The Aquitanian coast beaches exhibit longshore non-uniform multiple bars. We mainly focus here on the rigde and runnel system, in which ridges are regularly ...
A SIMPLE AND EFFICIENT WELL-BALANCED MODEL FOR 2DH BORE PROPAGATION AND RUN-UP OVER A SLOPING BEACH Fabien Marche1 and Philippe Bonneton2 We consider numerical solutions of the 2DH Non-Linear Shallow Water equations with a bed slope source term. To accurately describe the evolution of the shoreline over a strongly varying topography, we investigate a new model, called SU RF− W B. It relies on a reconstruction method for the treatment of the bed-slope and can handle strong variations of topography. The use of the recent VFRoe-ncv Riemann solver leads to a robust treatment of wetting and drying phenomena, which preserves the water depth positivity. An adapted “second order” reconstruction generates accurate bore-capturing abilities. This model is validated against analytical solutions involving a time-dependent moving shoreline and a study of mean-currents over a ridge and runnel system is investigated. This model should have an impact in time-varying numerical simulations in the surf zone.
INTRODUCTION
Several studies (Hibberd and Peregrine 1979), (Kobayashi et al. 1989), (Bonneton 2006) and (Bonneton et al. 2004) have demonstrated the ability of the 1D non-linear shallow water equations to describe cross-shore wave transformation in the surf and swash zones. However, recent studies (Brocchini et al. 2002) and (Peregrine et al. 2000) have shown that 2D bore propagation over a complex topography is a very difficult task from a numerical point of view. Dissipation in 2D bores crossing the surf zone leads to the generation of large scale vertical vorticity. Numerical investigations (Brocchini et al. 2002) revealed that study of vorticity demands higher accuracy than those of the classical shock-capturing methods. In particular, the treatment of the bed slope source term requires an accurate well-balanced scheme. In this paper, we present a new model which gives good results for 2D bore propagation and run-up over complex topography. The first difficulty which appears in the simulation of run-up phenomena is the treatment of the shoreline. Flow properties change very rapidly near the shoreline, as both water depth and velocity vanish and the position of the shoreline is always evolving. Various methods have been proposed to cope with these problems and there are two main approaches. The first one relies on coordinate transformations (which generate a map between the time-varying physical domain and a time-invariant computational domain) and uses a Lagrangian description of the flow (Prasad and Svendsen 2003). The second approach is based on the use of conservative high resolution bore-capturing schemes on fixed grids and the direct computation of flow properties is used to evaluate the shoreline position, without any tracking method (Hu et al. 1998), 1 2
Math´ematiques Appliqu´ees de Bordeaux, Universit´e Bordeaux 1, 351 cours de la lib´eration, 33405 Talence, France. D´epartement G´eologie Oc´eanographie, Universit´e Bordeaux 1, Avenue des Lucioles, 33405 Talence, France
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(Brocchini et al. 2002) and (Hubbard and Dodd 2002) for a 2D model. Our investigations lie in this context. The second difficulty which arises in the study of wave transformation on a sloping beach is the treatment of the bed slope source term. Most of the models introduced above rely on fractional step methods (FSM). This is well known today that FSM are not designed to preserve discrete steady states and are unable to compute the convergence toward a steady state or at least the evolution of small perturbations around these states. Thus, we need to discretize the flux gradient and the source term as a whole, achieving some particular properties, as the preservation of a large class of steady states. Since some early works (Greenberg and Leroux 1996), discretisations of the source term which preserve steady states and avoid the drawbacks of FSM have been pointed out. These methods are called well-balanced methods since they preserve balancing properties for steady states. In this context we investigate in (Marche et al. 2006) the use of a new “second order” well-balanced model, called SURF-SVWB, for the study of moving shoreline problems. This model relies on a robust bore-capturing and positivity preserving VFRoe-ncv homogeneous Riemann solver (Gallouet et al. 2003a). This property of positivity is rarely clearly exposed and enables the solver to deal with dry areas. The use of the recent “hydrostatic” reconstruction method (Audusse et al. 2004) for the treatment of the source term ensures the required well-balanced properties. Combined with a MUSCL reconstruction, it raises a stable and simple “second order” bore-capturing well-balanced scheme. This model is able to accurately simulate run-up and back-wash phenomena on various 2D topography and to compute the preservation and the convergence toward steady states. In this paper we will show the ability of the scheme to compute rip currents induced by 2D bores propagating over a complex topography. NUMERICAL MODEL Basic equations
The two-dimensional shallow water equations are written as follows : U,t + F (U),x + G(U),y = S(U), (1) hv hu h g 2 2 huv hu U= , F (U) = hu + h , G(U) = , g 2 2 hv 2 + h hv huv 2
and
0 S(U) = −g h dx , −g h dy
where ( ),x (resp. ( ),y ) stands for the derivate along the x direction (resp. the y direction), u = t (u, v) is the depth-averaged velocity with u and v the scalar components in the horizontal x, y directions and h is the local water depth. U 2
is the vector for the conservative variables, F (U) and G(U) stand for the flux functions respectively along the x and y directions and S(U) represents the bed slope source term with the bed slope ∇d = t (d,x , d,y ). A two-dimensional semi-discrete finite volume formulation of system (1) is given by : 1 1 d Uij (t) + (F∗ 1 − F∗i− 1 ,j ) + (G∗ 1 − G∗i,j− 1 ) = Sij 2 2 dt ∆x i+ 2 ,j ∆y i,j+ 2
(2)
where the cell-centered vector of conservative discrete variables is Uij = (hi,j , hi,j ui,j , hi,j vi,j ), F∗i± 1 ,j and G∗i,j± 1 stands respectively for the numerical 2 2 flux functions through interfaces in each horizontal direction and Sij is a discretization of the source term. For the sake of simplicity, we directly give here the expressions of these numerical flux functions and the reader is referred to (Marche et al. 2006) for further theoretical details :
t
= Fni+ 1 ,j + Sni+ 1 −,j , Fn,+ = Fni+ 1 ,j + Sni+ 1 +,j , Fn,− i+ 1 ,j i+ 1 ,j
(3)
Fni+ 1 ,j = F (Ui+ 21 ,j (0, Uni+ 1 −,j , Uni+ 1 +,j )).
(4)
2
2
2
2
2
2
2
2
2
and = Gni,j+ 1 + Sni,j+ 1 − , Gn,+ = Gni,j+ 1 + Sni,j+ 1 + , Gn,− i,j+ 1 i,j+ 1
(5)
Gni,j+ 1 = G(Ui,j+ 21 (0, Uni,j+ 1 − , Uni,j+ 1 + )).
(6)
2
2
2
2
2
2
2
2
2
In the following, we only detail the one-dimensional case for the sake of simplicity. Interface values
The interface value Ui+ 21 (0, Ui+ 21 − , Ui+ 12 + ) between cell i and i + 1 is obtained from a VFRoe-ncv linearized Riemann solver. This solver relies on the symmetrizing change of variable W(U) = t (2c, u) which preserves the water depth positivity at least for interface values (Gallouet et al. 2003b). The formalism of this solver is not recalled in details here. It relies on the exact resolution of a ˜ The eigenvalues of the linearized linearized problem around an averaged state W. convection matrix are ˜1 = u ˜2 = u λ ˜ − c˜, λ ˜ + c˜, √ where c = gh. Then, the exact solution of the linearized Riemann problem is ˜k : given by the sign of the eigenvalues λ ˜ 1 > 0 or λ ˜ 2 < 0, then the flow is super-critical and we recover a • If λ classical upwinding. The interface value is defined as follows : ˜k > 0 ∀k Wi if λ ∗ Wi+ 1 ,j = (7) ˜ k < 0 ∀k. 2 Wi+1 if λ 3
˜1 < 0 and λ ˜2 > 0 then the flow is subcritical and we obtain : • If λ 1 c∗i+ 1 ,j = c˜ − (ui+1,j − ui,j ) and u∗i+ 1 ,j = u ˜ − (ci+1,j − ci,j ). (8) 2 2 4 We can finally recover conservative variables, using the inverse change of variable. The key point is that this interface value is computed from values Ui+ 21 − and Ui+ 21 + issued from a combined limited linear/hydrostatic reconstruction. Variable reconstruction
More precisely, considering the cell i, we compute first linear reconstructions Ui,r and Ui,l respectively at i + 12 − and i − 21 +, using a minmod limiter. Values of Hi,l and Hi,r , where H = h + d, are also reconstructed, and we deduce reconstructions of di,l and di,r . Then, interface topography values di+ 21 ,j are defined with di+ 12 = max(di,r , di+1,l ). The positivity preserving hydrostatic reconstruction of the water height is thus defined as follows : hi+ 21 − = max(0, hi,r + di,r − di+ 12 ), hi+ 12 + = max(0, hi+1,l + di+1,l − di+ 21 ), and we deduce reconstructed values on each side of the interface : hi+ 21 − hi+ 12 + Ui+ 12 − = , Ui+ 21 + = . hi+ 12 − ui,r hi+ 21 + ui+1,l
(9)
Finally we introduce : Sni+ 1 − 2
=
0 g 2 g hi+ 1 − − h2i,r 2 2 2
!
, Si− 12 + =
0 g g 2 hi,l − h2i− 1 + 2 2 2
!
,
(10) and a centered source term discretization is added to achieve second order well-balancing : ! 0 Sc,i = . (11) hi,l + hi,r g (di,l − di,r ) 2 The extension to the two-dimensional framework on Cartesian meshes is straightforward. Wetting and drying procedure
No special tracking procedure is used in this model. A distinction is made between wet and dry cells with the introduction of an artificial threshold value htol = 10−10 m for the definition of a dry cell. The detailed wetting/drying procedure is simple. At the beginning of each time step, we search for dry cells, i.e. cells for which the water depth is less than htol , and set them to h = htol . The keypoint is to apply this “wetting procedure” to the quantity which are issued 4
from the hydrostatic reconstruction instead of the natural quantities, leading to a better accuracy at the shoreline and a better stability in practice. Thereafter, the shoreline is constructed from the mesh interface between wet and dry cells, without any interpolation. We have three possible types of inernal edge, which are wet/wet, wet/dry and dry/dry, the case wet/dry corresponding to the flooding or the drying of a specific area. Then, thanks to the robustness of the VFRoe-ncv solver, the computation is directly performed, without any distinction between these possible configurations. NUMERICAL ASSESSMENTS
We propose here some validations relying on analytical solutions. The periodic Carrier and Greenspan solution Numerical results Analytical solutions
0.15
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Figure 1. The Carrier and Greenspan solution. Numerical and analytical results.
We compare here numerical results (in solid lines) with the periodic Carrier and Greenspan 1D (Carrier and Greenspan 1958) analytical solutions (in dotted lines). This solution represents the motion of a periodic wave traveling shoreward and being reflected out to sea, generating a standing wave which is let run-up and 5
run-down on a plane beach. It has been observed that even though waves may break just before they reach the shoreline, the tip of the resulting swash motion has many similarities with the nonbreaking run-up of waves and this test case is generally acknoledged to be a suitable test case for the assessment of shoreline motions computation. We show on Fig. 1, a comparison between numerical and analytical solution, highlighting the ability of this new model in the computation of moving shoreline. There is a very good agreement between the two curves, both for the surface elevation profiles and time series of the shoreline motions. The analytical solution is used as a left inlet boundary condition to force the motion and we use ∆x = 0.01 and CF L = 0.7. The shoreline oscillations are accurately computed even after a large number of period. Evaluation of the L2 -error for this test can be found in (Marche et al. 2006). We stress out that many other well-balanced methods, highly validated in hydraulic situations, are unable to provide such results (Marche et al. 2006). The two-dimensional Thacker solution
We perform a comparison with the analytical solution introduced in (Thacker 1981), describing free boundary and free surface oscillations in a 2D parabolic basin (see Fig. 2). This case is perhaps one of the more difficult case to handle for a numerical model, since it involves a two-dimensional periodic moving shoreline with a no radially symmetric configuration. The shoreline is a circle in the (x, y) plane and the motion is such that the center of the circle orbits the center of the basin, while the surface remains planar with constant gradient at any given instant. We use here ∆x = ∆y = 0.01 m and CF L = 0.7.
Figure 2. Topography of the basin in the Thacker case.
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b) 0.3
0.25
0.25
0.2
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0.15
0.15 h (m)
h (m)
a) 0.3
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0 x (m)
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0 x (m)
0.5
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Figure 3. Thacker planar solution. Numerical and analytical results
We show on Fig. 3 a comparison between numerical (in solid lines) and analytical solution (in dotted lines) for free surface profiles during the fourth period. Tiny distortions can be observed but the planar aspect of the surface is preserved even after several periods. Moreover, we are able to compute a large number of period without any large instability (almost 10 periods have been effectively computed). Note that these results are almost as accurate as those presented in the literature, relying on more complex or expensive algorithms, like extrapolation and filtering (Lynett et al. 2002) or even adaptive refinement method (Hubbard and Dodd 2002). MEAN CURRENTS OVER A RIDGE AND RUNNEL SYSTEM
On natural beaches, accurate prediction of wave-induced current circulations which control the nearshore sediment transport, and therefore the morphological changes in the nearshore, remains an important topic. The classical approach for modelling ISZ wave-induced circulation is based on time averaged equations. 2DH numerical studies of ISZ mean currents using time-varying models is still not well developped. Few studies relying on Boussinesq-like models have been reported, like (Sorensen et al. 1998) or (Chen et al. 1999). These time-varying approaches may be very fruitful, providing informations not available with time-averaged models, like velocity oscillations, instantaneous vorticity fields or swash zone motions. The Aquitanian coast beaches exhibit longshore non-uniform multiple bars. We mainly focus here on the rigde and runnel system, in which ridges are regularly interrupted by down-currents oriented runnels, with a mean wave 7
length of about 400 m. Observations have shown that topographic feed-back may generate rip circulation cells. On the Aquitanian coast, these rip currents occur at each runnel outlet and their intensity depends on the length, the shape of the system, the offshore wave conditions and the tide level. The mechanism leading to the generation of a rip current is summarized in (Castelle et al. 2006). The main idea is that longshore variations of the topography are associated with similar variations in the pressure, due to mean water level changes and hence, lead to longshore pressure gradients. The longshore pressure gradients induce an intensification of the longshore current in the runnel. On the upper part of the beach, the longshore pressure gradients force the feeder currents. Water convergence inshore the runnel outlet results in the formation of a seaward oriented current, a rip current. Observations show that this seaward flow tends to be narrow and relatively strong but rapidly desintegrates outside the surf zone. For our analysis, we have neglected first the tidal variations, since it has been shown that tidal currents modulations can be neglected in a first approach, when compared to the wave-induced currents (Castelle et al. 2006). However, we have performed several simulations for various mean water levels to highlight tidal variations impact on the circulation. Fig. 4 shows the idealized ridge and runnel topography used for our simulations.
Figure 4. Contour plot of a ridge and runnel system
At this level of analysis, the study will only be qualitative. It is well-known that NSW equations lead to an anticipated steepening of the solution too far offshore. However, we have chosen to generate a periodic train of wave offshore of the surf zone. When this train of waves reaches the ISZ, the wave shape has change to the characteristic sawtooth shape. Hence, the simulation may be realistic in the ISZ enough. Such qualitative studies in the ISZ have been performed for instance in (Brocchini et al. 2002) for the study of macro-vortices 8
induced by single breakwaters. We set the wave period at Tw = 12 s. The offshore wave amplitude is the main characteristic controlling the rip velocities, as changes in the breaking patterns control the rip current characteristics. Our aim is to obtain a large enough bore-like wave amplitude in the area corresponding to the ISZ, taking into account the energy dissipation which occurs during all the propagation. For this purpose, we set the offshore wave height at Hw = 1 m. We use a variable mesh in the cross-shore direction. The largest cell size is ∆x = 2 m near the offshore boundary, while the finest cell size is ∆x = 0.05 m near the shoreline. In the longshore direction, the mesh is regular and the discretization step is ∆y = 1 m. The time step is constant and fixed at ∆t = 0.05 s. a) Surface elevation Mean water level Still water depth
3.4
Surface elevation (m)
3.2
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b) Surface elevation Mean water level Still water depth
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c) Surface elevation Mean water level Still water depth
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Figure 5. Section views of the the free surface elevation and set-up
Wave refraction and set-up
The refraction can be observed on Fig. 5 where profiles of the free water surface are plotted at three different locations in the longshore direction : a) y = 95.25 m, b) y = 190.5 m, c) y = 285.75 m. We have also plotted on the same Figure the three corresponding set-up profiles. Wave-induced mean currents
In the sequel hm stands for the value of water depth over the zero isobath. Fig. 6 and Fig. 7 show the two dimensional mean current patterns simulated during a normally incident swell, for respective values of water depth hm = 1 m, 9
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Figure 6. Two dimensional view of the computed mean currents in the system
hm = 1.5 m (Fig. 6), hm = 2.0 m and hm = 2.5 m (Fig. 7). Note that hm = 2.0 m corresponds to mid-tide and high-tide occurs around hm = 4.0 m. The steady state is reached after approximatively 800 s and the instantaneous velocity field is averaged over a wave period. A narrow seaward oriented current can be observed at the runnel outlet, associated with a large circulation cell, even when the whole system is not entirely submerged. For hm = 1 m, the maximum value of this current is about 0.14 m/s. For hm = 2 m, it is about 0.2 m/s. As expected, this current rapidly vanishes offshore of the surf zone. 10
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Figure 7. Two dimensional view of the computed mean currents in the system
As suggested in (Castelle et al. 2006), the maximum of rip velocity occurs at mid tide. The intensity of this maximum rip velocity then decreases with respect to the increasing mean water level. Still comparing with (Castelle et al. 2006), the second circulation cell near the lateral boundary is slightly shifted in the cross-shore direction, towards offshore. All these results are qualitatively similar to those shown in (Castelle et al. 2006) with only slight differences.
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CONCLUSION
From these validations and applications, this new model SU RF− W B appears as a promising and efficient tool for the study of many coastal hydrodynamic processes, including wave breaking, run-up and two-dimensional wave-induced mean-currents over complex topography. The two-dimensional physical processes involved in the last application can be accurately reproduced and the Finite-Volume framework, with high order shock-capturing abilities, clearly enables to handle the propagation and run-up of breaking waves, which was not possible with Finite-Difference approaches. Moreover, the shoreline tracking method proposed here is far more simpler to implement and less computationnaly expensive than many methods found in the literature. However, we need to validate the model with laboratory and field experiment data to assess the real improvements brought up by the well-balanced approach to simulate 2DH bore propagation and run-up. REFERENCES Audusse, E., F. Bouchut, M. Bristeau, R. Klein, and B. Perthame. 2004. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J.Sci.Comp., 25(6), 2050–2065. Bonneton, P.. 2006. Modelling of periodic wave transformation in the inner surf zone, Ocean Engineering,. in press. Bonneton, P., V. Marieu, H. Dupuis, N. S´en´echal, and B. Castelle. 2004. Wave transformation and energy dissipation in the surf zone: comparison between a non-linear model and field data, J. Coast. Res., 39. Brocchini, M., A. Mancinelli, L. Soldini, and R. Bernetti. 2002. Structure-generated macrovortices and their evolution in very shallow depths, Proceedings of 28th International Conference on Coastal Engineering, ASCE, 772–783. Brocchini, M., I. A. Svendsen, R. S. Prasad, and G. Belloti. 2002. A comparison of two different types of shoreline boundary conditions, Comput. Methods Appl. Mech. Engrg, 191, 4475–4496. Carrier, G. F. and H. P. Greenspan. 1958. Water waves of finite amplitude on a sloping beach, J. Fluid Mech, 4, 97–109. Castelle, B., P. Bonneton, N. S´en´echal, H. Dupuis, R. Butel, and D. Michel. 2006. Dynamics of wave-induced currents over an alongshore non-uniform multiple-barred sandy beach on the aquitanian coast, france, Continental shelf research, 26, 113–131. Chen, Q., J. T. Kirby, R. A. Dalrymple, A. B. Kennedy, and M. Haller. 1999. Boussinesq modelling of a rip current system, J. Geophys. Res., 109(9), 20617–20637. Gallouet, T., J. M. Herard, and N. Seguin. 2003a. On the use of some symetrizing variables to deal with vacuum, Calcolo, 40, 163–194. Gallouet, T., J. M. Herard, and N. Seguin. 2003b. Some approximate godunov 12
schemes to compute shallow-water equations with topography, Computers and Fluids, 32, 479–513. Greenberg, J. M. and A. Y. Leroux. 1996. A well balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J.Numer.Anal., 33(1), 1–16. Hibberd, S. and D. H. Peregrine. 1979. Surf and run-up on a beach: a uniform bore, J. Fluid. Mech., 95(2), 323–345. Hu, K., C. G. Mingham, and D. M. Causon. 1998. A bore-capturing finite volume method for open-channel flows, Int.J.Numer.Meth.Fluids, 28, 1241–1261. Hubbard, M. E. and N. Dodd. 2002. A 2d numerical model of wave run-up and overtopping, Coastal Engineering, 47, 1–26. Kobayashi, N., G. S. Desilva, and K. D. Watson. 1989. Wave transformation and swash oscillation on gentle and steep slopes, J. Geophys. Res., 94, 951–966. Lynett, P. J., T. S. Wu, and P. L. F. Liu. 2002. Modeling wave runup with depth-integrated equations, Coastal Engineering, 46, 89–107. Marche, F., P. Bonneton, P. Fabrie, and N. Seguin. 2006. Evaluation of well-balanced bore-capturing schemes for 2d wetting and drying procedure, Int. J. Num. Meth. Fluids,. in press, article online in advance of print. Peregrine, D. H., M. D. Patterson, and O. Bokhove. 2000. Large eddies and vorticity in the surf and swash zone, Book of Abstract, SASME. Prasad, R. S. and I. A. Svendsen. 2003. Moving shoreline boundary condition for nearshore models, Coastal Engineering, 49, 239–261. Sorensen, O. R., H. A. Schaffer, and P. A. Madsen. 1998. Surf zone dynamics simulated by a boussinesq type model. part iii. wave-induced horizontal nearshore circulation, Coastal Engineering., 33, 155–176. Thacker, W. C.. 1981. Some exact solutions to the nonlinear shallow-water wave equations, J. Fluid Mech., 107, 499–508.
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KEYWORDS – ICCE 2006 A simple and efficient well-balanced model for 2DH bore propagation and run-up over a sloping beach Fabien Marche and Philippe Bonneton Abstract number 127 Non-linear shallow water equations Finite volume Moving shoreline Varying topography Bore propagation Wave-induced mean currents Run-up Well-balanced
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