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Simulation of Taylor-Couette flow. Part 1. Numerical methods and comparison with experiment

Published online by Cambridge University Press:  20 April 2006

Philip S. Marcus
Philip S. Marcus
Affiliation:
Division of Applied Sciences and Department of Astronomy, Harvard University
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Abstract

We present a numerical method that allows us to solve the Navier-Stokes equation with boundary conditions for the viscous flow between two concentrically rotating cylinders as an initial-value problem. We use a pseudospectral code in which all of the time-splitting errors are removed by using a set of Green functions (capacitance matrix) that allows us to satisfy the inviscid boundary conditions exactly. For this geometry we find that a small time-splitting error can produce large errors in the computed velocity field. We test the code by comparing our numerically determined growth rates and wave speeds with linear theory and by comparing our computed torques with experimentally measured values and with the values that appear in other published numerical simulations. We find good agreement in all of our tests of the numerical calculation of wavy vortex flows. A test that is more sensitive than the comparison of torques is the comparison of the numerically computed wave speed with the experimentally observed wave speed. The agreements between the simulated and measured wave speeds are within the experimental uncertainties; the best-measured speeds have fractional uncertainties of less than 0.2%.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 146 , September 1984 , pp. 45 - 64
Copyright
© 1984 Cambridge University Press

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References

Alziary de Roquefort, T. & Grillaud, G. 1978 Computation of Taylor vortex flow by a transient implicit method. Comp. Fluids 6, 259–269.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.Google Scholar
Davey, A. 1962 The growth of Taylor vortices in flow between rotating cylinders. J. Fluid Mech. 14, 336–368.Google Scholar
Davey, A., DiPrima, R. C. & Stuart, J. T. 1968 On the instability of Taylor vortices. J. Fluid Mech. 31, 17–52.Google Scholar
DiPrima, R. C. 1967 Vector eigenfunction expansions for the growth of Taylor vortices in the flow between rotating cylinders. In Nonlinear Partial Differential Equations (ed. W. F. Ames), pp. 19–42. Academic.
DiPrima, R. C. & Swinney, H. L. 1981 Instabilities and transition in flow between concentric rotating cylinders. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub), pp. 139–180. Springer.
Donnelly, R. J. & Simon, N. J. 1960 An empirical torque relation for supercritical flow between rotating cylinders. J. Fluid Mech. 7, 401–418.Google Scholar
Eagles, P. M. 1971 On the stability of Taylor vortices by fifth-order amplitude expansions. J. Fluid Mech. 49, 529–550.Google Scholar
Eagles, P. M. 1974 On the torque of wavy vortices. J. Fluid Mech. 62, 1–9.Google Scholar
Gorman, M. & Swinney, H. L. 1982 Spatial and temporal characteristics of modulated waves in the circular Couette system. J. Fluid Mech. 117, 123–142.Google Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical Analysis of Spectral Methods. SIAM.
Haidvogel, D. B. & Zang, T. 1979 Accurate solutions of Poisson's equation by expansion in Chebyshev polynomials. J. Comp. Phys. 30, 167–180.Google Scholar
Jones, C. A. 1981 Nonlinear Taylor vortices and their stability. J. Fluid Mech. 102, 253–265.Google Scholar
King, G. P., Li, Y., Lee, W., Swinney, H. L. & Marcus, P. S. 1984 Wave speeds in wavy Taylor-vortex flow. J. Fluid Mech. 141, 365–390.Google Scholar
Kleiser, L. & Schumann, X. 1980 Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows. In Proc. 3rd GAMM Conf. on Numerical Methods in Fluid Mechanics (ed. E. H. Hirschel), pp. 165–173. Vieweg.
Keueger, E. R., Gross, A. & DiPrima, R. C. 1966 On the relative importance of Taylor-vortex and non-axisymmetric modes in flow between rotating cylinders. J. Fluid Mech. 24, 521–538.Google Scholar
Loncaric, J. 1981 The elusive strange attractor in two-dimensional Rayleigh-Bénard convection. B.Sc. thesis, MIT.
Mallock, A. 1888 Determination of the viscosity of water. Proc. R. Soc. Lond. A 45, 126–132.Google Scholar
Marcus, P. S. 1979 Stellar convection. I. Modal equations in spheres and spherical shells. Astrophys. J. 231, 176–192.Google Scholar
Marcus, P. S. 1981 Effects of truncation in modal representations of thermal convection. J. Fluid Mech. 103, 241.Google Scholar
Marcus, P. S. 1984 Simulation of Taylor-Couette flow. Part 2. Numerical results for wavy-vortex flow with one travelling wave. J. Fluid Mech. 146, 65–113.Google Scholar
Meyer, K. A. 1966 Los Alamos Rep. LA-3497.
Meyer, K. A. 1969a Los Alamos Rep. LA-4202.
Meyer, K. A. 1969b Three-dimensional study of flows between concentric rotating cylinders. Phys. Fluids Suppl. 12, II-165-II-170.Google Scholar
Meyer-Spasche, R. & Keller, H. B. 1980 Computations of the axisymmetric flow between rotating cylinders. J. Comp. Phys. 35, 100–109.Google Scholar
Moser, R. D., Moin, P. & Leonard, A. 1983 A spectral numerical method for the Navier-Stokes equation with applications to Taylor-Couette flow. J. Comp. Phys. 52, 524–544.Google Scholar
Orszag, S. A. & Deville, M. 1984 Submitted to J. Fluid Mech.
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Patera, A. T. & Orszag, S. A. 1981 Finite-amplitude stability of axisymmetric pipe flow. J. Fluid Mech. 112, 467–474.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 1–21.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar
Yahata, H. 1983 Temporal development of the Taylor vortices in a rotating fluid, V. Prog. Theor. Phys. 69, 396–402.Google Scholar