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Numerical Modelling and Analysis of Water Free Surface Flows∗

Fadi Dabaghi1, Abdellah El Kacimi1,2, Chakib Kada Kloucha1, Farouk Mezali1,3, Bassam Nakhle1,4

Abstract

Various environmental engineering applications related to water resources involve unsteady free surface flows. A full 3D models based on Navier-Stokes equations are a good description of the physical features concerning several phenomena as for example lake eutrophication, transport of pollutant, flood in rivers, watershed, etc. However these models are characterized by an important computational effort, that we aim to reduce in some case by the help of 2D models or by appropriate coupling models of different dimensions and by the use of the parallel algorithmic trough HPCN facilities. In this work, we present an overview of some approximations methodologies and techniques for an efficient numerical modelling of water free surface problems in a finite element context.

1. Motivation and problem setting In this section we describe the water free surface flows models developed in this framework and classified into two following categories: two phase flows based Navier-Stokes for eutrophication treatment through mechanical aeration, and shallow water flow for floods application.

1.1 Models for eutrophication The two phase flows model is used to simulate aeration remedial actions against eutrophication effects in lakes; a water reservoir is generally considered eutrophized when the concentration of dissolved oxygen reaches a low level : less than 3 mg/l. The main idea consists to inject compressed air in the reservoir bot-tom in order to create a dynamic and consequently oxygenate water. The numerical simulation for such resulting flow by conventional two fluids 3D Navier-Stokes models leads to many difficulties mainly due to their complexity and to the necessity of a fine grid needed for good representation of bubbles effect. Thus we suggest some cheap and realistic alternatives by considering one phase flow model, based on ve-locity-pressure semi-compressible Navier-Stokes equations, taking into account air effects through some correction terms representing the forces applied by the air bubbles on water (Abdelwahed/Dabaghi/Ouazar 2002), in addition to boundary conditions related to air injection velocity at the aerator position (Dabaghi 2000). In the same framework, a more general two fluids (air-water) model with a moving water free sur-face (Amara/Dabaghi/Kloucha 2004) is developed under some realistic assumptions on the wind velocity and the atmospheric pressure by using a convection equation describing the void fraction function of the water, which determines the wet domain.

∗ This work was supported by many Euro-Mediterranean Projects (WADI, ESIMEAU and CruCID) and French bilateral coopera-tion programmes (Morocco CMIFM, Algeria CMEP and Greece PLATON). 1 INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. fadi.el_dabaghi@inria.fr, abdellah.el_kacimi@inria.fr, chakib.kada_kloucha@inria.fr, farouk.mezali@inria.fr, Bassam.nakhle@inria.fr 2 EMI Ecole Mohamadia d'Ingénieurs, B.P. 765, Rabat, Maroc. 3 ENP Ecole Nationale Polytechnique, 10 Avenue Hassen Badi El Harrach, Alger, Algérie. 4 ESIB Ecole Supérieure d'Ingénieurs de Beyrouth, B.P. 11-0514, Riad El Solh, Liban.

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1.2 Shallow water free surface flows models Shallow water model describes flood in river, tidal fluctuation, bay and estuary flows, waves on shallow beaches, etc. It is derived from 3D incompressible Navier-Stokes equations, by depth averaging of the continuum mass and momentum balances. The derived Shallow Water models, known also as Saint-Venant, involve free fluid domain geometries characterized by their complexity and variability, and also by large scale computational needs for realistic simulations necessitating fine grid; to overcome these problems, we have used the a priori and a posteriori error analysis (Dabaghi/Kacimi/Nakhlé 2005) (Amara/Dabaghi/Guelmi 2004) to derive geometric indicators for improving the numerical simulations models by adaptive mesh techniques, in term of optimal quality solution, for a required precision, as well as in term of the computational cost. Moreover, we have integrated a conceptual hydrological model based on the HEC-HMS to compute runoff volume by subtracting the volume of water that is intercepted, infil-trated, stored, evaporated, or transpired from the precipitation in the catchments area. The resulting flood hydrogram is used as boundary conditions in the St-Venant model presented above.

2. Numerical Simulation of water free surface flows From the numerical point of view, the approximation of the models mentioned above is based on the char-acteristics method for the time discretization of the advection terms. This method leads to an upwind scheme, which has the double advantage, on a side to be closed physically to convection, and on another side to be an explicit scheme unconditionally stable in the finite element context, allowing therefore the use of large and reasonable time steps. At each time level, we have to solve a quasi-Stokes like problem, approximated by P1/P1 mixed finite elements (Dabaghi/Kacimi/Nakhlé 2005) for the velocity-height St-Venant formulation or by (P1 + bubble /P1) mixed finite elements (Abdelwahed/Amara/Dabaghi 2005) for the velocity-pressure Navier-Stokes formulation, to ensure the discrete LBB condition necessary in this context. For both formulations, the a priori error estimates established are verified numerically on many academic test cases before validation of the models on real cases such rivers flood or crossed 2D lake sec-tions. Moreover, for such simulations, the requirements in real time and large scale computing are very important, thus we used the HPCN facilities permitting the execution of the developed codes especially on real test cases requiring fine and more complex meshes (Dabaghi/Mezali/Abdelwahed/Nakhlé 2005).

2.1 Real simulation case of eutrophication remedial by aeration process Numerical simulations have been carried out on 2D cross section of Bouregreg lake in Morocco, under the assumptions of fixed wind velocity, no slip condition at the cross section delimitation, constant injection velocity and fixed injector position. The used mesh (figure 1) contains 10987 nodes and 21152 elements which represent 54113 unknowns. For illustrative purposes, we present in figure 2 the iso-values of the ve-locity in the cross section corresponding to the wind velocity effect at the time t=5s, 2min, 20min. This re-sult shows the wind velocity effect on the top surface of the lake as well as on the below thermocline level which indicates clearly that the technique is efficient in the sense that the air bubbles are trapped for long time in the poor oxygenated area and consequently able to dissolve a part of their oxygen in water.

Fig. 1: 2D cross section mesh, 10987 nodes, 21152 elements

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Fig. 2: Void fraction effect on the velocity, corrected one phase model at t=5s, 2min, 20min

2.2 Academic water free surface case governed by Navier–Stokes equations Another component of this work, is to take into account the top free surface by using a convection equa-tion describing the void fraction function of the water, which determines the wet domain. Numerical tests were done on a rectangular domain with and without a bump. The bump will generate naturally a free sur-face in addition to the wind constant velocity effect. The mesh is more refined in the neighbourhood of the free surface at time t = 0 (figure 3). The domain has a length of 250m and a width of 30m, the distance be-tween the initial free surface and the bottom is 25m. The global air-water mesh is constituted of 3450 nodes and of 6487 triangular elements in both cases. We plot on figures 4 the isovalues of velocity at times t=100, 550, 800s for both configurations and one can observe the good behaviour of the model.

Fig. 3: Meshes of computation domains

Fig. 4: Isovalues and vectors of the water velocity at times t=100, 550, 800s .

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2.3 Real simulation case of Foum Tillicht flood We present now a real simulation case, treating flood wave propagation in the watershed of Foum Tillicht (Morocco). The section of the river concerned with this study, located on the outlet side of Foum Tillicht watershed in the province of Errachidia. Figures 5 show the isovalues of the water depth in the flood pe-riod by indicating the potential inundated area and consequently to take the adequate safety measures.

Fig. 5: Mesh (107968 nodes, 215616 elements) and depth iso-values during Foum Tillicht flood, Morroco 1988.

2.4 Parallel algorithms and intensive computation Despite the various simplifications brought to the various models and the relative large computing facili-ties at our disposal, the numerical simulation of the aeration process on a real application case remains al-ways limited even in the 2D cases. An analysis of various cases tests showed that the part of Conjugate Gradient solver consumes more than 98o/o of the CPU time. This fact justifies to invest in the paralleliza-tion of the preconditioned conjugate gradient. The adopted strategy consists to equi-distribute contiguous lines package of the matrix on the available processors. One uses synchronous communication functions for the update of the total vector after each product matrix-vector. For validation purpose, we carried out the parallel code under MPI on a MIMD (HP-V 2250) machine, with 16 processors and 8Go of shared memory in Cross bar architecture. The tests were done on the same domain with 4 different meshes (Table 1) and execution series with a partition on 2, 3, 4, 6, 8, 10, 12 and 14 processors were performed.

Mesh Nodes unknowns

h1 1089 3267 h2 4225 12675 h3 16641 49923 h4 66049 198147

Tab. 1: Table of test meshes

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The results are presented according to two types of curves: - CPU and Elapsed Time (figure 7) with respect to the number of processors. - CPU and Elapsed Speed-Up (figure 8), to evaluate the performance efficiency of the parallel algo-

rithm, the influence of the communication between the processors and the granularity impact.

Fig. 7: CPU and Elapsed time curves

Fig. 8: CPU and Elapsed Speed-Up curves

Bibliography Abdelwahed, M., Amara, A., Dabaghi, F. (2005): Finite element approximation of a two phase flow oc-

curring in aeration process, in: International Journal of Scientific Computing, accepted. Abdelwahed, M., Dabaghi, F., Ouazar, D. (2002): A virtual numerical simulator for aeration effects in

lake eutrophication, in: International Journal of Computational Fluid Dynamics, 16, pp. 119-128. Amara, M., Dabaghi, F., Guelmi, N. (2004): Numerical Modelling and Adaptative Mesh Refinement for

Shallow-Water Equations, Proceedings IASTED –ASM Conference, Jun 2004, pp. 160-165. Amara, M., Dabaghi, F., Kada kloucha, C. (2004): A Numerical Model of Free Surface Incompressible

Flow, Proceedings IASTED–ASM Conference Jun 2004, pp. 270-274. Dabaghi, F. (2000): Numerical Aspects of Aeration Process Modelling in Eutrophised Water Basins, in:

Journal of Systems Analysis Modelling Simulation, 39, p. 1-23. Dabaghi, F., El Kacimi, A., Nakhlé, B. (2005): Characteristics time discretization and mixed finite ele-

ment approximation for shallow water equations, in: International Journal of Scientific Computing, accepted.

Dabaghi, F., Mezali, F., Abdelwahed, M., Nakhle, B. (2005): Performance parallèle d’un code E. F. Navier-Stokes 2D en vitesse pression pour la simulation d’écoulements diphasiques, Selected paper for publication in ARIMA

Copyright © Masaryk University Brno, Brno 2005. ISBN: 80-210-3780-6