 So, our next session will be by Professor Raghavendra Gupta of IIT Guwahati, he is a faculty in the chemical engineering department of IIT Guwahati and the session will be on turbulent flows. So, over to you, Professor Raghavendra. Thank you, Payal, for the introduction. I will try to discuss the basic features of turbulent flows and some glimpses of how the turbulent flows are modeled or what are the problems in modeling turbulent flows and how those problems are tackled. So, to start with, I have a picture here of turbulent flow behind a boat wake. So, when you are riding a boat and you see that the wakes are being left behind and the flow is generally turbulent. So, all of us have seen turbulent flows around us, most of the flows that are occurring in industry until it is micro-scale flows, they are turbulent in nature. And the first thing probably we use or first parameter we use to identify if the flow is turbulent or all laminar, we look at the Reynolds number. So, for example, all of us would have studied in under graduate that the flow is becoming or flow becomes turbulent in general for a pipe flow at about Reynolds number 2000, 2100, 2300. For boundary layer on a flat plate when we talk about, so the typical value when this transition for laminar to turbulent occurs, it's about 5 into 10 raised to the power 5 or for flow over a sphere of the Reynolds number is of about 3 into 10 raised to the power 5. Now, these are just typical values, but depending on how smooth the surface on which the flow is happening and how calm the environment is, for example, the pump from which the flow is happening, the surroundings, etc. Then these transition Reynolds number can change and people have shown that the flow can be laminar in a pipe flow for a Reynolds number, for a very high Reynolds number of the order of 10 raised to the power 5 even when they have taken special care that there are no disturbances present in the surroundings. So how do we characterize that the flow is turbulent? So turbulent as the name suggests, the turbulent flows are characterized by the random three dimensional fluctuations. So the randomness is that as the name suggests that when you say turbulent or not necessarily with respect to the flow when you use the term turbulent, it might mean chaos or randomness. So the same is true with turbulent flows that they are characterized by the random fluctuations in the flow and these fluctuations are generally three dimensional. So the fluctuations in what? The fluctuations in velocity and pressure. So for example, if you measure the velocity at a particular point in a turbulent flow, you are going to get a velocity like this, where you can see that the flow is not necessarily periodic. So turbulent flows are not generally periodic, the behavior is completely random. So randomness is what defines turbulence and then you have three dimensional fluctuations. And we all know that laminar flow where the name itself suggests the flow occurs in laminar which means in parallel layers where as turbulent flows where the flow can occur across the laminar. And in laminar flow as the Reynolds number is the ratio of inertial force and viscous force. So lower the Reynolds number, that means dominant the viscous force are so at low Reynolds number viscous force are dominant and the flow is laminar because the fluctuations that occur in turbulent flow are because of inertia. They die down because of the viscous effects. But when inertial effects become dominant and the Reynolds number become high, then it is not possible that all the fluctuations die down and you see the flow to be turbulent characterized by random fluctuations. So there's another picture of turbulent flows which we all see by smoke, for example by smoke from an incense stick. So the turbulence is characterized by randomness then diffusivity. So for example I am a chemical engineer and we are concerned with oftentimes with mixing. We want to do reactions in a reactor and in order that to happen two reactants need to be mixed first. So that mixing occurs or it can occur because of diffusion. Now diffusion can happen because of molecular diffusion. So molecular diffusion is that the molecules are moving around randomly. But this diffusion is very slow, especially in liquids. So in turbulent flows because there are a lot of vortices, a lot of adhesions and these adhesions can range from the smallest scale to a very large scale. So the largest scale can be the scale of your problem. So for example if you are considering the flow in a room, the largest scale can be the dimension of the room itself. And the smallest scale what we call the Kolmogorovland scale or the scales at which the fluctuations are dissipated by the viscous motion. So another very important property of turbulent flow is diffusivity which can be used. For example if you want to do heat transfer, not only reactions but if you want to do heat transfer, you get a lot of mixing because of the inherent diffusivity present in turbulence. Of course they are characterized by the large Reynolds number and as I said earlier that the large Reynolds number means that the viscous effects are not so important or are dominated by the inertial effects. Then you have three-dimensional vorticity fluctuations. So it suggests that in turbulent flows the vorticity effect or the vortices are present and these vortices, there are fluctuations in vorticity and these vortices are three-dimensional. So turbulent flow is three-dimensional in nature and then the energy that is present because there are fluctuations in turbulent flow and these fluctuations they eventually need to die or they eventually die down. So there has to be a continuous source of energy which causes the fluctuations in the flow and these fluctuations cascading happens. So these fluctuations move or the energy in the fluctuations is transferred from the largest scale, largest length scale to the smallest length scale. So that happens and it is called energy cascading and that the smallest length scale because as the scale becomes smaller and smaller, viscous effect can become important. So at the smallest length scale, the energy is dissipated by the viscous effects and this energy is converted to thermal energy. Now of course, so we said that there are number of scales present in turbulent flows but even the smallest length scale that is present in turbulent flows, the smallest length scale is bigger than the molecular length scale. So we can treat the flow to be continuum. I hope that everybody knows what is continuum because when the model fluid flow, the entire structure that you are modeling in open form, you saw Navier-Stokes equations, the momentum conservation equation and mass conservation equation. All these are differential equations and you can model a phenomena in using differential equations only when the continuum assumption is valid, only when you can treat the velocity fields to be continuous, density fields to be continuous, pressure fields to be continuous and that is possible only when the length scale of the problem is sufficiently large than a typical length scale which represent the intermolecular distance. In gases, this intermolecular distance can be a main pre-path and in liquids, the average molecular distance. So it is still continuum. Now these adhes can have a wide range of time and length scales. So as I said that the length scales or the size of the adhes, they will be the largest size that will be limited by the dimension of the system. How big? So if you're talking about the pipe, the diameter of the pipe can be your largest age adhes size and then this energy is being transferred from larger adhes to the smaller adhes which we call energy cascading. And this is this famous quote you see in many turbulence books that big worlds, worlds means the vortices, they have little worlds that feed on their velocity and little worlds have lesser worlds and so on to viscosity in the molecular sense. So this represents that there are bigger vortices from there, the energy or the turbulent kinetic energy is transferred to the smaller vortices and down to the smallest scale where the energy is dissipated by viscosity. So when the adhes become smaller and as such the smallest scale, the viscous effects are important and this kinetic energy is converted to heat. So this gentleman, Colmer Grove, who provided a lot of insight on turbulence by simple dimensional arguments and scaling arguments, so he suggested that the size of the smallest adhes or the length scale, velocity scale and time scales, all the things. So the first size means the length scale of the smallest adhes. So we know from here that the largest adhes can be the size of your system and then smallest adhes he suggested that there can be a universal correlation for the smallest adhes that will depend on the rate of energy dissipation. So energy dissipation, energy means the turbulent kinetic energy, which is basically in terms of fluctuations. So the rate of energy dissipation and fluid properties because the viscous effects will depend on mu and rho or if you want to write it in terms of kinematic viscosity then nu, which is mu over rho. So by just simply looking at dimensional arguments, the turbulent kinetic energy, which edge, I will have the definition in the later slides, but let me just write here of sigma u i dash squared. So turbulent kinetic energy is basically u i, v i and w i or u 1, u 2, u 3, they are the fluctuations, so the square of that divided by 2 represents turbulent kinetic energy. So depending on or using turbulent kinetic energy and the rate of this kinetic energy dissipation. So the unit of k is meter square per second square as you can see from this expression. So it should actually be called a specific turbulent kinetic energy and epsilon represents the rate of dissipation of turbulent kinetic energy. So the kinetic energy per unit time, which is meter square per second square, meter square per second square divided by second. So its unit is meter square per second cube. So you can construct length scale, velocity scale and time scale using simple dimensional arguments or the Buckingham pi theorem, which all of you would have studied in your undergraduate course on fluid mechanics. So this gives you an idea about the smallest length scale that is present in turbulent flows. So if you want to model from the modeling perspective, you will need to, because these scales can give you an idea that what is the smallest length scale, what is the smallest time scale that is going to be present in your flow. So you would like your mess to be smallest possible so that all these length scales are captured. So you can have an idea or you can calculate what L is going to be. Of course, you need to have some idea about what is epsilon, what is k, the fluctuations, etc. If you know that, then you will have an idea that what kind of mess or what size of mess is required. So typically those values will be very small. And that is the reason because as you would have learned that your solution will be as accurate if you are able to capture all the gradients. And to capture all the gradients, your mess needs to be sufficiently small in the regions where the gradients are large. So you need to have a very refined mess plus when you are resolving all the time scale, your time step also needs to be very, very small. And that makes modeling turbulence directly or solving the conservation equations directly using even with today's computational power that is at our disposal. It becomes very difficult to solve these Navier-Stokes equations for turbulent flows accurately so that you can capture all the length and velocity scales and time scales. So that is why you need to go or you need to develop some models which are generally heuristic models. So what can be done is one can decompose the velocity in mean and fluctuating components. So as you can see here that the velocity let us say u component of velocity with time, it is randomly moving. But if one can average this velocity over a sufficiently long time. So there are different averaging that can be done. If it is homogeneous turbulence, homogeneous meaning that it is same in all directions, then one can even use space averaging. If it is or one can do time averaging and one can even do the ensemble averaging by for state or ensemble averaging is that averaging over a large number of samples. So this averaging is done over a sufficiently long period of time so that your time average is independent of your T average. So you are going to get if you average over a sufficiently long period of time then you get a mean velocity which is represented here using u bar. So you can decompose your velocity in terms of u bar and u dash. So u dash is the fluctuating component. So for example, if I look at this particular at this particular time, this is my u dash or the fluctuating component and you have velocity u bar. Similarly, you can decompose the component of velocity v bar and v dash and the z component of velocity w in w bar and w dash. And the same can be done for pressure. So you can basically decompose the velocity field and pressure field in mean and fluctuating components and it is called Reynolds decomposition. Now this averaging can be done over or should be done over a sufficiently long time interval so that the fluctuations are averaged out. So if you average fluctuations or if you write say if you average fluctuations, this value is going to be zero because over a long period of time the sum of fluctuations will be zero. You can see from this relationship. But this averaging time scale will be sufficiently small from the time scale that is involved in your process. What is your flow time scale? Now, so u bar can still be a function of time that means your mean velocity in this case can be a time dependent velocity. Sometimes it might be steady, but if the flow is if you think that your flow is time dependent, then you can have a mean velocity which is time dependent plus fluctuations. So as I discussed in the previous slide that the fluctuations, they can be represented because if you average fluctuations over time, then you are going to get their value to be zero. So in place of fluctuating fluctuations, what is done is you take u dash square plus v dash square plus w dash square. So you take there the squares of u dash v dash and w dash and they are represented half of them, half of sum of fluctuations or a square of fluctuations you call turbulent kinetic energy. Again, so I would like to emphasize here that the unit of turbulent kinetic energy even though we call it kinetic energy, the unit is not meters per second square. Basically a specific turbulent kinetic energy or the kinetic energy of the fluctuations per unit density. Now, if you use those decompositions and write down in the Navier-Stokes equation. So if in your Navier-Stokes and continuity equation, if you substitute for example, for an incompressible flow, your continuity equation is del dot u is equal to zero. Now, if you substitute in place of u, u bar plus u dash, then so del dot u is equal to zero and you substitute u by u bar plus u dash, so mean component of velocity and the fluctuating component of velocity. And when you do that for the continuity and momentum equation, what you end up with is, so when you average the continuity and momentum equations, you are going to get similar looking equations. Del dot v bar is equal to zero, so it is continuity equation but now in place of v, you have v bar which represents the mean velocity. Similarly, you will get a momentum equation where you have the usual acceleration term, this is your time dependent or unsteady term and then you have convective term, pressure gradient. So in all these terms, what you have is mean velocity in place of the velocity or similarly mean pressure here. Then you have viscous term and the body force term. Now, there is one extra term that you end up with which is this, del dot minus rho v dash v dash, so v dash v dash is basically it represents the fluctuating velocity vectors. So this is an additional term. Now, when you are trying to solve a flow problem for an incompressible isothermal Newtonian flow, you have two unknowns and two equations, your unknowns are velocity vector and pressure. So you want to solve for velocity field and pressure field and you have two equations continuity and momentum equation. Now, when you are, when you have done the averaging, because in general, if you are an engineering student, if you are trying to solve some engineering problem, even if the flow is turbulent, your interest is in the mean flow. So if you will be fine, if you can find out the mean velocity and mean pressure field. Now, we have been able to convert using Reynolds decomposition and then averaging. So these equations, after averaging, they are called Reynolds average equations of RANS, Reynolds averaged Navier-Stokes equations. And the problem now is that we have, apart from v bar and p bar, we have another unknown which is minus rho v dash v dash. So we do not know this, what is, how we can find out because this is also unknown for us. So we have more number of unknowns than the number of equations. So the system of equations is not closed. So this is generally known as closure problem of terminals. Now, if you look at this term, rho v dash v dash, and so basically it is rho into v square or rho into velocity into velocity, which will give you a unit of pressure or unit of stress. So in analogy with this, it is called Reynolds stress. Because in a way, it is analogous to viscous stress or the viscous fluctuations or the molecular fluctuations. So in analogy with this, it is called stress and Reynolds stress or turbulent stress sometimes. And it is a tensor because you have three fluctuating components and you have their products. So you can have rho u dash, u dash, rho u dash, v dash, rho u dash, w dash and so on. So you can have nine components. And this term represents the fluctuations on the mean flow. So if you look at the system of equations, what this term is representing that the mean velocity field, how is it being affected by the fluctuations that is being represented by this term, which we call Reynolds stress. And now our task at hand is to model this Reynolds stress. So now we have an additional unknown rho v dash v dash. And this system of equations, as I said earlier, that it is not closed because we have more unknowns than the equations that are available to us. So this is what is called closure problem of turbulence. And that is why we need to develop turbulence models which can represent either you can have a transport equation where, which is for Reynolds stress or you can represent this Reynolds stress in terms of mean flow in its gradient, which is what we are going to discuss today. So there was one hypothesis or one assumption that was put forward that this Reynolds stress in analogy with molecular or kinetic theory of gaseous, where we get the concept of molecular viscosity. One can represent the Reynolds stress tensor in terms of a viscosity which is called turbulent viscosity or eddy viscosity or turbulent eddy viscosity. And the velocity gradients, which is say if you are talking about in xy plane then tau xy also tau xy here is only Reynolds stress. So Reynolds stress, tau xy r is equal to mu t into du by dy. So this Reynolds stress can be represented as the product of eddy viscosity and mean strain rate tensor. So when you write this completely then you will get strain rate tensor which is S ij here. So if you write this in three dimensions and it is convenient to write in index notations. So I can write this minus rho u i dash into u j dash is equal to 2 mu t where mu t is the turbulent viscosity into S ij where S ij is the strain rate tensor. Minus 2 by 3 rho k where k is the turbulent kinetic energy and delta ij is the chronicle delta which is equal to 1 if i is equal to j and 0 otherwise. So this represents the complete definition of Bussin's hypothesis that you can represent Reynolds stress in terms of S ij and mu t and of course this turbulent kinetic energy. Now what we have done that in place of Reynolds stress we have represented Reynolds stress in terms of a kinetic energy and the gradients of or the derivatives along the space of mean velocity. S ij is in terms of mean velocity so S ij you can write half of dou u i over dou x j plus dou of u j over dou x i. So that is your strain rate tensor and B jar in terms of mean velocity. Now we have got rid of or we can represent the Reynolds stress in terms of mean velocity gradients and turbulent viscosity but now this turbulent or any viscosity we need to represent in some of the turbulent properties. So this is what when you represent in terms of turbulent kinetic energy and epsilon which is if you remember it was the rate of dissipation of turbulent kinetic energy then it is called k epsilon model and when it is represented in terms of k and omega. So k is again turbulent kinetic energy and omega is the specific rate of dissipation of turbulent kinetic energy. So a specific rate of dissipation here means that the rate of dissipation of turbulent kinetic energy divided by turbulent kinetic energy so the unit of omega h second place to the power minus one. So again we can look at what these three terms mean k epsilon and omega so k we have already seen that it is turbulent kinetic energy represented by half u dash square plus v dash square plus w dash square where u dash v dash w dash are fluctuations and the unit is meter square per second square. Epsilon which is rate of dissipation of turbulent kinetic energy per unit mass so so the unit will be turbulent kinetic energy per unit time which is meter square second base to the power minus three. Now this epsilon h can be represented by this expression where nu is the molecular kinematic viscosity or mu over rho multiplied by gradients of fluctuation so dou u i dash by dou x k. So this represents this term represents the dissipation of turbulent kinetic energy and remember this dissipation happens because of the molecular viscosity so that is why we have nu in front of this term. Now omega which is called a specific dissipation rate or dissipation rate per unit turbulent kinetic energy so when you divide this turbulent kinetic energy dissipation divide by turbulent kinetic energy then you are going to get second inverse as your unit. So you can say that this omega also represents some mean frequency of turbulence probably that is why it is given this symbol to represent that because omega which is generally used to represent some frequency or second inverse. It can also be thought of a kind of one over omega can be a time scale and this is the time scale at which dissipation of k occurs or it can be also thought of the rate of dissipation because this rate of dissipation is happening from the largest at each to the smallest stage so this is rate of transfer of turbulent kinetic energy to the smallest at each. So then there are different models now this turbulent kinetic energy sorry if you go back what we have done is we have represented the Reynolds stress in terms of turbulent viscosity and mean strain rate tensor. Now we need to represent muti or we need to find out muti because unlike molecular viscosity it is not property of the fluid it is property of the flow and flow means it is property of the turbulent flow so it depends on the. turbulent kinetic energy and some representation of dissipation of turbulent kinetic energy so in k epsilon model we have this that we can write muti in terms of k and epsilon. And of course rho so if you do a dimensional analysis. You can easily see that muti will be rho k square by epsilon so you get turbulent kinetic energy is proportional to rho k square by epsilon and then you put a constant. Together so then you have a quality sign that muti is equal to rho c mu k square by epsilon. Now how do we find out k and epsilon initially when your flow is entering probably you know what you have some idea about what kind of fluctuations are present or what what is the turbulent intensity so you can put in some values of k and epsilon. And then how this k and epsilon is being transported. People have developed or one can derive the transport equations for k and epsilon directly from Navier-Stokes equation taking its different momentum. But then again you will have some unknown so which needs to be close so to cut matter sort what one can do is people have developed. Different transport equations for k and epsilon for the transport of turbulent kinetic energy and turbulent rate of dissipation of turbulent kinetic energy. So the first equation is for k and if you look at the first or the left hand side is exactly what you see for your say moment on the transfer of. Energy or specific enthalpy or the transfer of or the transport of momentum or transport of concentration so that is your material derivative term. This term basically represents the diffusion of turbulent kinetic energy so what you see here apart from mu mu is basically which represent the diffusion of turbulent kinetic energy because of. Molecular transport and mu t over sigma k it represents the diffusion of turbulent kinetic energy because of the turbulent fluctuations then you have. Another term pk which represents the production of turbulent kinetic energy and row of silent represents the dissipation of turbulent kinetic energy now this p term pk basically it is it comes from the interaction of the fluctuations so you see the fluctuations which is basically Reynolds stress. Multiplied by the. Mean strain rate tensor so this as we saw our hypothesis so it can be represented as mu t as square so so you have both of these can come as source term in your equation system of equations so basically a. Material derivative term or transient term and convective term diffusion term and then you have a source term similarly you have a transport of epsilon. So epsilon basically represent the rate of dissipation of turbulent kinetic energy again you have a steady term then then you have. Convective term diffusion term and then production of dissipation and dissipation of dissipation so so basically to cut matters or but you can say there is a transport equation for k there is a transport equation for epsilon. And. So so this k epsilon equation it generally useful or it can. Model correctly the free shear layer free shear layer flow so you have aware where there are no ball bounded flows. And the other condition is that pressure gradients are relatively small. It does not perform so well for ball bounded and internal flows except when mean pressure gradients of flow are small. And and it does not perform well in any case when there are large adverse pressure gradients so we need to know that what are the limitations on the model. Then there is another model which is called k omega model so using same dimensionless arguments one can write muti in terms of k and omega so rho k by omega and then again a transport equation for k. And the transport equation for omega you have all these three terms same unsteady convective and the future and then production of turbulent kinetic energy and dissipation as k basically represents that if there is some generation of turbulent kinetic energy and you may or may not have it. Okay. So so this k omega model is generally. Found to be quite accurate. For ball bounded flows where you will have boundary layers so it is very accurate accurate for boundary layers and the pressure gradients they might be favorable or adverse. But it has problem or it is sensitive to free stream boundary conditions. So so you have k epsilon model which is good in free shear flows whereas the k omega model which which is good in near wall flow so what people have done they have done blending of the two that if it is near wall then then one can use k omega model and if it is in the in the free stream then one can use. k epsilon model so again the hybrid of two or the models which are hybrid of two have come up and there are a number of variations of these two equation models. They are used they they have become overtime work costs for. For turbulent flow modeling especially in industry of course apart from that you have alias and DNS models. So when you when it comes to near wall near the wall if you look at the velocity profile for turbulent flows in general it might be internal flow or external flow this is the typical velocity profile or what you have that. Near the wall what you have on the x axis it's a non-dimensional term which is kind of a Reynolds number and it's called y plus so why you tau over new where you tau is basically. A velocity which is represented in terms of tau w over mu squared root and it is called friction velocity so why you tau over new new is the kinematic viscosity and at the y scale you have u plus so mean velocity divided by this friction velocity. So the viscous sub layer where the viscous effects near the wall viscous effects become important and then. At the center your. Velocity profile what is there for in the outermost layer whatever profile you had so this is from. For a pie flow so that is why you power law profile and you pie flow you might know that one seventh power why it's typically used and then you have a buffer layer or overlap layer where. Which which is basically connecting to that the transition happens so when you want. A one need to take into account that what is going to happen at the viscous sub layer and then to approaches because if you look at k epsilon model it is it is not so good near the walls. And and k omega model can can model accurately all the way down to the wall so there are different near wall treatment approaches that are used. So two approaches two different approaches one is that different wall functions have been developed so in such cases the viscous sub layer and buffer regions are not resolved so that means your mass is. Not defined near the wall it is such that that the y plus value is more than 30 so you are not modeling this viscous sub layer and overlap layer and in place of model a a velocity profile is provided at the wall using so and these functions are called. Wall functions another approach is near wall model approach so where you take care that you are modeling the near wall region so that you are actually your message sufficiently refined so your y plus is. It is of the order of one so so if you take the first. cell height which edge why and you calculate why plus than your wife was should be one or so or even less than that so that you will be able to capture viscous sub layer and buffer layer or overlap layer in your modeling so one need to take into account that what kind of. Near wall treatment you are using in your simulations. So if I summarize. That what is done in turbulence modeling. Spacially or I have only talked about two equation models here so one can decompose the velocity using green or Reynolds decomposition in a mean and a fluctuating velocity component and then one can average the equations the Reynolds average. Averaging and then it this averaging can be over time space or ensemble averaging. So by averaging you get what you get is Reynolds average behavior strokes or RMS equations and this gives an additional term in momentum equation which is called Reynolds stress. And and this Reynolds stress needs to be represented in terms of turbulent properties of the flow and the mean velocity so so this is done using. Using was in his hypothesis where Reynolds stress is represented in terms of eddy viscosity and and strain rate of mean velocity or mean strain rate. Then then what do you need to do you need to represent eddy viscosity in terms of turbulent properties and when you take K and epsilon as turbulent properties and you call it K epsilon model when you take K and omega H turbulent properties then you call it K omega model. So then then for you need to solve transport equations for K and epsilon in in the in in K epsilon model and transport equations for K and omega in K omega model. So I think that is what I had and hopefully it helped. So I will be happy to answer questions if you have any. So can you explain Bosnian hypothesis. So Bosnian hypothesis is basically. So if you look at what is this tau XY it's it's I have just represented it in terms of tau it is Reynolds stress so when you represent Reynolds you remember Newton's law of viscosity which tells that. The shear stress. Is equal to viscosity or dynamic viscosity multiplied by gradient of velocity right so along the same lines what is done is Reynolds stress tensor or or the Reynolds stress is represented as. Multiplication or the product of muti which is turbulent viscosity and. The gradient of mean velocity so do by dy multiplied by a viscosity or a viscosity like term which represents the turbulent diffusion right. So that is what is called Bosnian hypothesis or Bosnian assumption. Sir turbulent viscosity we can say turbulent viscosity is a volumetric viscosity because turbulent effect is a volumetric effect or three dimensional effect. Yeah so it is three dimensional effect but what do you mean by volumetric viscosity. Volumetric viscosity means it affect the volume of the infinitesimal element. Like when we say dynamic viscosity it is only related to the what is the moment transferred from first layer to second layer or layer to layer. But when we say volumetric viscosity it refers to volumetric effect on the elements. Probably you could say so but I think even if you talk about molecular viscosity you can have say a viscosity which is non isotropic then you will have layers along x direction layers along y direction so they are also molecular. Volumetric viscosity also molecular viscosity will be similar so. Albulent viscosity is actually a flow property and it is just that one need to represent it in some terms of some terms of turbulent property so you represent in terms of K and epsilon. So I would not diverge too much into that it is a surface phenomena and it is a volumetric phenomena because ultimately what you again said that it is an infinitesimal volume. So I don't see any merit that how we can benefit by saying it volumetric viscosity. Sir could you explain the near wall turbulence modeling like y y plus has to be one I did not understand that. Right so basically there can be two approaches one edge I mean there are two approaches that are being used one edge what is called wall function approach. So wall function approach means that you see this velocity this is this graph represents the velocity profile near the wall. Now you have viscous layer and overlap layer where the viscous effects are important and then outer layer where the viscous effects are not so important. So one can model this one can do first approach in the first approach one can have each or her the first element size so big that all this is coming into first cell. So the your cell is first cell is somewhere here so that your y plus is greater than 30 that means viscous layer and overlap layer you are not modeling and in place of modeling you suggest that they are each because this is an universal law. So so one can use wall functions or one can input in your model you can code such that a velocity profile is defined at the wall in this cell. Okay so that is your wall function approach now if you want to resolve no you want to model it then you need to have your mass so refined that you can capture you can resolve viscous layer if there are no wall functions wall functions built in in your mathematical model. Then you need to refine your grades such that you can capture this phenomenon because if your message such so big, let us say if in in near wall model approach if your mass or first cell height is like this then you will not be able to resolve viscous layer you are not going to find out the gradients in the first viscous layer and then overlap layer where the gradients of where you have a log log log coming into picture. So in order to resolve that you're in order to capture those gradients your mess should be sufficiently refined. So if you look at I think somewhere that that y plus is equal to five is where you have viscous sub layer. So if you want to resolve this viscous sub layer then you should have few elements. So typically say five six elements if you have there then you will be able to resolve viscous sub layer so that is why I said I said that you should have my plus is equal to one if if you want to model. Or if you want to capture the phenomena using your simulations. Of course it depends on what model or you are using or what is built in the code which you are using for near wall treatment. So probably take home message is that when you are modeling turbulent flow you need to look at that how the near wall treatment is being done in the mathematical model or the turbulent model that you are using. I hope this answers your question. Yes sir thank you sir and one more follow up question so then the if the first cell like it is very refined. Is there any restriction on the like upper self the second third can it be bigger like yeah you can you can change but but you should change it gradually so typically you will take the the ratio of first cell height and second cell height say say about 1.1 or 1.2. Because gradually you will change this so so there are no sudden changes and of course if you are capturing and then first cell height is y plus is equal to one and then second cell is quite big then again you will have only two cells in the viscous sub layer. So typically you change you start increasing the size but at the same time make sure that you have three, four or five cells so that you capture the viscous sub layer. Of course the viscous sub layer I think if you look at the profile it is u plus is equal to y plus. Okay so so it is almost linear velocity profile it does not appear to be linear here because this is log log scale right. Again when you go to overlap layer it is logarithmic phenomena so again the gradients are going to be large. So again you should have sufficiently sufficient resolution in this region also so that you are capturing the gradient here because the way you capture or if these the profile here is captured accurately then only probably you will have the mean flow profile and mean flow properties captured correctly. Sir in open form this near wall modeling is dependent on the solver or grid size. It will depend on I am not sure actually because I don't huge open form heavily. So I am also a learner of open form like you guys and I have not done any turbulence modeling using open form so probably I will not be able to answer that question. I will have to look at their help and then see so what you could do you can look at the literature where they have modeled K epsilon or K omega or any other model and see what is the near wall treatment. Okay thank you sir. Sir in K epsilon model initially in open form we are supposed to initialize some random value for K and epsilon so it's a function of velocity. So when there is inlet and outlet we can assume some velocity but there might be some cases where there is no inlet and outlet like we are just simulating a domain. So in that case how to initialize this K and epsilon value. Yeah so you can use the same thing what you are using when there is inlet and outlet right even if you have inlet and outlet you need to initialize the K and epsilon values in the entire domain right. So generally epsilon can be represented in terms of turbulent length scale and from the turbulent length scale you can represent turbulent length scale in terms of K and epsilon. So K typically you can take 3 to 5% the fluctuations or fluctuating velocity is typically you can initially assume that it is 3 to 5% of your flow so from that you can calculate the initial turbulent energy and then you have a length scale of your problem. So from length scale you can calculate what is the value of epsilon and that value you can use your initial gas as well as the value at the inlet boundary because you will also need to define K and epsilon at your inlet boundary. I think CFD online also have some calculator where one can use or one can input some parameters for the flow that you are solving and it calculates for you what is the value of K and epsilon and of course some similar formula are built up there. Yes sir but that is for some inlet and outlet boundary conditions. But I am asking like when there is no. Yeah so you can use the because it is just an assumption that what ideally you should know what is the value of K and epsilon in your flow initially. But then we do not know so and then same thing stands for for the inlet values for K and epsilon. So you can use the same thing for your flow and depending on how it evolves because what what you are going to get that with time how the value of K and epsilon are going to change. So eventually you should be able to get from equations how the values are changing until you introduce a lot of turbulence and then that may cause problems so you start with say very low turbulence and see how it grows. Yes sir sometimes what happens sir we initialize random values and what what we notice like the solution is not getting converse. Yeah so my suggestion will be to start with say typically three to five percent fluctuations. Okay sir thank you. Sir if the velocity profile is linear in viscous sublayer so why we need more number of grid points we can use one grid point at y plus is equal I mean y is equal to zero and another grid point at y plus is equal to five. You can try doing that and then you can have you make sure that you have that you have sufficient grid points in the overlap layer because the grid velocity profile is exponential. You can have two two elements here and then you can have more more elements in the overlap. Okay. Okay so if there are no further questions I think we can close the session. Thank you so much. It was a very informative session and many students were questioning you and you have been constantly answering them so thanks a lot. Thank you it's a pleasure to. Thank you thank you.