NANCY D. Calogine K.Chetehouna E. Lamorlette J. Margerit S. Ramezani N. Rimbert O. Séro-Guillaume Modèles de Combustion en milieux poreux Caractérisation de la végétation Modèles de propagation à grande échelle Modélisation de la lutte par largage

INTRODUCTION

The Scales Scale of Combustion Flames above vegetation Convection Radiation The wood is a porous medium with several phases and several components Scale of “chemical” reactions: Pyrolysis Drying Scale of Combustion Convection Radiation The vegetation is a porous medium with two phases

The modeling at different scales lvlv Vegetal Phase air + gaz Phase Mesoscopic scale Macroscopic scale 10 < Re < 10 4 Porous medium combustion model Gigascopicc scale 10 7 < Re Propagation model

VEGETATION MODEL

Vegetation Model CIRAD Software A.M.A.P. Vegetation has been considered as fractal Dedicated to virtual Agronomy Not well suited for computational purposes

Vegetation Model Construction by Iterated Functions Systems I.F.S. Barnsley, Hutchinson Deterministic Probabilistic p i probabilites Probabilityis p i

Example Vegetation Model Calogine et al. 1998

Vegetation Model Model 1 Order : 7 Model 2 Order : 4 Model 3 Order : 5 Other Types of Vegetals

Vegetation Model

Automatic generation of the mesh

7 millions cells h=10m Vegetation Model

écoulement z x yExperiment I.F.S. (order) NxNxNxNx NyNyNyNy NzNzNzNz N x xN x xN z tttt Simulation 1 6 ( loc =4)  t  0,1 Simulation 2 5 ( loc =7) 1812x  t  0,08 Simulation 3 5 ( loc =9) 2322x  t  0,06 Simulation n 7 (14)  t  0,03

Vegetation Model

FIGHTING MODEL

Dropping model Continuous Eulerian model Continuous Eulerian model New equation for Interfacial area density New equation for Interfacial area density New closure for Interfacial velocity New closure for Interfacial velocity Averaging procedure

Dropping model Stochastic approach Stochastic approach Figure 1: fitting of the drop spray p.d.f. V SL = m/s V SG = 35 m/s. From left to right the p.d.f. are log-stable p.d.f., Upper-limit Evans p.d.f., log-normal p.d.f. and log-Weibull p.d.f. Log-Levy stable distributions are better distributions than the usual ones

Dropping model Transport Equation for the interfacial density It can be shown that: It can be shown that: Where S c is a source related to curvatureWhere S c is a source related to curvature S e is a surface stretching termS e is a surface stretching term S fc is a term related to fragmentation and coalescenceS fc is a term related to fragmentation and coalescence

Dropping model Two –Fluids Model Mass Balance MomentumBalance Closure Equations Supplementary Constitutive Laws

Dropping Model 4.8 million cells used for the atomisation stageA 850,000 cells grid used in the dispersion stage

Dropping Model velocity field around the cloud of retardant in the airplane frame of reference t = 0.5 s isosurface a = 0.5 (2% gum mass percentage)

Dropping Model The results of part I are initial conditions for part II Data aliasing and averaging From atomisation modelling to spreading modelling

Dropping model Dropping Simulations

A 3D DETAILED MODEL OF COMBUSTION

Write all equations at Microscopic scale using distribution theory, so that boundary conditions are included in the equations. Example : Mass Balance in the gaseous phase Albini Model at microscopic level (1985) First Step: Extension of the equations Homogenized Medium

Convolute the equations with a compact support kernel of size h denoted m. In the gaseous phase Balance of Mass Balance of momentum Balance of Energy In the vegetal phase Balance of Mass Balance of Energy Second Step: Average procedure

Homogenized Medium For example And we define the following Average Quantities From the Balance of Momentum in the gaseous phase we obtain the relation Is the porosity

Homogenized Medium At Step 2 several new quantities has been defined. They must be related to average variables: It is the Closure operation. We used Rational Extended Thermodynamics (R.E.T.) Or Extended Irreversible Thermodynamics (E.I.T.) Jou, Casas-Vazquez, Lebon Third Step: Closure operation Equations for these new quantities are obtained by writing that the entropy production is positive. Müller, Ruggeri

Homogenized Medium The obtained model is very general and has the following characteristics 1°) Two Temperatures, one for each phases with a modified Fourier law 2°) Contains a model of pyrolysis for the vegetal with relaxation times 3°) The vegetal phase has an internal structure with possible modelling of chemical actions of retardant 4°) Extended Navier Stokes equation, with memory

Homogenized Medium Several Simplifications can be Made 2 Temperatures1 Temperature Generalised Fourier LawUsual Fourier Law Instantaneous momentum Transfer between Phases...

Homogenized Medium Energy Balance Mass Balance Solid Constituents Liquid Constituent Gaseous Constituent Mass Transfer Between Phases “wood and air” Simplified 3D Model PHASE W AS WOOD

Homogenized Medium Energy BalanceMass Balance Mass Transfer Between Phases “wood and air” 3D Model Continued PHASE F (AS FLUID) GAS Balance of Momentum + Thermodynamics Relations +Radiative Transfer Equation

15m 9m 6m 25cm Distribution exponentielle de porosité 1m/s Heat flux Bilan de quantité de mouvement Phase gazeuse Bilan d’énergie Litière+Gaz Bilans de masse Litière+gaz Équation du transfert radiatif Litière+Gaz Cinétique Chimique Litière +Gaz HH liti è re Numerical experiment in fire tunnels

AVI Numerical experiment in fire tunnels

A 2D PROPAGATION MODEL

Propagation Modeling Reduction 3D Combustion 2D Propagation Hypotheses Vegetation Height  small compared to the gigascopic scale “Gigascopic Scale L” Flow Almost Parallel to the ground

Propagation Modeling The Ratio  L<< 1, Small parameter L can be the size of the fire or the distance to the fire front Outer Part of the Asymptotic Expansion Inner Part of the Asymptotic Expansion = Propagation Model Quantities are continuous across the vegetation toward ambient air Use of Matched Asymptotic Expansion on the Whole Domain z > 0 (vegetation +ambient air) Asymptotic Expansion Asymptotic Expansion

Propagation Modeling Model I Only for phase W Propagation Model Are quantities deduced from experiments… outer expansion Mass Balance for the cellulose Mass Balance for the char… Energy Balance Radiative flux Temperature of gases Mass Balance for the water…

Avant le front d’évaporation Zone IDans la zone de séchage, Zone IIEntre la zone de séchage et la zone en feu, Zone IIIDans la zone en feu, Zone IV Dans la zone brûlée, Zone V Propagation Modeling

Modélisation de la Propagation Takes into account Drying, Pyrolysis, Wind, Slope Radiative flux with Is the inclination of the flame and K is the extinction function

Zone I Zone III Zone IV Zone V Zone II Highly non linear 5 free boundaries Propagation Modeling

Occupation density 0.5Moisture 0.07 Propagation Modelling

Wind effect, 30 mn, 45 mn, 1h15, 1h50 Propagation Modelling

Under a certain value of the occupation density the fire will not propagate Percolation Density : Propagation Modelling

FIGHTING MODEL

PESPECTIVES

Travail en cours Caractérisation de la végétation comme milieu poreux. Thermique et Radiatif Modèle de flamme dans le modèle de propagation Rendre opérationnel le modèle de propagation à très grande échelle. Amélioration de la convection. Pris en compte de l’effet du retardant sur la propagation Modèle simplifié de largage

NANCY

Travail en cours

Level Set or V.O.F. Formulation

What happens if the initial density is decreased ? Control on the initial density Simulation

3 QUESTIONS OF OPTIMIZATION AND CONTROL

What happens if the initial density is decreased ? Control on the initial density Simulation

Au dessous d’une valeur critique de de la densité d’occupation le feu ne se propage plus. Densité de Percolation : Propriétés à basse densité Expérience de Propagation-Percolation Fractalité du front ?? Mécanismes mis en jeu ??

Question 1: Fire Front Reconstruction Data Assimilation Is it possible to find a initial distribution of vegetation and a T>0 such that: Prevention for?

Aerial fighting by indirect attack The dropped fluid = Water + Ammonium Phosphate + Gum The Ammonium Phosphate decreases the activation energy in the pyrolysis law : Control on the retardant distribution Simulation

Fire Front Reconstruction Data Assimilation The flame is defined by: height tilt angle emitted power rate of spread

2°) Que se passe-t-il si la densité de combustible tend vers 0 ? Fire Front Reconstruction Data Assimilation Flame model %

Fire Front Reconstruction Data Assimilation Burnt zone Burning zone Unburnt zone. Extinction line Fire line Flux meters positions Data from satellite?

1 QUESTION OF STABILITY AND UNICITY

Uniqueness and stability. Let  the characteristic function of the burning zone

Uniqueness and stability. For homogeneous conditions of vegetation the straight line seems to be the only stable solution (Depend on wind and slope) The direction of propagation seems to be along the line of maximum rate of spread

f(T) une fonction non linéaire Wave solution Rate of spread is solution of the non linear egeinvalue problem Problème qui peut avoir plusieurs solutions ou pas du tout Rate of spread.

Dimensional analysis Small parameter Rate of spread. We look for a series solution

Radius of convergence of the series and uniqueness ? Rate of spread. is the unique solution of

1°) Much more questions than answers Conclusion 2°) Mathematicians are welcome 3°) Thank you

TYPOLOGIE DES MODELES

Modèles Géométriques Méthode de Enveloppes Hypothèse sur la forme géométrique du feu Méthode de construction du front à t + dt Méthode qui marche bien mais fournit des solutions très régulières

Modèles empiriques Modèle de Rothermel s surface spécifique, d épaisseur couverture végétale C charge ducombuxtible sec, r masse vol. sec. D H chaleur contenue dans la particule sèche V vitesse du vent à mi hauteur tan j pente Hu Humidité des particules de combustible He Humidité d’extinction R = f( s, d, r, tan j, D H, V, Hu, He) R = vitesse de propagation Hypothèse Chaque partie du front est déplacée avec la vitesse normale R P P+Rndt Front à l’instant t Front à l’instant t+dt n Méthode qui marche bien mais fournit des solutions très régulières

Modèles Physiques Modèle de Réaction diffusion Prototype Equation seule (1) Système d’Equations (2) Réactions chimiques Dynamiques des populations Modèles dits Physiques ont la structure

COMPARAISON DES MODELES

Modèles de Réaction diffusion peuvent faire aussi bien que les modèles d’enveloppe et de Rothermel Résultat des Courses Rothermel Ellipses Hamilton Jacobi Réaction- Diffusion Les modèles d’enveloppe se comportent très mal à basse densité ?

Conclusions 1°) Les modèles de propagation dits de Réaction-Diffusion pourraient fournir plus d’information que les modèles semi empiriques. 2°) Théoriquement ils peuvent faire au moins autant que les modèles d’enveloppe ou de type Rothermel. 3°) Ils sont plus faciles à calibrer Questions 4°) Peuvent ils faire plus ? Et rendre compte de comportements très irréguliers du front de flamme 5°) Si oui quelle stratégie de modélisation ? Conclusions partielles

Stratégie de Modélisation Modèle « Détaillé » Réduction Modèle « Souhaité » Sous modèle 1Sous modèle 2 Sous modèle N … Complexification Spécifications Echelle Objectifs…