Multi-dimensional shear shallow water flows: problems and solutions. (English) Zbl 1406.65068
Summary: The mathematical model of shear shallow water flows of constant density is studied. This is a 2D hyperbolic non-conservative system of equations that is mathematically equivalent to the Reynolds-averaged model of barotropic turbulent flows. The model has three families of characteristics corresponding to the propagation of surface waves, shear waves and average flow (contact characteristics). The system is non-conservative: for six unknowns (the fluid depth, two components of the depth averaged horizontal velocity, and three independent components of the symmetric Reynolds stress tensor) one has only five conservation laws (conservation of mass, momentum, energy and mathematical ‘entropy’). A splitting procedure for solving such a system is proposed allowing us to define a weak solution. Each split subsystem contains only one family of waves (either surface or shear waves) and contact characteristics. The accuracy of such an approach is tested on 2D analytical solutions describing the flow with linear with respect to the space variables velocity, and on the solutions describing 1D roll waves. The capacity of the model to describe the full transition scenario as commonly seen in the formation of roll waves: from uniform flow to 1D roll waves, and, finally, to 2D transverse ‘fingering’ of the wave profiles, is shown.
MSC:
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 35L67 | Shocks and singularities for hyperbolic equations |
Software:
HE-E1GODFReferences:
| [1] | Abgrall, R.; Karni, S., Two-layer shallow water systems: a relaxation approach, SIAM J. Sci. Comput., 31, 3, 1603-1627 (2009) · Zbl 1188.76229 |
| [2] | Audebert, B.; Coquel, F., Structural stability of shock solutions of hyperbolic systems in nonconservation form via kinetic relations, (Benzoni-Gavage, S.; Serre, D., Hyperbolic Problems: Theory, Numerics, Applications (2008), Springer: Springer Berlin-Heidelberg) · Zbl 1141.35420 |
| [3] | Baer, M.; Nunziato, J., A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials, Int. J. Multiph. Flow, 12, 861-889 (1986) · Zbl 0609.76114 |
| [4] | Baines, P. G., Topographic Effects in Stratified Flows (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0840.76001 |
| [5] | Barré de Saint-Venant, A. J.C., Théorie du movement non permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leurs lit, C. R. Acad. Sc. Paris, 73, 147-154 (1871), and 237-240 · JFM 03.0482.04 |
| [6] | Barros, B., Conservation laws for one-dimensional shallow water models for one and two-layer flows, Math. Models Methods Appl. Sci., 16, 119-137 (2006) · Zbl 1134.35377 |
| [7] | Benney, D. J., Some properties of long nonlinear waves, Stud. Appl. Math., 52, 45-50 (1973) · Zbl 0259.35011 |
| [8] | Berthon, C.; Coquel, F.; Hérard, J.-M.; Uhlmann, M., An approximate solution of the Riemann problem for a realisable second-moment turbulent closure, Shock Waves, 11, 245-269 (2002) · Zbl 0996.76039 |
| [9] | Besse, N., On the waterbag continuum, Arch. Ration. Mech. Anal., 199, 453-491 (2011) · Zbl 1229.35297 |
| [10] | Bouchut, F., Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources (2004), Birkhäuser: Birkhäuser Basel · Zbl 1086.65091 |
| [11] | Brock, R. R., Development of Roll Waves in Open Channels (1967), Caltech, PhD Thesis |
| [12] | Brock, R. R., Development of roll-wave trains in open channels, J. Hydraul. Div., 95, 1401-1428 (1969) |
| [13] | Brock, R. R., Periodic permanent roll waves, J. Hydraul. Div., 96, 2565-2580 (1970) |
| [14] | Castro, A.; Lannes, D., Fully nonlinear long-wave models in the presence of vorticity, J. Fluid Mech., 759, 642-675 (2014) · Zbl 1446.76077 |
| [15] | Coquel, F.; Hérard, J. M.; Saleh, K., A positive and entropy-satisfying finite volume scheme for the Baer-Nunziato model, J. Comput. Phys., 330, 401-435 (2016) · Zbl 1378.76117 |
| [16] | Dal Maso, G.; LeFloch, P. G.; Murat, F., Definition and weak stability of a non-conservative product, J. Math. Pures Appl., 74, 483-548 (1995) · Zbl 0853.35068 |
| [17] | Dremin, A. N.; Karpukhin, I. A., Method of determination of shock adiabat of the dispersed substances, J. Appl. Mech. Tech. Phys., 1, 3, 184-188 (1960), (in Russian) |
| [18] | Favrie, N.; Gavrilyuk, S. L., Diffuse interface model for compressible fluid-compressible elastic-plastic solid interaction, J. Comput. Phys., 231, 2696-2723 (2012) · Zbl 1430.74036 |
| [19] | Favrie, N.; Gavrilyuk, S.; Ndanou, S., A thermodynamically compatible splitting procedure in hyperelasticity, J. Comput. Phys., 270, 300-324 (2014) · Zbl 1349.74344 |
| [20] | Gavrilyuk, S.; Saurel, R., Estimation of the turbulence energy production across a shock wave, J. Fluid Mech., 549, 131-139 (2006) |
| [21] | Gavrilyuk, S. L.; Gouin, H., Geometric evolution of the Reynolds stress tensor, Int. J. Eng. Sci., 59, 65-73 (2012) · Zbl 1423.76137 |
| [22] | Gavrilyuk, S., Multiphase flow modelling via Hamilton’s principle, (dell’Isola, F.; Gavrilyuk, S. L., Variational Models and Methods in Solid and Fluid Mechanics (2011), Springer: Springer Wien) · Zbl 1247.76085 |
| [23] | Gavrilyuk, S. L.; Liapidevskii, V. Yu.; Chesnokov, A. A., Spilling breakers in shallow water: applications to Favre waves and to the shoaling and breaking of solitary waves, J. Fluid Mech., 808, 441-468 (2016) · Zbl 1383.76197 |
| [24] | Godunov, S. K., A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb., 47, 271-306 (1959) · Zbl 0171.46204 |
| [25] | Ivanova, K. A.; Gavrilyuk, S. L.; Nkonga, B.; Richard, G. L., Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel, Comput. Fluids, 159, 189-203 (2017) · Zbl 1390.76042 |
| [26] | Kapila, A. K.; Menikoff, R.; Bdzil, J. B.; Son, S. F.; Stewart, D. S., Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Phys. Fluids, 13, 10, 3002-3024 (2001) · Zbl 1184.76268 |
| [27] | LeFloch, P. G., Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves (2002), Birkhäuser: Birkhäuser Basel · Zbl 1019.35001 |
| [28] | Leng, X.; Chanson, H., Breaking bore: physical observation of roller characteristics, Mech. Res. Commun., 65, 24-29 (2015) |
| [29] | LeVeque, R. J., Numerical Methods for Conservation Laws (1992), Birkhäuser: Birkhäuser Basel · Zbl 0847.65053 |
| [30] | Liapidevskii, V. Yu.; Teshukov, V. M., Mathematical Models of Propagation of Long Waves in an Inhomogeneous Fluid (2000), Siberian Branch of the Russian Academy of Sciences: Siberian Branch of the Russian Academy of Sciences Novosibirsk, (in Russian) |
| [31] | Mandli, K. T., Finite Volume Methods for the Multilayer Shallow Water Equations with Applications to Storm Surges (2011), Univ. of Washington, PhD thesis |
| [32] | Mohammadi, B.; Pironneau, O., Analysis of the K-Epsilon Turbulence Model (1994), John Wiley & Sons: John Wiley & Sons New York · Zbl 1539.76001 |
| [33] | Montgomery, P. J.; Moodie, T. G., On the number of conserved quantities for the two-layer shallow water equations, Stud. Appl. Math., 106, 229-259 (2001) · Zbl 1152.76349 |
| [34] | Ndanou, S.; Favrie, N.; Gavrilyuk, S., Multi-solid and multi-fluid diffuse interface model: applications to dynamic fracture and fragmentation, J. Comput. Phys., 295, 523-555 (2015) · Zbl 1349.74377 |
| [35] | Ovsyannikov, L. V., Two-layer shallow water model, J. Appl. Mech. Tech. Phys., 20, 2, 127-135 (1979) |
| [36] | Ovsyannikov, L. V., A new solution of the hydrodynamic equations, C. R. (Dokl.) Acad. Sci. URSS, 111, 47-49 (1956), (in Russian) · Zbl 0073.41704 |
| [37] | Ovsyannikov, L. V., Lectures on the Fundamentals of Gas Dynamics (1981), Nauka: Nauka Moscow, (in Russian) · Zbl 0539.76079 |
| [38] | Richard, G. L.; Gavrilyuk, S. L., A new model of roll waves: comparison with Brock’s experiments, J. Fluid Mech., 698, 374-405 (2012) · Zbl 1250.76041 |
| [39] | Richard, G. L.; Gavrilyuk, S. L., The classical hydraulic jump in a model of shear shallow-water flows, J. Fluid Mech., 725, 492-521 (2013) · Zbl 1287.76089 |
| [40] | Richard, G. L., Elaboration d’un modèle d’écoulements turbulents en faible profondeur: application au ressaut hydraulique et aux trains de rouleaux (2013), Aix-Marseille Université, PhD thesis |
| [41] | Richard, G. L.; Gavrilyuk, S. L., Modelling turbulence generation in solitary waves on shear shallow water flows, J. Fluid Mech., 773, 49-74 (2015) · Zbl 1331.76034 |
| [42] | Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150, 425-467 (2001) · Zbl 0937.76053 |
| [43] | Saurel, R.; Gavrilyuk, S. L.; Renaud, F., A multiphase model with internal degrees of freedom: application to shock-bubble interaction, J. Fluid Mech., 495, 283-321 (2003) · Zbl 1080.76062 |
| [44] | Saurel, R.; Le Metayer, O.; Massoni, J.; Gavrilyuk, S., Shock jump relations for multiphase mixtures with stiff mechanical relaxation, Shock Waves, 16, 209-232 (2007) · Zbl 1195.76245 |
| [45] | Sedov, L. I., On the integration of the equations of one-dimensional gas motion, C. R. (Dokl.) Acad. Sci. URSS, 40, 753-755 (1953), (in Russian) |
| [46] | Teshukov, V. M., On hyperbolicity of long-wave equations, Sov. Math. Dokl., 32, 469-473 (1985) · Zbl 0615.35051 |
| [47] | Teshukov, V. M., On Cauchy problem for long-wave equations, (Neittaanmäki, P., Numerical Methods for Free Boundary Problems. Numerical Methods for Free Boundary Problems, ISMN 92, vol. 106 (1992), Birkhäuser: Birkhäuser Basel), 331-338 · Zbl 0817.35082 |
| [48] | Teshukov, V.; Russo, G.; Chesnokov, A., Analytical and numerical solutions of the shallow water equations for 2-D rotational flows, Math. Models Methods Appl. Sci., 14, 1451-1479 (2004) · Zbl 1149.76613 |
| [49] | Teshukov, V. M., Gas dynamic analogy for vortex free-boundary flows, J. Appl. Mech. Tech. Phys., 48, 303-309 (2007) · Zbl 1150.76335 |
| [50] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (2009), Springer: Springer Berlin-Heidelberg · Zbl 1227.76006 |
| [51] | Troshkin, O. V., On wave properties of an incompressible turbulent fluid, Physica A, 168, 881-899 (1990) · Zbl 0712.76050 |
| [52] | Truskinovsky, L., Kinks versus shocks, (Fosdick, R.; Dunn, E.; Slemrod, H., Shock Induced Transitions and Phase Structures in General Media. Shock Induced Transitions and Phase Structures in General Media, IMA Vol. Math. Appl., vol. 52 (1993), Springer: Springer New York) · Zbl 0818.76036 |
| [53] | Pope, S. B., Turbulent Flows (2005), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1075.76003 |
| [54] | Van Leer, B., Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136 (1979) · Zbl 1364.65223 |
| [55] | Wilcox, D., Turbulence Modeling for CFD (1998), DCW Industries |
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