 So far we have been looking at spray combustion we try to put a set up a formulation for how to spray combustion we looked at spray statistics and so on this is actually coming from droplet combustion and we also mentioned a little bit about coal combustion and maybe biomass combustion and so on so essentially we have covered something about heterogeneous combustion in this realm we should probably just touch upon a little bit about solid propellant combustion as well so we will just quickly talk about this the reason why we are actually going through all these things in quick succession is to essentially point out as far as this course is concerned that whatever we have learnt on laminar premix flames and laminar diffusion flames are actually applicable as we speak pretty much in the context of more complicated application oriented areas except that the situation is complicated by a lot of other things for example in sprays it is complicated by this spray statistics the diameter distribution and so on in the case of solid propellant combustion we will see what what else happens and then we will move on to turbulent combustion also to show that the basics that we have learnt will be applicable there and how it is complicated by the turbulence and so on. So in the case of solid propellant combustion what we are what we need to understand is essentially we have to look at two cases one is homogeneous propellants and the other one is heterogeneous propellants that the term propellant actually means that both the fuel and oxidizer are actually available in solid form together and the question basically is to what is the level to which they are really mixed together so in the case of homogeneous propellants they are actually mixed up to the molecular level. So fuel and oxidizer mixed at molecular level here that means when when you now have a a source of heat that ignites it and then this the combustion keeps going under steady state conditions essentially what happens is the solid gets heated up and then it is thermally unstable so it decomposes into the into gas phase products the gas phase products are essentially gas phase fuel molecules and oxidizer molecules which then in turn break down further to react in the in a gas phase flame and the gas phase flame releases a lot of heat that goes back to giving rise to the the condensed phase decomposition products of fear and oxidizer molecules in the gas phase. So that is how it happens as a matter of fact when we are looking at liquid droplets there are also I mentioned monopropellant liquid droplets okay so where you now can actually have a premixed flame that is surrounding the monopropellant liquid droplet and it pyrolyzes the droplet and the droplet decomposes into gas phase fuel and oxidizer molecules that react in the gas phase plane. So a similar idea essentially prevails for homogeneous propellants example of this is example you can think about is the most common thing is what is called as double base double base propellants it calls double double base because it is actually a combination of nitroglycerine and nitrocellulose so they are mixed together such that they actually form a mixture at the molecular level and then they will decompose into a gas phase fuel and oxidizer. On the other hand so obviously before we move on to heterogeneous propellants obviously what we are looking at is essentially is this should give rise to a premixed gas phase flame and so you can think about a premixed gas phase flame where the preheat zone now extends into the solid all right because the preheat zone essentially is where the heat conduction happens and the heat conduction now happens into the solid heats up the solid from its initial temperature up to a surface temperature that needs to be determined as part of the problem by supplying interface boundary conditions between the solid and the gas phase and in addition to that you call you still have the same problem essentially of an eigenvalue problem which is you have to find out the burning rate of the solid or what is the rate of which the surface regresses this is the that is essentially similar to the way we did the gas phase flame one-dimensional flame structure associated analysis in the case of heterogeneous propellants essentially the fuel and oxidizer or mixed at a particular level that means you typically have this is very typical it does not have to be exactly followed this way but typically we have a solid oxidizer powder which means like it is typically a salt chemical salt and the most typical one that we use is what is called as a what is called as ammonium perchlorate and so you can use these ammonium perchlorate salt so it looks like the common salt particles but common very various particle sizes and so essentially you have these particles that are now embedded in a in a polymeric fuel so this is essentially starting out with a polymer that is that is a highly viscous liquid into which you know mix things and then you set this polymer to become a solid and so the polymer that the fuel is essentially a polymer that is that also acts acts as a binder that binds the particles and you can also of course throw in metal particles to act as for the fuel to do this entire mixture but the metal combustion actually happens outside of the combustion between the fuel and oxidizer all right so effectively the binder in the oxidizer so effectively what we are looking for is the combustion that that propagates the flame into the solid that is caused by a flame that is between the fuel and oxidizer so this is going to actually look like now have a lot of solid particles so typically these are like about 200 microns to 300 microns in size the finer ones could be as fine as even 5 microns or even 1 micron in very extreme cases but typically it is of the order of about 50 to 70 microns 80 microns maybe at the most 100 microns and so on so essentially try to have a very tight pack of a lot of large solid particles and then you now try to pack further in between the crevices of form formed in between these big solid particles with further smaller particles and so now you get these particles and then you now have a binder that is binding the whole thing together so microscopically when you now have a flame that is trying to eat into this it is obviously not going to be a premix flame right so even at the microscopic level what happens is let us suppose that we can we can we can hypothetically say we just cut it with a surface like this and we now say that is now the burning surface obviously the burning surface is not going to be as flat as what it is and we will try to see what happens how the non-planarity develops as we as we speak but let us suppose that this is what we want to think about and now you have like an exposed area of oxidizer that is giving rise to oxidizer vapor and there is exposed area of binder that is giving rise to binder so this is actually let us say fuel here oxidizer here fuel here and this is if you now look at it like this within this region that is sort of like the inverse of the Berkshuman problem the Berkshuman problem is where we now had a fuel in the core and oxidizer in the sides but if you did not really care about exactly what chemicals that are that are being used we are essentially looking at two different constituents reactant reactant species that are coming out in this fashion so we expect to actually have a diffusion flame over here that is associated with this recall we had talked about flux boundary conditions so we can we can deliberately talk about where exactly the stoichiometric surface is going to attach is it going to be at the contact between the oxidizer and fuel or away these kinds of issues remain to remain to be discussed in these in these problems but we also talked when we talked about diffusion flames we talked about finite chemistry effects particularly near the leading edge and then we found that we need to have something like a leading edge flame or a triple flame structure and that is pretty much expected in these propellants because it turns out that the oxidizer actually is a mono propellant by itself it can burn by itself but we need the binder to hold the oxidizer particles together and the and push it closer to a stoichiometric situation therefore you are essentially having a oxidizer rich reactive mixture and in fact it can also go through a self deflagration so you do have a self deflagration flame of the oxidizer very close to the oxidizer particle out of which you now get oxidizer rich products of this flame which now makes with the binder fuel gaseous species and then obviously since you now have a oxidizer rich species you are now going to have more like a triple flame rather than rather than a diffusion flame that extends all the way. So typically modulars have identified that you now have like a mono propellant flame for the ammonium perchlorate then you have what is called as a primary flame in the diffusion flame that is coming from the triple flame kind of idea there the corners and then the trailing diffusion flame so a typical idea of multiple flames prevails for what is called as composite propellants heterogeneous propellants are also generally termed as composite propellants because it is a composite of particulates in a binder similar to composite materials and structures and the popular idea is you have essentially multiple flames of different structure that prevails on top and of course as I said you also have aluminum metal particles that are embedded aluminum is thermally very stable therefore as it comes out it is going to accumulate on the burning surface because it is just barely getting melted there at the time it is not ready to actually get evaporated so it and then so as it gets accumulated it now forms a sintered web and it is getting heated up by this gas place flame complex and ultimately it now reaches an ignition temperature by which it gets ignited and then it this is a sintered web actually rolls out and then goes up and then starts burning as a bigger agglomerate than the individual particles that we originally had so that is a complex problem that has a lot of ramifications and solid rocket performance where these kinds of propellants are used so all I want to point out is we have discussed many of these issues in trying to do laminar flames earlier on and all these ideas actually remain here as well so we now want to also do a similar kind of overview of turbulent combustion and to say that we want to just give an overview of turbulent combustion is extremely unfair because it is a subject that has actually spawned a huge amount of literature a lot of scientists have been working and turbulent combustion is one of the things that that has funded a lot of combustion research if you will in the last many decades that is primarily because most of practical combustion boring let us say some of these things that we talk about which are happening in very microscopic scales so that the Reynolds numbers are so small except for this most of the other kinds of combustion situations are turbulent turbulent in nature right so we are looking at let us say gas turbine combustors or liquid rocket combustors or furnaces or boilers and so on many of these things you are actually wherever you are going through most combustion situations you have turbulent flow so effectively what you have to do is here again just like how we did sprays first we have to get a hang of turbulent flows before we can actually get into the combustion so when we did sprays you remember you we had to first talk about spray statistics how do you characterize sprays how do you describe sprays and then we will worry about the combustion so similarly we have to actually go back a little bit and then give a get a quick overview of what is turbulent about turbulent combustion right so that means we have to look at what is turbulence in general we will we will try to restrict what we are trying to do here to the context of turbulent jet flames and when I say turbulent jet flames I will talk about both premix flames and non premix flames and let us see what that means so when we say turbulent jet flames so consider turbulent jet flames is what we want to do this is this is very very generic because what you can think about essentially is if you now have a little nozzle from which you now have a gas that is coming out at turbulent speeds and you know is used out like a like like a jet right so and then you have a lot of eddies here and there and so on it is a caricature that I do not want to really elaborate too much so so we now have a large turbulent jet so you can think about this is like like coming out of a burner that is fitted to a combustor or a furnace and so on so this is this is a very practical situation that come that they come across in many applications when you now say turbulent premix flames you could now think about a reactant mixture that is coming out of here and then letting in air from the sides entrained into the jet so or if you want to think about a turbulent non premix flame you could now say I have only fuel coming coming through this it is entraining air around and then you now have a flame so either way so depending upon how that is going to happen right so if you now have a premixed reactants you are now going to have a highly wrinkled a premixed flame that is going to burn and consume all the all the reactants within a very very short distance and once again recall so or if you now have or okay or if you now have only fuel right you now want to have a diffusion flame the picture that I am drawing there is essentially something like the stoichiometric surface so what you are essentially just like how we did the Berkshuman problem if it is possible for you to actually now mark the mixture fraction corresponding to the contour corresponding to the stoichiometric mixture fraction right you now get a turbulent flame and what is turbulent about it because you now have fluctuations that keep happening the next snapshot that you draw try to go through you are now not going to exactly follow the same contour this is an instantaneous picture so the next one is going to be slightly different and the third one is going to be slightly different then we keep on taking these snapshots and then put on top of each there you now get lots of these kinds of curves with which you now kind of form a a band and that is what you would call as a turbulent flame brush okay so it essentially causes a brush like structure on an ensemble averaged manner similarly for the premixed flame you again let us go back and think about what happens in a laminar situation we talked very significantly about the Bunsen-Bernard experiment where you could now have the hole at the bottom that allows for the air entrainment open and set up a premixed flame and then start closing the hole very gradually and then convert this premixed flame into a diffusion flame and we now find that the diffusion flame is a lot taller when compared to the premixed flame right a very similar situation would exist for for turbulent flames as well simply because in the case of diffusion flames the combustion is actually diffusion controlled whereas so the heat release is not going to be as intense or the chemical reactions are not going to be as intense because they are waiting for the mixing to happen right whereas in the premixed flame you don't have that kind of a constraint so the heat release is kinetically controlled and so you now have a lot of heat release that happens within a very short region so the volumetric heating load that you come out come out with a premixed flame is lot higher when compared to that of a diffusion flame for this reason for the same kind of thermal loading that means the amount of fuel that you are trying to burn at a particular rate your premixed flames are typically very compact when compared to diffusion flames this is true as well in laminar as in turbulent flows so if you are now thinking if you want to think about how so just like how I said you know you can now look at the stoichiometric surface for the diffusion flame also meant to point out you could now think about a local turbulent flame speed that is trying to match the flow and then trying to set itself up right and there we have a little problem you know am I going to actually say this instantaneous flame is the one that is traveling at a turbulent flame speed or is it like a turbulent premixed flame brush that is traveling at a turbulent flame speed so that the notion of what is a turbulent flame speed is a little bit difficult to think about and we will have to talk about it a little bit in greater detail shortly maybe the first thing that I am hinting is first of all for if you are completely ignorant we will now have to talk about something called a turbulent flame speed as opposed to a laminar flame speed we got a hint of it the moment we actually used SL where L is a subscript indicating laminar and I pointed out that when we do when we think about turbulent flames we will have to alter that and so we will discuss this the point again I am trying to make is the ideas that we had like the flame speed balances the flow the normal comfort of the flow those ideas will remain okay that is not going to change so if you want to now think about all these things the first thing that we have to do is forget about the flames for a moment we will now look at only the term turbulent jet right. So for the sake of looking at the turbulent jet let us not worry about whether this is premixed reactants or the fuel we will simply say it is just a jet of some gas right maybe air itself and in a quiescent air ambience that means you are essentially looking at a jet and in the jet you now have a region that is like where you have a unsteady shear layer where you now have two mixing layers you are talking about momentum mixing as well as species mixing and thermal mixing if they if it is not isothermal so all these things are happening within this region in an unsteady manner you could hope to actually view this part as essentially an unsteady laminar case if you are if you are trying to resolve these scales in your in your computation or whatever it is but pretty quickly you are now going to get into a region where you transition to turbulence right and then beyond this point you now have a fully developed turbulent jet when you are now having a fully developed turbulent jet we are now looking at a lot of eddies over here which are which are primarily homogeneous isotropic. So we are looking at situation where if you now pick a point which is given by x and another point which is given by x plus or distant distant or apart then we now characterize this by having a cross correlation between the two now if you are now looking at this region where it is fully developed turbulence for a homogeneous isotropic turbulence the location x does not matter location x is arbitrary and all you are looking at is essentially the actual distance rather than the displacement so that means you do not have to look at this vector form so the direction did not matter because it is isotropic so the location did not matter because it is homogeneous therefore and or can be replaced by mod R which was let us say simply or without the vector and then what you are looking for is a velocity correlation. So the starting point of thinking about turbulent flows is to think about a velocity correlation here that is a spatial correlation that means you are essentially looking at turbulent fluctuation at x, t correlated with the turbulent fluctuation that x plus r, t as an average right and then the then you normalize this so the normalized normalized correlation f of r, t then is just take the correlation r and divided by the rms so u prime squared t the whole bar and so why are we bothered about this because we want to now step into how this looks like so if you now look at how the f of r, t looks like then it is going to be versus r let us say so how does it look like versus r it typically goes it is normalized so we do not have to worry about the magnitude it goes asymptotically to 0 as you go further out in r and with respect to this we could now say somewhere along here we have a length scale eta which we would now call as the Kolmogorov length scale and somewhere in here we would now have a length scale L which we would call as the integral length scale we will talk about these things a little bit more carefully so let us not worry about them at the moment or trying to write down but what we are what I am interested in is if I can actually have a function that approximates this to good measure and that would be 1-3 fourths c over k epsilon r to the two thirds where k what we are talking about is k is what is called as the turbulent kinetic energy turbulent kinetic energy which is equal to 4 isotopic turbulence you can say this is equal to u prime 3 half u prime squared where u prime of course is a fluctuation over the mean of the velocity and epsilon is the dissipation and and c of course is what is called as the Kolmogorov constant Kolmogorov constant and then we want to talk about what is the integral length scale integral length scale L of t then is integral 0 to infinity f of r, t dr that is essentially the normalized correlation integrated over all all distance then you now get a measure of a scale that kind of caps the lot scale lot scale structures so and we will see what that means pretty soon and then you now have what is called as the Kolmogorov length scale which is the smallest length scale in your flow essentially what is happening is you now have a turbulent flow in which you now have a cascading that happens that that goes all the way down to the Kolmogorov length scale so eta is new cube divided by epsilon the whole to the 1 4 where new is the kinematic viscosity and of course epsilon is what we call as the turbulent dissipation sometime back all right we would also talk about a Kolmogorov time associated with the length scale and we will talk about these time scales and length scales and the velocity associated with these for many of these scales but should be at the moment we are just discussing the Kolmogorov scale so the Kolmogorov time t eta is new over epsilon to the half and Kolmogorov velocity Kolmogorov velocity Vn is new epsilon to the 1 4 right and between between the large scale and the Kolmogorov scale we could now think about a discrete sequence of eddies and we the reason why we are talking about these things is because ultimately we want to compare these things with a laminar flame thickness okay so when you now talk about combustion we now have length scales associated length scales and time scales associated with the flame and the and the flame speed that is associated with it so we want to be able to compare these scales with the those are the flame so we now talk about a discrete sequence of eddies discrete sequence of eddies within what is called as a inertial sub range I am going to explain this soon with a spectrum with the energy spectrum but let us first I mean it is kind of like a chicken and egg thing so we first have to talk about the cascade and then the spectrum or vice versa so so this is basically L divided by 2 to the n where of course until we reach eta where n goes as 1 2 etc that means for every eddy size it is it is half the size is the previous the bigger ready essentially that is how we keep on looking at a discrete sequence and what we can find out when in this inertial sub range essentially the dissipation remains constant you have a constant dissipation mechanism that works on these eddies therefore we could now say that epsilon is essentially like vn square divided by Tn where vn is actually the velocity associated with the nth eddy and the Tn is the time associated with the 10th nth eddy and this would be this would be the same as v eta squared by T eta where every eta and T eta are essentially the Kolmogorov scales so then the next thing that we want to think about is what is called as a wave number for some reason turbulent the turbulence community is very fond of k maybe because Kolmogorov they like Kolmogorov a lot so they keep using k for the turbulent kinetic energy as well as the wave number and so on so so this is typically like one over one over the length scale and what we are looking for is the energy density which is the kinetic energy density density per unit wave number k so we can now think about a spectrum right e of k versus k and usually we are looking at a spectrum that kind of looks like looks like this and you can now think about a division here where you are looking at this is the large scale this is the large scale and this corresponds to one over l which is the integral length scale and this corresponds to one over eta the Kolmogorov length scale and we are looking at this region as the energy containing eddies or energy containing you can say integral scales and this region is now referred to as the inertial sub range where you now have a constant slope of minus five third there is a famous law there and then below the Kolmogorov length scale is the viscous sub range so essentially what is going on is turbulent flows are of such high Reynolds numbers on the whole based on the large scale that you have a predominant inertial effect over the viscous effect as you keep on going down further and further to smaller and smaller scales until you reach the Kolmogorov scale beyond which the viscous effects begin to be important right so this is this is how that this thing pans out so it's kind of like a 10 15 minutes of turbulence or less than that maybe right just about enough to for us to think a little bit about what happens when you now throw in a flame on top of this right so we feel and then we want what we want to do is to think about premixed premixed turbulent combustion now the first thing that I would like to point out as a empirical fact or an observation is when you now have a turbulent flame it now propagates a lot faster than a laminar flame corresponding laminar flame how do I get there right how did I how did I get to the point essentially what it means is now you have a lot of these eddies in the flow that are trying to wrinkle or corrugate this flame and you now have on the whole a flame that looks thicker and wants to reach ahead of itself so it's like a very very crude way of thinking about us and then we have to think a little bit more refined manner but effectively it now gives rise to a larger flame speed when compared to the laminar flame and if you now thought that the laminar flame speed under typical conditions or of the order of a few tens of centimeters per second the the turbulent flame speed or three speeds are about an order order magnitude more so they actually are in the range of meters per second right still quite smaller when compared to most of the flow speeds that occur in let's say let's say a gas turbine engine so if you now think about a gas turbine engine where you're still looking at subsonic flows we are not talking about supersonic combustion here maybe not in this course we are looking at subsonic flows highly subsonic we went through this long ago when we wanted to show that the pressure remains more or less constant in in low Mach number conditions and we said that the Mach number is of the order of what point one and since the temperatures are pretty high in the combustion zone your speed of sound could be like of the order of 1000 meters per second for the kind of temperatures that we are looking at and so we are looking at flow velocities are of the order of a several men or many tens of meters per second 200 meters per second that kind of 100 that's kind of speeds we are talking about as opposed to which the the turbulent flame speeds are still quite low right and this is obviously nothing nothing to do with detonation speeds okay and they are like about three orders magnitude apart so you know you now have to think about the ideas that we had about flame shapes that are getting that are attain like for example if you had like bluff body v-gooders the flame still with the turbulent flame propagation it still tries to align itself at an angle to the flow such that the component of the flow velocity will match this flame speed so what you are talking about as a flame speed is this flame speed here is kind of like a time average flame speed so what happens in reality without without having to average right so for which what we want to do here is to actually look at the physics of what's going on instead of thinking about like let's say empirical correlations on how the turbulent flame speed vary or varies relative to laminar flame speed and so on that is for maybe a special course on turbulent combustion itself where you get quantitative about it but here we just want to qualitatively look at the different regimes of turbulent cells right so the idea of turbulent flame regimes comes from several investigators in the late 80s so the first of which is Borgi 1985 in fact the early researchers used to call this regime diagram as a Borgi diagram but it's been also put forward by Peters and define further and Bradley and co-workers it's like in 1989 on saw it all many of these people are still active researchers in the community and so you can see that this is not very very old it's about only about 20 years old idea so what we want to do here is to assume a unity Schmidt number that means we don't want to make a big distinction between viscous mixing and molecular mixing SC which is nu over D will now check this is one and we now go through a few definitions here so define or flame thickness flame thickness LF capital F refers to flame not fuel and we will stick to this when we are when we are doing diffusion flames as well equal to D over SL and you are using SL you see and that means we are actually thinking about essentially a laminar flame that is going through a fate of wrinkling and so on the way manner is subjected to the turbulent at ease right and then we want to now construct an idea of what a turbulent flame looks like from essentially what is a laminar flame right so we are using a laminar flame speed and then you have a flame time which is Tf equals D over SL squared let's see order of magnitude expression that you would get there so we want to define a turbulent Reynolds number turbulent Reynolds number which is defined as or E equal to let's say V prime L over nu where V prime is the turbulent intensity it's a measure of the fluctuation whether the turbulent velocity fluctuation it comes from how you can define your kinetic energy that means you can you can say you are looking at something like this under square root that means this is basically two thirds k power half okay so that that is how the turbulent intensity comes from and L is of course the integral length scale and nu is the viscosity and therefore since we are using Schmidt number equal to 1 we say this is actually V prime divided by V prime L divided by D and we just define the flame thickness is Lf equals D over SL so D is Lf SL so this is going to look like V prime L over SL Lf it's sort of like a velocity ratio times a length scale ratio right so the velocity ratio here is we are comparing the turbulent intensity versus the flame speed we are comparing the integral length scale of the turbulent it is with the flame thickness right so I am trying to sort of implant this idea right here because that's kind of the way we are going to think about this the next thing that we want to think about is a turbulent dump color number dump color number is something that we have come across earlier on which is essentially a flow time by reaction time so it compares your flow time scale by the reaction time scale and so typically it is quite large because it takes a lot of time for the flow to make a significant movement like over a characteristic length scale of the geometry when compared to the time it takes for the reactions to happen so dump color number is now defined based on this but you know you will find that it is a little bit different from the the way Reynolds number turns out to be this is essentially time scales right so this is actually SSL L divided by V' LF so it is actually coming from L over V' for the flow time divided by LF over SL for the chemical time so it is sort of like saying this this looks like this and this looks like that they kind of look very closely similar but then they are not right so keep in mind don't get confused about this and we now have a Karlovitz number kA so dump color number as the symbol da Karlovitz number has a symbol kA which is effectively Tf the flame time divided by the turbulence time so this is you can or you can say you can say LF squared divided by Eta squared in terms of length scales or you can say VN squared or V Eta squared sorry V Eta squared divided by SL squared whichever way you want to say this is essentially in all these cases we are looking at basically ratio of flame scales to Kolmogorov scale flame scale to two you can say flame scales because it could be time scale length scale whatever Kolmogorov scales right and with these we should be able to show kA sorry Re equal to da squared kA squared so that kind of sounds like da kA is equal to square root of Re so that is that is some way by which you could remember now what we have what we have been thinking about a laminar flame is pretty much the same as what we thought earlier which means it has a preheat zone and a reaction zone right and then the preheat largely the flame thickness is mostly preheat zone for a typically a high damn cold number or a high activation energy situation your reaction zone thickness is quite small when compared to the the flame thickness itself and it is given by what is called as a zeldovich number right of this zeldovich scaling so you can scale this scale the flame thickness by the zeldovich number in order to get you get the reaction zone thickness we have done this earlier when we were when we were talking about laminar flames. So we can now form a second kA number based on the reaction zone thickness alone the reason why we are thinking about this is first of all recall we had we were talking about these things as well as the turbulence length scales because we want to now begin to compare how the turbulent length scales are going to be relative to the flame thickness right and what you are thinking about is and you now keep on changing your regime you are now going to have a situation where the turbulent length scales are typically much larger when compared to the flame thickness but as you now keep on changing the regime you now get into a situation where the turbulent scales are going to be small small enough to be able to enter into the preheat zone but not the reaction zone right and then ultimately can we get it to be even smaller than the reaction zone and so on. So this is how we want to go through so therefore we also want to keep a track of the reaction zone thickness alone rather than just the flame thickness on the whole. So for reaction zone thickness reaction zone thickness L ? in the the premix frame a second corlovitz number k a subscript ? can be defined as L ?2 divided by ? we are still comparing this with the Kolmogorov length scale and of course we can show that this is equal to ?2 k a where ?2 is essentially the Zelovich scaling coming between the flame length and this is flame thickness and the reaction zone thickness L ? we will just stop today with showing or this is probably a good time to stop we think we finish these definitions we will pick up from here tomorrow.