 Good morning, welcome back. We are going to continue our discussion of secondary atomization to see what are the important parameters that govern this process. We went through two different kinds of analysis at the end of the last lecture. One using force scaling analysis and another using Buckingham pie theorem to give us a set of dimensionless groups that can be used to characterize the behavior of droplets in an air field, in a gas field. What we are going to do is start from there and say I could have the important dimensionless groups governing secondary atomization are we have a Weber number which we showed and we are going to like we said at the end of the last lecture that there are many combinations of these dimensionless groups. Weber number, capillary number, Reynolds number and you can see Onozoerga number which is formed by a combination of these Weber number and Reynolds number. These are all possible candidates for characterizing the behavior of drops in a gas field. So, typically for secondary atomization we found that this Weber number and Onozoerga number defined based on the liquid properties seem to play a big role and seem to be used in the literature to characterize the to characterize the regimes in which these drops break up. We will look at that in just a moment. So now just to sort of summarize Weber number as this group shows is it represents the competition and restorative surface tension force. Onozoerga number represents the relative importance of mu l and sigma. So essentially in this formulation the in the way the Onozoerga number is written here the liquid viscosity is trying to damp out any oscillations while surface tension is trying to cause the break up of the drops. So, we want to understand the different roles that these that the same physical properties surface tension place in the two in the two different phenomena. Let us take a typical example and see what these numbers look like. So let us take an example of a raindrop falling. So let us say D is about 1 mm rho a I am only going to do an order of magnitude analysis. So I will start by writing D is about 10 power minus 3 meters rho is about 1 kg per meter cube rho a. So we will assume for a moment that the drop has reached its terminal velocity condition roughly about 10 meters per second this is sort of an overestimation. We will see what this number looks like and sigma for water is about 10 power minus 1 Newton per meter approximately it is a it is about between 10 power minus 2 and 10 power minus 1. So given this the Weber number an order of magnitude of the Weber number. So the order of magnitude of approximately a 1 mm raindrop falling through air at a terminal velocity of about 10 meters per second gives you a Weber number of about 1. What it means is that the aerodynamic force. So if I take a raindrop falling at 10 meters per second the question I am asking is this likely to break up and if I have this air essentially in the frame of reference fixed around the drop there is air moving around it. So if I was to do this very sort of a rigorous fluid dynamic analysis this drop would have to deform into something like that that may not be it that it may further deform by flattening out into a pancake and if this were to happen this process continues till it breaks up. So let me draw those separately if I take a sphere initially under some air flow after some deformation this may become something like a pancake and the moment you have this sort of a flattening out this process is only going to be accelerated further until you start to get some kind of a break up. You can sort of imagine this and we will see more details on this in a moment but what the calculation that we just did showing us that the value of Weber number being order of magnitude 1 means this aerodynamic force that is trying to deform the drop and the surface tension force which is trying to bring it into a sphere are very close to being of the same order of magnitude. So the competition keeps the drop in a nearly spherical shape at this condition. So in other words if the aerodynamic force is much greater than the surface tension force say for example if I had a larger drop that would cause the Weber number to be higher or if I have a lower surface tension fluid that would cause the Weber number to be higher under those situations the drop is more likely to deform. Let us do the onus organ number calculation for the same fluid for the same situation onus organ number requires mu L which is about 10 power minus 3 kg per meter per second rho L which is about 10 power 3 kg per meter cube d which is 10 power minus 3 and sigma which is about 10 power minus 1 onus organ number for this would be mu L divided by square root rho L d sigma which in this case is about 10 power minus 3 divided by square root 10 power 3 into 10 power minus 3 into 10 power minus 1 which is on the order of 10 power minus 3 to 10 power minus 4 somewhere in there. What this shows is that the role of viscosity in damping out oscillations is very small because the onus organ number is on the order of 10 power minus 3 to 10 power minus 4 what it means is if the drop were to go into oscillations those oscillations would be sustained for a fairly long period of time. So imagine a real raindrop falling you would expect if you took images of this raindrop in the center of in the frame of reference fixed with the raindrop the drop would also show an oscillation it would not just be a fixed sphere because the onus organ number is very small the onus organ number was very large on or at least order of magnitude 1 I would expect the drop to be nearly spherical as it is coming down. So this is a situation of about order 1 Weber number and onus organ number being order 10 power minus 3 etc. So let us see what these really mean in the case of a spray and so we will first look at what Weber number means in the case of a spray. This is a simple classification from the work by Guilden Becher et al. from a paper titled secondary atomization what they show is that there are several different regimes of Weber number under which one could expect the single droplet to behave differently at over the range of Weber numbers. So the first and simplest is where the Weber number is between 0 to 10 let us say over order 11 is what they say this is the case where you would expect a drop. So in the vibrational mode I would expect a drop to undergo some kind of a deformation so initially it could be a small deformation so this leads into this and this in turn leads into this. Now this is not a process by which the drop is essentially going from going sort of quasi statically from a nice spherical drop into a dumbbell and then breaking up it is a vibrational break up mode that is this drop is undergoing a set of vibrations but those vibrations are amplifying in amplitude are increasing in amplitude. So the in other words I may have this vibration initially small but because the air is feeding energy into this vibrating drop this amplitude of the vibration could increase and one could imagine very simplistically that if the amplitude of vibration becomes on the order of the radius of the drop then you have a situation where from here you come to something that looks like that and it could pinch off. I could have a nice spherical drop but because of the modes of vibration of this drop it could undergo a break up by amplification of the vibrational amplitude. So and that is pictorially shown in this first part so essentially if you imagine in this particular instance there is very little preferential direction of the air motion so for example one would have to imagine in all these cases that the drop is introduced into an air stream the drop is introduced into an air stream of some velocity u and that gives rise to the Weber number basically. So initially the drop is at rest and you let go of it you have this relative velocity between the drop and the air that causes this kind of a behavior if your Weber number is very small. If your Weber number is slightly larger then the kinematics of what you expect would happen starting from here is that there is an initial deformation of this drop in the middle so you have an air that causes this nice and spherical drop to deform into what looks like a jelly bead and from there the progression is towards what looks like a bag so I will just draw this arrow to here just to I do not want to redraw the same figure. So this middle part could elongate with a thick rim on the outside here something like that so this is a thin sheet of liquid with an axis symmetric rim now you will have to imagine this is axis symmetric so now these are all sort of you know schematic representations but they are actually based on experimental observations so they are fairly accurate. So what happens at this point is that this drop that is being stretched out into this thin rim into a rim plus a thin sheet of liquid attaching to the rim the thin sheet of liquid is now looking more like a bubble which is sustained by this slightly higher pressure on the inside and that shatters into a bunch of drops so these drops you see are coming from the sheet breaking up and then these larger drops that you see are coming from the rim breaking up so you can straight away see that if my drop in the initial state so this is my initial state is in the Weber number range say about 10 or 11 to about 35 I am going to get this kind of a break up of the single drop and which means in each drop break up I get two different physics coming into play one due to the sheet breaking up and that sheet break up gives rise to very small fragments of drops and the second the rim breaking up now in our earlier lectures we discussed linear instability analysis. So if I take you know and we looked at the linear instability of a cylindrical column in I can look at the linear instability of a toroidal ring so if I were to create a liquid toroidal ring which is what this is and one can make these observations you know from drops breaking up that you can actually see a toroidal ring form and if I create a liquid toroidal ring and do a linear instability calculation of that liquid toroidal ring there is a certain preferential break up mode that will show up so this is my liquid toroidal ring there is a certain azimuthal mode let us say 2 pi by m is this theta. So there is a certain m which is preferred so there is a particular value of m that will have the maximum growth rate it is just like any other prefer it is like any other linear instability problem where a certain mode is likely to grow faster than others and that particular value of m now tells you what the so if I look at the rim again the particular value of m that you see causing the break up of this rim determines the number and size of the drops form from the break up of the rim. So you have two different physical phenomena causing the break up causing the final result impacting the final result in the case of the back break up process one the sheet which is this liquid sheet breaking up to give these fine fragments and then the rim breaking up to give you the larger drops and the result is that in this particular case you are more likely to get a bimodal distribution of droplets from a single droplet break up element because they are originating from two different physical processes. In the first vibrational break up you are more likely to see a unimodal break up that is the preferred break up process in the vibrational mode is for each drop to break up into two halves the preferred break up process does not mean it exactly splits into half but if I were to look at a population of drops at that same number condition then on the average I expect the drops to produce the mean diameter of the daughter drops produced from secondary process to be about half the size of the primary drop half the size in mass or you know when you do the calculation you will find what the scaling is in the diameter to direct. Now from there I will skip this multi mode because that requires a little more thought but we will move into sheet thinning now if you there is a supporting video to go with this paper that you can go to this journal website and look at where they have videos of this process so what I am describing is actually from an experiment from high speed imaging and experimental observations that you can see for yourself. The sheet thinning process also starts with the same initial condition so a drop of some diameter d at rest in a wind of velocity u what do I expect will happen in this if the Weber number is slightly higher and now in this range the first deformation process from here is where I start to see a thin rim form on the side so this is where the side of the drop is being stretched forward as opposed to the middle of the drop being pushed forward which was the case in the back break up mode. The natural progression from here on is where you create a very thin long stretch sheet and so it is almost like I am shearing drops of the sides this is more like our high Weber number diesel or jet atomizer and so I create this thin elongated liquid sheet breaks up into finer drops which are shown here and a few large drops from the break up of this disc like structure so I still may end up with what looks like a bimodal distribution but it is significantly more concentrated number wise in the smaller drop range. So the mean drop size of all the daughter drops formed from the break up events of a population of parent drops is going to be smaller as the Weber number increases that is the expectation and that is what is observed the last break up process is what is often called a catastrophic break up process which is where the Weber number is greater than about 300 and something. And this is actually very difficult to investigate and there is not much information available on how drops behave at this situation in fact the word catastrophic is descriptor of nothing more than lack of information lack of our information about this regime. But there is some experimental evidence of drop sizes that will form but the specific physics of how the drop breaks up at this high Weber number is something that is I think not completely understood as well understood as the others. Now this middle region that we skip as we went forward is called a multi mode break up regime which is where you have a transition from the middle part being stretched being pushed like we said the back break up so just to show so it is being this idea of the middle being stretched into like a balloon to the sides being stretched into a thin sheet. So the multi mode break up shows a little bit of both and therefore you see that this is often called the bag and stem break up so you have a stem to replace what would be expected to be seen from the sheet break up regime but you also have the bag from the back break up regime forming. So and this is again this gives rise to essentially bimodal distributions or in some instances you know other modes also but remember we are only talking of these distributions themselves in a statistical sense so the distributions themselves are formed from varying from just superposition of the different physical processes like the back breaking up and the rim breaking so we talked about that as an example. Now in all these processes we will have to physically understand how this is happening so like the way to sort of imagine this all of these kinds of stretching elongation and then break up processes are happening in time that is the drop should have enough time for one of these processes to take the atomization process forward. Take for example the sheet break up in the sheet thinning regime you have to have sufficient time to create velocity gradients inside the drop to stretch the drop out into a thin sheet and the time over which the drop is stretched into a thin sheet is much smaller than the time over which the drop can retract back into a spherical drop that is so you can think of this Weber number as being a combination of as being a competition between the time scales as well that is the viscous time scale or the surface tension time scale which is trying to restore this drop back into a spherical shape is much longer than the aerodynamic or the viscous time scale or aerodynamic time scale which is stretching this drop out into this thin sheet. So one process is so much faster than the other that it does not have time to catch up so that is the meaning of the Weber number. So what faith did in 1995 and this is sort of a collation of a lot of work that preceded faith structure and break up properties of space paper in the international journal of multi phase flow what faith did is to parse all this information into a regime chart. So we will start from this corner because it is the easiest to understand I believe if I have a an order one Weber number. So this is Weber number order one so this is our raindrop that we discussed earlier Weber number wise but instead of raining water if it was raining glycerin which is let us say a thousand times more viscous then this is essentially where our glycerin rain would fall order one on the owner's order number and order one on the Weber number and what this chart tells us is that under those circumstances the drop would hardly deform so the deformation of this drop would be less than 5 percent. So it would oscillate but the net deformation would be less than 5 percent of the radius so there is no expectation of any atomization. You can go towards higher and higher Weber numbers and even if you have somewhere a drop that is at a very high Weber number but correspondingly high owner's order number mind you these axis are all logarithmic. So a high owner's order number let us say a owner's order number about 10 power 3 would allow for a very high Weber number and still a stable drop. So if I want to create a nice and stable drop I have to have a high owner's order number even if I have a for every given Weber number. So let us just write down what these are so we know what we are meaning. Here is that diameter of the drop D occurs in the denominator in the owner's order number and in the numerator in the Weber number that means for the same fluid mu l rho l sigma and the air density being the same and u being the same the smaller drops tend to have a higher owner's order number and a lower Weber number. So we do not have to necessarily look at different property fluids even in the same spray some drops in other words smaller drops will tend to be in this corner here and the larger drops would tend to gravitate towards the other parts of this graph. So let us take our own little rain drop where we said the owner's order number is about 10 power minus 3 a Weber number is about 1 so this is where we were. This is our typical rain drop the deformation is expected to be between 5 to 10 percent but that means that it is likely to oscillate with the net oscillation amplitude being limited to about 10 percent. So no break up again as you go higher and higher up on the Weber number oscillatory deformation starts to occur when your Weber number is on the order of about 10 this is what we call vibrational break up mode in the previous slide and if I look at further increase in if you look go back let us say between 11 and 35 you expect a back break up and you can imagine what 11 and 35 would look like on the on a log scale this region over which one would expect back break up. Is that on the Weber number scale as I keep going up I transition into multi mode break up and then this shear or sheet thinning break up notice how so essentially the limits that we discussed in the previous slide correspond reasonably well with an axis if I were to draw the axis right through here. So these let us say start of the oscillatory break till right there then the back break up multi mode and then above that you have a little bit of sheet thinning and further up is your catastrophic break up. So these correspond reasonably well with the values that we discussed in the previous slide but as I increase my on a zorg number clearly these values are not going to be the same you can see where if I have on a zorg number order 1 these values for oscillatory deformation is now much higher back break up multi mode and etc. So the values that Gildenbecher et al gave in their paper are mostly for very low viscosity fluid which is typical of any sort of a fuel like gasoline or diesel or aviation kerosene these are all low viscosity fuels and in a given spray on a zorg number is expected to be usually small on less than order 1 for less than order 1 drops you expect that the sheet that the limits are what Gildenbecher et al the limits given by Gildenbecher et al will work but for larger on a zorg numbers 1 would have to look at the complete regime chart from faith. So this as you can see from the class of liquids that were studied this kind of information where experiments from various sources are all collated on to 1 graph is called a regime chart a regime chart regime chart like this is very useful because for my given situation I know what to expect in the what to expect for my behavior now at a given point in the spray I may have wide range of drops and these drops may be travelling with a range of velocities we have looked at joint pdf of size and velocity and when you have a pdf of size and velocity 1 can imagine that you actually have a pdf of Weber numbers and on a zorg numbers. So I can take the dimensional information from let say pdpa data that we looked at before and parse it into Weber number on a zorg number joint pdf. So at a given point in the spray given spatial location in the spray I have a whole distribution of Weber numbers and on a zorg numbers and these drops are now likely to break up by varying physics at the same point in the spray so the question is how do I model this break up process so I will say take 1 or 2 very trivial examples and then lead into something that is probably more applicable to a real situation the simplest model is what is called a Taylor analogy break up model. I can only start talking of modeling this process only after I know what the physical what the physics looks like so I do a bunch of experimental measurements parse it into a regime chart and now I am ready to look at modeling this process. So if I take the simplest model is actually the vibrational break up where I have this drop leading into sort of a dumbbell like mode and this dumbbell like mode is likely to break up due to oscillations. We will continue this in the next class and look at other models as well along with Taylor analogy break up.