[BLANK_AUDIO] Welcome back to Sports and Building Aerodynamics in the week on Computational Fluid Dynamics. In this module we're going to focus on turbulence modeling. And we start again with the module question. Which of these statements is or are correct? If applied according to best practice: The Reynolds stress model always gives better results than less extensive turbulence models. Or, no turbulence model is universally the best. Hang on to your answer, and we'll come back to this later in this module. At the end of this module, you will understand turbulence modeling for RANS. And you will understand the Boussinesq hypothesis and the gradient-diffusion assumption. Let's make a brief recall from the previous module where we had the 3D instantaneous Navier-Stokes equations that we subjected to the Reynolds decomposition. And then after averaging these equations we came up with the Reynolds-averaged Navier-Stokes equations in which the instantaneous variables have now been replaced by mean values. But in order to be able to do that we had to compensate for that by adding the so-called Reynolds stresses and the turbulent heat and mass fluxes. In this module, we're going to focus on how we can model these Reynolds stresses and these turbulent heat and mass fluxes. And actually to close the system. These models for the Reynolds stresses are called turbulence models. Let's look first at turbulence models for RANS and unsteady RANS. We have two main categories that have to be distinguished. So-called first-order closure models. And second-order closure models. In first-order closure models, we have the so-called Boussinesq eddy-viscosity hypothesis, which we use to relate the Reynolds stresses to the velocity gradients in the mean flow. The gradient-diffusion assumption, on the other hand, we use to relate turbulent heat fluxes to the mean temperature gradients, and turbulent mass fluxes to the mean concentration gradients. In second-order closure, on the other hand, we're going to establish and solve additional transport equations for these Reynolds stresses, and sometimes also for the turbulent heat and mass fluxes. Let's first focus a bit on the Boussinesq eddy-viscosity hypothesis. So as mentioned before, we're going to express the Reynolds stresses as function of the velocity gradients in the mean flow. So, this is the expression we are going to focus on. Here you see the Reynolds stress or the Reynolds stresses. We write them as a function of the turbulent viscosity, that can be interpreted by the simple train analogy. Then we have the mean strain rate here, indeed based on the gradients in the mean flow. We have the turbulent kinetic energy appearing here. And the so-called Kronecker delta. This turbulent viscosity is actually what the turbulence models will provide expressions for. And that's why they're also called eddy-viscosity models. There are linear eddy-viscosity models, but also nonlinear eddy-viscosity models. And examples are, well, the most well-know examples, at least in Building and Sports Aerodynamics are the standard k-epsilon model, variations of this model such as the RNG k-epsilon model, the realizable k-epsilon model. And then also the standard k-omega model and its variant the shear stress transport k-omega model are quite often used. And then the gradient-diffusion assumption. As mentioned before, we are going to relate the turbulent heat flux and the turbulent mass flux to the gradients, the corresponding gradients in the mean flow. These gradients are depicted here. And then the relation between those fluxes and the gradients are given by the turbulent heat diffusivity and the turbulent mass diffusivity, respectively. Often we do not know these, and often in computations they are actually inserted into the solution by specifying values for the so-called turbulent Prandtl number and the turbulent Schmidt number. Often these values are taken as constant, while in reality, they are usually not. They're actually also functions of the flow field. And again, indeed, they are not fluid properties, they are flow properties. Then in second-order closure, which is sometimes also called Reynolds stress modeling, and there the name Reynolds stress model comes from, there actually, we are going to establish and solve additional transport equations for the Reynolds stresses. And sometimes - I mentioned sometimes indeed - also for the turbulent heat and mass fluxes. This is usually done only for momentum when this model is applied, and almost never for heat and mass fluxes, at least not in Sports and Building Aerodynamics. Theoretically you could argue about the fact this model may be potentially superior. But this has definitely not yet been proven without any doubt in Sports and Building Aerodynamics. On the other hand, sometimes we find much less accurate results with Reynolds stress modeling. Let us focus on, well so-called, simplest of those models, the standard k-epsilon model, that as mentioned before is based on the Boussinesq hypothesis. Where the turbulence viscosity is calculated as shown on this slide, where you see k and epsilon appearing in these equations, and therefore we have the name k-epsilon model, where k is the kinetic energy of the turbulent fluctuations in the flow. And epsilon is actually the dissipation rate of the turbulent kinetic energy. Here you see then, these two equations that are added by the standard k-epsilon model. And in these equations actually, here, we find familiar terms that we already discussed also with the Navier-Stokes equations. So the first term indicates local variation with time. The second one is the advective term. The third one, the diffusion. The fourth one, generation of turbulent kinetic energy due to mean velocity gradients. Then, there's also terms actually, that can generate turbulent kinetic energy by buoyancy and finally there's also a dissipation term. You see some other parameters appearing in these equations. Sigma k and sigma epsilon are the turbulent Prandtl numbers for k and epsilon, respectively. And then there are some other constants. And the set of constants or values that you see here are actually most often used because they have been obtained by comprehensive fitting of computational results to experiments for a wide range of problems. The standard k-epsilon model has some advantages. It is certainly the most widely used and validated turbulence model and this is definitely also the case in Sports and Building Aerodynamics. It has excellent performance for many industrially relevant flows and reasonably good results for many flows, it is the simplest model for which only initial and boundary conditions need to be supplied. It is robust, and it's quite easy to implement and has a low computational cost. However there are also some clear disadvantages. Maybe the most important one is the assumption that this turbulent viscosity is actually an isotropic scalar quantity. While in reality it's a tensor. And this can have quite severe consequences for building aerodynamics. Let's look for example at this flow field, this schematic of a flow field, where you see the approaching neutrally-stratified atmospheric boundary layer, then actually stagnating at the building facade, splitting up in a flow going downward and a flow going upward. And on top of the roof you get flow separation which is a very important phenomenon, and recirculation, and then reattachment. Well what usually happens with the standard k-epsilon model and it can actually be mathematically proven is that the recirculation region on the roof is underestimated and sometimes even completely not predicted. And this, indeed, can give rise to quite erroneous results. Certainly in terms of pressure distribution on the building facade. The reason, indeed, for that is that we overestimate turbulent kinetic energy, also overestimate the turbulent mixing and that the weak separation bubble actually, will decline in favor of the large velocity gradients over the roof. Another typical problem, and also this can be mathematically proven to be related to the assumption that the turbulence viscosity is an isotropic scalar quantity, is that we underestimate the turbulent kinetic energy in the recirculation region. Therefore, this recirculation region, with RANS models in general, but especially with the k-epsilon model, is generally overestimated. So it will be too long, and this region of weak recirculating flow will be to large. Some important comments. There's definitely no turbulence model that is universally valid. And sometimes we see that even minor changes in the geometry of the flow already can really change which turbulence model performs best. Regardless of that if you want to assess the performance of turbulence models we can only do that after we have shown that the numerical errors are sufficiently small, quantified and negligible. But this is not always the case in turbulence model evaluation. There is a very nice quote here by Ferziger and Peric. And I would like to read it for you. Which model is best for which kind of flows (none is expected to be good for all flows) is not yet quite clear, partly due to the fact that in many attempts to answer this question numerical errors played a too important role so clear conclusions were not possible. In most workshops held so far on the subject of evaluation of turbulence models, the differences between solutions produced by different authors using supposedly the same model were as large if not larger than the differences between the results of the same author using different models. And this indeed indicates the extreme care that has to be applied in the evaluation of turbulence models. Let's go back to the module question. Which of these statements now is or are correct? If applied according to best practice, then, well it's definitely the case that no turbulence model is universally the best. In this module, we've learned about turbulence modeling for RANS, and about the Boussinesq hypothesis and the gradient-diffusion assumption. In the next module, we will focus on some basic aspects of the finite difference method, the control volume method, and the finite element method. We'll address the importance of high-quality grid generation. We'll discuss how discretization errors can be estimated. And how grid-convergence studies can be reported. Thank you again for watching and we hope to see you again in the next module. [BLANK_AUDIO]