Processus fractals et réaction chimique en milieux turbulents

Fractal sets were first related to the behaviour of dynamicalsystems with dissipation. Their evolutions are generally very sensitiveto initial and boundary conditionsand they have subsets of "state space" called "attractors". Representation pointslocated inside the domain of attraction are drawn towards an attractor. The pointrepresenting the system is free towander around the attractor at large times, consequently the ultimate behaviour ofthe system can be characterised by their study. The sensitivity to initial conditionsand the "deterministic randomness" may be considered as the two most essentialcharacteristics of dissipative systems like turbulence. The complexity of thelimiting surface of both the domains of attraction and the attractor introducesthe concept of fractal surfaces. These surfaces have an overall "butterfly" shape.Fractal sets cannot be represented by conventional smooth surfaces and consist ofintricate fine structures of ever decreasing scale. Fractal domains exhibit selfsimilarity in a range of scales, i.e. structures repeatedly occur with smaller andsmaller scales.At this early stage we have no need to lead the reader to more precise definitionsof fractal systems by introducing $H$ (self similarity at any point )or $K$ fractal sets (self similarity linked to the existence of singularities likeswirling points). The cascade process resulting from the advective terms of theNavier-Stokes equations are of course able to generate recurrent mechanisms which maybe examined by referring either to the space distribution of "structures" orto the shape of embedded surfaces. The smallest structures are distributed inrestricted areas of the flow and unfillingness becomes a characteristic of turbulentfields. As to the developments of surfaces embedded in turbulent fields we have tofirst recognize their fractal dimensions.\\A multifractal concept may be introduced by an intuitive route. At agiven time we identify turbulent structures by referringto an internal velocity law - for instance the differences ofvelocities $\Delta u$at two points $M$ and $M'$ located inside the structures - which satisfies Galileaninvariance. Taking into account the internal structure of the Navier-Stokesequations, this identification could be made by using a power law written as :$$ \Delta u_{r} \sim r^{h}$$where $h$ is a scale exponent. A given value of $h$ permits us to identify typicalstructures which are associated with fractal domains. They are not equallydistributed in space. Moreover the domains relative to typical structures randomlyevolve in time as do the shapes of immersed surfaces. The fractal character makesreference to a fractal dimension which is of course different from two ($D'=2$ forgeometrical surfaces and $D' \sim 2,36$ in usual turbulent fields ).For fractal surfaces the departure of $D'$ from its common value 2 iswritten as :$$ \mu_{2} = D'-2$$%$\mu_{2}$ permits a precise evaluation of the circonvolutions of the surface. Underconditions which are not too restrictive the evolutions of a material surfaceembedded in the flow is comparable with that of an iso-scalar surface disregardingthe role of molecular effects which intervene on the smallest structures (Batchelor).In other words the role of the Prandtl number or of the Schmidt number encounteredis obliterated in this sort of approach. With very high values of the Schmidt numberencountered in the cases of chemical species we can identify iso-scalar surfacesexhibiting very strong identations. $\eta_{B}$ (the Batchelor length scale) not$\eta_{K}$ is convenient to scale them.%\\[2ex]%If this kinematic approach is considered as loosely convincing for an introductionto fractal surfaces, reference may also be made to the cascade process inherent innon linear mechanisms which develop in the inertial range. The fractal process canalso be stated in another way by means of a statistical approach. Each stage of thecascade mechanism is characterised by a probability law and all of them areassociated with the global random processing in a multiplicative way. Provided thatthe elementary splittings are disconnected from each other the globalrandom mechanism is a log normal law.\\Whatever the chosen route, the cascade process is thepivotal point of all theories. Probability approaches only require a restrictednumber ofassumptions. However the details of elementary mechanisms are more or lessobliterated by the Central Limit Theorem which only requires each stage of theevolution to be independant of the other ones.%\\[2ex]%A fundamental property of fractal surfaces is the fact that their apparant areadepends on the length scale on which the surface is resolved. For the purpose ofpredicting the burning rate in the form of the flamelet theory, it suffices toconsider the real area $A$ of reaction zone whose geometrical properties are firstdescribed on scale $L$ (integral scale). This true area $A$ scales on $\delta_{l}$,the thickness of the reaction zone, that yields :$$ A=A_{0} \left ( \frac{L}{\delta_{l}} \right )^{\mu_{2}}$$%%\section*{The concept of the reaction zone}%%If we want to enlarge the concept of flame front to the case of flames where thepreheat zone is invaded by very small structures we must introduce the concept ofthe reaction zone which develops much as though it were a material surface. Whenassociating a speed with this zone a burning rate can be inferredand compared with that predicted by other theories. Contacts between fresh mixturesand hot points must be determined by the kinematic properties of the flow. In theEddy Break Up model, for instance, the characteristic time introduced in the sourceterm is related to the time of decay of the turbulence and consequently to thecharacteristic time of the cascade process. Therefore to be consistent with theEddy Break Up model, the time of decay of theturbulence has to be small compared with the chemical time.\\[2ex]In other words the cascade process may be introduced either by referring to thegeometrical behaviour of immersed surfaces through their fractal dimensions $D'$ ortothe characteristic time of decay. It is also possible to exploit in detail theproperties of surfaces immersed in turbulent fields by using the equation whichgoverns the surface itself (N. Peters, J. C. Vassilicos). We do not attempt to workout this promising method.\\[2ex]To predict combustion rates starting from the concept of fractal surfaces anevaluation of the true area of the reaction zone has to be made using the previousrelations. Moreover the speed of this reaction zone must be re-evaluated. To do sowe must account for the kinematic activity of a fine grained turbulence embedded inthe preheat zone. The chemical characteristic time is assumed unaltered, turbulentstructures being too large to invade the chemical zone. This leads to :$$\tau_{ch} \sim \frac{\delta_{l}^{\ast}}{u_{l}} \sim \frac{\delta_{l}^{\ast 2}}{\alpha^{\ast}}$$%The diffusion process is enhanced in the preheat zone as the consequence ofboth molecularand turbulence effects through the relation :$$ \alpha^{\ast} = \alpha+ \alpha_{T} $$Laminar and turbulent Prandtl numbers are assumed equal to one.$\alpha_{T} $ is conditioned by the motions of a class of turbulent structureslocated between $\delta_{l}$ and $\eta_{K}$ ; the spectrum must beevaluated in this range.\\For algebraic conveniences the Kolmogorov spectrum has been chosen, the use of a Paospectrumdoes not significantly alter the results. After some computations we find that :$$\frac{\dot{W_{B}}}{\dot{W_{0}}} \sim\left [ \frac{\delta_{l}u_{l}}{\nu} \right ]^{\frac{3}{2}(1-\mu_{2})}\left [ \frac{u'}{u_{l}} \right ]^{\frac{3}{2}(1-\mu_{2})}\left [ \frac{L}{\delta_{l}} \right ]^{\frac{3\mu_{2}-1}{2}}$$%$\dot{W_{0}}$ stands for $\rho u_{l}A_{0}$ which is the burning rate in the case oflaminar flames.\\When inserting the classical value of $\mu_{2}$ ($\mu_{2}=0,36$) one finds that :$$\frac{\dot{W_{B}}}{\dot{W_{0}}} \sim\left [ \frac{\delta_{l}u_{l}}{\nu} \right ]^{(0,96)}\left [ \frac{u'}{u_{l}} \right ]^{(0,96)}\left [ \frac{L}{\delta_{l}} \right ]^{0,04}$$%The burning rate is almost proportional to the turbulence intensity a situation whichrequires $\mu_{2}=0,33$. The ratio of the two length scales in the last term may bethought of as approaching one.\\[2ex]%Comparisons between predictions from the Eddy Break Up model and from fractaltheories arepresented in the figure \ref{zebulon2} of this report. The Eddy Break Up model usedfor this comparison has slightly beenimproved by introducing additional conditions for ignition. The results do not departfrom each other by more than about $10$ \%. This good agreement is presumably aconsequence of the common background of the two approaches which is the energycascade itself.%%\section*{Fractal dimension, another unexplored parameter}%Experimental and theoretical explorations have been made but they leave open thequestion as to which of the favorable arguments for the existence of a fractal dimension$D'$ is correct. At a first view we can suspect $D'$ to be a universal dimension inall the parts of the flow where the energy cascade hypothesis does exist. To clarify thisproblem, numerical experiments can be performed that requires large computerresources.\\Nonetheless fractal dimension has been determined in several reference cases. Themost important findings are reported herein.\\[2ex]In the case of an isotropic field with computations carried out in spectral spacewe tend towards the usual value $D'=2,36$, this is found whatever the orientations ofthe initial surface. The trajectories of the points belonging to the surface aredetermined during a convenient time and the fractal dimensions inferred from theindications of lines drawn on the distorted surfaces. The plane initial surface isconvoluted by the turbulent field step by step so that an assymptotic value of $D'$ isnot easy to determine. We can suspect the identations of the surface to be totallydevelopped after a time which is of the same order of magnitude as the time of decay.\\[2ex]We also recall that the introduction of a peak or trough in thespectrum alters the cascade process and consequently the time of decay. Such situations have been extensively studied by spectral analysiswithout making reference to fractaldimensions which should also be altered. Other investigationshave been made for instance where the surface is immersed in an homogenous shear flow.The presence of a wall significantly alters the fractal dimension which also depends on the orientation of the initial surface. Numericalexperiments have been made byusing Large Eddy Simulations in a channel flow. Computations havebeen carried out in physical space thanks to a code previously usedby Pr. Orlandi and Dr. Fatica this code is based on a method developedat Stanford University.As we ap\-proach the wall the fractal dimension decreases and$\mu_{2}$ ap\-proaches zero - the fractal method could be a successful way to discriminate kinematic effects from thermaleffects when a flame develops close to a wall.%%\section*{Conclusion}%%This approach makes a crude assesment of the possibilities opened byfractal theories in the prediction of burning rate. In some typicalcases it gives approximately the same results as more classicalmethods. In fact most of them are supported by thecascade concept which is a bold step indeed but not one we have serious reason to doubt. This new approach seems to be convenient to analysethe behaviour of a flame in the wall region but it fails where thedetails of chemical process must be accounted for.Cette th\`ese a pour objet l'\'etude des flammes de pr\'em\'elange enmilieu turbulent. Tout au cours de ce travail, l'aspect turbulences'estaffirm\'e primordial et les propri\'et\'es des surfaces immerg\'eesont d\^u \^etre analys\'ees avec un soin croissant.Comme tout sujet deth\`ese, cette \'etude pr\'esente de nombreuses difficult\'es. Priss\'epar\'ement, chacun des \'el\'ements\begin{itemize}\item m\'elange\item turbulence\item combustion\end{itemize}est en lui-m\^eme tout un programme. Leur couplage n'en est que plusdifficile \`a \'etudier. La formation du pr\'em\'elange ne sera pas\'etudi\'ee, les r\'eactants seront donc toujours suppos\'esparfaitement pr\'em\'elang\'es.\\Reste l'interaction turbulence-combustion. La formulation compl\`etedu probl\`eme n\'e\-ces\-si\-te\-rait :\begin{itemize}\item pour la turbulence \begin{itemize} \item les 3 \'equations de Navier-Stokes \item 1 \'equation de continuit\'e \end{itemize}\item pour la combustion \begin{itemize} \item 1 \'equation d'\'etat \item 1 \'equation de conservation par esp\`ece \item 1 \'equation d'\'energie \end{itemize}\end{itemize}Aucun cas industriel ne peut \^etre trait\'e \`a l'heure actuel\-le par lar\'e\-so\-lu\-tion d'un tel sys\-t\`eme.Dans l'approche par flammelette, l'efficacit\'e d'une flamme se mesuredirectement \`a l'\'evolution du volume de gaz br\^ul\'e. On est donctout naturellement amen\'e \`a s'in\-t\'e\-res\-ser \`a l'\'evolutionde la surface s\'eparant gaz frais et gaz br\^ul\'es. Plusieursapprochessont possibles, la plus directe consiste \`a \'ecrire l'\'evolution dela surface $\Sigma$ sous forme de bilan :\\[2ex]%\begin{tabular}{ccccccccc}taux de & &taux de & &surface & &surface & &surface\\variation de $\Sigma$&=&diffusion&+&produite& + &d\'etruite& + &d\'etruite \\ & &de $\Sigma$ par la& &par & &par& &par\\ & &turbulence & &\'etirement& &contraction& &extinction\\1 & & 2 & & 3 & & 4 & & 5\end{tabular}\\[2ex]On supposera que les effets $3$ et $4$ se compensent.Cette m\'ethode a \'et\'e utlis\'ee dans l'\'equation en $G$ parN. Peters ou J. C. Vassilicos.En ce qui nous concerne nous nous sommes plut\^ot orient\'es vers uneapproche fractale, qui bien que plus globalisante, permet elle aussicertains progr\`es dans la compr\'ehension des processus de combustion.%\\[2ex]{\large Remarques pratiques}\\[2ex]%Pour faciliter la lecture, ce rapport a \'et\'e divis\'e en troisparties. Chaque partie est \`a peu pr\`es autonome, elle commence parune nomenclature de ses notations et se termine par une bibliographiecontenant toutes les r\'ef\'erences qui y sont cit\'ees.Les \'equations importantes ont \'et\'e num\'erot\'ees ; chaquefois que l'on fait r\'ef\'erence \`a une \'equation son num\'ero estindiqu\'e entre paranth\`eses. Les num\'eros non entour\'es deparanth\`eses sont des rappels de figures ou de paragraphes. Lesr\'ef\'erences indiquent dans le texte le nom du premier auteur etl'ann\'ee de parution, dans chaque partie la bibliographie estclass\'ee par ordre alphab\'etique. Les annexes et la liste des figuressont donn\'ees en fin de rapport