to the case described in Section 6.1.2 but for two shocks. An R2 is needed on the right of the step to accelerate the ¯ow to critical conditions at its upper border leading to supercritical ¯ow down the step. The state-wave diagram takes the form S1

S2

R2

SST

Ul !Um !U2 ! U3 !Ur

…64†

Note that U3 must be a critical state …F3 ˆ 1†. It can be seen in Fig. 16. 6.2.3. 1-Shock, subcritical SST and 2-shock This corresponds to case 6.1.3 with two shocks, the ®rst one moving to the left and the second one to the right. An example is shown in Fig. 17, while the state-wave diagram reads S1

SST

S2

Ul !U2 ! U3 !Ur

Fig. 15. 1-Shock, 2-shock and supercritical transition (t ˆ 0:5 s).

…65†

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663

Fig. 16. 1-Shock, 2-shock and critical transition followed by 2-rarefaction (t ˆ 1 s).

Fig. 17. 1-Shock followed by a subcritical transition and a 2-shock (t ˆ 1 s).

6.2.4. 1-Shock, critical SST, 1-rarefaction and 2-shock This situation is a limiting case to the previous one when the step transition is choked as a consequence of increasing initial left velocity. Conditions to the left of the step lead to critical ¯ow to its right, thus F3 ˆ 1. An R1 must accommodate the ¯ow to meet the second shock S 2 . It is analogous to Section 6.1.2 replacing the left R1 with a S 1 leading to the following state-wave diagram S1

SST

R1

S2

Ul !U2 ! U3 !Um !Ur

…66†

The ¯ow is shown in Fig. 18. This ¯ow pattern may exhibit a dry bed situation if Ur is the null state, 0. The following statewave diagram then applies S1

SST

R1

Ul !U2 ! U3 !0 ˆ Ur

…67†

while U3 remains a critical state. Furthermore in case the shock intensity is small enough that h2 < Dz, the water cannot proceed over the step and the solution is reduced to S1

Ul !U2 ;

x<0

…68†

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Fig. 18. 1-Shock, critical transition, 1-rarefaction and 2-shock (t ˆ 0:7 s).

and 0 elsewhere, i.e. just a 1-D shock re¯ection. 6.2.5. Supercritical SST, 1-shock and 2-shock Finally when the ¯ow is supercritical from left to right across the transition the two shocks are pushed to the right of the step and one has the following pattern SST

S1

S2

Ul ˆ U2 ! U3 !Um !Ur

…69†

that can be seen in Fig. 19. 6.3. Solutions containing S 1 and R2 These solutions are analogous to those of Section 6.1 except for slight di€erences due to the asymmetry introduced by the step. 6.3.1. 1-Shock, 2-rarefaction and supercritical SST Supercritical ¯ow from right to left across the step pushing S 1 and R2 to the left. U3 coincides with Ur

Fig. 19. Supercritical transition, 1-shock and 2-shock (t ˆ 0:5 s).

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665

Fig. 20. 1-Shock, 2-rarefaction and supercritical transition (t ˆ 0:5 s). S1

R2

SST

Ul !Um !U2 ! U3 ˆ Ur

…70†

A practical situation can be seen in Fig. 20. If one has a dry bed as left initial condition, S 2 vanishes and the state-wave diagram is reduced to R2

SST

Ul ˆ 0!U2 ! U3 ˆ Ur

…71†

6.3.2. 1-Shock, 2-rarefaction, critical SST and 2-rarefaction Analogous to previous case when the ¯ow is not supercritical to the right of the step. The R2 is transcritical, containing the time axis and thus embracing the step. Note that U3 is a critical state …F3 ˆ 1†. The state-wave diagram takes the form S1

R2

SST

R2

Ul !Um !U2 ! U3 !Ur

…72†

An example is shown in Fig. 21. This con®guration also admits a dry bed solution in case Ul ˆ 0, the vacuum state. Again S 2 vanishes and the state-wave diagram is reduced to R2

SST

R2

Ul ˆ 0!U2 ! U3 !Ur

Fig. 21. 1-Shock and transcritical 2-rarefaction across the step (t ˆ 0:5 s).

…73†

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Fig. 22. 1-Shock followed by a subcritical transition and a 2-rarefaction (t ˆ 1 s).

and U3 remains critical …F3 ˆ 1†. 6.3.3. 1-Shock, subcritical SST and 2-rarefaction Completely symmetric to the simple solution of Section 6.1.3, the ¯ow pattern can be seen in Fig. 22. S1

SST

R2

Ul !U2 ! U3 !Ur

…74†

6.3.4. 1-Shock, critical SST, 1-rarefaction and 2-rarefaction Here the step transition is choked as in Section 6.2.4 and a third wave (R1 ) is needed to connect critical state U3 with the R2 . The state-wave diagram reads S1

SST

R1

R2

Ul !U2 ! U3 !Um !Ur

…75†

while a typical ¯ow picture is displayed in Fig. 23. 6.3.5. Supercritical SST, 1-shock and 2-rarefaction Flow is supercritical from left to right across the transition and both waves are pushed to the right of the step corresponding to the following state-wave diagram

Fig. 23. 1-Shock, critical transition, 1-rarefaction and 2-rarefaction (t ˆ 0:7 s).

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667

Fig. 24. Supercritical transition, 1-shock and 2-rarefaction (t ˆ 0:5 s). S1

SST

R2

Ul ˆ U2 ! U3 !Um !Ur

…76†

that can be seen in Fig. 24. A dry bed solution appears if Ul ˆ 0 and ur > 2cr > 0: SST R2

Ul ˆ 0! 0!Ur

…77†

which is just a dam break over dry bed solution with a superimposed translation to the right. 6.4. Solutions containing R1 and R2 In case only rarefactions are involved the solution to the Riemann problem is always made up of only two waves (R1 and R2 ). There is no need to introduce a third wave to connect critical ¯ow on top of the step with contiguous states. This is due to the fact that rarefactions have a ®nite width and can therefore accommodate the step within them reaching critical conditions if needed. 6.4.1. 1-Rarefaction, 2-rarefaction and supercritical SST Supercritical ¯ow from right to left across the step pushing R1 and R2 to the left as in Section 6.3.1. U3 coincides with Ur leading to the state-wave diagram

Fig. 25. 1-Rarefaction, 2-rarefaction and supercritical transition (t ˆ 0:35 s).

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Fig. 26. 1-Rarefaction followed by transcritical 2-rarefaction across the step (t ˆ 0:3 s). R1

R2

SST

Ul !Um !U2 ! U3 ˆ Ur

…78†

As can be seen in Fig. 25. Note that a dry bed solution appears if Um ˆ 0. 6.4.2. 1-Rarefaction, 2-rarefaction, critical SST and 2-rarefaction Analogous to case 6.3.2 when S 1 is replaced by R1 . The R2 is transcritical, containing the time axis and the step. U3 is a critical state …F3 ˆ 1†. The state-wave diagram takes the form R1

R2

SST

R2

Ul !Um !U2 ! U3 !Ur

…79†

A practical con®guration is shown in Fig. 26. In the limit, again, Um can be the vacuum state. 6.4.3. 1-Rarefaction, subcritical SST and 2-rarefaction The two rarefactions are separated by the step transition as it happens with the shocks in Section 6.2.3 R1

SST

R2

Ul !U2 ! U3 !Ur An example can be seen in Fig. 27.

Fig. 27. 1-Rarefaction followed by a subcritical transition and a 2-rarefaction (t ˆ 0:5 s).

…80†

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669

Fig. 28. Transcritical 1-rarefaction across the step followed by 2-rarefaction (t ˆ 0:7 s).

6.4.4. 1-Rarefaction, critical SST, 1-rarefaction and 2-rarefaction Here R1 becomes supercritical extending across the step what corresponds to the following state-wave pattern R1

SST

R1

R2

Ul !U2 ! U3 !Um !Ur

…81†

Note that U3 is a critical state. The ¯ow picture is displayed in Fig. 28. A dry bed solution appears in the limit Um ! 0. 6.4.5. Supercritical SST, 1-rarefaction and 2-rarefaction Finally supercritical ¯ow from left to right across the step pushes the two rarefactions to the right in complete analogy with previous 6.*.5 cases. The state-wave diagram is then SST

R1

R2

Ul ˆ U2 ! U3 !Um !Ur

…82†

An example of the ¯ow pattern can be seen in Fig. 29. This con®guration can also show a dry bed solution.

Fig. 29. Supercritical transition, 1-rarefaction and 2-rarefaction (t ˆ 0:35 s).

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7. Conclusions A set of Riemann solutions to the 1-D SWE with a particular bottom function has been presented. The problem is very speci®c and its interest lies more on the academic side rather than on its possible use in describing a natural shallow water ¯ow (although a dam break ¯ow with a bottom step has received some attention by the hydraulics engineering community [13]). The functions presented are built on the basis of the physical meaning of the SWE together with the theory of SCL in one space dimension and are classi®ed attending to their wave nature and position. The classi®cation procedure casts much light on the solution structure and, although no proof can be given, indicates that there exist no other con®gurations than those shown here. It is to be remarked the rich solution pattern found with at least 20 di€erent cases (in contrast with four in the ¯at bed Riemann problem). This set of solutions may be of help to the numerical computing community interested in solving the SWE in as much as source terms treatment is not a fully understood issue, despite the considerable progress made. The cases presented here can be used as a validation tool in testing source term discretisations or even help in ®nding appropriate integration schemes. Regarding this issue, it must be said that a direct Godunov type approach based on the exact solutions shown here does not seem practical at this stage. It should be borne in mind that there are 20 fully nonlinear di€erent patterns (plus the limiting dry bed cases) and the sorting and computing costs associated with embedding the solutions into a Godunov scheme could be prohibitive. Although work is being conducted to clarify this possibility, more success may be expected of using the ideas underlying the construction of the exact solutions to enhance classical source term discretisations.

Acknowledgements This work started while the ®rst author was a visiting professor at INSA de Rouen during Summer 1999. Financial support provided by INSA de Rouen and Programa Europa de Estancias de Investigaci on CAI-CONSID(DGA) is gratefully acknowledged. He also feels himself indebted to Prof. F. Benkhaldoun for the stimulating atmosphere and his help during that period. References [1] Alcrudo F, Benkhaldoun F. A new insight on shallow water equations with source terms. Internal Report August 99, Laboratoire LMI, Institut National de Sciences Appliquees de Rouen, Rouen, France, 1999. [2] Chow VT. Open-channel hydraulics. Kogakusha: McGraw-Hill; 1959. [3] Glaister P. Approximate Riemann solutions of the shallow water equations. J Hydraulic Res 1988;26:293. [4] Goutal N, Maurel F. Proceedings of the 2nd Workshop on Dam-Break Wave Simulation, Technical Report HE-43/ 97/016/A, Electricite de France, Departement Laboratoire National d'Hydraulique, Groupe Hydraulique Fluviale, 1997. [5] Greenberg JM, LeRoux A. A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J Numer Anal 1996;33(1):1. [6] Hwang NHC, Houghtalen RJ. Hydraulic engineering systems. NJ: Prentice-Hall; 1996. [7] LeVeque R. Numerical methods for conservation laws. Basel: Birkh auser; 1990.

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671

[8] LeVeque R. Balancing source terms and ¯ux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm. J Computat Phys 1998;146:346±65. [9] LeVeque R, Yee HC. A study of numerical methods for hyperbolic conservation laws with sti€ source terms. J Computat Phys 1990;86:187.  [10] Silber R. Etude et trace des ecoulements permanents en canaux et rivieres. Paris: Dunod; 1968. [11] Stoker W. Water waves. New York: Wiley; 1957. [12] Toro EF. Riemann solvers and numerical methods for ¯uid dynamics. Berlin: Springer; 1997. [13] UCL-Service d'Hydraulique. Meeting Report of the IAHR Working Group on Dam Break Modelling, June 1997, Louvain-La-Neuve, Belgium. Edited by Service d'Hydraulique, Universite Catholique de Louvain, Belgium, 1997. [14] Vazquez-Cend on ME. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J Computat Phys 1999;148:497±526. [15] Vreugdenhil CB. Numerical methods for shallow-water ¯ow. Dordrecht: Kluwer; 1994. [16] Watson G, Peregrine DH, Toro EF. Numerical solution of the shallow water equations on a beach using the weighted average ¯ux method. In: Hirsch C, Periaux J, Kordulla W, editors. Proceedings of the First European Computational Fluid Dynamics Conference, vol. 1. Elsevier Science Publishers, 1992. p. 495±502.