Lecture 61

Hello everyone, welcome back. So in the last class we derived the AdS equation for the incompressible flow and we also looked at the properties of the filter and in what case you basically I mean in how what are the ways in which this filter basically differ from the averaging process that we use in the RANS equation. We saw is that in this filters, so you can write pi equal to pi tilde plus pi prime and on the other hand let us say if you want write pi equal to pi bar plus pi prime, so in this case this is a turbulence fluctuations and this is as a sub grid scale fluctuations and we also saw that here in this case pi prime bar equal to 0, but here pi prime tilde is not equal to 0 because of the properties of the filter and there are some other differences also. For example, if you apply the filter twice it is not same as the applying filter once on the other hand this is if you apply your averaging twice it is equal to the averaging once. So this is your RANS and this is your LES and by applying this filter we derived the incompressible filtered Navier-Stokes equations and where we saw that we had an extra term which was tau s g s which needs to be model, but this is different from the tau a d or tau term, but we had in the RANS in a way that this only applies to the smallest grid that we have where we want the dissipation to happen. So, we are still resolving the spectrum up to great extent after that we just ask it to go down and we are not resolving some region which are for the responsible for dissipation. Now let us look at the compressible flow, so when we go to compressible flow we have some differences the first thing is that your density becomes also variable, so whatever operation that you are applying whatever filter that you are using density is not a constant. So, you have to have basically the rho tilde also in the case of incompressible flow density is a constant in that particular case if you apply this filter on rho this will be nothing but rho because of the properties of the LES filter the low pass filter. If it is applied on a constant it will give you the constant value this is not true for the compressible flow and if you try doing it for your advective term in your compressible flow equation in the momentum equation you will quickly realize like we had for the RANS equation. So, you have basically rho tilde plus rho prime u tilde plus u prime v tilde plus v tilde plus v prime and you can apply the filtering of the entire thing and you will have so many terms I think you will have around 9 terms 8 or 9 terms and this quickly makes the algebra too much cumbersome and the way we got rid of this in our RANS is that we use something called density weighted averaging or the power averaging. The same idea we are going to use in LES also and without going into much detail we are going to basically define an operation which is phi bar which is nothing but your filtered rho phi divided by filtered rho. So, your rho filtered phi bar is equal to rho phi tilde and this phi can be u v. So, if it is rho tilde u bar is equal to rho u tilde this idea is directly borrowed from the RANS derivation of compressible equation for the compressible flow because there is a typo here. So, if you do that for the continuity equation what we will get is this will become d rho tilde by d t plus del by del x i and you can write it as rho u tilde but this can also be written as rho tilde u i. By this property that will become rho tilde u i bar. So, this is your continuity equation this is easy to derive like we have done for the RANS equations and similar idea if you apply for the non momentum equation. So, we will have this rho u i u j tilde and this and then you can basically take rho u i as one variable and u j as another variable and like we have done for the RANS equations. If you do those derivations and plus you also write in terms of like rho u i u j is equal to like this rho u i u j plus rho u i u j tilde minus rho u i u j. If you do like this term can go to your s g s stress tau i j s g s and this term is what will come here and rho u i tilde can be written as rho bar rho tilde u i bar. This is how you can derive your compressible filtered equations and similarly I mean you can derive for the energy equations also the idea remains the same. You can apply the filter and wherever you have this multiplication of rho u you take this there are three variables you take this rho u together and another variable which is which may be e or which is energy or may be enthalpy or may be the velocity you can apply this and you can derive this energy equations. So, this I leave to you can go back and do this exercise as your homework and you can derive your energy equation write those terms one by one and you can see that what is the energy equation that you are getting. Now so far so good now what we need to do now is basically find out tau i j s g s what is my sub grade stress which is responsible for the dissipation of energy. You would in many literature you would see that this filtered equations are not derived and you directly use the complete Navier Stokes equation and plus this tau i j s g s term you would not have this tilde and bar sign and that is because in the implicitly filtered LES you are basically not applying any filter as we have been discussing for the last couple of classes that in the implicitly filtered equation the grid acts as a filter. So, essentially you have the Navier Stokes equations you are solving for the variable which you can call as a resolve variable u v w p rho and so on and so forth and you also add a sub grade stress tensor where the grid size is sufficiently small. So, that is what we are going to do. So, that is why this u i bar and rho tilde these are not the average equations either Fabry and Reynolds these are not average these are the filtered quantities it can be implicitly filter or explicitly filter. So, that is a very important difference that you need to realize despite of the fact that the form looks very similar you should not be confused that we are now solving for the instantaneous flow variable which is basically how the signal is varying in time and you still need to do the averaging while to calculate the average quantity you are not getting the average quantity directly in your LES like we discussed couple of classes back is that LES is first simulate and then average RANS is first average and then simulate. So, that idea you should not forget and despite the similarity in the form between RANS and LES you should remember that in RANS we are solving for the average Navier Stokes equations here we are solving for the filter Navier Stokes equations, but this filter can be an implicit filter also where we are basically not doing anything we are just ensuring that the energy gets dissipated at the smallest grid size that is available to us as long as we can ensure that the energy is dissipated and the build up is not happening we can basically run the simulation. Now, let us go to the SGS model. So, we have this subgrid stress tau i j SGS how do you find this value. So, you need to have a model this is a closed term and you need to have a model. So, let us go to the SGS model. So, again we will borrow the idea from the RANS model. So, remember I mean I am doing it for the compressible the incompressible is even easy. So, this is half the average in incompressible if you have tau i j R that will just be written as 2 mu t S i j bar this is what we have we had discussed and that will be also written as rho mu t into 2 S i j bar. These terms comes here because we need to ensure that when you take the trace it should give you the kinetic energy. So, this is for the Reynolds stress for the incompressible flow and this is for the compressible flow. So, I am deriving for compressible, but I hope that you can make the connection like what are the terms which are I mean. So, far it should have been it should be clear to you that what are the difference between compressible and compressible flow. So, we will borrow this idea and use apply in your LES SGS model. So, this is again the concept of eddy viscosity. So, you have lot many variables. So, tau SGS if you write this it will have like 6 independent tensors. So, you will have 1 1 tau i j SGS tau 1 1 SGS tau 1 2 SGS tau 1 3 SGS and then this is tau 2 1 tau 2 2 tau 2 3 tau 3 1 tau 3 2 tau 3 3 these are the SGS models and then because this is symmetric. So, 1 2 is equal to 2 1 and 1 3 is equal to 3 1, but still we have 6 variables that we need to find out. And what we did in the eddy viscosity model is that we know that we have the strain rate tensor available. So, and we know that for the stress quantity the natural stress of flow is related to the strain rate. So, why do not we make use of the strain rate here also. So, we can define this tau SGS is proportional to basically your strain rate tensor. And the proportionality constant like for the fluid stress we have this nu which is the natural viscosity or the molecular viscosity. In this case it will be a nu SGS. So, this is what is eddy viscosity model is that i s i j is known from your simulation this is a strain rate tensor and rho is known. So, the only unknown we have or the only thing that we need to model is the nu SGS which is your eddy viscosity. So, we had 6 unknowns and we basically simplified our problem to just one unknown and that we need to find out. So, for the compressor flow this will be your expression you have a strain rate tensor and then your viscosity. And this two terms I mean it comes out because I mean if you write it for a simple shear flow. Let us say if you have a simple shear flow like this and if you write your strain rate tensor. So, that will be right. So, del u i by del x j. So, half del u i by del x j plus del u j by del x i. So, here if you write this strain rate tensor for a simple shear flow where the only variable you have is del u by del y. So, you will have two places where it will come as half this is 0 half del u by del y 0 and then half del u by del y plus del u by del y. All other terms will be 0 this is how it will look like and then if you do S i j S i j basically you do term to term multiplication. So, that will be half del u by del y square and plus half del u by del y square and that will be nothing but and if you take the square root of that. So, you will have a 2 term coming here 2 S i j. So, that is how this 2 term come here because there will be 1 by root 2 term coming here and you can derive it and you can see that there will be a factor 2 coming in because of the consideration that you have this coming here also and here also. Now, just get back to the expression that we have. So, S i j the star why we use star here this is the old notation that we have been using for the range flows also is because we have this 1 by 3 delta u k by delta x k because for the compressible flow our del dot u is not equal to 0. So, that will come and you will have the consideration of the bulk viscosity in this particular case and that is why we denote it as S i j star. The other reason why we are using S i j star here that in some of the simulation some of the course what they do they absorb this term with the pressure. So, the pressure is also a normal stress. So, you will have pressure coming here all other components are 0 and this term is also acting normally. So, what you can do you can absorb this term into the pressure and then you will have a modified pressure and then all you need to basically use this SGS model applying only on this basically shear stresses. So, that is another way some of the SGS models some of the adiabatic models are used where they absorb this term into pressure that just another way of doing things. Now, the question is how do I find out the new SGS that is our variable use the same idea that we had in RANS the new SGS is the unit of this is meter square per second. So, in order to get that we need a length scale and you need a velocity scale this is your meter and this is your meter per second. For the length scale we will use the same idea that we had in RANS that we had this integral length scale. So, if you are and that was representative of the geometry that we had similar idea can be used here. So, here we call it as a sub grid length scale and because this model has to be made in such a way that it acts on a grid which is of a given size. So, that grid size has to come into the consideration so that is why you will have this C S multiply by this volume to one third this is nothing but your delta. So, delta is a representative of your grid size and as discussed earlier this volume can if it is structured grid this is delta x delta y delta z to the power one third. And C S is a small constant it is value is it can be determined by performing a D N S separately. So, what it is the value of this can be anything between 0 to 1 and different codes will have different values for example, I think fluent uses 0.1 some other codes use 0.17 and so on and so forth. But if you are developing a code for L E S and you want to find out what is the value of your C S what you can do as you can perform your D N S for your flow and then you perform an L E S for your flow with different C S value. And you can see that I mean for what C S value you have the best representation or best approximation to the D N S and that is how you can determine your small c constant. But it will be somewhere between 0.1, 0.2, 0.3 and so on and so forth. Now, we found our length scale we need to but I mean as we have done if you remember the our frontal mixing length model and one this mixing length model what we have discussed is that we have a log layer. If you plot this u plus versus y plus this is how it looks like in the log scale. So, this is somewhere your log layer and what we have derived is that your velocity changes as kappa y in this. If you use this you can see that your velocity changes as you use the log scale. So, what it does this kappa y it ensures that it is respecting the law of the wall when it comes close to the body. In all other regions you can use this C S delta to the power 1 by 3, but if it is a wall bounded flow you need to respect the how the velocity is satisfying the no slip condition when it comes very close to the wall. That is why this kappa y comes into the picture and you can take the minimum of this kappa y or this C S multiplied by delta. In fact, you can also use correction of the one which also basically matches the value more accurately in the log low region which is I think e to the power minus 1 minus y plus by a plus or something like that. So, we have discussed about this von Rist model earlier and multiply by delta. So, this also ensure that this is your von Rist model and then C S delta. So, minimum of these two. So, this will take care of your damping also very close to the wall e to the power 1 minus y plus by a plus. So, this is von Rist mixing model what we did the correction on the wall plantal model and you can refer to our earlier lectures we have discussed about this mixing model in more detail. Now, let us finalize our model. So, we got this length scale which I will just write here C S delta and what we need is the velocity scale and because this mixing model is inspired by this mixing length model. What happens in the mixing length model is that you basically look at the gradient and multiply by the. So, this let us say this is your del u by del y and multiply by this length scale that will give you the velocity scale. So, similar thing we will do here also. So, your del u by del y will be replaced by for a general 3 dimensional case 2 i S i j tilde S i j tilde and multiply by your length scale will be multiplied by delta. This is your length scale and this is your velocity scale and if you multiply that mu 2 will be u naught l naught it will become C S delta square square root of 2 S i j tilde S i j tilde. So, this is your the S g s viscosity. So, this is your new S g s I should not write new t this is and this will be new S g s because I have been using the same slide I wrote it new t this is new S g s. Now, this is the smierowski model and the value of C S has been basically calculated by many researchers this germano and this lily and many of the researchers actually use this homogenous isotropic turbulence to basically calculate C S and different values have been proposed over the years and as we discussed it can go to 0.17, 0.1, 0.23 and so on so forth. So, if you are using a code you can go back and see what is the smierowski constant that you are using. The last comment I want to make is on different models like we have this. So, when you want to use this LES model what happens when you go very close to the wall it requires some modification because you have this viscous sub layer where the LES I mean the S g s viscosity is not very important and then you have the law of the wall. So, it all needs to be respected. So, in order to basically address this issues we had this models we have the veil model what it does is basically it actually modifies the velocity scale. So, that it can address the value how it approaches the wall this modifies the velocity scale. Then of course, we have this one dish model which basically ensures that there is a damping in the viscosity when it approaches the wall and then we have some kinetic energy based model T K E based model what they do is that they change the equation itself when it approaches the wall. So, this is a very important aspect and that is why you have many models when it is a wall bounded flow. So, apart from this you can have wall model LES where it is a kind of hybrid model between the RANS and LES where you model the flow very close to the wall you may use a RANS equation and away from it you can use a LES equation. There is wall resolved LES where you have basically resolving everything and then you have a veil model we have T K E based kinetic energy based model we have one dish model and this these are the different ways to basically handle the viscosity very close to the wall. So, this is what I had to discuss about this LES and we have covered that how LES equations are derived what kind of filters that you can use and even if you need a filter and how do you use the viscosity to basically ensure that we have enough dissipation so that the solution does not blow up. The only remaining thing that is that needs to be covered is that how do you ensure that turbulent flux machines are given to the flow. So, what happens like if it is a RANS simulation what you do you give a turbulence length scale or the turbulence intensity and it calculates your K and epsilon we discussed in the RANS classes. It is not so easy in LES let us say let us go to our example of flow over a turbine blade let us say this is our geometry is and this is your inlet and there is certain turbulence level let us say 5 percent is your turbulence level. So, you cannot give an integrated quantity you cannot give let us say turbulence kinetic energy or the dissipation rate in the beginning what you need to give your u prime v prime w prime rho prime and p prime and so on. So, you need to provide the fluctuations in LES or a DNS. So, if I am an inlet turbulence coming in you need to provide the fluctuations and it is a very important thing in LES and because if you do not provide this then what you are doing is essentially your incoming flow is laminar and it I mean for it to develop turbulence it will require a very long length and typically you cannot afford those kind of things. So, there are ways to basically provide your input turbulence you can have a precursor simulation where you can find out these fluctuating quantities based on a turbulence level or there are approaches like digital filter there are approaches of where you basically recycle and rescale. So, and the other thing is that it needs to be changing with time. So, it has to obey certain statistical property of a turbulence and it should change in time every time you cannot have this like say fluctuations. So, it has to be basically correlated in time it has to be correlated in space right. So, that is another research area that how do you provide your input turbulence for a LES and DNS simulation and we will cover some of this aspect in future classes. So, with that we will stop here thank you very much.