 Good morning everybody. I hope you are all there and you had a nice weekend. So now let us just take recap of what we had studied on Friday. So I will be taking a fast detour on whatever we learnt on fluid mechanics. Essentially we spent time on fluid mechanics on Friday. Let me just take a recap of what we studied on the other day. That is we first said that there are two approaches. One is control volume approach and the differential approach and we said that complete velocity distribution, the pressure distribution and the temperature distribution would be obtained only in the differential approach. So differential approach is essential if you have to improve the design for example in aero foil or so and then we took a fluid particle and said that okay when it is in general function there are it can be visualized that it can undergo translation, linear deformation, rotation, angular deformation altogether or separately. So then we tried to mathematically represent each of these translation, linear deformation, rotation and angular deformation. Translation is nothing but velocity gradient. So it is the translation is very easily to visualize and then we went for linear deformation and we understood what the important thing is that del u by del x is stretching in the x direction and similarly del v by del y is stretching in the y direction and del w by del z is stretching in the z direction. So essentially for a fluid machination del u by del x is not just a velocity gradient I need to visualize that as stretching. So stretching or compression. So but then if it is positive I would like to remember that as stretching. So that is what essentially reduces to so that is if I were to consider net change in volume so I would have del u by del x plus del v by del y plus del w by del z that is why for an incompressible flow where there is no change in the volume we say that this del u by del x plus del v by del y plus del w by del z is equal to 0 that is what is called as volumetric dilation rate. So once we understand this linear deformation we went for rotation and angular deformation and that is essentially the velocity gradient del v by del x that is the gradient of the velocity not in the same direction but in the other direction. So del v by del x and del u by del y. So then we got the rotation as by taking appropriately counter clock wise as positive and clock wise as negative we got the rotations in z x and y directions as average of the both the velocity gradient that is half of del v by del x minus del u by del y is the rotation in the z direction. So is in x and y directions. So then we represented that in terms of curl and for deformation that is the angular deformation. So the net angular deformation is delta alpha and delta beta what is delta alpha and delta beta this is the net change in the deformation in angular deformation that is delta alpha and delta beta. If I represent that in the velocity gradient the net deformation would be del u by del y plus del v by del x this is nothing but my shear strain and that is what I kept telling in the last class that tau equal to mu del u by del y what we simplistically represent is nothing but the shear strain and another point I had missed out my students Sudhir just reminded me later on which I want to emphasize is that here this shear stress equal to viscosity into shear strain can be represented only for Newtonian fluids that is that shear stress is proportional to shear strain this statement can be made only for Newtonian fluids for non Newtonian fluid the function would be little different for example rheological fluids. So this I need to reemphasize that is for Newtonian fluids shear stress is proportional to shear strain and that proportionality constant is the viscosity and this is the shear strain. So this is what is fundamentally coming from so tau equal to mu del u by del y essentially represents this. So now that we have understood what is shear strain this is of course how do you represent the rotation we have solved a tutorial problem also on that so I need not have to spend time on that so we said that this is where incompressible flow field there is no net change of volume but incompressible flow field there is a net change of volume for a fluid particle. So then we went on a fast track and solved sorry derived the continuity equation and I will just represent the final equation that is d rho by dt plus rho del dot v equal to 0 and we said that d rho by dt is the total derivative that is that is the change of anything for example in this case it is density it can be velocity it can be temperature here when I say d rho by dt I have change in the density not only with time but also with space by virtue of yeah this is the total derivative not this sorry this is the total derivative by virtue of being moved in u v and w velocities that is we have velocities u v and w because of the convection we have the change of the density in x y and z and also change of density in time this is what we call as total derivative and substantial derivative I took the example for temperature also for moving into an ac rho. So that is this d rho by dt plus rho del dot v equal to 0 for incompressible fluids this d rho by dt is going to be 0 so I end up getting del dot v equal to 0 that is essentially del u by del x plus del v by del y plus del w by del z is equal to 0. So now we went for Navier-Stokes equation and we said that we have the rate of change of momentum has to be balanced by the surface forces in the body forces the surface forces considered are normal stresses shear stresses and pressure forces and body forces we usually do not take them in our closed form solutions but the body force generally which come into picture are gravity forces corallis forces centrifugal forces electromagnetic forces or any other body forces which you can think of okay. So now we took I am not going to get into the derivation because we have spent enough time on the other day so we went for we derived a complete Navier-Stokes equation and we ended up with this final equation so this is the rho du by dt that is the total derivative of the velocity u and this is nothing but the pressure force del p by del x which is coming from the pressure force and this is the viscous force and this is also the viscous force and plus the body force. So now we said that when the viscosity is 0 my the Navier-Stokes equation is going to reduce to the Bernoulli's equation or the Euler's equation so that is what we said and then we said that essentially there are four unknowns and we have four equations but we cannot get the closed form solutions under all situations because the equations are non-linear although it is well posed problem and of course we did it on a fast track how do we represent that is we I did not answer the question for example shear stress we can shear strain or shear stress it is very easily to look at that is sigma h y equal to mu del u by del y plus del v by del x but normal shear normal stress is very difficult to visualize 2 mu del u by del x but how did this come from so without deriving I had just stated that and complete derivation I said it is there here and I had requested you to go ahead and derive this yourself complete algebra is here with basic assumption I hope you have done over the weekend. So this is what is this derivation is going to lead us to that is it is going to tell that sigma xx is equal to 2 mu epsilon xx b is minus 2 by 3 del u by del x plus del v by del y so that is about the Navier Stokes equations okay so then we started off with what is turbulent flow we said that okay there is we define turbulence as an irregular motion which is generally makes its appearance in fluids but that is too difficult to mathematically handle so we redefined it as which way a random variation whenever the word a random comes I can take the recourse of statistics that is what we did so we said that okay if I have a velocity distribution varying with time remember this is for steady flow in mean mean steady flow mean flow it is steady that is my u bar does not vary with time so I have the fluctuating component u prime and the mean velocity u bar and then we went ahead Professor Arun taught us how do we how the fluctuating component averaged over time is 0 but if I square it and average I am going to have a finite value and that is this u prime squared bar and that is what we are going to tell that as normal stresses later today and this is the shear stress component that is u prime v prime bar that is I am taking at every instant the fluctuating component in the x direction of velocity and fluctuating component of y velocity component and that is v prime and multiply that and average that over a period of time that is u prime v prime bar and we said these are the rules and that is where we stopped in the last class so what we are going to do now is that we will take before we go to the energy equation although I wanted to start the energy equation but I thought that I should not be skipping this and going for energy equation I would just Reynolds average the continuity equation and the momentum equation I will just do it for 1 and I am not going to do it for this Reynolds averaging for the energy equation I will leave it to you for yourself of course it is not there in the notes maybe we will do that and put up in the notes. So this is the these are the continuity equations and the momentum equations of course I have taken for compressible flow sorry incompressible flow that is you would have noted here I have removed the rho by dt what is that we are going to do now so what I am going to do is that I am going to substitute in the above equations in these equations u as u bar plus u prime v as v bar plus v prime and similarly for w if I do that for the continuity equation I am not going to do all the steps very slowly I would request you to go through all these steps yourself because you have to get started with energy equation today so I intend to complete this Reynolds averaging within next 10 minutes so let us see how do we do that so now I have this is the continuity equation I substitute for u as u bar plus u prime and I am Reynolds averaging I am averaging this and then I have v bar plus v prime and I am averaging similarly for w bar plus w prime and had stated in one of the rules that the gradient that is if I have del f bar by del s gradient and if I Reynolds average that I can write that as del f bar by del s that is what essentially I am going to do here as you can see here this reduces to only del u bar by del x why because u prime bar we have just seen u prime bar is we have shown just in the last class that if you see here u prime bar is 0 so that is what we are doing here if I put that del u bar by del x plus del v bar by del y plus del u del w bar by del z equal to 0 this is what I get from my continuity equation now let us see if I do the Reynolds averaging for x momentum equation what does it happen so now so similarly I will take del u by del t so I have del u prime del u bar by t plus del u prime by t this tends to 0 so I get del u bar by del t but then for mean steady flows if I take this term also gets to 0 now let me take the next term but before I do I need to just do rearranging here so what am I doing here if you just watch carefully the x momentum equation I have u del u by del x plus v del u by del y plus w del u by del z I have rewritten this as del u squared by del x plus del u v by del y plus del u w by del z if you expand this you would see that the terms which are there other than these three terms if you sum them up they tend to be they appear as continuity equation and they become 0 just to watch this modification why because if I write this equation in this form Reynolds averaging is little easier that is the reason why I am writing these three these three terms of this equation in little different form okay so now I take this term I take this term and now first term of this is del u squared upon del x if I do the Reynolds averaging del u bar plus u prime whole squared whole bar upon del x if I do that I get del u prime squared bar by del x plus del u bar squared by del x so here again u prime u bar if I do the Reynolds averaging will become 0 that is also has been shown already in the last class that is u bar u prime if you see here this term u bar sorry u prime v bar u bar sorry u prime v bar if I Reynolds average that becomes 0 and v prime u bar if I Reynolds average that becomes 0 and okay this is also you see here so u squared bar you have u prime squared bar and u bar u bar so this term u bar u prime bar becomes 0 which is what we have two terms of that those two get they become 0 that is what essentially I am doing here so I get the first term as del u prime bar squared by del x plus 2 u bar del u bar by del x that is what I get for the first term first term and for the second term that is del u v by del y if I Reynolds average that I will be getting u bar del v bar by del y plus v bar del u bar by del y plus del u prime v prime bar upon del y similarly you can do for u v third term third term and similarly if I do for the pressure I will be ending up with del p bar by del x if I substitute all of this if I substitute all of this you would see that this is the additional term I have got del u prime bar squared and again this is the first term I have 2 u bar del u bar by del x plus u bar del v bar by del y plus v bar del u bar by del y plus del u prime v prime bar by del y plus u bar del u bar by del z plus w bar del u bar by del z plus del u prime w bar by del z plus all of these terms on the right hand side there is nothing extra which is coming in the right hand side because all the fluctuating components get vanish so but then if you see here something interesting is there in this equation so if you see this what is that I am just rearranging this if I rearrange this so this first few terms if you just take if you just see this okay I do not think I can show both the terms okay so if I rearrange this you have you see here 2 u bar del u bar by del x I am writing one of the terms as u bar del u bar by del x plus the next term u bar del u bar by del x here u bar del v bar by del y that is u bar del v bar by del y u bar if I take out second is v bar del u bar by del y and then again you have u bar del u del w bar by del z and w bar del u bar by del z that is here so this tends to become 0 because it is nothing but my continuity equation so I have the left hand side reduced to this and these terms which are there here del u prime bar squared by del x plus dou u prime v prime bar by del y plus dou u prime bar by del y plus dou u prime w prime bar by del z I am pushing it to the right hand side with a negative sign that is why I have minus rho del u prime squared by del x plus del u prime v prime bar by del y less dou u prime w bar w prime bar upon del z if you see here carefully these three terms were any way there in my earlier Navier stokes equation what is the additional term which is coming into picture that is the normal stresses del u prime squared bar by del x plus u prime v prime by del y plus del u prime w bar prime by del z the font size is less this will be the right way of looking at so these were the you see these three terms were any way there earlier but these are the terms which are coming up because of because of fluctuating components or because of the turbulence so this is this can be thought of if these three the Reynolds averaging if I do for y momentum equation and z momentum equation these are the nine terms which are coming new so these new terms if I write them in a vector these are the new terms which are coming out if I write them these are the three new terms if I write in determinant form if I write them in a determinant form you see here this is the normal stress this is the shear stress so these three terms that is minus rho u prime squared bar minus rho v prime squared bar and minus rho w prime squared bar can be visualized as normal stresses let me write this on the white board that is minus rho u prime squared bar minus rho v prime squared bar minus rho w prime squared bar can be visualized these three can be visualized as normal stresses but if you see the other three that is minus rho u prime v prime minus rho u prime w prime minus rho v prime w prime if you see that and minus rho w prime v prime that is let us write that minus rho u prime v prime bar minus rho u prime w prime bar and minus rho v prime w prime bar minus rho v prime which else that is it is that all is there anything else yes these three only these three are actually shear stresses which shear stresses these normal stresses and shear stresses now are not because of velocity gradients but because of fluctuation fluctuating components these are because of fluctuating components please remember that these fluctuating components are they there in amelior flow no these fluctuating components they are there only in turbulent flow this is what we need to remember and realize for the case of turbulent flow this is what is called as these stresses are called as Reynolds stresses why they are called as Reynolds stresses it goes without saying because Reynolds was the first person to do this Reynolds averaging and give us this approach when did he do he did it in 1895 what he did it 200 years back is it 200 years back no 100 and 120 or 110 years back we are doing it so what we are saying is that in turbulent flow we anyway had laminar stresses what were those laminar stresses they were essentially because of velocity gradients that is tau or sigma x y is because of mu into del u by del y plus del v by del x and normal stress was 2 mu del u by del x minus 2 by 3 mu del dot v these were essentially because of velocity gradient over and above this velocity gradient these fluctuating components if they are super imposed over and above the laminar stresses these turbulent stresses come into picture that is what we studied if you recollect if you recollect in turbulent this is okay in laminar flow they go in laminae in turbulent flow there is all sorts of mixing that is what is being said this mixing is because of turbulent stresses so what I wanted to say is that the shear stress is because of laminar stress and turbulent stress let us write this this is very important shear stress is equal to usually tau laminar plus tau turbulent what is tau laminar what is tau laminar laminar simple cases for simple cases I have mu del u by del y that means simple cases means for a flow or a flat plate why have I taken del v by del x equal to 0 here in this case because there will be no velocity gradient in v v there will be no that is what in a simple case if you want you can still go ahead and take no issue del v by del x you can take no issue plus what is this turbulent stresses is minus rho u prime v prime assuming that there is no w prime is that okay so now if you recollect in the turbulent boundary layer we had said that how many regions are there three regions what are those we have laminar sublayer and we have buffer layer. And we have turbulent boundary layer and we said that in laminar boundary layer it is going to be laminar shear streamline and in buffer layer laminar shear stress will be of the same order of turbulent shear stress and in turbulent shear stress it will be same of turbulent. Why in laminar boundary sublayer there are there is only the velocity gradients? Because fluctuating component u prime, u prime is 0 and v prime is 0, but here turbulent shear stresses it is not that velocity gradients are 0. It is just that turbulent shear stresses that is minus rho u prime v prime is much much greater than velocity gradient that is all I am saying. It is much much greater that is essentially what I am writing in my next transparency. So, as I said in a if you take a pipe wall I had taken earlier flat plate you have viscous sublayer near the wall and overlap layer or the buffer layer and the outer layer. So, you have laminar shear stress greater than turbulent shear stress. Why because fluctuating components are very very small and overlap layer both are same that is laminar shear stresses of the same order of turbulent shear stress and turbulent boundary layer laminar shear stress is much lower than the turbulent shear stress. Now I want to remind you every time there is a question by most of the participants what do I do with when I put ribbed n answers or n answers. What am I doing when I put an n answer my n answer height is such that this viscous sublayer is broken there will be all my shear stress contribution will be majorly by turbulent shear stress. So, that my heat transfer coefficient will be majorly by turbulent shear stress that is essentially what we are saying by putting ribbed n answers or enhancing surfaces. Now there is something called universal law let me tell this I would be reminding you when I go to what is called as for internal flows let me write this. So, what am I saying in the viscous sublayer I want all of you to write along with me in viscous sublayer I am saying that in laminar sublayer or viscous sublayer u plus is equal to y plus I will tell you in a minute what is u plus and what is y plus in buffer layer, but I want all of you to write along with me buffer layer it is take little time to understand buffer layer is u plus is equal to 5 log of y plus minus 3.05 5 log of y plus minus what is that 3.05 for this is for y plus less than 5 and this is for y plus greater than 5 and less than 30 this is less than 30 and turbulent boundary layer is u plus equal to 2.5 log of y plus plus 5.5 log of y plus plus 5.5 for y plus greater than 30 what is y plus I am not told y plus is y u tau by nu what is y y is as I move along the boundary layer that is y direction if this is my flat plate this is my y is that okay what is nu what is nu kinematic is u tau u tau is square root of tau wall by rho what is tau wall shear stress. So, it depends if I am in the laminar boundary layer laminar viscous laminar sub layer or viscous sub layer then what will be my tau laminar tau wall it will be velocity gradient mu into del u by del y if I am in the buffer layer it will be both gradient and the turbulent stresses if I am in the turbulent boundary layer it will be purely turbulent shear stress. So, now another thing u plus what is u plus u upon u tau u upon u tau okay how did I write these equations who gave me this equation these equations are written only when I can measure this shear stress right I have to measure the shear stresses that means I have to measure the velocity gradients in the laminar sub layer I have to measure the fluctuating components everywhere and then reduce this in this form if how did people do this actually they did the experiment and they found that and they found that this is you see I am plotting u plus on the y axis and y plus on the x axis. So, this is mistake here x axis is y plus it is not y u tau it is y plus if I take you see it is linear initially and later on this is actually logarithmic this is logarithmic plot. So, you have this is viscous sub layer this is buffer layer and this is turbulent boundary layer and there was another question on the other day why three layers why not two layers only as I was saying there was a discontinuity the viscous sub layer and the turbulent boundary layer there is a discontinuity that discontinuity you see if you extend this turbulent boundary layer it is not going to join the viscous sub layer. So, that is that discontinuity is covered by this intermediate buffer layer equation between y plus 5 and 30. So, this elaboration of representing u plus and y plus is what is called as universal profile or universal boundary layer profile or sometimes it is loosely called as universal velocity profile also why is it universal because it works not only for external flow, but also for internal flow whenever you come up with an experimental new experimental setup or whenever you write a numerical code you first check whether your experimental setup is right or wrong with this universal velocity profile with this universal velocity profile you have to first check if it is not matching this that means something is wrong in our setup or something is wrong in our numerical procedure what I am adopting. So, this is the mother of all that is why it is called universal, but it is not so universal as we would like it to be for example, for a jet impinging a plate this does not work it is only working till today it people have found that it works for internal flows that is flow inside a pipe and also for flow over a flat plate. So, it does not work for all situations, but still for most of the situations it work that is why it is called as universal velocity profile. So, this is this is what is the fundamental nature of turbulence. So, let me just take recap of what we did today in last half an hour. So, what we did we said that we have two components that is we have mean component and the fluctuating component we Reynolds averaged the continuity equation nothing new came out and we Reynolds averaged the momentum equation x momentum y momentum and z momentum equation in addition to the terms what we had earlier we came up with 6 new terms that is these three are the normal additional turbulent stresses and these three are these three are shear stresses additional turbulent shear stresses. It is like visualizing that in addition to laminar stresses we are super imposing the turbulent shear stresses because of fluctuating component velocity fluctuating component after understanding that we went to say that this is what it is shear stress because of laminar and shear stress because of turbulent and then we went for universal velocity profile we have three layers structure viscous sublayer overlap layer outer layer and in the viscous sublayer laminar dominate and turbulent layer turbulent shear stress dominate overlap layer or buffer layer both are important and this is represented by universal velocity profile or universal structure and that is represented by u plus and y plus I will come back to this y plus equal to 5 to decide what should be my trip or the enhancer thickness so that I am going to break my viscous sublayer. So please remember this when I come to internal flow for breaking the viscous sublayer.