Five-Mode Truncation of the Navier-Stokes Equations

​
t
2.
Re
46.36
x
1
(0)
-3.7
x
2
(0)
1.34
x
3
(0)
1.55
x
4
(0)
1.95
x
5
(0)
0.58
a
1
-10.
a
2
72.5
b
-9.2
c
-3.6
d
6.
e
19.5
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.

Details

The truncated Navier–Stokes equations are
x
1
'=-2
x
1
+
a
1
x
2
x
3
+
a
2
x
4
x
5
,
x
2
'=-9
x
2
+b
x
1
x
3
,
x
3
'=-5
x
3
-c
x
1
x
2
+Re
,
x
4
'=-5
x
4
-d
x
1
x
5
,
x
5
'=-
x
5
-e
x
1
x
4
,
where
Re
is the Reynolds number and
a
i
,
b
,
c
,
d
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.

External Links

Fourier Series (Wolfram MathWorld)
Reynolds Number (ScienceWorld)
Navier-Stokes Equations (ScienceWorld)
Attractor (Wolfram MathWorld)
Lorenz Attractor (Wolfram MathWorld)

Permanent Citation

Enrique Zeleny
​
​"Five-Mode Truncation of the Navier-Stokes Equations"​
​http://demonstrations.wolfram.com/FiveModeTruncationOfTheNavierStokesEquations/​
​Wolfram Demonstrations Project​
​Published: March 27, 2013