 In the previous lecture, we made two important observations. One was that the characteristics of turbulent flow are significantly different from those of laminar flow and secondly, turbulence once generated say inside a pipe somehow sustains itself right through the length of the pipe. It does not die away which means there must be some pumping action which feeds turbulence continuously to sustain itself while viscosity is trying to kill it. This lecture is the first of the two lectures which will try and explain how turbulence sustains itself. We will do this firstly by some orders of magnitude analysis by looking at and deriving actually the turbulent kinetic energy equation. Then, we will do scale analysis in which we will introduce the idea of length and time scales of turbulent eddies through spatial and auto correlation coefficients. The first question that arises is that turbulent fluctuations are extremely random and sharp as I showed you in the previous slide. Could it be so that these fluctuations actually split the fluid? For example, if I had a paper in my hand and if I stretch it and subject to very random motion, it is quite likely that the paper will actually split. Will the fluid split? That is the question because of stretching and torturing by the by the vortices. Can the fluid actually split? The question can be answered by considering orders of magnitude. So, the first of all in turbulent flows scales of velocity fluctuations vary from as high as that of the mean flow say in air it could be anywhere up to 1 meter per second to very low scales that are governed by the presence of this molecular viscosity. The associated length scales would vary from as high as the mean flow dimension say boundary layer thickness or radius of a pipe to a very small fraction of these quantities. These scales are associated with the turbulent fluid. Let us ask ourselves what happens at the molecular level. So, for example, molecular velocity in air would be of 50 meters per second much greater than 1 meter per second and the mean free path length would be of the order of 10 raise to minus 4 millimeters of delta and r and delta and r would be of the order of 1 millimeter to let us say 1 centimeter onwards. So, the velocity scales of molecules is much much greater than the mean flow velocity scales whereas, the length scales are much much smaller than the mean flow scales. Similarly, turbulence frequencies of fluctuations are of the order of 10 raise to 4 whereas, the molecular frequencies are of the order of 5 billion thus there is a vast difference between what happens at the molecular level and what happens in practical turbulence. You can see what this vast difference implies is that the two must be completely uncorrelated. Turbulence behaves in its own way, molecules continue to behave in their own way and therefore, we could safely assume that turbulence does not in any way destroy the basic characteristic of a fluid that the fluid will always remain as a continuum because of the presence of viscosity and no splits at the molecular level would occur. This is a very important observation about turbulence to make progress with theory of turbulence. So, the numbers of the previous slides suggest that the fluid viscosity will continue to influence events in turbulent flow in two ways. Firstly, by causing diffusion of the transported property, it can be momentum, it can be temperature or mass fraction or anything. Turbulence will cause because of viscosity causing diffusion and secondly, through dissipation of energy of the fluctuations to heat. Since, turbulent fluctuations are indeed killed by the action of viscosity and therefore, the continuum fluid continuum is maintained. In other words fluctuations are no longer allowed by viscosity to sustain themselves and they are simply killed by viscosity and therefore, the molecular behavior remains completely unaffected by turbulence. Having made this observation that the continuum is maintained, we now turn to the main point that the mechanism must therefore, exist that feeds energy from the main motion to sustain turbulence while viscosity kills turbulence. Study of this mechanism reveals that in vigorously turbulent flow, the diffusive role of viscosity is marginal that is the first one, but the viscosity plays its principal role through energy dissipation that the motions are killed and therefore, the kinetic energy dissipated. Now, this is in contrast to what occurs in laminar flow where the diffusive influence dominates over the dissipative one unless the fluid viscosity was very high as in oil flows that is something we have already studied while considering laminar flows. So, in order to explain whatever I have said in words through equations the Navier-Stokes equations, the first thing to appreciate that is that the Navier-Stokes equations written for an instantaneous velocity u cap are valid descriptions of turbulent flow that is the first thing because the continuum prevails all derivatives can be resolved for the instantaneous velocities and therefore, the equations are valid. So, that is the fundamental assumption on which we will proceed. So, for example, I can develop therefore, an equation for instantaneous kinetic energy this is the equation for u cap, u cap I square yes which essentially means u 1 square plus u 2 square plus u 3 square divided by 2 is the instantaneous kinetic energy is the instantaneous kinetic energy and how do I derive that? Well, recall that the instantaneous momentum equation is du cap I by dt equal to minus dp cap by dx I plus d tau ji cap dx j. If I multiply this equation by u I throughout u cap I and u cap I then you will notice this will be essentially rho times d u I u I cap divided by 2 by dt which is nothing but rho d e cap by dt and this will equal minus u cap dp cap by dx I plus u I cap d tau ji cap by dx j and therefore, rho times d e cap by dt would be equal to minus p absorbing u inside by dx I plus p cap d u I dx I plus again absorbing this inside d by dx j of u I cap tau ji cap minus tau ji d u I dx j this is what it will be but remember from continued equation which also applies the instantaneous velocities this term will be 0 and that is why you get this equation that is what I have written here d e by dt equal to d by dx I p by u I plus d by dx I of u j d by dx j minus mu phi v this is mu phi v the viscous dissipation term. Essentially what the equation says is the rate of change of instantaneous kinetic energy is the net rate of work done by pressure forces plus the net rate of work done by the stresses tau I j mu s I j s I j is the strain rate d u I dx j plus d u I j d x I minus viscous dissipation. So, turbulent energy increases because of these terms which are which can be positive or negative does not matter but when they are positive it increases but mu phi v by definition would being positive would always decrease instantaneous kinetic energy. So, viscosity plays the role of destroying instantaneous kinetic energy I do the same thing now to derive mean kinetic energy equation but in this case what I will do is I will begin by writing the Rans equations the Rans equations were d u I by dt equal to minus d p by dx I plus d by dx j of tau j I minus rho u I prime u j prime dx j plus the body forces which I am present in ignoring and these were the turbulent stresses. So, if I multiply this equation by u I throughout if I multiply this equation throughout by u I I would get again an equation for u I square by 2 which is the mean kinetic energy equation. So, here is that equation this looks very similar in many terms this is the pressure work term this is the work done by laminar stress this is the work done by turbulent stress and this would be the viscous dissipation due to mean velocity gradients and minus minus turbulent stress multiplied by d u I by dx j the mean velocity gradient. The equation essentially says that the rate of change of mean kinetic energy E is equal to the rate of work done by pressure forces rate of work done by viscous stresses rate of work done by turbulent stresses that is termed a minus the rate of energy dissipated by viscous action and minus this is the most important term F the rate of energy transfer to turbulence by mean motion d u I dx j. Now, why do I say rate of energy transfer to turbulence that you will appreciate from the next slide. So, now I want to derive an equation for the turbulent kinetic energy which is u I dash u I dash time average divided by 2 this is the turbulent kinetic energy is derived by first time averaging the instantaneous kinetic energy equation in other words this equation I time average each term then time averaging of this term would give me mean E plus turbulent E and so on and so forth and the equation would look as I have shown here the equation would look like this d rho d E E plus by d t plus d by d x j of u I rho u I prime u j prime plus d by d x j of rho u j u I prime u j prime u I prime a triple velocity correlation to appear equal to minus d p u I plus p dash u I dash plus d by d x j of tau I j I u I plus tau dash I u dash I minus tau I j d u I by d x j minus tau dash d u I dash by d x j where turbulent stress tau dash I j tau dash I j is mu times d u dash by d x j plus d u j dash by d x I this is the turbulent counter fact or the turbulent stress based on fluctuating velocity strain rates. So, from this equation which is time average form of the instantaneous kinetic energy equation I now subtract the mean kinetic energy equation which I derived on the previous slide I subtract this equation from our this equation then you will see that I would get an equation for rho d E by d t equal to minus d by d x j u I j j dash p dash plus this plus minus rho u I j prime u j prime d y d x j and then this term and this term. Notice that the C term here in the turbulent kinetic energy equation has exactly the opposite sign of the f term they are both identical terms, but in one case you have a negative sign here and this is the positive sign here you have the positive sign. In other what is loss of kinetic energy turns out to be gain of turbulent kinetic energy and what do these terms represent well that is what is shown on the left on the next slide rate of change of turbulent kinetic energy a which is the left hand side equals the rate of convective diffusion of total fluctuating pressure by velocity fluctuations to go back this is the term p dash plus rho u I square by 2 is the total fluctuating pressure this is the static pressure this is the dynamic pressure and therefore, the total term represents total fluctuating pressure and its diffusion due to velocity u dash j plus the rate of energy transferred from mean motion to turbulence by turbulent stresses which is the term C plus by the turbulent stresses it is the energy transferred to turbulence plus the rate of work done by viscous turbulent stresses this is the u j dash tau dash I j and I explain what the definition of tau dash I j is. So, that is the diffusion again of the stress itself or the stress work due to turbulent stress and finally, minus the rate of dissipation of energy by turbulent motion and that is the term e here is a product of fluctuating stress multiplied by fluctuating velocity gradient and therefore, this would be very much like the mu phi v term in which the phi v would now be formed from fluctuating velocity gradients equation for mean kinetic energy shows that e is lost in two ways firstly by viscous dissipation term e the mean energy is lost by viscous dissipation and secondly by work done by stresses on mean velocity gradients and e is lost in two ways firstly by viscous dissipation term e and secondly by term f which appears as a positive contributor to turbulent kinetic energy C term C. Hence, term C is called the production or generation term because it makes a positive contribution to rate of change of e. In laminar flow the mean energy is directly dissipated to heat in turbulent flow we can say that mean energy is first transferred to sustained turbulence before it is finally dissipated to heat through term e. So, the first mean kinetic energy goes to turbulence through some C which increases that but kinetic energy turbulent kinetic energy also decreases due to dissipation to heat before finally. So, this first and before seems to be there is a time lag or a space lag in a fluid flow whichever way we used to look at it. So, it is not an instant process it happens perhaps in stages now to do explain that we will have to make some further explorations into turbulence which I will take up in the slides to follow as well as the next lecture. So, remember mean energy is lost in two ways first by viscous dissipation and secondly by transfer to turbulence turbulent kinetic energy on the other hand is sustained because of this transfer from mean energy from the mean energy and secondly it is destroyed by the fluctuating counterpart of mu phi v which we call dissipation turbulent dissipation. So, beside dissipation and transfer mean kinetic energy and turbulent kinetic energy experience convective diffusion of energy through terms b c d b and these terms simply redistribute energy specially but make no contribution or zero contribution to integral energy balance as we would see now. So, for example, consider let us say flow between two parallel plates with an axis of symmetry and I shall now integrate I will now consider the kinetic energy equation. So, it will look like rho d e by d t equal to an assuming that the gradients of all terms are much bigger in y direction than they are in any other direction. We would write d by d y minus d by d y of v dash into p dash plus u dash square plus v dash square plus w dash square by 2 which is nothing but the kinetic energy minus or the plus minus rho u dash v dash into d u by d y plus d by d y of v dash into tau y x minus tau dash y x by d u dash d v dash by d y let us say where tau dash y x is mu times d u dash by d y dash. So, now if I integrate this from 0 to y that is over the volume of the channel then you will notice that this term will have v dash at the wall and v dash at the axis of symmetry and therefore, both of them would simply vanish. So, integral of that d v is simply 0 integral of this term will survive which we said was production term this likewise like this term this term will also vanish integral of whereas, this term integral of d v will survive because it is a product of velocity gradient and fluctuating stress that is what I show here. So, by wall and symmetry plane is considered to if the turbulent energy is bounded by wall where the velocity fluctuations are 0 because of no slip or by wall and symmetry plane tau i j tau dash i j is 0 is considered and equation for turbulent kinetic energy is integrated over the cross section then we would have d by d t e d v equal to net production minus net dissipation. What this tells us is following is that the net change in kinetic energy over a cross section would be positive when net production exceeds net dissipation. On the other hand if the net dissipation exceeded net production then e will simply die out the kinetic energy will simply vanish when there is near equilibrium that is production is very close to dissipation we would have transition. The equation then sets the conditions for sustainance of turbo that the net production over a cross section must exceed the net dissipation then you will the turbulent turbulence would be sustained at every cross section downstream there are situations in fact where even in a channel flow turbulence when generated can be made to made to relaminarize. One such possibility is to have a tube which is coiling around a steam with very high turbulent velocity fluctuations so that the dissipation term begins to dominate over the production term and the turbulent could then relaminarize inside a pipe but it is a coiled pipe very very special case not routine encounter in practical engineering when the production and dissipation are near balance you would expect a kind of flow in which for a while the flow is laminar when dissipation overtakes production and then the laminar flow will become unstable to produce in which production term to will take over from dissipation and little patches of turbulence and little patches of laminar fluid would appear in a flow and that is precisely what we call the transitional regime. Thus the turbulence derives its sustainance by drawing energy from the mean motion now how does this transfer actually take place that is what we want to ask how does this transfer take place now to understand that we must introduce the ideas of scale so in a laminar boundary layer for example is characterized by two length scales one is delta which is much much smaller the transverse dimension is much much smaller than the stream wise distance x and that gives us delta by x as being proportional to Reynolds x to the minus 0.5 and this is usually much much smaller than 1 so the relevant time scale however is t equal to x divided by u infinity and therefore if I substitute that in here I would say delta is proportional to nu t raised to minus 0.5 where nu is a small number therefore delta would be a very very small quantity again as shown earlier more importantly if you remember delta could be discovered only because of the inclusion of the transverse diffusion term mu d 2 u dy square in the laminar boundary layer equation. One way to interpret this fact is to say the smaller length scale delta is associated with the effect of viscosity with the effect of viscosity as shown here what about turbulent flow now in a turbulent flow turbulent boundary layer very close to the wall of course you have viscosity dominates viscosity affected region but the outer parts are certainly almost independent of viscosity independent of the effects of viscosity the laminar fluid viscosity. So, in this region if I were to say what brings about transfer of momentum well let us say it is a representative velocity fluctuation v dash mean let us say I call it v dash mean as the representative fluctuation which brings about a transverse momentum. Then you will see in turbulent boundary layer motions of several scales occur simultaneously and we are going to choose v dash mean as the representative velocity fluctuation in the direction of y away from the wall. Then the transverse momentum is carried out by minus rho u dash v dash time average which would be much much greater than mu d u by dy and d delta by d t that is the rate of growth of turbulent boundary thickness would be proportional to v dash mean essentially and therefore, delta would be proportional to v dash mean by t multiplied by t. But t as we observed would be x divided by u infinity even in a turbulent flow and therefore, v dash x by u infinity this would be v dash mean into x multiplied by mu. Thus the diffusion time scale delta by v dash mean would be approximately equal to mean time scale x by u infinity this is very interesting. The delta is a small length scale v dash mean is the representative fluctuating velocity in the turbulent core or the fully turbulent part of the boundary layer and the time scale associated with it is exactly same as the mean time scale and therefore, we shall regard v dash mean as a representative of the large scale motion. This is a very important idea that v dash mean would be taken as the representative of the large scale motion. Recall that the dissipation process mu tau dash d u dash i by d x j this was the dissipation process this I will represent as rho times epsilon where epsilon is called the dissipation rate of kinetic energy. Now, this actually kills turbulence rho into epsilon actually kills turbulence and smooths out velocity fluctuations due to action of viscosity and therefore, the length scales associated with it would be much smaller than the mean length scales and the time scales associated with it will also be much smaller than the mean time scale. At such very small scales of motion turbulent fluctuations in all three dimensions directions can be taken to be essentially statistically equal that is u prime square is equal to v prime square is equal to w. As well as the gradients will be 0 in other words the special variations will also be very small. When special variations of fluctuating time average quantities are 0 or this when the special gradients of the fluctuating quantities time average fluctuating quantities are 0 we say the structure is homogeneous and when the components of the velocity fluctuations are equal we say it is isotropic and then therefore, we would essentially have where the viscosity plays its dominant role we would have essentially a homogeneous and isotropic turbulent structure. It is characterized in association with epsilon by what are called Kolmogorov scales. So, Kolmogorov used the idea that very small scale motions are essentially characterized by the effect of viscosity and by the effect of turbulent dissipation and he brought in the quantity epsilon to represent the velocity scales of the associated with dissipation process as nu epsilon raised to 0.25 this is dimensionally correct. Similarly, the time scale T epsilon was taken as nu by epsilon raised to 0.5 and L by L sub epsilon was taken as nu cube by epsilon raised to 0.25 as the length scale. So, if I form Reynolds number based on length scale velocity scale I will get L v dash epsilon divided by nu and let us say it is of the order of 1 then it follows that it would be much much smaller than L v dash mean by nu which is the large scale motion and large scale length scale and that would be of the order of 100 or even more. Reynolds number associated with dissipative length scales and velocity fluctuation scale is much much smaller than the mean Reynolds number form from mean length scale and fluctuating velocity scales which would be of the order of 100 or more. Thus, we have provided relative estimates of the largest and the smallest scales. The largest scales belong to the mean dimensions, mean motion where at the smallest one belong to the dissipation scales. The most important to cut the story short the most important aspect of it is that whenever large scale fluctuations are present small scale motions are automatically created so that viscosity can play its major role via energy dissipation. The creation of this small scale motions is believed to be caused by the non-linear convective terms in the Navier-Stokes equations that this creation of smaller and smaller scales motion is not a one step process but takes place in a large number of continuous steps will be demonstrated shortly. It can be done in more than one ways and I will try to do it in as simple manner as possible. The large scale fluctuations thus create small scale fluctuations which in turn transfer their energy to produce even smaller scale fluctuations and so on till the scales are so small that non-linear terms become unimportant and viscosity takes over to produce an isotropic structure of turbulence. I mean that is the story that is the story of sustainance of turbulence. Now in order to explain these ideas little further it is customary to introduce the idea of a turbulence eddy. Now you can imagine that let us say I have two points in the flow and I consider let us say u dash here and v dash here at the same time t or let us say to begin with u dash and u dash itself. I consider the fluctuation in the x direction at the same time instant at two different points separated by a distance. Now we all know that if a fluctuation at this point will influence this point if they were close to each other. So, the fluctuation here would be influenced by fluctuation here. However, if this point was sufficiently far away say here then the u dash here will not be not influenced by u dash at x equal to 0. Let us say this is at x equal to 0 and this is the separation distance x 1 and this is the separation distance x 2 let us say. So at this point it is unlikely that u dash will sense what u dash at x equal to 0 is doing. In effect if I were to look at them in time at x equal to 0 then the fluctuation u dash would at x equal to 0 will look like that. At x equal to x 2 let us say they would look absolutely different and you can say that this form of u dash at x equal to 0 is completely uncorrelated with what is happening at large distance. What about intermediate distances? Here we can expect supposing x 1 which is very close to x equal to 0 I can expect something like that marginally good correlation. At least it will look somewhat similar but at very large distance it could be absolutely very very different. I can say therefore and of course if I took this second point to merge with this of course I will reproduce the same pattern which means a complete correlation exists between the two points when they collapse on one another. A complete disc correlation exists when they are very very far but a moderate correlation can exist for in between distances and of course we do not know what that distance x 2 will be in real turbulent flow and that is what we wish to find out. This is spatial influence but now if I take for example u dash at time t and u dash at time t and u dash at say another time t equal to t plus delta t separated by distance delta t time distance delta t then a similar situation would arise. If delta t was small I would expect reasonably good correlation between the two but if it was very very large then of course they would be completely uncorrelated. These ideas are expressed here in this figure where I show two points u dash 1 and u dash 1 at two points separated in x 1 direction and is define a spatial correlation coefficient as b i j under root b i i under root b i b j j where b i j is the u i prime u j prime at two different points but at the same time instant I do the same thing here for u 2 dash and u 2 dash separated in x 1 direction and then if I plot r i j from measurements of these quantities then it would be it would look like perfect correlation of 1 at separation distance equal to 0 and correlation would die out to 0 that means there is no correlation beyond some in a long distance. Similar thing would happen with respect to u 2 dash at the same time although it will go through a negative before going to 0 at infinity. This is called the longitudinal time scale correlation, this is called the transverse correlation and for example, if I were to integrate this over a over a long time of infinity then this would give me a a a length dimension a length dimension which would be representative the average dimension over which a fluctuation at a point is going to influence events that I would call as the integral length scale of the fluctuation or the spatial size of the eddy spatial size of the eddy in longitudinal direction. I can do the same thing for with respect to u 2 velocity and I would get a similar dimension in the transverse direction. I can say so I have estimated the size of the eddy physical size of the zone over which zone over which turbulence is going to influence events. Spatial correlation has 9 components R i j as defined there has 9 components in general and being a coefficient it would vary between minus 1 and plus 1 at these 2 extremes we say the correlation is perfect absolutely perfect because its magnitude is 1. When R i j is equal to 0 of course, no correlation exists between u i dash and u j dash which would understand the case when the separation distance r tends to infinity between 0 and 1 we say the correlation is moderate. It is tedious to measure R i j in a real non-homogeneous isotropic turbulent flow because 9 components must be measured in all directions for different values of separation distance r and the direction r 1, r 2 and r 3 and therefore usually only in the direction r 1 or x 1 is the only direction which is taken to measure whenever measured by and large correlation coefficients are extremely difficult to measure the spatial correlations are extremely difficult. I will stop here and continue with this lecture.