- Question:
- What's the deal on “61,506 dimensions?”
- Answer:
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A 3D state of the fluid at a given instant, i.e. the 3D velocity field at every point of
the cell, is a point in a state space. In principle this space is infinite-dimensional;
in practice it is always approximated by a finite-dimensional computer mesh. This dimension is bounded
from below by computational accuracy needs and the cleverness of numerical algorithms.
Solutions presented here are accurate to single precision accuracy, and for this
61,506 coupled ODEs turn out to be sufficient.
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What we really mean by “61,506 dimensions” is that accurate
description of Navier-Stokes requires discretizations of order of at least 100,000. As our exact invariant solutions require that many “dimensions,” truncated models do not capture
small scales that are essential part of turbulent dynamics. We do no modeling here - all our calculations are carried out in the full state space.
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