Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel. (English) Zbl 1390.76042
Summary: The formation of a periodic roll-wave train in a long channel is studied for two sets of experimental parameters corresponding to R. R. Brock’s experiments [Development of roll waves in open channels. Pasadena, CA: California Institute of Technology. (PhD Thesis) (1967); “Development of roll-wave trains in open channels”, J. Hydraul. Div. 95, No. 4, 1401–1428 (1969)], who measured permanent wave profiles by introducing periodic perturbations at the channel inlet. In both cases, a formed free surface profile is found in good agreement with the experimental results. Qualitative properties of solutions to the model are studied in the case where the perturbation frequency is lower than the experimental one, so longer waves are generated at the channel inlet. It is observed that the corresponding roll-wave train is strongly modulated. In the case where the waves of two different lengths are generated at the channel inlet, the coarsening is observed, i.e. the process where shorter waves disappear progressively by transferring their energy to longer waves forming later a regular roll-wave train. The coarsening phenomenon is always accompanied by a strong modulation. A comparison with the Saint-Venant equations is also performed. The formation of a single wave composing a roll-wave train is also studied in a domain with periodic boundary conditions (called “periodic box”) for the same sets of experimental parameters. The free surface profile is found also in good agreement with the experimental results. This allows us to justify the use of the “periodic box” as a simple mathematical tool for a qualitative study of stability of roll-waves. In particular, we study the stability of a single steady wave by increasing its length. It is shown that the wave becomes morphologically unstable after some critical wave length: it bifurcates into a system of two waves. In the framework of a simplified multi-D model of shear shallow water flows it is also proved that a single steady wave is stable under multi-dimensional perturbations.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
Software:
HE-E1GODFReferences:
| [1] | Barker, B.; Johnson, M. A.; Noble, P.; Rodrigues, L. M.; Zumbrun, K., Stability of viscous st.Venant roll-waves: from onset to infinite-Froude number limit, J Nonlinear Sci, (2017) · Zbl 1373.35040 |
| [2] | Boudlal, A.; Liapidevskii, V. Y., Stabilité de trains d’ondes dans un canal déecouvert, CRAS Mécanique, 330, 2-3, 291-295, (2002) |
| [3] | Brock RR. Development of roll waves in open channels1967, PhD Thesis, Caltech;.; Brock RR. Development of roll waves in open channels1967, PhD Thesis, Caltech;. |
| [4] | Brock, R. R., Development of roll-wave trains in open channels, J Hydraul Div, 95, 4, 1401-1428, (1969) |
| [5] | Chang, H. C.; Demekhin, E. A.; Kalaidin, E; Ye, Y., Coarsening dynamics of falling-film solitary waves, Phys Rev E, 54, 2, 1467-1477, (1996) |
| [6] | Dressler, R. F., Mathematical solution of the problem of roll-waves in inclined opel channels, Commun Pure Appl Math, 2, 2-3, 149-194, (1949) · Zbl 0038.38405 |
| [7] | Gavrilyuk, S. L.; Gouin, H., Geometric evolution of the Reynolds stress tensor, Int J Eng Sci, 59, 65-73, (2012) · Zbl 1423.76137 |
| [8] | Godunov, S. K., A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89, 3, 271-306, (1959) · Zbl 0171.46204 |
| [9] | LeVeque, R. J., Numerical methods for conservation laws, vol. 132, (1992), Springer · Zbl 0847.65053 |
| [10] | Liapidevskii V.Y., Teshukov V.M.. Mathematical models for a long waves propagation in an inhomogeneous fluid2000, (in Russian).; Liapidevskii V.Y., Teshukov V.M.. Mathematical models for a long waves propagation in an inhomogeneous fluid2000, (in Russian). |
| [11] | Press, W.; Teukolsky, F. A.; Vetterling, W. T.; Flannery, B. P., Numerical recipes in FORTRAN: the art of scientific computing, (1992), Cambridge University Press · Zbl 0778.65002 |
| [12] | 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 |
| [13] | Richard, G. L.; Gavrilyuk, S. L., A new model of roll waves: comparison with brocks experiments, J Fluid Mech, 698, 374-405, (2012) · Zbl 1250.76041 |
| [14] | 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 |
| [15] | Russo, G., Central schemes for conservation laws with application to shallow water equations, Trends and Applications of Mathematics to Mechanics, 225-246, (2005), Springer |
| [16] | Strang, G., On the construction and comparison of difference schemes, SIAM J Numer Anal, 5, 3, 506-517, (1968) · Zbl 0184.38503 |
| [17] | Teshukov, V. M., Gas dynamics analogy for vortex free-boundary flows, J Appl Mech Tech Phys, 48, 3, 303-309, (2007) · Zbl 1150.76335 |
| [18] | Toro, E. F., Riemann solvers and numerical methods for fluid dynamics: a practical introduction, (2009), Springer Science & Business Media · Zbl 1227.76006 |
| [19] | Tougou, H., Stability of turbulent roll-waves in an inclined open channel, J Phys Soc Jpn, 48, 3, 1018-1023, (1980) · Zbl 1334.76056 |
| [20] | Zanuttigh, B.; Lamberti, A., Roll waves simulation using shallow water equations and weighted average flux method, J Hydraul Res, 40, 5, 610-622, (2002) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
