Five-Mode Truncation of the Navier-Stokes Equations

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From a version of the three-dimensional Navier–Stokes equations for an incompressible fluid with periodic boundary conditions, a particular five-mode truncation was derived in [1]. The resulting set of nonlinear ordinary differential equations allows only a finite number of Fourier modes and behaves as a system with five degrees of freedom, thereby resembling the behavior of the Lorenz attractor.

Contributed by: Enrique Zeleny (March 2013)
Open content licensed under CC BY-NC-SA


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Details

The truncated Navier–Stokes equations are

,

,

,

,

,

where is the Reynolds number and , , , are empirical parameters.

References

[1] V. Franceschini, G. Inglese, and C. Tebaldi, "A Five-Mode Truncation of the Navier-Stokes Equations on a Three-Dimensional Torus," Computational Mechanics 3(1), 1988 pp. 19–37. link.springer.com/article/10.1007%2 FBF00280749?LI=true #.

[2] P. S. Addison, Fractals and Chaos, an Illustrated Course, London: Institute of Physics Publishing, 1997.



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