 To continue our discussion of special correlation coefficient, recall that I said that these correlation coefficients 9 of them are difficult to measure in a real non-homogeneous, non-isotropic turbulence usually when measured, they are only measured in one direction say R1. In homogeneous turbulence as we said earlier, all statistical correlations of time average fluctuating components are 0, their gradients are 0, but gradients of mean quantities can be finite that is the definition of homogeneous. Isotropic turbulence implies that any relation between the turbulence quantities must be constant or invariant under rotation of the coordinate system and under the reflection with respect to the coordinate system. As such turbulence cannot be isotropic unless it was also homogeneous. So, what this means is for a homogeneous isotropic turbulence only R11, R22 and R33 will be finite because all other quantities would involve special gradients of phi 1 dash phi 2 dash and therefore, they would all be 0 for i not equal to j and secondly for 180 degree rotation and let me explain this. So, let us say if I had x1, x2 and x3 and let us say I am considering at some point the fluctuation u dash and the fluctuation v dash let us say. Now if I turn this system through 180 degrees so that x2 takes this position and therefore, x3 would take that position this is the negative x2 and negative x3 that is the turning through 180 degrees. Then in this new coordinate system x1, x2, x3 you will see v dash will now appear negative v dash will be negative in the new system and therefore, you will see the time averaging of the product u1 dash u2 dash in the first system will equal u1 dash into minus u2 dash in the second system and this would essentially be minus u1 dash u2 dash. Now plus u1 dash u2 dash equal to minus u1 dash u2 dash can only be true if u1 dash u2 dash were identically 0 that is the meaning of homogeneous isotropic turbulence and for that only r11, r22 and r33 would be finite. Further r22 and r33 would also equal since the coordinate system is invariant under rotation about x1 axis that is what I showed. So, the r11, fr coefficient parallel to xn axis is called the longitudinal coefficient whereas, coefficient r22 equal to r33 equal to gr is called the lateral coefficient that is what I showed in the slide here. This is the lateral coefficient this is the longitudinal coefficient both fr and gr declined to 0 as r tends to infinity. For a given r, fr turns out to be usually of a bigger magnitude than gr. The coefficient curves are nearly parabolic near r equal to 0 and therefore, symmetric about r equal to 0. Expanding therefore, fr and gr in Taylor series about r equal to 0 you will see and retaining only the first couple of terms. You will see fr would equal 1 when r is equal to 0 minus r by what I have called Lf square and I will define Lf in a minute plus additional term. Likewise, gr would be approximately equal to 1 minus r by Lg whole square plus several terms where Lf square would turn out to be minus 2 times d2 f by dr square r tends to 0 raise to minus 1 and Lg square will be minus 2 d2g dr square. These are the projections of the second derivative of f and you will see for transfer for longitudinal correlation the Lf would appear as that and Lg would appear somewhere over there that is the estimate of Lg and this is the estimate of Lf. The Kolmogorov scales of course, would be very very small distance L epsilon would be somewhere here and here, but the integral scales would be of that order because there is the integral of this curve between 0 and infinity and likewise here. So, the Lf and Lg are somewhere between L epsilon and L integral the two length scales which we had earlier identified. We have now identified a third length scale which is in between these Lf and Lg are called Taylor micro scales and they are defined in this manner 2 u 1 square divided by du 1 dash by dx 1 whole square raise to 0.5. So, you can see this as a length dimension square raise to 0.5 and therefore, Lf and likewise Lg would be based on u 2 prime square separated by distance x in x direction. The special derivatives are very very difficult to measure because in a turbulent flow to measure fluctuating velocity simultaneously at two adjoining points it turns out to be quite a difficult task and therefore, the gradients of fluctuations in x direction would be quite difficult. We will see how to get over that difficulty, but before we do that we will make some important observations. Lf and Lg in Lf and Lg the derivatives of difficult to measure that is correct. Nonetheless, if these local spatial change is imagined to have been caused by the smallest scales of motion then Lf and Lg can be regarded as the average dimensions of the range of small scale motions because they are close to r equal to 0 and therefore, they can be considered to be range of small scale motions. Similarly, if we integrate Fr from 0 to infinity we will get integral scales L int f and L int g and thus we have four length scales Lf the micro scale in longitudinal direction, micro scale in the transverse direction, integral scale in the longitudinal direction, integral scale in the transverse direction in a simple homogeneous isotropic term. Besides of course, L epsilon at the smallest colmograph scales where viscosity kills turbulence and isotropic prevails. How do we estimate L epsilon? Recall that in slide 9 I showed that colmograph related L epsilon to new cube kinematic viscosity cube divided by epsilon raised to 0.25. Therefore, L epsilon can only be estimated if we can estimate epsilon the magnitude of the rate of kinetic energy dissipation. It can be estimated by noting that in isotropic turbulence and there is a wonderful book by Heinz. It is called turbulence and introduction to its mechanism and theory published by McGraw-Hill in 1959 where some properties of isotropic turbulence have been given. One of them is that du 1 dash by dx 1 whole square would equal du 2 dash by dx 2 whole square, but it would only be equal to half of du 1 dash by dx 2 whole square equal to du 1 by 2 du 2 dash by dx 1 whole square and so on and so forth. So, remember rho epsilon is actually equal to mu times 2 du 1 dash by dx 1 square plus 2 du 2 dash by dx 2 whole square plus 2 times du 3 dash by dx 3 whole square plus 2 times du 1 dash by dx 2 plus du 2 dash by dx 1 whole square plus du 1 dash by dx 3 plus du 3 dash by dx 1 whole square plus du 2 dash by dx 3 plus du 3 dash by dx 2 whole square plus d x 2 whole square. Now, we showed that all these will be equally in isotropic turbulence and therefore, I have mu times let us say 6 times d u 1 dash by d x 1 whole square and then, but d u 1 dash by d x 1 is equal to half times d u 1 dash by d x 2 whole square and therefore, you will see here d u 1 dash by d x 2 square. This will become d u 1 dash by d x 2 whole square plus d u 2 dash by d x 1 whole square plus 2 times d u 1 dash by d x 2 into d u 2 dash by d x 1 as the first term and likewise, there will be second term and third term, but if I make use of this relationship, then you will see this will become 2 times d u 1 dash by d x 1 whole square and then, d u 2 dash by d x 1 square would again become equal to 2 times d u 1 dash by d x 1 whole square and this would equal 2 times d u 1 by dash by d x 2 will be 2 times square root into square root into d u 1 dash by d x 2 square and therefore, you will see this is nothing but 2 plus 2 into 4 that is 2 plus 4 plus 4 is 8. So, I will get 8 times d u 1 dash by d x 1 whole square from this likewise, I can show I will get 8 times d u 1 dash by d x 1 square from these also and this is d u. So, I get essentially 3 into and therefore, all this will become 24 plus 6 is equal to 30 mu times 30 d u 1 dash by d x 1 whole square and that is what I have shown here that rho into epsilon remember this definition L f is equal to 2 times u 1 dash square over d u 1 dash by d x 1 square and that is what I have shown. So, epsilon would become essentially 15 times nu d u 1 dash by d x 1 whole square is equal to 30 times nu u 1 dash whole square by L f and likewise 15 nu times u 2 dash by L g whole square. This is how one estimates epsilon provided we know this quantity and this quantity we can also estimate L f. To estimate now integral length scale consider homogenous pure shear flow in which the strain rate S i j of the mean velocity gradient is constant. In the turbulent kinetic energy equation all spatial gradients of product quantities would vanish, but the mean quantities would survive and so would this survive, but that would go to d term will go to 0 b term will go to 0 and production and dissipation would survive. Then from turbulent kinetic equation and assuming steady state we would have minus rho u prime u j prime mean velocity gradient equal to tau dash i j d u dash by d x j which is equal to rho times rho into dissipation and or in other words production will exactly equal to equal dissipation. This is called equilibrium state. Another way of writing this is u i j u prime j equal to S i j by 2 remember equal to nu times remember this is tau dash as mu times strain rate S i j small S i j is d u i by d x j plus d u j dash by d x i whereas S i j is from the mean quantities by d x j plus d u j by d x i that is the mean S i j. So, in other words I get u i prime u j prime time average into S i j by 2 equal to nu times small S i j S i j divided by 2 and that is equal to epsilon because I have divided through by density. So, you get that as a very interesting result. Now, the left hand side of this equation is associated with large scale motion u i dash u u i dash S i j by 2 is really dimensionally because u i dash u j dash is essentially v dash capital V dash square divided by l integral essentially large scale motion because mean velocity gradients are involved and that would equal v dash cube divided by l integral and since that is equal to epsilon, epsilon would also be equal v dash cube by l integral. This is a very important result remember epsilon is associated with very small scale motion and action of viscosity and yet it can be estimated from the representative scales of the large scale velocity fluctuation and large scale integral length scale. Therefore, this is sometimes called the first law of turbulence that the ability to estimate epsilon i j or epsilon from large scale fluctuation velocities and integral length scale is a very welcome result because it helps us later on in economic computation of turbulent flow and this result is routinely used by modulus of turbulent flow equations. Another way of writing this same is that S i j, S i j strain rates of smallest fluctuating motion divided by the strain rates of mean motion would be v dash cube l in nu divided by v dash l in squared and that would equal v dash l in by nu or the turbulent Reynolds number formed from fluctuations of the mean and integral length scale. So, this is totally representative of the large scale. What it says is that and since R e t l in t is of the order of 100, it means that the strain rates of the fluctuating quantities at the smallest scales are much greater than the strain rates of the mean quantities like u, but the strain rates formed from u dash are much greater than these. Another way of saying is we can expect therefore that in terms of amount of straining, the small scale motions are totally again uncorrelated with the large scale motion. From the results of the previous two slides, we can now estimate and compare Taylor and Kolmogorov scales. So, time scale of t f would be L f by u 1 prime and that would equal 30 nu divided by epsilon and under root 30 t epsilon. That would equal under root 30 because nu by epsilon is square root of is really the time scale of the Kolmogorov scales. So, it shows that the Taylor micro scale time scale is under root 30 times Kolmogorov time scale in the longitudinal direction. Similarly, L f divided by L epsilon would be again that quantity. Time scale in the transverse direction would be root 15 times t epsilon. So, there is a considerable separation between time scales associated with Kolmogorov scales and the micro scales, but not as big as what we observed between integral scales and the Kolmogorov scales. So, integral and Taylor scales are related as follows. Since epsilon is v dash cube by L in cube is equal to 15 nu times u dash square by L g. If I take u dash as about a times v dash, then it follows that L g by L in would be a root 15 under root nu by v dash L in equal to that where Reynolds number of turbulence is of the order of 100. Therefore, t g by t integral would be under root 15 by that. In other words, the transverse micro scale would be much smaller than the integral scale and the same thing would apply also to the longitudinal time scale in comparison to integral time scale. The only difference being 15 would be replaced by 30. T epsilon by t integral as we observed is Reynolds t integral raised to minus 0.5. So, in summary we can say L epsilon is less than L f and g and is also less than L integral. Although the distance between this and this would be considerably smaller than the distance between this and this and the separation distance overall separation distance would be determined by the Reynolds number. Same story applies to the timescales. The Kolmogorov timescales would be much, much smaller than integrals timescale, but just about small from compared to the micro scale, Taylor micro scales. So, now we have discovered that there are three scales, the middle one being representative. As I said, spatial correlations are very difficult to estimate and therefore, that makes L f g very difficult to estimate. In order to do that, we undertake measurement of what is called as the autocorrelation in which as I said we consider the same point, but separated by let us say this is u 1 dash at t and this will be u 1 dash at t plus delta t as separated by time and we would define exactly in the same fashion as we defined the spatial correlation coefficient and this is what I show here. The value of u dash at t at the same point x k and the value of u dash j at t plus delta t this is essentially b i j and u i dash squared which is b i i and u dash j squared. This will be at t plus delta t and this would be at t and then again if I plot values of r i j for different values of delta t the separation time, then I would get a perfect correlation of course, when delta t is 0, but it will go on declining as I go on increasing delta t and beyond a certain delta t of course, the fluctuations at the later time would be completely uncorrelated with the fluctuation at time t equal to 0. So, like what we did earlier I can estimate a micro time scale tau micro associated with L F g as that equal to u dash i into d u i dash by d t at delta t equal to 0 raise to minus 1 taking advantage of the fact that the variation here is very nearly parabolic and therefore, the projected delta t or tau micro would be that likewise the integral time scale would be somewhere there where 0 to infinity r i j d delta t. Now, we can get an idea of what should be the smallest magnitude of t max required in Reynolds averaging in Reynolds averaging the remember phi cap average equal to 1 over t as tending to infinity 0 to infinity phi cap d t that is what we said and the really the for practical engineering measurements we would of course, need the estimate of epsilon. It has to be some finite time because we cannot go on measuring in for infinite time that estimate is now available all it says is that this from this expression we say continue for t max to be much greater than tau integral. This would ensure reasonably that phi dash bar would be equal to 1 over t max integral phi dash d t 0 to t max would be equal to 0 that is that is the meaning of the importance of the first importance of autocorrelation. There is also another very important thing as I said it is not possible to measure spatial gradients of u i dash to estimate r i j the spatial correlation coefficient. So, the time derivatives of fluctuations at a fixed point however, are easier to measure with a single instrument like a hot wire can enable us to measure that as a function of time and Taylor made a hypothesis that if mean u 1 is very much greater than u 1 dash which is usually the case then d 1 dash by d t can be taken as minus u 1 into d 1 dash by d x 1 which gives us the estimate of d 1 dash by d x 1 required to estimate l f the longitudinal Taylor micro scale. And therefore, we can say that r 1 1 x 1 d x 1 would be equal to u 1 1 r 1 1 delta t d t or in other words l integral would be u 1 times tau integral that is given here. Now, this is a very very important deduction there are two important deductions from autocorrelation. First of all they are much easier to measure than the spatial correlation the autocorrelation gives you the idea what T max should be usually 4 to 5 times the integral tau in is taken as a in practical measurements. And then we would we are also now able to estimate the spatial correlation coefficient and therefore, estimate the l integral l integral from tau integral which as I said is much easier to measure. So, my final comment on these two last two lectures would be that we have shown how turbulence once generated sustains itself by creating fluctuations of ever smaller and smaller length and time scales. This was shown firstly by observing terms in the kinetic energy equations. Secondly from transverse momentum transfer processes in a boundary layer where we got reasonably good idea of separation between the dissipation scale or viscosity affected scales and the large scale. And then thirdly we did the scale analysis we have also shown that although epsilon is associated with very very small scale motions magnitude can nonetheless be estimated from large scale characteristics of large scale motion. This factor is extensively used in turbulence modeling of Rans equations. The length and time scales of eddies are easier to measure from autocorrelation usually most people measure autocorrelation and from that derive the spatial correlation coefficients. Energy from mean motion is somehow transferred down down down to very small scales where viscosity takes over and kills turbulence. This story we have tried to understand in physical space with physical measurements what can be done with physical measurements. But by transforming equations in the wave number space it is possible to elucidate this story even more convincingly. That is called spectral analysis although the equations generated cannot be solved in physical space unless they are brought back again in the physical space. The equations in the wave number space are difficult to solve but nonetheless it reveals a story of what really goes on most likely going on in sustaining turbulence. Which are the terms which actually carry out energy production turbulent energy production. What is the role of the redistributive terms that on the cross section vanish and what is the contribution of the dissipation dissipating motion. That story I will take up in the next lecture where I will explain what is spectral analysis. There is also another possible explanation of this transfer process which can actually be shown figuratively by imagining stretching and torturing of an fluid element by vorticity dynamics equations. I will try to show you how both these actually tell the same story that we have revealed already through equations in the physical space and measuring capabilities in the physical space.