 Let us get started with the last session of the today's lecture, this is on finite volume method for the new stoke equations. So we started with pure conduction or pure diffusion moved on to pure advection, then we took the combination of the two, calling it as a convection. So in the last lecture we had talked about two dimension and steady state convection which is a diffusion advection phenomena. Now let us go to the name stoke equation. So in this objective of this topic, finite volume method is to show you the discretization of the full name stoke equation. I will start with the introduction followed by the finite volume method for the continuity equation and the transport equation which are momentum and energy term by term. By this time you already know that there are different types of discretization method, finite difference, finite volume and finite element and here we will discuss the finite volume method. We had already mentioned that there are two approaches to get the discretized algebraic equation. You can get this from the governing partial differential equation or you can also get it directly from the control volume from which this governing partial differential equation is derived. This is the method which I am discussing here because I feel that this is more easy for the students to understand. I call it a physics based finite volume method. Just to refresh you back about the new stoke equation, here this is a two dimensional new stoke equation for incompressible flow. This is the continuity equation, this is the x momentum, this is the y momentum and you have an edge equation. Now I have mentioned that if you look into the type of terms, basically you have unsteady term, advection term, diffusion term and I said that this is like filling the box type of problem. The small box represents the advected and the diffused variable. The big box represents the source term. In the energy equation also we have certain advected variable and source term. And in CFD we call this transport equation as having unsteady term, advection term, diffusion term and other than that whatever remains we call that as the source term. Now let me ask you a question that if you look into this equation, how many equations you have? Four equations. Continuity x momentum, y momentum and energy for two dimensional case. How many unknown variables you have? u velocity, v velocity, pressure and temperature. So now can you tell me which is the equation for u velocity, for v velocity, for temperature? Which equation is for u velocity? You have u in the other equation also, but why do you call this as an equation for u velocity? Because you have a term which gives you an idea that you can create a movie of u velocity by having this term, right? Because here you have a term which has, which represents the temporal change of x momentum and it includes temporal change of u velocity, okay? So using this time marching you can create a movie for u velocity. Using that this you can obtain the temporal evolution of v velocity and maybe you can create a velocity contour, a movie of velocity contour with respect to time. You can create movie for isotherms because you have this term here, okay? So x momentum for u, y momentum for v and energy equation for t, but so you are done with three variables. Which is the fourth variable? Pressure. So which is the equation for pressure? Tell me. You are left with one equation which you have not used and there is one variable which is left, okay? We are left with one equation and we are left with one variable. So let us say that continuity equation is an equation for pressure. This is the biggest challenge because when you look into the continuity equation there is no pressure. In fact, if you look into the history of CFD people were struggling. One of the main reason is that this continuity equation we call it as an equation for pressure, but right now you do not have any pressure in this. Somehow we have to convert this into as an equation for pressure. This gave rise to pressure correction method and so on. I hope you can understand and appreciate what I am saying. What is the main difficulty in solving this set of equations? Note that we do not have any conservation law which has a term for rate of change of pressure. This is for rate of change of x momentum, y momentum internal energy by which we can obtain temporal evolution of u velocity, v velocity, temperature respectively. But we do not have a conservation law which gives temporal evolution of pressure. In fact, this is one of the issue which causes lot of challenges in solving this system of equation. So somehow we have to convert this continuity equation as an equation for pressure. This I will discuss in more detail in the next topic where we will go to the solution methodology. But I just wanted to highlight this issue in this lecture. This finite volume method is basically discretization of advection term, unsteady term, advection term, diffusion term. It is lot of repetition. I had done this since yesterday evening. So I will do it very quickly because it is just a repetition. So I will just show it quickly. So let us suppose these are your grid points, yellow circles, interior grid points, boundary points as blue circles. Now let us start the, this is something which is slightly different as far as the discretization. Till now I had not shown you the discretization of continuity equation. I am showing this for the first time. So I will discuss it in little detail. Now what is, which law we apply? Law of conservation of mass or continuity equation. We represent it like this, but there are two levels of approximations which are involved. Let me go into it one by one. Actually in your undergraduate or let us say when you derive the continuity equation, you represent this total mass flow rate on this phase as a product of mass flux mx into delta y, mx plus delta x, small mx. So this is analogous to what I had shown for heat flux, advection, enthalpy flux and so on. The first level of approximation is the approximation which we use for fluxes. We had done this for conduction flux. We had done this for advection flux. We are doing the same thing here for mass flux. So this mass flux varies at point with respect to y here, with respect to x here, but we assume that its value is equal to the centroid value. So this is your expressing that this mx plus delta x is the value of the mass flux at centroid of the east phase multiplied by the surface area. This is surface averaging, which is the first level of approximation. Basically first level approximation is an approximation for the flux representation at a surface area at the centroid in general. The second level of approximation is that this mass flux is made up of product of density and normal velocity. So as far as this is concerned, it consists of ue. Normal velocity here is ue. So we need an approximation to calculate. Note that this type of interpolation of the velocity at the phase center I had shown you earlier in case of advection schemes also, but here let me tell you when you talk of momentum flux, there are two velocities which are involved. One is the velocity, normal velocity which contributes the mass flux and there is a second velocity which is the advected variable. So for the advected variable we use advection scheme, but whenever we want to calculate the mass flux, wherever we do, we do linear interpolation. Although this causes little trouble and there is one approach which is called as staggered grade, which I will discuss in the next topic. But I just want to highlight that when you discretize mass flux, normal velocities come and then you have to do some interpolation to calculate this mass flux at phase center in terms of neighboring cell center. Note that for this we do not use any advection scheme. Advection scheme is different. This is not, that is not used for, this is a normal velocity that is used for the advected variable. When you talk of the velocities, there are two velocities. One coming from the mass flux because when you look into the momentum flux is the product of mass flux, let us say x momentum flux is the product of mass flux into u velocity. y momentum flux is the product of mass flux into v velocity. But this mass flux also consists one velocity which is that velocity, surface normal velocity. For that surface normal velocity note that we do not use any advection scheme. We use advection scheme for the advected variable which is u velocity for x momentum, v velocity for y momentum and temperature for energy equation. To calculate the normal velocity, here I am showing you that we do linear interpolation. So, that was the discretization of the continuity. This unstudy I had talked many times. It is just the volume averaging first level of approximation. Second level of approximation is discrete representation of the temporal gradient and this is the general form of that. Advection term I had mentioned earlier that the term here, this is the mass flux multiplied by u velocity multiplied by surface area. This represents the rate at which this mass flux multiplied by surface area is mass flow rate. Mass flow rate multiplied by u velocity is what is mass multiplied by u velocity? x momentum. Mass flow rate multiplied by u velocity is rate at which x momentum is entering into the control volume from this surface and rate at which x momentum is leaving from this surface. So, it is entering from this two surface, leaving from this two surface. So, whatever is leaving you add this two minus whatever is entering you get the x momentum lost by the fluid in the control volume. Similarly, these are the for the x y momentum lost by the fluid in the control volume, this is for the internal energy lost or enteral energy or enthalpy lost by the fluid in the control volume. So, total advection I had discussed earlier that we can sum up I discussed this in pure advection as well and the advection advection flux I had mentioned time and again that it is momentum flux in momentum equation and enthalpy flux in energy equation. The first level of approximation in advection is the surface averaging to represent the fluxes at the face center this I had discussed earlier and the second level of approximation is to calculate the value of the advected variable using an advection scheme. I had discussed this in detail taking an example of temperature, but indeed that is also applicable for momentum flux. I had talked about what is advection scheme this is just a refresh of that this are the different advection scheme which we had discussed in the previous lecture and I had also mentioned that the total advection represents net x momentum, net momentum lost by the fluid in the control volume for momentum equation and net in enthalpy lost by the fluid due to advection transfer in energy equation. The diffusion is what we did in conduction for fluid flow analogous to conduction you have viscous stresses. So, whatever I am showing you are the viscous stresses in my first lecture I had shown you that for an incompressible flow although sigma x is 2 mu del u by del x sigma y x is 2 sorry mu del v by del y plus del u by del x, but we can simplify it and for an incompressible flow I had mentioned that this viscous stresses in next direction can be expressed in terms of normal gradient of u velocity. Sigma xx is in general it is 2 mu del u by del x this is mu del u by del y plus del v by del x, but in my first lecture I had shown you that out of the 2 if you take 1 and take this del v by del x and if you flag continuity equation it becomes 0. So, for an incompressible flow this is the simplified form of the viscous stresses. Now, if you look into the simplified form what you realize is here the gradient is with respect to x here it is with respect to y. Now, this x y is matching with which subscript that of the plane or that of the direction what is the first subscript first subscript of plane direction what is the direction of plane normal. So, that way I can say that viscous stresses are directly proportional is equal to expressed in terms of normal gradient of velocity viscous stresses in x direction in terms of normal gradient of u velocity viscous stresses in y direction in terms of normal gradient of v velocity and heat conduction in terms of normal gradient of temperature. So, note that this word normal gradient which I am repeating time and again. So, in CFD we have to come up with a procedure to calculate the normal gradient. Calculation of normal gradient is easy in this case in Cartesian coordinate system because in a normal gradient when you draw a normal line you get two yellow circles that is cell centers, but when it is a complex shape domain when instead of this vertical this is curved then in that case it becomes much more complicated as far as calculation of the normal gradient is concerned. So, in general in CFD to calculate the diffusion term we need normal gradient and for complex geometry we need more complex mathematics you can call say that to calculate this normal gradient, but there are methodologies to calculate the normal gradient. We discuss this in our regular course, but in this course I do not think time will permit us to do that there is some question. Sir, I am not able to understand sir that previous slide. She has said tau yx, sigma xy, there should be another term also not sir mu times dou b by dou x plus dou u by dou y. Correct. That I had done in if you look into my first lecture on introduction. That is fine sir, but here we are omitting one term y. No, no, the term which I am omitting and I am not only omitting from here I am also omitting one term from here, it was twice of this. Yes, yes. So, I have omitted one term from here, one term from here, but in my first lecture I had shown you that those two term if you do balancing it comes out to be 0 from continuity equation. I will give you the lecture slide you can go through it and if you have further question you can send me in. For incompressible flow it gets simplified that simplification procedure I had discussed in the first lecture. Any other question? This simplification actually makes things are more general or easy because the viscous stress in x direction, the viscous stress in y direction, the heat flux, all three we express in terms of normal gradients. No tangential gradients are involved. So, that is why I am intentionally taking this simplified rather than a general case, but when it is a general case we can it is not that it is we cannot handle it, we can handle it, but in this case for incompressible flow this course is limited to incompressible. So, at least I can make it more easy for you. So, total diffusion you can express in terms of normal gradient as I mentioned. I am not intentionally showing you the arrow here I had mentioned this earlier also because heat fluxes are always normal the viscous stresses in the x direction are normal in two planes and tangential in two planes. So, arrow I am not saying, but the expression remains same this is a diffusion coefficient which is mu in case of fluid flow momentum equation and conductivity in case of energy equation and this eta represent normal direction. Small d is a diffusion flux and what is this diffusion flux? This is viscous stress in momentum equation and heat conduction heat flux. It is a conduction heat flux in case of energy equation and this is the two levels of approximation for conduction term. First level surface averaging to calculate that diffusion at a phase in terms of centroid value and the second level of approximation is how to calculate the normal gradient. But piecewise linear approximation using a subsidiary law what was that subsidiary law? Newton's law of viscosity in case of momentum equation and Fourier law of heat conduction in case of energy equation. So, this is a total diffusion term. The total diffusion term represent by capital D represent net viscous forces in the x direction and y direction for momentum equation and net conduction heat transfer in case of energy equation. What is source term? Yeah this is something different. So, I am studied sorry continuity equation and pressure is something different which I had not discussed because till now we were discussing about unsteady term, advection term, diffusion term. Now, in x momentum equation we want pressure force in the x direction. So, I need to know the variation of pressure on the phases. So, there are two levels of approximations and in the y direction I need pressures in the horizontal phases. Let me go to the different levels of approximations which are involved. Like if you see pressure on this phase it is varying, it is a function of y in general. Here also it is a function of y, here is a function of x and pressure is also a flux per unit area term, force per unit area. The first level approximation is surface averaging again and representing this pressure in terms of the centroid value. And what will be the second level approximation? Expressing this pressure at phase centers in terms of neighboring stress. Note that always the clearly in this what I call as physics based finite volumeter always there are two levels of approximation and first level approximation is common. In general it is a surface averaging. You have different types of fluxes starting from conduction heat fluxes to let us say enthalpy flux, to mass flux, to pressure, to viscous stresses all this flux. When you study analytical fluid mechanics or heat transfer you express it as x, x plus delta x, y, y plus delta y where on the surface it varies. But here we are approximating as value at the centroid. So, this is in general first level of approximation. What is the second level of approximation? Second level of approximation is this fluxes when you write the expressions like conduction heat flux by Fourier law of heat conduction, viscous stresses by Newton's law of viscosity, mass flux you write as density multiplied by normal velocity, pressure force you write as pressure value of pressure. So, out of this different fluxes what you would realize is that in diffusion you need normal gradient of variable at the phase center. In all other fluxes you need value of the variable at the phase center. So, here again you need value of a flux that fluxes pressure and that we express in terms of value at the centroid of the surface. This is the first level of approximation and the second level is how to calculate pressure at phase centers in terms of neighboring cell centers. This is like a central difference linear interpolation. Actually the way I have shown you the two level of approximation for pressure is very much similar to what I had discussed for mass fluxes. And for total heat generation if there is a volumetric heat generation the expression is very simple volumetric heat generation is uniform. So, you just multiply by volume. This is a total heat generated by volumetric heat generation. So, what is the final x momentum equation? Viscuitized form is the rate of change of x momentum in discrete form. This is the total x momentum. This is the total rate at which x momentum is lost by the fluid in the control. All these terms are rate terms, note that. This is the total viscous force in the x direction. This is the total pressure force in the x direction. Similarly, in the y direction and in the energy equation rate of change of enthalpy of the fluid inside the control volume is a total enthalpy rate at which enthalpy is lost by the fluid inside the control volume. This is the total heat gained by conduction. This is the total heat gained by volumetric heat generation. So, I think I had reached almost towards the end of this lecture. Some of the things which I had discussed here, you will get, I had written one chapter in one of the edited book maybe 10 years back during my PhD. But at that time I was following the procedure which I saw in the books. It was governing equation based discretization. So, in this chapter you will get that thing, but the approximations which I am discussing here in detail you can get here also. But whatever I am teaching, things have improved a lot as compared to this chapter. But one good thing about this chapter is that the complex geometry formulation which I was saying, you can get the detail of that in this chapter. So, with this I had come to the end of this topic.