Numerical approximation of the 3D hydrostatic Navier–Stokes system with free surface
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 6, pp. 1981-2024.

In this paper we propose a stable and robust strategy to approximate the 3D incompressible hydrostatic Euler and Navier–Stokes systems with free surface. Compared to shallow water approximation of the Navier–Stokes system, the idea is to use a Galerkin type approximation of the velocity field with piecewise constant basis functions in order to obtain an accurate description of the vertical profile of the horizontal velocity. Such a strategy has several advantages. It allows

  • to rewrite the Navier–Stokes equations under the form of a system of conservation laws with source terms,
  • the easy handling of the free surface, which does not require moving meshes,
  • the possibility to take advantage of robust and accurate numerical techniques developed in extensive amount for Shallow Water type systems.

Compared to previous works of some of the authors, the three dimensional case is studied in this paper. We show that the model admits a kinetic interpretation including the vertical exchanges terms, and we use this result to formulate a robust finite volume scheme for its numerical approximation. All the aspects of the discrete scheme (fluxes, boundary conditions, ...) are completely described and the stability properties of the proposed numerical scheme (well-balancing, positivity of the water depth, ...) are discussed. We validate the model and the discrete scheme with some numerical academic examples (3D non stationary analytical solutions) and illustrate the capability of the discrete model to reproduce realistic tsunami waves propagation, tsunami runup and complex 3D hydrodynamics in a raceway.

DOI : 10.1051/m2an/2019044
Classification : 65M8, 65M12, 76M12, 35L65, 35Q30, 35Q35, 76D05
Mots-clés : Free surface flows, Navier–Stokes equations, Euler system, free surface, 3D model, hydrostatic assumption, kinetic description, finite volumes
Allgeyer, Sebastien 1 ; Bristeau, Marie-Odile 1 ; Froger, David 1 ; Hamouda, Raouf 1 ; Jauzein, V. 1 ; Mangeney, Anne 1 ; Sainte-Marie, Jacques 1 ; Souillé, Fabien 1 ; Vallée, Martin 1

1 
@article{M2AN_2019__53_6_1981_0,
     author = {Allgeyer, Sebastien and Bristeau, Marie-Odile and Froger, David and Hamouda, Raouf and Jauzein, V. and Mangeney, Anne and Sainte-Marie, Jacques and Souill\'e, Fabien and Vall\'ee, Martin},
     title = {Numerical approximation of the {3D} hydrostatic {Navier{\textendash}Stokes} system with free surface},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1981--2024},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {6},
     year = {2019},
     doi = {10.1051/m2an/2019044},
     mrnumber = {4036661},
     zbl = {1434.65147},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2019044/}
}
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Allgeyer, Sebastien; Bristeau, Marie-Odile; Froger, David; Hamouda, Raouf; Jauzein, V.; Mangeney, Anne; Sainte-Marie, Jacques; Souillé, Fabien; Vallée, Martin. Numerical approximation of the 3D hydrostatic Navier–Stokes system with free surface. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 6, pp. 1981-2024. doi : 10.1051/m2an/2019044. http://www.numdam.org/articles/10.1051/m2an/2019044/

[1] N. Aïssiouene, M.-O. Bristeau, E. Godlewski, A. Mangeney, C. Parés and J. Sainte-Marie, A two-dimensional method for a family of dispersive shallow water model. Working paper or Preprint (May 2019). | MR

[2] ANGE Team, Freshkiss3D home page. Available at: http://freshkiss3D.gforge.inria.fr (2017).

[3] E. Audusse, A multilayer Saint-Venant model: derivation and numerical validation. Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 189–214. | DOI | MR | Zbl

[4] E. Audusse and M.-O. Bristeau, A well-balanced positivity preserving second-order scheme for Shallow Water flows on unstructured meshes. J. Comput. Phys. 206 (2005) 311–333. | DOI | MR | Zbl

[5] E. Audusse and M.-O. Bristeau, Finite-volume solvers for a multilayer Saint-Venant system. Int. J. Appl. Math. Comput. Sci. 17 (2007) 311–319. | DOI | MR | Zbl

[6] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for Shallow Water flows. SIAM J. Sci. Comput. 25 (2004) 2050–2065. | DOI | MR | Zbl

[7] E. Audusse, F. Bouchut, M.-O. Bristeau and J. Sainte-Marie, Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system. Math. Comput. 85 (2016) 2815–2837. | DOI | MR | Zbl

[8] E. Audusse, M.-O. Bristeau and A. Decoene, Numerical simulations of 3D free surface flows by a multilayer Saint-Venant model. Int. J. Numer. Methods Fluids 56 (2008) 331–350. | DOI | MR | Zbl

[9] E. Audusse, M.-O. Bristeau, M. Pelanti and J. Sainte-Marie, Approximation of the hydrostatic Navier-Stokes system for density stratified flows by a multilayer model. Kinetic interpretation and numerical validation. J. Comput. Phys. 230 (2011) 3453–3478. | DOI | MR | Zbl

[10] E. Audusse, M.-O. Bristeau, B. Perthame and J. Sainte-Marie, A multilayer Saint-Venant system with mass exchanges for Shallow Water flows. Derivation and numerical validation. ESAIM: M2AN 45 (2011) 169–200. | DOI | Numdam | MR | Zbl

[11] E. Audusse, M.-O. Bristeau and J. Sainte-Marie, Kinetic entropy for the layer-averaged hydrostatic Navier–Stokes equations. Working paper or Preprint (2017).

[12] A.-J.-C. Barré De Saint-Venant, Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l’introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73 (1871) 147–154. | JFM

[13] Bathymetry & Relief, NOAA home page. Available at: https://www.ngdc.noaa.gov/mgg/global/global.htmlhttps://www.ngdc.noaa.gov/mgg/global/global.html (2017).

[14] O. Bernard, A.-C. Boulanger, M.-O. Bristeau and J. Sainte-Marie, A 2D model for hydrodynamics and biology coupling applied to algae growth simulations. ESAIM: M2AN 47 (2013) 387–1412. | DOI | Numdam | MR | Zbl

[15] F. Berthelin and F. Bouchut, Relaxation to isentropic gas dynamics for a BGK system with single kinetic entropy. Methods Appl. Anal. 9 (2002) 313–327. | DOI | MR | Zbl

[16] F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Stat. Phys. 95 (1999) 113–170. | DOI | MR | Zbl

[17] F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. 94 (2003) 623–672. | DOI | MR | Zbl

[18] F. Bouchut, An introduction to finite volume methods for hyperbolic conservation laws. ESAIM Proc. 15 (2004) 107–127.

[19] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Birkhäuser, Basel (2004). | DOI | MR | Zbl

[20] F. Bouchut and T. Morales De Luna, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. M2AN 42 (2008) 683–698. | DOI | Numdam | MR | Zbl

[21] F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography. Comm. Math. Sci. 2 (2004) 359–389. | DOI | MR | Zbl

[22] F. Bouchut and V. Zeitlin, A robust well-balanced scheme for multi-layer shallow water equations. Discrete Contin. Dyn. Syst. Ser. B 13 (2010) 739–758. | MR | Zbl

[23] A.-C. Boulanger and J. Sainte-Marie, Analytical solutions for the free surface hydrostatic Euler equations. Commun. Math. Sci. 11 (2013) 993–1010. | DOI | MR | Zbl

[24] Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles. Nonlinearity 12 (1999) 495–512. | DOI | MR | Zbl

[25] D. Bresch, A. Kazhikhov and J. Lemoine, On the two-dimensional hydrostatic Navier-Stokes equations. SIAM J. Math. Anal. 36 (2004/2005) 796–814. | DOI | MR | Zbl

[26] M.-O. Bristeau and B. Coussin, Boundary Conditions for the Shallow Water Equations solved by Kinetic Schemes. Research Report RR-4282, INRIA (2001).

[27] M.-O. Bristeau, B. Di-Martino, C. Guichard and J. Sainte-Marie, Layer-averaged Euler and Navier-Stokes equations. Commun. Math. Sci. 15 (2017) 1221–1246. | DOI | MR | Zbl

[28] M.-O. Bristeau, B. Di-Martino, A. Mangeney, J. Sainte-Marie and F. Souillé, Various analytical solutions for the incompressible Euler and Navier-Stokes systems with free surface. Working paper or Preprint (2018).

[29] M.-O. Bristeau, A. Mangeney, J. Sainte-Marie and N. Seguin, An energy-consistent depth-averaged euler system: derivation and properties. Discrete Contin. Dyn. Syst. Ser. B 20 (2015) 961–988. | DOI | MR | Zbl

[30] M.-J. Castro, J. Macias, C. Parés, A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. M2AN 35 (2001) 107–127. | DOI | Numdam | MR | Zbl

[31] M.-J. Castro, J.A. García-Rodríguez, J.M. González-Vida, J. Macías, C. Parés and M.E. Vázquez-Cendón, Numerical simulation of two-layer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195 (2004) 202–235. | DOI | MR | Zbl

[32] V. Casulli, A semi-implicit numerical method for the free-surface Navier-Stokes equations. Int. J. Numer. Methods Fluids 74 (2014) 605–622. | DOI | MR | Zbl

[33] Clawpack Development Team. Clawpack Software. Available at: http://depts.washington.edu/clawpack/links/nthmp-benchmarks/monai-valley/index.html (2019).

[34] D. Demory, C. Combe, P. Hartmann, A. Talec, E. Pruvost, R. Hamouda, F. Souillé, P.-O. Lamare, M.-O. Bristeau, J. Sainte-Marie, S. Rabouille, F. Mairet, A. Sciandra and O. Bernard, How do microalgae perceive light in a high-rate pond? Towards more realistic Lagrangian experiments. R. Soc. Open Sci. 5 (2018) 180523. | DOI

[35] E.D. Fernández-Nieto, G. Garres-Dìas, A. Mangeney and G. Narbona-Reina, A multilayer shallow model for dry granular flows with the u(I)-rheology: application to granular collapse on erodible beds. J. Fluid Mech. 798 (2016) 643–681. | DOI | MR | Zbl

[36] E.D. Fernández-Nieto, E.H. Koné and T. Chacón Rebollo, A multilayer method for the hydrostatic Navier-Stokes equations: a particular weak solution. J. Sci. Comput. 60 (2014) 408–437. | DOI | MR | Zbl

[37] E. Grenier, On the derivation of homogeneous hydrostatic equations. ESAIM: M2AN 33 (1999) 0965–970. | DOI | Numdam | MR | Zbl

[38] A. Gusman, S. Murotani, K. Satake, M. Heidarzadeh, E. Gunawan, S. Watada and B. Schurr, Fault slip distribution of the 2014 iquique, Chile, earthquake estimated from ocean-wide tsunami waveforms and GPS data. Geophys. Res. Lett. 42 (2015) 1053–1060. | DOI

[39] J.-M. Hervouet, Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method. Wiley (2007) | DOI | Zbl

[40] P.-L. Lions, Mathematical Topics in Fluid Mechanics. In: Vol 1 of Incompressible Models. Oxford University Press, Oxford (1996). | Zbl

[41] P.L.-F. Liu, H. Yeh and C. Synolakis, Advanced Numerical Models for Simulating Tsunami Waves and Runup. World Scientific Publishing Company 10 (2008).

[42] J. Macías, M.J. Castro, S. Ortega, C. Escalante and J.M. González-Vida, Performance benchmarking of tsunami-HySEA model for NTHMP’s inundation mapping activities. Pure Appl. Geophys. 174 (2017) 3147–3183. | DOI

[43] N. Masmoudi and T. Wong, On the HS theory of hydrostatic Euler equations. Arch. Ration. Mech. Anal. 204 (2012) 231–271. | DOI | MR | Zbl

[44] National Tsunami Hazard Mitigation Program (NTHMP). Proceedings and Results of the 2011 NTHMP Model Benchmarking. Workshop NOAA Special Report. Department of Commerce/NOAA/NTHMP, Boulder(2012).

[45] Noaa Center for Tsunami Research. Monai valley. Available at: https://nctr.pmel.noaa.gov/benchmark/Laboratory/Laboratory_MonaiValley/ (2019).

[46] L.V. Ovsyannikov, Two-layer shallow water models. Prikl. Mekh. Tekh. Fiz. 2 (1979) 3–14. | MR

[47] B. Perthame, Kinetic Formulation of Conservation Laws, Oxford University Press, Oxford (2002). | DOI | MR | Zbl

[48] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo 38 (2001) 201–231. | DOI | MR | Zbl

[49] J. Sainte-Marie, Vertically averaged models for the free surface Euler system. Derivation and kinetic interpretation, Math. Models Methods Appl. Sci. (M3AS) 21 (2011) 459–490. | DOI | MR | Zbl

[50] W.C. Thacker, Some exact solutions to the non-linear shallow-water wave equations. J. Fluid Mech. 107 (1981) 499–508. | DOI | MR | Zbl

[51] M. Vallée, R. Grandin, S. Ruiz, B. Delouis, C. Vigny, E. Rivera, E. Aissaoui, S. Alleyer, Q. Bletery, C. Satriano, N. Poiata, P. Bernard, J.-P. Vilotte and B. Schurr, Complex rupture of an apparently simple asperity during the 2014/04/01 pisagua earthquake (northern chile, mw=8.1). In: Vol. 18 of EGU General Assembly Conference Abstracts. EGU2016-8660 (2016).

[52] B. Van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. With an introduction by Ch. Hirsch, Commemoration of the 30th anniversary {of J. Comput. Phys.}. MR 1486274. J. Comput. Phys. 135 (1997) 227–248. | DOI | MR | Zbl

[53] C.B. Vreugdenhil, Two-layer shallow-water flow in two dimensions, a numerical study. J. Comput. Phys. 33 (1979) 169–184. | DOI | MR | Zbl

[54] S. Watada, S. Kusumoto and K. Satake, Traveltime delay and initial phase reversal of distant tsunamis coupled with the self-gravitating elastic earth. J. Geophys. Res.: Solid Earth 119 (2014) 4287–4310. | DOI

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