Lecture 59

Hello everyone, welcome back and in the last class we discussed about different solution models like RANS, LES and DNS and what are the grid strategy requirement for these different solution models. So, we have already discussed about RANS in great deal, so today we are going to discuss about the large eddy simulation, so we will take couple of classes on this large eddy simulation. So, let us go to the basics first, so the first thing that we have been talking about is the energy spectrum, so every model that you are going to use, every solution model whether it is RANS, whether it is LES or whether it is DNS, they can all be basically understood by using this energy spectrum. So, like we have been saying that if you are basically modeling the entire spectrum, then it is the RANS model, Reynolds Navier Stokes model. Of course, when you do little bit of unsteady RANS, you might basically be able to resolve some of the scales on the mean flow itself, so because U-RANS also solves the mean flow only. So, if the mean flow has got certain time scale involved, so that can be resolved using this approach, U-RANS approach. Then other extreme is the DNS, where you can basically resolve the entire thing from this end to this end, so you can basically at every wave number what is the energy, you can find out and in between is your LES. Now, if you make this spectrum here, this spectrum look like and this is your minus five third and this is E k versus k of course, in the log scale, so if you are resolving everything. So, like we discussed yesterday that you integrate this entire thing, you are going to get your kinetic energy. And in the and this k is the wave number, so when it is LES, you would be basically resolving the energy associated with large eddies, so on the left side of this spectrum, but you will be basically modeling very small eddies, which is on the right side of the spectrum. So, this is model, this one is resolve, now the question is how much you should resolve and how much you should model. And a good LES and a bad LES depends on how much you are resolving and how much you are modeling. So, typically a good LES is the one, where you can resolve at least 80 percent of the energy. So, if I make this curve again and let us say this is my 80 percent, so at least this part would be resolved using your LES to the direct solution by using the your appropriate grid. And this rest 20 percent, this can be model using either your sub grid stress model, which we will discuss shortly or may be in the next class, depending on how the time permits or in the implicit LES case, this can be taken care by your numerical viscosity. So, at least 80 percent of the energy should be resolved, if you resolve more than that, that is even better, but one thing one important thing that you need to understand it here is that like if you do range simulation. So, we have been talking about the grid convergence, that it should be independent of the grid, the LES it is very difficult to quantify what is your grid convergence, because as you change your grid size. So, let us say if you want to resolve 90 percent instead of 80 percent, let us say some you refined your grid and now it is able to resolve up to 90 percent, this is a 90 percent up to the blue line. So, this part now also is part of your LES, so your kinetic energy which is resolved divided by kinetic energy which is modeled is now different as you refined your grid. So, you are getting more and more scales resolved as you refine the grid, unlike in range where you are basically modeling all the scales, all the turbulence scales are modeled only mean flow is solved. So, it is very difficult to quantify the turbulence quantities, which are consistent as you basically refine the grid, so they are going to change, may be when you compare the mean flow then you can may be justify that as you resolve the grid, the mean parameters did not change. So, that could be one of the way to basically discuss about the grid convergence in the LES, but otherwise if you look at the turbulence quantities, the behavior of the eddies, they will be different as you work from a coarse grid to a finer grid, simply because you are resolving more and more spectrum. Now, the question is how does the LES work and how do you resolve some and how do you basically a model some, and so let us go there, so let us discuss in terms of grid, so this is my grid here, this is a grid and if you want to any resolve any resolve any eddy, you need this 4 grid points. So, if you have an eddy like this, so this can be resolved using this 4 grid points, because typically when you do a CFD simulation on the basically cell center, you have you store your velocity, your pressure all these quantities. So, when you basically connect the cell centers, you will be able to basically find the swirling motion as well, and that is what basically defines the eddy. So, if for example, in this particular grid, you can resolve a very large eddy which goes through all these points, and then you can also resolve the smaller eddies like this, and then probably smaller than this, and you can go and you can resolve the scales which is up to this, up to the first 4 points or any 4 points, any 4 grid cells, but any scale which is smaller than this will not be you will not be able to resolve. So, let us say if you have a grid like this, and if you have a eddy which is here, you are not able to resolve this grid, because simply you require at least 4 points. So, in this particular case, so if this is not up to your Kolmogorov scale, so it is not going to dissipate, because the dissipation happens at the Kolmogorov scale. So, what happens that these eddies are very unstable, so they form, so you have a large eddy, they will break into a smaller eddies, and then further smaller eddies, and then further smaller and smaller eddies, and so on and so forth, and finally, they dissipate, they convert into heat. Let us say you have a grid which can resolve only up to this scale, this is the finest grid that you have in your alias, so but the dissipation is happening at this scale. So, then what will happen is that you are not able to dissipate the heat that you need, and because of this if you are not able to dissipate, so energy will start building up. And that is what I was discussing in the last class, that if your grid is not sufficiently fine enough, and if you are not using a basically sub grid scale viscosity model or a numerical viscosity scheme with a numerical viscosity, your solution is going to blow up. And that is what happens in a typical D N S, so D N S has to be resolved up to the Kolmogorov scale, or something around that where the dissipation should be taken care of. Anything which is bigger than that grid size that is going to blow up, simply because it does not have the mechanism to go out. So, we discussed in the last class that you have a production of energy, and it is similar to dissipation of energy, so the production is happening, so this larger eddies are getting the energy from the mean flow denoted by u bar. So, the eddies are getting energy, but they are not able to dissipate. So, what will happen that the energy will keep on building up, if your grid is not sufficiently fine, and the scheme will blow up. So, in order to take care of that what we do, let us say we want a dissipation at this level only, because your grid is up to this point only. So, we want to have the dissipation at this level, this is our desire. So, how do you do that, is basically you can increase the dissipation rate, and the way to do it is basically you have this molecular viscosity, add some other viscosity to that, let us call that s g s, which is sub grid space viscosity, and we will discuss more about that. And let us say del u i by del x j del u i by del x j, so your dissipation rate which was epsilon, originally it was del u i by del x j del u i by del x j, viscosity times the velocity gradient. So, if you look at this expression, so if the denominator is delta x, some kind of size of the grid. So, if the delta x is higher, that means your grid size is higher, this entire quantity will be smaller, the del u of the velocity gradient will be smaller. And then this viscosity may not act or this entire quantity may not enough to dissipate the energy. Now, if we have the same delta x, but now you also add your new s g s here. So, what you are doing here, is that may be this was not able to, this one was not able to dissipate energy, because of the larger grid size, and the viscosity is your molecular viscosity. We will artificially increase the viscosity here, by adding some viscosity, so that it is able to dissipate the energy at this scale also, sufficient energy at this scale also. So, that is the crux of the matter, that in LES we are able to resolve as much as we can, and just to ensure that at least 80 percent is getting resolved, but beyond that we are going to use a model, which can basically take care of the dissipation, so that is the basics of the LES. So, this we have discussed that, now the kinetic energy spectrum has two parts, so the one part which is this part is resolved, and this part is model, it could be the s g s model. So, my kinetic energy of the resolved k r e s divided by total kinetic energy, which is k r e s, this is the turbulent kinetic energy plus k model should be greater than equal to 0.8 or 80 percent. So, that is what we are talking, but now how do we calculate this value, so k r e s if you go by the definition, so k r e s is coming from the simulation. So, we have our grid, which is resolving different sizes of grids from this end to up to this point, and this is giving us the resolved kinetic energy. So, if you look at the stress tensor, so you have basically u u, let us look at the WS quantities u prime u prime bar, and then u prime v prime bar, and u prime w prime bar, and then this is because it is symmetric, this will be u prime v prime bar, v prime v prime bar, and v prime w prime bar, and this will be u prime w prime bar, v prime w prime bar, and w prime w prime bar. So, this is your stress tensor, and you can relate it to the standard tensor, and these diagonal elements are what, they basically represents the turbulence kinetic energy. So, what do you want to find out, when we say we want to get this k resolved, so here the k is the kinetic energy. So, we want to find out half u prime u prime bar, v prime v prime bar, and w prime w prime bar, this is the turbulence kinetic energy that we are interested in. Now, remember in range simulations, we actually model this turbulence kinetic energy, so you get the k as an output of the simulation. So, let us say if you are using k epsilon, then you will get both the turbulence kinetic energy as well as dissipation rate as an output of the simulation itself, but here we are solving for u, v, w, p, rho without basically averaging the equation. So, we are not averaging the range, we are averaging the equation, so the question is how do we basically use this, how do you calculate this resolved kinetic energy. So, there is a way to do that, and I will just discuss that. So, we are let us look at how this works, let us say that you are doing a simulation in a lid driven cavity, this is a lid driven cavity, and you see the top lid is moving, and because of that there is a vortex that is forming. Let us say you are doing at a sufficiently high Reynolds number where the flow is turbulent, and if you put a probe somewhere here anywhere, what will happen, so initially you will start from some initial condition. So, your fluctuation it will vary, and then you will see that after some time the variation will be there, but it will be around a constant value. So, let us say this can be anything, this is u and this is time, this is at a probe point, this can be p also or this can be even density. So, if you plot a line connecting the average it will be like this. So, now the fluctuations are happening let us say after certain time when the transients are gone, this part will be the transient part where the flow is trying to develop, and after that this is called the statistically steady state. So, you can start taking the average after the transients are passed, you should not take the average from the beginning. If you take the average from the beginning here let us say you started the simulation, and at the same time you started taking the average. So, why we want to take the average, because we want to find the mean flow also. So, you will be solving for u, v, w, p, rho for example, but you are interested in the mean flow also that is u bar, v bar, w bar, p bar and rho bar. So, in order to get this quantity you need to basically start taking average, basically you need to start adding the value at every point of time, and then divided by the number of iterations. So, that is something when you want to take the average you should wait till the initial transients are passed, and then the fluctuations are around a mean line, and then you can take the statistically stationary, I mean it is a statistically stationary state, and then there you basically do the averaging. I think the more appropriate term will be stationary, rather than steady, because steady is used in a different term, terminology where d by d equal to 0. So, once you have taken this average, so this will give you the value of u bar, this is how you will get the value of u bar. So, you can easily calculate the value of u bar in any code by just adding the quantity over time, and taking the average. So, you will get the value of u bar, v bar, w bar, p bar and rho bar, but in order to get the turbulence counting energy, we need the quantity which is basically u prime, u prime bar, v prime, v prime bar, w prime, w prime bar. You might also be interested in other quantities stresses, you see a stresses like u prime, v prime bar and v prime, w prime bar, w prime, u prime bar and so on and so forth. So, how do we get these quantities in a LAS simulation. So, if we subtract this u bar from u, what we are going to get is the fluctuating quantity. So, we are getting the fluctuating, you can also get the fluctuating quantity every time. So, the best way to get your this quantities u prime, v prime bar is to use the identity that we have also discussed during when we are discussing RANS. So, let us say you are not only collecting this u bar, v bar, w bar, p bar, rho bar, but you are also collecting other bars which are u, u bar, v, v bar, w, w bar, u, v bar, u, w bar and v, w bar. What is this, but basically at every point of time you are multiplying u with u and you are taking that average as you go through the iteration. Similarly, you can do v, v, w, w, u, v, u, w and v, w, w. Now, so you have this quantity and this quantity both are available. Now, you can use the, you can do some math and find this u prime, v prime bar and so on and so forth. So, how do you do that? So, let us say if I have this u and v, if I multiply this, this can be written as u bar plus u prime and v bar plus v prime. Let us say that will become u bar, v bar plus u prime, v bar plus v prime, u bar plus u prime, v prime. Now, we take average of the entire thing u, v bar is equal to u bar, v bar because these are already average. So, these have become constant quantity and use the identity this u prime bar is equal to 0. If you have taken a good mean for a sufficiently long time, the average of the fluctuation should go to 0. Similarly, v prime bar, u bar, this will also go to 0 and plus u prime, v prime bar. So, now by using this you can get this u prime, v prime bar by just doing this u, v bar minus u bar, v bar. So, since you have stored all three quantities, you have stored this u, v bar and you separately stored u bar and v bar, you can calculate this u prime, v prime bar. So, similarly you can calculate u prime, u prime bar or it is also called u prime square bar. Remember, first you have to multiply and then take the average. So, that is a very important thing. So, this will be u, u bar minus u bar, u bar and similarly, you have v, v bar minus v bar, v bar. So, that will give you v prime, v prime bar and similarly, w prime, w prime bar. This will be w, w bar minus w bar, w bar. So, now you have access to all these three quantities. So, now that you got these normal stresses, just add them and divide by half and you got this visual kinetic energy. The other thing that we need is what is the kinetic energy through the model and this is a hard quantity to find out, especially if you are doing an implicit alias because there this quantity dictated by some kind of filter or some kind of numerical dissipation and it is difficult to find out. But, let us say if you are using a SGS model, any stress model which is inspired by the adivisory model, then you can get this quantity directly. For example, if you are using open form, so you can get this K SGS which is the kinetic energy that is through the SGS model. So, if you have that SGS model kinetic energy directly there and then you have the model from the alias, you can add them and divide by this quantity. So, K resolve by K resolve plus K SGS and you can check that whether it is giving you the value of more than 50 percent. If the value is more than 50 percent, then your alias is a good quality alias. So, that is a very good thing to find out when you are doing alias because there are many different kind of alias and you can claim anything to be your alias. But, if you do not check whether it is a good quality alias or not, it is very likely that the results that you are getting may not be the physically accurate. The other way is to check the quality of your alias is through the quantitical integral length scale. We have already discussed this in the context of range where we basically said that you have different kind of eddies in the flow and you choose a eddy. So, you can have eddy which is very big and then you have small, small, then medium. All these different kind of eddies are there and what you do is that you choose a scale which represents the average eddy size. So, to be more accurate in alias, this is the size of the eddy at an average kinetic energy of all eddies. So, if you have a spectrum like this, so somewhere here is your l 0 which gives you the average kinetic energy. Now, to confirm whether a grid size would be sufficient for alias or not, what people normally do is that they perform a range simulation. So, you can choose a grid size and on a grid size before you go to perform the alias, you perform a range simulation. Once you perform the range simulations on a grid and you intend that you can use the same grid for alias later. So, what you can do? Let us say you can choose a grid size which is basically type elements across l naught. So, what do you mean by that is this is my wave number and this is my energy. So, this is my let us say 2 pi by l naught. So, l naught is the scale which is basically gives you the average kinetic energy of all eddies. This is the typical size of eddy and if we choose a size delta which is basically l naught by pi. So, if we choose a size of delta which is pi l naught by pi. So, here we will have somewhere this is 2 pi by delta because delta is smaller than l naught. So, this region so the claim is or this not the claim, but a rather thumb rule is that if you could have a mesh which is resolving your integral length scale by using type elements or your element size is one fifth of your integral length scale, then it can resolve 80 percent of the energy. So, this region now can resolve the 80 percent of energy. Now, you know the definition of l naught. So, if you use any model k epsilon or k omega. So, you have the definition of l naught and we have discussed that in the last class because your eddy viscosity is basically multiplication of u naught plus l naught. So, by manipulating the value of by using the value of k and epsilon you can calculate the value of l naught and what is delta. So, delta is something which is given by Smigronski in a nice manner and we will discuss that probably in the next class. So, right now I will just tell you the definition the delta is like this is obvious that it should be a form of the grid size. So, it is c s cell volume to the power 1 by 3. So, you may have a grid. So, the structure grid. So, let us say this is my cell and in 3 d it will be in the third dimension also. So, you can get the average size of the cell by using this cell volume to the power one third and this is a quantity c s which is called Smigronski constant. So, delta is a filter. So, beyond that your energy is not resolved that is not resolved. So, the delta should be smaller than this cell volume. So, the c s value can be anything between 0 to 1 and what Smigronski has proposed that its value could be c s can be taken as 0.1. So, you can calculate the value of delta using this relation and l naught you know from your relation of k and epsilon or k and omega and you can calculate the value of l naught by omega and you should ensure that this value is basically should be greater than 5. If the value is less than 5 then it means that you do not have enough number of elements in a given integral length scale and you need to increase the grid size. So, this is one check that you can do when you are going to start the LES. So, before you do LES you do RANS get the value of this l naught from the RANS delta is a parameter which you can calculate using your basically geometrical calculation calculate the cell volume 2 power 1 by 3 multiplied by the Smigronski constant and calculate the value of l naught by delta and plot it. You can make a contour and see what are the regions where you need to resolve. There will be some regions where this value is less than 5 and that you need to resolve and once you do that then that mesh is will likely to be basically good for your LES simulation and then you can make it probably three dimensional and do the LES simulation. So, these are the two checks one is after the simulation and one is before the simulation. So, before the simulation you can do the grid check using this integral length scale parameter and after the simulation what you can do you have this k resolved which is the resolved kinetic energy and k modeled which is your kinetic energy based on your SGS model and take the ratio of k resolved by total kinetic energy and if the value is more than 80 percent then your LES is a good quality LES. So, with that we will stop here and we will meet in the next class where we will discuss about the Smigronski model. Before that maybe just one thing I want to discuss here before I close. So, as we discussed that we want to increase the dissipation at the smallest grid size. So, because why we are doing that is because let us say dissipation is happening at this grid size, but the LES has the smallest grid size this is the LES is smallest grid size and this is the grid size required based on Kolmogorov calculation type. We have discussed that what is the grid size that is required and the dissipation happens at this scale this is what is going to dissipate this is what is going to break down this is what is going to dissipate into heat, but your grid size is this much. So, we want the same dissipation which was happening here through the mechanism of the natural viscosity to happen here also. So, what we will do we will follow the logic that we had from RANS. So, we had what we did that in the RANS the main difference between the RANS and LES is that in the RANS we want the dissipation or the production the kinetic energy to be modeled everywhere regardless of the grid size. It may be very the grid size may be very small near the wall and very big I mean away from the wall, but everywhere we are going to model in LES we are not going to do that. In LES we say that you model only if the eddy is like smaller than certain size or in a more accurate way it is just bigger than the the smallest grid size that we have in the LES. So, we want our SGS to our model to work on this scale. So, this is where the SGS model would work. So, we can borrow the idea from the RANS simulation or the RANS setup instead the only difference is that now we do not have bar here that used to happen in the RANS calculation. So, this is our Navier-Stokes equation, but we add a new quantity here which is your tau SGS. And this is the same equation that you are going to get if you go through the proper mathematics of filtering the Navier-Stokes equation and you will arrive this quantity called tau SGS. So, this tau SGS is called the sub grid stress and this needs to model, but remember that this model should be applicable only on a grid which is the smallest one or in other words the eddy size which is just bigger than this the smallest grid. And of course, it is smaller than the smallest grid. So, all these things should be where this sub grid stresses should work and because of this addition and like we had this eddy viscosity model we can have the same argument that tau SGS can be written as 2 nu SGS rho multiply by some velocity gradient and so on and so forth. So, like we had this eddy viscosity model where we propose this eddy viscosity we can also have this sub grid scale based viscosity. And Smolenski was the first one to propose how to basically model this sub grid scale and we will discuss about that in the next class. Thank you very much.