A new numerical method for the simulation of three-dimensional flow in a pipe

  • A. Leonard
  • A. Wray
Contributed Papers
Part of the Lecture Notes in Physics book series (LNP, volume 170)


A new numerical method has been developed to investigate three-dimensional, unsteady pipe flows using a new velocity-vector expansion method. Each vector function in the expansion set is divergence-free and satisfies the boundary conditions for viscous flow. Other features of the general technique are as follows: (1) pressure is eliminated from the dynamics; (2) only two unknowns per “mesh point” are required; (3) there is rapid convergence of spectral methods; (4) there is implicit treatment of the viscous term at no extra computational cost; and (5) no fractional time-steps are required. In the present application of the method to flow in a pipe, the behavior of each flow variable near the computational singular point is treated rigorously and expansions in Jacobi polynomials have been shown to be particularly advantageous. The method has been tested on the linear stability problem for Poiseuille flow and has demonstrated rapid convergence of the eigenvalues and eigenfunctions as the number of radial modes is increased.


Poiseuille Flow Jacobi Polynomial Rapid Convergence Radial Mode Turbulent Channel Flow 


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  1. 1.
    Wray, A., and Hussaini, Y.: Numerical Experiments in Boundary Layer Stability, AIAA Paper 80-0275, Pasadena, Calif., 1980.Google Scholar
  2. 2.
    Orszag, S. A., and Kells, L. C.: Transition to Turbulence in Plane Poiseuille and Plane Couette Flow, J. Fluid Mech., Vol. 96, 1980, p. 159.Google Scholar
  3. 3.
    Moin, P., and Kim, J.: Numerical Investigation of Turbulent Channel Flow, J. Fluid Mech., Vol. 118, 1982, p. 341.Google Scholar
  4. 4.
    Moin, P.: Numerical Simulation of Wall-Bounded Turbulent Shear Flows, Proceedings 88th International Conference on Numerical Methods in Fluid Dynamics (this issue).Google Scholar
  5. 5.
    Patera, A. T., and Orszag, S. A.: Finite-Amplitude Stability of Axisymmetric Pipe Flow, J. Fluid Mech., Vol. 112, 1981, p. 467.Google Scholar
  6. 6.
    Schumann, U.: Subgrid Scale Model for Finite Difference Simulations of Turbulent Flows in Plane Channels and Annuli, J. Comp. Phys., Vol. 18, 1975, p. 376.Google Scholar
  7. 7.
    Orszag, S. A.:Spectral Methods for Problems in Complex Geometries, J. Comp. Phys., Vol. 37, 1980, p. 70.Google Scholar
  8. 8.
    Chorin, A. J., and Marsden, J. E.: A Mathematical Introduction to Fluid Mechanics. Springer-Verlag, New York, 1979.Google Scholar
  9. 9.
    Handbook of Mathematical Functions. M. Abramowitz and I. A. Stegun, eds., NBS AMS 55, Sec. 22, 1968.Google Scholar
  10. 10.
    Salwen, H., Cotton, F. W., and Grosch, C. E.: Linear Stability of Poiseuille Flow in a Circular Pipe, J. Fluid Mech., Vol. 98, 1980, p. 273.Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • A. Leonard
    • 1
  • A. Wray
    • 1
  1. 1.NASA Ames Research CenterMoffett FieldUSA

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