 The next topic is finite volume method. Till now I had been taking heat transfer examples because the first topic was computational heat conduction. Second topic was computational heat advection. Third topic was computational heat convection. The earlier topics were having the word heat in the middle and I was talking of temperatures mostly because when we talk of pure diffusion and pure advection it is easy to get a feel when we talk of heat transfer because in heat transfer pure diffusion is conduction whereas when you go to momentum conservation so I was saying that in heat transfer when you talk of pure diffusion in heat transfer you have pure conduction which is a real world situation. Now when you go to the momentum transport the pure diffusion is a case where only viscous stress is acting or do you have a fluid flow situation where there is only viscous stress and there are only viscous forces there is no momentum change there is no pressure force you do not have such real world situation. So when I wanted to talk of pure diffusion I had taken an example of heat transfer because heat transfer conduction is a pure diffusion phenomena and it is a real world phenomena. When you go to pure advection it is easy to get a feel of I can set an example that you are standing between ice and fire and I can talk of the temperature which you feel but it is difficult for me to take momentum transport for pure advection and then try to explain the advection transport that was the only reason I had restricted to heat but as I started this lecture saying that there are two types of transport phenomena and as also mentioned that the transport mechanism there is quite a lot analogy between the energy transport and the momentum transport. So whatever I had discussed all that taken examples from heat transfer conduction is basically a diffusion phenomena which is equally applicable in case of momentum transport advection as well as convection I had taken for heat transfer but in momentum transport I will show that indeed they are there. So in this topic it is basically using finite volume method for full nemistoke equation especially momentum equation I want to show you which is the term analogous to conduction that is it is a diffusion term which is a term which is analogous to enthalpy transport which constitutes the advection term. So the first subtopic is introduction I will start with an introduction followed by conservation laws and govern equations in fluid mechanics and then I will discuss the finite volume discretization. Note that here I am not doing discretization starting from the govern equations I am instead I am starting from conservation law which I am applying to control volume. So we will apply a finite volume method for mass conservation continuity equation is a differential equation here we will not talk about that but here we will obtain the linear algebraic form of linear algebraic equations from mass conservation linear algebraic equations from momentum and energy conservation which will constitute unsteady advection diffusion and source term. So all the methods to get a labric equations let me tell you whether you talk in fluid mechanics or solid mechanics in almost all the books they start with this equations and then show you how to obtain linear algebraic equations which I call as mathematics based procedure whereas here I am proposing the same end result through control volume applying the conservation law called as physics based for finite volume method. So there are three types of discretization method where people start from a differential equation and obtained algebraic equation finite difference method which is one of the oldest method in computational fluid dynamics. However when people started solving the more complex problem especially the complex geometry problem they found that finite volume method is better. So nowadays this finite volume method is more popular and it is there in most of the CFD software this finite element method is more widely used in solid mechanics as compared to fluid mechanics. So this is what I had emphasized in my earlier lecture that you start from a control volume apply conservation law and in your undergraduate courses you teach that from a control volume how to obtain differential equation where you use limit delta volume tending to 0 then only you get a differential term and in a most of the CFD books they start from a differential equation. So here volume is taken as 0 now in a CFD course or a book there is a volume integration which is done. So here you can say the differentiation is done and now here integration is being done and then algebraic equation is obtained. So rather than reducing the size of the volume and then integrating it why not start with the control volume and try to reach to same algebraic equation. You can disagree with me saying that the control volume approach which I am showing you right now is good for simple geometry problems. Actually I had done this same formulation for complex geometry also however I agree that when you start from a control volume and then when you do a discretization although it is easy to follow and understand and get a feel of what is happening but it becomes lengthy whereas if you follow this mathematical approach it seems to be very compact but it is difficult to assimilate and understand. What I would suggest is that especially if we are expecting that this course needs comes as an undergraduate course where all these formulations are being taught without where we do not start from differential equation and apply a Gauss-Tiebergen's theorem and we find that in most of the colleges in our country especially in undergraduate classes the level of mathematics is low. I believe it is a good idea to use this approach starting from the same control volume which the students see in the first class of heat transfer course even in the fluid mechanics course and then using simple mathematics use two level of approximation and get come up with the same algebraic equation. Once they get a feel and understanding then I think you can go ahead with this formulation it has its own advantage I agree with it but you are feel free to take your own decision whether you want to follow this approach or follow this approach but this is something which I have been which I had felt by teaching this course not only in IIT but I taught this course in NIT also I introduced this course in around may be 99 around 14 years back and taught this course two times so with teaching experience and doing research in this area I felt that this way of teaching the student have a better feel appreciation and along with the implementation detail I had seen the students here that most of them are able do not have too much of struggle and they can develop their own codes with this experience I am proposing the proposing this formulation and suggesting it. Now let us go to conservation laws and governing equations in fluid mechanics so here you will see that now I had put this slide which is different from the way I had discussed the conservation laws in my earlier slide I because this is an important thing conservation law so it needs to be presented proposed in different ways so whichever way you feel more connected to you can take your decision in taking it forward. So what I am doing is the rate of change I am converted as we are following a Ilarian approach where are two component are steady component and an advection component this is the rate of change term the rate of change of momentum is equals to force which is a momentum source for most x momentum is the force in the x direction or one in the this the force in the rising. This is the rate of change of x momentum this is the rate of change of 1 momentum this is the rate of change of enthalpy so for rate change of momentum, it is equal to force. For rate of change of enthalpy, it is equals to heat gained by conduction and volumetric heat generation. As far as the forces are concerned, in general there are two types of forces, surface force and body force. Surface force are those which are directly proportional to surface area of the control volume like viscous force and pressure force shown here, which are expressed in terms of fluxes, viscous stresses and pressure and the body force, which could be gravitational force. Electromagnetic forces, which comes into picture when you have a, when the fluid is in gravitational field, electric and magnetic field and this is the rate of change also as two component. The rate of change inside the control volume, which is unsteady component, there is an influence outflow across the control volume, which constitutes the advection component, the viscous force and conduction heat transfer constitute the diffusion component, the pressure force, body force and volumetric heat generation constitute the source component. So, in computational fluid dynamics, we classify, in books they classify the govern equations. Here, I am classifying the conservation law, having four component, unsteady, advection, diffusion and whatever is left other than this three is called as source. So, that was for conservation law. What I am showing here is the govern equations for incompressible flow. Note that, where I had mentioned earlier that momentum equation and energy equation are basically transport equations, where mass flux acts like a driver and the adducted variable acts like a passenger, which is u velocity in the x momentum, v velocity in the y momentum equation. And if you look into this equation, you can convert it like a fill in the box problem. So, there are two boxes, small box representing the advected variable, advected and diffused variable. And in the big box, you have what is called as source term. So, the advected and diffused variable, which is in small boxes u in x momentum, v in the y momentum and temperature in case of energy equation. Whereas, in the bigger box, which is source, it is minus del p by del x in x momentum minus del p by del y in y momentum and volumetric heat generation in energy equation. So, this is the general transport equation, where we use a general variable phi, which is the advected and diffused variable. We use a constant c, which is 1 for momentum transport and which is specific heat for energy transport. And the bigger box source we represent by s of phi. So, note that, whatever discretization we had done for conduction, it is also applicable for viscous force in the, this is the viscous force in the y direction, this is the viscous force in the x direction. Note that, the conduction heat flux is expressed in terms of normal gradient of temperature. The viscous stress in the x direction is expressed in terms of normal gradient of u velocity. Viscous stress in the y direction is expressed in terms of normal gradient of v velocity. So, whatever approximation we have used to calculate those normal gradient in conduction is indeed applicable for calculating the viscous forces viscous stresses and finally viscous forces. When you go to advection we have used certain advection scheme which for heat transfer represent enthalpy flux multiplied by surface area they are indeed applicable for momentum transport across the control volume for x momentum they represent x momentum flux transport for y momentum it represent y momentum flux transport in energy it was enthalpy transport. So I hope that just before when I started this topic there is a question on that whatever I am doing right now I am doing for temperature or heat transfer but they are indeed applicable for momentum transport I will show you more details in the coming slides. So this is the grid generation which I had showed earlier let us start with finite volume method for law of conservation of mass this discretization I had not shown you earlier because this is not an unsteady term this is not an advection term this is not a diffusion term so this is different from what I had shown you earlier but the procedure which we will be using is when you want to do mass conservation it is expressed as a product of mass flux multiplied by surface area. So for law of conservation of mass is rate of change of mass inside the control volume plus rate of change of mass across the control volume for incompressible flow rate of change of mass inside the control volume comes out to be 0 we are doing here everything for incompressible flow so I am not taking that term what I am taking is rate of change of mass across the control volume which is out minus in out flux minus in flux and the mass flux is expressed in terms of velocity multiplied by normal velocity density multiplied by normal velocity now in this so there is a here also there are two levels of approximations first level of approximation is the same which we had done for earlier for heat flux enthalpy flux that is on a surface any flux will vary on a surface so we are taking its value at the centroid of the surface which we call as surface area so this is the first level of approximation the second level of approximation is to calculate the value of the normal velocity which comes into the expression of mass flow rate in terms of neighboring cell center values so for this we do not use that idea of that example ice and fire this velocity you can say that there are two velocities one which comes in the mass flux and one which comes from the advected variable in momentum transport for the advected variable in momentum transport we use that idea of that example of ice and fire but note that the normal velocity which comes in the mass flux we do not use any advection scheme here if it is needed we do by linear interpolation however tomorrow I will discuss some method where we avoid this interpolation which is called as a staggered grid otherwise this interpolation creates some problem and there is no better interpolation okay but here I would just say that maybe here you can do a linear interpolation this is second level of approximation with this we can convert this mass conservation equation if you substitute this you can get an algebraic equation so this will be complete the finite volume discretization of law of conservation of mass now what I will do is that for momentum conservation and energy conservation that is the transport equation finite volume method so although we have five minutes so I would like to take few questions before we break for lunch and I will continue with this topic tomorrow morning from 9 to 11 Mirma University Ahmedabad please ask your question sir we want to have derivation of Peclet number advection upon diffusion is there so how it is derived that we want to know it is a good question the question is the derivation of the expression for the Peclet number honestly speaking I had not seen any derivation as such for the expression it is just that let us try to understand what does rho u so based on our physical understanding we take we have called some term as advection strength the idea something like this it seems quite intuitive also we are saying that the strength of that let us say advection phenomena is basically a transport phenomena so when we talk in terms of strength of a transport process so the strength in a transport process let us say in this fluid transport momentum transport the strength depends on the strength of the driver and who is the driver in this transport momentum transport or energy transport mass flux so that is why in the advection strength is expressed in terms of the driver strength which is mass flux and the diffusion strength is expressed in terms of ratio of the diffusion coefficient to that of the length scale so as far as I know there is no perfect derivation of this equation it is just that based on the physical understanding we come up with this expression thank you Kolhapur Institute of Technology please ask a question this is a question related to convergence criteria sir the effect of convergence criteria how it will the impact of that one how it will differ in case of linear type and non-linear type of problems and another one question sir in case of variable thermal conductivity function where that coefficient is B B coefficient how to take the value of that coefficient of B for a different materials and the third question that is what do you mean by that boundary element method these are the three questions I will start there are three questions I will start in the reverse manner the third question is boundary element method boundary element method I had not talked about this I had not talked about this word but this is a method which is used by people in solid mechanics where they do as far as I know some line integral I do not have much idea about it so I will not be able to answer it people mostly in CFD do not use that however I know that some chemical engineers did for interface tracking but I do not have any answer to that question the second question is in a multi solid heat conduction problem how do we decide the value of B now the thing is that for that for the material for which you want to calculate B you have to do an experiment where you calculate K as a function of temperature and then you do a curve fitting and then find out what is the value of B other for most of the material if there is a temperature dependence you go to a data book and you can find out what is the value of B now the first question is on convergence criteria now the specific question is that if you have let us say first linear equation and second let us say non-linear equation then that convergence criteria which we define as practically 0 so what magnitude of the convergence criteria is needed for a linear problem as compared to that for a non-linear problem I would say that in both whether it is a linear problem or non-linear problem in both the cases you have to take a value let us say to start with a larger value 10 to power minus 3 get a result decrease it and do a second simulation with 10 to power minus 4 and check whether the result there is change in the result or not so for both the cases you need a value and I had not done an experiment especially as come comparing the linear and non-linear problem and come to a conclusion this is what I can say about that question thank you so we will stop here.