On CABARET scheme for incompressible fluid flow problems with a free surface

V. A. Gushchin ; V. G. Kondakov

Matematičeskoe modelirovanie, Tome 30 (2018) no. 11, pp. 75-90.

Résumé

This paper proposes a new approach for the solution of problems of interaction of vortex structures with a free surface. The second-order accuracy finite-difference scheme based on the famous CABARET scheme is suggested for incompressible viscous fluid with a free surface. In the case of incompressible fluid the CABARET technique solves to further the task of solenoidal velocity field. Decision task involves decisions regarding the variable pressure SLOUGH and subsequent accounting pressure gradient in the calculation of the equations of motion. The decision of SLOUGH is a separate joint problem, which is not included in the description of the method of the CABARET in this paper the authors cite only the statement of the problem without specifying a particular method of solving systems of linear equations.

Détail

Mots-clés : vortex pair motion
Keywords: free surface, direct numerical simulation (DNS).

Bibliographie

[1] O.M. Belotserkovskii, V.A. Gushchin, V.N. Konshin, “The splitting method for investigating flows of a stratified liquid with a free surface”, USSR Comp. Mathematics and Mathematical Physics, 27:2 (1987), 181–191 | DOI | MR | Zbl

[2] S.W. Hirt, J.L. Cook, T.D. Butler, “A Lagrangian method for calculating the dynamics of an incompressible fluid with free surface”, J. Comp. Phys., 5:1 (1970), 103–124 | DOI | Zbl

[3] T. Buttler, “Development of LINC method”, Proc. of the second intern. conf. on numerical methods in fluid dynamics (Sept. 15–19, 1979, Univ. of California, Berkeley), Lecture Notes in Physics, eds. J. Ehlers, K. Hepp, H.A. Weidenmuller, M. Holt, H. Beiglbock, Springer-Verlag, Berlin–Heidelberg–New York, 1971

[4] S. Hirt, “Abitrary Lagrange-Euler numerical method”, Proc. of the second intern. conf. on numerical methods in fluid dynamics (Sept. 15–19, 1979, Univ. of California, Berkeley), Lecture Notes in Physics, eds. J. Ehlers, K. Hepp, H.A. Weidenmuller, M. Holt, H. Beiglbock, Springer-Verlag, Berlin–Heidelberg–New York, 1971 | MR | Zbl

[5] R.K.C. Chan, “A generalized arbitrary Lagrangian-Eulerian method for incompressible flows with sharp interfaces”, Journal of Computational Physics, 17:3 (1975), 311–331 | DOI | MR | Zbl

[6] C.W. Hirt, A.A. Amsden, J.L. Cook, “An arbitrary Lagrangian-Eulerian computing method for all flow speeds”, J. Comp. Phys., 14:3 (1974), 227–253 | DOI | MR | Zbl

[7] V.A. Gushchin, “Family of quasi-monotonic finite-difference schemes of the second-order of approximation”, Math. Models Comp. Simul., 2016, no. 8, 487–496 | DOI | MR

[8] V.A. Gushchin, V.N. Konshin, “Computational aspects of the splitting method for incompressible flow with a free surface”, J. of Computers and Fluids, 21:3 (1992), 345–353 | DOI | Zbl

[9] V.A. Gushchin, A.V. Kostomarov, P.V. Matyushin, E.R. Pavlyukova, “Direct Numerical Simulation of the Transitional Separated Fluid Flows Around a Sphere and a Circular Cylinder”, J. of Wind Engineering Industrial Aerodynamics, 90:4–5 (2002), 341–358 | DOI

[10] V.A. Gushchin, P.V. Matyushin, “Vortex formation mechanisms in the wake behind a sphere for $200 Re 380$”, Fluid Dynamics, 41:5 (2006), 795–809 | DOI | Zbl

[11] V.A. Gushchin, P.V. Matyushin, “Numerical Simulation and Visualization of Vortical Structure Transformation in the Flow past a Sphere at an Increasing Degree of Stratification”, Comp. Mathematics and Mathematical Physics, 51:2 (2011), 251–263 | DOI | MR | Zbl

[12] V.A. Gushchin, P.V. Matyushin, “Simulation and Study of Stratified Flows around Finite Bodies”, Comp. Math. and Math. Physics, 56:6 (2016), 1034–1047 | DOI | MR | Zbl

[13] V.A. Gushchin, “Large Scale Computations in Fluid Dynamics”, Large-Scale Scientific Computing. LSSC 2017, Lecture Notes in Computer Science, 10665, eds. I. Lirkov, S. Margenov, Springer, Cham, 2018, 491–498 | DOI | MR

[14] V.M. Goloviznin, A.A. Samarskii, “Raznostnaia approksimatsiia konvektivnogo perenosa s prostranstvennym rasshchepleniem vremennoi proizvodnoi”, Matematicheskoe modelirovanie, 10:1 (1998), 86–100

[15] V.M. Goloviznin, A.A. Samarskii, “Nekotorye svoistva raznostnoi skhemy Kabare”, Matematicheskoe modelirovanie, 10:1 (1998), 101–116 | Zbl

[16] V.M. Goloviznin, S.A. Karabasov, I.M. Kobrinskii, “Balansno-kharakteristicheskie skhemy s razdelennymi konservativnymi i potokovymi peremennymi”, Matematicheskoe modelirovanie, 15:9 (2003), 29–48 | Zbl

[17] V.M. Goloviznin, “Balansno-kharakteristicheskii metod chislennogo resheniia odnomermykh uravnenii gazovoi dinamiki v eilerovykh peremennykh”, Matematicheskoe modelirovanie, 18:11 (2006), 14–30 | MR | Zbl

[18] T. Sarpkaya, P. Suthon, “The interaction of a vortex couple with a free surface”, J. Experiments in fluids, 1991, no. 11, 205–217 | DOI

[19] W.T. Tsai, D.K.P. Yue, “Effects of soluble and insoluble surfactant on laminar interactions of vortical flows with a free surface”, Journal of Fluid Mechanics, 289 (1995), 315–349 | DOI | Zbl

[20] G.K. Batchelor, An introduction to fluid dynamics, University Press, Cambridge, 1970 | MR

[21] M.A. Lavrentev, B.V. Shabat, Gidrodinamicheskie problemy i ikh matematicheskie modeli, Nauka, M., 408

[22] J.J. Stoker, Water waves, Interscience Publishers INC., New York; Interscience Publishers, LTD., London, 1957 | MR | Zbl

[23] L. Orlanski, “A simple boundary condition for unbounded hyperbolic flows”, J. Comp. Phys., 21:3 (1976), 251–269 | DOI | Zbl

[24] V.M. Goloviznin, M.A. Zaitsev, S.A. Karabasov, I.A. Korotkin, Novye algoritmy vychislitelnoi gidrodinamiki dlya mnogoprotsessornyh vychislitelnyh kompleksov, MGU, M., 2013, 472 pp.