 Welcome back. We will continue our discussion of multi phase flow modeling. We are currently looking at taxonomy of multi phase flow models to look at what are the different approaches and what level of detail do they give us. So, we will start off where we left off. We started looking at phase as an entity that is separate from the discrete particles that make up that entity. So, we said each phase is governed by a unique property relationship either state relationship or and or a stress strain a stress strain rate constitutive relationship. And when we apply the different we start with the most exact model where we talk of resolving all the flow and stress fields in every phase writing a set of jump conditions to just to go from one phase to another. This is the highest level of detail we can get from a in a continuum model. So, where the water and air are both continue are this is the highest level of detail we can get. And this is typically what is employed in volume of fluid and level set methods where we want to look at every entity every drop or bubble or whatever is the multi phase flow problem on hand. And it will resolve the flow field inside every drop and every bubble. And if I write the set of the jump conditions appropriately like we said last time the I can also model break up and I can also model coalescence in some conditions ok. We will jump to the second choice and we will jump all the way to the other end of the spectrum. So, we are start we looked at exact modeling as our so called first choice. And the second choice I have here is where we look at all the way to the other end of the spectrum. And essentially which essentially involves using a single phase approach. So, if I take a spray a spray is composed of drop phase and air phase we have already defined what these terms mean. So, instead of talking of drop phase and air phase as being two separate phases we will look at the two as being as composed of one phase having one unique property relationship, but of having a variable density. So, say for example, in a certain part of the spray if the volume fraction of the drop phase is higher than the density correspondingly of that in that local part of this mixture of the two phases would be higher. I mean air is about 1 kg per meter cube and water is about 1000 kg per meter cube. So, if I have a larger volume of the in a certain space occupied by water drops or liquid drops that local region is going to look like a fluid having a density close to water or if it is if the volume fraction is close to 100 percent or if the volume I mean it never gets to 100 percent. But let us say the volume fraction is high locally that amounts to a high density fluid and in regions where the volume fraction of the drop phase is low it amounts to a region of low density. So, essentially we are replacing all the phases in the problem and in this case we are looking at a drop fluid or a drop phase and air phase and replacing the two phases together with one equivalent phase ok and that one equivalent phase has mixture properties. So, when I say properties the first thing that comes to my mind is density, viscosity of that phase are as we will see a little later on these are not trivial especially viscosity is not trivial to define. So, we will figure out a way to define it or at least we will figure out what the state of the art is in trying to define viscosity of this of this mixture phase. And then and then you can define other dependent variables like your velocity pressure etcetera at the mixture level. So, this is I have now defined a third fluid. So, we had two fluids my air phase fluid drop phase fluid and now I am defining a third fluid as a composition or a mixture of these two fluids. And we will only write balance loss mass balance and momentum balance. We will only write momentum balance for sure for only the third fluid. We will write mass balance for the first and second fluids ok. So, where we are the simplification here is that as far as momentum balance is concerned this is a single phase. But each phase has to balance its mass now I want to get the physics of how this taxonomy works to you first after I go through this we are going to look at the mathematics associated with each of these modeling choices ok. But our first objective is going to be to understand the physics of what is underlying each of these models ok. So, let us the first one was the exact approach. So, while we are talking of what is underlying the what are the assumptions underlying the modeling choice. We will also understand what are the computational requirements and what are the grid requirements for each of these modeling choices. So, let us quickly summarize the our modeling choice number one was where I choose a grid so fine that I am able to resolve the flow inside every drop including the smallest of drops ok. And I choose my time scale so small that I am able to capture the flow or all the droplet level phenomena whether it is coalescence breakup or whether it is some some kind of a circulation inside the drop whatever be it I want to capture all of that at the at I want to capture my time scale is such that I capture all of that that is my first modeling choice what we call the exact approach. The second modeling choice is a replacement of the two phases so I take the air phase I replace the two phases with a single phase. So, first let us first understand the resolution requirements the moment I use the word drop phase. The moment I use the word drop phase I am looking at a scenario where I do not see individual drops I am seeing this smeared white fluid in the case of a picture taken from a spray let us say ok. So, at what length scale and time scale would I see this smeared white fluid. So, it is like if I took a picture of a very fast moving spray I would just see these sort of streaks of whiteness at what length scale and time scale would I see those would I see that as the picture of my spray. If my length scale is larger than the largest drop first of all I cannot see if my length scale of resolution is so small that I am able to resolve any of the drops including the largest drop I am not in a at a level of picture where I am seeing this smeared into a phase. So, if the largest drop in my system is say 100 microns I cannot allow my grid to become much smaller than that 100 microns. I have to remain on the scale of 100 microns likewise on the time scale I cannot I am looking at this smeared entity. So, I do not want to go to the level where I am capturing every individual drop oscillation say for example. This is the multi phase model where you are looking at a phasic inter penetrating phasic continuum approach is only valid when I am looking at this smeared out picture ok. So, at that level I have the drop phase and the air phase and every position in the spray. So, if I take a little spray every little position in the spray has a volume fraction alpha of the drop phase and 1 minus alpha of the air phase. So, alpha D is the volume fraction of the drop phase. So, this volume fraction now becomes our defining variable in terms of telling us how much of each of the phases is in any given spatial location. So, if I know alpha D as a function of let say x and y simply to start with we will leave time out for a moment it really does not make any difference. If I look at this entire spray alpha D takes on a different value between 0 and 1 at different points in the in the spray. Now, I do not really know this red line spray is like a smeared out entity. So, in I know alpha D in some domain including the spray ok. Now, potentially alpha D here is very small is very close to 0 because there are hardly any drops in that part, but if I take a domain I know alpha D as a function of x and y in there and I can define a mixture density I mean it is just a marker right you can have alpha 1, alpha 2, alpha 3 if I have 10 phases also each one of the volume fractions has a bound 0 to 1. If I have a 3 phase problem I have at least 2 independent volume fractions alpha 1 plus alpha 2 plus alpha 3 equal to 1 plus rho A times 1 minus alpha D where rho A and rho D are the material densities ok. These are actual properties of the material that are measurable with any sort of a let us say water density can be measured by a hydrometer air density can be measured by other means. So, these are measured material properties, but my single phase which is a composition of the air phase plus the drop phase. Now, has a new property called mixture density in this mixture density I can write as rho D alpha D plus rho A times 1 minus alpha D for the case of a 2 phase problem. So, this rho M now becomes a function of alpha D ok. So, if alpha D is a new state property of my system. So, alpha D is different at different points and that is a property just like let us say temperature in a heat transfer problem alpha D is a property that I know at every point in my system and locally at every point I know rho as a function of alpha. So, this in some sense is rho is another property alpha is another property these 2 are related through this kind of a relationship this is looking a lot like our state property relationships are more more specifically like our ideal gas law. Ideal gas law relates pressure to temperature through density pressure density and temperature here is a simple way by which the density of my single fluid is related to another property of that fluid called alpha ok. So, I can now define other properties also similarly I do not want to go into that with before we go into more detailed mathematics associated with this model. But essentially what we have done is we have replaced these 2 fluids or we have replaced the spray with one continuous with one continuum fluid ok. Let us say I call this the single phase fluid, but you may call it the spray. This spray has a multiple has variable properties at different parts of the spray alpha D is different in different parts of the domain and rho the density of this is different at different parts of the domain. So, this is got all the look and feel of a compressible fluid that is essentially if you get down to the mathematics any sort of a multi phase problem modeled as a mixture essentially becomes a continuous continuum problem of a continuous fluid of a compressible fluid. So, once we do this I have replaced all of the properties I have replaced the air plus the drops and everything with the single fluid. Now I just have to solve my balance laws for that one single fluid ok. So, that is essentially you know where we solve a mass and momentum balance for that single fluid, but in trying to solve for the mass balance of that single fluid I have to account for the fact that I have 2 phases that are composed that make up this mixture and that I have a mass balance of the mixture, but the mass balance of the mixture is exactly the same as individual mass balances of the 2 phases separately ok. So, we will see what this means in terms of the mathematics a little later on, but for now we have replaced our fully complicated droplet laden flow problem as though it is a single phase with a variable density variable viscosity variable other properties, but what it means is by in doing the single phase single phase sort of assumption the single phase has a unique velocity at every point just like a single phase fluid flow problem ok. Now in all this discussion we are going to ignore that these ignore any kind of turbulence related effects that is a vast area in itself. I just want to introduce the modeling complexities and the modeling taxonomy in the context of laminar flows. So, if I have a laminar flow in a pipe single fluid say water flowing in a pipe I know the velocity uniquely at every point inside the pipe just like that if I know the single phase flow I know the velocity uniquely at every point and that velocity is the velocity of this single phase and that single phase being composed of the air phase and the drop phase means that the air and the drop phases are moving at every point in this domain with the same velocity ok. So, as far as our modeling assumptions are concerned this is the most restrictive in some sense that is that we are saying in writing this mixture model that at every point in my domain the air phase and the drop phase are moving with the same velocity that is the only way under which I can only manner in which I can make this assumption that I am dealing with a single fluid ok. So, before you embark on any kind of a modeling exercise you have to first understand the assumptions and see if those assumptions are valid for the physical system you are trying to model. So, we will look at that either later today or in the next class ok. So, this is our second choice we will come back and revisit the mathematics of each of these choices later on. The third choice is what we will call the particle ballistics approach. So, like for example, I know that I have a spray I will draw this one more time and I know that it is composed of drops and each particle is on its own. So, I can write for each particle I can write an equation like m x double dot equals sum of all forces. So, if m d is the mass of that drop I can write an equation that governs each particles motion of this form. This forces what are all the forces that could act on this particle one the drag force from the surrounding air. Two if there are other particles colliding with this particle and pushing it or creating an impulse force creating any kind of an additional force to accelerate or decelerate this in this my one particle under consideration ok or I could have other external forces gravity some sort of an electromagnetic force if I am looking at other effects etcetera. So, the idea is I take one particle and I know the set of forces acting on that particle alone I can write a set of equations for that particle and I write each I write an equation like this for every particle that is in my domain. As soon as I do that I have as many equations as I have a number of particles in the in the spray. So, typically again we go back to this number let us say 10 power 9 particles I have to write 10 power 9 such equations and solve them ok. Now at some level this is still intractable, but there are some assumptions that underlie that that are used to simplify this kind of a modeling approach where instead of solving 10 power 9 equations for 10 power 9 particles there are ways to group them into packets this is a term that is often used in this Lagrangian particle tracking literature where these packets are now allowed to propagate forward in time ok. So, a packet is not one drop, but sort of a collection of drops. So, the reason I do not I mean what I want to do here is understand the different limiting cases and I can always do sort of a combination of these models. So, a packet is like a volume fraction that is being advected by itself under the action of other external forces ok. So, this is like a phasic assumption super laid over a particle ballistics model ok. So, you can do all of these other combinations and every combination gives rise to a slight slightly different view of the spray, but what I want to do is restrict myself to where we know the limiting models the different exact and approximate models and how the other models arise out of combinations of this. So, essentially the heart of this kind of a modeling approach is the extent of this particle particle interaction. So, I am not looking at the packet approach that I discussed earlier I am looking at the individual particle level approach and at the individual particle level approach like I said each particle is on its own each particle is influenced by the flow field around it and possibly even vice versa. So, if I have considered the case of one particle sort of moving in a stationary medium. So, that one particle just like let us say a cricket ball thrown from the from the out field from near the boundary is on its way to the wicket keeper and the particle is influenced by the air, but the air is also influenced by the particle by the ball. So, the balls motion after the ball has completed its flight from the fielder to the wicket keeper you still have some remnant motion of air somewhere along its path ok. So, if I was to throw successive balls from the fielder straight to the keeper along the same trajectory the succession of balls do not take on do not experience the same force as the first ball thrown ok. This is essentially the meaning of the vice versa interaction that is the particle influences the air which in turn could influence other particles slightly differently ok. So, if I am interested in where the particles are going why am I interested in this vice versa interaction because that altered air field could affect the particles slightly differently than the first particle ok. So, I can write a set of equations for the particles of this form and the air phase is now modeled as a single phase fluid, but each particle is a source of momentum if it is moving and giving up some of its momentum. So, this is essentially the physics of how particle tracking works both one way coupling. So, that is each particle is influenced by the flow field alone would be called one way coupled model one way coupled particle tracking modeling. If you allow the vice versa thing to happen also that would be a two way coupled particle tracking model ok. I do not want to go into that go into this kind of a modeling approach in any more detail only because of the following reasons. If I want to look at the particle level model just like I have discussed it again I keep going back to my length scale time scale requirements at what length scale and what time scale can I say I can write a equation like m x double dot for that particle equal to sum of forces. First I need to be able to see individual particles right. If I do not see individual particles I cannot write it. What I mean by see is if I took an image of the spray at some resolution I can see individual drops at some spatial and temporal resolution we have already discussed this. Likewise I have to have a grid fine enough where I am seeing individual particles or rather fine enough that I am seeing individual particles but course enough that I am not going into every particle ok same goes with the photography requirement they are exactly analogous. So, if I now take this m x double dot equal to sum of forces the grid on the air phase associated with writing a model like this for the particle phase is such that I am able to see individual particles but not a collection as a fluid at the scale at which I am resolving and typically this is only valid far away from the nozzle where I can actually see individual particles. So, if I have a spray nozzle in the dense spray region very close to the nozzle the requirements the resolution requirements would be very stringent for me to be able to see individual particles. So, the region where a model like this is tractable is far away from that dense region where I have gotten into a region where particles are in individual flight particles or drops are in individual flight. So, this is in the rarefied spray region that this kind of a model is accurate and tractable ok but in most spray applications while this model is easy to implement and actually very useful in most spray applications especially if I am interested in the dense spray physics something like this would not be very useful. But if I am if it is sufficient for me to know where the particles are added after all the particles are formed and after the particles are in this rarefied region rarefied spray regime then a modeling like this would be quite ok. So, these are the sort of underlying assumptions we may talk of the mathematics of this a little later on but for now we will say we will we will go over to the fourth choice. The fourth choice is where I am looking at each phase as a continuum. So, here I have the air phase and the drop phase we have already defined these terms. So, the moment I write drop phase you know the sort of resolution at which you have to take this picture you also know the grid requirement it is at a level where I do not see individual drops I am only seeing this smeared fluid both temporally and spatially. So, this is a requirement this is a model the fourth choice is a model where each phase is allowed to be modeled as a continuum except that these two continua or interpenetrating that is at every point I have this alpha d a marker or a volume fraction that tells me the fraction of the region fraction of the spatial region that is occupied by the drop phase and 1 minus alpha d as the fraction of that spatial region that is occupied by air. And the again like I said that the dynamics of each particle are not important that is that is the whole idea of going to this phasic level view of my spray and a continuum velocity and stress field is postulated for each phase this is the distinction. If you go back to our mixture level or a single fluid level approach we said that each that the two fluids together are a mixture are mixed into a single fluid and you are only defining a velocity for that single fluid whereas in this fourth approach we are defining a velocity and a stress field for each of the two phases in this case we are only dealing with two phases air phase in the drop phase we are allowing each of the two phases to be moving with its own velocity field and we are not restricting them to be locally homogeneous locally moving with the same velocity. So what this actually does is now the drop phase can go where it wants to go and the air phase can go where it wants to go. So if I take a spray for example say for if I take a region here the drops probably want to go in this direction but the air may want to go in a slightly different direction. The reason for this is actually due to entrainment. So the air as the spray is spray is like a jet and the jet entrains air from the outside and essentially this is like a representative streamline and these representative streamlines mean that the air at this point may want to come in towards the middle of the spray but the drops may want to go in a slightly different direction. In our locally homogeneous flow assumption underlying the mixture model we are not going to be able to capture these two distinct velocity fields we are only going to get sort of a mixture level velocity field which may look like that which may look like that okay. So if I know for a fact that different parts of my spray have drops moving very differently from the air phase and I have reason to believe that we will see a little later on how to ascertain whether this is indeed the case before doing the modeling okay. So like if I have four choices for modeling we want to see without doing the modeling or without doing all four models which one is appropriate and we will see that model in choice calculations a little later but for now this allows each of the two phases to be moving with a different velocity okay. This is what are often called two phase models or also called Eulerian Eulerian approach Eulerian Eulerian models okay. So now I have a stress field the stress is of course the pressure field as well as the shear stress field. So I have both the normal and the shear components of the stress that I have that are included. Now for the drop phase what does this pressure component mean so if you think of just the drops without the air okay what does pressure mean in this drop phase. Let us be first very clear it is not the hydrodynamic pressure inside every drop because we have discarded that right up front we said the moment I define a drop phase I am not focused on individual drops I am in this mirrored out region. So when you put in this drop phase essentially at this level the drops may not be liquid drops they may be steel balls I have absolutely no problem thinking of just having steel balls as being the entities making up my drop phase. What does pressure mean in these drop phases in this drop phase is essentially is like a congestion of drops in one spatial region that causes a certain normal stress. So if I have locally at some point in the domain a high collision frequency of these drops high normal that results in like a normal stress in the drop phase that is what we mean by normal stress in this in this drop phase. It is not easy to imagine this kind of a normal stress in the context of liquid drops making up the drop phase but if you think of these as granular objects like think of these as really steel balls it is a little easier to understand that I have a lot of I have a collection of these steel balls that tend to are somehow compressed and the microscopic properties inside the balls are causing like a macroscopic stress in this mirrored out picture. The microscopic stress in each of the individual balls is resulting in some microscopic stress in this mirrored out phasic level picture. What does shear stress mean for the drop phase? Again if the drops are not at all interacting that is a case for no shear stress because the in the at the phasic level. So even though if I have drops that are not at all colliding or very low probabilities of collision then the shear stress of this phase is essentially very low because the whole idea of shear stress is where you immediately think of a simple quiet flow experiment. If I put this drop fluid in the in between two parallel plates and move one parallel one of the move the top plate what is the force required to move the top plate? If the drops are hardly colliding the force required to move the top plate would be zero. The bottom plate would have to somehow communicate its presence to the top plate for the force on the top force to move the top plate to become non-zero that only happens by this momentum diffusion process which happens through collisions. So if I have very low probabilities of collision that means the top plate can be moved at zero force which means it is a case of an inviscid fluid. So this idea this idea of viscosity of this drop phase is related to the collision frequency and the collision frequency as you can see is related to the overall volume fraction. So if the volume fraction itself is low they are not exactly related but you can imagine that if the volume fractions are low collision frequencies may go down but I can think of a case where the volume fraction is low but the collision frequency could be high it just depends on the local velocity non-homogeneity that exists at the droplet level. But for now all I want to stress is that this velocity field is at the phasic level. So this is the mean velocity of a collection of drops of this smeared fluid collection of drops that make up this smeared fluid and the stress field is also a property associated with this collection of drops at this phase level. So essentially if I have the phase level description I can define a normal stress and the shear stress of this continuum and that is related to the microscopic collision probabilities and collision frequencies. So these are this is basically what we would call a two phase model and this is of the also referred to in the literature as Eulerian-Eulerian modeling approach. We will stop here and we will continue our discussion on how to choose between these four models in the next class.