We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in
which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let
P
m
be the projection onto the first
m eigenspaces of
A =−Δ, let μ and α be positive constants with α ≥3/2, and let
Q
m
=
I −
P
m
, then we add to the NSE operators μ
A
φ in a general family such that
A
φ≥
Q
m
A
α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral
vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers
past a cutoff λ
m0
where
m
0 ≤
m, so that for large enough
m
0 the inertial-range wavenumbers see only standard NSE viscosity.
We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant
K
α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient,
l
0 = λ
1−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and
K
α are dimensionless and scale-invariant. The estimate grows in
m due to the term λ
m
/λ
1 but at a rate lower than
m
3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with
K
α and
c
α independent of
m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/
c within α orders of magnitude of unity, giving the estimate where
c
α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz
predictions for the number of degrees of freedom in 3D turbulent flow with
m so large that (e.g. in the distinguished-class case for
m
0 large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE
solutions. We would expect lower choices of λ
m
(e.g. with
a > 1) to still give good NSE approximation with lower powers on
l
0/
l
ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice
, motivated by the Chapman–Enskog expansion in the case
m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ
m
then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial
manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such
an of dimension
N >
m for the general class of operators
A
φ if α > 5/2.
The special class of
A
φ such that
P
m
A
φ = 0 and
Q
m
A
φ ≥
Q
m
A
α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension
m if
m is large enough. As a corollary, for most of the cases of the operators
A
φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension
m
0 for
m
0 large enough, though under conditions requiring generally larger
m
0 than the
m in the special class. In both cases, for large enough
m (respectively
m
0), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics,
and in particular trajectories on are controlled by essentially NSE dynamics.
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