 The other one is closure, and then the other one is RANS. A lot of my colleagues, when they say they're doing RANS, the RANS equation is an exact, low-order statistical equation associated with the Navier-Stokes equation. So when my friends that do DNS, they tell me they got this great result from DNS, I ask, does it agree with RANS? If it doesn't, something's wrong with your DNS. And what I mean by that is that the RANS equation is an exact, unclosed equation. So if they're doing DNS, they ought to be able to calculate every statistical property in that equation, and it ought to balance. If it doesn't, then there's something they need to look at in their DNS. So a RANS equation is something we don't need to apologize for. What we need to do is have a closure for it and see how we can go forward. This RANS equation is probably the most used equation ever. Engineers use it to do a lot of design and stuff, but they're using it with a theory that was introduced 150 years ago, which is the eddy viscosity theory. And so the eddy viscosity theory should be put away. And I have 19 slides, and I'll get going. Maybe you won't agree with me at the end of the slideshow. The other thing is the rental stress. Because of its position, well, I'm speaking to the choir now. I don't need to spend a lot of time on a closure. So let's get going and see what we have. Just a little tutorial, the first nine slides is giving some arguments of how we got to the closure model for a Newtonian fluid in the first place. And then we'll do four or five slides on turbulence. But one of the things I just want to remind the audience of is that this kinematics of continuum, so we're not just talking about discrete physics, we're talking about continuums, in the motion is a concept that we require our priori that a material element cannot occupy the same place as another material element at the same time. So we have this inverse hypothesis that the motion of a particle, there is an inverse. We rarely know what it is, but could be mapped back into its original position. So this leads us to the idea of a Lagrangian velocity field and an Eulerian velocity field. The equivalent of the motions, if you're in a different frame, so we have this definition of what is an equivalent motion. And if you as an orthogonal operator, it's arbitrary, we get to pick it. It's only time dependent, but it connects the motion in one frame to another frame. And so the star is just a different frame than the un-starred. And so this is the formal definition of what is meant in continuum mechanics by equivalent motions. So the Coriolis theorem, which transfers the motion to the, maps the, beginning with the mapping of the motion into its motion in the star frame, this is not an objective relationship between the chi and the chi star because of this factor here. This is the velocity. The velocity is not an objective property of the field. This is the acceleration. The acceleration is not an objective property of the field. And the acceleration is what enters into the equation of motion for linear momentum. So an objective property of a continuum, we do have a lot of objective properties of the continuum. An objective scalar field has the same value of the frames, in all frames. So if this were an objective scalar, that it's going to be the same number or the same scalar function in both frames. An objective vector is a vector which the magnitude does not depend on the frame, but its orientation does. If you have an objective operator, a dyadic valued operator, then this is an objective operator. And the eigenvectors and the eigenvalues of the operator are objective scalars. And the eigenvectors are objective vector fields. So not everything you touch is objective, but some things are really objective. For example, when we introduce classical assumption, is that the absolute pressure is related to the internal energy pre-enough mass. This is an objective scalar field, because we assume it is. And it works. This is the Cauchy stress vector is related to the normal vector of a material element. The normal vector of the material element is kinematically an objective vector field. We assume that the Cauchy stress vector is an objective. That's an assumption. That's not something that somebody gave us. And so as a consequence, you can then prove that the Cauchy stress is an objective operator. And incidentally, it may or may not be symmetric. That's an additional assumption. So what's shown here on slide seven is the unclosed equation of change for linear momentum and mass. So if you let the omega symbol be the angular velocity of the frame, then you can relate that to a rotational operator of the frame just by using the permutation triadic. So the equation is a mass balance or a continuity equation. So this equation is not closed. This equation needs to have a constitutive model for the density. So you have to say, because in continuum mechanics, we assume that every spacetime point is in thermodynamic equilibrium. And so you have to have a model for the density. It might be that you say that the density is a constant. If you say that the density is a constant, then the divergence of the velocity field must be equal to 0. If you say it's an ideal gas, then you get something else. So this is an unclosed equation, but we have to close it off. And this is the equation of change for the momentum per unit mass, which is called the velocity. So this equation is written in a frame of reference. So it's relative to a rotating frame of reference. And so you see the Coriolis term show up here. So this transformation allows you to gather the two omega together with the velocity gradient. This is the acceleration due to gravity. This is the Cauchy stress. Notice I put the transpose here to emphasize that it's not necessarily symmetric. There are fluids that have anti-symmetric stresses, not just symmetric. And then this is the effect on the pressure. So the deformations and strains, we have some important. This is the velocity gradient. The velocity gradient is related to the velocity gradient in one frame, the star frame, to the unstarred frame. And it has this factor here. This is not equal to 0. The velocity gradient of the velocity field is not an objective property of the motion. It's not an objective property of the motion. However, the symmetric part of the velocity gradient is. And you can prove that theoretically. That's not something you assume. That's just true, OK? So now we can sit here and make a hypothesis on what is a Newtonian fluid, and so we get to play the game. So if we assume that this Cauchy stress vector was an objective property, so this dyadic value operator, the Cauchy stress, must be an objective operator. And so if you're going to make a hypothesis that the left-hand side is objective, then the right-hand side has to be objective. And if you look around for a kinematic property that's connected with the Cauchy stress, this is the pressure that is objective because it's related to the thermodynamics. And then this is the deviatoric part of the stress. And this is the definition of a Newtonian fluid. So a Newtonian fluid is a mapping of the strain rate, which is objective, into tau. And tau is objective if the dilatational viscosity and the viscosity are objective. And you can prove that they are from the second law of thermodynamics. Second law of thermodynamics requires that. So this is the Newtonian model. This is a good equation, and it works, OK? Now, in 1897, Boussines needed a model, too, to close off the Rands equation. So he said, oh, just like they did in the continuum mechanics, going from the molecular scale to the continuum scale, why don't we have this idea of a viscosity? And that's where it came from. But that assumes that the fluctuating velocities at the subcontinuum scale are objective, and they are, because the only forces that are acting are objective forces. But it assumes that the fluctuating velocity at the continuum scale is an objective vector field, and it's not. It's not. But this might have been a good model. So that's how we got into the business of having an eddy viscosity theory and generalizations of that. And so I would have done the same thing, and so would you in 1897, OK? So this is the Rands equation, which is exact. So this is a decomposition. This is the Reynolds decomposition into a mean field. This is an ensemble operation on an ensemble of turbulent flows, and this is the fluctuating component. The Rands equation is an equation for the lowest moment. It's an equation for the average velocity, and it has this Reynolds momentum flux, which needs to be closed. It's not a closed equation. If you take this off the table, this is just a Navier-Stokes equation written in a frame of reference, in a non-inertial frame of reference. And the continuity equation remains the same, because we're looking at this at constant physical properties. So I have six more minutes. So hydrodynamic fluctuations, this is an exact equation for the hydrodynamic fluctuations, and just organized it in a special way to emphasize that this is a parabolic convective differential operator. This shows how the fluctuating velocity couples with the external fields. One of them is the velocity gradient. This has a symmetric part and an anti-symmetric part, and this is the Coriolis term. So this vector f prime is defined here. It's the divergence of a fluctuating stress as shown here. This is the fluctuating velocity multiplied by itself. So this is a dyad. This is the ensemble average of that dyad. And we introduce a operator called b, which is the normalization of this f prime dyad. Now, the important thing about b is that we know it must be a non-negative operator. This is a mathematical fact. The normalized rental stress must be a non-negative operator. That's a mathematical fact. When we do modeling in turbulence, this is something we think after we model it, and then we find out computationally that it's not, they call it realizable. So if it doesn't satisfy this property, then the operator is not realizable. So a normalized pre-stress b, you can make a hypothesis. And we did that in our work, but we just assumed that b was a dyadic value mapping of r into b. So we're shifting the closure question to b. And this is how we close off the equations. We say that b is a function of r, which is our focus for the Rands equation. Now, just using the Cayley Hamilton theorem, you can have an irreducible representation of this. You get two parameters. And these parameters can be functions of the invariance of r, but they have to be in this domain here for it to be realizable. We know that the b is a non-negative operator. We know that r is a non-negative operator. That's a mathematical fact. And so this constitutive model here is just a mapping of r into b. And if alpha and beta is in this domain, then that mapping is true for all turbulent flows, all turbulent flows, whether they're steady or unsteady, whether they're three-dimensional or whatever. It's just a mathematical fact. So here's the eddy viscosity. So let me get it. I've got three more minutes, Chairman? OK. Yeah, so I've only got, I have to scoot along here. So there's a mapping that we developed called a pre-stress mapping, where we connected this is the nonlinear equation for r. b is a function of r. a is an operator that came directly from the Navier-Stokes equation. But this shows that the solutions to this equation are going to depend on the Coriolis term. And that stemmed from the previous equation that I showed you. So here's Boussin's eddy viscosity theory. The question, why do we make this assumption? And I think it was a logical thing to do at the time. But what we know is that the real stress is not an objective operator. But they assumed that it was an objective operator. And that would work. So this was tested not too long ago in 2000 or so by some folks that did DNS simulations in span-wise rotating flows. I don't have time to give you a lecture on this. But let me show you the results of this calculation. So this is the predictions from anisotropic. These are the anisotropic energy states for fully developed channel flow with no ring rotation. We've known this since the 1930s. That the energy is not distributed equally among all the components. The Boussinist approximation assumes that the energy is distributed equally for this flow field in all components, everywhere in the flow field. All the way up to the wall into the center. Normalized normal components. The theory that I showed you and didn't talk with you about, well, this is what happens when you rotate it. So when you have a small rotation, then you break the symmetry. So this is the energy distribution in the channel with rotations. And of course, the rotation will affect that. The prediction from the Boussinist approximation and that it's still, the energy is still distributed equally among all the flows. When you have a hypothesis, you only need one experiment to show that it's wrong and you don't use that hypothesis anymore. You can never prove a hypothesis is correct. You can only prove that it's wrong. And this shows that the Boussinist approximation for this flow field is wrong. So why use it for any other flow field? So for the theory that we've developed at MSU, this is the DNS energy states and this is the energy states that we predicted a priori. We did not change any parameters to agree with this energy distribution. We've calibrated our model with non-rotating flows. But then when we took it to this data to compare it with the DNS, it doesn't fit the data, but it breaks the symmetry in the energy distribution in qualitatively the same way. So the question is, well, does the audience have any questions but this was way too fast probably for you to have any questions. But I just want to emphasize that we had this choice that this normalized rental stress had to be closed and so we selected the strain rate 150 years ago and we still use this. So the Eddie Wisconsin enclosure is objective because we assumed that it was, but it's not realizable and that's a disaster because the computer won't converge if you have a non-realizable behavior because the eigenvalues will become negative of the normalized rental stress. The theory that we published and we've developed over the last 20 years at MSU takes a different view as I've tried to explain that the normalized rental stress, this is a nonlinear algebraic equation for the normalized rental stress. A mapping of the normalized rental stress into itself but the parameter that you need is a characteristic time constant and this operator here that includes the Coriolis word. The thought for the day is that this we call it the URAPS closure because this means universal, realizable, anisotropic, pre-stress closure. We made up that word. But it's not objective unlike the Boussinesse approximation. However, it's realizable for every turbulent flow you want to compute. Every turbulent flow, whether it's statistically un-stationary or whether it's statistically stationary or it's three-dimensional and so forth. That is worth a chance on this type of mapping. The analysis, we don't claim that it's unique but we claim that it will work in a different way than the classical law. So I thank you for your attention. I apologize for going over time. I apologize for sitting down and I apologize that my mouth is dry but I'd be glad to answer any of your questions that you might have. No, the components of a tensor are not the eigenvalues that are... For instance, every scalar value of thermodynamic property is objective because that's the way we do thermodynamics. We assume that all the forces that act at an electric scale are objective forces. All right? And so whatever we derive from that is going to be objective. And so when I said that, the definition of an objective scalar, an example of that would be the absolute pressure, the absolute temperature, the specific energy of the internal energy and so forth that come out of thermodynamics. I don't know if that was your question but we could stand closer together and maybe I can answer your question. Yes.