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Fluid Dynamics: Solving the 2D Navier–Stokes Equations

In: An Introduction to Scientific Computing

Author

Listed:
  • Ionut Danaila

    (Laboratoire de Mathématiques Raphaël Salem)

  • Pascal Joly

    (Laboratoire Jacques-Louis Lions)

  • Sidi Mahmoud Kaber

    (Laboratoire Jacques-Louis Lions)

  • Marie Postel

    (Laboratoire Jacques-Louis Lions)

Abstract

The Navier–Stokes system of partial differential equations (PDEs) contains the main conservation laws that universally describe the evolution of a fluid (liquid or gas). In this chapter, we describe step-by-step the implementation of a numerical methods solving this problem using second-order finite-difference schemes for the space discretization. Time-integration is based on a projection method using a combination of Adams–Bashforth and Crank–Nicolson schemes. This method implies several basic algorithms that could be used for other problems: Thomas algorithm for solving tridiagonal linear systems, ADI methods for the time-integration of Helmholtz-type problems, Fourier decomposition for periodic Poisson problems, etc. The performance of the method is illustrated by simulating simple, but very nice flows: Kelvin-Helmholtz instability, vortex evolution and vortex dipole interactions.

Suggested Citation

  • Ionut Danaila & Pascal Joly & Sidi Mahmoud Kaber & Marie Postel, 2023. "Fluid Dynamics: Solving the 2D Navier–Stokes Equations," Springer Books, in: An Introduction to Scientific Computing, edition 2, chapter 0, pages 327-365, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-35032-0_15
    DOI: 10.1007/978-3-031-35032-0_15
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