 In the previous two lectures, we considered the formal aspect of turbulence and asked the question how does turbulence sustain itself. We showed this process of sustainance through breakdown of eddies and we explained this process in three ways. One was scale analysis, second was spectral analysis and the third was the vorticity dynamics. Now we must turn to more predictive aspects. After all, we wish to be able to compute or calculate friction factor and Nusselt numbers in turbulent flows. So, I am going to turn today to regions of flow close to the wall. If one analyzes experimentally, then one would find an inner layer as well as an outer layer and there are some special features of this inner layer which make it possible for us to determine the friction factor and Nusselt number. I will also show how Prandtl's mixing length idea can be employed to predict velocity distribution in inner layer. Let us turn to the main postulate. In our formal aspect, we dealt with turbulent flows whose structure is dominated by large eddies. That is where the production takes place and the diffusive influence of viscosity was rather small being confined only to carrying out dissipation at the smaller scales. Near the wall however, viscosity also plays its role in bringing about diffusion and what is the size of this inner layer? To begin with, let me say approximately y divided by delta in a boundary layer would be of the order of 15 percent and likewise y divided by radius where y is distance from the wall in a pipe flow would again be of the order of 15 percent. So, we are talking about a fairly narrow region close to the wall about 15 percent whereas away from the wall where diffusive influence of viscosity is very small would be greater than 0.15. However, it is the inner region which is of great importance to us because the greatest resistance to heat and mass transfer occurs close to the wall where or in the region in which viscosity plays its dominant role. That is where the fluid flow is sluggish and likewise therefore, the heat and mass transfer is also very sluggish and therefore, we are very much interested in this inner region. Now, it is of course, a very fortunate occurrence quite an accident of nature really that the most significant characteristics of this inner region are almost universal and we are going to exploit this universality of the inner layer to predict friction factor and asset number. So, let us ask ourselves a question what are the characteristics of this inner layer? This is the flow against a wall, this is the free stream velocity u infinity and this is the total boundary layer thickness delta let us say. Then the inner layer which as I said is about 15 percent of the total itself comprises of three characteristic layers as I have shown here. The innermost layer is often called the viscous sub layer is almost laminar like because that is where the effect of viscosity is so great that all fluctuations are almost killed and you get essentially laminar flow. In reality however, this layer is characterized by repeated but infrequent fluid burst. What happens is the laminar sub layer here grows a little becomes unstable and weak and at this point that lumps of fluid from the outer layer hit the inner viscous sub layer and up root fluid out into the outer layer again or the outer parts of the inner layer and this kind of fluid being flung out from the sub layer is quite visible if you did flow visualization of a typical turbulent boundary layer. Then this region is very intermittent but when I say it is intermittent but infrequently so for all practical purposes to begin with we might say the region is almost like laminar layer. The next is the transitional layer which may may liken to the inertial sub range that we identified during our formal aspects of turbulence and that is called the transitional layer. Now here both turbulent fluctuations as well as fluid viscosity both are equally dominant and then there is the fully turbulent part of the inner layer where essentially the flow is very much like fully turbulent flow. So, as I said inner layer has three layers in laminar sub layer transitional layer and the inner turbulent layer. The outer layer of course is definitely turbulent so we need not we will take up the outer layer towards the end of this lecture but presently we must concentrate on the inner layer because it is this part of the boundary layer that really offers significant rates resistances to heat transfer. It is also the region in which greater part of the temperature velocity and concentration gradients take place whereas the outer part has more or less uniform profiles. So, it is the inner layer which is of great importance to us. Now phenomenologically I may postulate that the velocity parallel to the wall U would be function first of all of the fluid property two properties being grow the density and viscosity. Of course, U must vary with distance from the wall therefore Y is included and tau wall would determine the shear stress at the wall that is the velocity gradient at the wall and therefore because tau wall is mu d u d y at y equal to 0 and therefore shear stress is also included here and then there are many other factors that are likely to influence the velocity profile in the inner layer. What are those other factors? The other factors would be the boundary layer thickness itself could well influence the nature of the velocity profile. The pressure gradient and its variation in the x direction could also affect velocity distribution. If there is transpiration or mass transfer then of course V w will also influence the velocity distribution and finally in heat transfer we often in order to enhance the rate of heat transfer particularly in gases we often in employ rough surfaces so even the roughness height would influence the nature of the velocity profile in this. Experimental evidence however shows that for a smooth impermeable firmness smooth meaning roughness height is 0 impermeable means V w is 0 the inner layer is almost completely free of all other parameters. Now this is very interesting that for V w equal to 0 and for a smooth surface the other parameters play a very minor role and I will explain why it is so. So for example independence from delta suggests that no information travels from the outer parts to the inner region. So inner region is sort of insular region that is not really affected by what happens very far outside into the outer layers. Independence from dp dx suggests that the inner region is also independent of the history of the flow except that the shear stress variation along the wall may influence a little bit the velocity profile but the influence would be expected to be not so great as far as the velocity profile is concerned. Structure of turbulence is thus presumed to be in local equilibrium that is the time scale of eddies in the inner layer are much much smaller than the time taken by the mean flow to change its structure appreciably in response to dp dx. You may like to think about this I often give the example of a city and a village. A city quickly responds to what happens in distant places for example Bombay would get influenced by what happens in New York or London but a village say 100 kilometers outside Mumbai would be hardly influenced by what happens in the world around I am here using an analogy so that you might remember what we mean by insularity of the inner region and the slowness of the inner region. This assumption of local equilibrium is valid for adverse as well as mildly favorable dp dx but not when real name anization is encountered at very high accelerated boundary layer. Of course we are talking about here very moderate range of plus and minus pressure gradients so highly accelerated boundary layers this pressure gradient parameter nu by u infinity square d u infinity by dx would be greater than 3 into 10 to the minus 6. So we are not talking about such highly accelerated boundary layer we are talking about only those which are more frequently encountered having moderate plus and minus pressure gradients. We included viscosity and distance from the y simply because as I said tau wall is equal to mu d u d y therefore we must include mu and y. The density rho is included due to the importance of momentum transfer resulting from velocity fluctuations in the transition and fully turbulent layers as I said there are bursts of fluid which come in break down the laminar sub layer and there is a burst of fluid coming out of the laminar sub layer and this requires momentum transfer from the outer layers to inner layer of the inner layers therefore density would definitely pay an important role and therefore we have included density. So if I were to go back to this slide so we are essentially excluded all others and only are going to concentrate on these four parameters because they define more or less the equilibrium of the inner layer. So if I were to carry out the dimensional analysis I would find that rho u square divided by tau wall would be function of rho y square tau wall by mu square. Now I define u tau equal to under root tau wall by rho this is often called the friction velocity square root of tau wall by rho has the dimensions of velocity. I will define a dimensionless velocity u plus as u divided by u tau and y plus would be defined as y u tau divided by mu you can see this is a kind of a Reynolds number based on distance from the wall and the friction velocity y plus. So both u plus and y plus are dimensionless so you will readily recognize that this is nothing but u plus square and this is nothing but y plus square or another way of saying is u plus will be a function of y plus and at least the phenomenology suggests that this relationship will be universal and it is often called the universal law of the wall. The exact forms of f y plus we must now find out what the form it will take in the three layers that is the laminar sub layer, the transitional layer and the fully turbulent layer of the inner term layer. So let us go layer by layer to do that you will recall that the Rans equations for actual momentum would look like this. These are the convection terms, this is the pressure gradient term, this is the total shear stress and there would be these terms which would arise only which are which are necessary to be included only when highly accelerated flows are considered but as I said we are not going to consider those so those terms will adopt. So very close to the wall u is very very small in the inner layer which is about 15 percent and u by du dx would be much much smaller than this quantity v du by dy and therefore this term can also be dropped as a result what we would get is d tau tot total divided by dy would be approximately equal to d p by dx plus rho v w du by dy. Now of course v equals v w this is the inner layer v w is present only here this is the turbulent this is the turbulent layer sorry and this is the transitional layer and this is the laminar sub layer. So we say that u by du by dx will be approximately 0 in all this region and v du by dy would be approximately equal to v w du by dy these are very very approximate in the sense that v w extends in effect in the inner 15 percent of the total boundary layer thickness. If I make these assumptions then you will see that I can non dimensionalize this by first of all I must integrate this. So tau tot would be d p dx into y plus rho v w into u and tau tot would be equal to tau wall at y equal to 0. So in other words this integration would result into tau tot divided by tau wall equal to 1 plus y divided by tau wall d p by dx plus rho v w u divided by tau wall. Now you can see this is nothing but this is tau wall divided by rho here in the denominator which is u tau square. So I can form a v w plus which is v w over u tau and u over u tau will give me u plus so this v plus u plus and I will define this term as p plus phi plus in which case p plus would be defined in this fashion nu divided by rho u tau cube d p by dx and v w plus would be v w by u tau these are the definitions. So hold this in your mind that the ratio of shear stress, total stress to the shear stress at the wall is 1 plus pressure gradient times y by tau wall and rho v w u by tau wall. So in effect this ratio is in fact influenced by the pressure gradient as well as the effect of v w as it should be. So now let us look at layer by layer to begin with we shall assume that d p by dx and v w are both 0 that means we are considering the case in which this is 0 and this is 0. So tau tot would be equal to tau wall throughout the inner layer and therefore shear stress is a constant. Now in the laminar sub layer tau tot would be equal to tau wall equal to tau l and tau t the turbulent stress would be 0 and that would be equal to mu times du by dy at y equal to 0. So integration of this would give me u into tau wall divided by mu times y plus constant but u is equal to 0 at y equal to 0 and therefore c is equal to 0 and therefore I get u equal to tau wall y by mu. If I multiply and divide this by rho then you will see I will get this tau wall divided by rho will become u tau square into y by mu and if I take one u tau on this side I will get u over u tau divided by u tau which is y u tau by mu in effect this is u plus is equal to y plus in the laminar sub layer u plus would be simply equal to y plus and that is what I shown here. So u plus is equal to y plus. Now when dp by dx is moderate the equation for d tau tot by dy now of course vw is still 0 shows there would be little bit of y dependence there will be y dependence plus pressure gradient that term is 0. So tau tot by tau wall would actually be influenced little bit by distance from the wall and therefore the second and third derivatives of u with respect to y will be nearly 0. Hence if we expand this u plus y plus relationship in Taylor series then about y equal to 0 then you will get u plus equal to y plus plus y plus 4 by 4 factorial d 2 u by dy plus 4 plus so on and so forth. Now this equation shows that for small values of y plus u plus equal to y plus holes which is the laminar sub layer but at some critical distance away from the wall u plus must abruptly depart from linearity. So this is a very useful little deduction that we will carry over to the next transitional layer. Now in the transitional layer there are no simple phenomenological arguments that one can give because both viscous and turbulent stresses are equally important in the transitional layer. There is however similarity between the inertial sub range of the energy spectrum and the transitional layer in that if we said that if u dash is a representative velocity fluctuation then the viscous length scale would be nu by u dash would be much much less than delta as we have already seen. If the turbulent Reynolds number u dash delta by nu is high and a layer covering a range of values of y can therefore be imagined in which the turbulent structure is independent of both delta the large scale as well as the viscous very very viscous length scale nu by u dash. This is how we characterize the inertial sub range as being relatively un influenced by either the very large scale or very very viscous scales. How should du dy vary then in this region? The du dy can only depend on u dash divided by y and if we for a moment say that u dash would be proportional to u tau then du by dy would be proportional to u tau divided by y and if I said that if I say that du by dy is proportional to u tau by y then I will get let us say du by dy equal to I am going to now call kappa transition u tau by y and if I were to integrate this I will get u equal to 1 over kappa transition u tau into ln of y plus a constant of integration which is c transition. Before I do that in fact I can say that if I make this du plus multiplied by u tau and make this dy plus multiplied by because y plus is equal to u tau y by nu. So dy being changed to that will become nu by u tau that will equal 1 over kappa transition u tau divided by y nu by u tau then you will see that this u tau gets cancelled with this this gets cancelled with that and I will have essentially du plus by dy plus equal to kappa transition 1 over y plus or I would get u plus equal to 1 over kappa transition ln y plus plus a constant of integration c t r. If this is the law that applies to the transition layer then it does indeed show that there is a clear departure from u plus equal to y plus which was in the laminar sub layer and this is now in the transitional layer and we had anticipated that there would be some distance y plus at which this transition would take place certain departure in the slope du plus dy plus which was equal to 1 now becomes 1 over kappa transition 1 over y plus. So there is a sudden change in the velocity gradient and also therefore the velocity itself. So the expected departure from linearity in the u plus y plus law is already attained. Now, incidentally this equation can also be recast as ln e transition y plus by k transition where c transition will be ln e transition by k transition this is simple. Let us turn to the fully turbulent layer now we what we are looking for is the u plus equal to f y plus for the turbulent layer. Therefore, this will be 1 over u tau du by dy equal to d f by dy plus into dy plus by dy or that would be 1 over u tau d u by dy equal to d f by dy plus into u tau by nu and therefore you see du by dy in the turbulent layer would be u tau square by nu d f by dy plus. Now of course in the fully turbulent layer mu t is much greater than mu so we do not expect nu to play any significant role in the fully turbulent part of the inner layer and therefore this expression must be independent of nu and therefore d f by dy plus must be proportional to mu d f by dy plus by mu equal to nu divided by u tau y so that dimensionally the two sides are correct or this is nothing but proportional to 1 over y plus and therefore we get d f du plus by dy plus as being proportional to 1 over y plus or this again gives u plus equal to 1 over kappa ln y plus now this will be for plus a constant of integration. So, the fully turbulent layer also suggests a logarithmic law but the values of kappa and c may be different from those of the transitional layer. Now, let us look at the experimental data because so far we have put up phenomenological arguments so let us look at the experimental data in which we look at this parameter which is the pressure gradient parameter delta 2 divided by u infinity du infinity by dx and I am looking at three types of flows one is the adverse pressure gradient boundary layer k equal to minus 1.434 into 10 to the minus 3 k equal to 0 is the 0 pressure gradient boundary layer because u infinity would be constant and k equal to 1.44 into 10 to the minus 3 which is the favorable pressure gradient what does this show? The experimental data are circles show 0 pressure gradient boundary layer, squares show favorable pressure gradient boundary layer and triangles show adverse pressure gradient boundary layer. Now, you can see right up to 5, 10, 30 or almost let us say up to 5, 10, 30 or almost let us say up to 5, about 100 there is a complete collapse of all experimental data on u plus versus y plus u plus versus y plus and u plus is a linear scale and y plus is a logarithmic scale and up to about 100 you will see that there is a complete universality irrespective of the pressure gradient. The favorable pressure gradient boundary layer data begin to depart from let us say somewhere around about 300 the 0 pressure gradient data seem to do quite well even up to Reynolds number of I mean y plus of almost 700 but the adverse pressure gradient data seem to begin to depart at about 300 and the favorable pressure gradient data seem to depart say at about 150 or so from the universality. So, we can say up to about 100 the velocity profile in these coordinates u plus versus y plus is almost universal and how does it fit u plus equal to y plus it seems is valid up to y plus less than or equal to 5 this is what we identify this is y plus equal to 5 here is identified as laminar sub layer. Then there is a change in slope as you can see and that is the transitional layer up to about 30 that is given by 5 l n y plus minus 3.05 this implies that 1 over kappa transition is 5 therefore, kappa transition must be 0.2 and that is region extends from y to about 30 y to about 30 that we identify as the transitional layer and u plus equal to 2.44 l n y plus plus 5.4 seems to apply for y plus greater than 30 and that we would say is the turbulent layer. So, the main observations are for k equal to 0 0 pressure gradient boundary layer these laws apply up to y plus equal to 700 and I have drawn these laws by the solid line that extends right till about 2000 y plus of 2000 for favorable pressure gradient y plus of 100 seems to be the upper limit of applicability whereas, again for adverse pressure gradient it seems to be about 300 y plus of 300 for pipe flow which is a very mildly favorable pressure gradient which I am not shown here. In fact, the experimental data show that again like k equal to 0 case experimental data for pipe flow would also fall on the universal laws till about y plus of 700, but it is safe therefore to say that in general irrespective of the pressure gradient that we would encounter the region y plus less than 100 seems to be almost certainly universal y plus less than 100 seems to be almost certainly universal. So, we have discovered that the inner layer in the absence of V w, but very moderate pressure gradients does actually have a reasonably universal structure, but the moment you exceed y plus of 100 the outside pressure gradient effects that is the others which we had ignored begin to play their role velocity profiles do depart from this universal law that we have identified. The main changes occur only in the fully turbulent part of the inner layer. The laminar sub layer and the transitional layer are somehow completely insular, they are not affected by the pressure gradient at all and the inner smaller region of the inner layer is also up to about 100 is also universal, but beyond 100 the pressure gradient starts playing its role. So, what is this special about y plus of 100? Well as I will show shortly it corresponds to say about 10 to 15 percent of the boundary layer thickness and we can explain this form for a pipe flow. So, for example, in a pipe flow F friction factor which is tau wall thereby rho u infinity or u bar square divided by 2 is 0.046 Reynolds raise to minus 0.2. So, if I take a Reynolds number of let us say 30,000 then what am I saying? This relationship actually can be shown tau wall by rho is u tau square divided by u bar square into 2 is equal to 0.046 into u bar into diameter divided by 0.046 into u bar into diameter divided by nu raise to minus 0.2. If I were to define here 0.046 into 2 u bar by nu into r u tau by nu into nu by u tau raise to minus 0.2 then you will see this is equal to 0.046 into this nu bar gets cancelled with this nu bar and I have 2 times u bar by u tau into r plus raise to minus 0.2 or you will see therefore this becomes 2 times if I take this term on this side then you will see I get u tau by u bar raise to 1.8 2 into u tau divided by u bar raise to 1.8 is equal to 0.046 into 2 raise to minus 0.2 into r plus raise to minus 0.2. What is u bar by u tau? This is the friction factor. The friction factor is actually equal to or the 2 times u tau square by u bar square and therefore u tau by u bar is actually under root of f by 2. So, in other words I get here 2 times f by 2 raise to 0.9 equal to 0.046 2 raise to minus 0.2 into r plus raise to minus 0.2 into r plus raise to minus 0.2 or r plus raise to minus 0.2 would be equal to 2 raise to 1.8 divided by 0.046 into f by 2 raise to 0.9. Therefore, if I take now Reynolds number of 30,000 then I can get the value of f from our usual relationship and therefore I can show that r plus will be about 811. That is what I have shown here. So, r plus would be about 811 that is at the axis of the pipe from the wall. r plus would be about 811 whereas the inner layer where universality exists y plus is about 100. So, y plus by r plus is approximately 100 by 800 at Reynolds number of 30,000. If the Reynolds number was bigger then this could go up to 100 by 1000 even or 1200 or something like that. So, we are essentially talking about a region of the order of 12 percent, 15 percent region which defines the inner layer. The constants that we identified as I show here the constants are kappa transition will be 0.2, kappa for the turbulent layer from 2.44 would be 0.41 and c transition would be 5.4. Of course, likewise e transition and e of the turbulent layer will be 0.543 and 9.512. This is just another way of writing these two. The three layer law of course is very nice, but as we said it has very sharp discontinuities and what we would really like is to have continuous law of the wall. How do we do that? So, we seek now a continuous law of the wall rather than this three layer description, mathematical description and to do that what I am going to do is allow for this bursting phenomenon as well as effects of D P, D X and B W. So, in analogy with Stokes law for laminar shear stress tau l equal to mu d u d y, we introduce a model due to Bosnianesque as tau t equal to minus rho u prime b prime equal to mu t d u d y and then Prandtl suggested in analogy with how laminar viscosity is defined in rho times l m v dash equal and v dash was likened to l m into d u by d y and therefore, tau t becomes rho l m square d u by d y d u by d y. These are these issues we considered in our formal aspects as well where v dash is the fluctuation responsible for transverse momentum transfer and l m is the mean eddy size in the inner layer sort of notional mean eddy size. Note that unlike mu turbulent viscosity mu t is a property of the flow whereas, mu is actually the property of the fluid. So, that is the difference between mu and mu t the turbulent viscosity. Now, the second question is how does mixing length l m vary? So, the transitional layer is characterized neither by delta nor by nu by v dash. So, the only relevant scale is y and therefore, Prandtl extended this argument to the entire region of the inner layer and propose that l m would be kappa times y some constant that is directly proportional to y in the inner part of the layer. Now, the experimental data for the measured from velocity profiles of l m show this is y axis l m divided by delta and this is y divided by delta and I am going from 0 to 1. Now, experimental data do show except for this little damping l m is in fact, quite linear till about let us say 0.18 or 0.1. So, 0.2 let us say this is what we call the inner layer, but beyond that point the l m begins to show lots of scatter. Now, we are considering here flows in adverse pressure gradient, favorable pressure gradient, zero pressure gradient as well as pipe flows and many other ducted flows and so on so forth. So, this seems to be quite peculiar about mixing length that it is nearly constant in the inner layer with I mean it is nearly linear in the inner layer say about up to about 20 percent or 15 percent and then it begins to show lots of scatter about a value of 0.09. Mind you it is somewhat difficult to accurately measure l m in the outer layer because d u d y in this region is also very small and therefore, it becomes difficult to measure this value of l m very accurately. Now, incidentally the boundary layers with different pressure gradients as well as v w as I said have been included for the 0.2 to 0.9 region the values of show a scatter about l m by delta equal to 0.09, but for y over delta less than 0.2 l m is nearly proportional to kappa 0.41, very close to the wall l m is somewhat lower damp than suggested by l m equal to kappa y. When Drist suggested that this damping actually occurs due to due to the effects of fluctuations in the transitional layer and therefore, he said the l m should be of Prandtl should be modified by introducing a damping function 1 minus exponential of minus 5 plus by a plus and therefore, from the previous slide mu t would therefore, be equal to rho into kappa y square 1 minus kappa exponential of y plus by a plus whole square d u d y, where from experimental data it is found that a plus equal to 26 for a smooth wall will bring about the amount of damping required in line with what is observed in experiments. Mu t of course, is 0 at the wall because y is equal to 0 there and in regions where viscosity is influential that is y plus less than 30 l m is smaller than the Prandtl's mixing land. The amplitude of fluctuation decreases thus exponentially as y tends to 0 and that is what we expect when viscosity begins to play its role in dissipation. So, in order to develop continuous law of the wall tau taught by tau wall which was shown earlier to be 1 plus p plus y plus n equal to v w plus u plus now can appear in this fashion where this mu t expression has been used to define tau taught by tau wall and this is a long expression. However, if the stress ratio is a fume to be unity then effects of p plus and v w plus can be absorbed in a suitably defined a plus. So, what we are saying is we are going to cheat on this equation we will say let p plus b equal to 0 and v w plus b equal to 0. So, that tau taught by tau wall equals exactly equal to 1 and would equal that relationship, but to account for effect of p plus and v w plus we would simply tune the value of a plus here that is the damping constant. If you take tau taught by tau wall equal to 1 we obtain d u plus by d y plus as a quadratic in d u plus by d y plus and the solution is simply u plus equal to integral 0 to y plus d u plus by d y plus into d y plus and the a here is given by that damping function. So, you need numerical integration to predict u plus as a function of y plus and that is what I have done here. Experimental data from different boundary layers with different d p, d x and v w are matched with predictions by tuning a plus in each case and k is in Crawford have proposed a plus to be 25 there by a into v w plus plus b into all this function plus 1. We will make use of this a plus later on in actual computations of friction factor and necessary number, but presently just see this that we do manage to predict the continuous law of the wall quite well predicts the experimental data in favorable pressure gradient, adverse pressure gradient as well as in 0 pressure gradient boundary layers up to 500, 700 in 0 pressure gradient, up to 100 in favorable pressure gradient and up to about 200 or 250 in adverse pressure gradient and therefore, we can say that we have now found a method for calculating the universality of the inner layer in a continuous manner. This is very useful when we when we do friction factor investment. Now, of course, the outer layers do not have any universality they are big they are significantly influenced by the by the pressure gradient and other effects and therefore, but nonetheless efforts have been made to shortcut methods have been made to universalize outer layers in this path. These are called the velocity defect law u infinity minus u tau equal to 1 minus cos y pi by delta and therefore, the total velocity function is given by this a equal to 0.55 k equal to 0.4 and c equal to 0.51. This of course, applies only to 0 pressure gradient boundary layer, but not in general there are other methods for outer layers, but I will not go into that at the moment. So, in summary I would say that we have shown that although the inner layer universality can be established for a wide variety of turbulent flows outer layer similarity is difficult to establish. For complete description of the outer layers we need to solve the Rans equations using turbulence models. The inner layer universality can be exploited in two ways and that is to derive approximate correlations for friction factor and Nusselt number and to specify wall boundary conditions for at y plus equal to a plus when outer layers are computed by Rans equations. This achieves computational economic, but this is an aspect that the CFD analysts essentially worry about. Now, to prepare the ground for studying turbulence models in the next lecture we shall explore the likely interaction between inner and outer layers. Thank you.