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

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 /λ1 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 e, 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.