Welcome back to Sports and Building Aerodynamics in the week on Computational Fluid Dynamics. In this module we're going to focus on near-wall modeling and we start again with the module question. If you would like to perform an accurate CFD simulation of surface heat and mass transfer in building aerodynamics, which statement is, or which statements are correct? A, wall functions should be used. B, low-Re number modeling should be used, or C, neither wall functions nor low-Re number modeling will give you accurate results. Please hang on to your answer again and we'll come back to this question later in this module. At the end of this module, you will understand how the complexity of near-wall flow can be taken into account in CFD simulations. You will understand the differences between wall functions and low-Reynolds number modeling, and you will also understand the advantages and disadvantages of these both approaches. Let's first recall the structure of the turbulent boundary layer, as we also discussed that briefly in week one. Far away from the wall, you will have large vortical structures that, however, cannot exist close to the wall. And as you move closer to the wall, actually, at some point, the turbulence vanishes, and you get the laminar sublayer, the laminar viscous layer. So there are indeed different regions in the turbulent boundary layer. The outer layer and inner layer. Where actually, the inner layer consists of the log-law layer, then the buffer layer, and then the linear sub-layer, where you only have viscous effects, so only laminar flow and these layers have their specific characteristics. Indeed, in the buffer layer, you have a more or less equally important share of viscous and turbulent effects. While in the log-law layer the turbulent effects are clearly dominant. This will be very important in how we are going to deal with near-wall modeling in CFD. The problem is indeed that many turbulence models such as the k-epsilon model, but are also many other models, are only valid for high Reynolds number flows, so they are valid for the turbulent core of the flow and not the region close to the walls, where indeed viscous effects can become important. So we have a different flow behavior in different layers. So we might also need a different treatment in these different layers. At least we need to know the extent, the height of every layer and, therefore, we're going to introduce two important dimensionless quantities. First of all, the y plus value. The y plus value, as you can see, is the product of density, friction velocity, and distance from the wall, divided by the dynamic viscosity and the friction velocity is defined as indicated in this equation. It's the square root of the shear stress of the fluid on the wall divided by the density. So that's the first dimensionless quantity, the other one is a dimensionless fluid speed. We're going to divide the fluid speed parallel to the wall by this friction velocity. When we have those two dimensionless numbers, dimensionless parameters, we can actually look at the so-called universal law of the wall. And that is the one that you see illustrated here, which is demarcated by different areas. You have the linear sub-layer where the equation u plus equals y plus holds. Then we have the buffer layer in between. And then we get into the so-called log layer or the log law layer where you get a logarithmic expression of the velocity as a function of this dimensionless y plus value. So in this semi-logarithmic diagram, indeed you see the logarithmic function as a straight line, and also the others indicated there, with in addition also measurement values. And you can see that in this log layer, they actually fit quite nicely to the logarithmic curve. This information now can be used in modeling the near-wall region. There are two main options, the first one is wall functions, and the other one is called low-Reynolds number modeling and this is schematically what a grid in both techniques can look like. In the wall function method often we use coarse cells, and to describe the flow behavior in the wall-adjacent cells, indicated here by the point p which is the cell center of this control volume. The behavior there, we're going to actually describe that by wall functions. By functions, empirical functions or semi-empirical functions that do not solve what actually happens in this layer between the wall and point p, but that actually describe what is more or less happening as a result of this layer in point p. The other approach is low-Reynolds number modeling. In this approach, we're actually going to apply discretization all the way down to the viscous sublayer, and inside the viscous sublayer, we are going to try to solve everything that happens inside the viscous layer, the buffer layer, and the logarithmic layer, by this higher spatial resolution. Both approaches have their advantages and disadvantages and let's have a brief look at those. Clearly, wall functions are an approximation. So we certainly make physical modeling errors there. Low-Reynolds number modeling, on the other hand, is more accurate. That is, if, indeed, like also indicated here and is the case in assumptions of low-Reynolds number modeling for Building and Sports Aerodynamics, the walls are smooth which actually, they never are. With wall functions, we can use a coarse mesh, so it's less computationally demanding than low-Reynolds number modeling, where we need quite a fine mesh to be able to fit some cells in this laminar boundary layer, the laminar sub layer. Because in this laminar sub-layer actually the thickness of that layer decreases with an increase of Reynolds number, it's also what we discussed in week one. Therefore, often these layers will be that thin, that narrow, that if you have a large urban model, but you also have to put cells in the viscous sublayer which is often a few millimetres or even less in thickness, well often it's not possible to apply low-Reynolds number modeling, because you can just not make a mesh like that, according to best practice guidelines. And that's why often in Sports and Building Aerodynamics, which are flows at high Reynolds numbers, we have to use wall functions. Wall functions, as mentioned before, are empirical and semi-empirical equations so you can actually fit roughness parameters into them which is very difficult with low-Reynolds modeling. There if you want to include roughness you should actually model all the details of the rough surface, which for small-scale roughness like present on building surfaces for example is clearly impossible. And then finally, wall functions are an approximation and they're not valid for near-wall heat and mass transfer. Because indeed, the largest resistance for heat and mass transfer between a fluid and a solid is situated in the laminar sublayer. If then with wall functions, we almost neglect this layer, we do not solve it, we just bridge it with a simple function, you cannot expect to get any reasonable or accurate results for this heat transfer in this laminar sublayer. On the other hand, we should use low-Reynolds number modeling but there we again have the problem of roughness modeling. So these are three advantages of wall functions which generally are used to motivate the use of wall functions. Let's look at the grid requirements. In wall functions, we're going to put, or try to put, the cell center of the first cell inside the logarithmic layer. There is that one here, and that means also that there are some requirements that we need to satisfy. These are some published requirements, y plus has to be between 30 or 60, sometimes 30 is preferred. But other sources give other limits, other boundaries. Sometimes it's said if you are between 30 and 1000 that this is already sufficient. Well, often in complex flows such as building aerodynamics with massive separation and recirculation it's not possible for every cell on a wall in the flow field to satisfy these requirements. Often you will have y plus values that are much, much larger and they are not necessarily causing very erroneous results, unless you're dealing with surface heat and mass transfer. Though, for low-Reynolds number modeling, on the other hand we want to fit cells into the laminar sublayer. This means we have to situate the cell centers into this layer, at least a few of them. And this means indeed, that y plus has to be smaller than 4 or 5, and preferably y plus equal to 1. However, one problem in this procedure is that the y plus value, or the y star value, which is also sometimes used. It's similar, but there's a slightly different definition. It is based on the friction velocity along the wall and the friction velocity you will only know after a calculation. So, only after a calculation, you will know if your grid was sufficiently fine. If not, you have to refine or coarsen it and you end up, indeed, in an iterative procedure. As mentioned before the thickness of the linear sub-layer, or the viscous layer, is inversely proportional to the square root of the Reynolds number. This means that in higher Reynolds number flows, as in Sports and Building Aerodynamics, we get very thin laminar sub-layers and for exceptional cases and exceptional simulations, for example, one building or one cyclist or well a limited number of cyclists, we can apply this very, very high grid resolution but for large urban environments this is clearly not possible. And that's why often wall functions in building aerodynamics are often the only option. Let's look at an exception to this. A study where we tried to investigate if it is possible indeed to make an extremely high- resolution grid to use low-Reynolds number modeling for building aerodynamics. And of course this is just a very, very simple building, a cube, with a height of ten meters exposed to neutral atmospheric boundary layer flow. We performed validation by comparison with wind-tunnel experiments of surface heat and mass transfer. And what you see indicated here is the convective heat transfer coefficient along two lines at the building's surface. You see the experimental ones in the horizontal axis and the numerical ones by CFD in the vertical axis and quite a good agreement overall, better than 10%. Based on that we made a full-scale model, so with a ten meter cube, with as you can see here, a very high-resolution grid close to the surface actually, the cell center point had to be put at 160 micrometer in order to be able to solve the flow down to the viscous sublayer, for the wind speeds that are indicated here. So one to four meter per second, then because we needed this high-resolution grid, we see that even for a very simple geometry we already end up with almost 1.9 million computational cells. Then the obvious advantage is that you can get quite accurate views on how this convective heat transfer coefficient varies along the facade for different wind directions, and that is indicated here. But we made the important assumption that the wall is smooth in our low-Reynolds number modeling approach. Then additional results can be shown where here we show for different wind directions, how the convective heat transfer coefficient changes along the wall. But then we can also compare what we would have got, if we would choose wall functions. And what you see here, standard and non-equilibrium wall functions being applied and you see that results are very far off from the more accurate low-Reynolds number modeling results. Again, assuming that we have a smooth surface. In addition, we developed some new wall functions, which bring the accuracy of heat transfer solved with wall functions actually much closer to the low-Reynolds number modeling approach. But it still has to be investigated how general these new wall functions can be applied. So, as mentioned a few times already, there are some problems here, because usually these studies and low-Reynolds number modeling in Sports and Building Aerodynamics is performed for smooth surfaces. And that applies to buildings for smooth building facades. However, there doesn't exist anything like a smooth building facade. The small-scale roughness, and this has been shown by measurements, can increase convective heat transfer by as much as a factor of two. And this cannot be accurately implemented with low-Reynolds modeling because we cannot just model these small-scale roughnesses and it can also not really be generalistically included in wall functions that will also be valid in for example separated flow regions of the flow. So this indeed remains a big problem. Let's come back to the module question then. If you want to perform an accurate CFD simulation of surface heat and mass transfer in building aerodynamics, which statement is, or which statements are, correct. Well it's actually, unfortunately, the last one. Neither wall functions nor low-Reynolds number modeling at the present state of knowledge, will give you accurate results. In this module we've learned about how the complexity of near-wall flow can be taken into account in CFD simulations. About the differences between wall functions and low-Reynolds number modeling, and about the advantages and disadvantages of both approaches. In the next module we're going to focus on errors and uncertainty in CFD and on verification and validation. Thank you for watching and we hope to see you again in the next module.
