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The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow
| Content Provider | Semantic Scholar |
|---|---|
| Author | Avrin, Joel D. |
| Copyright Year | 2008 |
| 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 Pm be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Qm =I − Pm, then we add to the NSE operators μ Aφ in a general family such that Aφ≥QmAα 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 λm0 where m0 ≤ m, so that for large enough m0 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 Kα 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, l0 = λ1−1/2 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 Kα are dimensionless and scale-invariant. The estimate grows in m due to the term λm/λ1 but at a rate lower than m3/5, 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 Kα and cα 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 cα 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 m0 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 λm (e.g. $$\lambda_{m}\thicksim a(1/l_{\epsilon})$$ with a > 1) to still give good NSE approximation with lower powers on l0/lε, 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 λm 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 Aφ if α > 5/2.The special class of Aφ such that PmAφ = 0 and QmAφ ≥ QmAα 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 Aφ in the distinguished-class case that we expect will be typically used in practice we also obtain an $${\mathcal{M}}$$, now of dimension m0 for m0 large enough, though under conditions requiring generally larger m0 than the m in the special class. In both cases, for large enough m (respectively m0), 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. |
| Starting Page | 479 |
| Ending Page | 518 |
| Page Count | 40 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s10884-007-9100-3 |
| Volume Number | 20 |
| Alternate Webpage(s) | http://www.cscamm.umd.edu/people/faculty/tadmor/spectral_viscosity/Avrin%203D%20SV%20Turbulence%20DyDEs2998.pdf |
| Alternate Webpage(s) | http://www2.cscamm.umd.edu/people/faculty/tadmor/references/files/Avrin%203D%20SV%20Turbulence%20DyDEs2998.pdf |
| Alternate Webpage(s) | https://doi.org/10.1007/s10884-007-9100-3 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
