Welcome back to Sports and Building Aerodynamics. In the week on Computational Fluid Dynamics. This module is going to focus on approximate forms of the Navier-Stokes equations. We start again with a module question. Which of these statements, is or are correct. If applied according to best practice A, Large Eddy Simulation always gives better results than steady RANS. B, Large Eddy simulation sometimes gives worse results than steady RANS. Or C, unsteady RANS always gives better results than steady RANS. Please hang on to your answer and we'll come back to this question later on. At the end of this module you will understand the difference between RANS, URANS and LES. And you will understand why steady RANS and LES are most often used in Sports and Building Aerodynamics as opposed to URANS and DES. Let's start from the instantaneous three-dimensional Navier-Stokes equations for a confined, incompressible flow of a Newtonian fluid. Like we saw earlier in the first week. What you see here is first the continuity equation. Then the momentum equations, the temperature equation, and the pollutant transfer equation. u indicates the instantaneous velocity, p the instantaneous pressure, theta instantaneous temperature, and c instantaneous concentration. The other parameters here are cp the specific heat, the thermal conductivity k, the molecular diffusion coefficient D and t the time. These are actually six equations. They have also six unknowns. So strictly you could say this is a closed system. However, they are nonlinear, coupled, partial differential equations and even with the above assumptions, being a confined incompressible flow of a Newtonian fluid, there's no analytical solution known for them for realistic geometries that we are interested in here. Several methods exist for predicting turbulent flows with CFD. The three most popular approaches in engineering are Direct Numerical Simulation, Large Eddy Simulation, and Reynolds-averaged Navier-Stokes. Let's briefly address each one of them. They will be schematically depicted by the figure here on the right side, in which you see a flow, for example in a tube or between two parallel plates with the straight lines indicating the mean flow. And the vortical structures, the turbulence superimposed on that. So you see vortices with a wide range of length scales present in this flow. Well in the approach of Direct Numerical Simulation we are going to solve the exact Navier-Stokes equations completely. Meaning we're solving all vortices, all eddies, and nothing is being modeled, meaning approximated. This of course means extremely fine grid resolutions, very small time steps. So therefore, this is very time consuming. It requires huge computational resources. Can only be applied for simple geometries at low Reynolds numbers. And it also gives you enormous amounts of data. Usually for Sports and Building Aerodynamics we do not even think about applying DNS. And we move actually one step further in simplification and in the approximation of the Navier-Stokes equations. We might move to Large Eddy Simulation. In this case, also schematically depicted by the picture on the right, we're actually only going to solve the so-called filtered Navier-Stokes equations. This means we're going to filter out small-scale turbulence. That part is not going to be solved. No, we're going to approximate only the effect of that part of the turbulence on the larger scale motion and on the mean flow. So this is an approach that is not exact. And it's much less computationally demanding than DNS. But still often, for Sports and Building Aerodynamics, it still is quite computationally demanding. And then we want to move one step further in approximating the Navier-Stokes equations. We might go to the Reynolds-averaged Navier-Stokes equations, where we're going to solve the averaged Navier-Stokes equations. We only solve the mean flow and all the turbulence is modeled. So we're going to approximate the effect of the turbulence on the mean flow by models. These are called turbulence models. So this is also not exact. It's less accurate, generally, than LES, certainly than DNS, but it's also generally applicable. So in the RANS approach, as just mentioned before, we're going to model the effect of turbulence on the mean flow. Schematically in one table you can see the difference between solving and modeling here. With DNS, we solve all the eddies in the turbulence spectrum and we're going to model nothing. With LES, we're going to solve the large eddies, we model the small ones. And with RANS we solve only the average flow, the mean flow and we're going to model all of the vortical structures. So this is exact and for the others we use modeling approximations and these are called turbulence models. Let's first have a look at RANS equations. We'll focus on the RANS equations in this module because these are the equations that we will most often use in Sports and Building Aerodynamics. Again starting from the instantaneous 3D Navier-Stokes equations where you indeed see instantaneous velocity, pressure, temperature and concentration. And let's then go back to what we talked about in the first week. The Reynolds decomposition. We're going to decompose every variable, whether it be a vector or a scalar in a mean part and a fluctuating part. And this is schematically depicted in the graph on the right side. Then we're going to insert this decomposition on the left side in the instantaneous Navier-Stokes equations. And we're going to apply averaging, which can be time-averaging or ensemble-averaging to these equations. And then what we end up with is this. So briefly going back. And then to the Navier-Stokes equations, and the Reynolds-averaged Navier-Stokes equations. What you see is that the instantaneous components or the instantaneous variables, have now been replaced by the mean variables. But in order to do that, we have to compensate for that, by the terms that are indicated here, that are encircled. And these are the Reynolds stresses and the turbulent heat and turbulent mass fluxes. So these actually do represent the influence of turbulence on the mean flow. And we have to take them into account in order to be able to justify this replacement of instantaneous variables by average variables. However now, this introduces new unknowns, these Reynolds stresses and the turbulent heat and mass fluxes. So we now have an unclosed system. So we have to come up with additional expressions, additional equations, and these are the equations of the turbulence model. Let's first have a look at the Reynolds stresses. Well, these, as mentioned before describe the effect of turbulence on the mean flow. And earlier in the first week we talked about the train analogy. And indeed we communicated that the exchange of mass and also momentum therefore between fluid layers moving at different speeds causes viscosity. In a laminar flow the exchange of molecules causes a momentum exchange that is responsible for molecular viscosity. In a turbulent flow we exchange groups of molecules, parcels. And that's what is called turbulent viscosity. Be careful, turbulent viscosity is not a property of the fluid, it is a property of the flow. So then indeed we need turbulence models to describe those Reynolds stresses. And that we will address in detail in the next module. Then another approach is unsteady RANS. So steady RANS is time-averaging of the Navier-Stokes equations. With unsteady RANS we do not apply time-averaging, but ensemble-averaging. Unsteady RANS therefore allows to take into account some unsteadiness in the flow. But it does not simulate turbulence, it only simulates its statistics. This can be a good option when the unsteadiness is very pronounced and deterministic. So just for example, von Karman shedding in the wake of an obstacle with a low-turbulence approach flow. And this is what we also saw in the first week with flow around the circular cylinder. However, this is not the case in atmospheric boundary layers. There we have an approach flow with a wide turbulence spectrum and a wide range of length scales involved. And this is a very different story then. So URANS has, probably for this reason, up to now almost never been used in Sports and Building Aerodynamics. So we are not going to focus on it further here. Then there's Large Eddy Simulation. In that approach, as mentioned before, we're filtering the Navier-Stokes equations. So we use a filter, which can be the grid size, not necessarily. We filter out the small eddies and we solve the large ones. The philosophy behind this is that the large-scale motion is responsible for most of the flow features and of the heat and mass transfer that we will see in this turbulent flow. However, this filtering also generates additional unknowns, so we need a sub-filter turbulence model. LES has superior performance compared to RANS and URANS. Because a large part of the unsteady flow is actually being resolved rather than only modeled. But it is more expensive and it is more complex. Then there are also hybrid techniques, where, for example, unsteady RANS is applied in a near-wall region, and LES in the rest of the domain. And the philosophy behind this is that LES application in the near-wall region would be very expensive, very computationally expensive. However, stand-alone LES can be used with wall functions. And a well-known example of a hybrid approach is Detached Eddy Simulation. Where LES is combined with the one equation Spalart-Allmaras model. However, combining these two approaches is certainly not straightforward. Actually, they are completely, fundamentally different approaches. They have specific grid requirements, and this requires specific care at the region where they have to be matched. So hybrid approaches actually have almost also never been used in Sports and Building Aerodynamics. And also here in this week we are not going to focus on them any further. So let's go back to the module question, which of these statements is or are correct? If applied according to best practice. A, Large Eddy Simulation always gives better results than steady RANS. And yes, indeed, this is correct. If applied to best practice with the right boundary conditions and the appropriate discretization, LES is intrinsically superior and can give much more accurate results. In this module, we've learned about the difference between RANS, URANS, and LES. We've learned why steady RANS and LES are most often used in Sports and Building Aerodynamics, as opposed to other valuable approaches, being URANS and DES. In the next module, we're going to focus on turbulence modeling for RANS. And we're going to explain the Boussinesq hypothesis and the gradient-diffusion assumption. Thank you very much for watching. And I hope to see you again in the next module. [BLANK_AUDIO]