 So, just we introduced you know little bit about turbulence and this fluctuating quantity mean quantity time average all these things we just brought in ok, but you know there is a complete derivation. So, what we will do is we will go through the derivation of the mass momentum conservation. Once we derive the momentum conservation equation then we will do this time averaging and putting it into turbulent form time averaging and then you will see the presence of this product of the mean and the fluctuating components and the derivatives of the same. So, instead of bringing it now we will bring it a little later point to note at this stage what you what you can take home at this stage it will come back again later is that because of the presence of turbulence because of this random motion of the particles of the molecules there is an additional contribution to the shear stress that additional contribution is we call it again as the turbulent shear stress just because it is of a form which is given by rho u prime p prime bar average. And we say that this I do not know to put it under anything else this is also increasing the shear stress in turbulence. So, this quantity we will call it by a name turbulent shear stress and therefore we will say du by dy times mu is shear stress definition. So, this also should have a similar definition we will say du bar by dy will keep the velocity gradient all the unknowns or all the randomness is put in this mu t this mu t is referred to as mu t mu is viscosity this mu t is referred to as some kind of a turbulent viscosity physically it is not viscosity is not becoming turbulent or anything it is this randomness which is manifesting itself in this mathematical thing which is given by u t. In fact, you see here t total is equal to mu du bar by dy which is the regular thing which we knew laminar part or whatever you want to call it plus the mu t du bar by dy because we like to put it in the same functional mathematical form we have expressed this rho u prime v prime time average as a mu t du bar by dy it is just the representation mathematically and then we say this mu is nothing but rho times mu that is definition. So, this is the part associated with mu du by dy and since again I want the same mathematical form I will call this mu t as a rho times em epsilon m and all the aspects of turbulence is put in this magical black box called eddy diffusivity of momentum ok who gives me this value nobody. So, I have I have not got any additional information to understand turbulence or to solve a turbulent problem by casting this term in this form all I have done is to make myself feel comfortable this looks familiar to me because it is in the nice form mu du by dy where this mu is now some additional term is also there who gives me this additional term nobody knows for that we have what we call as closure relationship closure relationship essentially see if I have a system of equation two equation two unknowns system is closed two equation three unknowns is a problem ok. So, when the number of unknowns become more than the number of equation our equations are conservation of mass momentum three directions and we will derive one scalar energy equation five equation I should have only five unknowns u v w p and p. Now, when I put this new turbulence aspect inside I have additional unknowns in the form of turbulence u prime v prime t prime p prime w prime ok I have massaged it all and make made it as epsilon m some quantity somebody has to give me this value ok. So, if those of you who have used fluent especially you would have seen turbulence model k epsilon all different kind of models which would have been a you click a button and it gets operation. Somebody has come up with these different kind of models which give a mathematical form or a functional form for these unknown quantities which is eddy diffusivity of momentum ok. So, this is nothing but mu t by rho and later on when we derive the energy equation we will also have minus k d t by d y represents the regular normal functional form for the heat flux this minus I mean k t d t by d y represents this turbulent contribution and again I will recast this k as rho c p times alpha because alpha definition we will use and I will say this k t turbulent thermal conductivity which has no physical meaning ok. It is just a mathematically created term just as heat transfer coefficient associated with radiation has no physical meaning h r we have we said h r is sigma epsilon t s square plus t surrounding square times t s plus t surrounding it is a function of temperature ok. So, such thing is not a unique quantity then it is a it is a fictitious quantity same thing I will say this k t this functional form is familiar to me I like to deal with this functional form therefore I will cast this as rho c p times epsilon h is equal to k t. So, that I bring the q into this form and this epsilon h is called as a eddy diffusivity of heat. So, contribution of the turbulent part is taken care by introduction of two quantities epsilon m and epsilon h problem is not closed because these two are unknown quantities therefore we have to make assumptions or input what we call as turbulence models to take care of these assumptions can be made epsilon m is equal to epsilon h that can be made but still one of them has to be input. So, the problem is you have you are simplified the equation so that it looks familiar and nice and you put in the one unknown which is by this turbulence model always we like to be in familiar territory anything new is always you know repulsing. So, we cast the turbulence equation it is very nasty in a form which is soothing to the eye nothing you know difficult to see but the unknowns are embedded in these quantities which have to be moded with this information we will stop with turbulence we will go back when we derive the momentum equations that time we will do the time averaging etc. Now, we will continue with some other concepts related to convection I just finish this eddy motion and thus eddy diffusivities are much larger than the molecular counterparts in the core region of the turbulent boundary layer. Eddy motion loses its intensity close to the wall and diminishes at the wall because of no slip condition these are all eddy is what just the mixing you can visualize the mixing was somebody gave me an example of washing machine it is a perfect thing everything is random total chaos. So, the mixing when we when you ask a student to draw the turbulent velocity distributed you just show some arrows in an all possible direction same thing the eddy is diminishing close to the wall because of no slip condition velocity and temperature profiles are nearly uniform in the in the core region of a turbulent boundary layer but very very steep in the layer adjacent to the wall what does this mean means if I draw the boundary layer and if it is a turbulent boundary layer I am blowing this up deliberately making it big this part close to the wall very very steep gradients are observed that is what is written there velocity and temperature profiles are nearly uniform that means in this part between this part and this part the almost constant velocity almost nearly uniform in the core region this we call as the core region of the boundary layer and very very steeply in the thin region adjacent to the wall resulting in large velocity and temperature gradient of the wall this will contribute to a larger du by dy at the wall greater than du by dy for laminate. So, this is the characteristic of turbulent flow. So, when I when I draw these velocity profiles it should be without a doubt turbulent it should be without a doubt laminar I cannot have a laminar profile which has almost constant velocity in this part I cannot it is wrong I have to draw laminar like this I cannot draw laminar like this it is not acceptable ok same thing for turbulent I have to have nearly uniform why because this whole mixing is causing the bulk fluid to move with roughly a uniform velocity that is what is meant by this velocity distribution and you can relate this to temperature also the fluid at all regions is roughly at the same temperature therefore the temperature profile is almost uniform therefore the velocity and temperature profiles are nearly uniform in the core region but very steep in the thin layer adjacent to the wall and the same point has come twice velocity and temperature gradients are large larger is the heat transfer and the associated wall shear stress. So, all of us are familiar with this characteristic diagram which is there in all textbooks if I have a flat plate velocity profile is laminar for some region what is the transition Reynolds number for flow over a flat plate not 2300 right. So, 5 times 10 to the power 5 roughly and turbulence transition does not happen at that number it happens over a range. So, all these misconception students will have these just by talking we can remove this yeah there is here we have put a sharp edge for a flat plate why is it so in fact in all of the textbooks it will be sharp why. If it is blunt what will happen? If it is blunt like this, what will happen? Yeah, flow may separate start because flow will start separating from here. I do not know whether it will become turbulent here or not that depends on my bluntness but I want my boundary layer to start of my plane. So, that is precisely why everyone in the textbook they put that as the sharp edge because the flow should separate there at the sharp edge. So, that little thing we need to notice and emphasize in the class why is that edge sharp. So, as the flow progresses downstream turbulence will certain slowly once the Reynolds number increases to a value where we are given a number 5 times 10 to the power 5 it can be before or after that over a range of Reynolds number and you will have books which show presence of transition region or some of them will just show transition to turbulence directly at a point both of them are okay I think students can draw either of them. Here I somehow do not like this too much because the velocity profile still looks laminar here but in a sense it should be this nature the turbulent kind of profile in the core region. Core region you see it is almost flat see here it is almost uniform velocity there is what we call as the laminar scrub layer I think all of us are familiar with this laminar sub layer is defined as a small region adjacent to the wall and there is this coordinate system U plus and Y plus which is there in fluid mechanics Y plus and U plus are given by wall shear stress by density under root and Y U star by mu or some such definitions are there. These are basically splitting the region into a laminar sub layer a buffer region that means laminar sub layer is the region where all it all the effects of turbulence are not there which is essentially behaving like a laminar layer buffer region is something which we do not know so we call it buffer where both can happen and the turbulent core okay and the coordinate axis instead of a Y we will define a coordinate as a Y plus and this will be typically 0 to 5 of Y plus 5 to 30 will be the buffer region and Y plus beyond 30 will be the turbulent core this is there in fluid mechanics but we are not concerned with that at this point yeah. So this does not tell anything about thickness it only tells that the turbulent boundary layer comprises of three things that is one is laminar boundary layer which is very much near the wall and away from the laminar boundary layer there is a buffer layer and above that there is turbulent boundary layer how do I differentiate these three how do I know that this is indeed where does where do I know that in turbulent boundary layer laminar sub layer is getting over it is not called as laminar boundary layer it is called as laminar sub layer or viscous sub layer how do I know that that is getting over when I start measuring shear stress the shear stress within the laminar sub layer will be indeed equal to mu del u by del y and what will be the turbulent shear stress in the turbulent boundary layer that is the turbulent portion in the top portion it will be minus rho u prime v prime what will be u del u by del y there what will be u del u by del y there mu del u by del y almost nil almost nil but what will be u prime v prime bar in my viscous sub layer yeah so what will be my minus rho u prime y prime rho u prime v prime bar in viscous sub layer zero because the flow is laminar there so shear stress is dominant by velocity gradients in the laminar sub layer and in the turbulent portion it will be only the fluctuating components that is minus rho u prime v prime and what is buffer layer buffer layer in which both the both the velocity gradients and the rho u prime v prime components will have the same order same order same order that is why it is called buffer layer so that is how we can differentiate coming back to your question it this doesn't tell anything about the thickness this doesn't tell you anything about the thickness so I don't know how to answer your question once again it is it will be function of Reynolds number and my location it will be if I have to answer for laminar boundary layer we know that delta is a function of Reynolds number similarly for turbulent also it will be function of Reynolds number Reynolds number yes it grows faster in turbulent boundary layer compared to laminar so turbulent boundary layer grows faster than in fact if you go back to that figure the rate at which it is growing there it is different yeah you see the way it is growing here is much x to the power of 0.5 and here it will be growing x to the power of 0.2 0.8 I think 0.8 0.8 so to answer your question the turbulent boundary layer size will be larger in turbulent portion compared to laminar did I answer your question no no what does this figure tell so that notion is that notion is why do you say maybe we will discuss there is a point perhaps you are trying to make why do you say that turbulent boundary layer thickness will be lower than that than that of the laminar boundary layer thickness huh why why do we prefer it that's a very good question that's a very good question that's very good yeah yeah heat transfer coefficient will be very high no it's not like that it's not like that see we cannot directly attribute to always to boundary layer thickness alone so what is the character of my boundary layer thickness decides my heat transfer coefficient if my boundary layer is turbulent it doesn't say that it is thin it only says that my shear stress is because of turbulent shear stress turbulent shear stress is way high compared to laminar shear that is indeed going to increase my heat transfer coefficient as professor has already showed us in fact just to add to what sir is saying see this is your laminar boundary layer and let us say transition has occurred right away like this we will see this a little later if I plot the heat transfer coefficient so to be there already it will go down here and then transition to turbulent it will increase and then start to decrease again so yes turbulent layer inherently has a larger heat transfer coefficient okay that is because of the additional component of heat removal capability which is there because of the turbulent aspect because of this u prime v prime w prime product with temperature so basically what was it k d t by d y at the wall represented the heat removal rate now I am adding another term to it right that is rho C p v prime t prime just make life easy k t this was what rho C p v prime v prime p prime bar so these v prime and t prime is contributing for my higher heat transfer coefficient similarly the way we define laminar boundary layer buffer layer turbulent layer similarly we can define this was for hydrodynamics similarly we can define it for thermal boundary layer how can I visualize in thermal boundary layer laminar thermal boundary layer will be k d t d y turbulent thermal boundary layer it will be rho C p v prime p prime and buffer layer will be both in comparable order of magnitude so to summarize it is not the thickness which is going to tell me why heat transfer coefficient is high it is going to tell the character of the boundary layer thickness only suggest whether my heat transfer coefficient is higher if that in fact you see as a thickness increases in fact books will give this explanation because boundary layer is offering a resistance to heat transfer h decreases as we move downstream this is probably what you have got confused with I think that is the reason wrong interpretation is the reason yes your heat transfer coefficient is decreasing with as I move downstream in the x direction but that is not thickness of the boundary layer is not the reason reason exactly for it it is because of these aspects which are going to change downstream so colloquially it is being put as the thickness of the boundary layer increases the heat transfer coefficient decreases so it has got ingrained in our system that is the cause thickness change is not the cause for the decrease just because actually because this nobody has asked us so far because this probably there is a doubt which nobody asked in even in our regular classes so we probably will bring this up to the students just to bring this into perspective I think out of memory I have written these this is the boundary layer turbulent boundary layer and it has been divided into three regions this is called as the laminar sub layer again it is a misnomer viscous sub layer you can call it books will call it laminar sub layer essentially where the viscous effects are the most dominant that is why it is called as a viscous sub layer above that above this region we have what is called as so called buffer layer and the turbulent core I think it is a picture is there so where does one region end and where does the other one begin that is has to be quantified mathematically so what has been done is some go ahead everything is there encyclopedia so this is your this is in case of a pipe it is going to be the same as where also this region is called as the viscous sub layer this is the velocity profile so one half of this is called as half the top half of the velocity profile for the pipe is gone this is the buffer or the overlap region and this is the outer or the turbulent core and what you see correspondingly here you see here this is the pipe center line this is the pipe wall this portion till here is the so called laminar shear stress this part is your turbulent contribution so in viscous sub layer or misnomer again laminar sub layer tau laminar or the viscous contribution to the shear stress mu du by dy at the wall is much much greater than the turbulence effect overlap layer these are of similar order of magnitude and for turbulent core we call it as a turbulent core it is much smaller compared to the turbulence contribution so why is it there why is it done like that so that we can neglect when we when I have an equation with 20 terms I would want to simplify it so when I divide it instead of using the 20 term equation all across the region I will say this 20 terms become 5 terms in this region because these these these go to 0 they are not 0 they are much smaller compared to the other terms in the turbulent core all the viscous terms I will drop off I will have a different set of 7 or 8 terms which is easier for me psychologically first and then mathematically to evaluate and in the buffer region I cannot do much I have to keep the whole thing so for a small region of interest I have to deal with the full equation otherwise for me both I can simplify the equations considerably if I split the flow into various regions and these are the definitions in the viscous sub layer we have defined these definitions are here u u tau in books it probably is called u star does not matter it is the same thing wall shear stress by rho square root is this so called u tau and a non-dimensional velocity here u plus is u by u tau and this is y plus is y u tau by mu so in the laminar sub layer which extends from 0 to 5 value of y plus so y plus is 0 means y is equal to 0 y plus going to 5 means the reference side is 5 corresponding y will be whatever number then 5 to 30 is this buffer or overlap region beyond 30 of y plus is this turbulent core in that 3 distinct regions I will give the equation for the non-dimensional velocity as u plus is equal to y plus u plus is equal to 5 log y plus minus these are all empirically fitted this is the overlap region and this one is the turbulent core this is done by the slope matching etcetera universal velocity profile actually why is it universal because no matter whether my flow is external or internal flow if I am building my setup first whether it is piv, hot wire, pitot anything no matter what measurements scheme I am using to check whether my setup is right or wrong right or wrong first thing I am going to do is to check whether my universal velocity profile is matching or not because you see left hand side also is not that left hand side or the right hand side do I know do I know right hand side what is y plus y plus is y u tau by mu what is u tau u tau is shear stress so within the laminar sub layer shear stress has to come from del u by del y which perhaps I have measured ok so again in the turbulent boundary layer what is y plus y u tau by mu professor said that u tau is again tau wall is that square root of tau wall by rho tau wall is what minus rho u prime v prime I should have measured that if I measure people have measured that and they have found extensively no matter what scheme or what sorry what measurement method I am using whether it is a wind tunnel or a pipe flow this works you see those are all values dotted circles are all measured values this is coming from Kharufit ok so but of course this is having theoretical background but still we are not deriving that we are just stating that but this can be derived from theory only thing that y plus 5 5.5 all these constants are fudged little bit here and there so they are sort of semi empirical relations ok so this can be derived from fundamentals ok so this is that is the reason why it is called as universal velocity distribution why is it universal no matter whether it is internal or external it works for us because actually we did not plan to teach all this but now that you ask the question so we came back here so I guess we have answered reasonably well what is various layers yeah please go ahead see when I am measuring all that yes if I understand your question I have got the shear stresses measured by how will I measure shear stress I will get the velocity profile completely with time and space fine in fact not only experimental one would check even in numerical whether this velocity profile distribution works or not for turbulence models so what will you do let me answer your question if I am right what will you do is if it is numerical then I have whole lot of information I have velocity profile velocity as a function of time and velocity of function of space so what I should do now take that data take that data and fit that in this form you got my point so it should work to decide what you are working in elaborate your question no problem numerical experimental are not different honestly yeah see ok now you have gotten into turbulence modeling ok I will tell you see if you use k epsilon model you what is the what is the influent or anywhere what is the warning it gives you what should be your y plus y plus it should be greater than 30 why why y plus should be greater than 30 because up to 30 I am going to assume my this velocity universal velocity profile distribution u plus equal to y plus for all y plus less than 5 and buffer layer y plus between 5 to 30 is buffer layer for all y plus greater than 30 only it is going to model as turbulence yes yes so that means inherent assumption is that universal velocity profile is going to work that is what is inbuilt in k epsilon turbulence model but let me let me complete for that universal profile is not valid for every case although it is said universal it is not truly universal it works for internal and external flows for smooth flows now let us say you all of a sudden put some disturbance here I have a disturbance here now flow takes place flow separates all of a sudden then this universal velocity profile does not work or for jet flows let us say free shear flows it does not work so that is why people do not recommend k epsilon turbulence model for highly separated flows or for free shear flows they do not separate so when they say y plus should be greater than 30 they mean that within the buffer layer and the laminar sub layer they are going to use these two relations that is that is they understand that is they understand that is what they call as high Reynolds number turbulence model and low Reynolds number turbulence model high Reynolds number is k epsilon low Reynolds number means my mesh has to be so small that my y plus should be less than less than 5 should be less than 5 how will I know whether my y plus is less than 5 or not I will choose a mesh as small as possible compute after computing only I will get my shear stresses then check back whether my y plus is less than 5 or not so that is how I have to iterate and get the mesh so until my fundamentals are right I cannot understand what I am doing in fluid so that is that is the understand so okay these are your you know relationships for wall shear stress skin friction coefficient and total frictional force okay now this brings me to one question which a colleague of mine would ask during fluid mechanics Viva what is the difference between this C F and F what is the difference between fanning friction factor and Darcy's friction factor I think the names are familiar so one is based on the wall shear stress other is based on the okay so which one is which one is a one is off by a factor of 4 right as long as it is an idea it is fine okay so I think we had drawn this thermal boundary layer earlier this is for a case where you have wall temperature smaller than the free stream temperature heat is flowing from the fluid to the wall okay so thickness of the thermal boundary layer at any location is defined as the distance from the surface at which the difference T minus Ts equals 0.99 times the maximum possible difference another important aspect he put surface rougheners for making our heat transfer coefficient high in heat exchangers we put surface rougheners why do we put surface rougheners why do we put surface rougheners to make it so turbulent so how do I decide my surface roughness height how do I decide that how do I decide that I should ensure that my boundary layer is thoroughly thoroughly turbulent not at all laminar that means I should have my surface roughness height should be such that my roughness height is greater than y plus of 30 or 5 30 let us take 30 no problem but usually we try to do 5 so I should measure the pressure drop I should measure the pressure drop such and get the shear stress and see whether my y plus is greater than 5 or not to remind you y plus equal to y u tau by nu u tau is square root of tau wall by rho how will I get my tau wall tau wall is my pressure drop or velocity gradient velocity gradient is difficult to measure tau wall is related to pressure drop so if I measure pressure drop so I will get my tau wall from that if I back calculate and get my y plus I should ensure that my y plus is greater than 5 you got my point so this is the physical significance of y plus whether I put surface roughness whichever type of surface roughness I should ensure that my y plus is greater than 5 then only my surface roughener whatever I have put makes sense it plays a role is that ok see this is a bone this is a contention see see point is as long as it is buffer there is a turbulence component already in it so you see putting a roughener does not come free when I put a roughener what will happen my pressure drop will increase higher the pressure drop sorry higher the height higher the pressure drop so we tend to we always try to optimize there are hundreds of papers on optimizing the height but usually the thumb rule is that we do not try to increase the height greater than y plus of 5 why because above 5 yes it will improve but already my boundary layer is going to be buffer the shear stress is reasonably high my heat transfer coefficients are also are going to be reasonably high so you try not to increase the height too much because you are your pumping power is going to go really high is that ok ok. So, we said thermal boundary layer develops boundary layer grows well on temperature profile is there all those things these two velocity and thermal boundary layer are not mutually exclusive they happen at the same time when there is a flow and there is a temperature difference between the fluid and the solid both this so called velocity boundary layer and the so called thermal boundary layer are going to be formed whether they form whether they are growing at the same rate which one grows faster which one grows slower are they of the same thickness at a given location all these things are directly related to the type of the fluid which is in use ok the shape of the temperature profile dictates the convective heat transfer coefficient meaning again we are coming back to gradient of temperature at the wall ok. So, temperature profile dictates the dT by dy which dictates the heat transfer coefficient ok. So, for flow over heated surface both thermal and hydrodynamic boundary layers will develop simultaneously noting that I am just reading it because it is a very well worded statement noting that the fluid velocity will have a strong influence on the temperature profile the development of the velocity boundary layer relative to the thermal boundary layer will have a strong influence on the heat transfer coefficient or convective heat transfer why again if the velocity boundary layer is different than what it is the corresponding du by dy will be different. du by dy will be different means corresponding temperature distribution because of the coupling between the fluid mechanics and the heat transfer we will see in the energy equation the temperature profile will get different because the temperature profile gets different the gradient of temperature at the wall will get different therefore your h will get different. So, these are you cannot diverse this one velocity and temperature they are married for life you cannot do anything about it whether you like it or no it is coupled ok. So, how I mean the rate of growth of the hydrodynamic and the thermal boundary layer is linked and this linkage is very very very important in deciding the heat transfer coefficient and what is this property of the fluid which decides which one grows faster or which one grows slower we call it as Prandtl number ok. Prandtl number again there is no definition it it is the relative thickness of the velocity and the thermal boundary layer is described by the dimensionless parameter called as the Prandtl number it is defined as molecular diffusivity of momentum to molecular diffusivity of heat ok. And we will say that it is nothing but nu over alpha nu because it is related to the viscous effect momentum transfer alpha because it is related to the heat transfer and we will call this by definition of alpha and this will be rho related to rho and mu this will be related to rho Cp by k you will get mu Cp by k as the Prandtl number. Now, what is the typical value of Prandtl number Prandtl number can be a very small number typically for liquid metals extremely small for oils very large glycerin etcetera it is much larger what is it going to play a role in I am not coming to this slide I just have to draw something. So, Prandtl number we have written this definition molecular diffusivity of momentum to molecular diffusivity of heat what do you understand by this how do you explain this to a student. So, that is that is you are telling me what is going to be there in the next slide this is what it is nothing to laugh this is correct because it is one of the most difficult concepts to explain to a student in case of liquid metal ok. So, if I want to if I have a flat plate and let me take this location x 1 at this location how far from see the effect of viscosity is felt why because of the presence of the solid surface otherwise it is a property of the fluid it is there with the fluid it is taking it there is no nothing happening, but because of a solid wall because of a solid surface all this headache of boundary layer is formed. So the presence of this solid surface has an effect on the velocity distribution which means momentum is getting transferred from one fluid layer to the other ok. So, when there was no solid surface there was no issue because velocity is uniform now because of the gradient in velocity because of this change in the velocity from one fluid layer to the other what is happening is that there is a transfer of momentum I have a larger momentum I give it to you. So that ultimately we are in equilibrium. So, this transfer of momentum takes place from the solid surface to the free stream ok. Now this is for velocity again if I had T infinity alone nothing would be a problem because I have T surface different from T infinity whether larger or smaller we do not care there is some transfer of some diffusion what is diffusion? Diffusion is just propagation correct. So, how much or how far over a given amount of time momentum is propagated versus how far in the same amount of time heat is propagated ok. So, that I introduce this time because that is the constraint we are imposing ok. So, in if the momentum has gone only to so much distance and heat has gone to so much distance that means that particular fluid is able to diffuse heat at a faster rate ok. So, my now when we say molecular diffusivity of momentum to molecular diffusivity of heat is directly related to the boundary layer thicknesses. Why? Because this viscous effects are felt where in the boundary layer region only beyond that I do not care about the viscous effect I do care, but they are comparably I mean comparatively smaller than the other effects in the boundary layer this viscous effects are very very strong. Therefore, what this means is for momentum this particular fluid is able to propagate to a smaller distance. So, this would represent my hydrodynamic boundary layer and larger the distance this would represent the corresponding thermal boundary layer and then what I say now forget these definitions I will say which property of the fluid takes care of this heat transfer alpha ok. Alpha is very very large therefore it is able to transfer heat very rapidly compared to therefore the two slides later what we have is Pr to the power n is of the same order as delta over delta t ok. So, Prandtl number of the order 1 would mean these two boundary layers are of similar thickness order of magnitude same ok then Prandtl number much smaller than 1 meaning delta t is much larger than delta and Prandtl number much smaller than 1 delta t is larger than delta liquid metal what he brought out carries away heat very very fast in liquid metal cooled reactor why because of this property alpha. So, what is happening in that momentum diffuses to a smaller extent heat diffuses much rapidly and if you go back to this table I am shifting back and forth sorry liquid metal Prandtl number is very very small nu over alpha is very very small or alpha is relatively very very large. On the other hand if you take a viscous fluid oil for example Prandtl number is of the order 52 God knows 1 lakh or something nu is much larger compared to alpha. So, your hydrodynamic boundary layer is very very thick compared to the thermal boundary layer ok. So, what it means is how good that particular fluid everybody likes certain thing everybody does not like everything. So, that fluid probably does not like to transfer heat very quickly it likes to transfer momentum very quickly ok. So, we it so we will give it a number we will call it a Prandtl number Prandtl number is very very small implies that fluid likes to transfer heat much rapidly than it is able to transfer momentum inherent disability you cannot do anything about it ok. So, this concept of momentum diffusivity to heat diffusivity is very very useful I do not I do not want to emphasize so much, but one of my pet things. So, if I have the momentum layer like this hydrodynamic boundary layer like this and thermal boundary layer like this exaggerated ok. My velocity distribution correct the fluid does not have any further change in the velocity in this part. For the velocity profile this part is where 0.99 has been reached the temperature if I had another color I would be able to plot there, but anyway this would be your temperature profile ok. Other case temperature profile free stream temperature T infinity ok. So, this is the velocity profile u of y this is u of y this is T of y T of y. So, this is T surface this is y this is x this is x this is y this is my velocity distribution this is my temperature distribution almost constant at a value T infinity beyond the thermal boundary layer. This is for Prandtl number greater than 1 Prandtl number less than 1 ok I have drawn these two. So, what is it merely a drawing exercise or is it more useful it is very useful why because if I am talking of a liquid metal I can for liquid metal this case the velocity is almost uniform except for a small region here temperature is going to change ok. Whereas other case temperature is almost constant is equal to T infinity for the entire part of the boundary layer except for the small part. So, when I am having the energy equation you will see the energy equation energy equation will have terms of the form u and T ok. For me to take both these variables and it is nightmare. So, if I am able to make one of them a constant it is only the other variable which is varying for me mathematically the equations get very simplified very is a strong word it gets simplified. So, with the assumption or with the fact that the Prandtl number is so in a very very small or very very large I can call the velocity is equal to u infinity here completely it is an approximation, but it is u infinity almost throughout the boundary layer except for a very small part. Here I can call temperature is almost equal to T infinity throughout ok an approximation again, but for me when I am dealing with derivatives etcetera derivative of a constant 0. So, terms can get vanished very easily life becomes not easy mathematical ok. So, not just on paper we are studying these things practically also for modeling or for being able to simplify equation to get some kind of a solution these are useful things. So, one of my favorite questions all times is draw this and draw the velocity temperature profile last year also I made a mass this year also. But natural convection where is it said that delta is equal to delta T no, no, no whatever theory we are studying here is valid even for natural convection for natural convection in fact I am going. Prandtl number there also it should affect no. Why I can I. No if I correct if I get your statement right you go ahead. I think what you are getting confused is most textbooks when they introduce natural convection will draw this, this is what they will draw in the derivation of the governing equation this is implying that Prandtl number is approximately 1 they are not bringing in additional concepts of these kind of differences that is probably the reason why this thought has come. Yeah if I am drawing like this this means velocity boundary layer and the so called thermal boundary layer or of same order of point force is active because of density difference. So, this concept boundary layer thickness is going to suggest that both are of same size it is they can be different as long as my Prandtl number is different. This is all fine but all that we are saying is that delta is going to be different from delta t based on Prandtl number where is that we are saying for vertical plate always delta is going to be equal to delta t where is the implication. So, there I am fixing my Prandtl number as 0.7 but if I change my fluid to water let us say if I consider natural convection in water with the Prandtl number of 6 we will discuss this. So, no issues we will record this question natural convection we are going to go we are going to non-dimensionalize the momentum and energy equations. So, we will realize that. That means governing equation non-dimensionalization of natural convection you know PR should be independent it should be independent of PR. PR will come it will be there okay we will see we will discuss. So, these are essentially what we have said delta is equal to delta t for PR equal to 1 greater than delta t for PR greater than 1 and less than delta t for PR less than 1 and I think students should be asked these are things which you can you have to test students of okay. So, I cannot overemphasize the importance of these things of course Prandtl number is given after Professor Prandtl again brief history we have this boundary layer theory essentially by Prandtl. So, he was the person who has contributed to fluid mechanics by giving us this concept of boundary layer and 1904 concept of boundary layer adjoining the surface of a body moving in the fluid which has led us to understand concept of skin friction and how stream lining is going to affect the drag in airfoil wings etc. Okay. So, what we will do real quick is try to talk about I think many of you have done are doing integral methods most of you are teaching integral approach to derivation of mass and momentum equation or energy equation whoever has done energy equation also yesterday we saw it was mostly integral method very hardly anybody is doing differential analysis any is there anybody who is doing differential analysis taking us you are doing one differential okay. So, by and large everybody is doing an integral approach. So, that is something which we are not going to do we will do a differential analysis of mass momentum and energy equation. No, because examination point of view is that emphasize there integral there is no derivation per se which is needed right okay I think so they just give the derivation and then I mean give the form and then cancel off terms for