 Good morning. So, about one hour behind schedule on conduction, but that is okay. Convection we have I think some amount of buffer time hopefully, but convection is again a topic which will have lot of questions, lot more questions I think than conduction. So, that is why we have kept about six modules, six sessions for convection. From what we had yesterday in your presentations, I think most universities when private institutions give convection to go by actually in some cases about eight lectures or ten lectures is what we have seen. I think that is a little bit too short for a topic as important as convection. And probably at the BTEC level students appreciate conduction a lot more than convection because convection it may seem actually and that is how I was taught and I think probably that is how you would have been taught. Choose the correct formula, apply it Reynolds number, Prandtl number, choose the correct formula, apply it, get Nusselt number, get H, 10 marks. So unfortunately the physics is completely lost and concept of it is a beautiful subject actually which combines fluid mechanics and heat transfer something which students just do not get the feel of at the end of the heat transfer course. And how to get that into the psych of the students I think that is a very challenging thing because fluid mechanics itself is a little bit difficult subject to teach and how it is taught we do not know, but I think looking at the way it was taught to us I think fluid mechanics also will be given by hand waving you know this is what it is, this is what it is kind of thing. So when fluid mechanics foundation itself is shaky, convection is shakier if there is a word like shakier. So students do not appreciate the point is that students do not appreciate convection, students find it difficult and probably teachers also find it difficult to communicate fluid mechanics they have not done why should I bother about it because it is unfortunately curriculum has become so compartmentalized that the connection is lost and I think we should ensure that the connection is given all the time and when connection is not there I do not think anybody will say what is the use of this every time we should say this is what you did in fluid mechanics see where it is coming here it is exactly the same thing why do not let us see these two together holistically when we are able to get connection between you know what we did in high school A B C D or you know L K G and see this adjective verb adverb all those things we are connecting all the time, but fluid mechanics heat transfer connection or for that matter I think it is the same in strength of materials and design all those things also. So it is very very important to establish that connection when we teach convective heat transfer. So fluid mechanics and these are like the two wheels of a train has to go together one is one cannot be separated from the other. So with that idea all of us I think will introduce convection the same way when there is a bulk motion of the fluid etc. This is what we say what would us to see every convection is one one subject or one topic which I think all of us including the students would have experience in some form or the other meaning h a delta t is that is the simplest equation that you can have q is equal to h a delta t product of three numbers how difficult can it be to get the solution what is so great about this equation why are we spending so much you know 10 lectures 15 lectures on it the problem is q is equal to h a delta t unfortunately that h is what is very very difficult to find. So if I have to get q correctly I have to get h correctly and getting that h correctly is next to impossible but of course nowadays things are much better. So the whole idea is to get that h correctly. So students you know we can say you know you have seen convection in so many applications in real life your hot water diesel you know flow or flat plate etc. is all you know very engineering concepts but whenever there is a flow cooling of some device or all you know cooling of t for example is a convection process you can just say leave it in a room where there is no fan it is going to cool much slower where where you turn on the fan it is going to cool much faster. You you start to feel better when you stand under the fan rather than when you are just without any kind of motion. So basically all these things you know people have experience convection and you can relate to it very very easily. One other thing which you can say is turn on the fan put it in one you do not feel that much better increase the speed you start to feel better why it is just that some quantity which is taking away the heat from the body is becoming better with the increase in the speed of the fan. So that quantity is what we will call by some name called as heat transfer coefficient. So when we give such examples and connect the dots I think it is it is beautiful. So convection for students is actually a scary subject. Now we have we have to remove the fear when we teach it. So all of us know classification this is the Newton's law of cooling. So H A T S minus T infinity T S T infinity are temperature of the surface and T infinity refers to the bulk fluid temperature. Now convection again this H for example is not a material property alone. So stagnant air or stationary air versus moving air will have a different convective heat transfer coefficient all of us know that. So I cool I cool the same cup of T in a stagnant air room where there is no motion of air it is going to take longer whereas I turn on the fan it is going to take a shorter amount of time. So it is the same air it is the same T it is the same T infinity what has changed is the H meaning the heat has been removed at a faster rate. So you get a colder T at a faster rate. So this H therefore is a function of many many many things type of the fluid. So that means all the property is associated with the fluid density viscosity specific heat and also the nature of flow of that fluid. Am I right? So if it is just stagnant air where it is just going around in the room without any pattern or with some kind of a slow pattern then it is a poor cooling. When you apply a force velocity by means of a fan or whatever external means it is a better means. So some characteristic velocity is there. So geometry I keep the same cup of T in a cup versus I pour it on a saucer and keep. So it is going to take a different shape. So the area exposed is also important. So geometry of the surface that is there it is important whether it is flow over a flat plate, flow over some other geometry all these things influence the heat transfer coefficient. Heat transfer coefficient can have values you know natural convection very very small numbers to tens of thousand when you talk of phase change applications. So this H going from 0 to a few thousand values we can classify it based on the type of the fluid. So typically you have values from single digit to a few thousands and what we are saying is heat transfer coefficient for natural or free convection for gases is about order of this is where this order of becomes very important. It is in order of tens you know 10 20 whatever. Liquids natural convection 50 onwards you know force convection slightly higher but gases are on the lower side force convection of liquids it is on the higher side again and boiling and condensation this the different topic anyway where you have the heat transfer coefficient very very high it is very common to get 10,000, 12,000 watt per meter square Kelvin. So with this idea let us just get back to so this H is a very very important concept and accurate measurement accurate value of H is very very important. So of course one other thing what we see in conduction we have there will be several problems which have been solved by the time we come to convection. Heat transfer coefficient of 25 watt per meter square Kelvin 50 watts these are things which were given as numbers to the students to solve the problem but till that time till we come to convection they do not know how it was obtained it is just a black box for them. What we are going to say is we will find out the value more importantly we will try to understand and appreciate the concept of an average value versus a local value of heat transfer coefficient what is this average versus local average means average over the entire geometry surface. So what we have given to the students during conduction course is only the average value H is 25 that is where assumptions become very very important one of the assumptions we will write is uniform heat transfer coefficient over the entire surface. What does it mean whether the plane wall is 10 meters in height or 1 meter in height H remains the same what is given in the problem H remains the same but now when we when we see we will appreciate that H is not going to remain the same what was given was an average value of the heat transfer coefficient. So heat transfer rate this airfoil diagram is very nice here. So it tells me that locally for a given area Das heat flux by convection is Q convicted that is given by H local times T surface minus T infinity. So H local times T surface minus T infinity times Das which represents the surface area associated with this element. Surface area would be this length this small length times width into the plane of the board. So Das times HL times delta T. So assuming the entire surface to be at constant temperature which we can make this assumption this comes out and what I get Q convection is nothing but H average. So I have probably just write this. So Q convicted is equal to H bar or H average surface area T s minus T infinity this equal to integral A s Q double prime convicted T s am I right one is the whole equation other represents an integral based on the heat flux locally. So this I am going to write it as T s minus T infinity A s H local Das. So this is H bar A s T s minus T infinity and I can cancel this off. So I will get H bar or average heat transfer coefficient is nothing but local value integrated with respect to. So till the student studies convection he or she is just going to get an average number which how it was obtained nobody cares we used to just give them a number but this local value. So this local quantity is something which we will now learn to calculate appreciate that it is going to change dramatically because of the nature of flow over the surface and the surface conditions. So this average value is what is used most of the time in engineering applications. Local measurements are very very important and of course lot of experiments are done nowadays to measure the local heat transfer coefficient by measurement of local surface temperature etc. But getting the local measurements is very very difficult. So you have taken that Q double bar convection D A s that is equal to T s minus T infinity into H local. So here you are putting up the assumption that the surface is having the uniform constant temperature. But in some application we are not finding out the uniform surface temperature and we are taking care of the average temperature of that. Like we are measuring the temperature at different points. Yeah it is here why we have taken this T s minus T infinity outside and understood your question. T s minus T infinity is because I have maintained the boundary condition here as constant wall temperature. Constant wall temperature. Now to come to your question let us say I devise a boundary condition constant heat flux. Then I cannot take this out. Yes. But what will be remaining constant? Q double dash. Q double dash will be same everywhere. T s will be different everywhere. But H also is going to be different everywhere. But so my question is when we are finding out the local heat transfer coefficient should we take the uniform temperature of the plate or? That is what I am telling you. It depends on the boundary condition what we have imposed in our experiment. If I am doing an experiment. But even if we are imposing the boundary condition say for example uniform heat flux which is getting out from the cylinder and we are having the embedded thermocouples over the surface and we are measuring temperature and then taking care of this averaging out. We should not average out. Yeah. We should not. If you average out you get some average heat transfer coefficient. Yes. But we should take local temperature and then get the local heat transfer coefficient. So this is with the assumption that it is a uniform constant temperature only. This is the constant wall temperature boundary condition is imposed here. If there is a variation of the temperature then it should not be T s minus T infinity. T local minus T infinity. That will be inside the integral. Yeah. So that is the expression would be a little bit more complicated. So you will probably have this way just to make it yes. Some T average minus T infinity integral a s h local. This will be T x or whatever coordinate you want to call it minus T infinity d a s. This d a s will also involve the characteristic dimension x. So yeah something like that. So it will not be a simple integral. But nevertheless you can do. We take a couple of temperatures even in our setup on the other day in the manual also says your 6 temperatures for flow over a pipe. I just take the average of those 6 temperatures and calculate the heat transfer coefficient. That is not right actually. That is not right because in Dittus-Polter all of our setups I am sure we would have been maintaining constant heat flux boundary condition. But we average all those temperatures. Averaging those temperatures has no meaning. I should compute heat transfer coefficient locally for everything. So that is what we will demonstrate when we do the experiments also. We should not average the temperatures and then take those temperatures. That is no meaning. We will understand that when we get to developing flow and fully develop the concept. Okay now I think we have to come to fluid mechanics whether we like it or not. So I will just put this slide and then take a recourse to the paper again. So whenever there is a flow everybody understands what no slip condition is. So first thing which we need to tell students. Do you understand what no slip condition is? I think that is very very important. Students probably know it but they do not know it with the correct terminology. So no slip condition does not mean the fluid is at rest at the flat plate. It means the fluid if the plate is moving the fluid also moves with the same velocity. So the relative velocity is 0 at that solid surface. So no slip condition just see fluid mechanics that is why it comes first. We are able to understand things related to velocity a little bit easier. So when there is a solid surface and there is a flow happening, you are going to have what is called as a boundary layer which is formed. What is this boundary layer? Boundary layer is nothing but a region which separates the flow into two parts. It is not a physically drawn plane or something. It is done for our convenience. Why is it done? We are going to see. Boundary what we say is within the boundary layer the effects of the fluid property called as viscosity plays an important role. Beyond the boundary layer viscosity effects are negligible. It is not 0 it does not vanish. It is negligible compared to the other forces which are going to be in place. So what happens in the boundary layer? If I have a flat plate and there is a so u comma t infinity I am just putting this t infinity for now. We will use this also a little later. So I have a flat plate which x is equal to 0 is at this location. This is the y coordinate and as the fluid flows over the surface let us say t s is greater than t infinity, t s is the surface temperature of the plate is greater than t infinity. First we will see the fluid mechanics then come to the heat transfer part. So there is going to be what we call as a boundary layer which is formed over the flat plate and locally we will give it a symbol delta within parenthesis x which tells me boundary layer thickness is a function of the x direction. What is this locus of points? This line which I have drawn represents the locus of points at which u over u infinity is equal to 0.99. So what it means is everybody understands and appreciates that the fluid first layer of fluid. I always like to give this example it is like a ream of paper. So the bottom most ream is glued to the solid surface. If the surface moves, the bottom most sheet of paper is also moving at the same velocity. If the surface is stationary, the bottom most layer is also stationary. So same thing is for the fluid. Then I am trying to pull the fluid because there is a flow because that is the boundary condition which you are imposed. So what is happening? The next layer of paper is constrained by the layer below it. So it says I cannot move with the same velocity u infinity. This is pulling me to 0 whereas somebody is pushing me to u infinity. So it will take a value much closer to the solid surface value which is slightly greater than 0 but much further away from u infinity. Subsequent layers will see the effect of the solid plate lesser and lesser. And as I move away from the solid plate which is stationary in this case, this effect or this pull is going to die down slowly. And that is what we say. If I plot this as the velocity profile something like this, next layer is like this, next layer is like this, so on and so forth. And somewhere it is completely free from the presence or the effect of the solid surface. That point we will say is called as the point where we say u is approximately equal to 0.99 times u infinity. Locus of several such points which I am marking by x is will be joined to form the hydrodynamic boundary layer. And I think all students should know this part. I do not think there is any, is there any question on this boundary layer part? Now what is happening within the boundary layer? What is causing this velocity gradient? This is caused because of the inherent property of the fluid which is called as viscosity. What is wall shear stress? Wall shear stress is nothing but tau is equal to mu du by dy at y equal to 0 or at the wall. So the velocity gradient at the wall dictates the wall shear stress. Larger the gradient, larger is the wall shear stress. And how is this calculated? We just have to put it like this in the reverse way and see it. I think this is easy to understand. So this vertical axis now becomes a velocity and this is the y coordinate. So du by dy in this way will give you the velocity gradient. Steeper the line, greater is the velocity. Turbulent flow therefore, we will come to that. I am going little beyond what I should be going. Turbulent flow we will draw a velocity profile which is almost touching like this. We will draw a velocity profile which is like this. And we will say oh steeper the gradient greater is the wall shear stress. The students do not appreciate it because they are not seeing it in the form du by dy. So all you have to do is flip the paper, rotate it and then you will see du by dy is much steeper slope for turbulent flow than for laminar flow. These are all things which we have suffered as students. So we should not make them suffer again. So now if I have this velocity boundary layer, there should be something similar in thermal boundary layer. Why should there be any difference? So what we are going to say is this flat plate. I am not drawing the velocity boundary layer now. I will focus on the thermal boundary layer. You can draw it on the same graph U infinity, T infinity. Surface temperature T s is greater than T infinity. So the plate is hotter than the fluid. So the plate is going to give heat to the fluid. Same logic that we have used for hydrodynamic boundary layer. The first sheet of paper, the first layer of fluid is by default going to be at the same temperature as the solid surface. So this value is T s. Next layer of fluid will see a slightly lower temperature than the solid surface and subsequent layers will see a lesser and lesser influence. This will end at a location which is given by T minus T s divided by T infinity minus T s is equal to point right away from the surface. There is a complication which we did not have in fluid mechanics. Fluid mechanics we said U by U infinity is equal to point 99, definition of boundary layer location. I had two velocities to deal with. One was the free stream velocity U infinity, other was the local velocity U. Now in heat transfer, moment I say there is a thermal boundary layer. We have not defined it as yet. I have three quantities which I have to deal with. Local temperature, surface temperature and one new thing which we call as T infinity which is analogous to U infinity. So I have all the three are going to take part in deciding how heat is going to flow through the fluid. Am I right? I cannot leave one or the other. The temperature locally is changing just as the velocity is changing locally. What is causing this? Surface temperature I cannot leave that. U infinity like T infinity is the imposed flow condition that also cannot be removed. So this ratio where it becomes equal to approximately point 99 that we will call as the locus of points which will form the so called thermal boundary layer. Thermal boundary layer is defined by the symbol delta T of x. Again it represents that it is going to go with increasing x the boundary layer thickness is going to increase. Now I told you the first layer of fluid. I am just going to shade this dark here. First layer of fluid. What is happening to that first layer of fluid? I am just going back to this fluid and the solid surface have the same temperature at the point of contact that is the first layer. This is called as the no temperature jump condition just as you had no slip condition, no temperature jump condition. Then what does this no temperature jump condition imply? It says for the first layer of fluid which is stationary or for the first layer of fluid which has 0 relative velocity between the solid and the fluid surface. There is no bulk fluid motion. So only conduction is going to contribute to the heat transfer. So heat transfer through that first layer of fluid is by conduction alone. Very good. How does it go out? E in minus E out plus E generated equal to E stored. Steady state E stored is 0. Single layer of fluid there is no generation of heat. E generated is 0. E in minus E out equal to 0 which means E in equal to E out. E in is coming by conduction. So if I take the control surface between the first layer of fluid and the second layer of fluid heat has to go out because of bulk fluid motion by convective heat transfer. Q convected from the first layer is equal to Q conducted through the first layer of fluid. This is very very important. Therefore I can write Q convected is what? H times T minus T infinity that is a heat flux convective heat flux T surface minus T infinity. Why is it T surface? Because of low temperature jump boundary condition that is what is given there. Low temperature jump boundary condition dictates that this layer of fluid is at T surface temperature only. So T s minus T infinity is the bulk fluid temperature. It has given the heat to the subsequent layers of fluid that is equal to minus K fluid. Why? We are doing the energy balance for the first layer of fluid. We are not doing the energy balance for the solid flat plate. The energy balance is for the fluid therefore it is K fluid dT by dy at the wall. Am I clear in this? So what I get because of this is a very very very important definition for the heat transfer coefficient which tells me minus K dT by dy at y equal to 0. dT by dy at y equal to 0. What does it represent? dT by dy temperature gradient at the wall. du by dy at y equal to 0 represented velocity gradient at the wall. So analogous, mu du by dy represented shear stress, K dT by dy represents identical in form. This is equal to h T s minus T infinity. And this is the definition of heat transfer coefficient. I do not have h was never defined so far. It was just h heat transfer coefficient. We never gave a definition for it. So it represents therefore h this is very very important. h is equal to minus K fluid dT by dy at y equal to 0 divided by T s minus T infinity. Larger the temperature gradient at the wall, larger the heat transfer coefficient. Now I have drawn this picture here. This is a snapshot of the temperature distribution at this location x1. Everybody remembers the velocity distribution and this is going to keep this here. Velocity distribution is going to be at another x2 something like this. This will be like this. x2 let us say x2 is here. What will happen to the temperature profile? Will it have the same slope or a different slope? Different slope. How many say different slope? It is okay if you do not have an opinion it is okay but we need to different slope. How many say same slope? See this is anchored T s. It is fixed. That is a boundary condition. Between x1 and x2 what has changed? Fluid has gotten more amount of heat. So it has to have a larger energy content. Energy content is manifested by temperature. So this being fixed the gradient I am drawing it bad really sorry about it. See like this. So it will become fatter than this profile. It is not drawn that well but it will be a flatter profile. That is the imposed boundary condition. Surface temperature fluid thermal conductivity value is less only. So naturally it will resist the heat flow through. Fluid thermal conductivity value is less. Whatever be the fluid thermal conductivity value correspondingly the heat transfer rate will change. It might not take away that much amount of heat. If one fluid has thermal conductivity value 0.7 and another has a very large thermal conductivity the magnitude of heat transfer will change will be smaller. But that does not prevent the surface boundary condition to be violated. I mean that does not allow the surface boundary condition to be violated. So T s has in no way related directly or T s should not guide the heat transfer rate. Same T s if I have honey versus oil versus liquid metal what will be different is the heat transfer rate. But whether honey flows or liquid metal flows the first layer of fluid will still remain at T s because that is a no temperature jump boundary condition. What will be different is this one minus kf dt by y. dt by dy will change to adjust to give you a smaller value of Q because k is very very small. So boundary condition is sacred. We cannot violate any boundary condition. So your question is liquids have some liquids I mean liquids have lower thermal conductivity it does not matter it will take away lesser amount of heat. But that does not prevent I mean that does not allow me to have a different boundary condition at the surface whether I am at x1 x2 or x infinity. So what I am trying to say is more heat has been added. So we will have a fuller fatter profile here at 2 and at 1. Now this brings us to a very very important concept. So if I have infinitely long plate what will happen to the temperature profiles later? Will it be a straight line ever? So it will become bulkier and bulkier velocity distribution. Will the nature of the velocity distribution change? If I maintain laminar flow it will become let us not bring turbulent formula. Let us say it is so long that it is still laminar then do not bring turbulent because it makes life difficult. Will anything change to the velocity profile? Nature will it does not become fat it will not become thin nothing will happen. So what does it tell me? Have you all heard of fully developed flow? What is fully developed flow? What is fully developed flow? What is the flow because at all we would find any kind of change in velocity profile. So if I take a snapshot of the velocity distribution at a given x location and move downstream to other x locations identical profiles are there. That is fully developed. Can I say the same thing about temperature? Because at both the end the temperature are fixed. The blood temperature is fixed and the fluid bulk temperature is fixed. So in nature of the profile. Nature will always be this kind of concave surface. I have added more heat to it right. Conditions have changed from this point to this point. So fully developed thermally is it possible? We will answer this question when we come to internal flow. I just want to leave this as a thought provoking question. It will become clear. Can it attain? If it attains wall temperature what happens to heat transfer? Fully developed. No, I just wanted to bring this issue of continuous change in the temperature of this profile as opposed to velocity profile. That is all I wanted to bring and this question what he raised that is what I wanted to trigger. So tomorrow when we do internal flow whether a flow can become thermally fully developed or no we will understand. This is a local heat transfer versus average value problem very straightforward. H is given as a function of x and you are asked to find the average heat transfer coefficient just obtained by integration. So it comes out to be 1.11 times the local value. So average value is about 11 percent greater than the local value and this is plotted there. So nothing difficult about it. I will take care of it. This Nusselt number we will define a little later. It is come here. We will define this a little later. This is Professor Prabhu likes to put in a historical perspective. So this is Professor Nusselt. Dr. Nusselt his thesis was on thermal conductivity of insulating materials and this contributed in the area of film condensation. This is theory of condensation then combustion of pulverized coal and of course heat transfer mass transfer analogy in evaporation and his work till the age of 70 and probably one of the pioneers in heat transfer we can call. Classification of convection I am going to leave there is nothing to teach here. So internal, external, laminar versus turbulent, natural versus forced all these things like that. All of us know definition of Reynolds number, rho, some characteristic velocity times some characteristic dimension divided by the viscosity. So what those characteristic dimension and velocity are those will change if the flow is internal or external and depends on the geometry whether it is a flow across a bank of tube, cylinder, sphere or flat plate correspondingly the velocities and the length scale is going to change. So physically it represents inertial forces to viscous forces and characteristic velocity distribution in laminar flow. I would like to draw this little bit to ask you a few questions. Now this is the pipe and what is drawn here is velocity distribution for laminar flow. Everybody agrees that this is for laminar flow? Now please draw this velocity as I have drawn and on the same graph that is at the same location X please draw the turbulent velocity profile. I need you to draw the turbulent velocity profile for the same mass flow rate. M dot is fixed, area of the pipe is fixed density it is an incompressible fluid. So I want you to draw the turbulent velocity profile one minute, drawn okay. How will it be? Laminar sub layer okay. I have drawn this something like this. How many of you have drawn like this? This is turbulent, this is laminar. Hope all of you have drawn like this. What does this tell me? What is the area enclosed by this velocity distribution? It represents the mass flow rate right and my way of looking at this is this way. So larger gradient at the wall is there for the turbulent flow. Therefore mu du by dy at the wall is larger for turbulent flow. Therefore wall shear stress is much larger for turbulent flow compared to laminar flow okay. So this is my arc coordinate okay which is y like this and this is the velocity vector direction okay. Mass flow rate is fixed. So there is apparent excess area here which is compensated by a change in the area there okay. Now what is the characteristic of turbulent flows? How would you explain turbulent flow to a student? Introduction. Random motion yes. Chaotic 3 dimensional. Chaotic 3 dimensional random. What are the words? Half hazard random okay. Eddies okay. Fluid chunks. What do you mean by fluid chunks? Group of molecules okay I have not come across this way okay. Any other? Discuss. Laminar also can be discussed. What else? Vortex. Mixing okay. Okay. All these are English. How will you tell real life turbulence example? Household. How many of us have seen honeycomb honeybees? Correct. What you are saying is correct. Correct. What else? Washing machine. Parade versus crowd okay. Very good example. So traffic in a I mean that is all correct example okay. So if I open that tap very very slowly small opening it falls down beautifully okay versus I am in a hurry I just open it fully it splutters splashes all around okay. So turbulence is there everywhere okay. We all have to deal with turbulence whether we like it or no and because of that we will say mathematically to quantify turbulence we will say any turbulent quantity be it velocity, pressure, temperature or any quantity can be defined in terms of see at every point there is some kind of chaos okay. So I will say let me define a hypothetical mean quantity so that that mean is there it is like a pendulum okay. I have a mean position and it is oscillation oscillating periodically about that position. Now in a pendulum it is the same amplitude both locations whereas in turbulent it can be small here large here next time small large here small here but if I plot these fluctuations with respect to time the actual velocity distribution could be something like this the signal would come out something like this this I say I do not like this let me just break this into a mean component which is u bar this is actual velocity u I break this into a u bar and a fluctuating component of velocity. So this is u prime and I say u local at any given instant of time and position is equal to u bar plus u prime similarly y component of the velocity v is equal to v bar plus v prime z component of the velocity is w bar plus w prime if I can do it for velocities I can do it for the pressure so I can say p is equal to p bar plus fluctuating component of the pressure temperature is a scalar quantity mean temperature plus the fluctuating component of the temperature. So what is this telling me there is always some kind of a mean about which these fluctuations in that particular quantity is going to occur these fluctuations I cannot quantify it is random that is what is the characteristic of turbulence when these fluctuations die out is going to 0 I get perfectly nice orderly flow this disorder is manifesting itself in the form of these quantities who told me this I am putting this because I like to make life simple mathematical if I want to quantify u v w p and t life is going to become difficult so I say this mean velocity plus this fluctuating component is what is equal to the actual velocity and these bars here represent the mean what is this mean have you heard of the word time average what is time average so time average mean what is time average 0 to some t any other definition of time average and one other question you can take over for t in your differential equation for momentum mass conservation we have not derived the energy equation you will see terms of this nature one of the terms typical terms you will see like this correct. So this if I want to put for the turbulent part it will be u bar plus u prime and similarly some such thing would be there so what it tells me is that a very very simple equation which I could write in about six lines now when I put things in turbulence it is going to have product of several terms which will have one mean component and one fluctuating component two fluctuating components derivative of those fluctuating components it is a nightmare to do it okay so why are we studying this then we will basically say these fluctuating components mean components all these things the product of these when I do a time average we will say the product of these components time average that is given here as one of the examples is given here I am not going to go into the definition right now u prime v prime bar that above bar represents the time average value so I write the full governing equation Navier-Stokes equation one of the terms I have written here u du by dx right so several such terms I will write I will expand the bracket take the sum of the derivatives multiplied by the term in front of it so I will get u bar du bar dx plus u bar du prime dx plus u prime du bar dx plus u prime du prime dx so I will get four terms each of which is a product of two quantities I will take the time average of the entire equation so that means I will have terms which will be u bar du bar dx time average each of these term will be time average and then we say time average of the mean quantity is essentially the mean quantity mean quantity does not change with respect to time so that is essentially equal to the mean quantity itself product of the mean quantity time average will give you the product of the averages okay then we will say u prime u prime time average u prime v prime time average these will not go to 0 okay and then you will say product of this mean and a fluctuating quantity time average of that so if I have terms of this nature u bar u prime and a time average of this this will be made equal to stop at this okay we have to write this and show and we will do that.