Variational bounds on the energy dissipation rate in body-forced shear flow

A new variational problem for upper bounds on the rate of energy dissipation in body-forced shear flows is formulated by including a balance parameter in the derivation from the Navier-Stokes equations. The resulting min-max problem is investigated computationally, producing new estimates that quantitatively improve previously obtained rigorous bounds. The results are compared with data from direct numerical simulations.Comment: 15 pages, 7 figure

Exact two-dimensionalization of low-magnetic-Reynolds-number flows subject to a strong magnetic field

We investigate the behavior of flows, including turbulent flows, driven by a horizontal body-force and subject to a vertical magnetic field, with the following question in mind: for very strong applied magnetic field, is the flow mostly two-dimensional, with remaining weak three-dimensional fluctuations, or does it become exactly 2D, with no dependence along the vertical? We first focus on the quasi-static approximation, i.e. the asymptotic limit of vanishing magnetic Reynolds number Rm << 1: we prove that the flow becomes exactly 2D asymptotically in time, regardless of the initial condition and provided the interaction parameter N is larger than a threshold value. We call this property "absolute two-dimensionalization": the attractor of the system is necessarily a (possibly turbulent) 2D flow. We then consider the full-magnetohydrodynamic equations and we prove that, for low enough Rm and large enough N, the flow becomes exactly two-dimensional in the long-time limit provided the initial vertically-dependent perturbations are infinitesimal. We call this phenomenon "linear two-dimensionalization": the (possibly turbulent) 2D flow is an attractor of the dynamics, but it is not necessarily the only attractor of the system. Some 3D attractors may also exist and be attained for strong enough initial 3D perturbations. These results shed some light on the existence of a dissipation anomaly for magnetohydrodynamic flows subject to a strong external magnetic field.Comment: Journal of Fluid Mechanics, in pres

Multiscale Mixing Efficiencies for Steady Sources

Multiscale mixing efficiencies for passive scalar advection are defined in terms of the suppression of variance weighted at various length scales. We consider scalars maintained by temporally steady but spatially inhomogeneous sources, stirred by statistically homogeneous and isotropic incompressible flows including fully developed turbulence. The mixing efficiencies are rigorously bounded in terms of the Peclet number and specific quantitative features of the source. Scaling exponents for the bounds at high Peclet number depend on the spectrum of length scales in the source, indicating that molecular diffusion plays a more important quantitative role than that implied by classical eddy diffusion theories.Comment: 4 pages, 1 figure. RevTex4 format with psfrag macros. Final versio

The Equity of Public Education Funding in Georgia, 1988-1996

A study of the effect of Quality Basic Education on the level of equity of public education funding in Georgia

Energy and enstrophy dissipation in steady state 2-d turbulence

Upper bounds on the bulk energy dissipation rate $\epsilon$ and enstrophy dissipation rate $\chi$ are derived for the statistical steady state of body forced two dimensional turbulence in a periodic domain. For a broad class of externally imposed body forces it is shown that $\epsilon \le k_{f} U^3 Re^{-1/2}(C_1+C_2 Re^{-1})^{1/2}$ and $\chi \le k_{f}^{3}U^3 (C_1+C_2 Re^{-1})$ where $U$ is the root-mean-square velocity, $k_f$ is a wavenumber (inverse length scale) related with the forcing function, and $Re = U /\nu k_f$. The positive coefficients $C_1$ and $C_2$ are uniform in the the kinematic viscosity $\nu$, the amplitude of the driving force, and the system size. We compare these results with previously obtained bounds for body forces involving only a single length scale, or for velocity dependent a constant-energy-flux forces acting at finite wavenumbers. Implications of our results are discussed.Comment: Submmited to Phys. Lett.