Lecture 57

Hello everyone, welcome back. So we have been discussing about Reynolds average Navier-Stokes equations and we saw what are the different kind of turbulence models and where we can use them, what are the limitations of the turbulence models and what are the situations where we cannot be sure of the results from the RANS or the turbulence models. So in those cases we need to go to some high resolution simulations, for example we have been discussing about the large eddy simulation which is LES and also we have been discussing about the direct numerical simulation which is DNS. So we have primarily three methods of simulating a fluid, one is a Reynolds average Navier-Stokes, we have studied a great deal about it. In the next few lectures we are going to see how we can perform LES and DNS and what are the practical considerations when we are using or performing LES or DNS. So one thing that we have discussed so many times is that compared to RANS the LES and DNS are very very expensive and one primary reason is that in Reynolds average Navier-Stokes equations or in the RANS simulations we solve for the average quantity and we do not get the fluctuating quantity, we model the fluctuating quantity. So we can have different kind of models, we can have the model which basically gives you the final output of turbulence kinetic energy or the dissipation rate or the specific dissipation rate. So like we have k epsilon model, k omega model, SST k omega model and then there are models which are based on stresses directly also, computation of stresses directly which are the Reynolds stress models which are not the eddy viscosity models. So because we are only interested in the mean flow so we need not go into the final scales of basically finding out the exact turbulence quantities but if a statistical value of the turbulence quantity we get that is good enough if we are doing for the mean flow. On the other hand is the DNS which is the direct numerical simulation and as the name indicates we are going to simulate the flow as it is numerically, we are not going to employ any model that is why it is called the direct numerical simulation. And because we are simulating directly we need to resolve all the scales of the turbulent flow and that is why it is very very expensive and today we will see that what are the expense that we are talking about like let us say we are doing RANS and the DNS. So what is the computational cost that is going to increase when we shift from RANS to DNS. So we will look about that very shortly. In between you have LES and if you recall the diagram that we have discussed some time back that in DNS we do not model any scale, we just simulate everything. In Reynolds we may model the entire turbulence and LES we basically if you talk in terms of eddies we know that turbulence flow is made of large eddies and small eddies. So in LES we simulate the large eddies but we model very small eddies. And there are multiple versions of LES also, there are wall model LES, there are very large LES, then there are hybrid methods of doing LES like hybrid RANS LES approach, there is well model. There are many variants of LES also, we will not go into those variants of LES into great detail but we would rather see that what do we mean when we perform LES. And once you have the physical picture of what we are trying to do using LES we can actually based on your interest you can go and look into different kind of LES models. So if you look at this picture this is a jet flow and you can find I mean this is a turbulent jet flow as you can clearly see and you can find different kind of scales in this flow. You can have very very small eddies okay you can for example you can see here there is a very small eddy and then you have this eddy and you have many different scales okay very small eddies like here like here okay. And in order to resolve this entire flow field you need to have grids which can for example resolve this also and you can resolve this entire big flow field also right. So you have variety of eddies and you basically when we say eddy we have a picture of like this in mind okay but it is a kind of flow which has got some kind of vorticity right and you can have large eddy and the small eddy but it is very difficult to basically identify or basically segregate a large eddy from very small eddy because these all eddies are these all different scales are mixed up okay. But in order to capture very small flow features okay which are very fine features we need to have that kind of grid only then we can capture that right. So look at any turbulent flow this is random the turbulent flow is random it is unorganized and despite being random it has got some kind of statistical properties okay it is chaotic and it has got multiple scale okay. So we can have very very small scale to very very large scale let us take the example of boundary layer for example okay. So you can have something which is happening very close to the wall very close to the wall let us say here and then you can have something which is of the scale of this okay. Similarly you can have let us say you go inside a channel okay so this is your large scale phenomena and then you have something inside the boundary layer okay. So you have to basically resolve both the smallest scale as well as the largest scale. If you talk let us say in terms of atmospheric boundary layer okay so this is let us say this is your big atmosphere okay. So you have to basically resolve it can go to the kilometers okay and then from the ground to the level where the ground effects are not very important so I am talking about that domain. So that can go up to kilometers on the other hand you can have few meters or centimeters which is very close to the boundary layer so both phenomena are important. So if you want to let us say resolve this atmospheric boundary layer using a diatomical simulation in order to have a grid which is basically very very small in size which can capture the boundary layer very close to the earth and it can also capture the effects of the earth outside okay. So this is one example I think we have seen this earlier also. So we have got a mean flow I mean this is a typical picture of a turbulent flow where you can basically divide your any flow field into a mean flow and a let us say fluctuating quantity okay. So this mean flow imparts energy okay to the eddies and you have large eddies so these are basically large eddies okay and these are very small eddies okay. So remember that famous Richardson poem that the large eddies impart energy to the next level of eddies and then they will basically impart energy to the next one and so on and so forth till they go to the smallest scale and when it is goes to the smallest eddy okay that is where the dissipation takes place right. So it is a entire cascade okay from the very large eddies to very small eddies. So here what is important is the dissipation right because here the viscosity basically plays a big role and here what is important is the production because these eddies get energy from the mean flow and in between you will have the transfer of energy. So same thing if you see in terms of the energy cascade right. So what you see here is wave number, the k is the wave number here okay this is not the turbulence kinetic energy okay. So this wave number can be related to the inverse of the eddy size. So if you have a eddy size let us say this is your eddy size typical eddy size whose length is let us say L. So then wave number could be written as 2 pi by. So if the wave number is increasing then your eddy size is decreasing. So this is very large eddy this side if you see this is very large eddies and this side is basically small eddies right. Now what you see in the y axis is basically the turbulence kinetic energy density it is not the kinetic energy but the kinetic energy density that means if you take the area under the curve this entire thing this will give you the entire kinetic energy right and this is plotted in the log scale okay and why it is plotted in log scale so that you can find some slope here and this slope has got some meaning okay if you have attended any course in turbulence you know that this slope is around minus 5 third and so on and so forth right. So when you plot in log scale it will be a straight line right so this is the large eddy and then this is the smallest eddy right and you can see this large eddy is represented by length scale L and the smallest eddy is represented by length scale eta. So right what happens is that in this region for example let us say somewhere around here up to this point the production is important okay and after this let us say the dissipation is important and in between is the transfer. So in the region where the production is important what are the quantity that are important is your turbulence kinetic energy and the dissipation rate right and where there is a dissipation of energy what is important is your viscosity because the dissipation is happening due to the viscosity right as well as your dissipation and in between where there is a transfer of energy what is important is the length scale L and the dissipation of energy right. Now what is our interest why we are discussing this in detail and this will be coming to this picture again and again because this is the crux of the matter which differentiates your RANS to DNS and LES okay in RANS you will not see this diagram because we do not segregate the turbulence kinetic energy based on the wave number or the eddy size okay we basically model it we basically model. So I will call it KE because for the turbulence kinetic energy also we use the same symbol as K so in order to avoid any confusion I will call it TKE okay it is called TKE. So in the RANS simulations okay we directly model TKE and we do not segregate based on the wave numbers we cannot get this picture but what we can get is the entire integrated quantity of this diagram right. On the other hand when we do LES or DNS so when you do DNS we will get this entire spectrum okay if you have a well resolved DNS we should get a picture like this we should get this five third curve we should get the dissipation range and we should get this integral range okay when you have LES so that is somewhere in between okay. So LES what you can do you can argue that okay I do not have that much computational resources to simulate the entire scale and I am not very much interested in what is happening at very very small scale right so your scales at which the dissipation is happening may be not that important to basically capture a mean flow dynamics right or even the some of the unsteady dynamics. So what you can do is that you can model this part let us say you can argue that okay I am going to basically capture my inertial scale which is the inertial range shown here on the integral range but very small scales remember when as we go to the smaller and smaller or the finer and finer grid size that is where the computational cost increases right so let us say I have something of 100 meter and then let us say I have something of 10 power minus 4 meter right. So what is the grid size so if this is my 100 meter and my minimum size is 10 power minus 4 meter let us say in this side or this side the grid size is if you consider a uniform grid is 10 power 6 right you will have 10 power 6 number of cells right in one direction right so if you decrease this value let us say you want to go to a scale which is 10 power minus 6 meter where your dissipation is happening right. So then in the same if you follow the same argument this will go to 10 power 8 right that many number of elements you will need to have if you want to go to this scale right let us say that your all the things that is happening the important events or important phenomena that is happening could be captured by this 10 power by scale which is up to 10 power minus 4 meter which let us say somewhere fall here okay which the remember the wave number is inverse of this scale okay. So then what you can do that between 10 power minus 4 and minus 6 you could use some kind of model okay what the model does it basically ensures that whatever thing which is coming on the right of this is dissipated so that is why the your models like SGS model which is called the sub grid scale model or there are some LES which is called implicit LES that takes care of this dissipation by actually considering a flux scheme which has the dissipation inbuilt or you can have some kind of filter okay. But essential idea is that you capture the initial inertial range to the extent possible and the integral range but model the dissipative range okay that is what we do in LES okay. So you will get a basically cascade up to the dissipative range one before the dissipative range and after that it is model you will not get the you will have the energy drop suddenly at this point okay. So now let us find out that what is the kind of number that we are talking about when we say that we want to resolve all the scale right. Let us say you want to do a DNS so what is the kind of numbers we are talking about depending on Reynolds number so what is the quantity that is basically dictating the flow in most of the fluid flow is the Reynolds number right which is some kind of density scale, velocity scale, length scale and your viscosity scale right. So for this Reynolds number let us say if I say that my Reynolds number on a wing because we have taken many examples for aerodynamics take another example. So let us say flow over a wing is happening at Reynolds number of 10 power 6 or 1 million let us say on a based on the incoming flow and the chord length or mean aerodynamic chord of the wing right so this is the scale. So the question is that if I want to do DNS of this wing what is the number of grid size or number of grid points or the grid size that we are talking about right. And it so happens that you can basically make some scaling argument to get into some approximate numbers of course that will not be a definitive number but you will know that what is the range of scale that we are talking about you will know the ballpark figure. And this was beautifully given by Kolmogorov who basically made some argument and made some assumptions and came up with a number which can be basically directly evaluated based on your Reynolds number. So what we are going to do in that so we will define two IDs if you go to the last picture so we will have the large ID let us say here we have the very small ID. So we will take these two IDs and we will say that the production is happening with the large IDs and dissipation is happening at the smallest ID and they will make some connection between the two and try to find out one number of the average grid size. So what I am doing here I am defining the largest ID with the length scale L. And it has the velocity scale which is U and for the smallest scale smallest ID which are responsible for the dissipation we take the length scale as eta and we take the velocity scale as V. Now we are going to look at the two extremes of the spectrum one is the production where the energy is taken by the largest scale from the mean flow and another side is where the dissipation is happening due to the smallest IDs. So this is my diagram this is my E k versus k this is on the log scale. So what we are saying that for the largest scale let us say somewhere here what is the rate of production of kinetic energy. So of course the kinetic energy is half M and U square and then rate of production of kinetic energy per unit mass that is M and then rate because this is rate this is T. And forget about so let us call it the production this is rate of production of kinetic energy which is happening due to the largest IDs M, M will cancel out and T you can define because we have defined the length scale and the velocity scale. So we can define the time scale also the time scale will be length by velocity length is meter velocity meter per second so that will give you the second. So this will become this will be of order of U square by L by U which is basically U Q by L so this is what is written here. Now let us look at the dissipation which is another extreme here. So what is the rate of dissipation? Now there are multiple ways to arrive at it what we are going to discuss is that what we have done in the RANS modeling also. So the dissipation is given by viscosity E ij E ij and this is the velocity tensor we have been discussing so many times and this is nothing but depends on the velocity gradient. So if you go by the basically a dimensionally argument so this will be nu this is viscosity and what is the velocity gradient for the smallest scale. So we have the velocity scale which is small v and then length scale is eta that will be v by eta v into v by eta do not get confused nu is like this v is like this so that will be square so you have nu v square by eta square so that is the rate of dissipation. Now what Kolmogorov did he made two assumptions and those are very fair assumptions one is that at any time in equilibrium for equilibrium turbulent flow the production the rate of production of kinetic energy is equivalent or similar to basically the rate of dissipation. So basically the production is like dissipation so you can say that u q by l which we derived here is equal to nu v square by eta square this goes as nu v square by eta square. Remember what we have started with we want to find out what is the smallest scale that we need to resolve based on a given condition let us say the Reynolds number. So this does not give you the enough information it relates the largest eddy to the smallest eddy in some sense in the sense that if you know the largest velocity scale and the length scale you can relate it to the smallest velocity scale and the length scale but we want something on the gross parameter like Reynolds number. So then the another argument that Kolmogorov made is that at the very smallest scale level which is V the viscous forces and the inertial forces are comparable what does it mean is that Reynolds number at that scale is equal to 1. So remember if the inertial forces are basically very high then dissipation will not happen because the things are getting dissipated at a very fast scale at the smallest eddy level. So that is why we argue that the viscous forces are very at least comparable to the inertial forces and as we know that Reynolds number is a ratio of inertial to the viscous force. So that means the Reynolds number at that scale is equal to 1 so that means V eta by nu is equal to 1. So that way we can say that V is equal to nu by eta. So we related our velocity to the viscosity and the length scale of this smallest eddy. Now if you use the second assumption which is basically V equal to nu by eta and we know the what is the dissipation rate what is the dissipation rate is V nu V square by eta square. So if you use that argument epsilon equal to nu V square by eta square and we have discussed that V equal to nu by eta. So you can replace that V thing so the viscosity and then V will be replaced by nu by eta square by eta square. So that will become nu Q by eta to the power 4. So the epsilon is nu Q by eta to the power 4 and that way you can find out eta is equal to nu Q by epsilon to the power 1 by 4. So we found out so what is nu this is basically the smallest eddy size this which we want to resolve. So we actually related it to the viscosity but there is another parameter which is dissipation rate and it is not easy to basically find out the dissipation rate. So what we are going to do now we will use the first assumption as well and what is the first assumption the production is basically very similar to the dissipation. So then nu Q by L is nu V square by eta square replace this V with nu by eta so that is nu by eta square by eta square which will become nu Q by eta to the power. Now we are going to do something here so we are going to rearrange some of these variables I am going to write it on the top. So we have nu Q by L is going as nu Q by eta to the power 4. So I will rearrange it so this is eta to the power 4 by L is equal to nu Q by nu Q. I am going to multiply L Q on both the sides so that will become eta nu by L to the power 4 is proportional to nu by U L. So this is like your largest eddy velocity largest eddy size. So what is largest eddy velocity and size so this you can relate it to let us say if you have a flow inside a channel so your eddy will be of this size so this is your largest eddy size for example and typically you define a Reynolds number on this kind of scales. So if you want to define a Reynolds number here so R U will be U L by nu and U is something like somewhere here at the mean flow at the center line. So if you use that definition you have eta by L to the power 4 is proportional to 1 by R U to the power Q and then going ahead you will have eta by L is basically proportional to R U to the power minus 3 by 4. So this is the scale of the smallest eddy to the largest eddy. Now what we have done we have related the smallest eddy size to the largest eddy size with the Reynolds number. So if you know the Reynolds number we know and we know the let us say your geometric size let us say here the L is given you can calculate the eta which is basically an R U to the power minus 3 by 4. So in the next class we will see some examples that when we have different kind of Reynolds number what is the kind of grid size we are getting when we want to resolve the smallest scale to the largest scale. So we will stop here. Thank you.