 In the previous lecture, we considered how we can treat the outer layers of a turbulent layer by differential equation or quantities called characterizing turbulence. We employed the wall function approach, so that the inner layer would not be computed at all. Instead, we would use the consequences of the inner layer that is the universality of the velocity profile there and apply the boundary conditions for kinetic energy and dissipation at the first node away from the wall. The node distance would be so chosen that the y plus of that node would be somewhere between 30 and 100. This kind of modeling technique proves very useful for moderately pressure gradient flow. There are situations where even the inner layers are influenced by situations or conditions far away from the wall. Such situations arise in highly strongly swirling flows or when a jet impinges on a flat surface and very strong strain fields develop. In such situations, one needs to go with this modeling technique even closer to the wall that is right at the wall and the boundary conditions will then be given at the wall itself. We are going to examine how to capture low turbulence Reynolds number region in turbulence models. That is the topic for today. Let us see how we go. I would first describe the low Reynolds number turbulence Reynolds number 2 equation model where the turbulent viscosity is calculated from kinetic energy and dissipation equations. Then the other approach as I said is not the ad viscosity model, but to actually solve the differential equation for Reynolds stresses. Again, there are high turbulent Reynolds number stress equation models and low turbulence Reynolds number stress equation models. Computationally, these two are very, very turn out to be quite expensive. Therefore, a shortcut approach is often adopted. It is called the algebraic stress equation model. This would complete our discussion on how to model a stress. Likewise, we must also adopt a strategy for modeling turbulent scalar fluxes. Again, there are the ad diffusivity models and the turbulent flux models analogous to the ad viscosity model and the stress equation model. Finally, I will show couple of slides on how to model interaction between combustion and turbulence. Let us consider the low turbulence Reynolds number forms of kinetic energy equation and the dissipation equation. We are going right up to the wall. The definition of epsilon is changed to epsilon star now, where epsilon star is defined here. Epsilon star is equal to the isotropic epsilon minus 2 nu d e raised to half d x i square. I will explain why this change is necessary. Then, there are the turbulent production terms and the dissipation term of the modified dissipation and likewise here also. But, the constants now are changed, are sensitized to distance to low turbulence Reynolds number regions. For example, one such term which is sensitized to distance from the wall and low turbulence Reynolds number region is 2 nu mu t d 2 u i d x d x l. The reference to all this will appear shortly after a few slides. The turbulent viscosity now is c d star raised to rho epsilon square divided by epsilon star and c d star is now the c d which was 0.09 in the higher Reynolds number form exponential of minus 3.4 1 plus r e t by 50 whole square. C 1 star is taken as c 1 itself and c 2 star is c 2 into 1 minus 0.3 exponential of r e t square. Epsilon star itself as I said is this quantity. Now, as you will see when r e t becomes large that is as we move away from the wall, all these constants would be rendered same as those that were used in the high turbulence Reynolds number model. The model constants and this is the reference Jones and Launder international general of Heading mass transfer in 1972. So, the model constants are sensitized to low turbulence Reynolds number region near the wall. They tend to height r e t values beyond the sublayers. The correction to c d is chosen to give a value of nu t in agreement with van der Roest mixing length formula that is the this correction or the damping factor accounts for what was observed by van der Roest. You will recall that we had determined c 2 from decay of turbulence behind a grid. Now, in the early part at small times the kinetic energy decays again as t raise to minus n, but n would be then of the order of 1.1 to 1.2, but at large times it becomes 2.5 to 2.8. C 2 is again determined exactly from the same procedure as before, but the value of C 2 is corrected as shown here. It is sensitized to turbulence Reynolds number. The correction to epsilon is introduced to account for non isotropic contribution to dissipation. As we will recall we said that as one moves close to the wall, the amount of dissipation in different directions is actually different. This term epsilon star epsilon minus 2 nu e raised to half accounts for that effect of non isotropic. Since, we are able to now go right up to the wall, wall functions are no longer necessary. In other words, we are not going to use the universal law of the wall at all. E and epsilon equations can now be solved with e wall equal to epsilon star wall equal to 0. To capture the effects of low Reynolds turbulence Reynolds number, in the region between if you recall this was the wall and in higher Reynolds number form we had chosen p over here. So, only one node to take care of entire 30 y p plus 100, but sorry 100 and now you will require to capture very sharp variations of velocity and other thing as many as 60 to 100 nodes and this makes computation. If it was a simple boundary layer, two dimensional boundary layer then it is not into with today's desktop computer is not a too much of an effort, but supposing we had the internal flow with two walls or three walls or whatever, then you will see you require 60 nodes here, 60 nodes here, 60 nodes here or a minimum 60 amine and then there are the nodes required in the core of the domain and that makes the number of nodes required for computation very, very large and such low turbulence Reynolds numbers models in as much as their validity is far greater than that of the high turbulence Reynolds number model tend to be very expensive and therefore, one needs to be little careful in using these models that means, we use them only when they are absolutely essential. So, extremely fine mesh usually greater than 60 grid nodes would be required in the y plus less than 100 region. So, that completes the discussion on ADB Scorsery models, we now turn to the stress equation models. So, the six transport equations for one point correlation u i prime u j prime are derived from the equation for b i j which we had discussed in 23, lecture number 23 in which the gradients with respect to separation distance psi k is set to 0 because we are now looking for one point correlation and only differentials with respect to x k would survive. This is the convective transport of one point correlation u i prime u j prime, this is the production rate of the stress, this is the diffusion rate of the stress and this is called the pressure strain term and I will explain the meaning of that in a minute and this is the dissipation or destruction of the stress itself given by epsilon i j and it would be different in different directions. Now, if you notice this equation is a differential equation mainly because of the convection terms and diffusion term, the rest of the terms are simply sources, the production term, the pressure strain term and the dissipation terms are simply sources. We will make use of this fact later in deriving the algebraic stress models where we will point out that these terms which make the equation a differential equation can actually be simplified. So, invoking the idea of local isotropy at high turbulence and also remember the destruction rate is equally distributed among all its components and hence epsilon i j is taken now as two thirds epsilon delta i j where epsilon is obtained from its equation, the epsilon equation. The pressure strain term acts in two ways. Firstly, it sustains the division of turbulent kinetic energy into its three components u i square and secondly, it destructs the absolute magnitude of the shear stresses. So, it works in two ways redistributes and destructs. Complete discussion of the pressure strain term would be quite lengthy. So, without further elaboration I will simply say how the final form of the pressure strain model looks like. So, the minus p s i j is equated to difference between a stress minus two thirds e delta i j. C p 2 likewise is production term minus p i i by 3 where C p 3 is p dash i j minus two thirds p delta i j and C p 4 is e times d u i d x j plus d u j d x i plus p s w. Now, this is called the wall reflection term and it is modeled in this fashion. P dash i j which appears here is the turbulent stress production term. So, let us see next slide how these terms get further simplified. So, this algebraic expression for p s i j is derived from its exact equation by Hengelich and Launder in an all stress model of turbulence and application to thin shear flows published in general fluid mechanics in 1972. The term containing C C p 1 is called return to isotropic. There is always a tendency for a stress to return to its isotropic and the difference between this quantity minus the two thirds of kinetic energy. Remember, when it is perfectly isotropic u i prime this should be u j prime, u i prime would be exactly equal to two thirds of e and therefore, it is simply in effect it is called the return to isotropic term. The p s w term is called the wall reflection term which accounts for the effects of pressure fluctuations or reflections from the wall as you can see in the pressure fluctuation term. Now, that becomes important close to the wall where the pressure fluctuations give the effect which is primary responsible for the presence of the wall. The recommended constants are C p 1 equal to 1.5, C p 1 dash equal to 0.12, C p 2 all these were constants were either taken derived from experimental data or tuned by numerical experiments. The l b the appearance of l b here is really the distance from the wall. If you had a situation where there are say more than two walls present then one would take the minimum of the two walls to take care of l b. The triple velocity correlation u i prime, u j prime, u k prime in the diffusion term d u i j is modeled from its exact equation. Again it is possible to derive an exact equation for this quantity from instantaneous forms of Navier-Stokes equations. Again it is a matter which I am not going to discuss how what the equation looks like, but it is good to remember that the equation for u i prime, u j prime gives the presence of u i prime, u j prime, u k prime that is the triple velocity correlation. Likewise an equation for u i prime, u j prime, u k prime would give rise to four velocity fluctuations terms time average of u i square, u j u k or u j square, u i u k and so on and so forth. So, this is the problem with the statistical approach to deriving equations for turbulence because there will always be terms of a order higher that need to be modeled and that is the closure problem and this is what presently the closure applied is that the triple velocity correlation is made proportional to double velocity correlation and its gradients as you can see here and C s is 0.08 to 0.11 from numerical experiments. So, the whole process of modeling a stress equation is quite involved, but nonetheless at the end of the day it produces very useful differential equation which in fact can be solved in the manner of all other transport equations. Now, the implementation of the stress equation models requires solution of six differential equations for u i prime, u j prime. It requires two equations for E and epsilon that is the kinetic energy and dissipation coupled with three Rans equations that is the original momentum equation and this makes it a fairly a formidable problem because six plus two for scalar equations and then three that that is six plus two plus eight plus three eleven equations and in a three dimensional flow this can be really really very formidable problem. The model forms presented above show that the spatial gradients of u i prime, u j prime occur only in the diffusion and convection terms and these terms make the equations differential. This is the observation we made earlier. So, the one way in which to get rid of this differential form is to say we will develop an algebraic stress model by using the idea that we can say that a u i prime, u j prime divided by E is approximately equal to convection minus diffusion of the stress divided by convection minus diffusion of the kinetic energy and that would reduce this right hand side to minus 2 by 3 1 minus C p 1 delta i j plus p by epsilon divided by p by epsilon minus 1 plus C p 1 multiplied by f here and f is equal to 1 minus C p 2 p i j by p minus C p 3 p dash i j by p minus C p 4 E by p into the strain rate. So, in other words you can now model the stress divided by kinetic energy by means of an algebraic expression though long it nonetheless reduces the amount of effort involved. So, in we need not solve the six stress differential equations for stresses instead we use simply the algebraic expression for stresses and that reduces the computational time considerably. Now, like the higher Reynolds number form of stress this is the higher Reynolds number form of the algebraic stress model. There is also a low Reynolds number form of the algebraic stress model and that is given as u i prime u j prime equal to minus two thirds E delta i j plus E multiplied by f and f here is a significantly long term it involves C p strain rates S as well as vorticity rates omega. So, simply accept for the moment that this model has a basis which I am not going to discuss here, but nonetheless it allows for the fact that the stress a turbulent stress is not only related to its strain rate, but is also related to its vorticity rate and that is what the next slide will show. So, omega i j is d u i d x j minus d u j d x i and S i j is equal to d u i d x j plus d u j d x i as we all know. Turbulent viscosity mu t is f mu C d star E square by epsilon star and f mu is sensitized in this way for the low turbulence and also number region. C d star on the other hand is sensitized to the maximum of the mean strain or the mean vorticity multiplied by a damping function and here S bar the mean strain rate is taken as E by epsilon star square root of minus 0.5 S i j S i j and mean vorticity is taken as E by epsilon star square root of omega i j omega i j. So, the constants that were introduced earlier C 1, C 2, C 3, C 4, C 5 are given here C 1 is minus 0.1, C 2 is 0.1, C 3 is 0.26 all these constants were derived from numerical experiments by Kraft, Launder and Suga at university of Manchester in 1996 and the model is tested for very complex strain fields like swirling flows, curved channels and jet impingement on a wall. Now, the great advantage is this model is algebraic and therefore differential equations for stresses need not be solved, but at the same time it captures the effects of the low turbulence and also number regions which are very important when you have very highly strained velocity fields and that is the great advantage of this model and as I said this model has been used to predict quite accurately highly swirling flows in ducts in curved channels and in the situation of a jet impingement on a flat or a curved wall. So, that completes our discussion on high and low Reynolds number forms of turbulent stress. Now, we turn our attention to turbulent flux and like you will recall in lecture 21, we had said that we begin by looking at the instantaneous form of the energy equation and then time average it and then we get this is the laminar flux and this would be the turbulent flux and the task now is to model this term rho m C p m u dash j t dash as the main turbulent flux term this is the turbulent counterpart of viscous dissipation and this is the turbulent dissipation itself. Now, usually as we know this quantity is much much smaller than that quantity. So, how do we model this term again we can adopt like the stress we can adopt the eddy diffusivity model or we can look for a transport equation for u j dash t dash. So, eddy diffusivity model goes like this analogous to mu t we define turbulent thermal conductivity k t. So, that minus u i prime u t prime would be written as minus k t times rho C p d t by d x i equal to alpha t the turbulent diffusivity d t by d x i which we can also write as mu t the kinematic viscosity turbulent kinematic viscosity divided by turbulent Prandtl number d t by d x i and fortunately the turbulent Prandtl number turns out to be an absolute constant about 0.9 when the turbulence Reynolds number is high. If you go close to the wall of course, you have to make turbulent Prandtl number a function of the turbulence Reynolds number itself. So, the energy equation would now look like d t by d t equal to d by d x k nu by Prandtl plus nu t by Prandtl t into d t by d x k plus q j n by rho C p where this is the generation term and as I said rho m epsilon usually is much smaller than the viscous dissipation term. So, what are the comments on the eddy diffusivity model? The model is very convenient because nu t is obtained from mixing length or one or two equation models of turbulence and Prandtl t is an absolute constant. So, everything is very well known and one can solve the turbulent energy equation in the manner of a laminar flow. The disadvantage however is that when nu t is equal to 0 alpha t will also be 0. Now, just imagine a situation let us say we are talking about a flow in an annulus let us say and this is the axis of symmetry then you will get a velocity profile which will be something like this and therefore, nu t will go somewhere equal to 0 over there, but now let us say this is hot and this is cold. So, we expect a temperature profile to be something of this type sorry not like this the temperature profile to be something of this type and clearly there would be temperature gradient here and which would cause the heat transfer in that direction or I mean sorry in that direction whereas, alpha t would be predicted 0 here and that is not acceptable to get practically correct predictions. So, in such situations the model really fails. So, but the advantage is like nu t alpha t is also isotropic again measurements of decay of non axisymmetric temperature profiles in a fully developed turbulent flow in a pipe suggest that very close to the wall of the tube the tangential eddy diffusivity is actually greater than the radial diffusivity. So, actually the turbulent eddy diffusivity itself is not isotropic as one moves close to the wall and that fact cannot be captured by the eddy diffusivity model and therefore, it is in general u i prime t prime must be obtained directly from its differential transport equations. So, like we said after realizing the limitations of eddy viscosity model we went to the stress equation model we would now go having realized the limitations of the eddy diffusivity model you would go to turbulent flux model. So, the equation for u i prime t prime is derived by multiplying equation for the instantaneous t by u i prime and equation for instantaneous u i by t prime the turbulent fluctuation adding and time averaging. Now, this process gives rise to a convection of u i prime t prime production of u i prime t prime and diffusion of u i prime t prime firstly by pressure fluctuations and secondly by velocity fluctuations this is the diffusion of the turbulent flux due to laminar diffusivity. This is the again like the pressure strain term this is the redistribution term and then there is the dissipation of u i prime t prime. So, again you get very similar terms to those you had up one had seen in the stress equation model like the pressure strain term the redistribution term is model as minus c t 1 epsilon by e u i prime t prime plus c t 2 u u dash k d t t dash d u i d x k. This should be in the bracket whole thing should be in the bracket equal to and this is further written in this manner for Prandtl greater than 1 and in this manner for Prandtl very very small that is liquid metal heat transfer then that is how the term is represented. At high r e t or Peclet number the task of redistruction term is performed by r e d t r d t and therefore usually the destruction term that we mentioned here is taken to be 0. In diffusion term effect of P dash is either neglected we either neglect this term altogether or take it to be some fraction of that term and finally the u i prime u k prime t prime is model as c t into stress multiplied by the gradient of the flux in the different directions. The model constants are c t 1 3.6 c t 2 0.266 and c t equal to 0.11 the required correlations are taken as from the eddy viscosity model wherever required and for complete range of Prandtl numbers Prandtl t is model in this fashion. So you will see that if Prandtl number was say of the order of 1 then this would be simply 2 and this term will be about 0.61 when added to that will give you about 0.92 or 0.91. But for very large Prandtl numbers the whole term would simply be about 0.88. So for Prandtl number greater than 1 Prandtl t equal to 0.9 can be taken to be fairly good representation. The term really makes a contribution for liquid metal heat transfer when the Prandtl number will be out there of 0.02, 0.03 and so on and so forth. Then the turbulent Prandtl number can exceed 1 itself. Like we derived the algebraic stress model, we can also derive algebraic flux model but to do that we need to first of all derive an equation for turbulent temperature fluctuations and that is shown here. So one can say t prime squared by 2 is really sorry there should be a bar on top is really some kind of a kinetic energy of the fluctuations if you like of temperature equals diffusion of the same quantity into laminar diffusion and turbulent diffusion the production of the same quantity and its destruction or the dissipation. Now it is this quantity which is usually model as being proportional to E by epsilon t prime squared that is the turbulent that is the dissipation rate of the turbulent temperature fluctuation is more or less proportional to E by epsilon t prime squared. So that term would be that E by epsilon t prime squared and now we can say that like the algebraic stress model we can say algebraic flux model we would postulate that the convection minus diffusion of the flux would be taken as production minus dissipation of the kinetic energy plus production minus epsilon of t prime squared divided by 2 some kind of averaging of the kinetic energy and turbulent fluctuation squared into u i u i dash u u t dash divided by E into square root of t prime squared. This is pure pragmatism in order to obtain an equation for u i prime an algebraic equation for u i prime u t prime and t prime squared would be C t dash E by epsilon u i prime t prime d t by d x k from production equal to dissipation production equal to dissipation that is ignoring the convective and diffusive part we would get that. So, one substitutes for t prime squared here minus dissipation gives you this quantity and therefore, one can obtain u i prime u prime all these quantities are known very well from these two expressions C t dash is taken as 1.6 for parental greater than or equal to 1 for liquid metals things are somewhat little more difficult and therefore, not discussed here, but algebraic stress model algebraic flux model again saves considerable computational time. Now, there is evidence from direct numerical simulation of heat transfer or the turbulent fluctuation equation in pipe flow. So, the computations were made for Prandtl number 0.1, 0.7 and 2 and I am showing here the inner region of less than 100. So, first of all a comment is in order to say that the mean t profiles for pipe flow agreed with the d n s data. So, therefore, we can say that at least the time average temperature profiles were in agreement with the measurements made at these three Prandtl numbers. The left figure shows that the peak of t prime squared shifts towards larger y plus as Prandtl number decreases as you can see and this is very much true that the production of the turbulent temperature fluctuations are closer to the wall at high Prandtl numbers than they are at low Prandtl numbers. These two are the budgets of this is the budget for u prime t prime, which looks very similar to the budget we had seen for turbulent kinetic energy and v dash t prime budget resembles the u dash v dash budget or the shear stress budget we had seen in the earlier and which justifies the ad diffusivity models. Lastly, I turn to the topic of interaction between combustion and turbulence. So, in combustion it is necessary to solve for reacting spaces and product spaces that are postulated in a reaction mechanism and the equation reads like this. For each space e k the mass fraction omega k convection of that into equals diffusion of that plus the rate of production or destruction of that depending on the sign of r k. So, d effective like mu alpha effective is the effective mass diffusivity and it is taken as nu divided by Schmidt number plus nu t divided by turbulent Schmidt number and at least in gaseous flows it is quite close to 0.9 like the turbulent Prandtl number. Now, the simplest postulate of the reaction mechanism is the simple chemical reaction SCR, which is written as 1 kilogram of fuel plus r s t kilogram of oxidant equals 1 plus r s t kilogram of product. So, essentially we have three spaces fuel, oxidant and product without defining what they are and r s t will be the air by fuel stoichiometric air by fuel ratio. Then the rate of consumption or generation of oxidant air would be r s t times the rate of consumption or destruction of fuel r product would be under the hand would be minus 1 plus r s t times r f u. Now, if it was a laminar flow r f u would be given by minus a exponential of e by r u t omega f u m omega ox n, where a e m and n are fuel specific constants. In a turbulent flow what we would like to do is to simply replace this omega f u by time average omega f u bar and omega ox bar, but there are difficulties and that is what we want to appreciate on the next slide. So, in turbulent combustion it is observed that the outer edges of the flames are very intermittent and jagged. Now, it is in this regions that really greater part of combustion takes place. Experimentally it is observed that even if the time average omega f u bar and omega ox bar are high, the actual r f u rates are not as high as would be expected from the Arrhenius formula that I showed on the previous slide. So, if you take bar values here, we do not get the value of r f u predicted turns out to be much greater than actually what is observed in the experiment. Now, why does this happen? This happens because the fuel and the oxidant at a given point are actually present at different times because of the turbulence velocity fluctuations which actually transport the fuel and although the time average values may be high, the actual reacting fractions tend to be smaller. Clearly therefore, the timescales of chemical reaction and turbulence must be considered simultaneously. Now, of course, the subject matter governing these issues is quite large, but here we would simply say that these timescales are S L by u dash in the RMS sense where S L is the laminar flame speed of the fuel under question. This ratio determines whether the reaction is dominated by combustion timescale or whether it is dominated by the turbulent timescales. Now, these ideas are captured in a model suggested by Spaulding. It is called the development of the eddy breakup model of turbulent combustion presented at the 16th symposium on combustion in 1976 and Spaulding said that instead of the Arrhenius form that is used here, what one ought to use is r f u equal to minus c, a constant into rho m into scalar fluctuation omega dash f u whole square which in practical computation is taken to be almost equal to omega f u bar. It makes very little difference, only thing is you must adjust cebu appropriately. As much as this model turned out to be quite good for turbulent combustion, some more refinements have been made and it is said that instead of taking omega f u bar that you saw here, what one should take is really the minimum value of a f u bar omega ox bar by r s t omega product bar by 1 plus r s t with certain proportionality constants and recommended ones are a equal to 4 and a dash equal to 2. This is simply again pragmatism derived from experiments and has proved to be quite good at least in the certain types of combusting flows. So, in the next lecture we shall discuss how two important aspects of turbulent flows that is the laminar to turbulent transition and the effect of wall roughness.