 Sir, my question is regarding topic number 1 slide number 73. In that slide we have taken a red spot for the field of study that is how do we decide upon the distance between the inlet and that red spot of study cylinder. So, the question is this is a circular cylinder shown here by red circle. How do we decide the position of this inlet boundary? I would like to mention that whenever we have to set up this position of this wall, let us say this left wall, right wall, we have to do a study which is called as a domain length independent study. In computational fluid dynamics there are different studies which we have to do. I will go back to that first slide of mind where I have taken the code from Malware, Einstein. A theory is something which nobody believes except the person proposing a theory. So, to propose that your results which you have obtained is correct, you have to do a various types of study. So, this is one type of study which is called as domain length independent study. You have to do grid independent study. You have to do time independent study if it is a transient problem. So, here to answer to your question what should be the position of this? This we decide, this varies from problem to problem. So, we have to do what we call as the domain length independent study. So, we take let us suppose we have a particular position, we do one simulation, then we increase the length and we start with let us say minimum size of this outer rectangle and we keep increasing it and we keep obtaining the results. Like in this case the results could be engineering parameters, lift force and drag force. So, you can start with a smaller size of this rectangle and keep increasing it in some let us say elemental increase and for each increase you obtain the engineering parameter and try to see how much is the difference between the results if you keep increasing. And finally, what happens is that this difference should asymptote, this difference should die down and that is what is called as the domain independent lengths of the domain. So, this is what is called domain length independent study. So, let us suppose inlet boundary is at 5 unit upstream, then you do one simulation at 7 unit upstream, then you do at 10 unit upstream and if you say that the results between 7 and 10 is negligible as compared to between 5 and 7, then you can say that 7 is good enough. Thank you. Thank you sir, we have one more question. Yeah please go ahead. Good evening sir. Sir, what is the difference between finite volume method and finite difference method? Any similarity or any advantages is that in finite volume method? The question is what is the difference between finite difference method and finite volume method? If you go back to the history of development of computational fluid dynamics, finite difference method was is the first method which was used in computational fluid dynamics. However, it was found that if you want to use finite difference for complex geometry problems, complex geometry problems are those problems in which case the boundary is not aligned along the standard coordinate system like Cartian coordinate or a cylindrical coordinate. So, in complex geometry problems, if you use finite difference method, people were struggling to solve the new stoke equations. So, this finite volume method was proposed which makes sure that the conservation laws are obeyed to a larger extent. So, and the basis of the finite difference method is a Taylor series expansion and in the basis of the finite volume method is that we start from the conservation law and apply to the control volume. So, this is much more accurate as far as the conservation law way is concerned. So, this is the in principle definition which I can highlight and application wise also finite volume is easily able to formulate for the complex geometry problem and most of the softwares nowadays are finite volume based. Thank you. Sir, if if unstructured grid which method we can go for? Question is in an unstructured grid, whether we use finite difference or finite volume. The answer is yes, we use finite volume method because finite difference method we cannot use in case of unstructured. Sir, if one more problem is there if you give in finite difference method then unstructured grid is beneficial for us. The question is whether I can give or talk about the unstructured grid. I am sorry to say that the syllabus of the course right now it is the first level course on computational fluid dynamics. The unstructured grid solver development is a much more advanced level. So, in this course it could not be possible. However, I look forward in future we can have an advanced course on computational fluid dynamics where we can take this. Sir, one more question sir. Good evening sir. Sir, my question is from exact solution slide number seven. Sir, actually here we applied the fully developed condition assuming that the boundaries are subjected to constant wall heat flex. Suppose let me consider a constant temperature boundary conditions where generally we will assume the fully developed condition as dou T dou x equal to zero assuming that the bulk mean temperature of the fluid will be equal to the temperature of the walls. And we came to know that the heat transformation rate from that point will be equal to zero. So, is there any such conclusion which we can draw on from the constant wall heat flex case by applying the thermally fully developed condition? So, the question is about the thermally fully developed condition that we have employed here for the constant heat flux situation. Let me actually point out that the implicit assumption in these problems is that there is a non-zero heat transfer from the wall to the fluid and therefore the surface temperature will be necessarily different than the mean bulk temperature. If it turns out that if surface temperature is going to be equal to the bulk temperature then of course things are going to be that there is no heat transfer. And I think you pointed out that in the case of constant wall temperature what ends up happening if I recall correctly is that the eventually the mean temperature will tend to the surface temperature so that the difference between the surface temperature and the mean temperature bulk mean temperature will keep on decreasing and will reach to zero thereby no heat transfer. In the case of constant heat flux situation actually that does not occur. What ends up happening is there is a constant temperature difference that is established between the wall temperature and the bulk temperature and this constant temperature difference remains as you move along the x direction downstream. So, there is always going to be that non-zero heat transfer maintained from the surface to the fluid which is actually the boundary condition that you are imposing that is the non-zero heat flux at the wall. Let Professor Sharma also say something on this so if you do not mind waiting for a minute. I would just like to add that in constant heat flux boundary condition what happens is that dT by dx at any horizontal level at any constant wire comes out to be a constant quantity. So, in constant wall temperature del theta by del x is equals to zero when theta is the bulk mean temperature where in this case dT by dx comes out to be a constant quantity. Thank you. Yes sir. One more question sir from. Sir actually we are getting the exact solutions for the assuming the flow is thermally and hydrodynamically fully developed. Can we able to get analytical solutions where the flow is hydrodynamically as well as thermally developing where we have a different kind of situation if I am right. Yeah so the question is we have worked out a solution here where the flow is hydrodynamically fully developed and thermally developing sorry both is fully developed I am sorry and the question is if we can if we can obtain an analytical solution when both the hydrodynamical as well as thermally developing flow is considered. So, one step between these two would be a situation where the flow is hydrodynamically developed but is thermally developing and that is what is popularly called in the literature as a grades problem and the grades problem is analytically worked out and you obtain a series solution for the grades problem. So, that is a situation where hydrodynamically developed but thermally developing flow is considered. It is much more involved solution in the sense that an advanced technique that we talked about the kind of technique namely the series solution is what is employed and you will find that in standard convection heat transfer books. I think I had written a couple of names during my lectures on convection heat transfer books and you will find that. I do not think the hydrodynamically as well as thermally developing situation is completely solved analytically at least I have not seen it. Let me ask professor Sharma if he has come across the solution of both hydrodynamically as well as thermally developing flow. No, I had also not encountered that and I do not think it is possible to obtain analytical solution for it. Thank you. Okay. Thank you sir. Thank you sir. KIT please go ahead. Hello sir, I have two questions to Puranik sir. One question is when the particle experiences some different kind of energy transactions and also that moments then is there any general considerations to couple these term to that momentum and that energy equations. If the any general considerations are there please elaborate that general considerations and under one question is there sir and can you repeat the difference between under relaxation and over relaxation technique sir. Yeah, so there are two questions. One is if we can talk about the under relaxation and over relaxation a little bit again. Let me take that question first. Yeah, so the concept of under relaxation or over relaxation is always employed with iterative methods of solutions. So remember the iterative method that we discussed. The iterative method is essentially you start with a guess solution field in the domain and then you keep on improving on that guess field using an equation such as what we had used here in the in the case of that two dimensional steady state situation. But fundamentally the idea of the under relaxation is described on our over relaxation also is described on slide number 17 in the finite different solution. So what we say is that we consider the change in the solution that we obtain through one iterative step and add that to the solution that we had available with us at the beginning of that iterative step and thereby you obtain the solution at the end of the present iterative step. That's the way to physically understand how the iteration is going. So if we if we are let us say if we are achieving a certain amount of change in the solution just for the sake of discussion through one iteration step. If we if we are permitting only a fraction of this change to be added to the to the solution that was available to us at the beginning of the iteration we will call this situation as under relaxing. Whereas if you are permitting larger than 100% change that we have available with us to be added to the solution that we have at the end of previous iteration step then we call that over relaxation. So this is how I actually would like to typically interpret this under relaxation and over relaxation from a purely physical understanding point of view. What ends up happening is that later on when you complete the discussion on Navier-Stokes solutions etc. you will realize that many times there is an iterative process of solution involved there and in the in the Navier-Stokes equations which are non-linear and coupled partial differential equations you will see that the requirement will always be using under relaxation because otherwise the solution will go out of control. In a very simple situation like what we described and worked on the lab yesterday this steady state temperature distribution it's a very simple equation in terms of its mathematical behavior also and here the over relaxation seems to also function well but in general for the fluid flow equations it's not going to work. So that's about your first or rather the second question. The first question was actually related to the governing equations in fact and if we have taken into account all factors that are affecting a fluid particle that's the way at least I interpret the question as the particle is moving whether we have taken into account the energy interactions whether we have taken into account any other interactions such as moments etc. And in fact all of that is built in to the governing equations which we have come up with. If you want to look at the moments on a differential fluid element we actually if you remember we actually took counterclockwise moments of the surface stresses which then were only in terms of the shear stresses about the center point for that elemental fluid mass that we were talking about. And we in fact obtained a result through some sort of a discussion that the result of that momentum angular momentum equation so to say for a differential fluid element is that the shear stresses occur in pairs. So tau ij was equal to tau ji is what the conclusion then that we obtain. Also the energy interactions occurring for a fluid particle are taken into account when we when we derived the energy equation. Only thing is as we were discussing earlier also in this Q and I session some of the details of the energy equations were omitted as part of the simplification. So the only part that was omitted was essentially the effect of viscous forces on the energy transfers and that is the only part that was omitted under the assumption that we are dealing with simple low speed flow situations where the action of viscous forces in the energy interactions is negligible. But otherwise all interactions are built into our governing equations without any trouble. So thanks thanks for that. There is another question sir. For Sharma sir there is another question and it is on topic number 2 slide number 29. What stability requirement exactly means to us? About stability requirement. What the stability requirement exactly means? The idea here is that if you want to understand that when you do an unsteady computation you march in time. What happens is that as time progresses there is some discretization error which is there at each time step. Now the idea is the error what happens to the growth of this error with respect to time. If the error grows with respect to time then what happens is that this temperature value approaches towards infinity and this is what we call as blowing up of the solution. So there is a stability analysis which we do more on a numerical analysis course and if you want to look into the details I would suggest you to look into the book computational fluid dynamics by J. D. Anderson where he has shown how we derive this equation. So this mathematically using von Neumann criteria we derive this expression which and this expression gives an expression that what is the minimum time step which we should use in an explicit method and this is a function of the grid size and the thermo physical property which is thermal diffusivity. So for more reference I would suggest you to follow the book. At this moment I can say that this expression comes by a mathematical procedure and to calculate the minimum time step which we should use. Maximum time step sorry in an explicit method. Thank you. Hello sir another one question from this side. Sir at the boundary the variables are highly sensitive that means what can we take care of that one in case of grid generation sir. The question is that the boundary variables are very sensitive can we take care in grid generation whatever I could understand from this question what happens in fluid mechanics is that the maximum gradient or let us suppose most of the action in fluid mechanics occurs near to the solid surface. So near to the solid surface especially if it is a turbulent flow there is a certain minimum grid size we should have near to the wall to capture the flow phenomena accurately near to the wall. So in grid generation the answer to your question is yes we have to take into account that the grid size should be small enough near to the solid objects. Thank you. Nitaminakshi Bangalore please ask your question. So regarding finite difference solution 7 how the spatial coordinates varies for the stretching and next question is the stability criteria is exactly equals to 0.5 in that case which method we have to prefer. The question is on slide number 7 of finite differencing and let me answer the stability part first here the stability condition is shown as the kinematic viscosity times the time step divided by the grid spacing squared should be less than half. In fact just before this question the stability criterion was being discussed and Professor Sharma also pointed out that in the case of explicit methods using a mathematical technique called the von Neumann stability analysis for simple situations like what we are describing here diffusion equation which is a linear equation we can find out that there is a restriction on the time step once we decide the grid size. In other words if we choose that the grid size is fixed at certain value delta y then we cannot arbitrarily take the time step value as anything but if you want to employ this explicit method the time step should be less than one half multiplied by the grid spacing squared divided by the diffusion coefficient which is the kinematic viscosity here. So this condition comes through a formal mathematical analysis which I haven't discussed the reason is that that forms under a numerical analysis part and as Professor Sharma mentioned a few minutes back if you want to refer to the book by John Anderson he has outlined the von Neumann stability analysis and specifically the criterion which is sitting on the slide here has been derived for a diffusion equation in that book so I would request that that's where you would like to go and look at it for detail may I ask you to repeat the first part of the question actually I didn't get the first part sorry. So in the differential sorry in the finite difference solution 7 so how the delta y values varies for the stretching So the question is on the grid spacing in slide number 7 on finite difference solution and I think if I understand your question you are you are asking whether the grid spacing is changing from one end to the domain to the other end of the domain using some sort of a stretching function which essentially means that you are talking you are talking about non-uniform grid sizes as you go from one end to the other in this particular case we have chosen to use a uniform grid spacing so that there is no stretching of the grid points or the grid size as you go from one location to the other this is the most simple situation where you have a grid generated with a uniform grid spacing so that delta y remains the same between any two pair of grid points I hope that is what your question was VIT Pune please go ahead with your question My question to Puranik sir slide number 39 and 40 Yes please go ahead My question related to slide number 39 As for temperature non-dimensionalization you have made it with this 39 slide number for differential analysis Yeah go ahead please So there you have made a non-dimensionalization of a temperature for the rest of the things you have taken as a direct values for here the difference of temperatures you have taken temperature difference for non-dimensionalization Yeah so the question is on the non-dimensionalization that we employ in the governing equations specifically the temperature non-dimensionalization is done in the form of ratio of temperature differences whereas all other are let us say the length scale divided by length scale so there is no differences elsewhere but only for the temperature this is actually following the standard treatment in non-dimensionalization of energy equation see typically in the case of energy equation we are talking about heat transfer and heat transfer will always occur only when there is a temperature difference between two bodies in the case that we are talking about it is the temperature difference between let us say a surface and the fluid that is flowing past the surface so it is just a standard convention that we talk about the temperature non-dimensionalization using a ratio of temperature differences and at least the understanding is that only through the existence of temperature differences we can talk about heat transfer so it makes sense in that fashion also many times what you will see is that this fully developed conditions also in case of internal flow with heat transfer will be coming in the form of the derivatives actual derivatives of such temperature differences being 0 so that is also one additional let us say feature because of which we end up using these temperature differences to find non-dimensional temperature at least that is what my understanding is let me ask professor Sharma if he has anything more to add because he works more in heat transfer than I do I would like to mention that whatever professor Pranik has mentioned about the constant wall temperature this is the way we do but the other boundary condition standard boundary condition is constant heat flux if that is the case then in the denominator we do not have a reference temperature which is shown here then in that case what we typically do is we take the temperature difference like let us say flow over a pipe in which pipe is having a constant heat flux boundary condition typically then we non-dimensionalize this T minus T infinity divided by the constant heat flux let us say q w into the diameter of the pipe divided by k so in a constant heat flux this denominator is different constant wall temperature is the way we non-dimensionalize so the next question is about the differential analysis slide number 40 you have made the non-dimensionalization of the equation number 2 we could not understand that second equation is concerned dou by dou x star into dx star by dx that part particularly so the question is on how this non-dimensionalization is proceeding in the case of this continuity equation so actually what has been done here which is projected on the slide here is simply look at the first term in the continuity equation which is partial derivative of u with respect to x just look at this because the other one is exactly the same the partial derivative with respect to x now we are trying to convert everything into these different coordinates so to say from the dimensional coordinates x and y to the non-dimensional coordinates x star and y star so in one way what we are doing is that we are coming up with a coordinate transformation and accordingly the differential equation is also going to change into a space where we are talking about the non-dimensional coordinates so if you look at partial derivative with respect to x all that we do is we express that partial derivative with respect to x using the chain rule where the chain rule will then say that it is partial derivative with respect to x star multiplied by the change in the x star with respect to x that is the standard expression from the chain rule of partial differentiation and then if you go back to the way the coordinate transformation is defined it is x star equal to x over lc where lc is that characteristic value or a reference value which is essentially a constant so therefore when it comes to evaluation of this derivative of x star with respect to x we simply go back to the expression with which we have related the two coordinates and if you differentiate this x star with respect to x what we will have is equal to 1 over lc so dx star dx meaning differentiate x star with respect to x and this will be simply equal to 1 over lc and that is what is substituted for this dx star dx where my highlighter is standing similarly then what remains is the u velocity inside the partial derivative and going back to our non-dimensional definitions u star is simply equal to u divided by uc that is the way we have defined our non-dimensionalization therefore u if you just do the cross multiplication will be u star multiplied by uc where uc is a constant value for the characteristic value of the velocity and that is what is written out here inside the bracket where you have u it is getting replaced with u star multiplied by uc uc is a constant so it's actually going to come out of the differentiation altogether dx star dx is equal to 1 over lc as we just described so that there is a constant factor of uc divided by lc which is going to then remain as a constant factor multiplying partial derivative of u star with respect to dx star and that is how the step is getting simplified exactly the same multiplying factor will come out of the non-dimensionalization of the second term where my highlighter is standing so there it will be a partial derivative of y which is expressed using the chain rule of derivatives and the v velocity is expressed as the non-dimensional v velocity multiplied by the characteristic velocity so this uc over lc will come out as a common factor from both terms and that's what is shown on the third step so it's this entire non-dimensionalization exercise whether you look at the continuity now or whether you look at the x momentum, y momentum and the energy is looking at each of these terms and utilizing the chain rule of partial differentiation first and replacing the velocities with their corresponding non-dimensional counterparts and then simplifying that's that's all it is