 So we have these definitions for pursuing the idea of regime diagram for pre-mixed turbulent flames of course in this process we have assumed somewhere along here that the Schmidt number equals 1 so let us suppose that for scaling purposes take the turbulent dissipation as we find cube divided by the integral length scale could only show typically that it is of the order of but let us suppose that we take it is equal to then then putting things together way essentially what you are looking for is an expression for V prime over SL in terms of non-dimensional quantities. So the non-dimensional quantities actually what we are looking for is expressions for V prime over SL in terms of L over Lf with the non-dimensional quantities involved in there and we should now be able to show that this is equal to RE times L over Lf to the minus 1 and that is also equal to Karlowitz to the two-thirds divided by sorry times Lf L times L over Lf time to the one-third why are we doing this because that is what the regime diagram is all about the regime diagram is one on V prime over SL on the vertical axis plotted against L over Lf what this basically means is that when you want to talk about turbulence you talk about it in by characterizing in two ways one what are the length scales involved in the in the turbulent flow that we are talking about the other one what is the intensity that is involved okay. So these are two things essentially it is sort of like in a time signal kind of thing what you are talking about is what are the frequencies involved and what are the amplitudes involved right. So these two things essentially indicate that and in both the cases and for both the parameters what you are looking for is to compare these two with their corresponding flame related quantities that means we are saying can we now look at the turbulent length scales in comparison with the flame length scale like the flame thickness right. So are we having typical eddy sizes that are that are bigger than or smaller than the flame thickness right that is on the one side as far as the scales is concerned and as far as the intensity is concerned when you now have a flame that is trying to propagate at a particular speed if the flow that is out there is now fluctuating right at a certain intensity if the fluctuation is very very small that as a flame is beginning to propagate it does not really go through any jitter it just this is so small that the flame can just go through without even realizing that it went through a fluctuation right then you do not really have a turbulent flow at all on other words if you now had significant amount of intensity that means the amplitude is a very large right then the flame is going to now go through fluctuations correspondingly therefore how are these fluctuations the fluctuation in turbulent fluctuation intensity how is it comparable to the flame speed itself right. So these are the two things by which we want to compare characterizes. So we will now have values thrown in on a log log plot this is actually a logarithmic diagram so we will now look at 10 to the – 1 10 to the 0 10 to the 1 10 to the 2 10 to the 3 etc maybe just one more space for 10 to the 4 and similarly over here could actually look for something like a squarish picture 10 to the – 1 10 to the 0 10 to the 1 10 to the 2 10 to the 3 and so on okay and what we want to point out here is the relationship between V prime over SL and L L L over LF through Reynolds number is one of a inverse relationship right that is this goes is one over that and whether Reynolds number is a coefficient and therefore we would actually like for a normal plot it should actually be like a rectangular hyperbola that is forming the relationship between V prime over SL to L over LF for constant Reynolds number but this is actually a log log plot so in a log log plot this actually shows up as a negative straight line with a negative slope and in in this particular exercise what we are trying to do is to have for example in this case this involves like the Zelda which scaling and we are using a Zelda which number value of about 10 and that is something that we have talked we have seen long ago when we are trying to scale the premixed flame thickness to find out how much is the reaction zone thickness going to be relative to the preheat zone thickness and so correspondingly we now expect that a Reynolds number of one a constant Reynolds number of one line is going to be a straight line in the log log plot with a with a slope that is inclined downward for r equals 1 and this regime now is essentially what we call as laminar flame that means with the turbulent Reynolds number defined this way this line corresponds to r equals 1 anything greater than r equals 1 forms this region to the right of and above of this line and anything that is to the left of and bottom below this line is laminar right so that means for turbulent flames we are essentially interested in all of this regime that is that is power point number one then the next thing that we have to look for is how is v' over SL related to L over LF through Karlovitz number so can we now plot constant Karlovitz number lines and the answer is this is actually going as one third of that and therefore in a log log plot it is going to be a straight line with a slope of 1 over 3 all right and the line is going to go upwards so we start the start doing this at this point corresponding to 1 over here and now draw a line that is Ka equal to 1 and Karlovitz equal to 1 we now start from this point so we now have this entire turbulent combustion regime at the moment divided into two parts right one which is having a Karlovitz number less than one and another one that where the Karlovitz number greater than one and in both the cases the Reynolds number is greater than one within this regime we now want to further differentiate between a well you could use actually a straight line in continuous straight line this is a horizontal line this is a horizontal line corresponding to v' over SL equal to 1 that means v' equal to SL right so this is a regime where RE greater than 1 Ka less than 1 but v' is greater than SL this is a regime where RE is greater than 1 Ka less than 1 and v' is also less than SL right and finally we want to now look at the situation where the second Karlovitz number is greater than one right so we now have a boundary and that is also going to go the same way because Karlovitz number that this is basically Delta squared Karlovitz right so they are directly related so v' over SL in terms of Karlovitz Delta Ka Delta is going to also have a one-third relationship for the L over Lf all right it is simply a different coefficient Delta squared therefore you will have the same slope but then it starts at around 10 because we are not looking at a Zeldovich scaling of 10 and so I mean these things could be a little bit off this way that way okay but essentially we are getting the picture like this depending upon the value of your Zeldovich number and this line corresponds to Ka Delta equal to 1 therefore this is RE therefore you now go back and say RE greater than 1 Karlovitz greater than 1 but Karlovitz Delta less than 1 and here we now have RE greater than 1 Ka greater than 1 Ka Delta let us just use this space Ka Delta greater than 1 so that means we now have four regimes that we have marked one based on Karlovitz less than 1 but v' greater than or less than SL okay we have got two and when Karlovitz is greater than 1 we still have Karlovitz Delta less than or greater than 1 so that is how we are splitting this so what is the significance of doing it this way the answer is we are now going to now call this the wrinkle flamelet wrinkle flamelet regime and we are going to call this the corrugated flamelet regime there we are going to call this thin reaction zone regime or reaction reaction sheet regime you can call it either way so you can say thin reaction zone or reaction sheet regime and this is what you would call as broken flamelet regime or the well stirred reactor regime it is differently called in different textbooks in this way well so that is kind of interesting so just to make it a little clearer we could probably use like a different color sharpies or the four lines that we are talking about that demarcate these five regions and so why are we the question is why are we calling these what they are right so now let us think of think about what happens in each of these cases so the first one is the wrinkle wrinkle flamelet regime and here we are having situation of re greater than 1 k a less than 1 and v prime v prime less than SL so when we say k a is less than 1 and look at what is k a k a is essentially the flamed time to the column of time all right so essentially it is comparing the flame scales with the column of scales or essentially what it basically says is we are still looking at the flame to be significantly thinner than the turbulent scales right so that is what that is what it basically means so you now have a flame that is let us say marked by two lines that we will try to do this for quite some time and then the point essentially is we will be able to do this for quite some time and that is what I would like to impart here these two lines are essentially this line is like the reaction zone including the thickness of the reaction zone and the dotted line is actually the upstream edge of the preheat zone so the preheat zone is spanning the thickness corresponding to this distance between these two lines and the thickness of this continuous line itself represents the reaction zone right so effectively in a one-dimensional flame you would have the reaction zone and upstream of it you will have the preheat zone so the upstream edge of the preheat zone is what is remarked by this broken line but we are now talk showing it as curved because what we are basically saying is this is now interacting with an eddy that is going around like this so it is tangential velocity is Vn depending upon the eddy size and this flame is now propagating locally normal to itself with a flame speed itself right and since you are having a situation where v' is still less than a cell the intensities are significant to make the turbulence felt by the flame unlike in a laminar flow but still not enough to penetrate the flame or do anything more all it does is to basically make the flame wrinkle around the eddy right the flame is too thin and then so the whole of the flame is too thin okay we will now progressively get to the point where the eddy sizes are smaller when compared to the flame thickness or the flame is becoming thicker when compared to the eddy sizes whichever way right and then things are going to get a little bit more complicated so the next regime for example would be the corrugated corrugated flame let regime where the Reynolds number is greater than 1 all right the Karlovits number is still less than 1 all right but v' is greater than a cell so this means the intensity is now larger than the flame speed so the the the eddies are now going to make their presence felt on the flame a lot more so what this basically means is the the the velocity here is going to be large and correspondingly the flame has to give way so that's how the flame always always tries to do things right it now when the flow is locally large it gives way by inclining itself to the flow such that only a component of the flow direction is balanced by the flame speed right so what then happens when this now begins to increase is so if you now think about a eddy that's now having a larger vn and I now have a flame that is interacting with it it now actually begins to curl up to encompass that so which which means in fact you should you should you could exaggerate this a little bit more by saying strictly speaking you could say that means you know the the the corrugation in the flame can now house an eddy comfortably right previously we don't have the flame corrugated so much but now you have a corrugation of the order of the eddy size right so that's the reason why it's called a corrugated flame then we now get into the thin flame reaction zone or the or the reaction sheet regime so here are e is greater than 1 and Karlovitz is also greater than 1 and of course v prime is greater than itself we have long past we have gone past that long back right and and therefore what we are talking about is Karlovitz is greater than 1 and that means the the the turbulent scales are now bigger than the the the the the turbulent look look at this if Karlovitz is greater than 1 then the flame thickness is now getting to be bigger than the turbulent scales right so what that really means is you are now going to have a a flame that is having a preheat zone starting out like that but you now have an eddy that is that is big enough to penetrate the the preheat zone and then the preheat zone thickens why does the preheat zone thicken is because you now have a turbulent transport of heat and mass the job of the preheat zone is essentially transport okay where heat from the reaction zone is being transported upstream to the reactants and mass from the reactants is being transported downstream to the due to the reaction zone right so you have a heat and mass transport that's happening in the preheat zone which is now in addition to the molecular transport that happens because of temperature and concentration gradients is also additionally happening because of the eddy that means this eddy is now taking the reactants and putting it into the preheat zone for the reactions to consume and taking the heat from the preheat zone and bringing it a priori ahead to the reaction zone and heating it up right so if you want to think about this in a statistical manner it's as if like you had a fairly thick preheat zone the eddy was doing the job of thickening the preheat zone right so you now can only talk about a reaction zone that is now trying to give accommodate the eddy but the preheat zone has been breached that is essentially what is going on so therefore that is the reason why we are calling this a reaction sheet regime that means all that we can talk about as a characteristic of the original premix laminar flame the remnants of it is only the reaction zone the preheat zone is now marred right so effectively the way we are thinking about this is can I hold on to my laminar knowledge right and that is just beginning to get breached here by the turbulence until then it is okay and there is a reason why we have been using this there is this term called flamelet all the time and there is this basic idea of what is called as laminar flamelets that means you can now think about in these regimes we can still continue to think about the turbulent flame or the turbulent combustion zone as being made of an ensemble of laminar flamelets right so locally where you have these things satisfied you can now say it is essentially a laminar flame but it is now interacting with an eddy and therefore it is getting curved or corrugated wrinkled or corrugated right so you can you can continue to preserve your idea because the original local structure of a one-dimensional laminar flame how we extended from a one-dimensional premix laminar premix flame to let us say for example a Bunsen burner the shoulder of the flame there or the base of the flame there or the tip of the flame there in all these things we try to accommodate the one-dimensional idea locally the flame is trying to match the flow right or it is shaping itself to match the flow and that happens everywhere still right so you could you could you could hold on to that that means locally if you now look at these gradients you can draw exactly the same pictures that you drew there is the concentrations are now falling down the temperatures is now growing up and all that stuff that structure is there right but here that structure is breached okay except only for the reaction zone and then finally you now get into the WSR or the broken flames the well-stured reactor the WSR regime here course the Reynolds number is greater than one of course the correlates is greater than one we got to go back and point out that we still have correlates delta less than one there for the previous case that means the eddy sizes are still not big enough to act sorry small enough to breach into the reaction zone but that is no longer the case here so here we have correlates delta is also greater than one right so in this case what happens is the even the reaction zone is torn apart so that means you now have the flow getting into the reaction zone and coming back again and locally it can strain the reaction zone and cause local quenching and reignition and so on so that means we now have the flame torn apart into shreds because of these eddies right so this is this causes local quenching and reignition events let's now talk a little bit about non-premise combustion so non-premise turbulent flames I am following Peters here and there is more work that's been done further out in more in more recent past but I am not covering that material as far as this is concerned now the problem in non-premise combustion regimes is we don't have a characteristic flame speed here right but instead what we want to actually look at is if you are looking for velocity gradients in the flow and of course turbulent flames will have velocity gradients in the flow so if you now looking at something like a strain rate strain rate let's say it is given by symbol a and you have like let's say something like do we infinity we do we infinity by do why for the flame it is also related to the pressure gradient in the flow that is like you can say dp by dy equals row infinity a squared where az strain rate and so if you now have a strain rate then the strain rate actually has the dimensions of time okay so this is meters per second this is meters right so it is more sorry strain rate has a dimension of 1 over time right so then with this we should now be able to come up with a diffusion thickness diffusion thickness or LD equal to da d over a the whole to the half so that is like a meter squared per second divided by 1 over second so the 1 over second gets cancelled you get meter squared per half yes you meter so now we are basically looking for something like a characteristic dimension associated with the flame in order to be able to compare with the turbulence that is that is what we were doing to the premix flame there it was rather easy to think about a time and space relationship through the flame speed but here we did not have a flame speed so we are going through this the next thing that we want to do is we do not want to actually think in terms of physical space because in in in diffusion flames the mixture fraction actually comes up as a very handy tool and we went through this when we did when we did diffusion flames you see so when you have diffusion flames it is one thing to actually get the flame shape and physical space but if you want to actually look at the flame structure you are better off looking at the mixture fraction space because the mixture fraction space all the concentrations and temperatures are all related to the mixture fraction and if you now supply the local mixture fraction you can find out how the how these things vary right so in the in the mixture fraction space we can now have a corresponding the diffusion thickness that is corresponding to the diffusion thickness that we have defined LD diffusion thickness in the mixture fraction space now is let us suppose we call this delta Z F again the the subscript F is corresponding to fling here because we are looking at the diffusion thickness around the stoichiometric surface right because you have the flame and then you now have a certain length scale associated with the flame that the eddy wants to interact with right so the in the mixture fraction space it is always about the stoichiometric mixture fraction where the flame is therefore what you are saying what we are looking for is the conversion from the physical space to the mixture fraction space goes through the modulus of the mixture fraction a modulus of the gradient of the mixture fraction at stoichiometric condition times LD and we also have the scalar dissipation rate so scalar dissipation rate chi we did design they will define this and that is twice D grad Z mod squared so obviously we can get an idea now we want to try to put these two together so that means delta Z F the diffusion thickness in mixture fraction space is chi st divided by 2a the whole to the half now if you want to just make this a bit simpler what you can do is you can now expand chi st about chi about chi st for small values of Z and then approximate let us say let us not say let us say the Z is not varying too far out from Z st right and then chi depends on Z therefore they expand chi about chi st for small values of Z that is like a Taylor series approach so expanding chi of Z for small values of Z about chi st and when you say small values of Z that means we are approximating so we can show that delta ZF is approximately 2 ZT right so this is effectively the counterpart of counterpart of LF in premixed flames and should be basically point out that this is mostly the preheat zone and we did point this out when we were doing diffusion flames if you recall right so when we did the laminar diffusion flames what we pointed out was the Berkschum and assumption was essentially saying that the flame is a sheet along the stoichiometric surface but that was essentially the reaction sheet right so it is sort of like saying in the preheat in the premixed flame you know if you now think about only the reaction zone as a sheet but consider the thickness of the preheat zone like the way we have been drawing pictures here right then what happens in the in the Berkschum and problem you have a flame that is a sheet which is like a reaction zone and you do have the transports going on that is mixing of fuel an oxidizer and heat all these things are going on outside of this sheet and if you want to think about what happens to the reaction rate the infinite reaction rate assumption is the one that says that the reaction rate is almost infinite here but if you now try to zoom in and then clarify this reaction zone like we would we would do in the preheat pre mix flame then we would find that it is essentially the reaction rate that is going to be picking only at this sheet similar to how we did in premixed flames right so effectively in most of these we have been mainly interested up to here we have been mainly interested in the premixed flame regime diagram what is the preheat zone doing relative to the eddies right so similarly if you are now thinking about a corresponding diffusion thickness for the flame we should be worried about this in the mixture fraction space this which is this how is that going to actually interact with the eddies and that is like the counterpart of mostly the preheat zone of the premixed flame and here again the laminar ideas are coming back to help us now the reaction zone thickness on the other hand right so this is the one where you have the fuel and oxidizer consumption zones and we would like to point out that the oxidizer consumption zone is actually quite larger in reality that is that is what we are saying is if you now do actual chemistry right the detailed chemistry for let us say methane our flames or something what you should find is the oxidizer significantly leaks out of the reaction zone to a further distance outside into the field region okay so keep that in mind and we would like to now call this delta ZR this is the counterpart of L delta that we had in the preheat zone in the premixed flame sorry and we would allow we would now like to relate this to epsilon delta ZF that means it is a fraction of the the diffusion thickness now what we know from literature on detailed kinetics like or diffusion flames with detailed kinetics is so we will just take this empirically here so it is seen that so let us just call what we wrote here as delta Z epsilon delta Z epsilon is going with respect to chi st power 1 4 this is actually from four-step methane air chemistry calculation and then we try to relate the epsilon here to the extinction epsilon so by scaling epsilon with epsilon Q at extinction then we can show that epsilon over epsilon Q is chi st divided by chi Q to the one-fourth and therefore our contrast right here delta ZR divided by delta ZF is epsilon Q chi st divided by chi Q to the one-fourth now the reason why we want to do this is this is actually the one that gives you L delta divided by L that is a counterpart in the premixed flames that we saw which is what is the length scale corresponding to only the reaction zone thickness sorry L delta divided by LF where so as I said this is the counterpart of LF and that is this and this is the counterpart of L delta okay from preheat zones so we are kind of doing something very similar here as what we did for the preheat zone so premixed flame so what we want to plot here is a fluctuation amplitude Z prime divided by the diffusion thickness in mixture fraction space and on the right answer on the on the x axis or the horizontal axis we want to plot something as a measure of turbulence which is now going to be the scalar dissipation rate at extinction divided by scalar dissipation rate at stoichiometric now strictly speaking we got to actually average this to indicate a forward average in fact this is what is called as a conditional conditional forward average scalar dissipation rate let us not worry about what it what it exactly means for the moment we will simply take it as of the order of the same order as the scalar dissipation rate at stoichiometric condition and in fact I should also point out what is what is meant by this Z prime so Z prime is nothing but Z what is called as Z Z double prime squared tilde to the half double prime essentially indicates a forward average fluctuation so this is a essentially a forward RMS okay so it is a root mean square of the forward average fluctuation okay so that is what it means and similarly when you now say Z prime ST what we really mean is the root mean square forward average fluctuations obtained at this stoichiometric condition right now if you are so now let us put some numbers here so we can say 1 or 10 to the 10 to the 0 10 to the 1 10 to the 2 etc there and then let us say so this is 10 to the minus 1 we start with 10 to the minus 2 and the y-axis 10 to the minus 1 10 to the 0 10 and so on and what we should be interested in is a region that is greater than chi q by k chi ST greater than 1 because when you now have a region that is less than 1 this is the extinction chi so essentially this corresponds to flame extinction and then what we should be interested in is when you have actually we should show this so we have Z prime ST divided by delta ZF going is chi q over chi ST I can see that here you have a this is the opposite so you have a minus 1 over 4 in a log log scale so starting from here you now go through a line that is like this so this is Z prime in fact I should say Z ST divided by delta ZR equal to 1 and this is slope minus 1 over 4 we are we are taking chi ST tilde as approximately same as chi ST here as I said and then you also have one more situation where you are looking at similar to previously where we had U prime was comparable to SL and ZST prime is comparable to delta S so that is here so this is ZST prime equal to delta ZF so that now breaks up things into four parts one is so the flame extinction is one region then we have what is called as separated flames connected flames and connected reaction zones right so let us let us again explain these things so when you now have so when you now have for example ZST greater than delta F right now what this means is you now have fluctuations that are actually greater than the delta ZF and delta ZF is actually corresponding to like the preheat zone and this basically means that fluctuations are extending to sufficiently lean and rich sides and therefore what this means is that you are now taking in taking some amount of so the fluctuations are now causing some amount of lean side of the flame to go to the rich side and vice versa so this actually begins to now separate the flames because the flames are effectively dictated by the what is near the stoichiometric surface for the reaction zone and if now you have these things going on you will now create intermediate regions of stoichiometric surfaces so the original flame is essentially broken so this this begins to cause separated flames so causes the reaction zone to get separated now for ZST less than delta ZF right we do have intense mixing but not enough to cause separation and sometimes you can also get partial premixing so what this what this means is you can have the reaction zone move around without really getting disconnected but you will now have the preheat zone gig have the flow a sense of fluctuations so effectively then you don't you don't really have disconnected reaction zones so this is essentially connected flames that is reaction zones are connected and finally for ZST less than delta ZR this is now beginning to look like fluctuations are still smaller than the reaction zone thickness itself then even the reaction zones I should say here reaction zones are disconnected but here even the core connected now what it turns out is actually you can sense these things completely in a jet diffusion flame so if you now think about a jet diffusion flame along its axis then the jet diffusion flame axis actually goes through these different regimes as you go along this and come down along here so you can now see these different regimes exist in a jet diffusion flame corresponding to this we will stop here and pick up from here some on some other day