186 research outputs found
On the optimal design of wall-to-wall heat transport
We consider the problem of optimizing heat transport through an
incompressible fluid layer. Modeling passive scalar transport by
advection-diffusion, we maximize the mean rate of total transport by a
divergence-free velocity field. Subject to various boundary conditions and
intensity constraints, we prove that the maximal rate of transport scales
linearly in the r.m.s. kinetic energy and, up to possible logarithmic
corrections, as the rd power of the mean enstrophy in the advective
regime. This makes rigorous a previous prediction on the near optimality of
convection rolls for energy-constrained transport. Optimal designs for
enstrophy-constrained transport are significantly more difficult to describe:
we introduce a "branching" flow design with an unbounded number of degrees of
freedom and prove it achieves nearly optimal transport. The main technical tool
behind these results is a variational principle for evaluating the transport of
candidate designs. The principle admits dual formulations for bounding
transport from above and below. While the upper bound is closely related to the
"background method", the lower bound reveals a connection between the optimal
design problems considered herein and other apparently related model problems
from mathematical materials science. These connections serve to motivate
designs.Comment: Minor revisions from review. To appear in Comm. Pure Appl. Mat
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
Internal heating driven convection at infinite Prandtl number
We derive an improved rigorous bound on the space and time averaged
temperature of an infinite Prandtl number Boussinesq fluid contained
between isothermal no-slip boundaries thermally driven by uniform internal
heating. A novel approach is used wherein a singular stable stratification is
introduced as a perturbation to a non-singular background profile, yielding the
estimate where is the heat Rayleigh
number. The analysis relies on a generalized Hardy-Rellich inequality that is
proved in the appendix
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