 So today we will talk about in just about one lecture this preferably brought rather interdisciplinary topic of combustion instability of course we include this because it is my favorite topic and I wanted to cover this as part of the lectures on combustion again the goal here is to show that whatever we have learnt in the laminar framework can be used to understand more complicated problems and one of them being combustion instability early on I think Williams said actually identified that there are three aspects to combustion instability one of them is intrinsic instability the second one is combustor instability the third is system instability so we will first briefly look at what these individually mean and then we will start looking at the details of each of these but before that we should also first of all look at the word instability regardless of combustion the idea of instability is not specific to combustion it is something that comes from what we call as perturbation theory so essentially it is an idea of course much more than that I think it is a much more common sensical idea there are lots of things around us which are which show inherent characteristics of instability so some extremely social examples would be like for example if I do not do exercise I grow way heavier and then that makes me do more makes it more difficult for me to do exercise and then like I keep growing heavier and so that is an instability right or the other possible the other example is if people stop using public transport then public transport run losses therefore they run less buses and therefore we still are going away from using public transport the buses now stop running and so on so lots of such social situations or you know human situations you can go through what is called as instability so the basic idea is if you now have a system which we think is an equilibrium and then we now try to make a lot of small perturbation question is a system coming back to the original position or is it it goes away from the original position so here there are a few different things that are kind of important you could actually have a dynamic system which is in equilibrium locally that means at a particular set of parameters so for example in a flow system you could have a the parameters could be Reynolds number in a combustion system the parameter could be equivalence ratio so you could now be running an engine at a set of conditions like air flow rate and that equivalence ratio pressure and initial temperature of reactants and so on and then for these set of conditions you now have some equilibrium and what we are talking about is perturbing about an equilibrium for this given set of conditions if you now move your set of conditions you now reach a new equilibrium and that need not necessarily be stable as the previous set of conditions for which you had an equilibrium so equilibrium does not necessarily mean stability you could have a unstable equilibrium or a stable equilibrium so for example the best example is for example before you now have like a rod that is hanging from a pivot and then if it is like this and then you now tap it it is supposed to go through a sinusoidal oscillatory motion like a pendulum and the what is called as a set point that is the equilibrium is where it is actually hanging down because of gravity but then if you now can also hold this rod up here then that is also actually in equilibrium the weight of the rod is actually being balanced by the support but now you try to tap it it now starts going further and further away from the equilibrium so the top position is an equilibrium as well but it is an unstable equilibrium and compared to the bottom position so the examples that we talked about where we are talking about the physical weight of a person versus the exercise is a situation where we were talking about something that was unstable so that is what is instability connoting there that means you have a situation and then any small perturbation keeps you going away from that position so that is what we really mean by instability to begin with intrinsic instability then is actually a instability where you do not have anything else coupled to the combustion it is only the combustion itself that is unstable right so the flame exhibits instability by itself a combustor instability on the other hand is a system of flame coupling with acoustics in the combustor. So this is this is a flame acoustic system where you have a feedback loop between the flame and the acoustics in the combustor we will talk in greater detail about this but essentially here it is not a standalone combustion alone a combustion that is going unstable it is combustion coupled with acoustics in the combustion chamber that is going unstable so that is the combustor instability the system instability is feedback with let us say fuel supply air supply or oxidizer supply etc that means it now is a bit more specific to how the system actually is configured which is going to cause this instability again we will talk in greater detail about so let us first talk about the intrinsic instability here we shall talk about premix flames plane prep or should say premix flames I should not say plane premix flames because that is that is the point of contention as far as the instability is concerned let us look at premix flames we have talked about the issues involved in here we could talk about two things one is the thermal diffusive instability and the other one is the Darius-Lando instability and we have actually discussed this previously when we were talking about corrections to the laminar flame speed okay and there we were basically talking about these effects of the flame curvature and the flow divergence effects right. So what we were basically saying is if you now have a planar flame and then that is supposed that you now perturb this right then we have the situation where the heat is actually going radially outward in a flame that is curved concave relative to the upstream reactance flow is going from left to right and in this case the heat is actually converging while the heat is diverging here and converging here the reactant flow or the reactant diffusion into the flame is happening the opposite direction that means here all the reactants are actually converging towards the flame and all the reactants here are actually diverging so you have a lesser concentration of reactants in this flame but they get heated up more but you have a greater concentration of reactants but they are getting heated up less because the heat is getting distributed right. So now the question comes what is the effect of Lewis number so the thermal diffusive instability is directly a consequence of non-unity Lewis number so when you have a non-unity Lewis number then the heat conduction upstream to the reactants and the reactant diffusion downstream to the flame they do not happen exactly over the same length scale right. So when you and we have gone through this we said when we have a larger Lewis number that means your conductive effects are actually more when compared to diffusive effects whereas it is opposite so depending upon this what you will find is there are ranges of Lewis numbers for which you will have something that is unstable right so this leads to unstable flame or oscillatory or stable flame what is what is meant by that is I mean again going back to the idea of instability in general if you now have a equilibrium point and then you now give it a perturbation and then you are now trying to track what happens to the amplitude of the perturbation with time right. If it comes down like this that is stable if it now grows like that that is unstable the amplitude keeps on exponentially growing that is unstable the other possibility is it looks like it is actually coming closer to the equilibrium point but it begins to oscillate right for clarity let us also look at a couple of other situations where you could you could it could look like it is coming back but then it oscillates and then its amplitude keeps growing right. So this is the this is also literally unstable and the third situation the other situation of course is when you now have something that comes back in an oscillatory manner right so then this becomes stable that means the amplitudes are dying down but in an oscillatory manner so there are many such possibilities what we are talking about here is a possibility where the amplitudes remain constant in fact this refers to like the way the pendulum behaves so when you now give it a tap it looks like it comes back but it kind of misses the equilibrium point because of inertia and then oops it went to the other side then it goes back and looks like okay it is going to be stable but oops it went up went to the other side and so on. So since you now look at a displacement and then at that at a small perturbation it looks like it is coming back you will tend to think that it is stable but that is what is called as only static stability but dynamically it now actually attains an equilibrium a neutral equilibrium of an oscillatory state okay and then of course in nonlinear dynamics if they would refer to it as a hop by vacation and so on we will not get into all that jargon here we will just say that for example you could lead to an unstable flame that means when you now have a plane premix flame that is perturbed it could now get become increasingly non-planar right. So if it now starts getting curved and curved more and more and more then that is an unstable flame a stable flame on the other hand is where you now have a perturbation and then it gets back to being planar right a oscillatory flame is it now starts actually oscillating with a with a certain constant amplitude right. So there are many such in fact the oscillatory flame is something that I will talk about in the more in the context of partially premix flames but right now you could say that for different Lewis numbers you can find that you can say the flame becomes unstable or the flame and typically the Lewis numbers of course Lewis number is actually a function of the reactant case because it involves the diffusivity of the reactant. So usually we are talking about non unity Lewis numbers of the deficient reactant because the deficient reactant is the one that is that the flame is more sensitive to okay. The other kind of instability that we are talking about is what is called as the Darius Landau instability or simply hydrodynamic instability and this is also something that we talked about in the context of corrections to the laminar flame speed in some sort of a steady state framework but not necessarily in a instability framework where what we are talking about is a premix flame can be thought of as a line of discontinuity between two different regions of different density right. So you have the cold reactants that are denser you have the hot products that are lighter right. So because of which you now have a flow what we talked about at that time was because of which you now have a flow divergence or a flow convergence behind the flame which could in turn accentuate the curvature. So the basic idea about the Darius Landau instability is if you now have any interface between two fluids of different densities it is inherently unstable intrinsically unstable or unconditionally unstable right. So that is the kind of mechanism that you will see prevailing in flames as well. So that is like a similar examples or for example you can look at like a surface of water water a surface of water with air being the other side so you now have two fluids of different densities separated by the interface. Now this interface now is susceptible to perturbation and when this perturbation happens it now starts forming waves right. So similarly flames also are actually interfaces between fluids of two different densities colder denser reactants and hotter lighter products. So you would face hydrodynamic instability as well in practice you have a combination of both and therefore it is not very unrealistic to expect plain premixed flames in the laboratory that is not because they do they are actually both thermal diffusively stable and Darius Landau stable they could be unstable but those instabilities are canceling each other alright. So where one of the instabilities is actually causing it to go away from the original position the other one is trending to actually get it back to the original position and so on. So one of them could be destabilizing the other one could be stabilizing the stabilizing force the stabilizing mechanism could overwhelm the instable unstable mechanism or if even if both of them are trying to be unstable they counteract in such a way that you now get a stable flame. So this this is possible now in the extreme case where actually you have a unstable flame and you now have this curvature keep on growing then you get into a regime called cellular flames. So this effectively leads to cellular flames where the the the the the the premixed flame is no longer planar but actually breaks down into many different cells and it now reaches a new equilibrium and a new stable equilibrium where it exists as these cells and therefore that that kind of a flame structure is what is called as a cellular flame a cellular flame is also kind of structure also observed for diffusion flames this is diffusion flame instability. So if you now begin to talk about diffusion flames under so you you you can also expect cellular flames under some conditions in fact you could you could expect that the thermal diffusive instability is predominant in diffusion flames because you are expecting diffusion effects to be important there Darius-Lando instability is not as common because you do not really see like the density jump happening as much as in a in a premixed flame. So premixed flames are more susceptible to that but in reality when you have a non-unity lowest number and you have varying density with temperature you can expect both both mechanisms to prevail to different extents. Then finally we have the situation of partially premixed flames and here there are some regimes of Lewis number greater than 1 where you get oscillatory oscillatory stable oscillatory flame that means you are you are looking at something like a triple flame or something that is more compact essentially like an edge flame so if you now look at a triple flame like this with a splitter plate and then you have fuel on this side oxidizer on that side you have a trailing diffusion flame and you have the the the fuel rich branch and the fuel lean branch there are some Lewis numbers for which this entire structure actually begins to oscillate back and forth at a constant amplitude okay and of course if you now change your Lewis number or change your damkohler number in fact the damkohler number dictates how much should be the standoff distance if you have damkohler number is effectively a indicator of the finiteness of the chemical reaction rates and so the more finite the chemical reaction rates are the or less close to infinite chemistry the damkohler number is more finite and correspondingly the standoff distance is increased and of course you even under stable conditions you will not get equilibrium flame established less than a certain damkohler number and the flame blows off that is a steady state there is a steady state the solution does not exist okay now for a steady state solution that exists this is susceptible to small perturbations for some grade some Lewis numbers greater than one where it actually undergoes oscillatory instability also so that is the case where we were talking about constant amplitude and of course the amplitude and the frequency or functions of Lewis number and damkohler number and they change continuously so that is something that and of course you can also think about the fuel fuel oxidizer ratio concentration ingredients that are that the flame is subjected to and so on so these are the parameters over which the oscillatory instability will manifest so that that is about intrinsic flame instabilities now let us look at the combustor instabilities here basically what is happening is if you now have a flame that is being perturbed the basic thing that we are looking at is what is happening to its properties and these properties mainly in this context could be the heat release rate the temperature downstream of the flame the density downstream of the flame of course all these things are related okay so the temperature rise is related to how much is the heat release and the density fall is dependent is related to the temperature rise right so if you now perturb the flame and then the flame moves around then the heat release rate undergoes a fluctuation and therefore the temperature goes undergoes undergoes a fluctuation and the density undergoes a fluctuation so when you now have these fluctuations they set out acoustic waves away from the flame and these acoustic waves will now go and get reflected at the ends of the combustor and then get reflected back and set up a standing wave and the standing wave is now going to actually fluctuate this flame further and when the flame fluctuates further then it sets out a stronger acoustic wave and so that the prevailing acoustics now begins to get amplified and this amplified acoustic wave sets up a stronger standing wave which now oscillates the flame even further and so on so now you begin to understand that the amplitude of the acoustic oscillations will keep on rising in fact the amplitudes of every oscillation that is there keeps on rising right so this is essentially the coupling of the flame with acoustics of course before I proceed further what I should say is this is the situation that I just described is something that is very simple to think about for a gas phase combustion alright that means you now have like a gaseous reactant there is gaseous fuel mixing with gaseous oxidizer may be a premix flame may be a diffusion flame right but essentially some gas phase flame that is undergoing this and then causing this situation but before before we proceed I also should point out that there are there are two ways of thinking about this depending upon what we are dealing with on the one hand you could now deal with gas phase systems gas turbine combustors gas turbine combustors could be gas phase systems or liquid phase systems there are land based gas turbines which use like natural gas for powering them or there are things like aviation gas turbines which use liquid fuels for powering them so you could have the liquid fuel as well in these and then you can also talk about liquid rocket engines liquid rocket combustors and so on but on the other hand you now also have solid rocket combustors and the way things happen in a solid rocket are slightly different from the way it happens in these so we will first discuss what happens over here and then we will go back and look at what happens here so let us look at look at the gas phase combustion for example gas phase combustors could have of course many times we are interested in gas phase combustors being premium having premix flames so you could think about a combustor in which let us say a typical geometry for us could be what is a what is a very simple geometry a simple geometry for us as far as premix flames is concerned let us suppose that we now take a bigotter if we now take a bigotter we would like to think that you can you can have a flame that is established like that so it is anchored at the edge of the bluff body and then it actually extends out we talked about this when we were dealing with flame shapes what we said was the flame is trying to trying to travel with a propagate with a turbulent velocity and then so it actually orients itself in such a way that the local normal velocity component matches the turbulent flame speed and so on but then the question is what happens when the flow velocity begins to oscillate right so in the flow bill the flow velocity begins to oscillate on top of a mean flow right so you have a mean flow but it is now going to go back and forth so why does the flow velocity oscillate to start your thinking on the loop let us suppose that there is a perturbation and then what we want to point out that this is going to actually give rise to more flow fluctuation okay so what we are saying is when you now have a flow velocity fluctuation you have a faster flow sometime and then a slower flow sometime so when you have a faster flow the flame is going to actually get more inclined and when you have a slower flow the flame tries to propagate still further in and so it is going to go like that so essentially now we think that the flame is going to flap back and forth like this right so when the flame tries to flap back and forth like this look at what is happening to the flame area so the flame area keeps fluctuating and since the flame area is along which the heat release is happening the total heat release in this region is now going to fluctuate because you had a larger area over which the heat was released versus a smaller area over which the heat was released and therefore you know going to get heat release fluctuations because of the velocity fluctuations and the heat release fluctuations now try to send out acoustic waves that amplify this oscillation and when the amplifier and when the oscillation gets amplified further this oscillation is going to become wilder right so that is the instability that we were talking about but unfortunately life is not as simple as this so what happens in reality is when you are now trying to actually send this fluctuation the flame simply does not move back and forth like this instead it actually moves more like that there is a perturbation that happens in bulk for the flow which the flame tries to match but there is also the perturbation that is being sent along the flame so there are two ways in which this perturbation is propagating one from the boundary along the flame and one in the flow field itself so depending upon how long the flame is locally the wave that is propagating along the along the flame starting from the boundary and the flow fluctuation that is happening anywhere in the anywhere at any part locally on the flame they are going to interfere constructively or destructively and big depending upon that you are now going to get a a baby pattern this now adds further to the flame area fluctuations and on top of it we have to consider now that you have a non-planar flame then we will have to consider the the flame curvature effects and the flow divergence effects on the flame propagation speed okay so there are all these other things and in reality the other problem that happens is the flame anchor point where the the it is actually being held at the flame holder that itself fluctuates because at the anchor point we went through this dynamics you now have a entrainment of non-reacting gases on the one hand and a heat loss to this and this heat loss begins to fluctuate the entrainment begins to fluctuate so where exactly the flame stops away from the burner that fluctuates and that is where that is where the the perturbation is being sent along the flame so if the flame itself were to move away and then you now send a fluctuation along the flame to constructively or destructively interfere with the bulk flow fluctuation then the flame shape gets altered correspondingly right so these things actually complicate this matter this is purely what we call as a kinematic effect the moment we are talking about the flame speed and a flame sheet and the flame is trying to balance the flow that kind of idea all the dynamics is considered within the flame we have we have long gone we have long passed we have long gone past the point where we have to worry about the dynamics of the energy balance and the species balance and so on that happens within the structure of the flame we only worry about the how the shape changes now so this is not even dynamics it is purely kinematical effects right so this these are some of the things that happen in the case of let us say premix flames similar things also happen in diffusion flames except that we do not really talk too much about it so let us suppose that you now had a simplified gas turbine construct where you now try to have a flame that goes like this and then you now adopt a Burke-Schumann approach to strictly get a diffusion flame because if you did not adopt a Burke-Schumann approach where you had a flame sheet assumption where the flame is flame sheet is coincident with the stoichiometric surface and so on all the all the ideas that we that we had then we will have a flame standoff you will have partially premix flame you will have flames that look like more like this that is that is what happens in reality but for the sake of thinking about diffusion flames let us think about like a Burke-Schumann flame and suppose that you now have a over ventilated Burke-Schumann flame and now let us suppose that we now oscillate the flow on top of a mean flow the mean flow was the one that gave rise to this flame flame shape alright but the moment you now oscillate this flow you now begin to actually have well what would you expect the first order effect of course is the first order effect as we saw earlier is if you now temporarily have a flow that is going faster then it should actually prolong the flame and for the for the other half of the cycle where the flow goes slower it should now short on the flame so the flame shape should now begin to change like this and therefore the flame area fluctuates again and the heat release is happening along the flame sheet very similar ideas to what we had just just talked about for the premix flame exists so we could we could think like that but that is a more simplistic idea what in reality happens is like what we talked about that means you also have like a a wavy structure that that goes on for this flame depending upon the frequency and the length scale of this flame and therefore you have to factor that in the and so typically what we are interested in is what is the heat release fluctuation as a ratio of the mean heat release for a given velocity fluctuation as a ratio of the mean velocity so you have now two things one is heat release fluctuation over the mean fluctuation mean heat release divided by the velocity fluctuation over the mean velocity right so this is a this is a quantity that is typically referred to as the flame transfer function or something like that and there is a lot of work that has been going on in these things essentially we are looking at what is the flame response to the oscillations and what we can understand in the gas phase flame context is these oscillations are essentially happening because of the velocity fluctuations so you do not expect a big effect for pressure fluctuations that are associated with acoustic waves it is a velocity fluctuations that are causing the havoc right so a quick fix for you would be if you if you find that there are huge oscillations that are happening is it possible for me to locate where the flame exists this the bluff body here or the flame the fuel injection point here in a place where I have what is called as a velocity node in the standing wave mode so you have a standing wave and you have velocity nodes then try to locate the flames there so that the flames are more silent right so they are not really subjected to huge velocity fluctuations no matter what the amplitude is because regardless of what the amplitude of the perturbation is the node is going to always have a zero velocity amplitude right so if you now try to locate your flames closer to the nodes of course the point is the presence of the flame itself changes the changes the the the mode the acoustic mode shape okay so you have to actually sort of dynamically chase the know the velocity node and then locate it so it is a little bit more involved than just saying that and similarly there are other problems like many times in fact one of the things that I have not talked about here is if you now have a bluff body you now have a vortex shedding that is associated with it and the vortex now begins to curl the flame and that increases or decreases the the the the flame area and these vortices are shed so when they vortices are shed they take away a part of the flame and burn them somewhere there and then a new flame is established and so on so you now have heat release oscillations that are associated with vortex shedding and now you have a new time scale in the problem relative to the water related to the vortex shedding frequency and how is this vortex shedding frequency going to relate with the natural acoustic mode of the duct that is satisfying the acoustic boundary conditions will now also begin to play a huge role in trying to dictate what the acoustic amplitudes are and the other the amplitudes of the other fluctuations so this is now getting to be quite complicated in the solid rockets on the other hand you have a very different situation what you have is now if you think about a solid rocket obviously we are talking about combustion chambers so combustor instability so we have to look at a solid rocket motor in which if you now have a propellant that is shaped like this then everything that is happening that that matters to us is within a very short distance from the from the surface of from the burning surface of the propellant and this and within this very short distance effectively if you now for example think about this region and then of course we talked about a homogeneous propellant which has like a a premixed flame or a heterogeneous propellant where you have let us say something like oxidizer particles and then you had some diffusion flame maybe some edge flame here and then a mono propellant flame there and so on the question is what is the effect of the acoustic oscillations in this chamber on this flame and here typically the way the solid propellant rocket design is done is to look for how is the burning rate of the propellant dependent on pressure okay so and usually you get a picture that looks like the R goes is P like that this is actually a log log plot okay and what we are talking about now is if I increase my pressure then I expect that the reaction rates should increase therefore the flames will now get attached closer to the surface and therefore they will send in more heat to the burning surface and then give rise to more gases that are coming out right so as the pressure locally increases we expect that the rate of rate at which gases are coming out should increase or the burning rate should increase that the instantaneous burning rate should increase and therefore it now puts in more gas into the chamber and that pressure rises the system more and if the pressure and when you now have a less pressure because of the acoustic perturbation then the burning kind of slows down and it puts less gas into the into the chamber so that relieves this and then the pressure the pressure decrease further decreases so when you now have an oscillatory pressure then the way the propellant burns could or most of the time does actually help the pressure to increase and decrease more and more. So here instead of looking at a velocity fluctuation based flame response as we did in the gas phase combustor we look at a pressure based propellant combustion response all right and here again we can now think about a response function where this is actually not in terms of heat release fluctuations and velocity fluctuations this is in terms of mass fluctuations the relative to pressure fluctuations. So the characteristic that we are using here to look at how the propellant response to the acoustics is slightly different from how we deal with in this class of problems. Finally let us now look at the system instability this again is not directly related to combustion but it affects combustion so this is mainly seen in gas turbines and liquid rockets particularly gas based gas turbines liquid rocket you can say fuel here you can say propellant here both of them together you are looking at the feed system so the best or the simplest example that we can think of is when you now have a situation actually it does not have to be fuel okay fine. So let us suppose that we have air coming in or oxidizer coming in and then you now inject your fuel on the side and your so that means in this region the fuel and air actually mix with each other and then you now have a premix flame that is set up there right. What we are now saying is that now if we have a perturbation okay this flame now perturbs and then gives rise to heat release fluctuations which now sets up acoustic wave so you now have a acoustic standing wave from here to there of course taking into account like a step change that is possible right. So in this acoustic mode where the fuel in let us actually located the pressure here is actually going to fluctuate because of the acoustics and with the pressure fluctuates you now have a ?p fluctuation for the fuel inlet and that is going to correspond to actually a fuel mass flow rate fluctuation. So when your fuel mass flow rate fluctuates for a given constant mass flow rate of air then you have what is called as equivalence ratio fluctuations so this leads to something called ? we are not even talking about the effect of u' or the velocity fluctuations we are talking about the effect of the equivalence ratio fluctuations so what basically happens you now have for one half of the cycle where the pressure is actually increasing here then you have less fuel coming in so this is likely leaner than design point okay so when it is leaner the flame now burns slower therefore it elongates but then when you go to the next half of the cycle where the pressure decreases you have more fuel that is coming in and then mixes with the air but the crucial thing is it is not an instantaneous mixing everywhere in the flow field in this inlet so you now have a fuel pocket that now convex down at the flow speed. So the question is when the flame is trying to come back if you now have a fuel pocket that arrives there with of course mixed with air and makes it less leaner then the flame will want to come back more that accentuates the instability so when we say when the fuel pocket comes here that means it has got to do with what is this distance relative to what is the velocity so there is a convective time delay for pockets of fuel to mix with air and come so the equivalence ratio now fluctuates because of the convective time delay and that can feed back into how the flame wants to flap back and forth and change its area and get along with the combustion stability because of that similarly in the case of liquid rockets typically what they do as far as design is concerned is in order to avoid something like this they have an injection injector plate which is designed to take a a delta P across the injector injector plate the pressure the injection pressure is essentially the pressure upstream of the injector plate relative to the combustion chamber pressure so it is essentially a pressure differential delta P and the way they design this is they want to have this delta P to be at least about 20 percent of the chamber pressure that means the injection pressure upstream of the injector plate should be 20 percent more than the combustion chamber pressure so much delta P so that any oscillations that are happening in the combustor is not really felt upstream now have a fairly robust high pressure over here which can push the propellants pretty much at the same rate regardless of the small fluctuations if this were only marginally more for example right then any fluctuations here will propagate upstream significantly and cause fluctuations in the flow rates of the liquid fuel and liquid oxidizer and therefore now when they mix and burn everything is going to oscillate and then that is good that could actually increase these oscillations even more and more and in fact many times where when this is not done right then these fluctuations can actually propagate all the way upstream to the tank that is where they get arrested so now your feed line has a certain acoustic characteristics anything that is confined has acoustic characteristics because of possible reflections and standing wave modes and so on right so the feed line can amplify the acoustics if it resonates with the oscillations that are there in the combustor so many times like the combustor could have something called radial modes for us the feed line could have a longitudinal mode that the same frequency and so on when these things coupled with each other then these oscillations become significant and then the whole system begins to have oscillations all over the place basically then telling us that system instability is a very big problem right but of course those are now getting into mechanical details a little bit moving away from combustion but you have to keep in mind that the heat release fluctuations in the combustion is like the primary driver for any perturbations to grow and all other things begin to couple with it. So as a matter of fact simple similarly when you now look at things like solid rockets where you have let us say what is called a segmented rockets which have inhibitors the inhibitors can now protrude into the flow when the propellant burns and you can you can now produce vortices and these vortices give rise to pressure oscillations but that is not significant many times when compared to what it can do to the propellant combustion response to be become significant so this is like a trigger and then the propellant combustion response now takes over and causes the combustion instability so many times there are a lot of other things in the system that we will go together with the combustion event once the combustion event is primarily inherently unstable or susceptible to acoustic instability. So the system instability typically is a secondary effect it is a combustion instability that we should mainly focus on trying to damp as much as we can so we do not want to get into how to damp these things just wanted to point out that just want to point out two things one such problems exist things are not as steady as what we have been going through they are inherently unsteady or they are susceptible to perturbations that is that is number one. Second we can understand many of these things based on whatever we have studied with laminar flows laminar combustion and steady state ideas and so on but the problems are lot deeper when compared to what meets I in this lecture thanks a lot.