Lecture 56 : LES

Hello everyone, welcome back. We look at the boundary conditions for the flow parameters like rho, u, v, w, t. Basically, all these boundary conditions can be applied to a laminar flow or if you are doing a range simulation or turbulent flow simulation for the mean flow. But, if the flow is turbulent, then there may be some free stream turbulence around the body or inside a domain. Also, there may be some turbulence which are getting, it can be turbulence or it can be any statistical turbulence that you need to put in there. So, typically let's say if you are doing a simulation in a wind tunnel. So, you would know that what kind of wind tunnel, I mean what kind of disturbance you expect in a wind tunnel. You may have 1 percent turbulence intensity or 5 percent turbulence intensity. If it is a internal flow, you may have the higher turbulence intensity. So, we need to convert this information for our CFD. So, in CFD, we need for example, the turbulence kinetic energy, dissipation rate or specific dissipation rate in the range simulation. As we discussed earlier, for LES and DNS, this is slightly harder because you need to give the information about the fluctuation, the turbulent fluctuation themselves like u prime, v prime, w prime, t prime, etcetera. We will not be discussing those here, but we will be discussing how we can provide information for the specific parameters that is required in the model. So, in the k epsilon, it is the k n epsilon. In the k omega, this is k n omega and so on. So, let us look at the turbulence boundary condition. So, we are basically the mean flow parameters are known in the sense that we know the boundary condition for them. Also, they are available from the calculation. Typically, the turbulence intensity is known. So, how the turbulence intensity is defined is basically i equal to u prime. I mean you can take the r m s of there and by u. So, if you want to talk about the kinetic energy, so kinetic energy is basically half u prime square plus v prime square plus w prime square. So, most of the range simulations, this is basically on the eddy viscosity model. We looked at the eddy viscosity model earlier. So, there we assume that basically the turbulence is isentropic. So, basically the scales of u prime and v prime and w prime are basically 1 is to 1 is to 1. So, this is how what you assume. Then, you can take this as basically kinetic energy is equal to half u prime bar square plus this also can be u prime bar square. So, you can actually call it u prime bar square. So, the scales are basically equal. There is no anisotropy in any direction. Then, it will become 3 by 2 u prime square and u prime square is basically nothing but u i square. You can use this definition u prime is u i. So, if you know the intensity, if you know the mean value, you can calculate the turbulence. That is the idea. So, that is typically known. The other known thing or other parameter that you need to give even if you do not have the information, you have to provide some estimate of these parameters because you have two equations. Mostly, I am talking about equation model now like k epsilon and k omega. So, you need to get boundary conditions and initial conditions of both k and epsilon or k and omega. So, from this intensity, you can get k, but how do you get the epsilon or omega? So, you need to get another information, which is the viscosity ratio or the length scale. The viscosity ratio represents that how much is the estimated eddy viscosity compared to the natural viscosity. This is mu t by mu. So, that ratio can vary anything from 1 to let us say 100. 1 is basically the eddy viscosity equivalent to your normal viscosity. Basically, dynamic viscosity and it is 100 means your eddy viscosity is 100 times higher than the dynamic viscosity in the boundary condition. Some information about length scale, so you do not have to give both either of the two. So, length scale is basically let us say if you have flow in a channel, flow is turbulent. So, you will have some expected size of the eddy. So, that can be if this is your hydraulic diameter. So, some fraction of the diameter will be your size of the eddy and that is the idea. So, we will see how we can use this information of viscosity ratio and the length scale to find this other parameter that is unknown. So, we need to basically get boundary conditions for the turbulent parameters using this information. So, these are used in most of the turbulence models. Here, I am sticking to eddy viscosity model. The RSTM model may require information about the different stresses with the boundary condition for different stresses themselves, which we will not discuss in today's class. So, k epsilon and all their variants, whether it is R and G, realizable or standard k epsilon, k omega and all their variants like SST, Wilcox k omega and there are Mentos k omega and then we have this SST k omega and this is a transition model. So, here we are not talking about the laminate kinetic energy here, but the turbulent kinetic energy n omega. This is the LKE model that we briefly touched upon. This is a transition model, where the laminate kinetic energy accounts for the energy in the laminar. So, this we discussed that when you assume the isotropy, it becomes your k equal to 3 by 2 into u i square. So, if you know the intensity, let us say intensity is 5 percent. So, that will be 0.05 and u is let us say 100 meter per second. Your kinetic energy will become 3 by 2 into u into i 0.05 into square. So, basically 3 by 2 into 5 square. So, that will be the value like 37.5 meter square per second square. So, that will be the value of the turbulent kinetic energy. So, this is how you calculate the turbulent kinetic energy based on the intensity and the mean flow. Now, we need to find the other quantity, which is dissipation rate or the specific dissipation rate. So, let us say your mu t by mu is known. As I told you that you need to give either this viscosity ratio beta or you need to provide another information, which is the length scalar. So, if the viscosity ratio is known, this is the definition of the viscosity in the k epsilon model. So, if you recall mu t by rho was equal to mu t, which is the adiabatic viscosity. This unit per meter square per second and to get that same unit, you have to it is proportional to k square by epsilon, where the k square is unit of meter square per second square. This is meter square per second cube whole square. So, that will give you meter square per second and the proportionality constant is C mu. Again, the small caution here in some of the models, you do not use C mu in this definition. Some of the models you use. So, you have to look at the manual that how they have implemented it. So, if you use this definition, you can directly calculate this. So, you know this mu t, right. So, mu t by mu is equal to beta. So, mu t will become b beta mu. This will become beta mu is equal to C mu into rho k square by epsilon. Then, you can get the value of epsilon from this, which will be C mu rho k square by beta mu. You know the viscosity from your Sutherland's law. From that, you can calculate your dissipation rate. So, this is how you calculate the dissipation rate. So, you need the information of the intensity that will give you the ketogenic energy and need the information about the viscosity ratio. From that, you can calculate the dissipation rate. You can assign this value at the boundary. In fact, in most of the course, for the turbulence, the boundary condition and initial conditions are the same. So, calculate the value, initialize with that value and assign that value at the inlet every time as the boundary condition. Sometimes, it is argued that you should have higher values of this quantity like k and epsilon in the initial condition than the boundary condition, right. So, let us say this is my domain. So, you will be initializing at t equal to 0. You need to provide something at t equal to 0 and then, you have to have something at the boundary also, right. Something at the boundary, something is coming in from here. So, once you calculate this value of k or epsilon or whatever quantity you have, in most of the simulations, you can use the same calculated value for the initial condition as well as the boundary condition. In some of the solvers, it is argued that initial condition requires a higher value because they dissipate. So, then you can have like a value which is like 10 times higher than the value in the boundary condition. Calculate the value of k and epsilon and multiply by 10 given your domain and doing the initialization. So, if the length scale is known, that was the second thing I was saying that let us say in your code, you have not only, you have the intensity and the length scale. Earlier, we saw intensity and the viscosity ratio. Now, if you have intensity and length scale, so how do you do that? Go back to our argument that the viscosity is basically proportional to velocity scale and the length scale. This is nothing but k to the power half. There is no doubt about that and this is the length. So, from this, you can calculate this. I mean you can compare this with the relation that we had which is mu t equal to rho c mu k square by epsilon. When you compare these two, you get l equal to this relation c mu k to the power 3 by 2 by epsilon. From that, you can calculate the value of epsilon. You know this k from the intensity c mu 0.09, l is the value that you are describing at the boundary. From that, you can calculate the value of epsilon. In some of the softwares like fluent, they use a different proportionality constant. You have to be careful what c mu you will be using here. Now, let us go to the specific dissipation rate. Now, we are going to talk about the SST omega or just the k omega model. So, there also the philosophy is the same. So, you know the viscosity ratio and you know that how your eddy viscosity is related to the turbulence kinetic energy and the specific dissipation rate. So, you can just use that specific dissipation rate omega and to calculate the value. You can use the value of the viscosity ratio beta. So, you can write this as beta mu is equal to rho k by omega. So, then your omega will become rho k by beta mu. You know the k, you know beta and rho and mu are also known. So, then you can calculate the specific dissipation rate. If the length scale is known, so we will be using this relation. So, this is coming from the k epsilon model, but this epsilon is related to k by this relation omega equal to epsilon by c mu k. This you can easily get because if you have mu t or let us call it mu t is equal to c mu k epsilon is equal to c mu k by omega. It depends whether you want to use c mu here or not. So, from there you can compare these two and this will become omega equal to epsilon by k. If you do not have c mu because in some of the models, you use omega equal to epsilon by k. Some of the models use epsilon equal to omega equal to epsilon by c mu k. So, this again you have to see which code you are using. So, by comparing this thing, you can get this relation. I mean the value of l equal to k to the power half by omega. This can also be you can also get it by using this relation mu t equal to c mu rho c mu k by omega which is equal to some velocity scale and some length scale. If you work this out, you will get this value of l equal to k to the power half by omega. Note that in some of the value, you will have c mu multiplied here or divided here. So, that you have to be careful. That depends on the definition in that particular software. I am just giving you a concept that if you know the turbulence kinetic energy and if you know the length scale, you can get the specific density and you get this value and assign at the inlet as well as the initial condition. So, this is how you do things. If you are using let us say 4 equation transition model like say gamma and Re theta. So, there you use gamma equal to 1 because you assume that there is some turbulence in the free stream even if it is not in the boundary layer. So, the gamma should always be equal to 1 and Re theta, you can use some analytical relation to basically calculate the momentum. So, this is for the 4 equation HST omega model which is used for the transition flow. So, there you do not have to initialize only the k and omega, but also gamma and Re theta. So, the analytical relations are available in the literature and you can use those relations based on the mean flow parameters. One last thing I want to say that there are so many calculators available online where you can provide your intensity or the length scale intensity or your viscosity ratio and it will give you all this value that you want. You can also write a little MATLAB code or maybe in excel also you can calculate by using the definition that we just discussed. Only thing that you need to be careful is that how do you define your relation between mu t and your k epsilon or k omega whether you use the C mu here or you do not use C mu here. Some software use C mu as a constant, some software just define it as C by omega or k epsilon. Now, some estimates because not always you will have the values available with you like even the turbulence intensity value. Let us say in some experiments you did not measure that then you do not have the precise value. So, you have to make some estimates. So, how do you make those estimates? So, for example, if the flow is fully developed in a duct or pipe typically you can take the intensity as 5 percent and the length scale may be equal to the hydraulic diameter. You will get likely to get robust result in this case even if your length scale is lower than the hydraulic diameter. So, this is how you do it for pipe and duct and if you do not have any information about the turbulence intensity. So, typically 1 percent to 5 percent is a good estimate for the turbulence intensity, but let us say if you are dealing with a very disturbed flow for example, exhaust turbine then the intensity can go very high and can go up to 20 percent and these flows are very sensitive also. So, I mean you have to do some kind of sensitivity analysis to get the parameters right. For external flows the turbulence intensity can be lower than taken as 1 percent or even lower than that and the viscosity ratio can be taken as 10. So, basically mu t by mu is equal to 10. So, remember even if you have the information about the turbulence intensity you may not have the information about the viscosity ratio or the length scale. So, you have to basically keep this parameter in your code to calculate the value. So, these are some of the estimates that I am telling you. For internal flow the viscosity ratio is higher and you can use the value let us say between 10 to 100, for external flow it is below 10. Now, having said that there are many flows which are extremely sensitive to the choice of input turbulence parameters like what intensity you are giving, what length scale you are providing, what is the viscosity ratio that you are giving. So, in those cases you have to basically be very careful and what you need to do is that you need to do a sensitivity analysis. So, you can change the length scale and see whether the flow results are changing a lot or it is not that sensitive to the parameters. If you take for example, the flow over a turbine blade or a compressor at relatively low Reynolds number or may be the transitional Reynolds number and there is high curvature on the turbine blades and then you have lot of disturbance because the flow that is coming in is from another turbine they are highly disturbed. So, in those cases the flows are extremely sensitive to what kind of length scale you are giving or what kind of intensity you are providing. So, in those cases you have to basically do the sensitivity analysis and try with different intensity, different length scale and see whether your results are changing. Now, so this is all I had to say about the boundary conditions. One last thing that I want to mention that let us say you have a wall, you need to give this boundary conditions on the wall also. So, typically the zero gradient boundary condition on the wall works fine for most of these parameters. For inlet and outlet you are basically assigning the value, but at the wall you can just provide the zero gradient boundary condition for all of these like k, epsilon, omega, gamma, r theta and so on and so forth. In the next class we will start discussion about this large eddy simulations and we will discuss about some of the models in the large eddy simulations also and that will basically finish this course about CFD. Thank you very much.