Revision as of 21:22, 7 April 2016

Solving Asymptotics-based Evolution Equations !

Evolution Equation Modeling for wrinkled fronts

For propagating fronts, considered as density surface discontinuities, a class of EEM (Evolution Equation Modeling) can be derived by analytically solving the set of

• Euler equations
• Rankine-Hugoniot jump relationships
• a local kinematic relation defining the local front velocity (possibly depending on local curvature).

when the density contrast between 'hot' and 'cold' fluids is asymptotically small.

This Sivashinsky-type perturbative approach leads to non-linear non-local EE, with easily identified meaningful terms.

FLAMEX is a suite of solvers able to compute high accuracy solutions to asymptotic expansion based evolution equations (EE), of 1rt or 2nd order in time.

Main features of the EEM Module

• Solving first-order or second order EE in the Fourier or Fourier-Legendre basis.
• Time resolution using ETDRK1 and 4, using contour integrals to avoid cancelation errors (Trefethen et al. 2005)
• EEM for 2D or 3D planar, 3D expanding/converging fronts, acoustics, fast-transients, tangential velocity, gravity effects...
• Laminar or turbulent configurations (EE with additive noise, e.g. Passot-Pouquet, Kraichnan-Celik, Von Karman/Pao or 'DNS turbulence')

Examples of first-order in time obtained EEM: for 2D planar, 3D planar, or 3D expanding fronts

Sample Results for EEM

We show below a short gallery of pictures obtained using the FLAMEX solver.

Sample results for 3D planar fronts

Sample results for 3D expanding fronts with variable turbulence intensity,

Notice the 'soccer-ball' and 'cauliflower' aspects of the front, as described by Zel'dovitch in the 40's.

• See also some quantitative comparisons between FLAMEX and DNS (HALLEGRO) results at [1]
• More details (and comparisons with experimental results) in [2], [3]

Modules for asymptotic modeling of Flame-Balls

Some modules of FLAMEX have been devoted to the numerical solution to flame-balls (FB) dynamics. Using a Batchelor approximation for the surrounding Lagrangian flow and high activation energy asymptotics, we derived a nonlinear forced (stochastic) integrodifferential equation for the current FB radius. This gives access to the FB response to the ambiant Lagrangian rate-of-strain tensor \(g(t)\). For a diagonal \(g(t)\) deduced from random Markov processes of the Ornstein-Uhlenbeck type, or linearly filtered versions thereof, extensive numerical simulations and approximate theoretical analyses agree that

• flame balls can definitely live for much longer than their time of spontaneous expansion/collapse;
• large enough values of lifetimes are compatible with Poisson statistics;
• the variations of the lifetime with the characteristics of \(g(t)\) mirror the latter’s statistics, more precisely that of trace(\(g^2\)).

System of equations, EEM and sample results of FB radius dynamics

Modules for percolating fronts

In solid-like pre-mixtures, when the initial reactant content and/or reactivity are maxima along the axis of straight channels, curved flames are able to propagate under conditions. Such individual channels (and the conditions for propagation) can be lumped as building blocks of a larger percolation network.

Possible applications deal with wildfires propagation, sprays, inhomogeneous reacting media.

LEFT : FD DNS of a single-channel flame, showing pulsating instabilities (iso-temperature). a layer of heated unburnt material is formed then very quickly ‘shaved’ by a secondary channel- flame. This may even take place on top of the secondary flames themselves.

RIGHT : Numerical Simulation of the large-scale propagation (percolation) of a reacting front in non-diffusing disordered pre-mixtures (as a network collection of single-channel flames). The density of passing links fraction is increased from \(p=0.51\) to \(p=0.8\).

More details in [4].

Participants

Yves D'Angelo, Guy Joulin, Eric Albin, Rui Rego, Olivier Esnault, Gaël Boury, Lancelot Boulet, Viphaphorn Srinavawongs, Yosifumi Tsuji.