 So, in the previous lecture we looked at effects of low turbulence and also numbers. Now, this low turbulence and also number can occur close to the wall that is one type of situation, but it can also occur along the length of a flat plate where the boundary layer is growing and where the laminar flow turns turbulent through an intermediate region which is which we normally define as the transitional region and in that transitional region again the turbulence and also number would tend to be low and therefore, the models that we just discussed are very much applicable to such situations, but as I said these turbulence models with roll turbulence and also number require fairly heavy computation expense because of the very large number of grid nodes required close to a wall and therefore, at earlier times simpler approaches to modeling transitional region regime were adopted and my purpose for the in this lecture is to really look at laminar to turbulent transition in little bit more detail. The second issue concerns rough wall. So, much of our discussion till now has been about smooth walls, but in engineering in order to enhance heat transfer we often employ rough walls deliberately structured or naturally rough walls and we want to see how to capture the effect of a rough wall in either in a wall function or ultimately in generating the universal law of the wall for the inner layer of a turbulent boundary layer on a rough wall and finally, we will look at like we derived the universal velocity law for the inner layer. Is there a temperature law for the inner layer and that is what we will these are the three topics we will take up in this lecture. So, for example, in a duct when Reynolds number based on hydraulic diameter or in an external boundary layer when the Reynolds number based on actual disc in x is increased laminar flow undergoes transition before turning into fully turbulent wall. This transitional phenomenon occurs actually occurs over a range of Reynolds number although in the undergraduate work we often believe that laminar to turbulent transition is very abrupt, but actually the transition occurs over a range of Reynolds numbers. Although the range is very small compared to the normal values of fully turbulent Reynolds numbers that we normally encounter, but nonetheless the range exists and we are going to account we want to know how to account for that range. The lower end of the range of Reynolds number indicates end of laminar conditions whereas, the upper limit of the range signifies establishment of fully turbulent conditions and both F and N U demonstrate unique features in the transitional layer. For example, you will recall that the friction factor in a pipe for example, decreases linearly in a up to Reynolds number of let us say about 2300 and then the friction factor actually increases a little to say till about Reynolds number of 5000 and then the so this is the fully turbulent, this is fully laminar and in the transitional region here the friction factor Reynolds number relation is quite reverse of what happens in the laminar and turbulent regions in that friction factor actually increases with Reynolds number and therefore, this curious phenomenon needs to be explained in somewhat greater detail. Now, many heat exchange equipment operate in the transitional region particularly in process industries this is often found that the Reynolds number inside the tube is often in the transitional region. Similarly, on a gas turbid blade the pressure gradient varies in the range as low as minus 10 to the minus 8 2 of this parameter rho by U infinity square d U infinity by 2 as high as 10 raise to minus 5 both these are considering both pressure slide and the suction slide and the free strip turbulence intensity is vary from as low as 2 percent to 10 percent. Now, in this conditions transitional range of r e x may well occupy as much as 50 percent of the chord of the blade what is meant here is for example, in a flat plate initially you will get laminar flow and then a small transitional region with the x and then x transition n and then you will get a turbulent boundary layer. So, this occupies a much smaller length of the total length of a boundary layer, but on a turbine blade the this is the chord length and you could get transitional Reynolds number as transitional range occupying almost 50 percent of the chord length. So, that is why we need to study turbulence somewhat in a transitional region in somewhat more carefully. So, now, let us first of all get our ideas of transition in a pipe flow cleared. So, in a pipe flow if for Reynolds number 2300 f increases with Reynolds number and if you were to measure u velocity at a fixed point in the flow at any radius of the pipe then you will see a picture something like this. A flow will be show turbulence for a while at a point then for a certain period it will be quiet laminar like then it will be turbulent again then it will be very quiet again then turbulent and then so on and so forth. If we measure this for a certain total time and divide this time period equally in steps of delta t then in each delta t a small fraction of it delta t sub t delta t sub t would be occupied by turbulent patch likewise here this much would be the delta t t whereas this is the total is delta t. Now, we define intermittency as so intermittency itself is defined as gamma delta t of turbulent patch divided by the total time step delta t and this would be the k th patch let us say and then if we sum up all this over k equal to 1 to n time steps and divide this by n then that is called the intermittency. This is the intermittency definition it tells you about the fraction of the time for which the turbulent patches occur. Now, obviously it would it can only be vary between 0 and 1 and that is what I have shown here that intermittency gamma would vary from 0 to 1. Now, of course it would the value of the gamma would vary in a pipe flow from axis to the wall at different points, but usually we speak of some kind of an average gamma. So, we do not worry about the radial variation too much here is what I show the value of average gamma versus x by d of the pipe flow at different actual distances. This is the measurement made at Reynolds number of 2300 and you will see that the intermittency does not exceed 0.2 even after going well beyond 500. So, in this case the flow will never turn turbulent at all in practical lengths of pipes which would be usually 100 D or 200 D at best, but flow will become fully turbulent at very very long distance from the entrance. At 2400 x by d of 500 it has reached about 0.4, but at 2500 you will see it has reached about 0.8 and at 2600 it has reached almost 1 at x by d of 500. Now, if you had a Reynolds number of 5000 then of course, it will reach the fully turbulent gamma equal to 1 in a very very short distance and it is such flows that we say are turbulent right from the inlet itself. Between 2300 and 5000 the transitional flow will continue for all practical lengths of the pipes, but for 5000 and above the flow will be turbulent right from the inlet itself in a pipe flow. Similarly, if you were to look at the external boundary layer you will see if the flow is coming from the left then there will be a laminar layer here with gamma equal to 0, but then from here to here which is the start of transition to end of transition intermittency will go on increasing with distance. How it will increase? We will see shortly and then fully turbulent conditions will be reached here at gamma equal to 1. Pictorially, if I were to look at the plan of this it will look like this. If the flow is coming in from here you will get very steady flow up to start of transition. Occasional spots of turbulence will be seen and then tend to grow engulfed by laminar fluid which kills the spots, but it is this energy transfer which gives rise to new spots of turbulence and they would grow, but they are again engulfed by laminar fluid surrounded by laminar fluid and therefore they get killed, but the process of energy transfer creates more spots and so on and so forth. Ultimately, they grow not only in number, but also in strength and ultimately at the end of transition you will get a completely turbulent boundary layer or the turbulent fluid. The transition is identified with the occurrence of intermittent turbulent spots surrounded by laminar fluid in an external. This is quite visible. You can do a flow visualization experiment to see this. Variation of intermittency with x in a boundary layer has been correlated and the one of the first correlations was by Rodemner Sima, an Indian scientist working at the University of Science. It was 1 minus exponential of minus 412 beta square where beta itself is a dimensionless quantity x minus start of transition divided by x where the value of intermittency is 0.75 minus x where the value of intermittency is 0.25. This is of course very well to use in an experimental setup because you can see where the intermittency is 0.75 and 0.25 and therefore determine the denominator quite exactly. Computationally, this is not very convenient and therefore more recently in the 80s, Abu Ghunaman Shaw have come up with this expression gamma equal to 1 minus exponential of minus pi times xi cube where xi is x minus x T s divided by n of transition minus start of transition. So, all that remains is finding ways to determine x T s and x T e and this is there are several suggestions. So, for example, for determining start of transition where there is a sudden departure in laminar variations of delta C f x and stenton x, one can C B C is one author, Fraser is another author. So, C B C has suggested that at transition at the start of transition, the Reynolds number based on momentum thickness delta 2 is balanced by this expression where the Reynolds number x is based on start of transition. So, what one does is if you were doing let us say applying integral method or then during the laminar flow, you will calculate r e x and r e delta 2 and see whether the that expression that I showed you is actually satisfied. You will see at a certain point x T s, the r e x and r e delta 2 would be related by this expression and that would identify the start of transition. On the other hand, this expression of course, does not take into account the effects of turbulence intensity which can vary quite a bit particularly in gas turbine and compressor applications and therefore, Fraser has suggested this formula r e delta 2 s is related to this f m and turbulence intensity T u and m is the pressure gradient parameter and includes d u infinity by d x. So, when this equation you go on calculating r e delta 2 s and when it matches with this for m greater than 0 and m less than 0, then you say that is the start of transition. m equal to 0 of course, make it f m equal to 6.91 itself. If turbulence intensity is not accounted and taken as 0, then this is simply exponential of 6.91 plus 163 and you take the value of x T s where r e delta 2 is 163 into exponential of 6.91 for 0 pressure gradient boundary layer and turbulence intensity equal to 0. For all other cases, the x T s must respond to the pressure gradients. Now, how about end of transition? For end of transition, you would say you define r e sigma as u infinity x T e minus x T s divided by nu. So, it is based on the transitional length if you like r e sigma. r e sigma naught is a function only of the turbulence intensity 4.6 into 1 plus 1710 m raise to 1.4 into all this. Of course, this term is valid only when m is greater than 0, which means that term is negative and that would occur in an adverse pressure gradient. We can locate the end of transition in the adverse pressure gradient as would be expected. So, this is the recommendation of Fraser and Milne and the paper was published in 1986 in institution of engineers proceedings. There is another recommendation and that is by Sibesi who say that r e sigma which is based on transitional length should be when it equals 60 times r e x T s which was identified on the previous slide r e x T s raised to 2 by 3, then that would signify the end of transition. Then there is yet another one which is by Deuchy and Zierke, which again does not take into account turbulence intensity, but they recommend that the end of transition r e delta 2 at end of transition should satisfy this expression, where now instead of m which was based on like m which was based on delta 2 square d u infinity by dx. Here also you see this is really m 1 plus m at start of transition. So, that is the most important part this is at the value of start of transition. So, there are three expressions particularly in adverse pressure gradients all of these work very well. If you wanted to use the integral method for solution of the integral equation through the transitional layer, you would need u over u infinity in the transitional layer. Now, simply by observing how the experimental data look like. So, this is u over u infinity divided by y given in inches here in the transitional region the profile would go something like that. As the r e x increases and here r e x increases in this fashion and finally, you would end up with a fully turbulent profile like that. In u plus y plus coordinates it would appear something like this and it is quite customary therefore, to say that u over u infinity at transition would be 1 minus gamma times u over u infinity laminar plus gamma times u over u infinity turbulent. So, having identified the values of x t s and x t e you can determine the gamma distribution with respect to x and therefore, you can use this formula and this is pure pragmatism there is no big theory, but it seems to work this kind of pragmatism and actually predicts transitional boundary layer quite well in using integral method. Now, we turn to how to account for effect of wall roughness. Now, effect of wall roughness as you will recall even from your undergraduate days when the roughness height divided by the diameter is 0.001, then we say that the tube is smooth and you will get this kind of profile for smooth pipe turbulent region, but this would have the height roughness y r of d less than 0.001, but if you increase y r by d then you will remember that you get friction factor which goes on like that. So, much so that you may even get that and this is where the roughness is increasing. So, the effect of roughness is to enhance the friction factor or the really the pressure drop and this is what is observed in experimental data that you will recall from your undergraduate days. You want to explain how what are the other features of rough surface. For example, cement pipe is quite rough quite naturally, but many a times in as shown here. So, it will have a jagged surface experimentally such a surface is produced by pasting sand grains on the various sizes on a surface of equivalent or average height y sub r, but many times we actually have a rough surface which is deliberately structured by providing ribs for example, like this or studs three dimensional studs can be provided of width w and pitch p and it becomes quite difficult then to develop any universal law for surfaces like this. So, in order to circumvent this difficulty what is done is we always take sand grain roughness as a benchmark and determine the friction coefficient for that and for a structured surface if the friction coefficient agrees with that of the sand grain roughness then we say that the structured surface had an equivalent roughness height equal to the sand grain roughness height. So, sand grain roughness are done in a laboratory measurements and they provide the benchmark. What we would expect is near the law near wall law to be u plus now to be not only function of distance from the wall y plus, but also the y r plus the roughness height and since we do not expect much of a laminar contribution, we will essentially have a turbulent layer developing and therefore, we expect that u plus would be 1 over kappa l n y plus a plus c which would be function of the roughness height y r plus. Usually laminar sub layer does not exist when y r plus is greater than 70 and we say such a surface is fully rough surface, but if y r plus is less than 70 then you can still get effects of viscosity present and we have to account for those. So, that is what we have done in the next slide. So, here is a plot of C r which is a function of y r plus and the sand grain roughness data was generated by a person called Nikuraj, a very well known scientist from Soviet Union and it is quoted this data is quoted in the book by Schlichting called the boundary layer theory about which I mentioned even in the during the laminar boundary layer discussion. So, you will find C r versus log 10 of y r plus when it is perfectly linear and rising it is a perfectly smooth surface, y r plus itself is very very low and therefore, it is not really a rough surface. So, we say up to here the surface is smooth then the effect of roughness begins to come in, but C r would then decline and up to about say log y r plus to the base tan of about 1.8 which is really y r plus above over 70 and then it is found that the experimental data remains really constant with increasing y r plus. So, this is taken as the indicator of the fully rough surface and C r value for that is 8.48. So, for a discrete roughness in equivalent sand grain roughness y r e is defined and we say u plus will be equal to 1 over cap l n y by y r plus C r dash and the equivalent roughness height then would be y r exponential cap pi into 8.48 minus C r dash where C r dash is a function of the width of the rib or the pitch of the rib and is determined experimentally. So, that is how one would get the universal law of the wall for a given structured surface. One can also modify the mixing length here that instead of y plus one would now take y plus plus delta y naught plus a divided by a plus and likewise instead of y would take here y plus delta y naught and delta y naught is correlated in this way by K's and Reynolds this approach is no longer used now, but nonetheless it provided excellent measurements excellent predictions which compared with experimental data very well. Finally, we turn to near wall heat transfer and like we develop the law of the wall for velocity from phenomenology where we postulated u as a function of here we will say t minus T w the wall temperature is going to be a function of firstly y tau wall mu and rho which really determine the four things which really determine the velocity profile and in addition now you will have a q wall because that would determine the temperature gradient at the wall C p is the specific heat of the fluid and K the conductivity of the fluid and therefore if one carries out the dimensional analysis one would find that a quantity t plus quite unusual looking, but what you have here is a t plus equal to minus t minus T w over q w into rho C p u tau t plus would be function of y plus a Prandtl number and q w plus q w plus is really q over rho u tau cube and that is really the effect of friction at the wall which generates heat, but usually compared to the actual heat transfer this term is very very small and therefore it is not of great relevance effect of that can be taken up through property variations at a later time, but therefore this term is really drop and we say that the t plus will be a function of y plus n Prandtl only. So, let us see what is the form f y plus Prandtl will take in different layers of the inner layer. Now to generate the t plus law we first of all look at the differential equation where we say that for a smooth impermeable wall at any rate the convection term would be very very small and therefore the total diffusion of heat flux would be equal to 0 or essentially q total divided by rho C p would equal q wall divided by rho C p equal to minus alpha d t by d y plus v plus y m t prime and all this would be equal to a constant. In the sub layer q wall over rho C p will be minus alpha d t by d y in the laminar sub layer q wall over rho C p would be equal to minus alpha d t by d y and therefore q wall is constant so if I integrated that I will get t minus t wall equal to q wall minus q wall y divided by rho C p into alpha. Now if I said that the minus t minus t w divided by q wall minus divided by rho C p would be equal to y by alpha and if I divide this by u tau here then this will be u tau and this is nothing but the definition of t plus and that will be equal to y by nu u tau into nu divided by alpha and therefore this is nothing but our y plus and this is Prandtl number. So, in the laminar sub layer it is straight forward to show that t plus would be equal to Prandtl times y plus and Prandtl times y plus is also equal to Prandtl times u plus. So, laminar sub layer is fairly clear. Now in the turbulent layer which is characterized by a maximum of 30 and 30 Prandtl number less than y plus and less than 0.2 delta plus or the thermal thickness would be determined by the Prandtl number in the intermediate transition and sub layer regions and therefore we take maximum of 30 to 30 Prandtl to 2 delta plus which is the end of the transitional layer. The relationship d t by d y would read as d t by d y equal to q wall over rho C p u tau equal to d f by d y plus into d y plus by d y which is u tau by nu and this expression must be independent of nu which gives me that d f by d y plus must be equal to d t plus by d y plus equal to now we say 1 over kappa t y plus or t plus equal to 1 over kappa t l n y plus all this for a fixed Prandtl number and therefore the constant of integration here c t would be a function of Prandtl number and kappa t turns out to be kappa divided by 0.9 equal to 0.44. In the fully turbulent layer you will again get a logarithmic law for the temperature in the laminar sub layer you will get this. The transition region is somewhat complicated not easy to define but the two laws that we have got t plus equal to y plus into Prandtl for laminar sub layer and t plus equal to 1 over kappa t times l n y plus plus c t Prandtl for the turbulent layer. These two laws suggest that it is possible to generalize this as t plus equal to sigma times u plus plus P f. Now to understand this consider the turbulent law for example the turbulent law says that t plus the turbulent law says t plus t plus t plus t plus t plus t plus t plus t plus is equal to 1 over kappa t l n y plus plus c t Prandtl but u plus is 1 over kappa l n y plus plus c which is or 5.4 if you remember. So I can substitute for l n y plus here then I will say t plus equal to kappa over kappa t u plus minus 5.4 plus c t Prandtl or another way of writing is k kappa over kappa t into u plus plus another quantity called kappa over kappa t minus 5.4 divided by kappa over kappa t and if I were to call this quantity p f for the timing then you will see this and if I call this to be sigma or then it will look like sigma u plus plus p f. Similarly in the laminar sub layer you had t plus equal to Prandtl y plus is also equal to Prandtl u plus. So I can write this as Prandtl times u plus plus 0 and say that this sigma is now Prandtl u plus plus 0. The laminar Prandtl number. So it is possible to generalize both these laws by an expression like so t plus u plus plus p f where p f and sigma take different values in different layers. So we are going to say now that this kind of representation t plus equal to sigma u plus plus p f applies across from laminar sub layer onwards to fully turbulent layer provided we represent we interpret sigma correctly and p f correctly. Now sigma which is kappa over kappa t in a way represents as you will see shortly the slopes of the velocity and temperature profiles in the fully turbulent layer and therefore it would amount to essentially turbulent Prandtl number and therefore I have replaced kappa by kappa t by Prandtl t. So p f for the fully turbulent layer would be that whereas for the fully laminar sub layer it would be 0 and sigma would be equal to laminar Prandtl number. So for the entire layer now entire inner layer tau taut is equal to tau wall mu effective du dy and q tau taut is equal to q wall minus k effective by dt dy and therefore if I define Prandtl effective equal to c mu effective by k effective then it would simply amount to dt plus by du plus by the definitions that we have. To understand this let say q wall is equal to minus kt dt dy or k is equal to minus kt dt dy or k is equal to minus k effective rather which is k plus kt and tau wall is equal to mu effective du dy which is equal to minus k plus kt dt dy and this is equal to mu plus mu t du dy then if I take a ratio of this you will see I get q wall divided by tau wall divided by rho say I say this then this will be rho here. So tau wall divided by rho would be equal to minus k plus kt divided by mu plus mu t equal to dt by dy into 1 over du by dy and therefore this will become q wall divided by u tau square equal to minus k plus kt divided by mu plus mu t dt dy by du by du. If I divide this now by rho cp then I will get q wall divided by rho cp u tau square will be equal to minus alpha plus alpha t over mu plus mu t dt by dy. I can write this now as minus dt divided by q wall divided by mu plus mu t dt by divided by rho cp u tau is equal to so dt by dy divided by du by dy so dt by dy divided by du dy so dt by dy it will equal 1 over u tau into du by dy into du by du by du by dy into alpha plus alpha t divided by mu plus mu t which is nothing but 1 over prandtl effective and this can be written as this can be written as dt plus by dy and this can be written as du plus by dy and therefore I get the expression that dt plus by du plus equal to prandtl effective and that is what I have shown here and therefore you will see here prandtl effective is equal to dt plus by du plus and therefore t plus would be 0 to infinity prandtl effective du plus and this would give us a continuous temperature law as a function of u plus or function of y plus depends on how one wants to read it and by comparison of this if I compare this expression t plus equal to a 0 to infinity prandtl effective du plus with sigma times u plus plus pf then I can show that pf would be simply 0 to u plus prandtl effective divided by prandtl t minus 1 into du plus where prandtl effective is prandtl times 1 plus nu t by nu plus 1 plus alpha t by alpha and nu t by nu will be given by this why because remember tau wall is mu plus mu t du divided and therefore tau wall over rho which is equal to u tau square will be nu plus nu t into du by dy and that will be equal to nu into 1 plus nu t by nu du divided by dy and therefore you will see that if I 1 will equal nu by u tau square into du by du by du by du by dy into 1 plus nu t by nu and this is nothing but du plus by du by plus into 1 plus nu t by nu and therefore you will notice that nu t by nu will be simply 1 over du plus by du by du plus minus 1 so that is what I have shown here so we say nu t by du by du is equal to du plus by dy plus raise to minus 1 minus 1 and alpha t would be simply nu t divided by prandtl t this expression shows that remember du plus by dy plus is equal to 1 in the laminar sub layer and naturally therefore this will be 1 minus 1 so in the laminar sub layer nu t by nu goes to 0 and prandtl effective would simply go to prandtl number yes because that d t plus by dy plus there is equal to prandtl number and du plus by dy plus is 1 and therefore now the task is simply this how do we determine pf as a function of u plus well we need the values of du plus by dy plus so either we can take it from take them from the three layer law or by which which applies only to smooth surfaces or we can use the van risk mixing length model so that we can also allow for pressure gradient and suction and blowing for du plus by dy plus and use it here to integrate this expression to obtain pf as a function of u plus so you will see that t plus in a way because of its definition of u plus t plus which is equal to minus t minus t w rho cpu tau divided by q w which I can also write as t w minus t into rho cpu tau divided by q wall or I can speak of this as 1 over rho cpu tau as h up to distance y q w by q w by q w. q over t w minus t can be thought of as rho cpu tau divided by heat transfer coefficient up to distance y in a boundary layer and therefore t plus being inverse of the conductance can now be thought of as resistance to as the resistance to heat transfer and therefore the relationship that we have t plus equal to u u plus plus pf multiplied by sigma is really what it indicates is since u plus is related to resistance due to momentum transfer and t plus is resistance to heat transfer we can say pf is really the excess resistance over the resistance to momentum transfer and that is the interpretation one can now give to pf. pf will be equal to 0 for Prandtl equal to 1 as a rule what I have done is I have integrated using du plus by dy plus from Wander's mixing length model and found the following. So for example here is the value of pf and here are the values of u plus up to u plus equal to phi you have laminar sub layer and you do get the resistance to heat transfer is in excess of momentum transfer even in the laminar sub layer and then at this is at Prandtl number of 50 this is at Prandtl number of 5 at Prandtl number 1 there is no it is all 0 there is no excess resistance and as we observed earlier you will have perfect analogy between heat and momentum transfer for 0.7 which is air you have negative pf which means the resistance to heat transfer is less than the resistance to momentum transfer and in liquid metal the resistance is even lesser than that to momentum transfer. The transitional layer would occur somewhere around y u plus of 13 end of transitional layer corresponding to y plus of 30 would occur somewhere here and from here onwards you have really fully turbulent layer fully turbulent layer and it so happens that in the fully turbulent layer the pf value reaches almost constancy it remaining only a function of Prandtl number. So, we can say pf in the limit is always a constant for a given Prandtl number and that these values of pf at large values of u plus have been correlated and the most well known among these is pf infinity is equal to 9.24 divided by Prandtl t dashed 3 by 4 and into another function of Prandtl divided by Prandtl t. We can also interpret our Ct Prandtl in this manner and pf infinity would be Prandtl by Prandtl t 5 minus 5 by 4. This correlation for Ct Prandtl was obtained by Russian scientist Iqadar and Yaglom and it covered the range of Prandtl numbers from liquid metal range to a very way viscous oils. For rough surfaces pf infinity is given by both by Dupriy and Sebesky and also by Jatilikey who gave this formula and therefore, I have given here the references to both of them and for a rough surface it is 5.19 Prandtl raise to 0.4, why are equivalent roughness height raise to 0.2 minus 8.48. Jatilikey's formula requires experimental information on A which is meant essentially for structured surfaces. So with this I end the lecture which covered the topics of transition. It covered the topic of temperature law and we found that a continuous temperature law is also possible to be derived for the inner layer.