Dispersion in Shallow Moment Equations

doi:10.1007/s42967-023-00325-2

Abstract

Abstract: Shallow moment models are extensions of the hyperbolic shallow water equations. They admit variations in the vertical profile of the horizontal velocity. This paper introduces a non-hydrostatic pressure to this framework and shows the systematic derivation of dimensionally reduced dispersive equation systems which still hold information on the vertical profiles of the flow variables. The derivation from a set of balance laws is based on a splitting of the pressure followed by a same-degree polynomial expansion of the velocity and pressure fields in a vertical direction. Dimensional reduction is done via Galerkin projections with weak enforcement of the boundary conditions at the bottom and at the free surface. The resulting equation systems of order zero and one are presented in linear and nonlinear forms for Legendre basis functions and an analysis of dispersive properties is given. A numerical experiment shows convergence towards the resolved reference model in the linear stationary case and demonstrates the reconstruction of vertical profiles.

Key words: Shallow flow, Free surface flow, Non-hydrostatic model, Dispersive equations, Moment approximation, Hyperbolic systems

Ullika Scholz, Julia Kowalski, Manuel Torrilhon. Dispersion in Shallow Moment Equations[J]. Communications on Applied Mathematics and Computation, 2024, 6(4): 2155-2195.

TrendMD

References

1. Airy, G.B.: Tides and waves. In: Rose, H.J., et al. (eds) Encyclopaedia Metropolitana (1817-1845), Mixed Sciences, Vol. 3. London (1845)
2. Baker, J.L., Barker, T., Gray, J.M.N.T.: A two-dimensional depth-averaged μ(I)-rheology for dense granular avalanches. J. Fluid Mech. 787, 367-395 (2016)
3. Barros, R., Gavrilyuk, S.L.: Dispersive nonlinear waves in two-layer flows with free surface part II. Large amplitude solitary waves embedded into the continuous spectrum. Stud. Appl. Math. 119(3), 213-251 (2007)
4. Barros, R., Gavrilyuk, S.L., Teshukov, V.M.: Dispersive nonlinear waves in two-layer flows with free surface. I. Model derivation and general properties. Stud. Appl. Math. 119(3), 191-211 (2007)
5. Barthelemy, E.: Nonlinear shallow water theories for coastal waves. Surv. Geophys. 25(3), 315-337 (2004)
6. Bonneton, P., Chazel, F., Lannes, D., Marche, F., Tissier, M.: A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model. J. Comput. Phys. 230(4), 1479-1498 (2011)
7. Casulli, V.: A semi-implicit finite difference method for non-hydrostatic, free-surface flows. Int. J. Numer. Methods Fluids 30(4), 425-440 (1999)
8. Chanson, H.: Environmental Hydraulics for Open Channel Flows. Butterworth-Heinemann, Oxford (2004)
9. Christen, M., Kowalski, J., Bartelt, P.: RAMMS: numerical simulation of dense snow avalanches in threedimensional terrain. Cold Reg. Sci. Technol. 63(1), 1-14 (2010)
10. Cienfuegos, R., Barthelemy, E., Bonneton, P.: A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I: model development and analysis. Int. J. Numer. Methods Fluids 51(11), 1217-1253 (2006)
11. Escalante, C., Fernandez-Nieto, E.D., Morales de Luna, T., Castro, M.J.: An efficient two-layer non-hydrostatic approach for dispersive water waves. J. Sci. Comput. 79(1), 273-320 (2019)
12. Escalante, C., Morales de Luna, T.: A general non-hydrostatic hyperbolic formulation for Boussinesq dispersive shallow flows and its numerical approximation. J. Sci. Comput. 83(3), 62 (2020)
13. Escalante, C., Morales de Luna, T., Castro, M.J.: Non-hydrostatic pressure shallow flows: GPU implementation using finite volume and finite difference scheme. Appl. Math. Comput. 338, 631-659 (2018)
14. Favrie, N., Gavrilyuk, S.: A rapid numerical method for solving Serre-Green-Naghdi equations describing long free surface gravity waves. Nonlinearity 30(7), 2718-2736 (2017)
15. Fernandez-Nieto, E.D., Parisot, M., Penel, Y., Sainte-Marie, J.: A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows. Commun. Math. Sci. 16(5), 1169-1202 (2018)
16. French, R.H.: Open-Channel Hydraulics. McGraw-Hill, New York (1985)
17. Garres-Diaz, J., Castro Diaz, M.J., Koellermeier, J., Morales de Luna, T.: Shallow water moment models for bedload transport problems. Commun. Comput. Phys. 30(3), 903-941 (2021)
18. Garres-Diaz, J., Escalante, C., Morales de Luna, T., Castro Diaz, M.J.: A general vertical decomposition of Euler equations: multilayer-moment models. Appl. Numer. Math. 183, 236-262 (2023)
19. Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331-407 (1949)
20. Gray, J.M.N.T., Edwards, A.N.: A depth-averaged μ(I)-rheology for shallow granular free-surface flows. J. Fluid Mech. 755, 503 (2014)
21. Green, A.E., Naghdi, P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78(2), 237-246 (1976)
22. Hagemeier, T., Hartmann, M., Thevenin, D.: Practice of vehicle soiling investigations: a review. Int. J. Multiph. Flow 37(8), 860-875 (2011)
23. Houghton, D.D., Kasahara, A.: Nonlinear shallow fluid flow over an isolated ridge. Commun. Pure Appl. Math. 21(1), 1-23 (1968)
24. Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, Cambridge (2003)
25. Kazolea, M., Delis, A.I., Synolakis, C.E.: Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq-type equations. J. Comput. Phys. 271, 281-305 (2014)
26. Kern, M., Bartelt, P., Sovilla, B., Buser, O.: Measured shear rates in large dry and wet snow avalanches. J. Glaciol. 55(190), 327-338 (2009)
27. Koellermeier, J., Rominger, M.: Analysis and numerical simulation of hyperbolic shallow water moment equations. Commun. Comput. Phys. 28(3), 1038-1084 (2020)
28. Kowalski, J., McElwaine, J.N.: Shallow two-component gravity-driven flows with vertical variation. J. Fluid Mech. 714, 434-462 (2013)
29. Kowalski, J., Torrilhon, M.: Moment approximations and model cascades for shallow flow. Commun. Comput. Phys. 25(3), 669-702 (2019)
30. Lannes, D.: The Water Waves Problem: Mathematical Analysis and Asymptotics. Mathematical Surveys and Monographs. American Mathematical Society, Providence (2013)
31. Le Metayer, O., Gavrilyuk, S., Hank, S.: A numerical scheme for the Green-Naghdi model. J. Comput. Phys. 229, 2034-2045 (2010)
32. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems, vol. 31. Cambridge University Press, Cambridge (2002)
33. Lin, P., Li, C.W.: A σ-coordinate three-dimensional numerical model for surface wave propagation. Int. J. Numer. Methods Fluids 38(11), 1045-1068 (2002)
34. Ma, G., Shi, F., Kirby, J.T.: Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Model. 43-44, 22-35 (2012)
35. Madsen, P.A., Murray, R., Sorensen, O.R.: A new form of the Boussinesq equations with improved linear dispersion characteristics. Coast. Eng. 15(4), 371-388 (1991)
36. Madsen, P.A., Bingham, H.B., Liu, H.: A new Boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech. 462, 1-30 (2002)
37. Mergili, M., Jan-Thomas, F., Krenn, J., Pudasaini, S.P.: r.avaflow v1, an advanced open-source computational framework for the propagation and interaction of two-phase mass flows. Geosci. Model Dev. 10(2), 553 (2017)
38. Miles, J., Salmon, R.: Weakly dispersive nonlinear gravity waves. J. Fluid Mech. 157, 519-531 (1985)
39. Miles, J.W.: The Korteweg-de Vries equation: a historical essay. J. Fluid Mech. 106, 131-147 (1981)
40. Nadiga, B.T.: An adaptive discrete-velocity model for the shallow water equations. J. Comput. Phys. 121, 271-280 (1995)
41. Nadiga, B.T., Margolin, L.G., Smolarkiewicz, P.K.: Different approximations of shallow fluid flow over an obstacle. Phys. Fluids 8(8), 2066-2077 (1996)
42. Nagl, G., Hubl, J., Kaitna, R.: Velocity profiles and basal stresses in natural debris flows. Earth Surf. Process. Landf. 45(8), 1764-1776 (2020)
43. Noelle, S., Parisot, M., Tscherpel, T.: A class of boundary conditions for time-discrete GreenNaghdi equations with bathymetry. SIAM J. Numer. Anal. 60, 2681-2712 (2022)
44. Panda, N., Dawson, C., Zhang, Y., Kennedy, A.B., Westerink, J.J., Donahue, A.S.: Discontinuous Galerkin methods for solving Boussinesq-Green-Naghdi equations in resolving non-linear and dispersive surface water waves. J. Comput. Phys. 273, 572-588 (2014)
45. Parisot, M.: Entropy-satisfying scheme for a hierarchy of dispersive reduced models of free surface flow. Int. J. Numer. Methods Fluids 91(10), 509-531 (2019)
46. Peregrine, D.H.: Long waves on a beach. J. Fluid Mech. 27(4), 815-827 (1967)
47. Phillips, N.A.: A coordinate system having some special advantages for numerical forecasting. J. Meteorol. 14, 184-185 (1956)
48. Pudasaini, S.P., Hutter, K.: Avalanche Dynamics: Dynamics of Rapid Flows of Dense Granular Avalanches. Springer, Berlin (2007)
49. Ruyer-Quil, C., Manneville, P.: Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys. Fluids 14(1), 170-183 (2002)
50. Ruyer-Quil, C., Manneville, P.: Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357-369 (2000)
51. Sanvitale, N., Bowman, E.T.: Using PIV to measure granular temperature in saturated unsteady polydisperse granular flows. Granul. Matter 18(3), 1-12 (2016)
52. Savage, S.B., Hutter, K.: The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199, 177-215 (1989)
53. Schaefer, M., Bugnion, L.: Velocity profile variations in granular flows with changing boundary conditions: insights from experiments. Phys. Fluids 25(6), 063303 (2013)
54. Scholz, U., Kowalski, J., Torrilhon, M.: GitHub Repository: Dispersive Shallow Moment Models. In: Source Code www.github.com/ShallowFlowMoments (2023)
55. Stansby, P.K., Zhou, J.G.: Shallow-water flow solver with non-hydrostatic pressure: 2D vertical plane problems. Int. J. Numer. Methods Fluids 28(3), 541-563 (1998)
56. Steffler, P.M., Yee-Chung, J.: Depth averaged and moment equations for moderately shallow free surface flow. J. Hydraul. Res. 31(1), 5-17 (1993)
57. Su, C.H., Gardner, C.S.: Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation. J. Math. Phys. 10(3), 536-539 (1969)
58. Torrilhon, M.: Modeling nonequilibrium gas flow based on moment equations. Annu. Rev. Fluid Mech. 48, 429-458 (2016)
59. Wei, G., Kirby, J.T., Grilli, S., Subramanya, R.: A fully nonlinear Boussinesq model for surface waves: I. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71-92 (1995)
60. Whitham, G.B.: Linear and Nonlinear Waves. Pure and Applied Mathematics: a Wiley Series of Texts, Monographs and Tracts. Wiley, Hoboken (2011)
61. Zhang, Y., Kennedy, A., Panda, N., Dawson, C., Westerink, J.: Boussinesq-Green-Naghdi rotational water wave theory. Coast. Eng. 73,13-27 (2013)

[1]	Laura Río-Martín, Michael Dumbser. High-Order ADER Discontinuous Galerkin Schemes for a Symmetric Hyperbolic Model of Compressible Barotropic Two-Fluid Flows [J]. Communications on Applied Mathematics and Computation, 2024, 6(4): 2119-2154.
[2]	Michael Herty, Niklas Kolbe, Siegfried Müller. A Central Scheme for Two Coupled Hyperbolic Systems [J]. Communications on Applied Mathematics and Computation, 2024, 6(4): 2093-2118.
[3]	Jean-Luc Guermond, Bojan Popov, Laura Saavedra. Second-Order Invariant Domain Preserving ALE Approximation of Euler Equations [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 923-945.
[4]	Matteo Bergami, Walter Boscheri, Giacomo Dimarco. A High-Order Conservative Semi-Lagrangian Solver for 3D Free Surface Flows with Sediment Transport on Voronoi Meshes [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 596-637.
[5]	M. Semplice, E. Travaglia, G. Puppo. One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 143-169.
[6]	Ohannes A. Karakashian, Michael M. Wise. A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear, Dispersive Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 823-854.