Abstract
The paper presents an integral technique simulating all phases of a landslide-driven tsunami. The technique is based on the numerical solution of the system of Navier–Stokes equations for multiphase flows. The numerical algorithm uses a fully implicit approximation method, in which the equations of continuity and momentum conservation are coupled through implicit summands of pressure gradient and mass flow. The method we propose removes severe restrictions on the time step and allows simulation of tsunami propagation to arbitrarily large distances. The landslide origin is simulated as an individual phase being a Newtonian fluid with its own density and viscosity and separated from the water and air phases by an interface. The basic formulas of equation discretization and expressions for coefficients are presented, and the main steps of the computation procedure are described in the paper. To enable simulations of tsunami propagation across wide water areas, we propose a parallel algorithm of the technique implementation, which employs an algebraic multigrid method. The implementation of the multigrid method is based on the global level and cascade collection algorithms that impose no limitations on the paralleling scale and make this technique applicable to petascale systems. We demonstrate the possibility of simulating all phases of a landslide-driven tsunami, including its generation, propagation and uprush. The technique has been verified against the problems supported by experimental data. The paper describes the mechanism of incorporating bathymetric data to simulate tsunamis in real water areas of the world ocean. Results of comparison with the nonlinear dispersion theory, which has demonstrated good agreement, are presented for the case of a historical tsunami of volcanic origin on the Montserrat Island in the Caribbean Sea.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
http://tsun.sscc.ru/hiwg. Accessed Apr. 18, 2016.
Pelinovsky, E., Gidrodinamika voln tsunami (Hydrodynamics of Tsunami Waves), Nizhny Novgorod: Inst. Prikl. Fiz. RAN, 1996.
Langford, P.S., Modeling of tsunami generated by submarine landslides, PhD Dissertation, New Zealand: Univ. of Canterbury, 2007.
Rabinovich, A.B., Thomson, R.E., Bornhold, B.D., Fine, I.V., and Kulikov, E.A., Numerical modelling of tsunamis generated by hypothetical landslides in the strait of Georgia, British Columbia, Pure Appl. Geophys., 2003, vol. 160, no. 7, pp. 1273–1313. dx.doi.org/ [REMOVED HYPERLINK FIELD] doi 10.1016/0142-727X(89)90003-9
Fine, I.V., Rabinovich, A.B., Bornhold, B.D., Thomson, R.E., and Kulikov, E.A., The Grand Banks landslidegenerated tsunami of November 18, 1929: preliminary analysis and numerical modeling, Mar. Geol., 2005, vol. 215, nos. 1–2, pp. 45–57. http://dx.doi.org/ doi 10.1016/j.margeo.2004.11.007
Papadopoulos, G.A. and Kortekaas, S., Characteristics of land-slide generated tsunamis from observational data. Submarine mass movements and their consequences, Adv. Nat. Technol. Hazards Res., 2003, vol. 19, pp. 267–374.
Fedotova, Z.I., Chubarov, L.B., and Shokin, Y.I., Simulation of surface waves caused by landslides, Vychisl. Tehnol., 2004, vol. 9, no. 6, pp. 89–96.
Watts, P. and Grilli, S.T., in Proceedings of the 13th International Offshore and Polar Engineering Conference, Honolulu, Hawaii, USA, May 25–30, 2003, vol. 3, pp. 364–371.
Heinrich, P., Schindele, F., Guibourg, S., and Ihmlé, P., Modeling of the February 1996 Peruvian tsunami, Geophys. Res. Lett., 1998, vol. 25, no. 14, pp. 2687–2690. http://dx.doi.org/ doi 10.1029/98GL01780
Imamura, F. and Imteaz, M.M.A., Long waves in two-layers: governing equations and numerical model, Sci. Tsunami Hazards, 1995, vol. 13, no. 1, pp. 3–24. http://library.lanl.gov/tsunami/00394724.pdf.
Dutykh, D. and Dias, F., Energy of tsunami waves generated by bottom motion, Proc. R. Soc. A, 2009, vol. 465, pp. 725–744.
Gusev, O.I., Shokina, N.Yu., Kutergin, V.A., and Khakimzyanov, G.S., Numerical modelling of surface waves generated by an underwater landslide in a reservoir, Vychisl. Tehnol., 2013, vol. 8, no. 5, pp. 74–90.
Sælevik, G., Jensen, A., and Pedersen, G., Experimental investigation of impact generated tsunami; related to a potential rock slide, Western Norway, Coast. Eng., 2009, vol. 56, no. 9, pp. 897–906. http://dx.doi.org/ doi 10.1016/j.coastaleng.2009.04.007
Fritz, H.M., Mohammed, F., and Yoo, J., Lituya Bay landslide impact generated mega-tsunami 50th Anniversary, Pure Appl. Geophys., 2009, vol. 166, no. 1, pp. 153–175. http://dx.doi.org/ doi 10.1007/s00024-008-0435-4
Horrillo, J., Wood, A., Kim, G.B., and Parambath, A., A simplified 3-D Navier-Stokes numerical model for landslide-tsunami: application to the Gulf of Mexico, J. Geophys. Res.-Oceans, 2013, vol. 118, no. 2, pp. 6934–6950. http://dx.doi.org/ doi 10.1002/2012jc008689
Mohammed, F. and Frits, H.M., Experiments on tsunamis generated by 3D granular landslides. Submarine mass movements and their consequences, Adv. Nat. Technol. Hazards Res., 2010, vol. 28, pp. 705–718.
Mohammed, F., Physical modeling of tsunamis generated by three-dimensional deformable granular landslides, PhD Dissertation, Georgia: Georgia Inst. of Technol., 2010.
Beizel, S.A., Chubarov, L.B., and Khakimzyanov, G.S., Simulation of surface waves generated by an underwater landslide moving over an uneven slope, Russ. J. Numer. Anal. Math., 2011, vol. 26, no. 1, pp. 17–38. http://dx.doi.org/ doi 10.1515/rjnamm.2011
Harbitz, C.B., Lovholt, F., Pedersen, G., Glimsdal, S., and Masson, D.G., Mechanisms of tsunami generation by submarine landslides: a short review, Norweg. J. Geol., 2006, vol. 86, pp. 255–264.
Pelinovsky, E.N., Analytical models of tsunami generation by submarine landslides. Submarine Landslides and Tsunamis, NATO Sci. Ser., 2003, vol. 21, pp. 111–128. http://dx.doi.org/ doi 10.1007/978-94-010-0205-9_12
Didenkulova, I., Nikolkina, I., Pelinovsky, E., and Zahibo, N., Tsunami waves generated by submarine landslides of variable volume: analytical solutions for a basin of variable depth, Nat. Hazards Earth Syst. Sci., 2010, vol. 10, pp. 2407–2419. http://dx.doi.org/ doi 10.5194/nhess-10-2407-2010
Macías, J., Vázquez, J.T., Fernández-Salas, L.M., González-Vida, J.M., Bárcenas, P., Castro, M.J., Díaz-del-Río, V., and Alonso, B., The Al-Borani submarine landslide and associated tsunami, a modelling approach, Mar. Geol., 2015, vol. 361, pp. 79–95. http://dx.doi.org/ doi 10.1016/j.margeo.2014.12.006
Okal, E.A. and Synolakis, C.E., A theoretical comparison of tsunamis from dislocations and landslides, Pure Appl. Geophys., 2003, vol. 160, no. 10, pp. 2177–2188. http://dx.doi.org/ doi 10.1007/s00024-003-2425-x
Lynett, P., Hydrodynamic modeling of tsunamis generated by submarine landslides: generation, propagation, and shoreline impact. Submarine mass movements and their consequences, Adv. Nat. Technol. Hazards Res., 2010, vol. 28, pp. 685–694.
Watts, P., Grilli, S.T., Kirby, J.T., Fryer, G.J., and Tappin, D.R., Landslide tsunami case studies using a Boussinesq model and a fully nonlinear tsunami generation model, Nat. Hazards Earth Syst. Sci., 2003, vol. 3, pp. 391–402. http://dx.doi.org/ doi 10.5194/nhess-3-391-2003
Cecioni, C. and Bellotti, G., Modeling tsunamis generated by submerged landslides using depth integrated equations, Appl. Ocean Res., 2010, vol. 32, no. 3, pp. 343–350. http://dx.doi.org/ doi 10.1016/j.apor.2009.12.002
Grilli, S.T., Vogelmann, S., and Watts, P., Development of 3D numerical wave tank for modeling tsunami generation by underwater landslides, Eng. Anal. Boun. Elem., 2002, vol. 26, no. 4, pp. 301–313. http://dx.doi.org/ doi 10.1016/S0955-7997%2801%2900113-8
Ma, G., Kirby, J.T., and Shi, F., Numerical simulation of tsunami waves generated by deformable submarine landslides, Ocean Model., 2013, vol. 69, pp. 146–165. http://dx.doi.org/ doi 10.1016/j.ocemod.2013.07.001
Liu P. L.-F., Wu T.-R., Raichlen, F., Synolakis, C.E., and Borrero, J.C., Runup and rundown generated by three-dimensional sliding masses, J. Fluid Mech., 2005, vol. 536, pp. 107–144. http://dx.doi.org/ doi 10.1017/S0022112005004799
Kozelkov, A.S., Kurulin, V.V., Tyatyushkina, E.S., and Puchkova, O.L., Application of the detached eddy simulation model for viscous incompressible turbulent flow simulations on unstructured grids, Mat. Model., 2014, vol. 26, no. 8, pp. 81–96.
Kozelkov, A.S. and Kurulin, V.V., Eddy-resolving numerical scheme for simulation of turbulent incompressible flows, Comput. Math. Math. Phys., 2015, vol. 55, no. 7, pp. 1232–1241. http://dx.doi.org/ doi 10.1134/S096554251507009X
Lynett, P. and Liu, P. L.-F., A numerical study of the run-up generated by three-dimensional landslides, J. Geophys. Res., 2005, vol. 110, no. C3. http://dx.doi.org/ doi 10.1029/2004JC002443
Volkov, K.N., Deryugin, Yu.N., Emelyanov, V.N., Karpenko, A.G., Kozelkov, A.S., and Teterina, I.V., Metody uskorenuya gazodinamicheskikh raschetov na nestructurirovanykh setkakh (Methods of Acceleration of Gas-Dynamical Calculations on Unstructured Grids), Moscow: Fizmatlit, 2014.
Kozelkov, A.S., Deryugin, Yu.N., Lashkin, S.V., Silaev, D.P., Simonov, P.G., and Tyatyushkina, E.S., Implementation in LOGOS software of a computational scheme for a viscous incompressible fluid using the multigrid method based on an algorithm SIMPLE, Vopr. At. Nauki Tekh., Ser.: Mat. Mod. Fiz. Protsessov, 2013, no. 4, pp. 44–56.
Kozelkov, A.S., Shagaliev, R.M., Dmitriev, S.M., Kurkin, A.A., Volkov, K.N., Deryugin, Yu.N., Emelyanov, V.N., Pelinovsky, E., and Legchanov, M.A., Matematicheskie modeli i algoritmy dlya chislennogo modelirovaniya zadach gidrodinamiki i aerodinamiki (Mathematical Models and Algorithms for Numerical Modeling of Hydrodynamics and Aerodynamics, the School-Book), Nizh. Novgorod: Nizhegor. Gos. Tekh. Univ., 2014.
Hirt, C.W., and Nichols, B.D., Volume of Fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 1981, vol. 39, no. 1, pp. 201–225. http://dx.doi.org/ doi 10.1016/0021-9991%2881%2990145-5
Ubbink, O., Numerical prediction of two fluid systems with sharp interfaces, PhD Dissertation, London: Imperial College of Science, Technology and Medicine, 1997.
Zhainakov, A.Zh. and Kurbanaliev, A.Y., Verification of the open package OpenFOAM on dam break problems, Thermophys. Aeromech., 2013, vol. 20, no. 4, pp. 451–461.
Volkov, K.N. and Emelyanov, V.N., Techeniya gaza s chastitsami (Flows of Gas with the Particles), Moscow: Fizmatlit, 2008.
Ferziger, J.H. and Peric, M., Computational Methods for Fluid Dynamics, Berlin: Springer, 2001.
Chen, Z.J. and Przekwas, A.J., A coupled pressure-based computational method for incompressible/compressible flows, J. Comput. Phys., 2010, vol. 229, no. 24, pp. 9150–9165. http://dx.doi.org/ doi 10.1016/j.jcp.2010.08.029
Rhie, C.M. and Chow, W.L., A numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J., 1983, vol. 21, no. 11, pp. 1525–1532. http://dx.doi.org/ doi 10.2514/3.8284
Jasak, H., Error analysis and estimation for the finite volume method with applications to fluid flows, PhD (Mech. Eng.) Dissertation, London: Imperial College of Science, 1996.
Volkov, K.N., Deryugin, Yu.N., Emelyanov, V.N., Kozelkov, A.S., and Teterina, I.V., Raznostnye skhemy v zadachakh gazovoi dinamiki na nestrukturirovannykh setkhakh (Difference Schemes in Gas Dynamic Problems on Unstructured Grids), Moscow: Fizmatlit, 2014.
Rouch, P., Computational Fluid Dynamics, Albuquerque, NM: Hermosa, 1976.
Waclawczyk, T., Remarks on prediction of wave drag using VOF method with interface capturing approach, Arch. Civil Mech. Eng., 2008, vol. 8, no. 1, pp. 5–14. http://www.acme.pwr.wroc.pl/repository/177/online.pdf.
Samarsky, A.A., Teoriya raznostnykh skhem (The Theory of Difference Schemes), Moscow: Nauka, 1989.
Kozelkov, A.S., Deryugin, Yu.N., Tsibereva, Yu.A., Kornev, A.V., Denisova, O.V., Strelets, D.Yu., Kurkin, A.A., Kurulin, V.V., Sharipova, I.L., Rubtsova, D.P., Legchanov, M.A., Tyatyushkina, E.S., Lashkin, S.V., Yalozo, A.V., Yatsevich, S.V., et al., Minimal basis tasks for validation of methods of numerical simulation of turbulent flows of incompressible viscous fluids, Tr. Nizhegor. Tekh. Univ., 2014, no. 4 (106), pp. 21–69.
Volkov, K.N., Deryugin, Yu.N., Emelyanov, V.N., Kozelkov, A.S., and Teterina, I.V., An algebraic multigrid method in problems of computational physics, Vychisl. Metody Programm., 2014, no. 15, pp. 183–200.
Kozelkov, A.S., Kurulin, V.V., Puchkova, O.L., and Lashkin, S.V., Simulation of turbulent flows using an algebraic Reynolds stress model with universal wall functions, Vychisl. Mekh. Splosh. Sred, 2014, vol. 7, no. 1, pp. 40–51. http://dx.doi.org/ doi 10.7242/1999-6691/2014.7.1.5
Kozelkov, A.S., Kurkin, A.A., Pelinovskii, E.N., and Kurulin, V.V., Modeling the cosmogenic tsunami within the framework of the Navier–Stokes equations with sources of different types, Fluid Dyn., 2015, vol. 50, no. 2, pp. 306–313. http://dx.doi.org/ doi 10.1134/S0015462815020143
Kozelkov, A.S., Effects, accompanying entering of asteroid in the water medium, Tr. Nizhegor. Tekh. Univ., 2014, no. 3 (105), pp. 48–77.
Kozelkov, A.S., Kurkin A.A., and Pelinovsky, E.N., Tsunami of cosmogenic origin, Tr. Nizhegor. Tekh. Univ., 2014, no. 2 (104), pp. 26–35.
Eletskii, S.V., Maiorov, Yu.B., Maksimov, V.V., Nudner, I.S., Fedotova, Z.I., Khazhoyan, M.G., Khakimzyanov, G.S., and Chubarov, L.B., Simulation of surface wave generation by moving a part of the bottom along the coastal slope, Vychisl. Tekhnol. with Vestn. Kazan. Univ., 2004, vol. 9, no. 2, pp. 194–206.
Fedorenko, R.P., The relaxation method for solving difference elliptic equations, Zh. Vychisl. Mat. Mat. Fiz., 1961, vol. 1, no. 5, pp. 922–927.
Kozelkov, A.S., Kurulin, V.V., Lashkin S.V., Shagaliev, R.M., and Yalozo, A.V., Investigation of supercomputer capabilities for the scalable numerical simulation of computational fluid dynamics problems in industrial applications, Comput. Math. Math. Phys., 2016, vol. 56, no. 8, pp. 1506–1516. http://dx.doi.org/ doi 10.1134/S0965542516080091
Pelinovsky, E.N., Zahibo, N., Dunkly, P., Talipova, T., Koselkov, A., Kurkin A.A., Nikolkina, I.F., and Samarina, N.M., Tsunami, Tsunami induced by volcano eruptions on Montserrat, Caribbean Sea, Izv. Russ. Acad. Eng. Sci., Ser. Prikl. Mat. Meth., 2004, no. 6, pp. 31–59.
Pelinovsky, E., Koselkov, A., Zahibo, N., Dunkly, P., Edmonds, M., Herd, R., Talipova, T., and Nikolkina, I., Tsunami generated by the volcano eruption on July 12–13, 2003 at Montserrat, Lesser Antilles, Sci. Tsunami Hazards, 2004, vol. 22, no. 1, pp. 44–57.
Kozekov, A.S., Evaluation of tsunami hazard of Caribbean coast, PhD Dissertation, Nizh. Novgorod: Alekseev State Tech. Univ., 2006.
Goto, C., Ogawa, Y., Shuto, N., and Imamura, N., Numerical method of tsunami simulation with the leap-frog scheme (IUGG/IOC Time Project), in IOC Manual, New York: UNESCO, 1997, no.35.
Watts, Ph. and Waythomas, C.F., Theoretical analysis of tsunami generation by pyroclastic flows, J. Geophys. Res., 2003, vol. 108, no. B12. http://dx.doi.org/ doi 10.1029/2002jb002265
Kirby, J., Wei, G., Chen, Q., Kennedy, A., and Dalrymple, R., Fully nonlinear Boussinesq wave model documentation and users manual, Research Report, Newark DE: Center Appl. Coast. Res. Departm. of Civil Eng. Univ. of Delaware, 1998, no. CACR-98-06.
https://www.ngdc.noaa.gov. Accessed April 20, 2016.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.S. Kozelkov, 2016, published in Vychislitel’naya Mekhanika Sploshnykh Sred, 2016, Vol. 9, No. 2, pp. 218–236.
Rights and permissions
About this article
Cite this article
Kozelkov, A.S. The Numerical Technique for the Landslide Tsunami Simulations Based on Navier–Stokes Equations. J Appl Mech Tech Phy 58, 1192–1210 (2017). https://doi.org/10.1134/S0021894417070057
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0021894417070057