Well Balanced Adaptive Simulation of Pollutant Transport by Shallow Water Flows: Application to the Bay of Tangier

International Journal of Hydraulic Engineering 2014, 3(1): 10-23 DOI: 10.5923/j.ijhe.20140301.02 Well Balanced Adaptive Simulation of Pollutant Transport by Shallow Water Flows: Application to the Bay of Tangier E. M. Chaabelasri1, I. Elmahi2,*, R. Abdellaoui2, N. Salhi1 1Laboratoire de Mécanique et Energétique, Faculté des Sciences, BP 717, 60000 Oujda, Maroc 2Equipe de Modélisation et Simulation Numérique, ENSAO, Complexe universitaire, B.P. 669, 60000 Oujda, Maroc Abstract A well balanced adaptive scheme is proposed for the numerical solution of the coupled non-linear shallow water equations and depth-averaged advection-diffusion pollutant transport equation. The scheme uses the Roe approximate Riemann solver with upwind discretization for advection terms and the Vazquez scheme for source terms. It is designed to handle non-uniform bed topography on triangular unstructured meshes, while satisfying the conservation property. Dynamic mesh adaptation criteria are based on the local pollutant concentration gradients. The model is validated for steady flow over irregular bed topography, recirculation due to a sidewall expansion in a frictionless channel, and pollution advection in a flat-bottomed channel. An idealized application to the simulation of pollutant dispersion in the Bay of Tangier, Morocco is presented, which demonstrates the capability of the dynamically adaptive grid model to represent water quality scenarios in a bay of non-uniform bed topography and complicated shoreline. Keywords Shallow water equations, Pollutant transport, Finite volume method, Roe solver, Dynamic mesh adaptation, Unstructured meshes the user to estimate pollutant transport, concentration 1. Introduction distribution, and basin residence time. The hydrodynamic module solves the two-dimensional Pollution of the Strait of Gibraltar has increased depth-averaged shallow water equations and hence is used to significantly in recent decades as a by-product of the growth investigate the forcing mechanism responsible for of maritime transportation activities. Pollution is particularly circulation patterns in the Bay of Tangier. Pollutant transport hazardous to ecologically sensitive coastal regions, such as is modelled by means of an advection-diffusion equation for the Bay of Tangier located on the southern coast of the Strait the depth-averaged concentration of substances of Gibraltar. The environment of the Bay of Tangier is contaminating the seawater. Herein, the resulting system of subjected to human impacts from nearby urban development, equations is formulated so that it constitutes a hyperbolic industry, agriculture, fisheries, and ports (including the system of non-linear conservation laws with source terms. newly operational Tangier Mediterranean Port). These In recent years, there has been increasing interest in the activities release toxic effluent that is causing ecological design of numerical schemes based on non-linear damage to the bay. conservation laws. A particular challenge is to obtain Predictions of the risk posed to the water quality of the high-order accurate solutions in space and time for flows Bay of Tangier due to pollution from different sources could over complicated bed topography. Various finite volume play an essential part in establishing guidelines for schemes developed for general systems of hyperbolic environmental remediation and protection. In particular, conservation laws have been applied to the non-linear numerical models of flow hydrodynamics could aid decision shallow-water equations (NLSWEs), utilising upwind makers in establishing effective countermeasures in order to methods based on approximate Riemann solvers. Such reduce the pollutant discharges from particular sources. The solvers include Roe’s method[1], monotonic upstream numerical model presented herein solves the bay hydrodyna schemes for conservation laws (MUSCL) in curvilinear mics in conjunction with passive species transport, enabling coordinate systems[2][3], essentially non-oscillatory (ENO) schemes[5][6], the weighted essentially non-oscillatory * Corresponding author: [email protected] (I. Elmahi) method[6], and the Harten, Lax and van Leer (HLL) solver Published online at http://journal.sapub.org/ijhe [7]. Most of these methods are capable of capturing Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved shock-like behaviour to a high degree of accuracy and International Journal of Hydraulic Engineering 2014, 3(1): 10-23 11 perform particularly well for steep-fronted flows like 2 ∂+()hu2 gh discontinuous, transcritical flows over flat bed topography. ∂∂()hu 2 ()huv ∂Zb + + =−−gh ghS fx (2) However, for spatially varying bed topography, special ∂∂txy ∂ ∂ x treatment is required in order to discretise the component of and the source term stemming from the bed gradient so that it gh2 properly balances the relevant flux gradient term and ensures ∂∂ ∂+()hv2 ∂ that water at rest remains so. In a well-balanced solver, the ()hv ( huv ) 2 Zb ++ =−−gh ghS fy (3) discrete source terms balance the discrete flux terms. For ∂∂tx ∂ y ∂ y example, Bermudez and Vazquez[8] proposed an upwind where h is the total depth from the sea bed to the free surface, method for the non-linear shallow water equations with bed u and v are the depth-averaged velocity components in the slope source terms, which was applied by Vazquez- Cartesian x and y directions, Z is the bed elevation above a Cendon[9] to a range of shallow water flow problems. b fixed horizontal datum, g the acceleration due to gravity, and LeVeque[4] proposed a Riemann solver that balanced the S and S are the bed shear stress components, defined as source terms and flux gradients in each cell of a regular fx fy computational mesh. However, the extension of this scheme uv22++uv22 for unstructured meshes is not trivial. Hubbard and = = Sfx nu M 44, Sfy nv M (4) Garcia-Navarro[3] proposed a further numerical treatment, 33 in which the upwind method of Bermudez and Vazquez is hh used for source terms. Numerical methods based on surface where nM is the Manning’s coefficient. Assuming the water gradient techniques have also been applied to shallow water pollutant mixture is fully mixed in the vertical direction, the equations by Zhou et al.[24]. depth-averaged pollution dispersion equation may be written The present paper describes a high-resolution finite in advection-diffusion form for a passive contaminant as, volume shock-capturing scheme for the solution of the ∂()()()hC ∂ huC ∂ hvC ∂ ∂∂C ∂C non-homogenous hyperbolic system of equations that ++=(Dhxy )( + Dh )(5) represent shallow flow pollutant transport processes. The ∂t ∂ x ∂ y ∂ x ∂∂ xy ∂ y computational mesh is unstructured, triangular, and where C is pollutant concentration, and Dx and Dy are dynamically adaptive. The Roe’s approximate Riemann pollutant diffusion coefficients in the x and y directions. solver is used for advective fluxes. Time integration is Using matrix-vector notation, the coupled pollutant transport performed by means of a Runge-Kutta algorithm. The system can be written method is of high resolution and aimed at representing 2DH domains with complicated boundary and bed geometry, such Wtx++ FW1212() F () W yx −− FW () F () W y as the Bay of Tangier. (6) The paper is organized as follows. Section 2 briefly =SW() + Sf () W outlines the governing equations. Section 3 deals with the construction of an efficient well-balanced high-order finite where W is the vector of dependent variables, F1, F2 are the volume scheme on unstructured meshes. Section 4 describes inviscid flux vectors, F1 , F2 the diffusive flux vectors, S is a dynamical adaptive procedure based on multi-level the vector of source terms describing the bed variations, Sf refinements and unrefinements by monotiring the pollutant describes the friction terms, and the subscripts x, y, and t concentration in the computational domain. Section 5 denote partial differentiation. In full, the vectors are provides details of the model validation. Section 6 presents the demonstration study of hypothetical pollutant dispersion h hu hv in the Bay of Tangier. Conclusions are summarised in 2 hu 2 + gh huv Section 7. hu 2 WF=,,12 = F= 2 , hv hv2 + gh huv 2 hC 2. Governing Equations huC hvC In conservation form, the two-dimensional non-linear 0 0 shallow water equations are given by Equations (1) to (3) below[25-26]. The depth-averaged continuity equation is 0 0 FF= ,,= (7) ∂∂h() hu ∂ () hv 120 0 ++=0 (1) ∂∂tx ∂ y ∂∂()hD C ∂∂()hD C ∂∂xxx ∂∂yyy The depth-averaged momentum equations in the x- and y- directions are and 12 E. M. Chaabelasri et al.: Well Balanced Adaptive Simulation of Pollutant Transport by Shallow Water Flows: Application to the Bay of Tangier 0 all dependent variables are represented as piecewise constant 0 in the cell as follows, − ∂Zb gh ∂x −ghS fx 1 WWi = dV S = , S f = ∫ ∂Zb Vi − −ghS fy Vi gh ∂ y 0 0 3. Numerical Model The flow domain is partitioned into a set of triangular cells 2 or finite volumes, Vi ⊂ℜ . Let Γij be the common edge of two neighbouring cells Vi and V j , with Γij its length, Ni() is the set of neighbouring triangles of cell Vi , and nij= (,nn x y ) is the unit vector normal to the edge Γ ij and points toward the cell V j (see Fig. 1). A cell-centred finite volume method is then formulated where Figure 1. Cell-centred control volume For the triangular elements used here, the integral around the element is written as the sum of the contributions from each edge, such that + ()WWnn1 − iiV + FW(nn ,)d n Γ− FW ( ,)d n Γ ∆ i ∑∑∫∫ t j∈∈ Ni()ΓΓj Ni() ij ij (8) nn = ∫∫SW( )dVV+ Sf ( W )d VVii n where WWi= (,,)xxt i inis the vector of conserved variables evaluated at time level tn = nt ∆ , n is the number of time n steps, ∆t is the time step, and Vi is the area of cell Vi . To evaluate the state Wi , an approximation is required of the convective and diffusive flux terms at each edge of the cell.

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