 Along with Professor Puranik welcome all of you the question answer session we will start with Fakkam Ja College Hyderabad. I have one question with respect to the number of grid points because either in the laboratory or even today's discussion it is found the selection of number of grid points is quite critical, but generally the higher the number of grid points we get more accurate the results. So how do we decide about the number of grid points for a given problem? That is a very important question, question is in a particular simulation let us say today we have done a lab where you have used a grid size so how do we decide about the number of grid size for a particular problem. Now the answer to this question is that we do a study which is called as grid independent study into Varun's lab session we had given you a problem where we have mentioned what grid independent study is. So you know that there are certain approximations which we are using like surface averaging, volume averaging, piecewise linear approximations. So for this approximations to be accurate enough the size of the control volume needs to be small enough. Now when we say small enough it is a qualitative information so we do not know what is the level of small enough we need to capture the results accurately for a particular problem. So because we exactly do not know what is how much is the action in a particular problem how much is the gradient of temperature and velocity in a particular problem. So when we take up a problem we start with a grid size which we believe that the grid size results in a control volumes which are small enough however we are not sure then we need to do a second run where we use a larger number of grid size and then check that the results which we have got from the larger number of grid size as compared to the smaller number of grid size whether the change in the qualitative as well as quantitative results are negligible if that is so we say that our grid our the grids which we have taken earlier is good enough for that particular problem but the number of grids varies from problem to problem it varies from the value of the governing parameters which you are taking in a problem. So this is decided based on a grid independence. Thank you I have just another clarification today morning if I remember you have discussed about this slug flow where the temperatures at the wall and at the entrance are given and the temporary profiles have been drawn but there should be a condition that the entrance temperature is higher than the wall temperature otherwise if it is less the profiles would come in reverse manner. So today morning I have taken this problem so the question is that the inlet temperature I had mentioned that the wall temperature is greater than the inlet temperature so the question is that this does not seem to be correct because in that case if the inlet temperature is less than the wall temperature then this profile should not come that is not the case here in this case let me remind you that here we are not plotting the temperature profile what the dimensional temperature profile what has been plotted in the next slide is theta not t theta is a non-dimensional temperature and I would like to draw your attention that theta has been defined in this way. What you are saying is correct if it is temperature dimensional form whereas we are plotting the non-dimensional form of the temperature thank you. Water room is the case that we should mention the condition whether t infinity greater than t w are not based upon that only the profile will change is not. Actually the profile which I am showing you corresponds to what it will also corresponds to a case if t infinity is greater than t w it is just that it is independent whether it is which one is greater but the way it has been non-dimensionalized this is the results which you will get but just for physical explanation I had taken one of the case saying that the inlet temperature is greater than wall temperature but if you do non-dimensionalize this way the results are indeed applicable whether t infinity is greater than t w or whether t w is greater than t infinity. Another question sir can you suggest any book which deals with the numerical analysis in CFD because anyway most of the books they deal with the numerical methods so can you suggest any book on numerical analysis which is normally not discussed here. The question is in this course mostly we are discussing numerical methods not the numerical analysis if you see most of the CFD books they discuss numerical methods and partly they discuss numerical analysis also like if you follow the book of JD Anderson where he has done the stability analysis he has shown what is numerical diffusion also if you follow the book of Perriken-Fersinger you can see the order of convergence which is a part of a numerical analysis he had shown so these are the two books which are coming to my mind may be professor Huranik can add to it yeah so there is one book by Tannehill Anderson and Plecher which is called computational fluid mechanics and heat transfer let me just write this on the tablet here at least I will write the author's names and this is available in Indian edition this book has sufficient amount of numerical analysis but let me just add quickly that typically the numerical analysis that you will see in either this one or the Anderson's book will be usually with respect to the finite differencing more than on finite volume so for finite volume technique the little bit that you will see is in the book by Fersinger and Perriken so let me write that also and I think the title of the second book is computational methods for fluid dynamics I think or something of that sort but the authors are correct sir what is the book for fluid mechanics to go through so the the question is what say what say reasonable good book for fluid mechanics if you if you go back to my set of slides in part one I had mentioned I think three good undergraduate fluid mechanics books but what I will request is that if you just hold on to hold on to it for a few days at the end of the workshop we are going to upload one document on moodle which that file will contain actually a much exhaustive list of textbooks for both fluid mechanics as well as for the CFD and we will also try to write maybe one line about that book as to what level it is and typically what is good about that book and so on so if you do not mind just waiting for a few more days at the end of the workshop we will upload one one file on moodle where we will include all this information thank you. Yes I am S. Pillai now new Panvel please ask the question. Good afternoon professor Sharma my question is regarding length scale selection of length scale please refer to topic two slide number five on grid generation my question is how to select delta x and delta y which is a control volume size basically and which indirectly decides the computational resources so how do you exactly decide on that is there any limitation on the length scale. The question is how to decide the grid size delta x and delta y for a particular problem now this decision based on our as I say that CFD is basically guided by the fluid mechanics principles so when we want to decide the size of the grid we need to have an understanding of the fluid mechanics for that problem we need to have understanding of in which region most of the fluid dynamic action is happening let us say for flow over a flat plate most of the action occurs close to the flat plate so we need to have a very fine let us say delta y we can have a larger delta x because in the axial direction the change in the velocity is the velocity gradients are much smaller as compared to the velocity gradients in the transverse directions so this is the way for so this answer to this is the decision making varies from problem to problem because each problem has its own fluid dynamic story and based on that understanding we try to place we decide the size of the grids thank you. Okay sir I have one more question in problems where the boundary layers are addressed where in practically solving the problems we need to give some thickness of boundary layer y plus condition is to be given so what are the guidelines for that. So the question is that especially when you are solving turbulent flow simulations let us say for flow over a flat plate then there are specific theory or as far as CFD is concerned for turbulent flows where we based on the y plus values we decide that the number of grid points so the question is that how do we decide in case of turbulent flow sorry I will not be able to answer this question because I had not worked on turbulent flow simulations but whatever limited I know I know that based on y plus we make a decision about the number of grid points may be professor Puranik will add to what I had. Let me just quickly add to that see the decision on the y plus as to how much refinement you want near the solid surface let us say to resolve the boundary layer it actually depends upon the turbulence model that you are employing usually if you decide to choose one particular turbulence model normally there is sufficient literature available on how to choose the grid size that is suitable for that particular turbulence model. So unfortunately there cannot be any universal answer to that it depends somewhat on the turbulence model that is being used but roughly speaking we want to resolve as close to the wall and roughly speaking y plus of about 5 to 7 is what you need to go to in order to resolve the boundary layer completely it may not be possible to do it computationally the sources may be required very high but in general the objective is roughly to resolve up to a y plus of about 10 is what I have typically seen. PSG Coimbatore. I have a question the topic number 3 slide number 8 it is regarding the special topics in computation of the heat conduction in that in the implementation details for grid points for the heat fluxes and temperature the heat flux along the tangential direction at the node j, i max minus 1 is equal to tangential heat flux at the node j, 1 will you please explain me that using the white board. So the question is we have taken a problem on this cylindrical coordinate system cylindrical coordinate system cylindrical polar coordinate system r5 coordinate system and I had mentioned that this is a physical domain we solve in this physical domain however for ease in scrolling through the grid points we convert this into a computational domain which seems to be in quotient coordinate system but this is only for the programming purpose we have converted into this so that we can scroll through all the grid points easily. Now in this let me explain you the question here is that in the pseudo code I had mentioned that the heat flux in the angular direction at i max minus 1 is equal to the heat flux at 1, i is equals to 1. So let me show you what is heat so I here in this case is representing the angular direction j is representing the radial direction so what we do is that if you go to the next slide first we calculate the heat flux and the angular direction at i is equals to 1 and the way we calculate is so the heat flux at i is equals to 1 to calculate that we take the temperature here which is at i is equals to 2 which is shown here temperature at i is equals to 2 minus temperature at i max minus 1. So you can say that we are taking the difference of the left border cell and the right border cell because in this problem there is continuity in the angular direction however for programming simplicity we have broken it we have done what we call as branch cut but to calculate the gradient we take the left border cell we subtract the left border cell minus the right border cell value and divided by the distance between the this two border cell and let me remind you that the distance comes from the physical domain and that way we calculate heat flux in the angular direction at 1. Now once you have calculated this in the this phase is actually physically same as this phase I will show you in the next slide the green line sorry the green line which I am showing you here is where what is the branch cut you can see physically it is the same line computationally we have broken into two parts so what I mean to say that this point is same as this point that is the reason that in the next slide we are putting heat flux at i max minus 1 is equal to heat flux at i is equals to 1. Thank you sir the same problem if you do it in a finite difference technique finite difference method there again if we have the continuity condition in the circumferential direction for example instead of temperature and if we have pressure for how to approach that problem that is the boundary condition fixing the boundary condition. So the question is that instead of temperature if we are solving this problem for fluid flow then instead of temperature let us suppose if there is a variable pressure let me tell you that in the Neyvastok equation although I had not discussed it till now but the pressure equation which we solve which we will discuss may be tomorrow is a Poisson equation and in that equation is also like a heat conduction equation with volumetric heat generation. So in that case also we calculate this way only the way whatever has been discussed here this is what is called as a periodic type of boundary condition because there is a continuity along the angular direction. So the procedure which is discussed here similar procedure is used as far as handling pressure is concerned. Thank you sir excuse me that here in the when you convert this the radial coordinates and that is in the circular domain when you convert this circular domain into rectangular domain what is that the left hand most the squares that is with green color indicates what is the right most line with squares the green color indicates. So the question is that what is the green square on the left boundary of this domain and the right boundary of the domain represent I had mentioned that just for simplicity in deciding the running indices of the phases of the control volume. Because the way we are programming the codes we have given we are doing what you can say physics based formulation and programming. So on the phases you know there are two phases two types of phases in a standard coordinate system like in a Cartian coordinate system there is one there is vertical phase centers and horizontal phase center we have heat flux in the x direction on horizontal phase vertical phase center and heat flux in the y direction on the horizontal phase center. So the green square which is shown here is just for you that from this picture you get an idea that these are the phase center you have to scroll through to calculate heat fluxes in the x direction for Cartian coordinate system and heat flux in the angular direction in cylindrical polar coordinate system. Now in this case I will repeat again that as there is a angular continuity when you make a branch cut and convert into this form this boundary physically is same as this boundary is just like if you take a paper and rotate the two ends you get a circle okay. So this is the grid point for heat flux in the angular direction. Sir one more question here we have equated the tangential heat flux at the j-1 sorry i max-1 with the heat flux in the tangential direction at the node 1 that is i is equal to 1 in between there is a heat flux in the tangential direction at the node i max what is the relation between the heat flux in the tangential direction at the node i max with the heat fluxes in the tangential direction at the nodes i max-1 and 1. So the question is what is the relationship between the q at i max with the q at i max-1 let me tell you the way we have used the convention the running indices for the east face of a control volume is equal to the running indices of the centroid of the control volume. So the last green square which you see is this one and what is the running indices of the corresponding centroid of the control volume it is i max-1. So in your in our program if you go and see in the angular direction q at i max does not come into the expression q at i max-1 is the last value. So q at i max is not needed thank you VPCOE erode. My question is purely application oriented this I want to check that whether this problem can be applied by CFP or not ok I hope you are aware of what the cooling towers then the conventional cooling towers what's happening the hot water is sprayed from the top of the cooling tower and water is drawn from the the ambient temperature water is drawn from the bottom of the cooling tower so that two phase flow is taking place but in case of the three phase fluidized to bed cooling tower I hope you are aware of what this three phase fluidized to bed cooling tower in which in addition to the hot water and ambient temperature air in between these two we are creating a fluidization zone where some plastic balls of different diameters are introduced into the fluidization bed so that the hot water while sprayed from the top of the cooling tower the heat transfer surface area being increased by the plastic balls so that the heat transfer enhancement is taking place this is the problem my question is whether since it is a multi phase flow that is three phase flow hot water that is liquid solid and vapor multi phase flow is taking place is it possible to find out the temperature profile of the water across the bed height fluidization bed height number one number two is it possible to get the fluidization velocity profile across the fluidization zone this is my question my question is that's the second next question is can we apply the CFP concept CFP analysis to these problems sir please help me thank you the question is different from what is there in the lecture slide it is more of an application the question is whether the CFD tools can be applied in case of three phase flow where there is a solid there is a liquid there is a vapor and the system which is being talked about is the fluidized bed although I had not done that type of simulation I had done a multi phase flow simulation where vapor and liquid is involved but whatever literate limited literature had seen in this regard I can say see at seeing that CFD is a capability to do this so I will stop with this limited answer thank you VP COE Bharamati sir my question is what is the effect of Reynolds number and Prandtl number on the steady state temperature conditions contour or what is the significance of these two numbers so the the question is what is the effect of Prandtl number and Reynolds number on temperature contours so I need actually more specification on the on the problem meaning if you are if you're talking about a conduction type situation or in a in a fluid flow so can you please can you please say little more about the problem this is regarding the convection type of flow ok so it's related to convection type of flow so the the nature of the Reynolds number sorry the the value of the Reynolds number rather would decide what sort of a flow situation that we are dealing with whether it is a turbulent flow or whether it is going to be a laminar flow as we know and the Prandtl number usually decides the the extent of the boundary layers that are formed for the hydrodynamic situation as well as the thermal situation so the ratio of roughly the ratio of the boundary layers for hydrodynamic to thermal situations is what is given by the the Prandtl number in that sense the temperature contours you cannot explicitly tell what's going to be the effect of Reynolds number and Prandtl number but the the the two parameters will definitely determine what sort of a temperature field and a flow field is going to be decided in your situation and thereby the eventual temperature contours will be decided in that sense it's not very sensible for me to say that it's going to be only this kind of effect or that kind of effect they are going to be definitely affecting the flow as well as heat transfer but specific effect on the temperature contours is is somewhat difficult to to answer let me let me ask professor Sharma if he has any comment on on on this to add the product of the Reynolds number and Prandtl number comes out as a what we call a speculative number and when you look into the non-dimensional equation in heat transfer convective heat transfer one by Reynolds number into Prandtl number that is one by Peclet number comes as a diffusion coefficient in the equation so if the product of the Reynolds number multiplied by Prandtl number is more then the inverse of that is less so in that case diffusion is less or conduction heat transfer is less as compared to advection heat transfer or vice versa thank you so it has a relationship with the relative magnitude of conduction heat transfer with advection heat transfer Amrita school of engineering column. Sir this is regarding the convergence criteria that we said we are forced to set the convergence criteria as the armors value of the error sir is it possible to if I need the error at a point to be limited for for a particular reason I need the value at a location then can I set the convergence criteria that the error at that point or at that plane is in a limit. So I think there are the question is on the way we define the convergence criteria I had defined in a some way where I had calculated the RMS value I think Professor Puranek had defined in a slightly different way people define that term in a different ways but that is fine it is just that that maximum difference between each grid point the temperature difference which Professor Puranek has mentioned it is a more stringent criteria this so it may happen that in that case if a solution is converged by let us say at 10 to power minus 3 if you take the RMS value it may be 10 to power minus 4 but both are considered good enough so but there are certain problems where taking the RMS values is considered better so that is why as per my experience I had used this proposed this type of convergence criteria. Yeah I would like to mention that in your question you have said that take the difference of the temperature at one point I would like to point out it is not at one point if there are let us say 25 grid points inside the domain then we get the 25 difference we take the temperature difference of those 25 grid points between the two consecutive time step and among those 25 difference values we pick up the maximum difference and check whether that maximum difference is practically 0 thank you. Under free convection over the object like sphere how to take boundary conditions and how to decide the size of the domain. So the question is that for a free convection situation that to an external flow in an open domain free convection let us say across a cylinder in an open domain how to decide the size of the domain typically in such situation very large domain is taken and in that case the boundary conditions which are used are the at the boundary there is there will be some depending upon a particular problem we will come to know that let us suppose if you take a hot cylinder due to buoyancy the fluid will rise and so the top boundary of the domain will have an outflow boundary conditions and below we try to take a much larger upstream distance and however the domain size which we take as I said that the size of the domain there are various problems in CFD especially in external flow problems we take a size of the domain we get a results however we are not sure that whether the size of the domain is correct enough to give the accurate results so we increase the size of the domain and do a second simulation and check whether the difference between the results is whether there is a difference between the results in two different domains and if it is not then we say that the grid size the domain size is good enough thank you. Hello sir one more question is there topic number five slide number four topic number five slide number four in which direction fluid is flowing and what is the direction of heat flow. So the question is in which direction fluid is flowing and in which direction heat flow is occurring this had taken a convection convective heat transfer problem and I had taken four different cases case one where the flow is from the ice side had taken an example you are standing between let us say ice at zero degree centigrade fire at 100 degree centigrade and I had taken four different cases. So in the first case flow is from left to right in the second case flow is again from left to right in the third case and fourth case you can see that the velocity is in the negative direction so it has a negative value so the flow is in the negative direction so the answer to the first part of the question is in first and second case flow is in positive x direction and in the second case flow is in the negative x direction as far as the direction of the heat flow is concerned the direction of heat flow is always from higher temperature to a lower temperature although when you have a flow it carries along with it the enthalpy and it is transported in the flow direction thank you. Sir from the same topic slide number 15 is there any relation between delta and delta three is there any correlation which defines the two or they are totally independent variables. Question is whether there is a relationship between the boundary layer thickness and velocity boundary layer thickness and thermal boundary layer thickness yes there is a relationship and in fact it has a relationship with a non-dimensional number Prandtl number okay Prandtl number is defined as a ratio of kinematic viscosity divided by thermal diffusivity and I had mentioned that this is a direct correlation with kinematic viscosity and this with thermal diffusivity okay so as a relationship with growth of this so when the Prandtl number is more then the boundary velocity boundary layer thickness is more as compared to thermal boundary layer thickness and vice versa. Again one more question hello sir under free convection you said that size of the domain we have to take very large size but I have not followed how to take boundary conditions which exactly boundary conditions we have to exactly select. So the question is what boundary condition one has to take in a natural convection problem actually the boundary condition varies from problem to problem natural connection convection problem I had been very generic in answering that question I had taken a case where let us suppose there is a sphere or there is a cylinder which is heated and due to buoyancy there is a plume which is formed and it tries to rise upwards so then I had said you can take a top boundary where you can use an outflow boundary conditions and I had said you take a bottom boundary where you take u is equals to 0 and v is equals to 0 and you can take the side boundary where you can use the far field boundary conditions thank you. Professor Puranik will add to this. Yeah just a quick addition to that and typically these natural convection problems when you are solving you end up taking a sufficiently large domain and the object size is much smaller than the domain and having chosen such a large domain usually the free stream conditions you can safely assume at least on three of the four boundaries and the top boundary is usually taken as an outflow type situation but the side and the bottom boundary if it is sufficiently far from the object which is losing heat by natural convection I think it is reasonably safe to take free stream conditions as the boundary conditions I think that is what that is what makes sense to me thank you. Sir the question is about up winding how that can be good for compressible flow especially even I just interested in how that up winding is better for compressible flows. So let me try to answer that because I have some experience in compressible flows see usually so the question is how is up winding good for compressible flows and what I'll do is I'll try to give a purely physics based answer so if you see compressible flows typically what we are dealing with is that the advection is much stronger than the diffusion so that there is a clear direction established by the flow as far as the momentum transfer is concerned so if you see let us say from a computational standpoint you are looking at the finite volume situation and you consider a a cell phase between two adjacent cells so as far as this cell interface or cell phase is concerned depending on which side is the more dominant in terms of the fluid flow you can choose the up winding so that if the flow is strong enough from left to right or right to left you want to you want your numerical method rather to mimic that as as cleanly as it can and that's the reason we end up choosing up winding when we go for compressible flow situations so the the answer really is in the fact that the convection sorry the advection is very strong compared to diffusion in case of typical compressible flows thank you Periyar Tanjavur please ask your question sir throughout this kinematics throughout the kinematics we have come across some stream function and all other topics but we don't cross anything about potential function it's a less important in the state of desire that is we never come across anything about potential functions we have come across the stream functions and all so wouldn't we talk about the potential function yeah so the the question is on the use of potential functions what is getting pointed out is that during the kinematics we have talked about stream function but we haven't talked about potential function so the the answer is that the potential function has a restricted restricted use in the sense that only under irrotational conditions you can use the the potential function to define the velocities with the use of a gradient of potential function but it is restricted to only irrotational flows and in general irrotational flows forms a very small class of fluid mechanics and that's the reason we don't really have to deal with potential functions much on the other hand stream function is something that has slightly more general usage at least as far as two-dimensional situations are concerned in the sense that you can utilize that to to plot streamlines as well and perhaps that's the reason stream function is is used more but potential functions is restricted only to some cases which are these irrotational flow cases nirma university and about please go ahead yes sir I have question in topic number four slide number 38 to sir master what is the reason for unbounded solution for SOU and quick see the question is that what is the reason for unboundedness in second order upwind scheme and second order upwind and quick scheme you remember that I had taken a lecture of study state one-dimensional advection sorry you study state one-dimensional convection and where I had shown you that when you use a central difference scheme and when the pecklet number is greater than 2 then there for that scheme I had shown you that there is an unboundedness okay or there is an oscillation so the so that is similar thing happens in case of quick scheme and second order scheme which results in unboundedness solution so the relationship between the coefficients which are involved the neighboring coefficients is such that it results in oscillations unboundedness on a code circuit thank you okay one more question you mentioned about numerical diffusion today what is the significance or reason for that the question is today I mentioned that first order upwind has a numerical diffusion now to do the question is that what is the mathematically we can understand as far as the numerical diffusion is concerned I would had not shown taken the background of those that is typically done in a numerical analysis however I would suggest you that if you go follow the book of computational fluid dynamics by J.D. Anderson there they had shown that if you use a first order upwind scheme there are certain terms which comes into the expression which gives rise to the numerical diffusion may Professor Puranik will add to it yeah let me just add to that numerical diffusion part and if you read this Anderson's or something similar again the discussion would be based on a finite difference technique and depending on the Taylor series expansions and the number of terms taken or refused in the Taylor series expansion and so on what I would like to point out on a purely physical basis since the question was the significance of numerical diffusion the reason I am trying to point this out is because this is something that we routinely encounter in compressible flows so for example in a compressible flow there is an occurrence of something called a shock which is essentially a discontinuity in flow variables so your numerical method ideally would like to capture a shock as a discontinuity which is essentially a sharp change over practically no distance however if you end up using first order upwind schemes what happens is that this discontinuity gets smeared over a certain distance and this is what we call numerical diffusion so the numerical method is unable to capture exactly the kind of discontinuity that occurs in a real-life situation rather it smears it over a sufficient distance and that is what on a purely physical basis if you want to look at the effect of diffusion on flow solutions I think that's a good example to look at thank you. Ambritha Coimbatore please ask the question. Sir my question is regarding topic number four slide number 38 where you have given comparison between all the schemes to capture this shock wave but if you see the comparison between the first order of winding and quick schemes especially for a coarser grid quick scheme can capture the shock but it was not accurate especially away from the shock so that case why can't we use the quick scheme near the shock and why can't you couple the quick scheme with first order of winding away from the shock wave. So the question is to use a scheme which is something like a hybrid scheme which is being said that let us use quick scheme near to the sharp change and the oscillation is the quick scheme it is being pointed out is able to capture the sharp change better than first order of wind scheme. So what is being suggested is that why not use this quick scheme in this region and wherever oscillation is occurring why not use first order of wind scheme I had not used it and I have a serious doubt whether you will get an accurate solution because first order of wind may spear the solution all through the system. Let me just add to it based on my experience with compressible flows see usually when you are computing a solution where situations such as shocks are expected a priori or beforehand you don't necessarily know where the shocks are going to come so in that sense it is essentially impossible to design a hybrid scheme such as what you are referring to however there is a very standard approach that is employed along with these quick type schemes and that is what is called as a flux limiter approach which results into what is called as a total variation diminishing or TVD type schemes that is usually a slightly more advanced topic than what we have been covering. However a reasonably good discussion of these TVD approaches using the flux limiters to essentially limit the oscillations that the higher order schemes like quick are introducing near discontinuities you can find in the finite volume book by Versteeg and Malala Sekara but I think I would like to limit my answers to only that because it is slightly out of the scope but your point is well taken but there is a well known fix for it and that is what is done in terms of these TVD schemes using the flux limiter approaches. So thanks. KIT go ahead please. Sir I have three questions first one is in between the two finite volumes the variation of that variables we are taken a linear if the non-linearity in the variables are present can we catch by making some modification in the schemes this is my first question and second question is when we are operating the course in case of the labs when the iterations are going on one word I observed that is unsteadiness continuously what is the meaning of that unsteadiness and the third question is I want one suggestion from your side sir if the chemical kinetics is present in the process and which scheme is better. So there are three questions first question is in an advection scheme you remember that we had use linear variation in case of second order of wind it is a linear extrapolation scheme central difference is a linear interpolation scheme so the question is that when we assume locally linear variation in fact not only in this advection scheme for conduction or diffusion also we have used piecewise linear approximations so the question is that if the variation is non-linear how good is this linear approximation I would like you to point out to the figure which I had drawn a simple figure a non-linear function f of x as a function as x varies if you remember I had drawn a very non-linear function and I had pointed out that however non-linear a function may be if you take a very small delta x within that small delta x even the function is highly non-linear even then locally linear approximations are good enough note that in computational fluid dynamics anyway we have to do a grid independent study anyway we have to take a very small grid size and in a very small grid size this linear variations are very common in computational fluid dynamics and they indeed work well this is the answer to the first question in the second question in the lab codes there is a word which I had used un-studiness I think this is the right word yesterday it was pointed out that in my lecture slide I am using a word convergence for the un-study state problem so I think this is the right word instead of using word convergence criteria I should say it is an un-studiness criteria so in the code our term has been defined un-studiness so what we do is that we calculate the root mean square value of temperature and compare with that un-studiness value that un-studiness value we ideally want to reach to 0 but computer does not know what 0 is so we say that un-studiness is 10 to power minus 3 or 10 to power minus 4 so as you know that in un-study state code reaches stops when this un-studiness criteria is fulfilled or un-studiness is of the order of 10 to power minus 3 or 10 to power minus 4 regarding the third question chemical kinetics what are the types of schemes which are used honestly speaking I do not have any experience and exposure to this so I will not be able to answer this Professor Puranik will I have a very limited exposure to chemical kinetics specifically related to hypersonic flow where the air is undergoing dissociation type reaction so that type of chemical kinetics is what I have some experience with but in general the characteristics of other combustion type reactions also are more or less similar so the answer to your question is that see whenever you are including chemical kinetics in your CFD type simulation usually you will end up adding convection diffusion equations with a source term which will describe that chemical kinetics so finally you are still dealing with an advection term a diffusion term and a source term so there is absolutely no difference between numerical treatment of those terms than what you have seen here so typical advection that you have already seen such as quick or first order upwind or second order upwind you can utilize for even the use of discretization of the chemical kinetics equations similarly the diffusion part in the chemical kinetics equation is usually different in the central difference form so whatever you have seen already you can utilize it for chemical kinetics also thank you here I have a doubt about the analogy in the topic number 4 slide number 30 so and in that quadratic upwind interpolation convection kinetics how some doubt about forgetting the analogy whereas 5e is equal to 35e plus 65p minus 5w whole by 8 if possible can you try to please elaborate this and this is my first question and my second question and my second question is and if possible can you provide one example about fluid structure interface by using of cfd analogy the first there are two questions first question is on this slide I had shown that the while using the quick scheme these are the expression these are the usual expression when there is a uniform distance between the grid points here let me come to this expression and talk about it here we have taken a case where mass flow rate is in the positive x direction and the positive y direction and in this case so effectively there is a flow is inclined upward so that if you take its component you get mass flux in the x positive x direction mass flux in the positive y direction now in the phase you know if the flow direction is positive x direction then this is the downstream neighbor and in a quick scheme if I would like to remind that in a quick scheme the weight which we give to the downstream neighbor is 3 by 8 which is shown here this is the upstream neighbor and the weight which we give is 3 by 4 or 6 by 8 which is given here and the weight we give to upstream of upstream neighbor is minus 1 by 8 this is the usual expression which had whose derivation I had shown in the previous slide that is being used here and when you go to the west phase the downstream neighbor is 5P upstream is 5W and upstream of upstream is 5WW that is what is shown here similarly it is done for the south phase center and the north phase center and regarding the second question, second question is different from what has been taught here for the fluid structure interface the question is how in CFD we take care of those there is a method which is now becoming quite popular which is called as a immerse boundary method where instead of a body fitted grid Cartesian grid is laid down and when the object moves the good thing about this Cartesian grid is that you do not have to do remaching after each time step then in that cases as far as fluid structure interface is concerned across the when the solid moves it transports certain momentum to the fluid which needs to be accounted for so the answer to your question is that as far as handling fluid structure interface is concerned whenever a solid moves in a fluid it not only transports momentum but it also leads to a mass transfer across a control volume which is fixed in space so these are the two things which needs to be taken in account as far as the fluid solid interface is concerned thank you Pruba call it in the please ask the question answer my question is what is the difference between discretization error and round off error so the question is what is the difference between discretization error and a round off error now let me answer this question in a finite difference for finite difference method in a finite difference method if you go back and see how the central difference approximation was applied there in finite difference method a Taylor series expansion is done and from the Taylor series expansion we try to get difference equation for let us say delphi by del x by forward difference or backward difference so when we use Taylor series expansion and when you try to get a finite difference equation other than the finite difference equation you will see the remaining terms which have the higher order derivatives so that those terms are called as truncated terms and in those truncated term the lowest power of let us say if it is a Taylor series expansion in the x direction then the lowest power of delta x which is there it is called as a discretization order of discretization error discretization error and the truncated terms gives the discretization error now as far as the round off error is concerned so the round off error comes that when you are using computer and then computer you know that it has certain precision single precision or double precision so after a certain decimal it truncates the value so this gives rise to the round off error so overall I can say that discretization comes when we do the discretization whether it is a finite difference discretization or a finite volume discretization there is an approximation which is used which leads to a discretization error where round off error just come from when you operate with the computer which is a finite precision thank you yeah Athol I have one question this is regarding the iron osmar flows so when we take this iron osmar flows usually when we try for any kind of this advection schemes we try to get a numerical advection term and there is one turbulent turbulent diffusion term so you like when you go for a higher accuracy solutions usually the numerical diffusion term should be lower than your turbulent diffusion term so what usually we do is we try to increase the or refine the computational grid which will increase your computational power or we will go for higher order schemes so when we go for higher order schemes what we end up with what like the higher like disposed error we are going to get wherever the gradients are high so in this case like what exactly we need to do actually whether we need to go for a refining a grid or going for higher order schemes which will be a better option so the question is at higher Reynolds number there are two strategies which are being followed in fact that is true for any Reynolds number one is that you refine the grid for using a particular advection scheme or use the same grid size and use a higher order scheme now the answer to this question is that ultimately the answer is decided by the computational cost which is involved if using the same grid size and with the higher order scheme you are able to achieve higher accuracy then you have to see what is the computational cost or computational time it takes on the other hand if you follow the second strategy where you use the same scheme which you are using and define the grid and then see what is the computational cost which is involved so this two strategies are there and it varies from problem to problem so it is difficult to answer that which one is better because you yourself have to do the comparison and then come to a conclusion for a particular type of problem which you are solving. Thank you I have one more question this is regarding the first level or second level approximation what we take whenever we are dealing with either an unsteady term or an advection term so how we click whether we are working on a surface averaging to calculate the enthalpy term or when you come to unsteady this thing whether we will go for volume averaging so can you explain little bit on this like first level of approximation or second level approximation. The question is on the approximations which are used in the finite volume method can I elaborate on this approximation whichever term we take whether it is an unsteady term or advection term or diffusion term or even source term there are two approximations which are used and that I had called as two levels of approximation that is just classified as two levels of approximation because these are the two approximations which are involved now either the term which we are considering either it is directly proportional to volume like unsteady term or a source term or there are other terms which are directly proportional to surface area or the terms which have fluxes like advection term and diffusion term so the terms which have fluxes so the terms which have fluxes are the flux is directly proportional to the surface area the terms which are directly proportional to volume then we need certain terms within a volume in case of fluxes we need the flux on a particular surface area now when you take a volume or a surface area and the flux varies on the surface point to point if you take the source term like if you take an unsteady term which is directly proportional to the volume the rate of change of internal energy or rate of change of temperature varies from point to point within the volume so we need to have variation of that term in a volume or a surface so the way we had said averaging or surface averaging what I mean is that anyway here we are following a procedure where we have to take very small control volume very small surface area so rather than considering its variation let us take one point value which is accurate enough which is second order accurate so we say that in a volume let us take the centroid of the volume in the surface area let us take the centroid of the surface area and let us take the rate of change of temperature at the centroid of the control volume and let us take the fluxes at the centroid of the surface area this is what I mean by volume and surface averaging respectively and this is what I called as first level of approximation regarding the several level of approximation what happens is whenever you try to express the let us say flux like conduction flux or an enthalpy flux actually this flux are at the faces of the control volume now the expression for this fluxes consist of let us say either velocities or temperature in case of diffusion it consist of the gradient of velocity or temperature in case of advection it consist of value of the velocity or temperature at the face center so we need an approximation because finally we want to end up with a algebraic equation where this variables are not at the faces but we want expression where they are they should be expressed in terms of the cell center value so we need to come up with an approximation to calculate the value of the this variables at the face center for advection the normal gradient of this variables for diffusion term at the faces for which we use a piecewise linear approximation that is what I called as a second level of approximation because by using this two approximation we start from a control volume apply conservation laws and end up with an algebraic equation thank you I thought I have one more question like whenever we are dealing with this Peselt number is equal to 0 so usually what we take is like usually the advection or convection neglected can we taken as a pure conduction case whenever we have that velocity being very small or Peselt number is equal to 0 will it be a pure conduction case over and out yes you are right the question is that when Peselt number is 0 or very close to 0 can we take it as a pure conduction yes that is the correct answer because in that case advection is 0 almost 0 so it will be pure conduction and I had shown my lecture slide that the variation is linear in that case and I had in that course sir I have a question regarding what is a slug flow and sir how does the slug flow the region develops with respect to the thermally developed and how does the hydrodynamic boundary layer develops in a slug flow the question is regarding the slug flow ok so this example had taken for the slug flow so the question is that how the hydrodynamic development occurs in the slug flow there is no hydrodynamic development in this slug flow because we have taken the flow in such a way that there is no velocity gradient everywhere u is u infinity and v is 0 so there is no velocity gradient in the vertical direction there is no velocity gradient in the horizontal direction so that way there is no action in the flow this situation is hypothetical because in real world situation in a plane channel let us say if you have a flow there will be boundary layer growth I agree that we have taken non-physical flow field but this is an analytical solution and this is a test problem for which we have taken so I would say do not try to get into the physics of the flow as far as this is concerned and as far as heat transfer is concerned so from this I had shown you the temperature profile at the end so in this case it just happens that if let us suppose the wall is hot then this fluid which is entering at ambient temperature slowly gets heated from the wall and as it keeps moving on the center line temperature increases continuously and if it is very long finally it should reach to the wall temperature that is the way the thermal development occurs thank you Sir one more question sir regarding plane Poiseuille flow sir how does the regime is defined when the flow is fully developed or the developing region also is defined under Poiseuille flow regime yeah so the question is on plane Poiseuille flow and whether we take both the developing as well as developed regions in the plane Poiseuille flow so the plane Poiseuille flow simply means that we are talking about flow in a parallel plate channel in that sense the name only suggests that we are talking about a two-dimensional channel and the flow going through it so at least my interpretation would be that both developing as well as developed regions can be clubbed under one name that is the plane Poiseuille flow I would also agree with that Sir one more terminology I want to ask sir like when we talk about thermally developed hydro dynamically developed flow and thermally developed flow one great flow regimes come in the name I have heard I don't know much about it can you just throw some light on it yeah so the question is on something called the grades flow situation and what is exactly that so the grades flow is actually a hydro dynamically developed but thermally developing situation okay so I will repeat once again it's a hydro dynamically developed but thermally developing region Professor Sharma has to say something if you look into this slug flow the grades flow what Pranik had just mentioned you can understand with this slide where instead of this uniform velocity profile at different actual location if you have a fully developed parabolic velocity profile which is a hydro dynamically fully developed velocity profile then the same slug flow will become a grades flow thank you does that mean that it happens only when the pecklet number is high will it be in low pecklet number not pecklet Prandtl number sorry for I am talking about Prandtl numbers yeah so as I mentioned that the Prandtl number has a relationship with growth of the velocity boundary layer and thermal boundary layer so let us take one case where if Prandtl number is greater than 1 then in that case the velocity development will be so velocity boundary layer will be above the thermal boundary layer so in that case what will happen what happens in case of plane channel flow is the boundary layer grows from the bottom wall boundary layer grows from the bottom from the bottom wall as well as top wall and it meets at the center and thereafter after some length flow becomes fully developed so if your Prandtl number is greater than 1 which means your velocity boundary layer will be above thermal boundary layer in that case the hydrodynamic development length will be smaller so the flow will develop before the thermal development and vice versa thank you yeah let me just quickly add to that the greats problem is possible for all Prandtl numbers is just that for a Prandtl number equal to 0 it will become the slug flow that what professor Sharma has shown so I would say that greats problem is for all Prandtl numbers there is no restriction on the Prandtl number in that sense sir one last question from our side sir in the implementation of certain schemes you refine the grid size initially you said it is a coarse grid and then subsequently you said it is fine grid so the question is sir how do we decide upon what is the fineness and what is the coarseness like what is the limit beyond which we say now we are into a regime of say finer grids or fine that is the question normally asked in case of solid mechanics related problems as well over to you sir thank you so the question is whenever we so this is a question which has been asked earlier also that how do we make a decision as far as grid size is concerned and so the point is that we start with the grid and then we refine the grid and the grid which we have taken in the beginning we call that grid as a coarse grid this is just a qualitative way because we let us suppose I have taken 100 by 100 and 200 by 200 but this is just a qualitative information that 100 by 100 is coarse and 200 by 200 is fine but let us suppose if I run both the my simulation in both the grid point and if by 100 by 100 and 200 by 200 are giving almost same result then I can say that may be 50 by 50 is a coarse grid and 100 and 100 is as fine as 200 by 200 because it is giving me the accurate solution so what I am trying to tell you here is that the answer to the question that what grid we should take in a particular problem it varies from problem to problem and we have to do a study which is called as grid independent study then only we will come to know what is coarse grid what is fine grid because once we get a result where further refinement in the grid does not show any change in the result then we can say that the grid which we are using are fine if it is leading to large change in the result then we will say that the grid which we had taken is a coarse so this word is a qualitatively and the answer also varies from problem to problem the grid size which we have to take also varies from problem to problem thank you NIT Warangal good evening sir my doubt is regarding this topic number 3 slide number 8 sir if we are doing the CFD analysis of any circular rod typically like spend nuclear fuel rods or some nuclear fuel rods with volumetric heat generation we are not considering the central point we are starting i j is equals to 1 and the circle that is next to the center and if we substitute the equation the governing equation at the center it will blow up to infinity since r is equals to 0 now if we are interested to find the temperature at the center how are we going to deal this this is a very interesting question the question is that in this slide I had taken let us suppose a square plate with a circular hole but instead of that let us suppose if there is no hole then what is pointing out that if you draw the control volume when let us suppose there is no hole or it is a solid rod such as he has given an example of nuclear rod then near to the center what you find is the control volume has only 3 faces it becomes like a triangle it does not have fourth face because at the center maybe I will show in the whiteboard so let us suppose you have a circular domain and when you draw the concentric circles and concentric radial lines also what will happen that near to the center you will get a control volume which is like this so here the surface area is 0 surface area is 0 so here what is happening is that you are getting a control volume where one of the surface area is 0 now the way I suggest to work this situation is that so when let us suppose if it is a conduction problem as surface area is 0 here so the conduction heat transfer in the radial direction on this surface will be 0 and there will be here you will have let us say q r plus dr here you have let us say q phi here you have q phi plus d phi so this way you do an energy balance then your solution will not blur thank you sir if we want to find out the temperature at the center of the nuclear fuel rod where we typically expect the maximum temperature so how are we going to evaluate it sir this is that if I want to calculate the temperature at the center of the nuclear rod how we will evaluate it let us come back to the whiteboard again I would like to make a correction that this is not a small q but it should be capital Q because the surface area is 0 heat flux is not 0 now let me say that this is the control volume first control volume near to the boundary second control volume near to the boundary and this is the third control volume near to the near to the center so you have one grid point here one grid point here and one grid point here and his question is that when we use the finite volume method we calculate the temperature here let us say this is T1 let us say this is T2 and let us say this is T3 so in a finite volume method we do not calculate temperature at face center at centroids of this control volume now the answer to this question how to calculate temperature at this point so in this case we do extrapolations like you can take this two point and extrapolate here if you want to use higher order explorations so that you get a more accurate result in that case you may have to take three points or four points so we do one sided extrapolation to calculate the temperature temperature is calculated as a function of T1 T2 T3 and so on depending upon the order of extrapolation thank you so we will stop here