Showing papers by "Peter Constantin published in 2003"

Flame enhancement and quenching in fluid flows

Abstract: We perform direct numerical simulations 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 the 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 ν ∼ aU + 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 ν ∼ U 1/4. We also study the flame extinction (quenching) for an ignition-type reaction law and compactly supported initial data for the scalar field. We find that in a...

Numerical study of the Eulerian-Lagrangian formulation of the Navier-Stokes equations

Abstract: An Euler–Lagrangian analysis of the Navier–Stokes equations is performed with use of numerical simulations. On this basis we propose a new method for capturing vortex reconnection. It is found that the diffusive Lagrangian map becomes noninvertible under time evolution and requires resetting for its calculation. This sets a time scale and its frequent resetting corresponds to vortex reconnection. Another time scale defined by the connection coefficients, responsible for noncommutativity of Euler and Euler–Lagrange derivatives, is shown to be on the same order during reconnection. This introduces a novel singular perturbation problem of connection anomaly underlying reconnection.

Fronts in Reactive Convection: Bounds, Stability, and Instability

Abstract: This paper examines a simplified active combustion model in which the reaction influences the flow. We consider front propagation in a reactive Boussinesq system in an infinite vertical strip. Nonlinear stability of planar fronts is established 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. © 2003 Wiley Periodicals, Inc.

Chapter 4 Near Identity Transformations for the Navier-Stokes Equations

Abstract: Ordinary incompressible Newtonian fluids are described by the Navier–Stokes equations. These equations have been used by engineers and physicists with a great deal of success and the range of their validity and applicability is well established. Together with other fundamental systems like the Schrodinger and Maxwell equations, these equations are among the most important equations of mathematical physics. Nevertheless, their mathematical theory is incomplete and requires cut-offs. This chapter discusses some results reflecting research concerning diffusive-Lagrangian aspects of the Navier–Stokes equations. There are two distinct classes of approximations of the Navier–Stokes equations that are considered. First, the energy dissipation is treated exactly but the vorticity equation is not exact. This class contains the Galerkin approximations and mollified equations, and also treats the vorticity equation exactly although the energy dissipation is approximated. This is the class of vortex methods and their generalizations. This class is related by a change of variables to a class of filtered approximations of the formulation; the models are a subclass of these. The Navier–Stokes equations and their various approximations can be described in terms of near identity transformations. These are diffusive particle path transformations of physical space that start from the identity. The active velocity is obtained from the diffusive path transformation and a virtual velocity using the Weber formula. The active vorticity is computed from the diffusive path transformation and a virtual vorticity using the Cauchy formula. The path transformation and the virtual fields are computed in Eulerian coordinates.

Filtered viscous fluid equations

Abstract: We analyze an Eulerian-Lagrangian description of filtered incompressible fluid equations. A viscous Cauchy formula is derived for such equations. The models discussed are related to vortex methods and to filtered Kuzmin-Oseledets equations

Transport in rotating fluids

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). We obtain bounds for the vertical gradients of the Lagrangian displacement that vanish linearly with the maximal local Rossby number. Consequently, the change in vertical separation between fluid masses carried by the flow vanishes linearly with the maximal local Rossby number.

Bounds on Turbulent Transport

Abstract: : The subject of "Bounds on Turbulent Transport" was introduced in a series of ten lectures. The six lecturers constitute almost all of the contributors to this subject. The subject was introduced and foundations were laid by five lectures by F. H. Busse. In the middle of the first week, L. Howard reviewed his historical first approach to this subject and described more recent advances. Additional lectures by P. Constantine, R. Kerswell, C. Caulfield and C. Doering provided modern advances. We trust that the lecture notes will constitute a timely review of this promising subject. The following weeks had many highlights with approximately 40 additional lectures. The mini symposium on rotating convection in early July included presentations of experimental, ocean, atmospheric, and planetary observations. During the rest of the program, participants and visitors who have studied turbulence, convection, and instability in numerous geophysical situations with application to the ocean, the earth's atmosphere and planetary circulation made numerous contributions.