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The numerical experiment in fluid mechanics

Published online by Cambridge University Press:  21 April 2006

Hassan Aref
Hassan Aref
Affiliation:
University of California, San Diego, La Jolla, CA 92093, USA
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Abstract

Several aspects of the use of digital computers to generate solutions of equations of interest to fluid mechanics are discussed. The inter-disciplinary nature of the field of computational fluid dynamics (CFD) is emphasized: the dependence on strides in computer technology, the impact of advances in algorithm development, the continuous interaction with laboratory experiment and analytical theory. The particular role of that mode of computer usage usually referred to as the numerical experiment is highlighted. ‘Experiments’ of this type have played a central role in establishing concepts such as the soliton and the strange attractor as paradigms within fluid mechanics. The ambitious goal of providing digital counterparts to laboratory equipment such as the wind tunnel is considered. The possibility of abandoning the Eulerian representation of flow fields in favour of following swarms of Lagrangian particles on a computer is stressed. Issues arising from and results of using this methodology are reviewed. Computer simulations are contrasted with computer generated animation. The paper concludes with speculations on future developments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 173 , December 1986 , pp. 15 - 41
Copyright
© 1986 Cambridge University Press

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References

Amsden, A. A. & Harlow, F. H. 1964 Slip instability. Phys. Fluids 7, 327–334.Google Scholar
Aref, H. 1982 Point vortex motions with a center of symmetry. Phys. Fluids 25, 2183–2187.Google Scholar
Aref, H. 1983 Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345–389.Google Scholar
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.Google Scholar
Aref, H. 1985 Chaos in the dynamics of a few vortices—fundamentals and applications. In Theoretical and Applied Mechanics, Proc. 16th Intl Congr. Theor. Appl. Mech. (ed. F. I. Niordson & N. Olhoff), pp. 43–68. Elsevier.
Aref, H. 1986 Finger, bubble, tendril, spike. An essay on the morphology and dynamics of interfaces in fluids. Polish Acad. Sci. Fluid Dyn. Trans. 13, (in press).Google Scholar
Aref, H. & Balachandar, S. 1986 Chaotic advection in a Stokes flow. Phys. Fluids (in press).Google Scholar
Aref, H. & Siggia, E. D. 1981 Evolution and breakdown of a vortex street in two dimensions. J. Fluid Mech. 109, 435–463.Google Scholar
Baker, G. R. 1982 Generalized vortex methods for free-surface flows. In Waves on Fluid Interfaces (ed. R. E. Meyer). Publ. MRC Univ. Wisconsin-Madison, vol. 50. Academic.
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1980 Vortex simulation of the Rayleigh—Taylor instability. Phys. Fluids 23, 1485–1490.Google Scholar
Ballal, B. Y. & Rivlin, R. S. 1976 Flow of a Newtonian fluid between eccentric rotating cylinders: Inertial effects. Arch. Rat. Mech. Anal. 62, 237–294.Google Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12 (Suppl. 2), 233–239.Google Scholar
Beckmann, P. 1971 A History of . The Golem Press.
Bensimon, D. 1986 Stability of viscous fingering. Phys. Rev. A 33, 1302–1308.Google Scholar
Birkhoff, G. 1954 Taylor instability and laminar mixing. Los Alamos Sci. Lab. Rep. No. LA-1862; appendices in Rep. LA-1927.Google Scholar
Birkhoff, G. 1962 Helmholtz and Taylor instability. In Hydrodynamic Instability, Proc. Symp. Appl. Maths, vol. 13, pp. 55–76. Providence: Am. Math. Soc.
Calogero, F. 1978 Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations and related ‘solvable’ many-body problems. Nuovo Cimento B 43, 177–241.Google Scholar
Campbell, L. J. & Kadtke, J. B. 1986 Stationary configurations of point vortices and other logarithmic objects in two dimensions. Los Alamos Nat. Lab. preprint.Google Scholar
Campbell, L. J. & Ziff, R. M. 1978 A catalog of two-dimensional vortex patterns. Los Alamos Sci. Lab. Rep. No. LA-7384-MS, 40 pp.Google Scholar
Charney, J. G. 1963 Numerical experiments in atmospheric hydrodynamics. In Experimental Arithmetic, High Speed Computing and Mathematics, Proc. Symp. Appl. Maths, vol. 15, pp. 289–310. Providence: Am. Math. Soc.
Christiansen, J. P. 1973 Numerical simulation of hydrodynamics by the method of point vortices. J. Comput. Phys. 13, 363–379.Google Scholar
Chuudnovsky, D. V. & Chuudnovsky, G. V. 1977 Pole expansions of nonlinear partial differential equations. Nuovo Cimento B 40, 339–353.Google Scholar
Cochran, W. G. 1934 The flow due to a rotating disk. Proc. Camb. Phil. Soc. 30, 365–375.Google Scholar
Couder, Y. & Basdevant, C. 1986 Experimental and numerical study of vortex couples in two-dimensional flows. J. Fluid Mech. 173, 225–251.Google Scholar
Deem, G. S. & Zabusky, N. J. 1978 Vortex waves: Stationary ‘V states’, interactions, recurrence, and breaking. Phys. Rev. Lett. 40, 859–862.Google Scholar
Degregoria, A. J. & Schwartz, L. W. 1986 A boundary integral method for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 164, 383–400.Google Scholar
De Josselin De Jong, G. 1959 Vortex theory for multiple phase flow through porous media. Water Res. Ctr. Contr. No. 23, UC Berkeley, 80 pp.Google Scholar
De Josselin De Jong, G. 1960 Singularity distributions for the analysis of multiple-fluid flow through porous media. J. Geophys. Res. 65, 3739–3758.Google Scholar
Emmons, H. W. 1970 Critique of numerical modelling of fluid-mechanics phenomena. Ann. Rev. Fluid Mech. 2, 15–37.Google Scholar
Feigenbaum, M. J. 1980 The metric universal properties of period doubling bifurcations and the spectrum for a route to turbulence. In. Nonlinear Dynamics (ed. R. H. G. Helleman), Ann. N. Y. Acad. Sci. 357, 330–336.
Ford, J. 1978 A picture book of stochasticity. In Topics in Nonlinear Dynamics: A Tribute to Sir Edward Bullard (ed. S. Jorna), AIP Conf. Proc. 46, 121–146.
Fox, G. C. & Otto, S. W. 1984 Algorithms for concurrent processors. Phys. Today 37, 53–59.Google Scholar
Frisch, U., Hasslacher, B. & Pomeau, Y. 1986 A lattice gas automaton for the Navier—Stokes equation. Phys. Rev. Lett. 56, 1505–1508.Google Scholar
Ghoniem, A. F. & Sherman, F. S. 1985 Grid-free simulation of diffusion using random walk methods. J. Comput. Phys. 61, 1–37.Google Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical analysis of spectral methods: theory and application. CBMS-NSF Reg. Conf. Ser. Appl. Maths vol. 26, 170 pp. SIAM, Philadelphia.Google Scholar
Hartree, D. R. 1937 On an equation occurring in Falkner and Skan's approximate treatment of the equations of the boundary layer. Proc. Camb. Phil. Soc. 33, 223–239.Google Scholar
Hele-Shaw, J. H. S. 1898 The flow of water. Nature 58, 34–36.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Ann. Rev. Fluid Mech. (in press).Google Scholar
Hopcroft, J. E. 1984 Turing machines. Sci. Am. 250, 86–98.Google Scholar
D'Humieres, D., Lallemand, P. & Shimomura, T.1985 Lattice gas cellular automata, a new experimental tool for hydrodynamics. Los Alamos Nat. Lab. Preprint LA-UR-85-4051.Google Scholar
Jameson, A. 1983 The evolution of computational methods in aerodynamics. J. Appl. Mech. 50, 1052–1070.Google Scholar
Kadanoff, L. P. 1986 Fractals: Where's the physics?. Phys. Today 39, 6–7.Google Scholar
Knuth, D. E. 1973 The Art of Computer Programming. Addison-Wesley.
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Landau, L. D. & Lifshiftz, E. M. 1959 Fluid Mechanics. Pergamon. 536 pp.
Leith, C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11, 671–673.Google Scholar
Leonard, A. 1980 Vortex methods for flow simulation. J. Comput. Phys. 37, 289–335.Google Scholar
Leonard, A. 1985 Computing three-dimensional incompressible flows with vortex elements. Ann. Rev. Fluid. Mech. 17, 523–559.Google Scholar
Lichtenberg, A. J. & Lieberman, M. A. 1983 Regular and Stochastic Motion. Springer.
Lin, C. C. 1943 On the Motion of Vortices in Two Dimensions. Toronto University Press. 39 pp.
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water II. Growth of normal mode instabilities. Proc. R. Soc. Lond. A 364, 1–28.Google Scholar
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
Mandelbrot, B. B. 1977 Fractals, Form, Chance, Dimension. W. H. Freeman & Co.
Margolus, N., Toffoli, T. & Vichniac, G. 1986 Cellular—automata supercomputer for fluid-dynamics modelling. Phys. Rev. Lett. 56, 1694–1696.Google Scholar
Metropolis, N., Howlett, J. & Rota, G-C. (eds) 1980 History of Computing in the Twentieth Century—A Collection of Essays. 659 pp. Academic.
Mcwilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.Google Scholar
Morikawa, G. K. & Swenson, E. V. 1971 Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14, 1058–1073.Google Scholar
Novikov, E. A. 1983 Generalized dynamics of three-dimensional vortical singularities (vortons). Sov. Phys., J. Exp. Theor. Phys. 57, 566–569.Google Scholar
Olfe, D. B. 1986 Fluid Mechanics for the IBM PC. McGraw Hill.
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento 6 (Suppl.), 279–287.Google Scholar
Orszag, S. A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41, 363–386.Google Scholar
Orszag, S. A. 1972 Comparison of pseudospectral and spectral approximation. Stud. Appl. Maths 51, 253–259.Google Scholar
Orszag, S. A. & Israeli, M. 1974 Numerical simulation of viscous incompressible flows. Ann. Rev. Fluid Mech. 6, 281–318.Google Scholar
Orszag, S. A. & Yakhot, V. 1986 Reynolds number scaling of cellular—automaton hydrodynamics. Phys. Rev. Lett. 56, 1691–1693.Google Scholar
Patterson, G. S. 1978 Prospects for computational fluid mechanics. Ann. Rev. Fluid Mech. 10, 289–300.Google Scholar
Pullin, D. I. 1982 Numerical studies of surface-tension effects in nonlinear Kelvin—Helmholtz and Rayleigh—Taylor instability. J. Fluid Mech. 119, 507–532.Google Scholar
Reichenbach, H. 1983 Contributions of Ernst Mach to fluid mechanics. Ann. Rev. Fluid Mech. 15, 1–28.Google Scholar
Robinson, A. L. 1985a When are viscous fingers stable?. Science 228, 834–836.Google Scholar
Robinson, A. L. 1985b Fractal fingers in viscous fluids. Science 228, 1077–1079.Google Scholar
Rogallo, R. S. & Moin, P. 1984 Numerical simulation of turbulent flows. Ann. Rev. Fluid Mech. 16, 99–137.Google Scholar
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond. A 134, 170–192.Google Scholar
Saffman, P. G. 1986 Viscous fingering in Hele-Shaw cells. J. Fluid Mech. 173, 73–94.Google Scholar
Saffman, P. G. & Meiron, D. I. 1986 Difficulties with three-dimensional weak solutions for inviscid incompressible flow. Phys. Fluids 29, 2373–2375.Google Scholar
Seitz, C. S. 1985 The cosmic cube. Commun. ACM 28, 22–33.Google Scholar
Tkachenko, V. K. 1964 Dissertion. Inst. Phys. Probl., Moscow, USSR.
Tryggvason, G. & Aref, H. 1983 Numerical experiments on Hele—Shaw flow with a sharp interface. J. Fluid Mech. 136, 1–30.Google Scholar
Tryggvason, G. & Aref, H. 1985 Finger interaction mechanisms in stratified Hele—Shaw flow. J. Fluid Mech. 154, 287–301.Google Scholar
Van Dyke, M. 1984 Computer-extended series. Ann. Rev. Fluid Mech. 16, 287–309.Google Scholar
Wilson, K. G. 1983 The renormalization group and critical phenomena. Rev. Mod. Phys. 55, 583–600.Google Scholar
Worthington, A. M. 1908 A Study of Splashes. Longmans Green.
Yamaguti, Y. & Oshiki, S. 1981 Chaos in numerical analysis of ordinary differential equations. Physica D3, 618–626.Google Scholar
Yarmchuk, E. J., Gordon, M. J. V. & Packard, R. E. 1979 Observation of stationary arrays in rotating superfluid Helium. Phys. Rev. Lett. 43, 214–217.Google Scholar
Zabusky, N. J. 1981 Computational synergetics and mathematical innovation. J. Comput. Phys. 43, 195–249.Google Scholar
Zabusky, N. J. & Kruskal, M. D. 1965 Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243.Google Scholar