 Let us go to the next topic, so with this topic I had talked about how to obtain the algebraic equation for full Navier-Stokes equation, so discretization is done now let us go to solution methodology, how to solve this system of equation, which equation we should solve first. The title of this lecture is solution of Navier-Stokes equation on a staggered grid, when people started solving the full Navier-Stokes equation initially they were struggling to get the solution, so if you go back to the development of CFD, people were in the beginning solving this using stream function vorticity formulation, where what they do as I mentioned that there is no explicit equation for temporal evolution of pressure, so handling pressure was causing lot of problem, so if you cannot handle something what you try to do you try to eliminate it, so in that stream function vorticity formulation mathematically you can at least eliminate del p by del x and del p by del y, how? You differentiate x momentum by del y del y and y momentum by del y del x cross differentiation and then subtract the two equation, so del square p by del x del y cancels down, so using cross differentiation in the x and y momentum for 2D situations only, they came up with stream function vorticity formulation, with that they but in that there is an issue that the boundary conditions you have to give in terms of stream function vorticity and many times it creates lot of trouble. And another problem with that formulation is that if you want to get pressure after getting the velocity field through stream function and vorticity, you have to solve one more equation which is a pressure Poisson equation to get the steady state pressure distribution. Later on maybe around 40 years back there were two groups which were mainly working in the developing a solution methodology using not the stream function vorticity but the actual new stoke equation, so what one was in let us say Los Almas lab and second Prussia-Paternker and Prussia-Spalding group in Imperial College London and they came up with a procedure and that is called a staggered grid where what they said is that if you want to handle pressure properly we need to avoid interpolations of pressure and mass fluxes. You know that when we discretized pressure as shown you earlier, if we had pressure grid points if we have let us say grid point for pressure let us say yellow circles for pressure different from that of velocity then only we can avoid interpolation. You know that if the fluxes comes at the face center and if your grid point is not at that face center then you have a trouble and you need to do interpolation, extrapolation. But if we define the points for let us say u velocity v velocity and pressure not at the same point but let us stagger it cleverly. So here note that what is mean by staggered grid is let us not use same spatial location for grid point for u velocity v velocity and pressure. In fact somehow I had introduced staggering directly or indirectly where I had introduced I had already introduced staggering in some sense when I said that the grid point for temperature qx and qy do you remember yellow circle red sorry green squares and red inverted triangles that is the type of staggering I am talking about where was the green square on vertical face center where was the red inverted triangles horizontal face center that is what is mean by staggering. There we did for qx, qy here we will do that green square we will say that it is a grid point for u velocity red inverted triangle we will say it is for v velocity anyway inverted triangle is similar to v shape and square is similar to u so there is a relationship and circle is for p but if you do that then your control volume is different because your control volume needs to be drawn in such a way that circle should lie at the centroid for pressure green square should lie at the centroid for u velocity and red inverted triangle should lie at the centroid of the control volume so your control volume is staggered note that not only because here you are using a finite volume method but with this staggering the interesting thing is that when you apply continuity equation which is in pressure control volume like you see qx and qy we are at the face center directly so here instead of that you will have u and v velocity so to calculate the mass fluxes you will not need any interpolations which I will show in more detail later on this is what I had discussed earlier let me these are the grid points for we are saying that we will stagger it okay now let me tell you what the problem is if we do linear interpolation for pressure let us do a linear interpolation you know that this is the pressure force in the discretized form correct on the west face pressure is compressive it acts in positive x direction on east face it acts in negative x direction so this is the total pressure force in the x direction now if the grid is uniform if you do linear interpolation this pw can be written as pp plus pw divided by 2 and this p can be expressed as this then the difference gives you this if you look into this what you are seeing is that if you use the same grid point for pressure u velocity and v velocity then when you are calculating u velocity for this point that expression involves pressures of the alternate cells not of the same cell you are getting where do we use this in x momentum equation x momentum equation is an equation for which variable u velocity so when you are calculating u velocity the discretized form of the pressure does not involve small p subscript capital P this is what is called as pressure velocity decoupling you will see this word in many CFD books there is breakage in the communication between the velocities and pressure at the same grid point if you want to avoid that you need to have if you want to avoid this breakage of communication you need to avoid this interpolation if you want to avoid this interpolation you should have grid point for u here and v here but if you have grid point for u here then the u control volume will be staggered by delta x by 2 and if you have v velocity here it will be staggered by delta y by 2 control volume is staggering I will show this later on with figures okay let me tell you what is the consequence people initially when they try to solve with this type of discretization many times they ended with this exact pressure distribution this exact pressure distribution obeys this but this is not a realistic pressure distribution so they got a conservation law obeying pressure field and velocity field but they were not physical there was no convergence issue it was converging it was obeying conservation law but the numbers were not not physical so we not only try to avoid the interpolation for the pressure but there is again thing even the velocity also if you if you calculate the mass fluxes what you realize is that in the continuity equation if you do linear interpolation then in the continuity equation that u capital P does not come into the expression you are doing a discretization of the mass conservation of a particular control volume and the velocity at that particular node does not come into the expression okay so there is a this causes a pressure velocity to decoupling and we so the basic idea is let us try to avoid the interpolation to calculate the mass fluxes and pressure at phase center so this is a staggered grid so this is the grid point for pressure this is the grid point for u velocity it is the same green square which I had shown earlier for qx and this are the grid points for v velocity note that inside the domain this green square are 4 in the x direction 5 in the y direction and then red inverted triangles are 5 in the x direction 4 in the y direction and this are the boundary points for pressure this is for u velocity this is for v velocity similarly on the other walls this is at the top wall these are the running indices so we have different grid points for grid points for u velocity v velocity and actually this is circle is not only grid point for pressure but this is also used for temperature that is why it is called as scalar control volume let me show it separately so this is a control volume for pressure it can also it could also be for temperature for non-isothermal flows and where is the control volume for u velocity this is the control volume because if u grid point is staggered control volume will also be staggered because this grid point has to light the centroid of that grid point centroid of that control volume and this is the control volume for v velocity so let me show it separately pressure control volume okay now when you look into the pressure control volume what do you realize that you have u velocity in the east phase let me tell you we are using the same convention which I discussed earlier but there I had written i,j here I am writing capital P that is the only difference but when you coded this P means i,j of u capital E means i plus 1 capital W i minus 1 capital N j plus 1 and so on so this is the control volume which is used for discretization of continuity equation and energy equation for non-isothermal flow and what you need to note here is when you want to do mass balance you can see that the grid points which are sitting at the phase center they corresponds to normal velocity so when you discretize mass conservation you need only first level of approximation second level you do not need because at the phase center you already have grid points for normal velocity this is one thing so we avoid interpolation for normal velocity but we also avoid interpolation for pressure you can see if this is the control volume this is P capital P this will be P capital E so when you saw you use this control volume where for law of conservation of momentum in x direction x momentum equation when you use that you need pressure at the phase center and now you have grid points of pressure at the phase center so you also avoid interpolation of pressure set phase center so you note that you avoid two things two interpolations interpolation of normal velocity at phase center interpolation of pressure at phase center not only next momentum equation also in y momentum equation I am also showing you the other grid points in this control volume because at the corner for you control volume you get v velocity and for v velocity control volume at the corner you get the u velocity is this clear any question on this in case of temperature the energy equation when we will try to solve also we will not require interpolation for you yes because the same control volume this question is that if we solve for temperatures this control volume will be used for temperature so for temperature you know that for temperature when you are solving you have advection term when you want to calculate enthalpy flux at the phases you need mass fluxes into temperature here you avoid mass flux but yeah that's a but not for temperature no interpolation is needed but when you want to calculate let us say momentum flux here or here you need mass flux in this at this phase center I would like to point out that for temperature you avoid interpolation for mass flux but not for x momentum and y momentum you know that mass flux not only comes in the let's say x momentum equation or energy equation this is the same control volume it is used for that mass flux will be a interpolation will be avoided but this mass flux also comes in the momentum flux x momentum flux y momentum flux to calculate mass flux here at this phase centers you have to do interpolation I'll show you what type of interpolations we do I'll show you later on but that's a good point said non-uniform velocity field has resulted because of the interpolation between the adjacent grid points that is because of the interpolation advection scheme we had used not the interpolation of the advection scheme but it is due to the interpolation of the pressure and the normal velocity or mass flux I'll go back and show it to you it's not due to advection scheme it is due to the interpolation of pressure mainly this later you also had showed interpolation for not very velocity normal velocity yeah normal velocity normal velocity which constitute mass flux but there is a second velocity advection scheme is not used for this normal that is used for advected variable so you have got feel of the picture of the different control volumes idea of the grid points okay so let us now go to the formula the solution methodology I am calling it is a finite volume method because there is some new interpolations which has come up so I wanted to discuss that those interpolations and I wanted to point out also that in this case you only need first level of approximation which is surface averaging of mass flux at the face center the second level of approximation is avoided in this case because at the face center you directly have this velocity okay now here there is something important this is the unsteady term unsteady term is there is nothing this phi is basically in this case you this in this case it is V and in temperature note that pressure control volume is used for temperature also so unsteady term is we use a cell center value so no problem can use it directly okay now here when you want to calculate the total rate at which x momentum is lost by the fluid in the control volume what you realize is that the mass flux you need here you do not have a grid for normal velocity note that you had the grid for normal velocity for the pressure and the temperature control volume so here you have to do some interpolation and what interpolation is done here is this it is just a linear interpolation so for these two phases vertical phases you need u velocity so you do linear interpolation of u velocity for north or north and south space center mass flux involves V velocity okay so this is interpolated as up plus now when you want to calculate V I would like to point out that we use these two velocities this you can see this is on south face you need V velocity here this is the south face center so you take linear interpolation of VSE plus VS divided by 2 for north face it is V capital E plus V capital P divided by 2 why because this is lying at the center this is the centroid of the north face so we do this linear interpolation to calculate the normal velocity similarly in case of V velocity sorry in case of rate at which y momentum is lost you need mark note that this mass flux in previous slide it was I was using a subscript u in this slide with along with mass flux I am using a subscript v and u velocity here is calculated as mean of these two values here mean of these two values here and v is calculated taking vp vn and vs linear interpolation when you go to the temperature you do not need any interpolation for mass flux note that why because u velocity is lying here v velocity is lying here so you do not need any interpolation note that to calculate the mass flux you avoid interpolation in continuity equation and energy equation but not in x momentum and y momentum equation is that clear any question on this you can see no interpolation I started with an study term moved on to adduction term now let us go to diffusion term what is diffusion in x momentum it is the viscous force acting in next direction where we need normal gradient in quotient coordinate system it is very easy because along the normal direction you have grid points so you use piecewise linear approximation and get the viscous forces note that on two phases it is in normal direction and two phases is in the shear direction this is to calculate the viscous forces in quad direction okay and this is to calculate the conduction heat fluxes note that I am using here I am not using here minus k so that way when it is minus k the sign are always positive actually to have a generality I have tried to avoid this minus k because maybe I can improve it because earlier I am talking of minus I will improve this slide I think it is getting a confusion actually it should be in positive x and it is little older slide so it remains I will correct this what I mean is that when I was teaching this I took diffusion coefficient as plus k for energy equation but I can reverse that sign of this arrows and I can say minus k is the diffusion coefficient so I will improve I will change this slide now another thing is that when you want to calculate pressure force so we did unsteady term advection term diffusion term then let us go to the source term what is source term in x momentum pressure force so this is the control volume for you used in x momentum equation here what you see is that when you want to calculate the total pressure force in the x direction the pressure grid point is lying at the face center of u control volume okay so you and similarly in case of v so you avoid interpolation of pressure so to be more specific you avoid where you avoid interpolations you avoid interpolation of mass fluxes at two places where in continuity and energy okay and you avoid the interpolation of pressure in x momentum as well as y momentum equation so this is the finite volume discretization of the x momentum equation this is the discretization of y momentum equation and this is the discretization of energy equation I think with this I completed the discretization I have yet to start the solution methodology now let us go to the solution methodology till now we were discussing discretization now let us go to solution methodology till now we had talked about explicit implicit method for conduction pure advection advection diffusion now this terms x momentum y momentum equation energy equation have a temporal derivative continuity equation has a temporal derivative in case of compressible flow but what is incompressible there is no temporal derivative but when you want to solve the system of equation you need to give time level in case of continuity equation also so note that in any CFD approach you have to take the continuity equation in an implicit form so this velocity normal velocity should be at the new time level okay so the continuity equation always needs to be implicit I will discuss it more little later then it will be more clear to you when you look into the full nearest equation there is no what you can call it as a fully explicit or fully implicit approach what we have is that close to explicit and close to implicit so one which is close to explicit we call it as a semi explicit one which is close to implicit we call it as a semi implicit approach let us first start with a semi explicit approach in this approach we will take the advection and diffusion term at the old-time level so but when you want to solve so in a semi explicit approach actually I will write the equation original proposition then here in this the solution methodology which we are adopting is called as predictor character method because we have a starting trouble note that our original proposition should always be implicit for continuity equation and it should be implicit for pressure otherwise you land into trouble this is the original proposition for semi explicit method if you go to semi implicit method the only difference is this advection term and this total diffusion term instead of n it will be n plus 1 so original proposition can not be pressure at n value but if you use this there is a starting trouble how many unknowns you have you want to calculate velocity of the new time level which involves pressure at the new time level which you don't know there is a starting trouble so you need a starter what is a starter instead of taking pressure at the new time level let us take pressure at the old time level but when you take starter the velocity which you will get will not be the correct one they will not obey the continuity equation note that finally we need a velocity field which obeys the momentum equation as well as continuity equation that is mass conservation so that's why this is called as a predictor step so this taking the pressure at old time level we get just got started now so this is equation 1 this is equation 2 the difference between this two is called as correction this is the original proposition this is a prediction you know the correct value is equals to predicted plus correction so correction will be correct value minus prediction so this is our correct value original proposition this is prediction so correction will be original proposition minus prediction so if you subtract this to what will happen this adduction term will become 0 this diffusion term if you subtract become 0 and you get an expression like this now the difference of this pressure we will call that as pressure correction so actually the correct value of the velocity we represent in terms of u star plus u prime so we will use a we use a superscript star for predicted value and prime for correction so the correct value of u at n plus 1 is equals to predicted value which is u star plus correction which is u prime similarly pn plus 1 is equals to pn plus p prime so this is the difference of this two pressure we will represent it as okay as I said that this is your velocity correction u velocity correction this is your pressure correction we subtract this two and we define un plus 1 is equals to u star plus u prime so the difference is u prime pn plus 1 is equals to pn plus p prime so note that here we are getting an equation for velocity correction as a function of pressure correction let me do one thing maybe you you are getting some lot of equations and maybe you are not getting feel of what I wanted to explain before showing this let me go into a slide which from which I used to give you a feel before I go into the formulation let me talk philosophically then go to equation the philosophy philosophy something like this let me discuss in words let us suppose you have pressure suppose let us say in a control volume the pressure is 1 atmosphere this is the previous time level pressure what I am saying is that let us say I took the previous time level pressure and predicted velocities so let us suppose I predicted velocities like this because this is not the correct way of doing I should take pressure of the new time level so whatever I predicted as I said this is a predicted this is not the correct why it is not correct because 4 kg is going out and 2 kg is going coming in this is not physical note that this picture is only numerical picture this corresponds to a particular iteration numerical iteration in real world you will not have this situation okay so let us suppose in one iteration we got this now let me ask you based on your understanding that if 4 kg is going out 2 kg is coming in and let us suppose you have radio knob to control the pressure inside this control volume so let us suppose you rotate the radio knob in the clockwise direction to increase the pressure and in other direction to decrease the pressure you should understand if you increase the pressure or decrease the pressure what will happen to mass coming in if you decrease the pressure what should happen to mass more mass should come in if you decrease the pressure more mass should come in if you increase the pressure less mass will come in because if you increase the pressure it will create a resistance for the incoming flow so more mass what is happening here, more mass is going out. So, what do you want to balance it? You want this incoming flow to increase. So, what you should do? You want to reduce the pressure. Let us suppose from 1 atmosphere to 0.5 atmosphere. Basically, what you are doing? I showed you philosophically, but numerically there is an equation which says how much tuning you need to do. So, there is an equation for pressure correction as a function of mass imbalance, let me tell you. What is the pressure correction in this case? 0.5. And what is the mass imbalance? Out minus in 4 minus 2, 2 kg per second. So, I am talking right now philosophically, but later on I will show you specific equations. So, that is why before showing the equation I wanted to discuss here philosophically, because we do not have explicit equation for pressure and pressure causes lot of problem. So, there is a pressure correction method which first I want to discuss philosophically, then I will discuss the equations. But when you do that, because this equation which is there it is not very accurate, because right now I am showing you one control volume, but in actual case you have many control volumes and it is a coupled system and the equation which we have consist of neighboring values only. So, this prediction, so even if you reduce it by 0.5 based on the pressure correction, you get a new velocity field at the faces, but is it guaranteed that it will converge, it will over the continuity equation? Let us see. So, let us suppose with that we got new mass fluxes, new mass flow rates. Now, what has happened? Now, more mass is coming in, this is again not a physical picture, this is a numerical picture. Let us call this as second iteration for pressure, 6 kg is coming in, 2 kg is going out. So, the mass imbalance is minus 4 kg per second. Now, what we will do? We will increase the pressure from 0.5 atmosphere to 2 atmosphere. I would like to point out that based on this mass imbalance, we have an equation to calculate the pressure correction as a function of mass imbalance. Once we get the pressure correction as a function of mass imbalance, then this new velocity fields which I am showing you, these are, we have another equation to calculate the, this change to 3. So, there is a plus 2 which is happening, correct? This change from 2 to 1. So, there is a minus 1 from where that minus 1 is coming, plus 2 minus 1 comes from velocity correction as a function of pressure correction. Maybe I had written that equation, I will show you. So, there is an equation for mass flux correction as a function of, initially let us suppose, actually what we do is that we first predict the velocity using the pressures of the previous time level. Once we predict the velocity, then we calculate the mass source, such as in this case it is 2 kg. Mass source means mass imbalance, out minus in 4 minus 2. This is mass imbalance. Then we have one equation, pressure correction as a function of mass imbalance. Let us suppose in this case, you give it point minus 0.5. Once you get this pressure correction, then we have one more equation, mass flux correction as a function of pressure correction, which gave this numbers, 1 plus 2, 2 minus 1. So, mass flux correction is how much? Plus 2 and minus 1. So, here I am discussing philosophically, but there are, as I showed here, the mass flux correction as a function of pressure correction, pressure correction is minus 0.5. We get this new mass flux correction. You get new mass imbalance and then this goes on and on. Till you get a picture, where mass imbalance is close to 0. Now, I can go to equation and show you. Basically, I need to show you how many equation, mass flux prediction, mass, how to calculate mass. So, this is just summation. That is nothing but mass flux correction as a function of pressure correction. Pressure correction as a function of mass imbalance and mass flux correction as a function of pressure correction. But at least you can get a feel that there should be a relationship. And note that with the mass imbalance is positive, the pressure correction has to be negative. So, when we will derive equation, I will show you that this pressure correction will be, there will be a negative side on the right hand side. Because the relationship between the pressure correction and mass imbalance is that, if more mass is going out, then the pressure correction has to be negative. What is pressure correction here? Minus 0.5. So, from here, we understand philosophically and it is just not that I am showing you some number distribution. Indeed, we have mathematical equation that what is, how I got this minus 0.5, how I got from 1 to 3. Is that clear? Any question on this? Right now, I am showing you one control volume. But just imagine, let us suppose I showed you 25 yellow circles. So, there are 25 radio knobs type of thing. And if you rotate one, it affects all the other, all over the domain. But here in CFD, as we take some neighbors, nearby neighbors only. So, that is why this equation of the pressure correction is not exact, there are some approximation. So, that is why it goes iterative. So, this is the first iteration, this is the second iteration. So, if this picture will keep on changing, you can create a movie out of it, mass imbalance with respect to time. But your, but fortunately, the system of equation is such that indeed you get a picture, where mass imbalance is close to 0. So, note that this is something like a, here it marches iteration by iteration. It is not a steady state and finally converges. So, note that this is a loop for solving the continuity equation. So, note that till now we are talking of unsteady problem, where in explicit method, we had only one loop that is for time marching. But here in solving the continuity equation, we use a predictor-corrector approach. Why we are using it? Because we do not have explicit equation for pressure. We convert the continuity equation into a equation for pressure and to solve it, it is a couple system, we have to solve it iteratively. So, there is for explicit method, there is one outer loop for time, there is one inner loop for continuity equation, solving continuity equation. So, what equations we are looking forward? Mass flux prediction as a function of predicted velocities and sorry velocities and pressure of previous time level. We look for second equation on pressure correction as a function of mass imbalance, third equation for mass flux correction as a function of pressure correction. Actually by staggering the velocity, can you call that grid point, staggered grid points at grid point for mass flux? Because although these velocities are staggered, but actually they act as a mass flux for continuity equation. So, this mass flux prediction is basically prediction of staggered velocities. Is that clear? Whether you are getting this? Where do you do mass balance in pressure control volume? And in pressure control volume, u velocity sitting in vertical phase center, v velocity sitting in horizontal phase center. So, the normal velocities are sitting in pressure control volume at the phases. So, mass flux prediction is basically the staggered velocity prediction. Now, I can go back and show you that those equation, then you can appreciate in a better manner. So, this is an original proposition. Note that we are predicting u velocity, but you can also call it as, because this are the velocity which will be used to calculate the mass flux in continuity equation. You can also call this as mass flux prediction, because once you know the normal velocity, you can just need to multiply by the surface area to convert and density to obtain the mass flow rates. So, you subtract these two. What is this equation? I discussed the philosophy where I was talking of different equation. Which is this equation? Velocity correction is a function of pressure correction. So, that in that slide, earlier slide I showed that mass flux prediction changed from 1 to 3 kg per second. It changed from 2 kg per second to 1 kg per second on the horizontal phases. So, that correction plus 2 minus 1 come from this equation. This is the equation for velocity correction as a function of pressure correction. That mass flux changed only when you change the pressure from, let us say, 1 atmosphere to minus 1 atmosphere to 0.5. So, you did some pressure correction. You did some pressure tuning. That pressure tuning give rise to some normal velocity change, which contributed to mass flux change. Similarly, for v velocity, original proposition prediction and correction. v velocity correction, pressure correction. So, these two slides I had shown you expression for velocity correction as a function of pressure correction. In any way, here you are predicting velocities. This predicted normal velocity will be used to calculate mass imbalance, because this velocity contribute to mass flux. This I had discussed earlier that there is no explicit equation for calculation of pressure. There are two class of method for calculation of pressure from the continuity equation. One is called as the pressure correction method and the second is called as the projection method. In the projection method, instead of pressure correction, we use pressure. This is a slightly different method. I will see if I have time, I can discuss this also, but the philosophy is more or less same. This I had discussed earlier. So, we predict the velocity using the previous time step pressure, but it does not satisfies the mass balance. Thus, we need to correct the velocity. It is only possible by correcting the pressure inside the control volume based on the mass imbalance. This is in words, because if I give you the lecture slide, you can go through it to understand what I had said. So, if more mass is getting out as compared to coming, the pressure needs to be decreased. Pressure correction should be negative and vice versa. By tuning the pressure in each cell center based on the magnitude and direction of mass imbalance, the predicted velocity and new mass imbalance will be generated. The pressure and velocity correction is continued iteratively till mass balance occurs and a divergence free velocity field is obtained. So, I had shown you expression for prediction. Now, we will go to correction. I had shown you how many equations, how to predict the normal velocity or mass fluxes and then second equation for velocity correction as a function of pressure correction. That second equation we got from by substituting by subtracting the original proposition from the predicted equations. This is our proposition. Correct velocity is equals to predicted plus correction. Now, I will show you how to convert this continuity equation to a pressure equation. In this case, it will be pressure correction equation. Actually, this pressure and velocity correction is continued iteratively till a divergence free velocity field is obtained. Then only, it is actually strictly speaking, this is the new value of u star using the old value of u star plus u prime. This new value of u star becomes u n plus 1 only if it obeys the mass balance. Is that clear? This velocity, you do not get this velocity correction in one iteration. When it is iterative, this is the new value of u star using the old value plus velocity correction. When this new value obeys the mass balance, then that star value becomes, that iterative star value becomes u n plus 1. Now, let us try to understand how to convert this continuity equation to an equation for pressure correction, because here we will get an equation for pressure correction as a function of mass imbalance. Earlier, we had got velocity correction as a function of pressure correction. So, what we will do? You are clear with this equation. This is just a discretized form of continuity equation. Now, what we will do? You will write wherever u n plus 1 is there, u star plus u prime, v n plus 1, v star plus v prime. This u star plus u prime, basically earlier I had shown you expression for, what is this? This is the u velocity correction. This is the v velocity correction. Note that we subtracted the original proposition with the predicted and we got an equation for this u prime and v prime in the previous slide, which is given by this. So, u n plus 1 is equal to u star plus u prime. For this u prime, v prime, earlier slide, I had shown you this expression. There is a mistake here. This equal sign should not be there. Now, what we will do? We will substitute this into this. Right now, I am showing for u p, v p, but similarly you can obtain u w and v s also. When we substitute into this, then what you get is this, which can be further simplified into. Actually, if you look into this, what is the nature of this equation? Let me ask you. Today morning, I showed you some discretization. Do you see any hominality between that discretization and this? You remember, I did finite difference discretization and then said, let us multiply by volume, because when you do finite volume, it is basically you do volume integral into the governing equation. So, when you multiply by the volume, what happened to that del square t by del x square? In the finite difference, you got delta x square in the denominator. When you multiplied by delta x, delta y, you got delta y by delta x. Just see here. Here, you are also getting that delta y by delta x for this two. Here, you are getting delta x by delta y. There, instead of pressure, it was temperature. Do you see any common thing between that equation and this equation? That was a diffusion equation. That was an unstudied diffusion equation. What I am trying to say, there is no unstudied term here, but there is something which seems to be like a diffusion. What he is saying basically is that this is a second order derivative. If you look and if you try to convert this into differential equation from the discretized equation, actually this is del square p prime by del x square. The discretized form will give you this two term. This two term will give you the differential equation is del square p prime by del y square. But actually, this is not equal to 0. It is actually a Poisson equation. It is not a Laplacian equation. So, if you try to convert this into differential equation to get the nature of, to understand the nature of equation, it is del square p prime by del x square plus del square p prime by del y square is equals to mass imbalance. This b, this u star into rho into delta y is what? Rho star into rho, u star into rho is mass flux multiplied by delta y, sorry. This u star multiplied by rho delta y is mass flow rate in the x direction. v star multiplied by rho delta x is mass flow rate in the y direction and you can see there is a negative sign here, which finally comes down to this b. What is this b? This is mass imbalance. So, it is like a steady state heat conduction equation with volumetric heat generation. What is the diffusion coefficient in this case? What is analogous to diffusion coefficient in this case? Time step. There you have, here in conduction, you have conductivity. Here analogously, you have time step. Any question on this? Just try to understand how we have converted the continuity equation to an equation for pressure correction. And I had started with the philosophy. Philosophy prediction, mass, pressure correction, mass imbalance, pressure tuning. The same philosophy is right now being shown mathematically. I am doing it towards the end of the day. See you tomorrow morning.