Showing papers by "Peter Constantin published in 2002"
TL;DR: This work provides a rigorous a priori bound E(k)
Abstract: We consider the energy spectrum of a quasigeostrophic model of forced, rotating turbulent flow. We provide a rigorous a priori bound $E(k)\ensuremath{\le}C{k}^{\ensuremath{-}2}$ valid for wave numbers that are smaller than a wave number associated with the forcing injection scale. This upper bound separates this spectrum from the Kolmogorov-Kraichnan ${k}^{\ensuremath{-}\frac{5}{3}}$ energy spectrum that is expected in a two-dimensional Navier-Stokes inverse cascade. Our bound provides theoretical support for the ${k}^{\ensuremath{-}2}$ spectrum observed in recent experiments.
77 citations
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TL;DR: In this article, the authors considered the stability of planar front propagation in a reactive Boussinesq system in an infinite vertical strip and established nonlinear stability for planar fronts for narrow domains when the Rayleigh number was not too large.
Abstract: We consider front propagation in a reactive Boussinesq system in an infinite vertical strip. We establish nonlinear stability of planar fronts for narrow domains when the Rayleigh number is not too large. Planar fronts are shown to be linearly unstable with respect to long wavelength perturbations if the Rayleigh number is sufficiently large. We also prove uniform bounds on the bulk burning rate and the Nusselt number in the KPP reaction case.
14 citations
TL;DR: In this paper, direct numerical simulations of an advected scalar field which diffuses and reacts according to a nonlinear reaction law were performed to study how the bulk burning rate of the reaction is affected by an imposed flow.
Abstract: We perform direct numerical simulations (DNS) of an advected scalar field which diffuses and reacts according to a nonlinear reaction law. The objective is to study how the bulk burning rate of the reaction is affected by an imposed flow. In particular, we are interested in comparing the numerical results with recently predicted analytical upper and lower bounds. We focus on reaction enhancement and quenching phenomena for two classes of imposed model flows with different geometries: periodic shear flow and cellular flow. We are primarily interested in the fast advection regime. We find that the bulk burning rate v in a shear flow satisfies v ~ a*U+b where U is the typical flow velocity and a is a constant depending on the relationship between the oscillation length scale of the flow and laminar front thickness. For cellular flow, we obtain v ~ U^{1/4}. We also study flame extinction (quenching) for an ignition-type reaction law and compactly supported initial data for the scalar field. We find that in a shear flow the flame of the size W can be typically quenched by a flow with amplitude U ~ alpha*W. The constant alpha depends on the geometry of the flow and tends to infinity if the flow profile has a plateau larger than a critical size. In a cellular flow, we find that the advection strength required for quenching is U ~ W^4 if the cell size is smaller than a critical value.
2 citations
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TL;DR: In this article, the authors considered uniformly rotating incompressible Euler and Navier-Stokes equations and employed a formalism that relates the total vorticity to the gradient of the back-to-label map (the inverse Lagrangian map, for inviscid flows, a diffusive analogue for viscous flows).
Abstract: We consider uniformly rotating incompressible Euler and Navier-Stokes equations. We study the suppression of vertical gradients of Lagrangian displacement ("vertical" refers to the direction of the rotation axis). We employ a formalism that relates the total vorticity to the gradient of the back-to-labels map (the inverse Lagrangian map, for inviscid flows, a diffusive analogue for viscous flows). The results include a nonlinear version of the Taylor-Proudman theorem: in a steady solution of the rotating Euler equations, two fluid material points which were initially on a vertical vortex line, will perpetually maintain their vertical separation unchanged. For more general situations, including unsteady flows, we obtain bounds for the vertical gradients of the Lagrangian displacement that vanish linearly with the maximal local Rossby number.
1 citations
01 Jan 2002
TL;DR: In this paper, the incompressible Euler equations can be written as the active vector system (AVS), where the initial data is A(x,0)=x, so for short times this is a distortion of the identity map, using the new u instead of u 0 in the Weber formula.
Abstract: The incompressible Euler equations can be written as the active vector system
$$ (\partial _t + u \cdot
abla )A = 0$$
where u=W[A] is given by the Weber formula
$$ W[A] = P\{ (
abla A)^* \upsilon \}$$
in terms of the gradient of A and the passive field v=u 0 (A). (P is the projector on the divergence-free part.) The initial data is A(x,0)=x, so for short times this is a distortion of the identity map. After a short time one obtains a new u and starts again from the identity map, using the new u instead of u 0 in the Weber formula. The viscous Navier-Stokes equations admit the same representation, with a diffusive back-to-labels map A and a v that is no longer passive.