 I welcome all of you to the fourth day of the lecture on series of lecture on computational fluid dynamics. The first two days were taken by professor Puranic where he had laid down the foundation for fluid mechanics before we get started with the CFD. I started what you can call as hardcore CFD with an introduction followed by grid generation and then we had a lab session in the afternoon which consisted of problems in grid generation. There were basically two types of problem, algebraic grid generation and elliptic grid generation. In the elliptic we had further two types of grids one where it was a grid generation in an inclined plate and the second we had a square plate with a circular hole and there were different types of grid generation and I had also mentioned if you go through the problem set carefully that O type, C type, H type can be applied to some specific problems and towards the end the last lecture yesterday was on computational heat conduction where I had shown you that we use two levels of approximations and we start from a control volume and end up with the same elliptic equation which you see in most of the CFD books where they follow a mathematical approach of the derivation where they start with the governing equation and take the volume integral apply Gauss divergence theorem and end up with the same elliptic equation. What I had thought today is that let me start with whatever I had thought in the yesterday lecture. I will first start with grid generation what is the key idea in grid generation because I could realize that yesterday many of you had certain confusion as far as grid generation is concerned and especially this is a 5 day workshop finally we will get 10 day workshop. So whatever I am delivering now I have double the time to deliver and it is the final thing so I am little fast I understand but I am also trying to cut some syllabus based on the yesterday's feedback. At least I am trying to although initially I thought that I will try to cover almost 100% but based on the yesterday's feedback or I could see that I am going little fast. So I had cut down some syllabus but at least I am trying to cover 80 to 90% of the total syllabus. So it will be fast but I am trying to be little slow to at least judge your understanding and then proceed. So I will start today's lecture explaining the things which I had done. I will take two three pages to explain this. So I will do it here so that I go slow and you understand each and everything clearly. So let us start with grid generation what was the key idea in grid generation. Let us suppose you want to generate grid for flow across a car. So I will take a car model. Let us suppose I take a 2D car model. Let us suppose I represent the car as this is a simple car model and let us suppose typically the height of this domain is 20. The upstream length, this we call as the upstream length and this we call as the downstream length. This is the upstream length, this is the downstream length. So let us suppose this is 10 and this is 20. So total length is 30. I have to generate grid on this. I solve note that we solve Navier-Stokes equation in this domain. This is which type of domain? This is physical domain using finite volume method. Note that this is a physical domain. This is the job which you have to do. So the first thing when you want to solve a CFD problem you have to generate grid. For that first you will create a geometry. So this is the geometry and the domain which you will create. Now what I had taught yesterday is that now my job is to generate grid and I had mentioned what is grid generation? Right now this domain has two boundaries. So the fluid is lying between the two boundaries. Which boundaries are there? Outer square and the inner solid surface. So there is inner boundary, there is an outer boundary and the fluid is in between two. So I have two boundaries which are not connected, simply connected. Now grid generation means I have to divide into certain fixed number of grid points. And I said that in structured grid generation analogous to series of horizontal and vertical line in Cartian coordinate system we have curvilinear grids here. And let us suppose here I want to have grids of 101 by 101 which will make it 10,201 grid points. So I want this many grid points. Let me tell you the grid points which I mean here will be the intersection of the two family of lines. And the methodology which we will be using is we will use the finite difference method for grid generation. And in finite difference method I have mentioned that the grid points are defined at the intersection of the grid lines. So I will have two family of lines xi and zeta. There will be 101 xi line, there 101 zeta line. I will start with the computational domain. My computational domain is this is my computational domain. Note that this I am using only for grid generation. It has nothing to do with final solution procedure. Final solution we will do it in this domain. Note that. So in computational domain this is only used for grid generation. We have xi zeta coordinate system and the xi is 1 by 1. Now if I want 101 by 101 grid points then I will draw 101 equispaced horizontal and vertical lines. I would suggest that you can note down because this is something which will help you in understanding later on. So we draw 101 equispaced horizontal 101 equispaced vertical lines. Once we draw that we get 10201 intersection or you can also call those as grid points. Is that clear? Now the third step is you can calculate delta xi as what is the length in the xi direction 1. If you draw equispaced if the total line vertical lines are 101 then the spacing will be 100 equispacing. So it will be 1 divided by 100 which is 0.01 and delta zeta will be 1 divided by 100 again. Is that clear? Once you know delta xi delta zeta you can write a very small program because xi will vary in the x direction as 0, 0.01, 0.02 till 1. So you can determine so fourth step will be determine xi and zeta at 10201 points. Is that clear? Fourth step will be map the boundary of the computational domain to the boundary of the x direction sorry or okay you can write it also to the boundary of the physical domain. How do we do that? The physical domain is not simply connected. So we have to use branch cut. I may use a branch cut like this. So I will start may be probably a, b as this. Then let us suppose I will go like this and come back and may be my c is also here. Let me put it as like this a is at the top, b is at the top. So I go around, come back, c comes here back again, d comes here then d to a goes like this. So I have to make a branch cut so that I connect the inner boundary with the scar surface to the outer boundary which is the outer boundary of the domain. This is my branch cut which I had shown by blinking line in the yesterday's slide so that there is one to one correspondence between the computational domain and the boundary of the computational domain and boundary of the physical domain. Is that clear? Then as a user you have to make a decision how many number of points you want here. You do not know the dimension of the car. You know the dimension of the domain and once you make a decision the number of divisions which you will make you can know the x and y coordinates. So once you map it properly then you calculate x and y coordinates at the boundary of the physical domain, map it to boundary of computational domain. So let us suppose if I decide that between a and b I will have four points. So there will be one to one correspondence between the points here and there. So with this you know x and y coordinates on the boundary. You know xi and zeta at all the, I had mentioned here, you know xi and zeta after you have gone through this step at all the 10201 points you know xi and zeta. With this step you know at the boundary points at least what is x and y. Once you know so what you know xi and zeta at all points and second thing x and y at boundary points this is what you know. So let us suppose you use some black box. You give this as an input and what you get as an output is x and y at interior points and what is this black box? This is an equation which you solve by finite difference method. What is this equation? It is definitely a PDE but you may also need to use algebraic equation for clustering. Because to solve a PDE you need the dependent variable that for independent variable which is xi and zeta at all the grid points which is known here and you need boundary condition for the dependent variable which is x and y. Note that this equation is such PDE is such that x and y are your unknowns and xi and zeta are your independent variables. x and y is dependent variable. Any question on this? Feel free to tell me if you do not understand this thing. If I have missed anything to explain you can feel free to tell me. Now let me go to the second. Please use my case. I am sorry but can you please elaborate on point number 4? Point number 3 is a question on elaborating point number 4. What is point number 4? The boundary of the computational domain to the boundary of the physical domain. Actually I wanted to write this, map the boundary of the physical domain to the boundary of the competition. By mistake I wrote it but I continued. But actually what I mean is that you start with the physical domain and you have to map. So this boundary of the cars it should lie at the boundary of the competition. In this physical domain we have inner boundary, outer boundary. Both the boundaries should lie on the boundary of the competition. That is what I mean by mapping. If you want both to lie then they need to be simply connected. Because this is simply connected, computational domain boundary is simply connected. Whether that answers your question? That is what I mean by mapping. That we do by using a branch cut because in computational domain boundary is simply connected. So to make the physical domain simply connected I need to draw a branch cut. Such as I said A to B is this, B to C is complete rectangle, then C to D is this, then D to A is goes over the car surface. Are there any rules for this procedure for branch cuts? Or it is our convenience? Question is procedures for branch cuts. I discussed different types of branch cuts yesterday. Let us start from there. O type, C type and H type. And I had mentioned that normally depending upon the flow problems we know we have some idea based on our understanding in fluid mechanics what will be the pattern of the stream line. And based on the pattern of the stream line if your grid lines, one of the grid line either Xi or Zeta is along the stream line then it is considered to be a good quality of grid. So this branch cut or which part of the domain I should map with which part, this is decided based on the family of line which is generated by O, C and H type. Any other question? This can be done directly without mapping. Is it possible? You can do unstructured grid also. For structured grid this is widely used in all the software, this type of method. So you can come up with a new method but this is commonly used and this is what we want to do. And this inner boundary that shape, geometry how to get the equation from the car outer profile. This is given, actually what happens is that whenever you want to do a CFD modeling the first part is solid modeling. So those geometrical information is given by solid modeling. See now in the grid we have done mapping of both the boundaries of physical domain and the computational domain. And for getting the interior mapping we are going to use elliptic equation. Correct. So that means boundary condition, here we have got a directly boundary condition we are going to have. Yes. And in the last class yesterday problem you told one is human boundary condition. Yes. That is the case, how do you know this boundary mapping, how will you do? We start with a, let me first pose this question in simpler way. What he is asking is that right now I am saying that here I had written that we map the boundary. So we had written the second point we know x and y at the boundary points. So this x and y the way we are giving it is a Dirichlet boundary condition. Correct. So now he is asking that if I give this Dirichlet boundary condition I will get a non-orthogonal, the grid lines will be non-orthogonal at the boundary. I had mentioned in the yesterday lecture and in the problem sets which we had given to you in the second problem we had used that orthogonality condition at the boundary and we had shown you that the grid lines are indeed hitting normally at the inclined wall of that parallelogram shaped plate. So his question is that if I am using Dirichlet boundary condition how can I ensure the orthogonality? Is it correct? Yes. So the answer to this is that we start the first iteration with this Dirichlet boundary condition. But if we want to ensure orthogonality there is a separate Neumann boundary condition which I had shown in the yesterday lecture slide. We have to implement that. So when you use the Neumann boundary condition after each time step the boundary point x and y will change. Actually I intentionally avoided it due to there are more important things which I will do later on. In the final workshop I will definitely include. I will discuss that. Any other question? While preparing this competition domain now in this particular problem we have taken 101 number of horizontal and vertical lines. So is there any limit that or how to decide this number? Let me tell you CFD is not like any software. It needs some intelligence of the user. So this decision of the grid the user has to decide. Because even if you use the software it will ask you how many grid lines you want. It will not automatically say because what happens is that and that is this something the software people or any developer cannot decide. There are certain things which a user has to decide. So this is one decision which you have to make based on your understanding that what is the complexity of the problem which you are solving. Whether you are solving let us say flow across a circular cylinder or flow across a car or flow across an aircraft. And based on that you have some you propose some domain size and based on the domain size you try to get an estimate maybe from analytical fluid mechanics that near to the wall what should be the grid size. And based on that you do some rough calculation and may you come up with some grid numbers. So that way I said just I gave some number 101 first. I gave some number but this may not be the right one. You may have to do a separate grid independence study. Maybe 1000 by 1000 also you try. So there is separate I will show you have a test problem in the tomorrow's in the afternoon lab session we have a grid independence test problem also. So then you will understand that it is just we start with some grid number and anyway we have to do grid independence study to make sure that our results are accurate. Yes because of by increasing the grid points. Is there any effect on your results? It may have it may not have. You have to run it then only. I cannot say that it will have or no. Yes or no it is decided only if you run it. Because CFT is not so intelligent that it can say. Can I just start? Sure. See the comment I would like to make is exactly what he was just talking about. Usually you have to do a grid independence study when you are doing these CFT simulations and more often than not I can tell you that the results will actually change when you are doing the grid independence study. So a priori meaning beforehand you start you may not have a good idea as to what good a grid you want. So you start with something as exactly what he said and keep on refining it until a certain level that you realize that the results are not really changing much. That's one part of it. Other part of it is what he had already pointed out but I will just reinforce that which is you look at the physical situation and let us say that you are trying to resolve a boundary layer situation for example. So there is a flow going past a solid surface and you are actually interested in knowing the shear stress distribution on the surface. So in that case what is important is that you want to capture the boundary layer profile correctly which means that perhaps you will have to do a grid refinement close to the surface and then away from the surface it doesn't have to be as dense. So such things are obviously based on some experience and that is why I think what he is saying is that right at the beginning you cannot really decide that this is what it is. It has to be an iterative process in some sense. Thanks. Good. I am happy that Professor Puranik is also pitching in to the questions because I may give an answer in some with some flavor he may have a different flavor of. So I would suggest that you can keep interrupting whenever you need. Sir, can you suggest some good reference book for grid generation? There are two books from which I had learned. So that is what I can suggest. One by Professor Sundarajan. There is a chapter by Professor Sundarajan in a book on computational fluid flow and heat transfer. This is an edited book. So various chapters are written by various faculties of different ideas. So in this book Professor Sundarajan has written one chapter on grid generation which I felt is very good. And there is a full book on grid generation by Thompson, Wartsey and Martin. Sorry if I am wrong in naming them. But Thompson is the first name. In fact maybe you can get the soft copy of this book also. It is a full book on grid generation. Yes, actually that is right. Thompson and Wartsey's book is considered to be the most standard book for grid generation. I do not remember. I will find out and let you know. But also I will add one more book to that list and it is called Computational Fluid Dynamics for Engineers by Hoffman and Chiang. There are three volumes actually, 1, 2 and 3. The first volume last chapter is on grid generation and that is also very well written. I forgot to mention actually some of the material which I have taken especially the algebraic grid that I had learnt from that book only. Hoffman and Chiang. H-O-F-F-M-A-N-C-H-I-N-G Unfortunately perhaps that book is not easily available in India. But some libraries I am very sure will have it. Algebraic grids whatever I am teaching here I have taken from the questions from there. Thank you. So now any other questions? Is there any restriction on the size of the computational domain with respect to the physical domain? This question is size of the computational domain. So I think the answer is similar to what was the previous question. The previous question was grid independence. Now you are asking question on domain independence. So the answer is similar to what was there. Here also we take a domain size. We are not sure it is good enough. We need to increase the domain size. Let us say in this problem I started with 30 by 20. Then I increase it to 40 to 30. Length is 30 in this case and height is 20 for the physical domain. So to know that if I generate a result whether this domain height, the upstream length, downstream length are good enough. If you run you will not be sure. So you run one more case. With length you increase to 40, height you increase from 20 to 30. Then run it again, get the results like how much difference is in the results. So if the difference is negligible then you can say this is good enough. So this is what is called as domain size independent study. So you ask the size of the computational domain with respect to the physical domain. Computational domain size does not change. It will always be one by one. Whatever may be the physical domain size it is independent of that. Do we have to do any Jacobian transformation when we do mapping? When you do the elliptic equation, when you do at that time it comes. I think what your Jacobian you are saying is that from if you look into the book of J. D. Anderson, there is a Jacobian which he shows. But actually in that what he does is that he does not solve in this physical domain. He solves in the computational domain. So that is why there it comes, Jacobians comes. But here what we are saying is that that is not commonly used nowadays. What is used is that once we get a grid here, forget about this computational domain and this is our domain in which we solve. Sir, I am tempted to ask one more question. It may be a premature question also. No, no. This governing equations we are going to solve on the physical domain now. See for example, we have got other systems. Say for example, we have got the X, Y, Cartesian grid system. In this we are going to have some two curvilinear systems. So when you discretize, we are going to follow any, there is going to be any difference or it is going to be the same discretization technique. Discretization technique what I discussed in the class, finite difference method yesterday lecture. Same technique is used. Even for curvilinear also you can have east wall, west wall like that. Yeah. But velocity, see if we take u velocity, it will not be in the u direction. See for example, suppose at the east phase u velocity, then we are going to resolve into two. So let me pose this question. What is asking is that if I have a curvilinear grid. So let us suppose I have a grid, one grid I am drawing. This is one line, this is one line. So this grid if you see, it is not aligned horizontally, vertically or any standard coordinate system. So we need to come up with a finite volume formulation for this type of control volume. This is called as complex geometry formulation, which teach this to our regular course. But here in this course, I do not think I will have time to teach this. Okay, but there is a formulation which is there. Okay, I will stop here because I understand because right now we are having 50 people. In the final workshop, we may have around 1500 or 2000 people. And then there is a really people start asking questions. We do not want to discourage asking questions. But after a point we have to make sure that we cover syllabus within time also. So let me just warn you. Myself and Professor Pranik, we have taken note of it. And at some point of time, we will say that you can have further question through Moodle. But in live interaction, we can do question answer till certain time only. Okay, but feel free to ask your questions. Let us leave the judgment to us when to stop. I have Professor Pranik address this. Okay, so let us go to the next thing. Now what I am thinking yesterday, Professor Rupesh Shah from NIT Surat had suggested that if I can show the discreet difference between the discretization of finite difference, finite volume, because people have certain confusion what is the difference between finite difference and finite volume. And what I thought is that I take one example where finally, whether you use a finite difference or finite volume, you end up with same algebraic equation. Okay, so with that idea in mind, I added one more thing is that when I say finite volume, right now the way I had proposed there are two types of finite volume. One is what I call as equation based and second what I call as control volume based. So what I will do is that now I will show you the formulation of the discretization for finite difference, finite volume and finite volume of two types. So let us start with that. Okay, discretization. We have three types of discretization techniques. Let me divide this paper into three parts. Here I will discuss finite difference method. Here finite volume method, governing equation based. And here finite volume method, control volume based. I had taught to mostly in this way and if you look into all the CFD books, they taught each in this way. And the idea is we start with the governing equation. With governing equation we use. Here I will take a governing equation which is two-dimensional unsteady state heat conduction with no volumetric heat. Volumetric heat generation is very easy to discretize. So I am not discussing that. So what is unsteady state? Two-dimensional rho Cp del capital T by del small t is equals to k del square T by del x square plus del square T by del y square. Let us discretize this using finite difference method and two different types of finite volume method. Let us first start with finite difference method. From finite difference method, the way we discretize is let us start with unsteady term. This would discretize as rho Cp. For del capital T by del small t, we use a forward difference key. So it becomes t i comma j to the power n plus 1 minus t i comma j to the power n divided by delta t. For the diffusion term we discretize as k into t i plus 1 comma j minus 2 t i comma j plus t i minus 1 comma j divided by delta x square plus t i comma j plus 1 minus 2 t i comma j plus t i comma j minus 1 divided by delta y square. This is which type of scheme? Central. What is the order of accuracy? Second order. So this is the discretization. What I will do is that to show that this equation is same what we get in finite volume method. In finite volume method what we do? We do volume integral. What is this equation? Right now this equation is per unit volume. Govern equation is per unit volume. In finite difference method we do not do volume integral. So this final discretize equation is also per unit volume. Let us do one thing to show that this finite difference equation is same as finite volume. Let us multiply this discretize equation with volume. So multiplying by volume, what is the volume in 2 d? Delta x delta y. So here I will get this will be multiplied by delta x or let us say delta volume I would write. And when you multiply this by delta x delta y what you will get? A into t i plus 1 minus plus 1 minus 2 t i plus t i minus 1 divided by delta x into delta y. Is this okay? Plus p j plus 1 minus p i comma j sorry there is a 2 here. Plus p i comma j minus 1 divided by delta y into delta x. Is this okay? Why I am multiplying? Because in finite volume method we do volume integral. So to show that this equation is same as finite volume I have multiplied by a volume. Just to show that however in finite difference method we solve this equation. Anyway the right hand side is 0 both side we are multiplying by volume so it does not matter left side as well as right side. Now let us go to finite volume method which I had taught in the previous lecture. What we do? We have used 2 levels of approximation. The idea which we use here is we take the control volume, Cartesian control volume of line delta x delta y. This is the grid point p this is the neighboring grid point e this is the neighboring grid point w. There is a neighboring grid point n and let us say this is the neighboring grid point s. We have defined the width of the control volume delta x delta y. This distance we have defined as delta s e I had defined but it can be also called as delta x e. This can be called as delta x w. This can the distance between the north and e can be delta y north and the south it can be delta y south. So these are the geometrical parameters which will come into the expression. And the way we started is let us first start with the unsteady term. The unsteady terms comes out to be how do we do unsteady term? We start with the volume integral rho c p del capital T by del small t d v. And in this volume integral density can specific it if you take it is not function of temperature then anyway it is uniform in the volume. So anyway we can take this outside the integral. But this rate of change of temperature within the volume we have done a second order approximation. And we have assumed that this is approximated as rho c p del capital T by del small t at the centroid of the control volume into delta v p. Is this ok? So for this we have to use the first order approximation. So for this we have first or first level of approximation. And then we use second level of approximation to represent this as rho c p t p n plus 1 minus t p n divided by delta t into delta v p. Is this ok? What is the second level of approximation? Discrete representation of the temperature gradient at the centroid of the control volume. Now let us look into the finite difference equation and this equation. Whether they are same they are exactly same ok. If you use the governing equation finite volume based governing equation. Let me tell you the same this same idea is used here also. So this unsteady term treatment is what I discussed here is the same here. There is no difference into that. There also it is done in this way only. Professor You can correct me if I know ok. So this is same exactly same here. So unsteady term here comes out to be approximated. Unsteady term approximated as rho c p t p n plus 1 minus t p n divided by delta t into volume. This is the same for unsteady term. You can see that this term this term and this term they are same ok. Now let us go to the next term diffusion term. How do we do here? Diffusion term what is the first level of approximation? We say that the heat flux let us say q x here varies on this surface. But we want to represent as one point value. So we do surface averaging. So we say that we will represent it by centroid. This is centroid this is centroid this is centroid. So that way we represent this as and then we say that q actually we take direction of heat transfer in the positive direction. So the way you do in under gadget heat transfer force q is on the west face q is going inside and east face it is coming out. And here we apply the conservation law. What is the conservation law? This represents the rate of change of internal energy and on the right hand side we have total heat gained by conduction. So when you talk of gained by conduction gain is in minus out. So this west face and south face heat is going in. So we express it as this is again approximation q w minus q e into delta y delta y is the surface area of the vertical surface plus q n minus q s into delta x. This is coming from first level of approximation that q x q x plus d x q y q y plus d y it varies over the surface. So this is the surface averaging we do for the heat flux representation one point value represents. And what is the second level of approximation? Second level of approximation is discrete representation of Fourier law of heat conduction. Where we represent q w s minus k T p minus T w divided by delta x e and q e s minus k T e minus T p minus k T p minus sorry T e minus T p divided by delta x w. Similarly you can do for q n and q s. Now we substitute this to this equation from the second level of approximation whatever equation we have got in the first level of approximation we substitute. Once you substitute you will get this equation exactly. I am not showing you because it is just a substitution and checking it out. And I wanted to finish in this one page. So you substitute this discrete representation of Fourier law of heat conduction into this equation and you will see that you will get you indeed get this equation. You can do the algebra on your own. It is just a substitution and checking it out. This is the control volume based. What is done in equation based which you see in all the CFD books? What they do is that they start with vector form because it is a mathematical mathematics based. So they start with a vector form k del square T and they take the volume integral. Then they apply what they call as Gauss divergence theorem. Using Gauss divergence theorem they convert the volume integral to surface integral. But then they convert what happens? This is del T dot n. What is this n? It is a surface normal unit vector into d s. Actually this can be converted into actually this surface is simply connected surface which has consist of four surface. So this surface integral over all the four surface can be converted into two vertical and two horizontal surface. Now as far as this unit vector n is concerned these are surface outward normal. This normal we take as surface outward normal. Let us suppose i is the unit vector in the x direction and j is in the y direction. So which one is? So you will get a plus minus sign. So this will be in the plus i direction. This is minus i direction. This is plus j direction and this is minus j direction. So if you substitute this if you calculate del dot n here you will get positive sign. Here you will get negative sign. So what you get is is equals to k del T by del x minus del T by del x on east face sorry plus del T by del x on east face minus del T by del x on west face plus k del T by del y on north face and del T by del y on south face. Sorry I am not multiplying by area. There should be an area term here. Actually this is also an approximation. Why this is an approximation? Because here I am introducing this east face center, west face center, north face center and south face center and actually this has to be multiplied by surface area and the surface area here is delta y and the surface area here is delta x. Is that clear? Note that when we go from here to here there is an approximation. This surface variation you can convert this into four different integral exactly. There is no issue into that. There is no approximation in that. But each of those four surface integral when you convert into face center values there is an approximation. And then it is dT by dx e is approximated by using another east face linear approximation minus T e minus T p divided by delta x e minus of T p minus T w divided by delta x w into delta y plus k into T n minus T p divided by delta y n minus of T p minus T s divided by delta y s into delta x. This again you can see compare these two. Actually this will what will happen to this? This I can further write as minus k T e minus 2 T p plus T w divided by if delta x e and delta if you have uniform grid, uniform grid point is delta x e and delta x w is equals to delta x. And again this will come out to be T n minus 2 T p plus T s divided by delta y into delta x. Now compare this with this. Sorry there is a plus sign here. Here there was a plus sign. Compare these two. They are exactly same. So with this example you can get the difference between the finite difference and the two different types of finite volume method. So each one each person can have its own perception. So you one may feel that this finite volume method is good. It is short. I feel that this is easy to teach to the students. I may be wrong also. So whatever you feel you make your own judgment. But these are the two different procedure. Sir in finite difference technique you multiply it by delta v on both the sides. That is like I feel in order to compare with the finite element you sorry finite control volume you multiply. Otherwise it is not required. Yeah it is not required. I am multiplying only to compare and show that. Yeah compare and show. Sir and the second thing is the third method is a special case of second method. Actually more. Yeah yeah yeah. It is the same actually. If you look it is I am not using Gauss-Tierbacher's theorem. Yeah. No without Gauss-Tierbacher without Gauss-Tierbacher's theorem also we can integrate volume integral. We have to do on the control volume. Yeah yeah yeah. Yeah yeah. And I think that the second method is more general method. Yeah it is more general. I agree. I perfectly agree. It is more general because when you go to complex geometry formulation you can use lot of mathematical concepts. Yeah yeah. But first for first course in CFD I felt that this is at least to get started the student because you say that you are having CFD courses and the students are not taking because it is too mathematical. So it is just an alternative approach. It is up to you to decide. There is I will not say this is good or this is bad or. Okay. So we have spent almost one time one sorry one hour before we are starting the next lecture. But I think this one hour interaction might have cleared lot of doubt for you.