Lecture 60

Hello everyone, welcome back. The last few classes we were discussing about the G-score resolving simulations and we discussed about the like DNS and LES and how they differ in terms of computational cost and compared to RANs, okay. So in the last class we started discussion about LES and we said that in terms of energy spectrum, if this is your energy spectrum, this is E k versus k, a good LES is the one where it resolves at least 80 percent of the turbulence kinetic energy, right. So if this is 80 percent let's say, okay, so the LES should resolve this region, okay and you can use a model for this region which is mainly for the dissipation, right. So this is one check you can do in LES, you can, we discussed about how to calculate this resolve energy, resolve kinetic energy, basically the k-RES, okay and also the k-SGS, so SGS can be directly calculated if you are using an explicit LES, it can directly be an output of the model that you are doing, model that you are using and k-RES can calculate based on, so you have this u v bar and u bar v bar and just you can subtract them and you will get this u prime v prime bar, okay, we had this derivation yesterday and you can calculate what is the ratio of k-RES by k-RES plus k-SGS or k-model whatever, so and if the value of this is greater than 0.8, then it is a good quality LES. So this you can check well, you are basically, you have already performed LES, the other thing we discussed regarding LES is like how do you decide the grid size before starting LES, so typically what you do, you perform a range simulation, right and from the range simulation depending on what model you are using, so let us say you use the K-Float model, okay, so you can calculate the length scale from this K-Float model and how do you calculate that, remember like we have discussed in the eddy viscosity model, all these models have this new turbulent or new eddy, okay, eddy viscosity is basically some factor of L naught into u naught, okay, some length scale and some velocity scale because this, it is unit of k is meter square per second and this is meter and this is meter per second. So if you are using a k-epsilon model for example, then this thing will be, so your L naught will be I guess k to the power 3 by 2 by epsilon and you can calculate it also because see k is, unit of k is, so meter square per second square, right, this is k to the power 3 by 2 that will be meter cube by second cube, right and epsilon is the dissipation rate, so how much energy is being dissipated per unit time, so meter square per second cube, right, meter square per second square is the kinetic energy and per second of that. So if you do k to the power 3 by 2 by epsilon that will give you the unit which is meter, which is the length scale, right. So you can calculate the length scale from the model and what we also discussed that you can have an integral length scale, so what is the meaning of integral length scale in this case? So that is the length scale at which the average of kinetic energy is obtained. So this can be your, so this can be your 2 pi by L naught, right and L naught you can calculate from this, from any turbulence model, I have given example of k epsilon model but you can use k omega or any, any model that you use based on adiabatic viscosity model that will have this length scale, okay. And what we also argued that typically how do you find out if m s is good enough for LES is that we should have at least 5 elements to resolve the integral length scale and when it is that it is likely that it is going to resolve the centripetal kinetic energy. So if we are using 5 let us say we have base scale which is delta which is L naught by 5. So this is my 2 pi by delta, delta is the length scale for LES, okay and we are arguing that this typically resolves centripetal kinetic energy. Now delta you also know from your grid because typically delta is C s into volume to the power 1 by 3, okay and for the let us say if your mesh structure and this is C s multiplied by delta x, delta y, delta z to the power 1 by 3, okay. So this is some kind of determination of the length or your grid size, right. If it is unstructured then you can use any method to calculate the cell volume, right and C s is a constant whose value vary from 0 to 1 and typically it is taken as 0.1 or you can some codes take 0.17, 0.2 and so on and so forth, right. So you know the value of delta from your grid, okay, you know the value of L naught. So from your range simulation and what you can do? You can plot this L naught by delta in your entire region, okay and you should see the regions where this value is less than 5, okay. Typically what we want? This L naught by delta should be greater than 5. If it is greater than 5 that means we have at least 5 elements in each integral length scale and then the mesh is good enough for LES and if the value of this L naught by delta is less than 5 then that is the region you need to make refinement, right and you can do that and this is how you basically decide the size of your LES grid. So typically because you do a RANS in, RANS you perform in two-dimensional framework, right. So based on your two-dimensional framework you can also decide that if your grid is 3D what is the value that you should use in the span-wise direction also because you will get this minimum delta x delta y in 2D and accordingly you can decide out delta z also and it also depends how much variations you have in the span-wise direction. Some flows are nominally 3D that means that the third dimension is for the teloponnes to develop but you can have the mean flow in 2D that means there is no change in the mean flow in the span-wise direction, okay. In that case you can relax the grid size slightly in the span-wise direction compared to what you have in the boundary layer on the face, right but if the grid is, if the flow is truly three-dimensional then you have to make sure that you are resolving in the span-wise direction very well, okay. So this is what we discussed in the last class, okay. So today we are going to discuss about the LES equations, okay. So in the last class we had a very brief discussion about LES equation towards the end when we said that like we had this like Reynolds stress or the Fabry stress we also have this sub-grid stress in the large eddy simulation and the only difference was is that in the RANS we are using for the mean flow, ER we are using for the actual flow variable, okay which is U V W and the second difference that we had is that in the Reynolds average noise stress simulation we use this turbulence model everywhere, okay. In the LES we want this sub-grid scale model or the sub-grid stress to act only on the grid size which is smallest grid that means that if you go to the chi-schedule structure so it can resolve this grid and this grid, this eddy, this eddy, right but it cannot resolve eddy which is just fitting in here, right and suppose the dissipation is happening at a scale lower than this which is what typically happens in LES in that particular case what we need to do is that we need to do something so that the dissipation happens at this grid size, okay and for that we need to basically increase your viscosity to increase the dissipation rate and you have molecular viscosity and you will add some kind of this sub-grid stress, okay the viscosity for the sub-grid stress. So let us because there is a lot of analogy between the RANS equation and the LES equation in terms of this sub-grid stress, let us go through the derivation of the LES equation systematically to see how these equations differ from the RANS equations, okay. So when we think about LES so we can think about a sieve, right for example what we use to basically filter out anything, right so when you put something in a sieve so the larger particles will remain the smaller will go down, right. So the LES is in LES also we apply filter, this filter may be explicit or may be implicit in the sense that the grid itself may act as a filter or you may put a filter mathematical filter to basically filter your Navier-Stokes equations, right. In the most common practice we do not apply explicit filter and this is due to Smigronsky who said that your grid can be used as a filter, right because ultimately what we are trying to do is that for the smallest grid size we want to basically model some kind of dissipation, right. So even if you are using the explicit filter in LES it is a good idea to basically go through how these equations are derived, okay and the same concept will be useful even if you are using an implicit filter or explicit filter, okay. So filter is basically a kind of a convolution operation which is what is written here in G so if you have this phi can be any quantity it can be u, v, w, p, rho etcetera, okay. In the incompressible flow it is only u, v, w, p and if it is flow is compressible then you will do as well. So what it does what this convolution operator or the filter does is that it acts on this function phi, okay and it basically changes the nature of phi basically it acts as a low pass filter so it will allow the low frequency but will not allow very high frequency and it can be also written as so what we are doing we are denoting the filtered one as tilde and the normal one as phi, right. So this can also be written in a short form as G star phi, okay. So G is the convolution operator acting on phi to give you phi tilde, okay. Now these are some examples of the filter. So one is for example this box filter, okay like this or you can have the Gaussian filter, okay or you can have the spectral filter. Typically this box filter and Gaussian filter are used in the explicit filter alias approach and all of these have this same function that it acts on a signal, okay and it removes the high frequency content. Without going to details just let me show you one example. So let us say this is your original signal, okay this blue line that you see is an original signal and below is the convolution operator this is Gx, okay and this Gx is acting on this signal and you can see that as it acts on this signal, okay and the region starts getting smooth. So this is a box filter that is working acting on this, okay. So this is already smooth because this the filter is already acted on that and as it the filter goes through this signal and you can see that this becomes smoother and smoother which is shown in the right side, okay. So this will become smoother and smoother and like this, okay and at present it is in this region, right. So here also it has made the signal smooth up to this point and then as it goes through you can see that it becomes more and more smooth, okay and it basically loses the high frequency signal there, right. So if you do truly a DNS then you will resolve the original one but in the case of the LES let us say for applying a box filter what you are resolving is this signal and I gave you an example of box filter but the similar filters I mean similarly you can think about the Gaussian filter or the spectral filter and this is how different filters work, okay. So this one is for DNS, okay let us say if you are not using any filter and if you use a cut off filter this is the red region is what you see when you use it you basically cut off at a particular frequency, okay or a wave number. So you can see that the energy is just going down at the place, okay and the green one is the top hat filter what you see here these are the top hat filter, okay and then the blue one is the Gaussian filter and this is what we have been discussing when we are talking about LES that when you have this EK versus K, okay you are not resolving everything but you are cutting off at certain point of time and as long as you ensure that it is capturing an 80 percent of energy this region does not matter because this is primarily responsible for the dissipation of energy which does not participate in the key dynamics of the flow, okay. So now that we have filtered our variables, okay so we have filtered our variables u, v, w, p etc., okay so we can write the full variable or the full signal, okay phi is equal to phi tilde plus phi prime, okay I want to differentiate here this phi prime is not like what we had in the RANS simulation, okay. So RANS simulation these were like all turbulent fluctuations phi prime in the case of RANS, okay so these are basically turbulent fluctuations, okay. In the case of LES it is only in the sub-grid scale it is not that all the turbulent fluctuations have been captured here, okay because some of this turbulent part is also there in phi tilde, okay. So this is only the sub-grid scale fluctuations, okay and now going through the similar argument like if you recall we had discussed the properties of a Reynolds operator, right. So what were the properties of Reynolds operator for example equilibrium mean, okay so what was the Reynolds operator if you think in terms of incompressible flow, okay in Fabry we had the rho also but in the incompressible flow you had basically d t, okay and you basically started t equal to I mean t naught 2, t naught plus t, okay and I think divided by d t, okay t naught 2 t, okay. So this was the our Reynolds average or the time average, okay and we had discussed about some properties that if it is a good time average the mean of the fluctuation will be 0, okay and the other thing was that if you take the average of average this is nothing but the average, okay and then we had like if you have this d u bar by d x because a linear operator you can write it d u bar by d x similarly the summation will commute if you have phi plus psi bar it will be phi bar plus psi bar, right. So like this this is for RANS and like this we can have some properties for the filter also for the LAS filter also and what is the what are the properties? So one thing is that the integral of this the filter g x d x equal to 1 that means that if you have a constant A and you apply this convolution on that, okay A you will get the same value A, okay it will not change and like the Reynolds operator it is also you can if you are if the apply the filter on a summation of two variable you can break it or you can rewrite as the you can basically commute the operation you can first apply the filter and then do addition later and similarly if the operator is linear you can write it like this, okay. It is different from the Reynolds averaging and the reason is because we had discussed earlier that our fluctuation u prime in the LAS is not as the u prime for RANS it is not as u prime for RANS this is the turbulent fluctuation and this is the LAS fluctuation, okay LAS or SGS fluctuation that will be a proper term, okay and you can show it mathematically also using this filter. So let us say if you apply this filter twice you have a variable phi and apply this filter twice then you are basically this convolution is this is g convolution operation and then another convolution on phi and which is not equal to just one convolution. So here we had like u bar and then you take another bar because it is already mean so then it will be equal to just the mean, okay no matter how many times you are applying this averaging operation but here in the case of LAS, okay once you filter and if you filter again it will be different from so the variable which is filtered twice is different from a variable which is filtered only once and if you use this property here, okay this what is phi prime, okay so phi prime is basically phi minus phi tilde which is phi minus g into phi, right? So this is a variable, okay and then you take the apply the convolution on top of that so that will become g into phi minus g into g into phi and since this is not equal to this, okay so this value will not be 0, okay. So phi prime tilde will not be 0. So in general the LAS filter do not basically satisfy these two properties. Now that we know the filter we can basically derive the equations but before that I just want to discuss a little bit about the implicit and explicit filter. So in the implicit filter, okay we have discussed about the box filter and Gaussian filter and cut-off filter and so on and so forth, right? But they will not be basically explicitly used while you are writing a code, okay or while you are doing with the simulation and what we assume that your our grid itself is the LAS low pass filter, right? So the grid size acts as a filter, okay and this is a more common approach like an SMAGOL and STA and all. So where there they use this implicit filter but some researchers also use an explicit filter where they basically typically use the box or Gaussian type and this is applied to the grid, okay and both have got some advantage and disadvantage, okay and in terms of advantage what you have in explicit filter is that you basically have the control of what is going through your sieve, right and you know what is happening there but what are the disadvantage in explicit filter LAS is you basically lose some of the numerical resolution. So your scheme has got certain resolution but you are applying filter on top of that so you may lose some of this resolution, okay but this is I mean more tedious to write if you in terms of developing a code the explicit filter code is slightly tedious to write but implicit you make a grid and you ensure that the grid satisfies the property that we discussed earlier and that grid can be used as a low pass filter, okay. Now let us go through the derivation and like we did for the RANS case in this case also first we will derive for the incompressible and then we will make some comments for the compressible flow, okay and why we do that because as we know from RANS is that in the case of compressible flow your rho also becomes a variable and when you have the conserved variable like rho u and if you try to basically break it in terms of fluctuations and the mean flow we will get terms, okay if you have rho u v for example, okay you get triple product and there are many terms which are basically which complicates the equation a lot, right. So that is why it is easy to visualize in terms of rho where rho is a constant and u v are basically you can easily handle with two variables, okay. So this is your Navier-Stokes equations once again this is your temporal term then you have advection term, the pressure term and the stress term and in this case tau i j will be nothing but nu del u i by del x j, okay and that is why the system will become in the vector form del square u, right for the incompressible flow. So if we apply filter on this so we are applying filter on this Navier-Stokes equation, okay so basically you are multiplying this by g so what will happen the first term will become you will get a filter here, okay here you get a term which is u i, u j and filter on both of them, okay and then p will get a filter, okay and then finally you will have a filter on the tau i j term the shear stress term also, okay so which is basically nothing but nu del square u filter, right. Now what we will do we will write rewrite this term, okay u i, u j tilde and what we will do is that see we are solving for u i tilde and u j tilde, okay so we will rewrite this expression in terms of u i tilde and u j tilde separately, right. So if you want to do that then it can be written as u i tilde, u j tilde and plus u i, u j tilde minus u i tilde, u i tilde, u j tilde, right. Now this following the Reynolds stress argument that we had in advance we can write this in terms of the stress, okay so that is your sub grade stress, okay because we are dealing with the fluid flow problem so these forces basically lead to some stress and they can be expressed in terms of our known gradients the velocity gradients and that is why it is easy to write in terms of the sub grade stress, okay so that is how we will do so this will become tau i j s g s and if you take that out and put it back here what you will get is the momentum equation for your incompressible LES, right. So you have this term tau i j tilde and then you have tau i j s g s and what we will do in the next class we will derive about the derive compressible LES equations and we will also see that how we can apply this our adiviscosity model in this tau i j s g s, tau i j s g s, okay and this can be written as mu s g s, okay multiply by some kind of your so your gradient field the velocity gradient field, okay like we have done for the RANS, okay maybe there will be rho here, okay. So this we are going to discuss in a next class and we will see that for the compressible flow also we have basically the similar arguments we will do the power averaging because the density is also a variable there and we will have some more complicated terms of course we will have to we have to have the energy equation also but that is what we are going to do in a next class, okay. Thank you very much.