 analysis of heat transfer in a fully developed channel flow. This is an interesting simulation, a really clever simulation by Sergio Pirozoli, where he sets a boundary condition such that the flow mimics the momentum field case in many respects. A lot of this work was done by Aang Zhao, who's a second author, but I'm lucky to give the talk. So in light of that the the talk is going to proceed by first showing some of the elements of the analysis for the mean momentum equation, which it's a little more simple to understand, and then we'll move on to the same method for the mean equation for scalar transport, which ends up being a two-parameter problem with the Reynolds number and peranol number. And then we'll do an interesting comparison of the peranol number for one case, because that allows you to look at two mean flow equations that have the same form, same boundary condition, but give different solutions because of the indeterminacy of the mean equations. So to start with let's just consider turbulent channel flow. So this is a two-dimensional channel flow, infinitely wide, infinitely long. The walls are spaced too dealt apart, we have no-slip walls, and it's driven by a spatially uniform pressure gradient force. The mean equation of motion is given here, so this is just a regular old Rands equation, and we'll go ahead and simplify that for this flow configuration. And if we do that, then we get the following equation here, so this is just a mean momentum balance or mean force balance for turbulent channel flow, and what I've done is I've internormalized it, so I'm using nu and utal, the utal being the friction velocity, determined from the pressure gradient in these fully developed flows to normalize, so that's what the superscript plus refers to. And epsilon squared is a small parameter equal to one over the Reynolds number, and Reynolds number is based on the channel half height. This epsilon squared comes about by doing integral momentum balance for the channel flow, and you can show that this term does indeed become that. So this equation is exact for the given flow configuration we have, and the complication here relative to say scaling laminar flow, where we're going to try to develop an invariant form of the equation, and then once you have an invariant form, you can essentially get a similarity solution because you only have to integrate it once, you get one single solution. That's attractive if you believe that there's, for example, a universal log law in turbulent channel flow, because that should be the solution to an invariant equation. The issue here, however, is that even though this equation holds for the entire channel, across the channel there are different regions where the leading order balances change. And so what'll become important is understanding the transition between these leading order balances, and in particular across this layer three. So what I'll go ahead and do is describe how we determine these leading order balances, and it's empirical. Okay, so anytime we want to go ahead and do an analysis of the mean equations of motion, we run into the closure problem. And in the closure problem, to address that, either you have to make some sort of hypothesis, make a guess, pose something, some assumption, or you have to use some level of empiricism. In this present approach, we use a level of empiricism right at the beginning, where we determine the leading order balances. Once we have those leading order balances, everything else is just using the equations of motion and the boundary conditions. Okay, so this is distinct from many other approaches. All right, so here's the empirical step. This is a sketch, but I'll show you some data later on that look basically like this. And to determine the leading order balances of a three-term equation, we just take the ratio of two. So here we're taking the viscous force to the turbulent inertia, which is nothing more than the Reynolds stress gradient. And you can see a layer one, which is basically the viscous sublayer. Layer two is a balance between these two stress gradients. Layer three, which we'll focus on primarily, is where all three terms become leading order again, because there's an exchange of balance where in the layer three, we have the viscous force and turbulent inertia terms balancing. And out of layer three, we have the pressure gradient and turbulent inertia terms balancing. Okay, so across there, we have all three terms come into balance. And let's go ahead and take a look at that. And we'll look at it, first of all, at low Reynolds number. So at low Reynolds number, the reason I do this is so you can actually see the pressure gradient term, because remember it's one over the Reynolds number. So the green curve is our turbulent inertia. The red curve is the viscous force. And then the pressure gradient, which is constant everywhere, is the blue curve. So you can see in layer two, we have dominance of the viscous force and turbulent inertia. But as we approach layer three, they both become small. And in fact, the turbulent inertia crosses zero because the Reynolds stress itself goes to repeat. Okay, so it's in this region where we need to rescale. And the reason we need to rescale is that if we write the equation in its inter-normalized form, then you have a small parameter, epsilon squared. Formally, one would expect that term to be small. But clearly, it isn't in layer three. It's of the same order. It's of leading order. Okay, so this requires some rescaling across layer three. And the way we do this is we say, well, we want all terms in layer three to be of leading order. So the mathematical representation of the equation is in accord with the empirical observation that all terms are leading order there. So we rescale to these hat variables with stretching coefficients, gamma and alpha. Turns out this is the simplest form and you don't have to rescale the velocity itself. And we require that all terms be order one. And in fact, we set the Reynolds or the pressure gradient term equal to one. And by doing that, it allows you to solve for gamma and alpha. And if you do that, you get a scaling for alpha and gamma that's related to the square root of the Reynolds number. Okay? And if you plug that rescaling into the equation, then you get the following equation, which is invariant. So across layer three, we have this equation where all terms are formally order one in magnitude. So have the right magnitude. And if you consider a new coordinate system that starts at the peak in the Reynolds stress, you just have to really pick some place in layer three that's rational as a new origin. Then you have this new stretch coordinate, the hat variables. And this analytical work leads to the scaling layer width for layer two, three, and the position of the peak in the Reynolds stress because that always occurs in layer three. This scales like one over the Reynolds number or square root Reynolds number. This has been previously observed by folks like Srinivasan and Afsal. And what we show here is you can get it analytically from the equations. And just to further this point, if you take pipe flow data, which has the same form of equation as a channel, you confirm this square root Reynolds number dependence out to at least Reynolds number 10 to the fifth, delta plus 10 to the fifth. So that's a pretty good confirmation there. Okay, so we can do this for the mean equation for the overall flow in general. But what I want to focus on now is a discussion of what the equation emits across an entire hierarchy of scaling layers. And to get at this, we begin with this transformation where we now introduce this small parameter beta. And I'll tell you what beta is here in a minute. But suppose we just have this transformation and we stick it into the equation's emotion, which we had up there previously. Then we get an equation that looks just like the original equation, except we've replaced this one over delta plus term with beta. Now beta, we call the, or this equation here, we call the adjusted Reynolds stress. And the interesting thing here is when the Reynolds stress, this adjusted Reynolds stress goes through a peak, then this derivative of the adjusted Reynolds stress is zero and it allows yourself for beta at that point. Okay, so what beta is, is actually corresponds, or what it corresponds to is a position along the curve of this turbulent inertia profile. And at every position, depending on the value of beta, you get a peak in the Reynolds stress and the d beta dt dy plus goes through zero. And so you have a layer three transition with a width that occurs continuously across this curve. But the layer widths get wider as you move outward. Okay, so now we have a hierarchy of finite width scaling layers that is continuous on this decreasing portion of the curve. And if we play the same game now and rescale the equation on each one of these layers, you get this scaling. Okay, so we play the same game we did for layer three, where we can then show that each layer width is ordered beta to the minus one half. And in fact, you can just use beta to the minus one half, which we have a formula for here. And the analysis further shows that you get exactly self-similar behavior from layer to layer if this parameter a, which is order one, approaches a constant as a Reynolds number goes to infinity. In fact, analytically, you can show that that is expected. Okay, well the theory allows, or the calculations allow you to go ahead and just, if you have DNS data, compute the layer width distribution. And the theory tells us that on this layer width distribution, the derivative of the layer width with respect to y plus, or the layer width itself, well the derivative approaches a constant, or the layer widths become proportional to distance from the wall. So this is an analytical justification for distance from the wall scaling that's often assumed in turbulent flow analysis, or turbulent wall flow analysis, but never really proven. And so this is an analytical justification for that. And on the, the domain here, this dW plus dy plus, the slope of the width distribution, is actually approaches a constant as the Reynolds number goes to infinity. And if you invert that equation, it's the stretching coordinate, or it's the stretching equation for the y coordinate. It is the coordinate stretching that allows you to write an invariant equation on each of these scaling layers. And in fact, the analysis, I'll show in a few slides, that's directly related to the von Karman constant. Okay? Okay, and the way you can show this is, if you go back to that equation for a, and a is related to phi, the stretching coefficient, okay, and instead of writing it in your hat variables, you convert it back to plus variables. And then you invoke the vorticity equation for the channel flow, which is exact here. Then you get a single equation for the mean velocity that's nonlinear, and that holds on this inertial domain where the length scale distribution approaches a linear function or its gradient approaches a constant. Well, you can go ahead, especially if you have a friend like Martin Oberlach, who can help you, and integrate this. And so we say if f is equal to the negative second derivative of the velocity, plug it into that relation, separate variables and integrate, squaring and inverting, and then integrating two more times to get a log log. Okay? So what have we shown? We've shown that for this self-similarity, we can go ahead and just transform the equations, integrate it, and get a log log. And notice that the slope coefficient is, in fact, the stretching coefficient squared. And so this gives an alternate way of evaluating the von Karman constant that's based on the analysis here. Notice also that we have a linear correction here in the analysis of Oberlach that I did with Martin Oberlach. We show that b has to approach zero as y plus becomes large or Reynolds number becomes large either or. And in fact, it has to decrease faster than 1 over delta plus in the DNS data actually show this. And of course, then once you have u, you can go ahead and solve for t by once integrating that equation. Okay, so that's the mean momentum equation analysis. Now what I want to do is turn to the scalar analysis. So what Sergio did was he considered the same channel flow instead of having a, well, and with a pressure gradient. Okay, so it's fully developed in the velocity field. And it's also fully developed in the scalar field in the same way as in the velocity field. Not that the scalar is linearly increasing in amplitude as you go downstream, but what happens is that you have a constant flux interior everywhere, uniform generation, I'm sorry. And thus you have a constant flux outward. Okay, and he did these computations over a pretty big Reynolds number range and for scalar variations between 0.2 and 1, which is really a remarkable set of simulations. Here, q is the uniform heat generation, theta is your mean scalar, and alpha is your diffusivity, and our pranial number is new over alpha. Okay, well, if we take this equation and manipulate it a bit, what we can show is that there's a constant outward scalar flux that acts like the wall shear stress in the momentum case. Okay, so this flow is really nice for investigating the scaling behaviors because we don't have the complications of this heat flux from the walls. We have this uniform heat generation and thus for the pranial number equal one case, we get an equation that's identical to the momentum equation. Pardon me? It's a constant pressure gradient, right? Constant heat flux generation. Yeah. Okay, well, now we can go ahead and internormalize this. And so we end up with the same epsilon squared Reynolds number. And you can go ahead and use this friction temperature definition here. You plug that in and you get your internormalized equation. And now it's a two parameter problem where we have the Reynolds number here and we have the pranial number here. So it's a little more complicated than the previous case. And what we want to do then is go through the same procedure we did for the momentum, except now for the scalar in this more complicated two parameter problem. Okay, well, the first step is to go ahead and try to determine the leading order balances. And as one might expect, we get a four layer structure similar to what we see in the momentum case, except now we have pranial number dependencies. And so here's the ratio of the mean diffusion. So it's analogous to the viscous force to the gradient transport. That's what I'm called it. Okay, so we see the same four layer structure. We have a layer one, a layer two, this transition layer, layer three, and then into layer four where its diffusional effects are negligible. Now in this slide here, this image here, we've done a empirical determination for the scaling in layer one. And into the first part of layer two. This is empirical. We've tried to do a little bit of analysis, but the analysis gets complicated near the wall because we don't have its analogous to layer three in terms of rescaling, except it gets truncated by the wall. And so we don't have enough information to actually analytically determine the scaling. But we believe this is the scaling for layer one under this square root pranial number, Reynolds number scaling, because it does cause the data to collapse in this region. Okay, what I want to consider now is the analysis across layer three here and see if we can combine the Reynolds number and pranial number to get a scaling in that region in a single invariant equation. Okay, you can probably guess if we're successful, otherwise I wouldn't be giving the talk. So as before, we go ahead and we pose stretching coefficients. So here they're alpha and beta. This is different beta than previously. And we go ahead and plug those into the equation. We demand all terms be order one. So this is the rescaling across layer three. And you can go ahead and solve for beta and alpha. And now you see that it has a combined Reynolds number, pranial number dependence. And you can get the same coordinates that are centered about layer three. Okay, so we're rescaling across layer three. And we have these same stretch coordinates there. And if you plug these stretch coordinates in, you get the same invariant equation that we had previously, where all terms are of leading order, which has to be the case across layer three. So now we've got an analytical expression that's in accord with the empirical observations. Okay, well, if we go ahead and rescale the momentum balances or the, I'm sorry, the scalar balances that I just showed previously under this stretch coordinate, you see that the layer three, well, two into three and into four now becomes invariant with some low Reynolds number deviations here. In fact, you can see that if we look at the scaling of the peak in the turbulent heat flux here. So this is y plus theta max is the peak in the turbulent heat flux, you have some deviations down here at small Reynolds numbers. But once you get up into the higher actually Peckley numbers, once you get into the higher Peckley number regime, you get a nice scaling that goes with the square root of Reynolds number over pranial number. Okay, just to maybe convince you more that the scaling actually works for the turbulent heat flux. Here's the of the of the conditions that Sergio computed. So four different Reynolds numbers, three different pranial numbers. Here's the curves of the turbulent heat flux under normal y plus scaling. And here's the same set of curves, all the same curves under the rescaling that was analytically determined by rescaling the equations of motion or the equations for the scalar, I should say. Okay, with that we can go ahead and do the same analysis for the layer hierarchy. Okay, so we play the same game as we did before where we now have adjusted turbulent heat flux. Previously we had the adjusted turbulent or turbulent inertia term. Plug it into the scalar equation and we get that this lambda, which acts like the beta previously, appears. And then we have, so this is similar to the momentum case where we can go ahead and solve for lambda for each value of the gradient of the turbulent heat flux. So it's on the decreasing portion of that curve. I don't have the curve here. So then we can go ahead and use the rescaling that we get from that. And you plug it into that and you get an invariant equation on each of the layers of this hierarchy. So now we have a scalar layer hierarchy that mimics the momentum layer hierarchy, except it's a pranal number dependence. And that will show up in its behaviors. Okay, and if we plug these rescalings into that on each layer of the hierarchy we get an invariant equation. So this will also lead us to the log law for the scalar. Okay, so let's go ahead and take a look at the layer width distributions for the scalar for different pranal numbers and different Reynolds numbers. Okay, so in the first case we have, let's see, fixed pranal number. So here we have fixed pranal number and varying Reynolds number. And these layer width distributions look very much like the momentum case. Okay, so we have a minimum here, a maximum here. And then once you're on the non-diffusional domain you get a linear slope here. And that linear slope is directly related to the scalar von Karman constant. If we allow the, look at fixed Reynolds number, but varying pranal number, you see that you see deviations in the domain where diffusional effects, whether momentum or scalar are important, but they go away once you get on the inertial domain or the domain where the scalar diffusion becomes small, a high peckly number regime. Okay, so that shows up beautifully in the layer width distributions or their behaviors. And now what I want to do is focus on the slope of this region here and take a little look at that. Okay, or well, let's first of all go ahead and get the scalar log law. Okay, so as before, we can write our equation or the condition for self-similar behavior as 2 over, now it's a phi theta sub c. So this is like 1 over phi or 2 over phi as previously. We get this nonlinear equation and we go ahead and use the same mathematics to integrate it twice or transform it, integrate it twice and you go ahead and get your log law. So it has the same form, same linear term that goes away. In fact, we've confirmed that c2 decreases faster than 1 over Reynolds number, which it must for that to asymptotically go away. This offset as it is for the momentum case becomes negligible as the position of the inertial domain, the y plus position of the inertial domain becomes large. And once again, the von Karman constant here, now for the temperature profile, is directly related to the stretching parameter of the similarity solution. Okay, well, there's been numerous observations in the past that the von Karman constant for the scalar is different for the momentum case and turns out even in the Prandtl number one, Prandtl equal one situation. And from Sergio's simulations, he found empirically just by measuring the slope of the mean temperature profile that the scalar von Karman constant was something like 0.46. If we measure it according to the present theory, looking at the slope of this W profile, we see that at low Reynolds number, low Peckley number cases actually, that we get variations from that. But as we get higher and higher in the Peckley number situation or Peckley number values, then we get values that agree very well with his purely empirical, well, or his direct empirical measure of that. And so we're quite confident that the theory and the actual measurements are in accord as the Peckley number becomes large. This is an asymptotic theory, so this is what we would expect. Okay, so to finish up the talk, what I'd like to look at is the interesting case of the momentum equation and the Prandtl number one, or Prandtl number equal one case. Okay, so under that case, the situation, we have a momentum equation that looks like this, where we have our viscous force, turbulent inertia, and pressure gradient term, one over Reynolds number. And for Prandtl number equal one, we get equation that's in form identical. Okay, for this particular situation where we have uniform heat generation and an outward flux such that the mean temperature profile is invariant as you go downstream, then you get exactly the same boundary conditions. Okay, so we have exactly the same boundary value problem, but we have a different solution. Okay, now if this were regular determinant mathematics, this would be a problem. But we have indeterminacy here because the equations are time averaged and we have things like the Reynolds stress or the turbulent heat flux. Okay, and in this case, the momentum case, we get a von Karman constant equal 0.39 and in the passive scalar case, we get a von Karman constant of 0.46. Okay, so this is all due to the indeterminacy of the equations and it turns out it has to do with the difference between the u fluctuations and the theta fluctuations. Okay, and the best, well let's go ahead and first see how this shows up in actual visualizations of the flow field. So these are two snapshots at the same instant in time across the channel and in the top case we have the u fluctuations and in the bottom case you have the theta fluctuations and you can see these two figures mimic each other except the u fluctuations are blurred. Okay, so there's a blurring of these u fluctuations and this has to do actually with the pressure strain. Now we know that it has to be the u fluctuations because it's turbulent channel flow and the v fluctuation profiles are identical. This is a passive scalar and therefore the only thing that can change the turbulent heat flux and make it different from the Reynolds stress is the u fluctuations. This blurring actually has to do with the pressure strain correlations and I'll identify that in a minute but if we go ahead and look at the variances of the u versus the theta, so we're looking at the, and you see out here in this region that at same Reynolds number, remember the Prandtl number is equal to one, we get variations. Okay, you'd expect those to be the same if the two flow fields were identical but notice here that the Reynolds stress and turbulent heat flux profiles are nearly identical. Now they're not identical because the Karman constant is different and the theory tells us that it's the behavior of the decay of that turbulent heat flux or Reynolds stress that leads to the different values of the von Karman constant but visually you can see that they're very similar so it has to be in the u fluctuations and in fact you can go to the transport equation for the u fluctuations and compare it to the scalar transport equation so here's the one for the u fluctuation transport and you see you have these pressure terms whereas in the scalar one you don't. Now other people have noticed this but not in the context of the solution to the mean flow equation which we're able to show here in this talk and so in conclusion I'll just state that we've analytically determined the scaling behaviors for the two parameter combined Reynolds number and Prandtl number problem for the fully developed channel flow with uniform heat generation again I think this was a very clever simulation and owing to the uniform heat generation versus the more common uniform surface heat flux these two the momentum in the scalar equations look identical for the Prandtl number equal one case they however give different formulations in the origin of this difference is traceable to the pressure strain transport in the u equation so with that I'm done thank you