Undular bores and secondary waves - Experiments and hybrid finite volume modelling

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Revision received January 2, 2001. Open for discussion till August 31, 2002.

JOURNAL OF HYDRAULIC RESEARCH, VOL. 40, 2002, NO. 1 33

Undularboresandsecondarywaves–Experimentsandhybridfinite-volumemodel-ling

Ondes secondaires de Favre – Mesures expérimentales et modélisation par volumesfinis hybrides

SANDRA SOARES FRAZÃO, research fellow Fonds National de la Recherche Scientifique, and YVES ZECH, professor, Departmentof Civil Engineering, Université catholique de Louvain, B 1348 Louvain-la-Neuve, Belgium

ABSTRACTSecondary free-surface undulations (Favre waves), appearing for example after the opening of a sluice gate or at the head of a bore, cannot be repro-duced by numerical models based on the hydrostatic pressure assumption. The Boussinesq equations take into account the extra pressure gradients butare difficult to integrate due to the high-order derivative terms. The paper describes the physics of wave initiation and proposes a demonstration of theBoussinesq equation based on relatively wider assumptions than usually adopted. A linear stability analysis is developed in finite-difference frame tohighlight some potential source of numerical instabilities. These conclusions are transposed in a new hybrid finite-volume / finite-difference scheme,which reveals a better accuracy in period and amplitude when evaluated against experiments.

RÉSUMÉLes ondes secondaires de surface (ondes de Favre) apparaissant par exemple à l’ouverture d’une vanne de fond ou associées à la tête d’un ressautmobile, ne peuvent être représentées par des modèles numériques basés sur l’hypothèse d’une distribution hydrostatique de pression. Les équationsde Boussinesq tiennent compte des pressions non hydrostatiques mais elles sont difficiles à intégrer avec précision à cause des termes contenant desdérivées partielles d’ordre supérieur. L’article décrit physiquement la naissance des ondes de Favre et propose une démonstration des équations deBoussinesq fondée sur des hypothèses plus larges que celles qui sont généralement utilisées. Une analyse de stabilité linéaire, développée dans le cadrede différences finies, a permis de mettre en évidence quelques sources potentielles d’instabilités numériques. Ces conclusions sont transposées à unschéma nouveau utilisant de manière hybride les volumes finis et les différences finies. Cette méthode, confrontée à des résultats expérimentaux, serévèle plus précise, tant pour la période que pour l’amplitude des ondes.

1. Introduction

The Saint-Venant shallow-water equations, widely used for mod-elling free-surface flows, rely on an important assumption: as thevertical velocities are assumed to be negligible, the resulting pres-sure distribution is hydrostatic. This representation has proven tobe of satisfying accuracy for representing a wide range of situa-tions like flood flows, or even dam-break induced waves. How-ever, some features like secondary free-surface undulations can-not be reproduced by numerical models based on the hydrostatic-pressure assumption.In nature, such undular wave trains can be observed for exampleafter the opening of a sluice gate. Depending on the flow condi-tions, waves with amplitude up to twice the initial bore amplitudecan be generated. Those waves were investigated experimentallyfor the first time by Favre [4] in a rectangular channel. The prob-lems addressed by Favre, and which led him to his important ex-perimental work are linked to hydropower plants and navigationlocks. Rapid operation of those systems can induce waves withheights becoming so important that they can damage the channeland river banks. His experimental set-up allowed Favre to mea-sure the development of a wave train induced by a rapid openingor closing of a gate.More recent experimental work of the same kind has been per-formed by Treske [17] in channels of rectangular and trapezoidalcross-sections, and by Marche et al. [10] in the dam-break frame-

work.The description of Favre waves in a depth-averaged mathematicalmodel requires an extension of the shallow-water equations, suchas proposed by Boussinesq [2]. By adopting certain realistic as-sumptions on the distribution of vertical and horizontal velocitycomponents, extra-terms can be added to the momentum equa-tion, accounting for the changes in the pressure distribution. Morerecent developments also contributed to the understanding ofthose weakly non-hydrostatic wave propagation phenomena(Peregrine [12], Schröter [15], Steffler and Jin [16], Prüser andZielke [13], Marche et al. [10], Nadiga et al. [11]) and are stillbased on the outstanding initial theoretical work by Boussinesq.Several finite-difference schemes have been developed to solveBoussinesq equations. Peregrine [12] derived such a finite-differ-ence scheme to compute the development of an undular bore,starting from a smooth initial water profile. However, it must bestated that the proposed scheme is unable to compute waves witha steep initial profile (discontinuous free surface).More recently, Schröter [15] developed a finite-difference schemefor the Boussinesq equations written in variables (h, q) instead of(h, U), h being the water depth, q the unit discharge and U themean velocity. To ensure the stability of the scheme, the Courant-Friedrichs-Lewy (CFL) number is limited to 0.7, and an addi-tional numerical filter (spline smoothing of computed results) isintroduced. This scheme was used by Prüser and Zielke [13] andchecked against measurements by Treske [17]. Although good

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34 JOURNAL OF HYDRAULIC RESEARCH, VOL. 40, 2002, NO. 1

Fig. 1 Initial waveFig. 2 Extra horizontal pressure gradients due to flow filaments curva-

ture, resulting vertical acceleration and free-surface deformation

agreement was obtained for the wave height, significant discrep-ancies appeared in the wave celerity.A study on Favre secondary waves associated to a dam-break-induced bore was carried out by Marche et al. [10]. Those authorscomparedtheirexperimentalmeasurementswithcomputationsbya finite-difference model with artificial viscosity (Gharangik andChaudhry [5]).All those numerical models present the disadvantages to be eitherunstable for initially steep wave profiles, or to need the explicitintroduction of artificial viscosity to avoid spurious oscillationsand instabilities. An attempt is made here to propose a stable fi-nite-volume scheme to compute secondary oscillations even aris-ing from a steep front.New experimental work has been carried out and is also describedin this paper. The results were compared to those obtained byTreske and Favre and showed a good coherence. In consequence,they were used to validate the proposed numerical scheme

2. Physics of secondary wave initiation

Favre waves, investigated in the laboratory of the Civil Engineer-ing Department at the Université catholique de Louvain, weregenerated in two different ways. One way was to induce a suddenincrease of the discharge in the channel, by raising up rapidly agate separating two regions of initially constant water depths.This is similar to the experiments carried out by Favre himself,where secondary undulations immediately grow on the initiallysteep front. Another possibility was to increase the discharge ina flume initially at rest by starting up a pump. The latter methodcannot lead to an abrupt rise of the discharge and to a steep initialwater profile, as the discharge increase is limited by the inertia ofthe pump. In this case, what is observed is a rather smooth in-crease in water level, leading to a wave profile like that illustratedin figure 1. This initially smooth positive wave will steepen whiletravelling in the downstream direction, and undulations slowlygrow at the front head. The growth of these undulations, occur-ring in both cases of sudden and progressive increase in dis-charge, is linked to the departure from hydrostatic pressure distri-bution near the bore front, and is best explained when startingfrom an initially smooth wave profile. In that case, the assump-tions of small curvature of fluid filaments and hydrostatic pres-sure distribution hold at least until the wave has reached a statewhere the steepness is such that curvature effects can no longerbe neglected.Peregrine [12] shows how the wave steepening leads to local ex-tra pressure gradients due to the vertical acceleration of the water.However, some aspects of his description appear unsatisfactory,

and an attempt to clarify the situation is made here. Consider along wave in shallow-water like that in figure 1, resulting for ex-ample of a progressive increase in discharge. The initial phase ofthe motion is well described by the Saint-Venant equations, as thechange in water level is smooth enough and the pressure in thewater is effectively close to a hydrostatic distribution. When trav-elling, the wave steepens because its local celerity depends on thewater depth in such a way that the top of the wave (point A)moves faster than the toe (point E). The water surface curvaturebecomes then sufficiently important to affect the pressure distri-bution significantly. Suppose the wave in figure 1 has reachedthis state.Points B and D are the points of maximum water-surface curva-ture, and thus of maximum divergence with the hydrostatic pres-sure distribution. Curvature at B is such that the pressure there isless than hydrostatic, while at D it is greater than hydrostatic.Those alterations of vertical pressure distribution will induce hor-izontal pressure gradients as sketched in figure 2. Those, in turn,will generate additional horizontal currents, which, by continuity,will result in vertical displacements of the water surface. The freesurface will be raised at B and lowered at D. Points E and A willalso undergo the same phenomenon, but in a less significant way.This process will continue and a sequence of waves is formed,which will grow in amplitude until reaching an equilibrium, orbreaking.

3. Boussinesq equations

The classical shallow-water depth-averaged equations are ob-tained on the basis of the hydrostatic pressure assumption. How-ever, the process of vertical integration of the three-dimensionalequations of hydrodynamics is by no means limited to nearly hor-izontal flows and can be extended to flows where some verticalacceleration is allowed. One of the simplest and most widely ap-

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JOURNAL OF HYDRAULIC RESEARCH, VOL. 40, 2002, NO. 1 35

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plicable extensions is the one proposed by Boussinesq where theinclination of the fluid filaments is supposed to increase linearlyfrom zero at the bed to a maximum at the free surface.The demonstration we propose here slightly differs from the clas-sical textbooks (Liggett [9], Peregrine [12]) in the sense thatirrotational-flow assumption is not really required for derivingBoussinesqdepth-averaged equations.These areobtained by inte-gration of the continuity and movement equations of inviscidfluid.Assuming two-dimensionalflow(widerectangularchannel)these equations read

To highlight the influence of the vertical velocity component it isconvenient to define ϕ = w/u as the inclination of the fluid fila-ments regarding the bed, i.e. a kind of measure of the non-paral-lelism of the streamlines. Making use of the continuity equation(1), the momentum equations (2) and (3) can be rewritten as

The depth-averaged continuity equation is obtained by integrating(1) over the total water depth h. Using Leibniz’s integration rule,and making use of the bottom boundary condition w0 = 0 and ofthe free-surface kinematic condition ws = h,/ t + u h/ x the inte-gration of (1) from z = 0 to z = h yields the classical form

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dzuhU0

The following assumptions are needed to integrate (4) and (5) :(1) the streamline inclination ϕ increases linearly from zero at thebed to a maximum ϕs at the free surface; (2) the local velocitymay be written u = U + u where U is the depth-averaged velocityand u the deviation with depth from this mean value, supposedsmall compared to U ; (3) the velocity deviation u , the surfaceinclination ϕs and the various partial derivatives / t (progressiveevolution) and / x (long waves) are small enough to consider thata product of at least two of these functions is negligible comparedto only one of these. Applied to the water surface (kinematic con-dition), these assumptions yield

in such a way that (6) is rewritten, taking into account the linearvariation of ϕ

Integrating (8) from z = 0 up to z and making use of the continuityequation (6) yields

Let us now consider the momentum equations (4) and (5). Inte-grating (5) from z to the free surface z = h where p = ps gives anexpression for the pressure distribution with an additional termcomplementing the usual hydrostatic distribution :

We can now integrate the equation of motion (4) from z = 0 toz = h, replacing the pressure p by (10) and considering that thedepth-integration of the velocity deviation u is zero :

Equation (11) is similar to the expression proposed by Boussinesqin 1877 and differs from the Saint-Venant momentum equationby the third-order derivative terms. Multiplying (11) by h, addingthe depth-averaged continuity equation (6) multiplied by U, andmaking use of the continuity equation (6) to transform some ofthe terms in brackets yields

The latter form of the momentum equation is called ‘quasi-con-servative’, as the left-hand side is exactly the classical homoge-neous and conservative Saint-Venant equation, From this point ofview, the non-hydrostatic terms (expression in brackets) on theright-hand side of (12) appear as source terms.

4. Numerical model

Solving the Boussinesq equations (6) and (12) requires a some-what different scheme than the classical finite-volume integrationof the Saint-Venant shallow-water equations. The motion equa-

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36 JOURNAL OF HYDRAULIC RESEARCH, VOL. 40, 2002, NO. 1

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tion (12) involves third-order derivatives in time and space lead-ing on the one hand to a relaxation of the time-step CFL conditiondue to the dispersive character of (12), and on the other hand tothe unfeasibility of a pure finite-volume scheme which requiresa fully conservative form of all the partial derivatives. Thedispersive character of (12) can be demonstrated in the case ofsmall-amplitude waves. The same assumption yields a frame fora linear stability analysis of a finite-difference scheme followingDeleersnijder [3], which highlights the need to express someterms in implicit formulation. The numerical approach used hereconsists in applying a finite-volume technique for the conserva-tive terms on the left-hand side of (12) while finite-differences areused for the non-hydrostatic terms on the right-hand side. Thishybrid scheme takes advantage of the accurate finite-volume for-mulation for the predominant convective and pressure terms with-out neglecting the secondary-wave terms. As an implicit treat-ment of some terms is required for ensuring the numerical stabil-ity, a predictor-corrector technique, consistent with the conclu-sions of the finite-difference stability analysis, is proposed.

4.1. Dispersive character of the Boussinesq equations

Additional assumptions are needed for a linear stability analysisof the system (6)-(12). Considering long waves, we can split thewater depth h in

where H is the mean water depth assumed constant in time andspace and η the elevation of the free surface above this referencelevel. The long-wave assumption implies that the wave amplitudeis small compared to the mean depth η << H. Moreover, we onlyconsider shallow-water cases where the water depth is small com-pared to the wavelength H << λ. This latter assumption may beproved as equivalent to the condition U / x << / t which impliesthat the water velocity is small compared to the wave speed (seee.g. Lamb [8]). The continuity equation (6) then reads

The motion equation (11) can be handled similarly after simplify-ing the terms in brackets in the same way as in (12) :

where ψ is a constant equal to unity introduced to ‘mark’ the non-hydrostatic correction. The system (14)-(15) admits a plane-wavesolution of the type

consisting in one term of a Fourier-series expansion of an oscilla-tory wave. In (16), Re is the ‘real part’ operator, A and B are com-plex constants, λ = cT the wavelength, c the wave celerity, T theperiod, k = 2π/λ is the wave number and ϕ = 2πc/λ = 2π/T theangular frequency.Substituting (16) into (14) and (15) yields a homogenous systemof algebraic equations

which has a solution only if its coefficient matrix is singular. Thislatter condition gives the dispersion relation

i.e. a relation giving ω as a function of k, leading to the followingexpression of the wave celerity

If the celerity is a function of the wave number k and thus of thewavelength λ, the waves are called ‘dispersive’, because wavesof different lengths, propagating at different speed, ‘disperse’, orseparate (Kundu [7]). We can see that the non-hydrostatic termhere is responsible for the dispersive character of the waves. In-deed, neglecting this term (ψ = 0), we recover the shallow-waterequations in which the wave speed or celerity is inde-Hgc =pendent of the wavelength. This dispersive nature of the waveschallenges the numerical scheme to reproduce accurately thewave periods.

4.2. Linear stability analysis for finite-difference scheme

In a space-time grid where n is the time (n∆t) and j the location(j∆x), equations (14) and (15) read in finite-difference form

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Note that the second term of (21) g η/ x is defined at time n+1,which is required for the stbility of the numerical scheme.Assuming a plane-wave solution

with θ = k∆x and where α = e-iω∆t is the time-dependent part ofthe solution. The system (20)-(21) can be rewritten as

Using the relations e±iθ, the dispersion relation reads

The scheme will be stable if the amplitude of the solution (22)does not grow with time, i.e. . The stability condition is1≤αfulfilled if -1 β 1, leading to the amplification factor 1=αand to the following condition on the time step ∆t

which is less severe than the classical condition prevailing forhydrostatic case with ψ = 0 (see e.g. Abbott and Basco, [1])

Applying this analysis to a scheme where the second termg η/ x of (21) is defined at time n instead of n+1 shows at thestability condition would never be fulfilled. This result, strictlyascertained for the present finite-difference scheme, will also pre-vail for the hybrid finite-volume scheme proposed in the next sec-tion.

4.3 Hybrid finite-volume scheme

The idea is to apply a standard finite-volume scheme to the con-servative form (6) and (12) of the Boussinesq equations, com-bined with a finite-difference treatment of the non-hydrostaticterm, considered as a source term.Equation (6) and (12) can be rewritten in vector form as

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The inter-cell fluxes and may be calculated as func-2/1−iF 2/1+iFtions of initial vectors Ui and Ui+1 using for example mean values(superscript *) at the interface

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where fU(Fr) and fD(Fr) are respectively upwind and diffusionfunctions of the Froude number Fr. The well-known Roe scheme[14] uses (29) with the following definitions

In order to take into account the stability condition pointed out forfinite-difference schemes, a special treatment of the pressure termis required. The following predictor-corrector scheme is thus pro-posed to solve (28).

Predictor step for the continuity equationThe continuity equation in (28) is solved explicitly in a finite-volume scheme to find a provisional value of . Let this value1+n

ihbe . We obtain( )1+n

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where is the mass flux at the interface between cells i andprediq 2/1"

i+1 calculated using all variables at time n according to (29)-(30)

Implicit step for the momentum equationThe momentum equation in (28) reads, using a finite-differenceapproximation for the source terms

This can be rewritten as

The momentum flux is evaluated by Roe’s model, accordingimpli 21+σ

to (29)-(30), but with the hydrostatic-pressure term calculatedwith the provisional value from (31), to ensure the stability( )1+n

ih

condition

where the mean interface values (with superscript *) are com-puted from (30) using the variables at time n.

Corrector step for the continuity equationThe continuity equation is solved again to find :1+n

ih

The mass flux in (36) is evaluated with time-averaged val-corriq 2/1+

ues of the mass fluxes q

This scheme is first-order accurate in space, and it was observedthat the intrinsic numerical diffusion significantly damps the ef-fects of the non-hydrostatic source term. To avoid this inconve-nience, second-order accuracy in space has been introduced usingthe MUSCL scheme (Hirsch [6]), consisting in a linear recon-struction of the free-surface which is thus no more composed ofcells with constant water depth.

5. Experiments

5.1. Experimental set-up

An experimental flume was built in the laboratory of the CivilEngineering Department at the University of Louvain (UCL).Figure 4 shows the channel configuration. A gate separates tworegions of initially constant water depth : the upstream reservoirand the channel itself, both at an initial rest state. Water-levelgauges (C0…C5) were placed in the downstream part of thechannel, to measure the time evolution of the water level.When opening the gate rapidly, a bore travels into the down-stream channel. The wave Froude number Fr, defined later inequation (41), will determine the bore aspect. In the supercriticalcase (Fr >> 1), the bore will consist of a steep front. On the otherhand, in the near-critical state (Fr 1), undulations will grow at

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JOURNAL OF HYDRAULIC RESEARCH, VOL. 40, 2002, NO. 1 39

Fig. 4 Channel configuration (plane view and elevation) - Dimensionsin m

Fig. 5 Favre waves in the LGC channel

Fig. 6 Computed free surface

(38)$ % $ % 1122 HaUHaU �

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21

12

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22

22

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gHaUH

gHaU "�"

(40)$ %121

21 2

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HgUa "�

(41)$ %1221

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the bore head, leading to a travelling wave train (see figure 5).The picture is taken from the downstream end of the channel. Theundulations growing at the front of the wave can be clearly identi-fied. Figure 6 shows the water surface computed with the above-presented numerical model. The qualitative agreement with thepicture feature is good. The first wave has the maximum ampli-tude and is followed by waves of decreasing heights. For higherFroude numbers, the waves will break and the wave train willtake the form of a steep bore.

5.2. Comparison with other experiments

In this kind of experiments, the Froude number is defined as the

ratio between the wave speed a - U1 and the celerity ,1Hgc =where a is the absolute speed of the discontinuity (see figure 7).The following continuity and momentum relations can be writtenacross the discontinuity

giving the relative speed of the discontinuity

The Froude number can thus be expressed as

As already mentioned, the Froude number is the important param-eter for classifying the waves. Its value will range between 1 and1.25 … 1.3. For higher Froude numbers, the first wave will breakor even the wave train will completely disappear and become asteep travelling bore. Figure 8 shows a comparison of the maxi-mum zmax (upper point series), the minimum zmin (lower series)and the mean value zm = H2 - H1 (intermediate series around thecurve corresponding to equation 41) of the wave height for three

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40 JOURNAL OF HYDRAULIC RESEARCH, VOL. 40, 2002, NO. 1

Fig. 7 Definition sketch for the Froude number

Fig. 8 Comparison with experimental series by Favre and Treske (�

Favre, Treske, Q present study)

Fig. 9 Development of an undular bore, hydrostatic theory,Peregrine finite-difference scheme (upper curve) and hy-

brid finite-volume scheme (lower curve)

experimental series : Favre, Treske and the experiments describedpreviously. For all series, the measurements are in good agree-ment with past investigations.It clearly appears that for a certain limit value of the Froude num-ber, the maximum relative wave height decreases, which denotesthe breaking of the wave.

6. Validation of the numerical model

Figure 9 shows computed profiles during the development of anundular bore, starting from an initially smooth wave profile, cor-responding to a smooth increase of the discharge in a channelinitially at rest. This is equivalent to the situation illustrated in thesecond section to explain the physics of the growth of undulationsin a weakly non-hydrostatic system. The initial wave profile isproposed by Peregrine [12], and is such that the initial motion canbe described by a hydrostatic theory. Results computed with theSaint-Venant shallow-water equations are compared to theBoussinesq equations solved by a finite-difference scheme(Peregrine [12]) and by the proposed hybrid finite-volumescheme (see figure 9). It must be outlined that Peregrine’s schemesolves simplified equations where the non-hydrostatic term islinearized, assuming small amplitude and U / x << / t, which isnot the case for our scheme.For the clarity of the figure, the successive profiles are shifted.The steepening of the wave computed with the hydrostatic theory(Saint-Venant equations) clearly appears, while with the

Boussinesq equations undulations grow at the head of the bore.No significant differences appear between both non-hydrostaticcomputed results, indicating that in this case of smooth initialwave profile, the assumptions chosen by Peregrine [12] tolinearize the non-hydrostatic term are valid. This is however notalways the case as will be shown in the next comparisons betweencomputed results and experimental measurements.As already mentioned, Peregrine’s finite-difference scheme wasunable to reproduce the experiments starting with a steep front.However, with the hybrid finite-volume scheme, a discontinuousinitial profile was successfully used in the following computation.Figure 10 shows the measured and computed water depth evolu-tion at the 6 gauging points in the experimental channel. Theagreement is good, in particular the amplitude of the first crestand the wave propagation celerity are well represented.Figure 11 shows close-up comparisons for different Froude num-bers. Agreement is good; the wavelengths are well reproduced,with a slight overestimation of the dispersive character of thewaves. The main discrepancies with the measured profiles lie inthe damping of the wave : the computed undulations vanishquicker than the measured ones. However, figure 11d clearlyshows that the complete wave train is well represented, with agood accuracy of the computed H2 water depth. The small pertur-bations in the measured water profile after t = 12 s are due to lo-

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JOURNAL OF HYDRAULIC RESEARCH, VOL. 40, 2002, NO. 1 41

Fig. 10 Water level evolution at the gauging points – Fr = 1.104,experiments, hybrid finite-volume scheme

a) Point C4, Fr = 1.081 b) Point C3, Fr = 1.104

c) Point C2, Fr = 1.138 d) Point C4, Fr = 1.192

Fig. 11 Water level evolution at some gauging points for differentFroudenumbers, experiments, hybridfinite-volumescheme

Fig. 12 Consequences of linearizing the non-hydrostatic term,experiments, hybrid finite-volume scheme with linearizedand non-linearized non-hydrostatic term

Fig. 13 Influence of the time step, experiments and hybridfinite-volume scheme

(42)1max

CFL F??"

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cU

cal 2D reflection effects induced by little imperfections in theexperimental flume walls.The good agreement of the wavelengths is not affected by theFroude number. However, although the first wave amplitude iswell computed, a trend to underestimate the subsequent wave am-plitudes is observed for increasing Froude numbers. This mightbe considered as an intrinsic limit of the Boussinesq equations, assome essential assumptions made in their derivation are less validin this case. Indeed, as ϕs, / t, / x become larger, their productsare no longer negligible.The consequences of linearizing the non-hydrostatic term did notclearly appear on figure 9 when comparing Peregrine’s schemeto the present hybrid finite-volume scheme on a smooth wavecase. Figure 12 compares results computed with the hybrid finite-

volume scheme, on the one hand with the complete non-hydro-static term, and on the other hand with this term linearized in thesame way as in the equations solved by Peregrine.The figure shows that linearizing the non-hydrostatic term in-creases the error on the wavelength, i.e. the dispersive characterof the computed wave. This highlights the need to take all termsinto account since the assumptions used to lin.earize the non-hy-drostatic term are not really fulfilled here (especially η << H).Finally, figure 13 shows the influence of the time step, repre-sented here by the CFL number:

This definitionstrictly applies for shallow-water explicit schemes.Fornon-hydrostaticwavepropagation, the linear stabilityanalysisconducted in a previous section resulted in the time-step condition(25), taking into account the implicit treatment of the non-hydro-static terms. The hybrid finite-volume scheme combines both ap-proaches with the consequences that the CFL number has to besmaller than 1 for stability, but also that the numerical accuracyincreases for still smaller CFL numbers. So, as illustrated in fig-

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42 JOURNAL OF HYDRAULIC RESEARCH, VOL. 40, 2002, NO. 1

ure 13, although the computations are stable for any CFL numbersmaller than 1, the accuracy in amplitude increases when decreas-ing the time step.

7. Conclusions

A new numerical scheme for computing weak non-hydrostaticflows is presented. This scheme solves the Boussinesq equationsby a hybrid finite-volume method, consisting in applying a finite-volume scheme for the conservative part of the equations andfinite differences for the non-hydrostatic terms.The propagation of an undular bore, studied for the first time ex-perimentally by Favre, is a typical example of such weak non-hydrostatic flows. It was observed that undulations appear if theFroude number (41) ranges between 1 and 1.28. For higherFroude numbers, the waves break, leading to a steep front.New experiments have been carried out, and appear to comparewell with similar series by other authors. The results were used tovalidate the numerical scheme. In the range of validity of theBoussinesq equations, the computed and measured profiles are ingood agreement, although the computed wave length accuracycould be improved.Linearizing the pressure term, as proposed by Peregrine, workswell for mild waves. For steep initial conditions, the proposedscheme constitutes an improvement of the existing finite differ-ences schemes in that it is able to accurately predict the wave ce-lerity, height, and wavelength, without any artificial viscosityusually needed to increase the numerical stability.

Acknowledgements

The authors wish to thank François Lederer and Grégory Bodartfor their significant contribution to the experimental work. Theyalso wish to express their gratefulness to Prof. Eric Deleersnijder,Institut d’Astronomie et de Géophysique Georges Lemaître(UCL, Belgium) for his interest and valuable contribution to thestability analysis of the proposed scheme.

Notations

a absolute speed of the discontinuityc wave celerityCFL Courant numberF flux vectorFr Froude numberg acceleration due to gravityh water depthH mean water depth (constant)k wave number 2 π/λp pressureq unit discharge and mass fluxS vector of source termst timeT wave periodU depth-averaged velocityU vector of hydraulic variables

u, w horizontal and vertical velocitiesu’ deviation from the depth-averaged velocity u – Ux, z horizontal and vertical co-ordinatesα amplification factorη free-surface elevation above Hθ phase angleλ wavelengthρ water densityσ momentum fluxϕ streamline inclinationι non-hydrostatic ‘marker’, equal to unityω wave angular frequency

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