Lecture 58

Hello everyone, welcome back. So, in the last class we started talking about the scale resolving simulations and we basically made a scaling argument using Kalmogorov's assumptions. And we found that the smallest study that we want to resolve, let us say we are using a direct micro simulation and divided by the largest study. So, that is basically if you use eta by L, so that goes as R e to the power minus 3 by 4. And this is purely the by considering the scaling argument and we had two assumptions, one is that the rate of production of energy is basically equivalent to the rate of dissipation, that is the one thing. And the second assumption was that at the smallest level or at the level where the dissipation takes place, the viscous force is basically equal to the inertial force. That means that the viscous forces are so high that the eddies can dissipate. So, that were the two assumptions and by considering those assumptions and by considering different velocity and lengthy scales for the largest study and the smallest study we arrived at this number. Now, what we are going to do today is basically check for different conditions, what is the actual grid size that we are talking about and which is very easy to calculate. So, let us say I mean before you go to the atmospheric motion which is mentioned here, let us think of a pipe flow and let us say the diameter of the pipe is around 5 centimeter. Let us take a very moderate Reynolds number R e equal to 10,000. So, if you want to resolve in one dimension also, so eta by L would be R e to the power minus 3 by 4 for this case, which will be say 1 by R e to the power 3 by 4 and this is nothing but R e 10 power 4. So, that will become 1 by 10 to the power, so this is 5 centimeter. So, this is R e to the power 3, this one. So, if you want to calculate eta will be L into 10 power minus 3. So, L is 5 centimeter which is 0.05 meter into 10 power minus 3 which is basically 5 into 10 power minus 5 meter. So, this is what will be our smallest grid size to resolve the Kolmogorov scale also. So, if you want to make a grid for a DNS, the first grid point will be some of this scale 5 into 10 power minus 5 meter or somewhere here also. Of course, as you go away from the wall and where the basically the shear is not very high, you can basically have some relaxation and that depends on your grid topology and your consideration of the physics, where you can have very fine grid and where we can have very coarse grid. So, those consideration you can take, but in general like this is the kind of length or the dimension that we need to have to simulate this flow. Now, this is for a very moderate Reynolds number 10 power 4. If you go to 10 power 6 for example, the same thing will become 5 into 10 power minus 7 meter. So, this is the kind of dimension that we are talking about. Now, what will be the typical grid size? Remember, this is in one dimension. Now, if you go by this same argument eta by l is equal to r e power minus 3 by 4. So, your grid size in 1 d will be somewhere like l by eta, the largest to the smallest, the largest due to the smallest study. So, that will be r e to the power 3 by 4. Now, also we have discussed so many times earlier also that l s and d n s you cannot do in 2 d. I mean you can, but see turbulent flows are inherently 3 dimensional. So, if you want to have a physical meaning of what is happening in the flow, it is recommended that you always perform these simulations in 3 d, at least in a practical physical situation. So, this is if you want to take the grid size, consider the grid size for 3 d. So, this will be a typical grid size for each dimension. So, r e to the power 3 by 4 multiplied by r e to the power 3 by 4 and then multiplied by r e to the power 3 by 4, then this will become r e to the power 9 by 4. Now, this is the 3 d spatial dimension. We also have the time dimension and as you know that the time is delta t is basically related to the delta x through the Courant number. So, lambda is equal to c delta t by del x. So, we can say that the delta t also goes similar to let us say if you take Courant number is equal to 1 and speed of sound is also around 1, then the typical time scale is the smallest time scale is also equal to the typical smallest grid scale. So, that is why we have been saying that if you reduce the grid size, you also have to reduce the time step size. Otherwise, your solution will blow up because what will happen that if you use a very large time step for a very small grid. Basically, you are jumping across the grid and the solution will blow up that we have seen in some of the simulations also. So, if you consider the time scale also. So, that 3 d plus time it will become r e to the power 9 by 4 and if the time also goes by around length scale this is 3 by 4 that will become r e to the power 12 by 4 which is r e cube. So, imagine let us say typical Reynolds number on a flow over a wing in a practical situation is 10 to the power 6 per million. So, to get to simulate that 1 million Reynolds number you need a grid point which is r e to the power cube which is 10 to the power 18. So, that many grid points you need to have to basically in your domain to simulate that kind of flow and that is why. So, that is kind of excise scale. So, that is why it is very difficult to simulate or do a practical DNS of those kind of realistic Reynolds number. If you have something which is going around number 7 it will become the grid size will 10 power 21. So, that is why the variation of the scale as you increase the Reynolds number the scale ranges increases and that is why your grid size increases because you want to resolve all kind of scales right. So, that is why DNS in practical situation is very very difficult only for very limited flows let us say one example is your turbine blade flow. So, if you talk about a let us say low pressure turbine. So, they normally fly at a high altitude where the density is low and typically in the cruise condition the Reynolds number can go very very low it can go as low as 25000. So, then if your Reynolds number is very low then the grid size will be somewhat manageable and you can perform the DNS of that kind of situations. So, that is why you will see some DNS studies on flow where turbine blade, but you cannot do DNS of let us say full blown 3 dimensional wing at full length scale in a practical situation. You can do DNS on a Reynolds number which is let us say much lower than the practical Reynolds number let us say 10 power 4 or so, but if you want to do at 10 power 6 and all you need to have that kind of grid and for that you need to have that kind of super computer and even on the fastest super computer you cannot simulate that kind of flows. Atmospheric is the similar situation here. So, the large eddies as we discussed is the order of kilometers and the smallest eddy is order of millimeter right where is the boundary layer. So, then you can take the ratio of that like 1 millimeter to 1 kilometer right that is the scale we are talking about and accordingly and that is in 1 dimension you have to go in the 3 dimension plus the time marching and all those things. So, you can imagine the kind of grid size that you will have in the in the practical DNS simulation and if you do not obey this argument that you have to resolve the very small grid. What will happen is that either your solution will blow up or you will come up with a solution which may be even worse than the Reynolds model and why is that that if you do a very coarse DNS we are not really going to the scale of the convoluted scale or at least at a scale where the viscosity taking place then the energy will start building up and at some point of time the your numerical solution will blow up it will not even proceed and if it is proceeding that means that it is getting dissipation from somewhere either the numerics may have very a lot of dissipation and that is why it the solution is continuing, but in that case that dissipation you may not know that where it is cutting off I mean if you look at this e k versus k it might be cutting off here which is maybe right in the inertial range right and then it may give you completely wrong result it may basically kill the scales which are very very important right. So, the coarse DNS is very very dangerous if you have a very coarse mesh you can apply some kind of model there and make sure that you know that where the energy I mean we are cutting your energy spectrum else you can try with some range simulations there are many advanced models nowadays and at least you can have a fair idea of where the solution can go wrong right you know the limitations of different models and you can use those models for finding out where could be the separation or transition or or elimination. So, based on that we will just go over this approach is once again in the context of LES and DNS that we discussed. So, RANS as we know it is a steady state and U-RANS is basically unsteady RANS. So, we are doing the still solving the Rayleigh-Notes average Navier-Stokes equation except that you have d by d t term also there. So, in the RANS U-RANS also we are solving the average Navier-Stokes, but you will have the d by d t term. So, in the RANS case because we are solving the Reynolds average Navier-Stokes equations. So, we basically first average and then simulate we average the equations we got a equation which is an average quantity and then use the simulation of course, when we averaged it we have this closure problem that we have already discussed. So, you will have this U prime V prime bar that you need to basically model and that is why the you can use employ your eddy viscosity model or your Reynolds stress model to overcome that closure problem and simulate it. So, this is typically considered to be valid the time scale of the turbulent fluctuations is basically much less than the physical time step and that physical time step should be much less than the global energy time scale. So, we will come back to this assumption when we have some more practical applications which we will discuss later. On the other hand if you go to the large eddy simulation. So, here we are going to basically resolve the large eddies, but model a smaller ones which may be responsible for only for dissipation. So, in this case we basically in the RANS we was first averaged and then simulated here we simulate using the Navier-Stokes equations and in fact, more accurately it is a filtered Navier-Stokes equations why it is a filter is because what the filter does is that if your grid size is above certain level basically it is resolving the eddies which is above certain level. So, basically it is filtering all the eddies which are smaller than that scale and putting some kind of model there. So, but we will not get into that argument of how to derive the filter Navier-Stokes equations. So, it is basically for all practical purposes it is a Navier-Stokes equations and for the eddies which cannot be resolved by the grid we use some kind of model. So, we simulate we do not average the equation in the beginning. So, we simulate the equation as it is and after that we do averaging if you want to get the mean flow. So, when you simulate you will get the fluctuating quantities unlike RANS where you get the average quantities even in U-RANS you get the average quantity. So, you simulate and then average. So, that is a main difference RANS average then simulate LES and DNS simulate then average. LES is difficult to analyze it can be easily complicated by numerics and the post processing is also not that easy and the reason for that is that the grid size is very large in LES it is three dimensional and you do not get the mean quantities directly. So, there let us say if you want to get the turbulence kinetic energy in RANS you directly. So, you are solving for that equation dk by dt. So, you are getting the turbulence kinetic energy directly here you will get this quantity u prime v prime w prime and so on and so forth. And if you want to calculate this turbulence kinetic energy then you will have you have to do you calculate the u prime u prime bar plus v prime v prime bar plus w prime w prime bar and so on and so forth. It requires little bit of involved calculation which I will discuss how to calculate in LES also. And if your flux scheme has got lot of numerical dissipation then also there are issues because as I discussed that if you have let us say the spectrum like this and this is where your dissipation starts. But if your flux has got lot of your numerical scheme has got lot of dissipation then the dissipation will start here and this is basically affecting the accuracy of the flow. So, this dissipation can so there are 2 schools I will not get into that discussion here one is called implicit LES other is called explicit LES. What you do in explicit LES you actually put a model to basically dissipate the flow here. In the implicit LES what is the argument is that the numerical scheme itself has got the dissipation and the dissipation can basically take care of the dissipation of the energy at the lower scale which we are not resolving. So, the numerical dissipation can basically add to the dissipation that you have or the numerical viscosity can add to the natural viscosity and it can give you the dissipation even at the higher scale and that is why the simulation will go on. So, there are 2 these different schools and I will without going into that you can in the literature you will find all kind of simulations in LES and as I discussed in the beginning that there are different ways of doing LES. So, they have like WM LES which is wall model LES there is well model and there are very large LES VLES and then there are some hybrid model. WM LES is also kind of hybrid model there is WR LES which is wall resolved LES and even to do LES you need to think that your numerical scheme is good enough that it does not have enough dissipation. In RANS you can go do away because you are using the WNS model but here the accuracy becomes a concern. So, in that case you should have a scheme which has got very least dissipation. This is what I was saying that when you have the implicit large eddy simulation which is IELS you do not use a SGS model and you make an argument that the numerical dissipation is basically provides filtering needed at the smallest scale and it models the effect of this smallest eddies and it prevents the artificial buildup of energy at the smallest scale. Because right see if you want to talk about the dissipation, so dissipation is happening at viscosity nu and then you can have let us say del T u by del x square. Now, if your x is not small enough, so then action of this viscosity will not be good enough. So, what is the argument in implicit LES is that because you have the numerical viscosity nu plus nu n, so even if your x is not small enough in LES for a sufficiently large x also you can get the same effect as this one. So, that is the argument that your numerical dissipation accounts for the dissipation that is not happening by natural viscosity. So, this is what is called the implicit LES and then in the DNS finally, is the direct simulation of numerical equations and by implication and by standard definition it requires that all spatial and temporal scales are resolved down to Kolmogorov scale. Of course, as I told that there are some regions where the shear is high and where you require very very fine grade like in a boundary layer and there are some regions which are away from the boundary layer where you may not require that kind of fine grade. So, those arguments hold here also and if you do coarse grid DNS it is similar to implicit LES because implicit LES also we do not use any kind of sub grade scale model. So, it boils down to basically implicit LES, but you need to be very careful that you should know that what kind of dissipation is happening in the implicit LES due to the numerics. And finally, this picture in the RANS we model everything, so we do not get the variation of energy with the wave number all you get is integrated quantity the total turbulent kinetic energy in the LES we resolve and a good LES is the one where you at least resolve 80 percent of the turbulent kinetic energy. So, that is what is good LES and we will discuss more about that in the subsequent classes and rest 20 percent is model here. So, 80 percent is resolved in a good LES and 20 percent is modeled and in the DNS we basically resolve the entire thing we do not model anything. And this is for example, one picture of like RANS versus LES and DNS is very common you can find in easily find if you search in Google RANS LES versus DNS you will see that. And you can see that in the DNS you are basically resolving all kind of scales very small scales here and the RANS is just the mean flow as the flame is going up. In the LES we are resolving the larger scales like this may be like this like this there and there are some scales here, but you are and this is unsteady, but you are really not resolving very small scales as you are doing in the DNS. And therefore, the requirement of the grid size is highest in the DNS is smallest in the RANS and in between you have this LES. So, with that we will stop here and we will have some more discussion about LES in the next class till then. Thank you very much.