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Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow

Published online by Cambridge University Press:  29 March 2006

Gareth P. Williams
Gareth P. Williams
Affiliation:
Geophysical Fluid Dynamics Laboratory, ESSA Princeton University, P.O. Box 308, Princeton, New Jersey 08540
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Abstract

A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.

Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.

As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 37 , Issue 4 , 28 July 1969 , pp. 727 - 750
Copyright
© 1969 Cambridge University Press

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References

Arakawa, A. 1966 Computational design for long term numerical integration of the equations of fluid motion. Two-dimensional incompressible flow. Part 1. J. Comput. Phys. 1, 119.Google Scholar
Blackman, R. B. & Tukey, J. W. 1958 The Measurement of Power Spectra. New York: Dover.Google Scholar
Bryan, K. 1966 A scheme for numerical integration of the equations of motion on an irregular grid free of non-linear instability. Mon. Weath. Rev. 94, 39.Google Scholar
Cooley, J. W. & Tukey, J. W. 1965 An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297.Google Scholar
Danielson, G. C. & Lanczos, C. 1942 Some improvements in practical Fourier analysis and their application to X-ray scattering from liquids. J. Franklin Inst. 233, 365.Google Scholar
Eady, E. T. 1949 Long waves and cyclone waves. Tellus, 1, 33.Google Scholar
Hamming, R. W. 1962 Numerical Methods for Scientists and Engineers. New York: McGraw Hill.Google Scholar
Harlow, F. H. & Welch, J. E. 1965 Numerical calculation of time dependent viscous incompressible flow of fluid with free surface. Phys. Fluids. 8, 2182.Google Scholar
Henrici, P. 1962 Discrete Variable Methods in Ordinary Differential Equations. New York: Wiley.Google Scholar
Hockney, R. W. 1965 A fast direct solution of Poisson's equation using Fourier analysis. J. Ass. Comput. Mach. 12, 95.Google Scholar
Kurihara, Y. 1965 On the use of implicit and iterative methods for the time integration of the wave equation. Mon. Weath. Rev. 93, 33.Google Scholar
Lanczos, C. 1961 Linear Differential Operators. London: Van Nostrand.Google Scholar
Lilly, D. K. 1964 Numerical solutions for the shape preserving two-dimensional thermal convection element. J. Atmos. Sci. 21, 83.Google Scholar
Lilly, D. K. 1965 On the computational stability of numerical solutions of time dependent non-linear geophysical fluid dynamics problems. Mon. Weath. Rev. 93, 11.Google Scholar
Miyakoda, K. 1962 Contribution to the numerical weather prediction: computation with finite difference. Jap. J. Geophys. 3, 75.Google Scholar
Phillips, N. A. 1959 An example of non-linear computational instability. The Atmosphere and Sea in Motion. New York: Rockefeller Inst. Press and Oxford University Press.Google Scholar
Platzman, G. W. 1958 The lattice structure of the finite difference primitive and vorticity equations. Mon. Weath. Rev. 86, 285.Google Scholar
Richardson, L. F. 1922 Weather Prediction by Numerical Process. Cambridge University Press (Reprinted by Dover).Google Scholar
Richtmyer, R. D. 1957 Difference Methods for Initial Value Problems. New York: Interscience.Google Scholar
Shuman, F. G. 1962 Numerical experiments with the primitive equations. Proc. Int. Symp. on Numerical Weather Prediction. Tokyo, 1960. Meteor. Soc. of Japan.Google Scholar
Smagorinsky, J. 1958 On the numerical integration of the primitive equations of motion for baroclinic flow in a closed region. Mon. Weath. Rev. 86, 457.Google Scholar
Smagorinsky, J., Manabe, S. & Holloway, J. L. 1965 Numerical results from a nine-level general circulation model of the atmosphere. Mon. Weath. Rev. 93, 727.Google Scholar
Williams, G. P. 1967 Thermal convection in a rotating fluid annulus, Parts I and II. J. Atmos. Sci. 24, 144, 162.Google Scholar