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The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow

Content Provider	Springer Nature Link
Author	Avrin, Joel
Copyright Year	2007
Abstract	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 $${\mathcal{A}}$$ (respectively $$\dim_{\rm H}{\mathcal{A}}$$ and $$\dim_{\rm F}{\mathcal{A}}$$ ). For a constant K $_{α}$ on the order of unity we show if μ ≥ ν that $$\dim_{\rm H} {\mathcal{A}} \leq \dim_{\rm F} {\mathcal{A}} \leq K_{\alpha} \left[\lambda_{m}/\lambda_{1}\right]^{9\left(\alpha - 1\right)/(10\alpha)} \left[l_{0}/l_{\epsilon}\right]^{(6\alpha+9)/(5\alpha)}$$ and if μ ≤ ν that $$\dim_{\rm H} {\mathcal{A}} \leq \dim_{\rm F} {\mathcal{A}} \leq K_{\alpha} \left(\nu/\mu\right)^{9/(10\alpha)}\left[\lambda_{m}/\lambda_{1}\right]^{9\left(\alpha -1\right)/(10\alpha)} \left[l_{0}/l_{\epsilon}\right]^{(6\alpha + 9)/(5\alpha)}$$ where ν is the standard viscosity coefficient, l $_{0}$ = λ 1 −1/2 represents characteristic macroscopic length, and $$l_{\epsilon}$$ is the Kolmogorov length scale, i.e. $$l_{\epsilon} = (\nu^{3}/\epsilon)$$ where $$\epsilon$$ 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 $$l_{0}/l_{\epsilon}$$ is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition $$\lambda_{m} \leq (1/l_{\epsilon})^{2}$$ , the estimates become $$K_{\alpha} \left[l_{0}/l_{\epsilon}\right]^{3}$$ for μ ≥ ν and $$K_{\alpha}\left(\nu/\mu \right)^{\frac{9}{10\alpha}}\left[l_{0}/l_{\epsilon}\right]^{3}$$ 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 $$\mu \thicksim c\nu$$ for 1/c within α orders of magnitude of unity, giving the estimate $$c_{\alpha}K_{\alpha}\left[l_{0}/l_{\epsilon}\right]^{3}$$ 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. $$\lambda_{m}\thicksim a(1/l_{\epsilon})$$ 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 $$\epsilon \thicksim \nu^{\alpha}$$ , motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes $$\lambda_{m}\leq \nu (1/l_{\epsilon})^{2}$$ , 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 $${\mathcal{M}}$$ for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such an $${\mathcal{M}}$$ 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 $${\mathcal{M}}$$ 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 $${\mathcal{M}}$$ , 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 $${\mathcal{M}}$$ are controlled by essentially NSE dynamics.
Ending Page	518
Page Count	40
Starting Page	479
File Format	PDF
ISSN	10407294
e-ISSN	15729222
Journal	Journal of Dynamics and Differential Equations
Issue Number	2
Volume Number	20
Language	English
Publisher	Springer US
Publisher Date	2008-01-03
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Smoothness and regularity of solutions Ordinary Differential Equations 3D turbulent flow models degrees of freedom inertial manifolds Attractors Inertial manifolds Applications of Mathematics Partial Differential Equations attractor dimension Navier-Stokes equations Nonlinear parabolic equations
Content Type	Text
Resource Type	Article
Subject	Analysis

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