skip to main content


Title: Algebraic bounds on the Rayleigh–Bénard attractor
Abstract The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.  more » « less
Award ID(s):
1818754
NSF-PAR ID:
10337299
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
Nonlinearity
Volume:
34
Issue:
1
ISSN:
0951-7715
Page Range / eLocation ID:
509 to 531
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Motivated by the search for sharp bounds on turbulent heat transfer as well as the design of optimal heat exchangers, we consider incompressible flows that most efficiently cool an internally heated disc. Heat enters via a distributed source, is passively advected and diffused, and exits through the boundary at a fixed temperature. We seek an advecting flow to optimize this exchange. Previous work on energy-constrained cooling with a constant source has conjectured that global optimizers should resemble convection rolls; we prove one-sided bounds on energy-constrained cooling corresponding to, but not resolving, this conjecture. In the case of an enstrophy constraint, our results are more complete: we construct a family of self-similar, tree-like ‘branching flows’ whose cooling we prove is within a logarithm of globally optimal. These results hold for general space- and time-dependent source–sink distributions that add more heat than they remove. Our main technical tool is a non-local Dirichlet-like variational principle for bounding solutions of the inhomogeneous advection–diffusion equation with a divergence-free velocity. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’. 
    more » « less
  2. null (Ed.)
    In inertial-range turbulence, structure functions can diagnose transfer or dissipation rates of energy and enstrophy, which are difficult to calculate directly in flows with complex geometry or sparse sampling. However, existing relations between third-order structure functions and these rates only apply under isotropic conditions. We propose new relations to diagnose energy and enstrophy dissipation rates in anisotropic two-dimensional (2-D) turbulence. These relations use second-order advective structure functions that depend on spatial increments of vorticity, velocity, and their advection. Numerical simulations of forced-dissipative anisotropic 2-D turbulence are used to compare new and existing relations against model-diagnosed dissipation rates of energy and enstrophy. These simulations permit a dual cascade where forcing is applied at an intermediate scale, energy is dissipated at large scales, and enstrophy is dissipated at small scales. New relations to estimate energy and enstrophy dissipation rates show improvement over existing methods through increased accuracy, insensitivity to sampling direction, and lower temporal and spatial variability. These benefits of advective structure functions are present under weakly anisotropic conditions, and increase with the flow anisotropy as third-order structure functions become increasingly inappropriate. Several of the structure functions also show promise for diagnosing the forcing scale of 2-D turbulence. Velocity-based advective structure functions show particular promise as they can diagnose both enstrophy and energy cascade rates, and are robust to changes in the effective resolution of local derivatives. Some existing and future datasets that are amenable to advective structure function analysis are discussed. 
    more » « less
  3. The large deflections of panels in subsonic flow are considered, specifically a fully clamped von Karman plate accounting for both rotational inertia in plate filaments and (mild) structural damping. The panel is taken to be embedded in the boundary of the positive half-space in ℝ3 containing a linear, subsonic potential flow. Solutions are constructed via a semigroup approach despite the lack of natural dissipativity associated with the generator of the linear dynamics. The flow–plate dynamics are then reduced—via an explicit Neumann-to-Dirichlet (downwash-to-pressure) solver for the flow—to a memory-type dynamical system for the plate. For the non-conservative plate dynamics, a global attractor is explicitly constructed via Lyapunov and recent quasi-stability methods. Finally, it is shown that, via the compactness of the attractor and finiteness of the dissipation integral, all trajectories converge strongly to the set of stationary states. 
    more » « less
  4. Wootters, Mary ; Sanita, Laura (Ed.)
    The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to the global nature of its updates. We present optimal bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree. Our bounds extend to the non-uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known as Variance Mixing and Entropy Mixing, introduced in the study of local Markov chains by Martinelli et al. (2003), imply Ω(1) spectral gap and O(log n) mixing time, respectively, for the Swendsen-Wang dynamics on the d-ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish Θ(log n) mixing for the Swendsen-Wang dynamics for all boundary conditions throughout the tree uniqueness region; in fact, our bounds hold beyond the uniqueness threshold for the Ising model, and for the q-state Potts model when q is small with respect to d. Our proofs feature a novel spectral view of the Variance Mixing condition inspired by several recent rapid mixing results on high-dimensional expanders and utilize recent work on block factorization of entropy under spatial mixing conditions. 
    more » « less
  5. Abstract

    The Swendsen–Wang algorithm is a sophisticated, widely‐used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to its global nature. We present optimal bounds on the convergence rate of the Swendsen–Wang algorithm for the complete ‐ary tree. Our bounds extend to the non‐uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known asvariance mixingandentropy mixingimply spectral gap and mixing time, respectively, for the Swendsen–Wang dynamics on the ‐ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish mixing for the Swendsen–Wang dynamics forallboundary conditions throughout (and beyond) the tree uniqueness region. Our proofs feature a novel spectral view of the variance mixing condition and utilize recent work on block factorization of entropy.

     
    more » « less