 In the last lecture, we were trying to length scales of the different size eddies. So before going to a formal assessment using the scaling relationship that we have established, maybe let us now look into some visual demonstrations of how these eddies might look like. So just look into these, some of these are simulated flows but just look into the rotating structures and you will see that these rotating structures are having a wide range of length scales. So if you see that they are really having a wide range of length scales and they fluctuate over a wide range of time scales. So we will just go on looking into some of these types of visual demonstrations to figure out the roles played by the eddies. So just see the roles played by the eddies which are not there in the laminar flow. So these eddies are turbulent eddies, maybe couple of more ones. So these are 3 dimensional visualizations. So all these have been generated by computer simulation. So you can see, visualize the structure of these eddies. We will just pass it a bit fast and try to figure out. So this, if you want to see the eddies in different planes, so you can see that the structure in eddy, structure of the eddy is changing from one plane to the other. So it clearly gives us an indication that there is nothing called a 2 dimensional turbulent flow. Turbulent flow is always 3 dimensional and unsteady fundamentally. So that is the first understanding that we develop out of this. So at all different sections and at all different planes, you see these different characteristics of these rotating structures. And these rotating structures are continuously evolving with time. That is also one of the important things. So you have not only a wide range of length scales but a wide range of time scales. And we will try to have an estimate of these ranges of length scales and time scales. So the whole idea of this understanding was to have an appreciation that you may have wide range length scales of the eddies. And to quantify that, let us say that we are now interested to get a feel of the difference between the system length scale and the Kolmogorov length scale or the smallest eddy length scale. So the system length scale or the largest eddy length scale, sometimes known as integral length scale, so let us see that what is this. So if let us say that L is of the order of 1 meter, this is an example. That we are trying to take good numbers so that we come up with easy estimates. So the system length scale, say you have a 1 meter system length scale, the largest eddy is also of that length scale. Let us say that the Reynolds number in that is 10 to the power 4. So then what will be eta? So 1 meter into Reynolds number to the power – 3 4. So 1 into 10 to the power – 3 meter, right. So if you make the Reynolds number larger and larger, the disparity between L and eta becomes more and more. 10,000 is not a very large Reynolds number. It is just like moderately large. So if you make the Reynolds number really very very large, this disparity will be more and more and you have eddies at almost all intermediate length scales between these. So that is what we say that the existence of multiple length scales or not only multiple, a wide range of length scales differing in order of magnitude by at least 1000. So you see the order of magnitude difference. Similarly if you look into the time scales and the velocity scales. So the velocity scale in the smallest eddy. So velocity scale in the smallest eddy, how do you estimate? The velocity scale is of the order of nu by eta from the Reynolds number scale equal to 1. So the kinematic viscosity, it is roughly like say 10 to the power – 6 meter square per second for water, nu by rho, 10 to the power – 3 divided by 10 to the power 3. And if you take eta as say 10 to the power – 3 meter, then you come up with a V of the order of 10 to the power – 3 meter per second. These are small velocities and not only that, if you look into the system scale velocity that is quite large. So the system scale velocity that is u0 that is governed by the system scale Reynolds number that is quite large. The time scale, so the time scale for the system it is l by u0 or the large eddy for the large eddy is of the order of l by u0. And for the small eddy that is the Kolmogorov time scale, so this V is Kolmogorov velocity scale. So if you consider the time scale for the small eddies that is sort of eta by, so it is possible to have an estimate of the time scales and the length scales and the velocity scales. The other important aspect of the large eddy and the small eddies or the distinctive aspect is that the large eddies have a sort of directionality or a directional preference because they are large and they have some preferred directions over which they have their activities. On the other hand, smallest eddies have no directional preference and the distinction therefore is that the largest eddies are very much anisotropic. So they do not have like isotropy or direction independence type of behaviour. On the other hand, if you go to smallest eddies they are virtually isotropic. So it is not that they are actually isotropic but they are approximately very much isotropic. So the transition of paradigm from the largest eddy to smallest eddy is also in the form of big anisotropy to a reasonably good state of isotropy. And that is possible because the eddies will tend to become more and more isotropic as and when they are able to dissipate whatever energy is being transferred to them through viscous effects because viscous effect sort of tries to equilibrate it in all possible directions. So viscous effects are stronger and stronger for smaller and smaller eddies and that is why as you go towards smaller and smaller eddies the dissipation effect makes it more and more isotropic or direction independent. Now we will try to understand another important thing that see these eddies are having rotations and when they have rotations they must have vorticity. So we will try to see that how these vorticity is evolve for these eddies. We will try to develop a sort of governing equation for vorticity and we will try to understand that qualitatively by understanding the relative interaction between the large eddies and small eddies and so on. So let us say that we start with the vector form of the Navier-Stokes equation. So let us say that we have no, so this is like the momentum equation in a vector form which we derived. Now what we are interested to do is to get expressions for vorticity out of that. So we know that vorticity is the curl of the velocity vector. So let us take curl of both sides of this equation so that we have a chance of coming up with the vorticity. So just take curl of both sides. So if you take the curl of both sides then what happens? First term, so curl is a vector operator with respect to the special gradient. So with respect to time you may just take it inside outside without any problem. Now let us try to simplify this. So for simplifying this clearly we understand that this is equal to the vorticity vector. Let us call it zeta. There is another term which we can, of course this is also zeta. There is another term which we can clearly simplify. What is this? This is 0. This is the curl of gradient of a scalar. So curl of gradient of a scalar by a vector identity this is 0. So we have now to concentrate on this term. So it is curl of, now let us write u dot del u. So what is this? This is by the vector identity this one. So this just we have written the vector identity. So when we have written this vector identity, so let us first consider the first term. So this is like, this is a scalar u square by 2, u square. So when you write, so this is a dot product of a vector with a vector right. So when you write this, one important thing that you are getting out of this is whatever it is, it is a scalar. So you have curl of gradient of a scalar. So the first term becomes 0. Therefore, this boils down to minus of curl of this one because this is the vorticity vector and this is as good as, so the left hand side becomes what? Rho, one of the terms that is this term we retain in the left hand side, the other term we bring in the right hand side. So the right hand side becomes, so can you identify what is the term which is there in the square bracket in the left hand side? This is the capital d dt of zeta, the total derivative of zeta. So we can write the, so we have got a transport equation of vorticity by starting with the Navier-Stokes equation. And let us just write it in terms of the kinematic viscosity. If you divide both the sides by the density, then this one is equal to, you can clearly see that this is what if you have a vortex element, that is an element within which there are elements of vorticities, then this vortex element may have a change in vorticity. So the total derivative is representing what? It is a change in vorticity of an element because of a combined effect of change in time and change in position in going to a different place where the velocity field is different. Therefore it is subjected to a different velocity gradient at a new location and with respect to time also there has been a change, the total effect is a combination, sorry 1 by the total derivative of the vorticity is what? You have right hand side, this is quite straightforward to understand. That is the second term. The second term represents what? It represents the viscous dissipation of vorticity, so to say. So there is some total rate of change of vorticity. It is related to something which is viscous dissipation but something else also which is not viscous dissipation and we will try to understand that what is this something else or what is the implication of this term and that we will do in a very simple and qualitative manner. So when we do it in a simple and qualitative manner, we will again go back to the picture of the large eddies and the small eddies. So when you have a large eddy, for a large eddy, the Reynolds number is large, right. So for a large eddy, the inertia force is much, much greater than the viscous force. For smaller eddies, the viscous forces are also there. Now if you just simply in a rough way model an eddy, you can say that a vortex element, the rate of change of angular momentum of a vortex element, let us say i omega, when we say the total rate of change, it is like we are writing the total derivative because the rate of change may be because of many things. So this is as if we are tracking a fluid element which is going from one place to the other because of change in time and change in position. Combine effect is the some net rate of change of angular momentum and that must be equal to the viscous, the torque due to viscous forces. So here because whatever force what we are seeing is a viscous force, that is what is the forcing parameter. So this you can write as i d omega dt plus omega di dt is equal to the viscous torque. I am just writing it therefore d omega dt is equal to minus omega by i di dt viscous torque by i. Let us consider the large eddies. So what happens for the large eddies? Those are very special cases when the viscous effects are very very negligible because for the large eddies the Reynolds number is very large. So for the large eddies the angular momentum is conserved. Now when the large eddies are extracting energy from the mean flow what is going to happen? Their angular velocity will increase. So omega will increase but if i omega has to be conserved then i should decrease. So i should decrease means their sort of radial length scale should decrease and therefore if they were more or less, if the large eddies were more or less like this the small eddies I mean their subsequent transformation to preserve the angular momentum will be of a shape which is if you consider this as a radial dimension. This is R2 say this is R1. So R1 will come down to a lower R2 but if it is the same large eddy the volume has to be conserved. So if the radial length scale has decreased the lateral length scale should increase and therefore the vortex element or the eddy has got so called stretched. This is known as vortex stretching. So vortex stretching, vortex stretching is one should not create a misunderstanding that vortex stretching is only for the large eddies not like that but we are giving an example of a large eddy where very large eddy where viscous effects are negligible just to give a clear relationship between what is the change of the moment of inertia how it is related to the angular velocity. So if you have now eddies which are rotating at a higher speed they must sacrifice it in terms of having a lower moment of inertia so that i into omega is conserved and when you have that that means it is sort of stretched to have the same volume but now with a lower radial scale. So that the moment of inertia is now less. Now what happens for a case when viscous effects are present qualitatively same because if you see that when this viscous effects were not there you see d omega dt. So d omega dt is the rate of change of the angular velocity and you see that is related to – of di dt. So one increment is another decrement and now the viscous torque will also play an additional role. Now if you come back to this equation see there is a lot of similarity between this equation and this. This is a differential equation. This is a very qualitative this is also like some sort of differential equation but not rigorously derived. So it is just by putting terms qualitatively and what we see here is that if you consider the left hand side see this is the total derivative of vorticity. It is almost very closely related with the total derivative of the angular velocity because the angular velocity is half of the vorticity vector. So these 2 are related. The viscous terms these are related therefore whatever is this term and whatever is this term these 2 must be carrying the same meaning. That means so what does this term indicate this term represents the effect of vortex stretching. With a larger omega you have a decrement in i therefore this term in the vorticity transport represents a vortex stretching. So vortex stretching is one of the very important activities that is taking place in a turbulent flow structure and it can be shown that this effect is important or there only for a flow with a 3 dimensional structure and a turbulent flow has only a 3 dimensional structure. On an average it might be 2 dimensional, 1 dimensional whatever but fluctuations are there in all possible direction. So even if the mean flow is 0 in a particular direction but you still have fluctuations in all possible directions. The question is how we quantify these means and the fluctuations. The next thing where we will go to is the statistical description of turbulent flow. When we say statistical description that means there is some uncertainty which we want to show which we want to express by some sort of averaging or finding the standard deviation these types of parameters. So one of the most important or most fundamental and sometimes considered to be the easiest statistical description is averaging. So we are now going to discuss about some concepts of averaging in a turbulent flow. So to have a qualitative picture let us say that you are plotting the velocity as a function of time. So if you are having a turbulent flow maybe you are having this type of random fluctuation in velocity as a function of time. It might so happen that you are not interested about this randomness. You are interested to see that how it is on an average but the question is if you want to find out average over a given period of time what is that time. So the average over a given period of times if you want to find out say u average. u average is what? You must integrate u with respect to time over a time interval say a time interval of say time equal to 0 to time equal to capital T divided by t and take some limit. Formally it is written as limit capital T tends to infinity. What is the meaning of this infinity? That we have to understand. So this is a formal definition of something known as time average. So time average at a given location. So time average at a given location that means this u0 is a function of a given position say x0 and this u what we are writing inside is a function of both position and time. So u as a function of x0 and time. The time effect has got nullified by integration with respect to time. So this is only at a fixed position. So now this time scale what is this time scale? See the turbulence fluctuations have very small time scales. So you can see that over very short time it is having a very rapid fluctuation. So if you want to average it out you must take a time scale which is much larger than the characteristic time scales over which the turbulent fluctuations are there. So that turbulent fluctuations are averaged out or smoothed out and that means that with respect to this turbulent fluctuations this time scale is like infinity, very large. So this infinity is not in a literal sense. This is with respect to the locally fluctuating time scales. But this should also be much less than the system length scale, a system time scale. The system time scale says from here to here there is some change. So now if you consider this time scale as the entire system time scale then you will not be able to capture the transiences in the average sense. So that means you will lose all the information and come up with a single value. If you come up with the, so if you average over the entire time. So the time period over which you are averaging is very critical. It should not be too small so that still it is within the range of the individual turbulent fluctuations. But it should not be too large so that you can resolve the transiences on an average. So it should be something in between. It should not be as large as the system time scale but it should not be as small as the individual turbulent fluctuation time scale. So if you now make a sort of this type of averaging and plot u bar then it is possible say you get the u bar like this. And u bar maybe in this example u bar is not changing with time. So this type of case where the actual thing is time dependent but on a statistically average sense the average is not a function of time. This is known as stationary turbulence or steady turbulence. Usually we call the use the term stationary turbulence because steady is a misnomer. Turbulent flow is never steady. So but the meaning is like stationary is like time independent. So this means time independent average behavior. It does not mean time independent actual behavior. Only on a statistically average sense this is not a function of time. But it is also possible to have it a different way. Let us say that you have an average like this. And on the top of that say you have fluctuations like this. So the line which is drawn with a blue is an indicator that you may have a turbulent flow which may be having a time average which is not a constant which is varying with time. So the solid line which is going through the middle is an indicator or maybe let us just mark it with a different color. So if you mark it with this particular color. So this example is a case where it is not a stationary turbulence. So this is a stationary turbulence. But the other is not. So after time averaging you may still get time dependence. But that time dependence is over a time scale which is important for the system. And that is something we need to keep in mind. So this is called as time averaging. Similarly you may also go for a space averaging. So what is space averaging? Very very similar. So space averaging is averaging with respect to space or position at a given instant of time. So very similar just swap the space and the time variables. So now what we are doing at a given time we are taking the data at different spatial locations and finding an average out of that. In experiments we usually are not very careful about time averaging and space averaging. But in experiments whatever averaging we do intuitively is known as ensemble averaging. So let us see what is an ensemble average. What is an ensemble? So if you do a large number of experiments with identical conditions. So that is called as an ensemble. So let us say that we are doing an experiment where we want to find out the velocity variation as a function of time and position. So what we do? So let us say that we have a pipe at a given location. So there could be many locations. Let us say that this is one location. At this location we want to measure velocity as a function of time. So we are doing one experiment where we are doing it. Again we are doing another experiment. Then we are getting a reading at this point. These experiments are all done at identical condition. But because of the randomness the output is not identical. Because there could be slight variations in the experimental conditions and those got amplified. That is like an instability that is there in a turbulent flow. So therefore if you repeat such experiments for each experiment you will get some sort of data or information at your identified points at a given instant of time and you can make an average prediction out of all experiments. And that average prediction out of all experiments is known as ensemble average. You have to keep in mind that these experiments must be performed under identical conditions. So identical conditions you are but the irony is you are believing it is identical condition but there is always a slight perturbation or slight difference from one condition to the other which is making it deviated from the exactly identical condition. So when you have ensemble average it means averaging a large number of experiments conducted under identical conditions. So when you say averaging from a large number of experiments under conducted under identical conditions we will also try to understand the implications of this averaging and the relationships with time averaging and space averaging for certain special cases. We will come to those special cases subsequently. Now when you have an average you also have a deviation from the average in a statistical sense that is represented by the RMS or the standard deviation. So now if you write say the x component of velocity as an example say as u average plus u prime. So this is the average. So when we are saying this as average we are not committing what sort of average. It could be time average, space average, ensemble average whatever but with respect to average there is always a fluctuation. So this is average and this is fluctuation. In most of the textbooks the average is written by an upper case and the fluctuation is written by a lower case or something like that but like when we write in the board it is very difficult to distinguish between the upper case and the lower case. So we are going to use this the prime for the fluctuation and the bar for the average. In some cases for the average this type of breast symbol is also used. So this is a typical symbol for ensemble average but again I mean one may use any either this sort of symbol or the over bar I mean either is fine as a notation for averaging. So if you want to find out what is the RMS. So first of all you want to find out that so what is RMS? So root mean square deviation. So first is the deviation. So deviation from what? Deviation from the mean. So deviation from the mean is u-u bar that is u prime. So you have to find out the summation of this and squares basically squares of the individuals and the summations and then so you sum it up divided by the number of data and that will give you the square of the standard deviation or the variance and the square root of that is the standard deviation. So basically you are making an averaging of this. So all those expressions we are representing by this. So basically summing up of this data over a number of data and dividing by the same number of data is averaging. But why we are using this symbol is because the averaging may not always be on a discrete data. It may be on a statistical sense with a probability distribution function. So we do not know that whether it is on a discrete set of data that you are doing it or you are fitting the data with a probability function and then finding out an average with that probability function. So we are not committing with any specific definition of averaging but just marking it with this one. So when you do that it is like the mean square deviation and when you make a square root it is the root mean square deviation. So the RMS of u is like this one. Similarly you will have RMS of v and RMS of w for the velocity components along the 3 directions. There is some important terminology called as isotropic turbulence. What is isotropic turbulence? Isotropic turbulence means that the turbulent statistics are independent of direction. That means the RMS values of the velocities should be direction independent. That means whatever is RMS of u, same should be RMS of v and same should be RMS of w. So that means you must have, so this is directional independence of turbulent statistics. Why the directional independence of turbulent statistics is going to be important? Because it may enable us in simplifying certain considerations when we are mathematically modelling a turbulent flow. So we have to keep in mind that when you consider the averaging, so if you consider the averaging of say u prime, what is this? Say you want to find out the average of u prime. Let us say time average. So what will you do? This one. But what is its value? Let us say we want to find out what is the average of u. So average of u is limit as, with the limit and other things we will write later on. So u is u bar plus u prime with whatever limit. So the first term see u bar is like an average which is not varying with this time. So the first term becomes u bar. So u bar integral of dt from 0 to capital T by capital T. So u bar into t by t that gets cancelled out. So plus this one. That means the fluctuation of this must be 0, average. So average of the fluctuation is 0. That is why it is a fluctuation, random fluctuation. Its average over that time must be 0. But the average of u prime square or v prime square or w prime square is not 0. So the average of product of the 2 fluctuations that will not be 0. That we have to keep in mind. Now, so we have introduced a terminology of isotropic turbulence and similarly we may introduce a terminology of homogeneous turbulence. So what is homogeneous turbulence? Just by the name it is clear that the turbulence statistics are independent of position. Just like isotropic it is independent of direction. So this is direction, position independent turbulence statistics. Turbulence, it is better to say turbulence statistics. Let us consider a stationary turbulence. So that is also another terminology and stationary turbulence means you have the mean is independent of time or the average value is independent of time that is stationary turbulence. So for a stationary turbulence what we may say is that if you are doing a time averaging, if you are doing a time averaging then the time average value is what? The time average value if you repeat a many number of experiments then the time average value is as good as ensemble averaging. That means at a given point if you record the data and average is the data at different times. So that doing the data at different times is as good as doing the identical experiment at different conditions because the average should not vary with time for a stationary turbulence. So for a stationary turbulence if you are doing the experiment at different time only you are allowing the random fluctuations to be there as a function of time. Average is independent of time and the random fluctuations are those which make one experiment different from the other at a point. Therefore if you do identical experiments and if you get an average of that at a given location with respect to number of experiments that is as good as time averaging for a stationary turbulence. That means for a stationary turbulence you have the time averaging same as ensemble averaging. So stationary turbulence will have a conclusion that the time average is equal to ensemble average. If you have a homogeneous turbulence, homogeneous turbulence is where you have the turbulent statistics position independent. So when you have turbulent statistics position independent that means at a given time if you do the experiment at different positions. So if you do the experiment at different positions then what happens? So at a given time you are doing the experiment just at different positions but different positions have the same behaviour in terms of the average. That means doing the experiment at a time at different positions is just like doing different simulated experiments at different positions. So that means for homogeneous turbulence you must have the space average equal to ensemble average because varying over space is just like having identical experiments in terms of the mean characteristics. The mean characteristics should not be function of position for homogeneous turbulence. Only fluctuations are functions of position. So if you have a stationary plus homogeneous turbulence then you must have time average equal to space average equal to ensemble average. So homogeneous plus stationary turbulence will imply that you must have the time average equal to space average equal to ensemble average and this is known as ergodic hypothesis. So we have looked into the averaging and the finding out the rms of different quantities and of course one may do certain other statistical operations using the features of turbulent flow and one of those important features that we will consider is by developing a correlation in the turbulent flow. So a correlation and correlation coefficient for turbulent flow. Let us say that you have a random variable capital X. So when you have a random variable we say what is a random variable? The outcome of a random experiment which depends on chance. So if there is a variable which has its outcome dependent on chance or probability. So that is a random variable. Now if you have a random variable at a time say t1 and another random variable say y at a time t2 then if you just make a product of these and find the average it sort of represents the average correlation between the random variables X and Y at times t1 and t2. If X and Y are the same random variables then it is called as a autocorrelation. So autocorrelation when you have X and Y same random variables. So what it basically tries to represent? So if you have the feature of a random variable at a particular time and if you have the feature of the same random variable at a different instant of time you are trying to see that how these 2 features are correlated. That means say what is happening now and maybe say what is happening after 20 seconds or 10 seconds or 100 seconds. So if those outcomes are there and if you make a product of those outcomes and average it over all possible data sets then what average information you get is that on an average how the events are correlated and if the same event is there that means if the random variable X and Y are same then how the outcome of the random experiment at time t1 is related to the outcome or correlated with the outcome of the random experiment at t2. If it is a stationary turbulence then it does not matter what is the origin of this t1. So you may have say X at t and say X at t plus an additional time tau. So this t is immaterial the origin of this t is immaterial if it is a stationary turbulence that means the turbulence statistics are not function of time. So then for stationary turbulence for stationary case you are able to write the correlation in this way. Now what is this random variable? The random variable that we are looking for here is mostly the velocity. So let us say that we are looking for the velocity or maybe one of the velocity fluctuations. So this is known as the autocorrelation of u prime. So when you have the autocorrelation of u prime the important thing is that this may be normalized or this may be expressed in terms of a so-called non-dimensional manner because it is like velocity square. So if you normalize it the normalization is with respect to the mean square deviation. So that means if you want to write it in terms of a coefficient so this if you call as say if you give it a name say capital R which is a function of tau then you may have small r which is again a function of tau which is capital R divided by maybe this one. This is known as autocorrelation coefficient. So if you have a autocorrelation coefficient this is just like having a normalized way of writing the correlation coefficient. And it may be shown that its magnitude is always between 0 to 1 by a Schwarz inequality which is commonly used in statistics. So this will be normalized always between 0 to 1. Now it is also possible to have a sort of Fourier transformation of the autocorrelation coefficient into some frequency domain and that is possible like if you have say for example if you consider a transformation like this. So this is known as energy spectrum of the turbulence and we will see that why it is so and with the inverse Fourier transform it is also possible to recover the r. So when you have this one so if you this is minus and this is plus then you consider the special case of tau equal to 0. So if you consider tau equal to 0 and say you are since it is stationary turbulence you do not care what is this t. So you may also consider t equal to 0 because the statistics will not be function of time. So when you have t equal to 0 and also say tau equal to 0 then your r tau will become u prime square right mean of u prime square. So u prime into u prime at 0. So then what does this represent s omega at tau equal to 0 or s0. So s0 will be nothing but represented or let us say you put omega equal to 0. So just put omega equal to 0. So if you have s0 what does it represent? To understand that let us first put what is r equal to what is r at tau equal to 0 that will be easier for you to follow first. So first let us say what is r0. So what is r0? Minus infinity to infinity s omega d omega right and that from the definition of r is this one right. Therefore what we may conclude out of this that whatever we consider as energy spectrum the integral of the energy spectrum over all possible frequencies is an indicator of the rms of the velocity fluctuations right. So that is how. So the Fourier series analysis is important because you may mathematically get this easily if you know the power energy spectrum distribution and from that directly you get a statistical behavior of the rms of the correlations. Now the big question is that why we are going for all these statistics? Why it is necessary? So that question is the ultimate question for analyzing turbulence. See the turbulent flow may be approached or solved by the Navier-Stokes equation. So if somebody asks you a question are the Navier-Stokes equation valid for turbulent flow very much valid. But what makes it almost intractable for solving the turbulent flow problems? One of the important things is the wide range of length scales and time scales. So if you want to resolve such a big range of length scale and time scale by a solution strategy that is very very tedious. Not only that there is a whole lot of uncertainty about the sensible dependence the sensitive dependence on the initial conditions or the experimental conditions or the boundary conditions. So if there is a slight perturbation the perturbation will get amplified and therefore if you exactly know what is the initial condition and boundary condition your Navier-Stokes equation will still give the exact solution. But if you do not know then the Navier-Stokes equation will not give the solution based on the experimental condition that you are simulating no matter whether you are dissolving whatever scales. So in principle equations are applicable but because of the uncertainties and strongly sensitive dependence to slight perturbations from the initial and boundary conditions the outcome is not something which is acceptable and that acceptability becomes more and more vulnerable as you are not able to resolve all these length scales and time scales. So what is the alternative? Alternative is you go for a statistical description. But the thing is that when you go for a statistical description we will see that the in terms of statistical behaviour turbulence will be deterministic that is you may have randomly fluctuating variables but the statistical descriptions are not random they are deterministic. But we will see that a problem will come that the governing equations will not be closed. So this is an irony that in terms of the statistical description you have the governing equations which are sort of deterministic but they are not apparently closed. On the other hand in terms of the actual variables the governing equations are perfect they are closed but if they are not deterministically solvable. So this type of dilemma is there and that is why understanding turbulence through mathematics is one of the very difficult things and it has been till now not solved and in classical physics this is considered to be the last unsolved problem in classical physics that is understanding of proper mathematical description of turbulence. So whatever description of turbulence we will be having will be very very elementary just to give you a qualitative picture and that we will do in the next class. Thank you.