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Finite Element Modeling of Free Surface Flow in Variable Porosity Media

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Abstract

The aim of the present work is to present an overview of some numerical procedures for the simulation of free surface flows within a porous structure. A particular algorithm developed by the authors for solving this type of problems is presented. A modified form of the classical Navier–Stokes equations is proposed, with the principal aim of simulating in a unified way the seepage flow inside rockfill-like porous material and the free surface flow in the clear fluid region. The problem is solved using a semi-explicit stabilized fractional step algorithm where velocity is calculated using a 4th order Runge–Kutta scheme. The numerical formulation is developed in an Eulerian framework using a level set technique to track the evolution of the free surface. An edge-based data structure is employed to allow an easy OpenMP parallelization of the resulting finite element code. The numerical model is validated against laboratory experiments on small scale rockfill dams and is compared with other existing methods for solving similar problems.

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Notes

  1. The hydraulic gradient is the measure of the variation of the hydraulic head for unit length [20].

  2. Eq. (3) is by definition the volumetric porosity \(n^{v}\) whereas in Fig. 1 a cross section of the control volume is considered and a sectional porosity \(n^{a} := A_{E}/A \) should be defined as the ratio between the area of pores and the total cross section area. Consequently, a lineal porosity can also be defined as the ratio between the length of pores over the total length (\(n^{l}:= l_{E}/l\)). Fortunately Bears in [3] demonstrated that in a porous medium this distinction is unnecessary being

    $$\begin{aligned} n^{v} = n^{a} = n^{l}. \end{aligned}$$

  3. The Reynolds number is the dimensionless coefficient that, being the ratio between inertia and viscous forces, quantifies the relative importance of each one for a given flow [20]. It is defined as \(\frac{\rho u \, l}{\mu } \) where \(\rho \) is the fluid density and \(l\) is a characteristic length (in pipes it coincide with the diameter).

  4. Eq. (1) and all the alternative non linear formulations that are presented in the next sections are commonly called resistance laws because they measure the resistance made by the porous matrix to the fluid flow.

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Acknowledgments

The research was supported by the FP7-Capacities ULITES project GA-314891 and ERC Advance Grant SAFECON project AdG-267521. The authors wants to acknowledge Prof. Miguel Angel Toledo, Dr. Rafa Moran and Mr. Hibber Campos of the Technical University of Madrid (UPM), Mr. Angel Lara and Mrs. Pilar Viña of CEDEX for the experimental results provided for this work.

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  1. Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Barcelona, Spain

    Antonia Larese, Riccardo Rossi & Eugenio Oñate

  2. Universitat Politècnica de Catalunya, UPC Barcelona Tech, 08034, Barcelona, Spain

    Antonia Larese, Riccardo Rossi & Eugenio Oñate

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Correspondence to Antonia Larese.

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Larese, A., Rossi, R. & Oñate, E. Finite Element Modeling of Free Surface Flow in Variable Porosity Media. Arch Computat Methods Eng 22, 637–653 (2015). https://doi.org/10.1007/s11831-014-9140-x

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