 Good evening. My name is Germano. I am at Duke University. It's almost an honor reposition because I live in Italy now. I am retired and I would like to speak about blend and nudge Navier-Stokes equation, what it means. Okay. First of all, demotivation. We have today two interesting and, in a sense, similar topics that are the integration of CFD and FD, and hybrid runs, yes. From the point of view, it means how we can dialogue between different topics and different modeling techniques. The simplest mathematical approach engineering approach, pragmatic, pragmatic approach are always being blending and nudging. In a sense, a simple way of mixing is the simple way. And so I will analyze these approaches and then I will speak about blended and nudged Navier-Stokes equation and now I will show what it means. First of all, motivation. We have computational fluid dynamics, experimental fluid dynamics. In an interesting preface about data stimulation in fluid dynamics, we found this statement. It's probably fair to assert that AFD and CFD have been independently developed for a long period of time, the last several years or whatever. More attention has been paid to the integration of AFD and CFD to compensate mutual weaknesses. Then another motivation. We have Reynolds-Severage Navier-Stokes equation runs. We have larger dissimulation, LES. Also in this case, from the practical point of view, it could be very interesting to couple LES and runs. It's arguably interesting paper about hybrid less runs. It's arguably the main strategy to drastically reduce computational cost for making LES affordably in a wide range of complex industrial applications. You see that the motivations are theoretical and practical at the same time. We have a practical motivation in order to reduce the cost. I am speaking obviously of very non-homogeneous flow, not anisotropic flow or non-homogeneous flow. In this case, but we have also very interesting theoretical motivation about that. Let us look at the way in which we can analyze the problem. We have the Navier-Stokes equation. LES has been by Leonard formalized from the point of view of filter and Navier-Stokes equation. To filter results is very simple. To filter equation is not so simple. I have a lot of friends, mathematicians that always say what are you doing at this stuff. Anyway, I would introduce a linear operator. I will indicate with these symbols the filtered quantities. I apply this filter to the equation. I filter the Navier-Stokes equation. What happens? It happens that in Navier-Stokes equation we have a lot of new forces. The divergence is not zero. There is a force in the momentum equation. The main term is that term. It appears in the Reynolds equation. It appears in LES. It is a stress. We have the force that is due to the non-linearity of this term. Then we have also two other quantities due to another point that usually the filter does not commute with the derivatives. If there is no commutation with the derivatives, space derivatives, time derivatives, we have this term. The divergence is not zero and there is a force induced by the non-commutativity of the filtering operators with derivative and the derivative of the filtering operators. Fortunately, the operators, the most important operators, the LES average, the Leonardo average, they are commuting with space and with times. So we have only this term in the equation and this term is the so-called subgrid stress. The Reynolds average, I use the Reynolds average from the point I assume that there is the ergodic, that the Navier-Stokes ergodic, obey the ergodic principle so that the Reynolds average intended as an infinite time average also is commuting with time and space so that we have also in this case we have only this is the famous Reynolds stress. So from this point of view, LES and Reynolds are the same. It's only a question of scale because formally they have a term that is stress and that is induced by the non-linearity of the equation. Let us go on. The procedure for hybrid around less are the detached integration, global modeling. There is a lot of way in which, as we have seen in the previous presentation, model, very complex k-epsimum model are modified in order to become zonally LES by introducing the grid as a particular length, a significant length and so on. I don't enter in all that that is so complex, so difficult that they're real. A very simple way is blending. Blending is a very simple way to blend the model. To blend the model means, okay, there is the blended RANS LES model is a blended, there is a blending function, k, function of XI and T, and this is the demodulated. This is the way of managing the coupling of LES and RANS from some zones to another zones. Okay, let us look at this blending formalism from the point of view of filter at the Navier-Stokes equation. It's very interesting because if we look at the problem from the point of view that it's probably better to introduce a blended filter and to look to the model that is associated to the blended filters following the idea of Leona. If we introduce a blended filter H, and we define obviously the quantities like that, we have this result. Well, okay, obviously the stress are always algebraically defined in the same way. This is the Reynolds stress. This is the subgrid stress. This is the hybrid RANS LES stress. And the result is that, okay, the pragmatic blended model is right. If you have another contribution, there is another contribution. This is the model associated to the blended filters. And this contribution is important in order to interface RANS and LES in different regions. And from the point of view, it's always a very pragmatic way of looking at the problem to blend, but at least I blend filters. And from this point of view, that is a filter. Obviously there are other quantities related to this filter because this K depends on X and T. And so there are other contributions as we have seen. But the most important contribution, the contribution due to the stress, the hybrid stress is like that. So that the hybrid stress is a composition of Reynolds stress, a weighted composition of the Reynolds stress, and there is a... Let us go rapidly to the integration of CFD and AFD. Obviously I only give a very simple consideration. The data simulation is a topic that you know is so complex and there are variational methods, the regression. Interesting, from the point of view, are very interesting the sequential methods. The sequential methods, you imagine you have a computer that is produced by an experimental facility coupled with a computer, and they dialogue. And they dialogue and they give results, they improve at least. We hope that they improve the simulation. Obviously it could be the opposite. But anyway, nudging is the typical way in which sequential methods are introduced, I don't know, in meteorological provision. To nudge, to nudge what it means exactly. Nudging in a sense is the Newtonian relaxation. It's an idea solved that it is attributed to Newton. And it means, okay, we have this equation. The CFD, AFD, nudging algorithm that has been used, is used in some application, is to add this term. And this term is a feedback force driven by the measured velocity field that in some sense controls the computation by enriching, by driving the computation with what in the... This is a very simple way of looking at that, but it was now used in meteorological simulation. We continue to do computation and we enrich our computation with real data, real data that arrives to the computer by the real data provided by meteorological station. If we look also at nudging from the point of view of the filter at the Navier-Stokes equation, we have the same, okay, nudging obviously is intuitive, is easy to implement. It lacks a firm theoretical foundation like blending. The nudging coefficient is based on pure empiricism. If the nudge coefficient is too strong or the computation is driven so rapidly by the data that it becomes unstable, if it's too little, the effect is... So there are a lot of problems as exposed in this very interesting paper by Lashmi Varan and Lewis. But let us look from the point of view of a filtered algorithm. Let us... We have a computation and we have an experimental simulation. And we assume that they are solutions of the Navier-Stokes equation. Okay, the hybrid CFDF filter can be defined in the same way of the hybrid filter that provides the dialogue between LES and RANS so that now here I use these abbreviated symbols. K is a blending factor, function of time, such that at the beginning is one. One means, okay, I have my computation, now I have real data and I try to introduce this data in order to nudge my computation. And obviously the limit of KT is infinity. What happens? It happens that the filtered equation for the hybrid CFDFD are given by that. And now we have two terms. We have the first term is that we have as for RANS less an hybrid stress. But we have also another term that is due to the fact that my... my blending factor K is function of T so that this filter does not commute with time. And the non-commutativity with time produces now this term here. And it's very interesting because feedback is a... feedback blending force. The term is also in this case very similar to the blending. And the coefficient lambda now is directly related to the function K. One, three copies of feedback blending stress given by that. In a sense, it's a non-linear blending stress. This is very new. It's not a relaxation. And it's due to the non-linearities, obviously. Very rapidly I conclude. Well, if the blending factor is exponential, we have the same results as usual nudging. If exponential, this value of one over K is lambda B is lambda. And that is the result that I have published recently. So that conclusion. Modified version of the Navier-Stokes equation has been recently proposed where a nudging term or a blending model has been added. Mind, I have obviously reduced it to the minimal form of that. They are very complex, but from the point of view of the impostation of the problem, that is what happens in order to improve the dialogue of different techniques of different theories. The filtered from the point of view of the filtered Navier-Stokes equation associated to a hybrid blending operator, they are similar as regards to the nudging term and the blending model, but different for the appearance of a forcing stress. And this term could be relevant in the formulation of hybrid runs, LES approach, and this CFD integration. As usual, the conclusion is that there is a lot of work to do. I am very old, probably. I hope that young people that are here will be a little interested by all this stuff. Thank you very much for your attention. No, no. People who nudging based on the mean points, whether in mean areas with mean... This seems to be much more successful because what happens with this term which you show the implementation term is because there is one region where there is a lot, and then this term is gone, and then we can look at mathematical rigorous assumptions because there is a layer which becomes a bit languageable. Is that... What's your proposal now? There is a lot of work. You are right. Obviously, experiments are... In many cases, you are with particle, image, velocity, symmetry. You have 2D slides. Or you have... Or you have average values and LES. Okay, but this is only the beginning. The idea of managing all that from the point of view of introducing filters. Here, I have only imagined that you have really... The true are the results from the starting point that I have the true. The model is not so correct. The hope is that the model can be driven by the truth in order to... That is the fantasy there. But in many cases, I have seen that there is people, particularly in Japan, that are interested in the problem. There is a lot of post-processing in order to improve the dialogue. It's not so simple. The problem of having to couple, to nudge LES and runs is, in a sense, easy to do because I introduce more complex filters. And in any case, it's interesting, in my opinion, at least to look at what happens from the formal point of view if we introduce filters, which is the model associated to the filter. They give some hints at least in order. But anyway, you are right. It's very complex, the problem of data simulation. So what is being solved by... I think there is a piece of chalk. No. We are very... I think the classic way of looking at, if I have understood your question, of joining LES and runs. Excuse me. Yes, yes. Okay. Okay. So, two problems. There is a wall. Runs LES. This is the usual way. So, I put my lens here. But another way is, I don't know. Here, I have to put the lens. Here, runs what there is in a channel. A southern Spanish. So, in many cases, everything is known from the point of view. But here, we have the problem. The vortices and so on. So that the way in which the reason why, in many cases, I have the problem of joining LES and runs. I don't know. In Stanford, now they have done a hybrid LES and runs in which there is a complete turbine and there is the compressor. And the runs and the expansion is LES. And so, that is not so intuitive that I have runs near the wall and LES in the back. The first time I spoke about that, people said, why? It could be the, no. We have a deboundary layer. Now, we have a lot of very good runs models near the wall that manage very well accelerated boundary layers, pressure-driven boundary layers. And obviously, LES means the eddies. We have no eddies near the wall. We know that so that we are obliged near the walls to use runs models. But I don't know if I rightly understood your question. Well, that is probably more than a joke. You are right. It's more than a joke. But what about DNS? It was following this. Yes. No, no. From the physical point of view, I mean, in the last, in my activity, I published a paper in which I was very proud about the filtering invariance. The equations are the same. The equations don't know that you are filtering the interference operator or with LES operator. The problem was due to the fact that at first people was using defluxation. Also that defluxation are very easy and very nice for statistical fluctuation because the mean of the statistical fluctuation is zero. But the mean of the LES fluctuation is not zero. It is better to introduce formally the sub-gradient stress as you are right. I think, but, and today it's not so long. Now you don't see paper in which they speak of LES fluctuation. At least that is a good improvement in my opinion. So the, the stresses are the final like that. And all that, and also the triple correlation are defined algebraically. If you define algebraically the moments, they are the same from the algebra, so that the difference is in how you model, as you have posted this morning, the problem of the, I beg your pardon, the, the, you were speaking this morning about the fundamental quality of a model that is the realizable, but not only realizable. The, the invariance, the framework, it applies to LES, it applies to the Leonhardt filter. So it's model, in the model you have the length, you have all that, but from the formal point of view, the equation is same, excuse me. Yes, I think so, global modeling. Yeah, but that is a practical point of view. But from the theoretical point of view, you could, global modeling. I think they're doing, they're using hybrids and then what you're solving may be a very narrow class of flows and you can use it to do interpolation, not predictions. So there is such a, Probably. The fundamental is that in the RANS, you forgot the structure. In the LES, the RANS is the structure. So they are completely different things. Oh, they are different. So the energy spectrum, that is the, building things in turbulence is forgotten in the RANS. So it's something else. Yeah. That's it. And also the scale, if you go years close to the wall, it's no longer mostly boundary tension. But if you can, if you could model boundary conditions, you don't look back here. But no more than that. So I had a practical, typically, we do LES, and many times the grill is the filter. Yeah, an adaptive mesh. But I think it's... So then you know something like speciality, you know, I'm trying to play with myself now. And they have this magic contribution function that mixes the turbulence model with LES. Yes, I know very well the paper by special... It's kind of a phenomenological function. They've used, I mean, some of the specialities started using it. Everybody keeps doing their own... The famous paper on the IA88 paper. So basically you have two pieces of art there. You have the adaptive mesh refinement, which is usually totally unharmed. It's based on whatever you think is important.