 Very good morning to all of you so the last four classes we will be looking at a new topic a very important topic for practical engineering applications which is the turbulent convective heat transfer okay so although we do not have too much of time since we were covering other topics most of this semester so we will spend at least four lecture hours to emphasize the nature of turbulent convective heat transfer because most of the practical applications that you are all dealing with I am sure are turbulent in nature and I am also sure that many of you are already taking a course on you know some introduction to turbulence or turbulence modeling or part of this is already being covered in a fluid mechanics course okay but the approach in heat transfer is very similar except that for simple applications simple arguments we will use certain analogies and one analogy which we have already shown for the laminar external force convection is the Reynolds analogy so we saw that since the structure of the momentum and energy equations are very similar so once you get the expression for skin friction coefficient you can directly derive the expression for Nusselt number using the Reynolds analogy so to start with the simplistic kind of arguments in heat transfer turbulent heat transfer state that we can actually use certain analogies like these like the Reynolds analogy we can extend these analogies to turbulent flows also okay and reasonable accuracy they give you know in a very less time you can get a good prediction of the local Nusselt numbers and things like that without having to solve them rigorously using computational methods okay so let us quickly look at the corresponding range of the flow parameters which will classify the flows as either laminar or turbulent and as you all know in the force convection case this is the Reynolds number so if you look at external flows okay so we use the definition of Reynolds number so let us say local Reynolds number and generally if this local Reynolds number is greater than 5 into 10 power 5 this is for the flat plate so the flow tends to change from the regular streamline laminar pattern to a more chaotic turbulent pattern okay so similarly if you go to internal flows so let us say flow through a duct so we use the Reynolds number based on the diameter of the duct and typically if this is greater than say 2300 so the flow is said to be turbulent and what about the natural convection so a natural convection how do we classify what will be the parameter we can either use Grashof number or Rayleigh number okay so generally people use Rayleigh number which is the product of Grashof and Prandtl number and if you take the case of a flat plate or a cavity an enclosure so typical values of Rayleigh number greater than 10 power 7 these are classified so this could be a vertical plate or enclosure cavity this is in the natural convection in a cavity so that is why in the project I have asked you to work at Rayleigh numbers less than 10 power 7 okay up to 10 power 6 it is still laminar now what happens if you cross these thresholds either Reynolds number or Rayleigh number so when you put a probe okay so typically it could be a hot wire probe which is used to measure the local velocity instantaneous velocity at a particular point suppose you have the flat plate okay so let us say that till certain position where your Reynolds number is less than 10 power 5 so this is where our blasher solution and everything is applied okay so now after 5 into 10 power 5 you have a small transition zone in which instabilities will start will change the pattern of flow okay so once these instabilities start you will find that now the turbulent region where Reynolds number is greater than 5 into 10 power 5 you will see motions which are definitely not streamlined and not only that you will see several structures several large eddies of all length and time scales okay so large eddies typically and of a definite structure these are called coherent structures okay now you can have length scales which are quite disparate right from you know order of magnitude to 5 or 6 orders of magnitude in the length scales depending on the Reynolds number okay so the smallest eddy will be naturally lying closer to the wall okay so as you go from the turbulent boundary layer edge towards the wall the size of the eddies will naturally become smaller and smaller and finally close to the wall the flow will be only laminar okay so essentially near the wall you have the effect of viscosity which is molecular viscosity which is predominant as compared to the turbulent motion and therefore this will be referred to as the viscous sub layer or laminar sub layer okay so now the action of the wall therefore is to damp the effect of turbulence okay so without the confinement without the wall you do not have a laminar sub layer therefore and you only have a range of scales in turbulent flow so in the case of a typical boundary layer profile the effect of wall is to damp the large scales and finally the large scales cascade to smaller and smaller eddies and finally it becomes laminar close to the wall okay now only in this laminar region you have the significant effect of molecular viscosity so outside this it is all turbulent and the turbulent motion will dictate the velocity profile temperature profile and the characteristics of flow field and temperature field okay so the effect of molecular viscosity and thermal conductivity will not be significant outside the viscous sub layer okay so this is a very very interesting you know physical phenomena mainly because it is very challenging to observe so many length scales and also we have therefore different regimes different regions where the effect of the molecular viscosity can be significant and in other places where it is insignificant okay so now if you therefore look at these structures so what do we how do we capture these different length scales so if you put a probe for example let us say a hot wire probe and try to measure the velocity at a point here so how do you see the variation with respect to time okay so let us plot velocity versus time so you might start with some value at time t equal to 0 and then you might see some fluctuations could be anything but you will definitely see a lot of fluctuations like this it could be periodic need not be periodic okay now this measurement is called the instantaneous velocity that means this is at a particular point and over a time instant that you keep monitoring continuously okay and you do not apply any filter to this so whatever comes out to the probe and if the probe is quite sensitive you have to be important that pay attention that the probe is sensitive to all the time scales of turbulence and therefore it should be able to resolve these fluctuations okay now these fluctuations represent that there is inherent turbulence in the flow field okay in a laminar case you would not see this fluctuations it will be a constant value at that point in time okay so initially there could be some unsteadiness but finally once it reaches a steady state condition it will not change but in the turbulent case there is nothing like a true steady state there will be always an inherent fluctuation that you will observe and therefore so this represents the nature of turbulent flow regime now that you can capture this the how do we then post process so what we do usually is take an average of this readings okay so we can classify this entire instantaneous data into an average and a fluctuation about the mean okay so that we can assume there is a fluctuation which will represent as u prime it could be positive or negative about the mean and therefore we can decompose the instantaneous velocity as a fluctuation which we will represent is an average which will represent by an over bar okay this average can also be a function of time we will see that plus you have the fluctuating components so this is a simpler way of basically representing the instantaneous velocity because the instantaneous velocity is too much of data which we may not need okay so in order to simplify the unpost process and also solve the equations we look at the mean velocity and fluctuation about this mean and therefore we assume saying that the instantaneous velocity is equal to some of the mean and the fluctuation so now this mean velocity also can be a function of time depending on that kind of filter that you apply so when we say mean velocity here how do we calculate this mean velocity this is nothing but integral some time t2 t plus ? t u of t dt 1 over ? t okay so we can do some kind of an averaging in this case I have done what is called as a time average okay so we can average the instantaneous component over a period of time with a certain filter with this which is ? t here and we can if you do this average you get what is the what the mean velocity we have represented here okay now if you apply the filter carefully enough for example if you have an instantaneous velocity something like this so some kind of waviness which comes in time but it is periodic and repeatable so if you apply the averaging filter carefully you should be able to construct a pattern which is repeating and periodic understand so these are not small fluctuations so you should distinguish the waviness which is actually the time variation of the mean velocity from the fluctuation which is your rms velocity or root mean square velocity so if you plot the instantaneous velocity here and if you carefully apply the time filter okay such that this time filter is smaller than the waviness okay so suppose this is your time scale of the unsteadiness so your ? t should be definitely smaller than this if you apply a ? t larger than this time scale you will be only getting something like this right so it has to be smaller than the waviness time scale but it should be larger than the fluctuating time scale okay so these are the time scales of fluctuation which you want to smooth out so in order to smoothen the fluctuations you need to apply a filter which is larger than the fluctuating time scale but smaller than the time scale of the unsteadiness okay so if you select that ? t appropriately then you should be able to reconstruct you mean which is actually a function of time so this is showing that there is a periodic unsteadiness in the flow pattern the mean flow itself and this is coming after you apply the appropriate smoothing therefore the correct representation will be to say that an instantaneous velocity can be decomposed into a mean which is a function of time okay plus the fluctuating component now there are different ways of doing this averaging so what I have described here is time averaging you can also do what is called as ensemble average that means you take certain data at a certain location you take it repeat it again you repeat it again you repeat it again so you have several sets of data okay and then do a statistical averaging of all these different repetitions so that is called as ensemble average okay anyway so for the simple cases of and also considering incompressible flow regime people say that does not matter whether you take a time averaging or ensemble averaging they both lead to the same set of equations okay so we do not have to worry too much about this for incompressible flows for compressible flows we do what is called as fabric average that means we also have to account for the variation of density okay with time and also the fluctuation of density whereas in incompressible flow we assume the density is constant although there is a turbulence the turbulence is affecting only the velocity and temperature not the density the density as a property is constant okay whereas in a compressible flow the density also has fluctuations so therefore we have to average row you for example in compressible flow not just you okay so this is called fabric averaging which we will not be worrying about here you can also do what is called as spatial averaging so you can use a filter which is not time but the grid size for example so this is typically done in what is called as large ad simulations okay so you use a spatial filter which is actually the grid size which can resolve the which can actually resolve the smallest ad possible beyond that all these smaller ad's have to be modeled so you apply typically in alias the spatial filter is the grid size delta okay so like this you can talk about different ways of doing this average but nevertheless after you average you say that the instantaneous component is therefore a superposition of the mean component and the fluctuating component so like this if you do the averaging so how should you model or how should you capture all these scales so as you can see that these fluctuations are a result of the small eddies okay so the time scale of these eddies are resulting in this fluctuation so therefore how do you really resolve this so for this what we do is general the most generic case we solve the Navier-Stokes equation as it is okay and we should be able to when you solve Navier-Stokes as it is meaning there is no analytical solution to that you only do it numerically but we use a grid size which can actually resolve all the length scales length and time scales okay so this kind of treatment is called direct numerical simulation so popularly referred to as the DNS so this is the precise most precise way that means we just take whatever Navier-Stokes equation because they are valid irrespective of whether the flow is laminar or turbulent and solve them exactly but the only constraint is how much you can actually physically resolve numerically you can resolve okay so that is limited by the grid size but you should understand that this is not very trivial because most of the times the ratio of the largest eddy to the smallest eddy can be greater than 10 power 3 okay so the smallest length scale is called the Kolmogorov length scale the largest length scale could be of almost the size of for example here it could be almost comparable to the boundary layer thickness itself okay the smallest length scale is the length scale before which it finally gets dissipated into heat okay so therefore this is called the Kolmogorov length scale and if you have if you if you plot what is called the energy spectrum so usually when you are doing a DNS the first thing that you are asked to show is the energy spectrum that means it tells you how much turbulent length scales turbulent energy spec scales have you resolved okay with your grid so you will be able to show here energy cascade okay so that shows that the energy is kinetic energy turbulent kinetic energies transferred from the largest scales finally to the Kolmogorov scales okay and finally that energy is dissipated as heat at the wall due to the viscous dissipation finally the all these turbulence turbulent momentum coming from the larger eddies is actually handed over to the smaller and smaller eddies and finally the smallest eddy will dissipate this there is nothing else to dissipate take away this energy from the eddy so but it dissipates itself into heat due to viscous dissipation effects at the wall okay so generally when you do a DNS you plot the energy spectrum and show that you have resolved all the energy scales of turbulence right from the largest scale to the smallest scale and this length scales can be at least 10 power 3 the ratio of the largest to the smallest length scale so that means you should have a grid size which is 3 orders of magnitude difference so if you are therefore doing domain which is 1 meter the smallest grid size should be of the order of 1 millimeter okay this is an example right so that means you should have enough grids to resolve anything from 1 millimeter all the way to 1 meter okay and sometimes it even falls sub millimeter scale okay 10 power 4 if it goes 10 power 5 it goes you are actually resolving micron scale eddies okay so this becomes a very very challenging task especially in three dimensions this I am talking about in one dimension okay if you need resolution in turbulence is inherently three dimensions so if you are doing a three-dimensional simulation then it will be really a humongous task you are talking about meshes of 1 billion typically okay so to do even basic turbulent simulations so which will be very very computationally challenging to therefore the alternate option is doing some other ways like larger dissimulation or the more practical method which is called the Reynolds average so I think what we will do is spend the next lectures only discussing about the practical method which is the Reynolds deriving the Reynolds average Navier Stokes equation and then how do we find simple solutions to the Reynolds average Navier Stokes equations so now I mean the Reynolds averaging assumes first you decompose your instantaneous velocity into the mean and fluctuating velocity so this is called as Reynolds decomposition so this is the starting point of deriving the Reynolds averaged Navier Stokes equation which is also popularly referred to as the Rans equations. Now before we do this we also have to decompose all the flow parameters not only the velocity but also what are all the other things in the Navier Stokes equation pressure for example and finally in the energy equation we have temperature you can do the same thing for V velocity also okay we do the Reynolds decomposition for all the flow parameters here by constructing a filter which could be either a time wise filter on an ensemble wise filter and then we decompose that into a mean and fluctuating component now before we do the derivation there are certain rules that we have to list down so I will just state what are called the rules of averaging okay. So these have to be satisfied when we apply these averaging to the entire Navier Stokes equation the first thing is that when you take the average of the fluctuating component that is you would have a mean here now if you want to find the average of this what it will be it will be 0 okay so that is because you take the mean of the instantaneous that should be the same as the mean velocity itself so therefore the mean of the fluctuating component should be 0 so you have a positive fluctuation negative fluctuation so when you apply this over average it over the delta T it should cancel out so that is the first rule and therefore what does it tell you now when we apply a mean to a mean when we average the mean component over the same delta T it should give back the same mean so it does not matter how many times you average the mean component it still get back the same mean that means you still apply the same delta T over that okay so therefore it does not make any difference to the average okay now there is a product rule which states that if you take the product of mean and an instantaneous quantity okay and then you average this component you can break this into a mean of the mean which will be nothing but the mean times the mean of the instantaneous what is the mean of the instantaneous it is nothing but the mean again the mean component okay so since because the instantaneous average is 0 so if you can substitute as u mean plus u prime okay so this will be simply u prime this u bar u bar plus you have u bar u prime since u prime average is 0 that will be 0 okay so the fourth rule is the summation rule which says that if you have for example two mean values you add them and then you take a mean of that it is the same as taking the the summation of two means okay so that means if you take the sum of two means and again take a mean of that it will be the same as just the sum of the two means independent okay so number 5 apply for example mean to the integral so that means I am averaging the entire integral of an instantaneous component so I have say integral u dx and I want to average this so this averaging operation applies only to this you are not to the integral okay so this can be written as therefore what so if I apply the averaging to the instantaneous component it returns the mean so u bar dx the same thing can be applied to the differentiation operator also so if I say d by dx of you and then I apply the mean to this entire thing it will return d by dx of u bar okay so now the other component the other rule is an important rule it says that if I take for example a product of the mean and the fluctuating component and if I apply the averaging of this what do you think will happen 0 because this can be split as according to the product rule u bar x u prime bar and since u prime bar is 0 so this should be 0 okay so these are the rules that you have to keep in hand when we start deriving the Rans equation so let us quickly go over the derivation part so once you apply the rules it becomes much easier so you please write down the two dimensional steady state incompressible navier stokes including the energy equation okay so first is the continuity all in dimensional form then the x momentum so we are writing this only for force convection right now okay without the body force term but later on you can also derive this for the natural convection case right so now what we will do first is apply the Reynolds decomposition okay so we can decompose all these are instantaneous quantities interval and flows okay so we have to therefore do a Reynolds decomposition of the instantaneous quantities and then we have to average these equations so we have to apply the averaging operation over the entire equations so then only we get what are called the Reynolds averaged navier stokes equation okay so can you quickly do that for the continuity equation and tell me what you get so I have written down the Reynolds decomposition here so you should be able to so you can write the continuity equation as d by dx of u bar plus u prime d by dy of v bar plus v prime and now average of this entire equation so usually some textbooks follow some conventions for example time average is given by an over bar ensemble average is given by this kind of parenthesis okay so you write this means it is an ensemble average okay so if you write like this this is time average so some textbooks follow this convention so now how do you write this use the rules of averaging and tell me so this will be so this operator d by dx of u u bar plus u prime bar so that will be what d by dx of u bar so we can of course write this as like this plus d u prime bar by dx plus d v bar by dy plus d v prime bar by dy so according to our rule this is what this is also 0 right so therefore you have a continuity equation for the mean flow field so there is a continuity equation which satisfies the mean velocities and therefore that should also be a continuity equation satisfying the fluctuating component because since you have a continuity equation for the instantaneous flow field and there is a continuity equation satisfying the mean so if you substitute back you can actually write a continuity equation satisfying the fluctuating component also so the velocity fluctuations u prime v prime also should satisfy their own continuity equation now please do the same thing for the x momentum equation do the Reynolds decomposition and averaging and tell me what are all the terms so I will parallely do it but you also do it on your own and then verify so therefore if you look at the right hand side this is okay everybody must have written this step on the right hand side this will be this is incompressible so there is nothing there is no averaging operator on the density so 1 by rho d by dx of p bar plus p prime bar p prime bar will be 0 similarly if you do the averaging here this is u bar plus u prime bar so u prime bar is also 0 similarly here right so therefore on the right hand side it is straightforward what do we have after averaging the equation you have dp bar by dx plus d square u bar by dx square plus d square u bar by d y square so the instantaneous quantities are now replaced only by the mean quantities so let us look at the left hand side first term okay so u bar plus u prime into d by dx so how can we write this we can split into four terms so one is u bar d by dx of u bar averaging operator applied to that plus you have u bar d by dx of u prime averaging operator applied to that okay please remember how I am writing it okay so it is very careful I have not because we cannot now split the averaging operator separately to this term and this term okay so we have to first multiply these two terms and then apply the averaging operator to understand it okay it is not simply something like this so you have this term plus into multiplied by d by dx of this okay there is no rule which says that this can be written as the averaging operator of over this multiplied by d by dx averaging operator applied over this okay we have separate rules for d by dx and separate rule for the product not a combined rule okay so therefore we have to first multiply the terms and then apply the individual averaging operator so now we will go back to the averaging rule which says that we can apply the averaging operation to these independent things okay so now we have two more terms left one is what u prime into du bar by dx whole bar plus we have u prime du prime by dx whole bar okay so therefore now we can apply the averaging rule so what it will be for this case u bar d by dx u bar which is fine plus this will be 0 so you have u bar d by dx of u prime bar which will be 0 so when it comes to this now we have the last term okay we have u prime bar multiplied by du prime bar by dx but we do not have an averaging rule for that okay which says that this should be 0 okay in fact when you say this can be actually returned you look at it as d by dx we will see that u prime u prime and then apply the average so when you take the product of the fluctuating components multiply it with it itself with either u prime u prime or u prime v prime and then take the average this is not 0 you have to be very careful because this is the one which comes as turbulent stresses so in this case this is nonzero quantity we will just write it as u prime into du prime by dx okay we can put the averaging there but that that does not matter so we can actually leave it as it is okay so therefore we can just keep the averaging operator like this so if you take the second term we have v d by dy of so what do you think will come out of this second term on the LHS so we have v bar dou u bar by dou y that is correct the mean components will come as they are now fluctuating component when they are multiplied with the derivative of the mean components they will be gone and you have another component which is v bar du bar by dy and the mean of this which will not be 0 okay so therefore if you write the combined equation on the left hand side so you have u bar du bar by dx plus v bar du bar by dy so this looks similar to the instantaneous terms convective terms plus additionally you have the term which is u bar u prime du prime by dx averaged plus v prime du prime by dy averaged the right hand side you have all the mean quantities so now we are going to introduce the continuity equation for the fluctuating component which says that du prime by dx plus dv prime by dy is equal to 0 so therefore we can write this as d by dx of u prime u prime bar plus d by dy of u prime v prime bar minus what do you have d by dx of what do you have one case you have u prime du prime by dx plus what is the other term u prime dv prime by dy okay if you take u prime common then we have actually du prime by dx plus dv prime by dy which will be satisfying the continuity for the fluctuating component so therefore this will be 0 you understand so similarly I ask you to repeat the same thing for the V momentum also okay so therefore from the V momentum what do you get you have if you repeat the same averaging exercise you will have the mean components in the advection term plus you have additional terms you have u prime v prime bar plus d by dy of v prime v prime bar the other terms will be 0 on the right hand side you have dp bar by dy plus nu into d square v by dx square so all these are mean components okay so now therefore we will stop here so in the turbulent trans equations you have similar terms in the advection component as the instantaneous but they are replaced by the mean quantities apart from that you have two additional terms on the left hand side okay so these are the u prime u prime derivative and u prime v prime derivative so if you write the y momentum equation also you get similar derivatives of the product of the fluctuating components so these can be actually taken towards the right hand side okay and they can be clubbed to the existing shear stresses the shear stresses are nothing but tau xx tau xy and so on apart from that you also have u prime u prime bar u prime v prime bar and so on so these are called as now turbulent stresses okay so although the nature of these stresses are originating from the advection term okay they are named as turbulent stresses just for the ease of grouping these together so they are grouped along with the viscous stresses and then they are called together as the total stress so it seems that there is an additional stress which is coming in turbulence but that is originating from the inertial terms and that is why when they are taken to the other side they have given a negative sign okay because they enhance the momentum so they are not in the conventional sense that they are not stresses which impede the flow so they are the ones which are actually promoting the exchange of momentum and energy therefore so they have a negative sign because they are originating from the inertial terms so these are called as turbulent stresses so what I suggest you to do is do the same thing for energy equation also and see what is the Reynolds averaged energy equation so tomorrow we will list down all these Rans equations together and then see how we treat these turbulent stresses okay thank you