Abstract
The runup of nonlinear surface gravity waves is numerically simulated in two and three dimensions on the basis of the Navier-Stokes equations. The three-dimensional problem is formulated, and the boundary and initial conditions are described. The splitting method over physical processes is used to construct a discrete model taking into account the cell occupation coefficient. The runup of nonlinear surface gravity waves is simulated in two dimensions for slopes of various geometries, and the numerical results are analyzed. The structural features of the simulated three-dimensional basin are described. Three-dimensional models for the staged runup of nonlinear surface gravity waves breaking on coastal slopes in shallow water areas are considered.
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References
Y. Watanabe and H. Saeki, “Three dimensional large eddy simulation of breaking waves,” Coast. Eng. J. 41, 281–301 (1999); doi: 10.1142/S0578563499000176.
P. Lubin, S. Vincent, S. Abadie, and J.-P. Caltagirone, “Three-dimensional large eddy simulation of air entrainment under plunging breaking waves,” Coastal Eng. J. 53, 631–655 (2006).
Z. I. Fedotova, “Justification of a numerical method for the simulation of long-wave runup,” Vychisl. Tekhnol. 7(5), 58–76 (2002).
N. M. Borisova, “On modeling of hydraulic bore propagation at incline bank,” Sib. Zh. Vychisl. Mat. 10(1), 1–18 (2007).
A. I. Delis, M. Kazolea, and N. A. Kampanis, “A robust high resolution finite volume scheme for the simulation of long waves over complex domains,” Int. J. Numer. Methods Fluids 56, 419–452 (2008).
F. C. K. Ting and J. T. Kirby, “Dynamics of surf-zone turbulence in a spilling breaker,” Coastal Eng 27, 131–160 (1996); doi: 10.1016/0378-3839(95)00037-2.
O. Kimmoun and H. Branger, “A PIV investigation on laboratory surf-zone breaking waves over a sloping beach,” J. Fluid Mech. 588, 353–397 (2007).
I. B. Abbasov, “Numerical simulation of nonlinear surface gravity waves transformation under shallow-water conditions,” Appl. Math. 3(2), 135–141 (2012); doi: 10.4236/am.2012.32021.
I. B. Abbasov, A. I. Sukhinov, and A. E. Chistyakov, “Numerical simulation of the runup of nonlinear surface gravity waves based on the Navier-Stokes equation,” Proceedings of the 14th All-Russia Conference-School on Modern Problem in Mathematical Simulation, Abrau-Dyurso, September 12–17, 2011 (Yuzhn. Fed. Univ., Rostov-on-Don, 2011), pp. 10–15.
I. B. Abbasov, I. S. Semenov, and V. V. Tsarevskii, “Numerical simulation of the runup of nonlinear surface gravity waves on a gently sloping beach,” Izv. Yuzhn. Fed. Univ. Tekh. Nauki, No. 6, 19–22 (2012).
P. J. Roache, Computational Fluid Dynamics (Hermosa, Albuquerque, 1976; Mir, Moscow, 1980).
C. A. J. Fletcher, Computational Techniques for Fluid Dynamics (Springer-Verlag, Berlin, 1990; Mir, Moscow, 1991), Vol. 2.
F. H. Harlow and J. E. Welch, “Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface,” Phys. Fluids 8, 2182–2189 (1965).
O. M. Belotserkovskii, V. A. Gushchin, and V. N. Kon’shin, “The splitting method for investigating flows of a stratified liquid with a free surface,” USSR Comput. Math. Math. Phys. 27(2), 181–191 (1987).
A. I. Sukhinov, E. F. Timofeeva, and A. E. Chistyakov, “Construction and investigation of a discrete mathematical model for computing coastal wave processes,” Izv. Yuzhn. Fed. Univ. Tekh. Nauki 121(8), 22–32 (2011).
A. A. Samarskii, Introduction to Numerical Methods (Nauka, Moscow, 1987) [in Russian].
N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (BINOM, Moscow, 2006) [in Russian].
I. B. Abbasov, I. S. Semenov, and V. V. Tsarevskii, “Code for three-dimensional simulation of surface wave runup in a shallow bay,” Russian Federation Inventor’s Certificate No. 2012617087, 2012.
I. B. Abbasov, I. S. Semenov, and V. V. Tsarevskii, “Code for two-dimensional simulation of surface wave runup in a shallow bay,” Russian Federation Inventor’s Certificate No. 2012616206, 2012.
USSR Seas Project: Hydrometeorology and Hydrochemistry of USSR Seas, Vol. 5: Sea of Azov, Ed. by N. P. Goptarev et al. (Gidrometeoizdat, St. Petersburg, 1991), pp. 75–88 [in Russian].
V. A. Mamykina and Yu. P. Khrustalev, Coastline Zone of the Sea of Azov (Rostov. Gos. Univ., Rostov-on-Don, 1980) [in Russian].
V. K. Debol’skii, R. Zaidler, and S. Massel’, Dynamics of Channel Flows and Nearshore Lithodynamics (Nauka, Moscow, 1994) [in Russian].
Q. Zhao, S. Armfield, and K. Tanimoto, “Numerical simulation of breaking waves by a multiscale turbulence model,” Coastal Eng. J. 51, 53–80 (2004).
I. B. Abbasov, “Modeling of nonlinear surface gravity waves in shallow water with regard to dispersion,” Dokl. Earth Sci. 429, 1605–1607 (2009).
Building Code No. 33-01-2003: Hydraulic Facilities (Gosstroi Rossii, Moscow, 2004) [in Russian].
T. G. Smirnova, Yu. P. Pravdivets, and G. N. Smirnov, Bank Protection Facilities (ASV, Moscow, 2002) [in Russian].
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Original Russian Text © I.B. Abbasov, 2014, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2014, Vol. 54, No. 5, pp. 871–886.
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Abbasov, I.B. Three-dimensional simulation of the runup of nonlinear surface gravity waves. Comput. Math. and Math. Phys. 54, 900–914 (2014). https://doi.org/10.1134/S0965542514050030
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DOI: https://doi.org/10.1134/S0965542514050030