 good morning all of you so we have completed nearly about 10 classes so far I hope most of you are following what are what is happening in all these 10 classes and being the first tutorial is due on 14th of this month and I will also most likely early next week upload the second tutorial on similarity solutions okay so most likely we will complete the similarity solutions in the next couple of classes and start the integral analysis or external flows okay so we have quite a few topics to cover so I have to move a little bit fast and I hope your earlier knowledge of fluid mechanics will help you in covering all the things that we won't be discussing here okay so let us continue with the Falkner scan similarity solutions that we had derived yesterday for these are for flows with pressure gradient and final similarity differential equation appears in this particular form and these are the boundary conditions and once again you have to use the shooting method to solve the ODE numerically I have also given you the three first order OD is to which it can be reduced and the same boundary conditions like that of the Blasius equation so once you solve this for okay so one more thing is that so these are some of the velocity profiles if you solve the ODE by the shooting technique and this is how the velocity profiles look and these are the similarity profiles I am plotting eta on the y-axis and f prime which is nothing but non-dimensional velocity on the x-axis and I can substitute different values of M into this and for each value of M I can solve the ODE by shooting method and I can plot the profiles for example M equal to 0 this is the flat plate case you get a profile something like this exactly similar to the Blasius profile and for positive values of M greater than 0 some say like 0.33 something like this so one it is something like this so if you look at M equal to 1 this is the stagnation point flow right and this is your flat plate all right okay so therefore if you look at your values of M which are basically increasing okay so the profiles are in this particular fashion and for the values of beta the wedge angle I mean let me once again draw the representation so that you can understand okay so if can you tell me how the M and beta related or beta y is your wedge angle beta by 2 minus beta okay so far negative values of beta that is I was giving an example when you roll this in the anti-clockwise direction so that they both coincide that is the M equal to 0 case and if you roll it further roll this particular surface further down it becomes negative okay for the negative values typically if you make it more and more negative you will get adverse pressure gradient flows right now in the example that we have seen yesterday and for a particular value for M which is exactly minus 0.091 you will see the flow is separating okay so at this particular location the boundary layer theory will not be valid anymore so there is no point in going further less than minus 0.091 so this is the limit of M where you can up to which you can go negative and the value of M up to which you can also go on the other side for typically we are more interested up till the stagnation flow so we can go up to M equal to 1 okay so what is happening to the velocity gradients so typically if you want to calculate say du by d eta as you keep increasing your M what is happening to du by d eta smaller so typically if you are saying okay so I say this is my eta range okay if you compare 0.32 so this is my change here okay this if I if I look at f for this value so in which case your gradient is higher whether is it for 0.32 or whether it is for so he in this case your eta values are this is your d eta here for this value of df prime and this is your d eta here okay in which case the d eta will be more for this case so therefore in which case the gradient will be higher for this case okay so as you keep increasing your M okay the velocity gradients become more and more steeper okay so therefore so this is to do with the particular kind of flow that you are looking at so typically what you can do I will give you some problems where you can substitute the corresponding configuration according to the value of M and you can get these profiles and you can try to see how the variation in the slope appears to you okay so this is all with respect to the solution we are more interested in the derived quantities like the skin friction coefficient for the flow as far as the flow is concerned so we can calculate the skin friction coefficient locally so this is nothing but you du by dy okay now if you substitute everything in terms of the similarity variables so you will be getting an expression which is like this in terms of f and eta okay so this depends on the curvature at the wall this by now you all know for the case of flat plate the value of this is 0.332 so this leads to the familiar expression CF is 0.664 by square root of Reynolds number okay so now all we need to know is for a given configuration what is this curvature at the wall once we know that we can calculate the skin friction coefficient locally for that particular configuration so once you solve the equations by the shooting method for different values of M we can have a nice tabulation where we can tabulate the curvature terms for different configurations so the for the value of beta this is the value of M and this is d square f by d eta square at eta equal to 0 and this is the particular configuration or case that it corresponds to okay so beta equal to 1.6 okay so 1.6 times pi so you will find that this is something like this okay or maybe you can say it becomes like this you have a flow which is like this right it is much more than one so it is much more than your right angle is more than pi all right so in this particular case you can you can have flows like this and the corresponding value of M will be equal to 5 if you substitute into this expression okay if you compute the slope the curvature at the surface this will be 2.6344 exactly in fact you can do it in fact I will give you an exercise where you can do it and compare with these values okay should be getting the same values and now the case of beta equal to 1 M equal to 1 and 1.2326 what is this case stagnation flow all right so beta is 0.5 M is 1 over 3 and the corresponding value is 0.75746 beta equal to 0 M equal to 0 what will be the value 0.332 this is the flat plate and point 0.14 the corresponding value of M will be minus 0.0654 and the corresponding value 0.16372 okay and finally minus 0.1988 okay corresponding value of M can you guess what is the corresponding value of M minus 0.091 okay that is this particular case that we have so what do you think will be the curvature for this this is a separated flow okay at the point of separation what is the slope what will be du by dy for separated flow so if you have a if you have a flow suppose you have a gradient like these and here you plot you have profile like these okay now at the point of separation the profile becomes like this after that in fact so this is dp by dx greater than 0 okay this is separation point this is separated flow so you can visualize the flow coming like this and at this point detaching and then you have a nice big separation bubble here okay so what what how do you check that flow has separated or it is about to separate what is that not curvature first you go to the first derivative before you go to second derivative du by dy for the separate for a condition what is the condition for separated flow du by dy at y equal to 0 equal to 0 this is the case okay once it is separated what is the condition there it should be negative okay therefore what will happen to curvature when it is at the separation state 0 right so this has to be 0 and this is a separation case separated flow all right so I think this is giving you final summary and special case where we can apply these stagnation flow if you look at flow past a circular cylinder okay so this is your radii are and this is your approaching free stream velocity which is constant and then the free stream now when it travels over the surface it becomes a function of your local coordinate X and where X is defined in this particular fashion okay that is the sector location sector distance starting from this point where it is 0 okay so if you look at the case of flow past the circular cylinder from the potential flow theory you can actually calculate how the U infinity is varying locally okay so the profile is given I think you must have studied this in your incompressible flow course so U infinity by U any guess how it varies if you go along the periphery or the circumference of the particular cylinder how does the local free stream vary what is the value of velocity here it has to be 0 and where does it reach maximum where where your ? equal to ? by 2 okay and X is equal to R ? basically so X by R should be equal to ? by 2 at that location it becomes a maximum okay so it should be a sinusoidal variation okay only you have sin 0 is 0 sin ? by 2 is 1 so if you say U infinity X by U infinity it has to be a sin variation and what should be the variable X by R all right maybe I can use capital R because this is your radii of the cylinder and what should what factor should come here at X by R equal to ? by 2 this will become 1 does you be U infinity become U infinity there it should be 2 because it becomes exactly twice the because it has to accelerate again from here once it accelerate it has to go more than the free stream velocity there okay so this is your inviscid velocity profile for a circular cylinder now if you are looking at region values of X which are which are very small that means close to the stagnation region okay so then you can approximate this sin X by R as simply X by R for small values of X okay so for small values of X by R that is close to the stagnation you can write your U infinity by constant free stream velocity as 2 X by R okay so therefore if you look compare this with your Falkner scan form of velocity profile which is like something like U infinity of X is C X power M so what can you deduce what should be value of M what should be the value of C if you compare these two C is equal to 2 U infinity by R okay and what is the value of M one okay so therefore this profile will be something like okay so what kind of flow does it mean stagnation point flow okay so when you are looking at region close to the stagnation even for a curved surface like cylinder okay you can approximate the flow pattern to be similar to the stagnation point flow for which we have already calculated the profiles and the curvature at the wall okay so this is a very important thing so it does not limit the Falkner scan solution does not mean it is only applied to a wedge configuration like this it can even apply to any stagnation flow even for a bluff body like this not it does not need to have a sharp corner okay provided you are looking at only region close to the stagnation region so if you simply use C equal to 2 infinity by R so this is nothing but the stagnation flow okay the same solution will hold for the cylinder also as well okay so this is a very important useful correlation because to calculate for example the heat transfer in the stagnation region of a cylinder you can solve the energy equation from the Falkner scan solution and you can apply that to get the local Nusselt number profiles for the cylinder okay okay so for small values of X your sin ? can be replaced as ? right okay so therefore now we will move on to the heat transfer problem so the boundary layer energy equation when you write it down for the flows with pressure gradient or without pressure gradient they are both the same okay so without the viscous dissipation term this is your energy equation where your ? is defined how T-TW by T-TW so I want my temperature profile to look identical to the velocity profile all right and the boundary conditions at Y equal to 0 your ? should be it should be 0 okay if I had defined my ? this way okay it has to be similar to the velocity profile right so at Y going to 8 ? should be 1 okay now what I can do from the definition of the similarity variable that I use also from the definition of stream function which is a function of the similarity variable I can plug in for U V convert all X and Y in terms of ? okay the same way that we did for the blushes energy you can you can apply that here because it is no different and except that when you write the similarity a variable ? here this is a function of U 8 of X by ? X rather than simply U 8 in blushes so when you differentiate this with respect to X you have to be careful now you have to account for the variation of this so you can substitute C X power M okay and then you have you can differentiate it okay so for example I will if you say that this is Y square root of C X power M by ? X so this can be written as Y square root of C by ? x power M-1 by 2 okay therefore if you say your D ? by DX so this will be Y square root of C by ? x M-1 by 2 x X power what for differentiate what should I get exponent M-3 by 2 so I can write that as M-1 by 2-1 okay so once again Y C by ? X power so this entire thing is what ? so this will be M-1 by 2 into ? by X okay so you should take care when you are differentiating and transforming the variables now that your free stream velocity is a function of X okay so appropriately you do all the substitution and transform this in terms of the similarity variable and everything in terms of F and you will get the similarity equation for energy D square ? by D ? square okay so this is your energy equation okay for the case of M equal to 0 this reduces to the flat plate energy equation similarity solution okay so so on to additionally here you have M plus 1 by 2 because of the factor of M which comes in in the free stream velocity okay if you substitute all of that you will definitely be able to reduce this and the boundary conditions as ? going to 0 ? equal to 0 ? going to to ? equal to 1 all right okay so once you know the flow you know the value of F you plug it in for a corresponding value of M you can solve this once again by shooting method okay the same way that we have been doing and you can get the profiles for ? as a function of ? all right so the same procedure repeats here now what I am going to give is just the way we tabulated the curvature at the wall for the velocity profiles I am going to give you once you substitute and get the values you can get the slope at slope of the temperature at the wall for different values of M okay for different configurations how does it look because this is required to calculate the nusselt number nusselt number depends upon this temperature slope at the wall okay so if you do the tabulation so d ? by d ? at ? equal to 0 okay so M your Prandtl number now you should realize the temperature profiles are now function of your velocity profile your Prandtl number and your M okay so far a given value of M for a given value of M you know the velocity profile put that function value of M and also the Prandtl number which you want to calculate so both of all the three have to be simultaneously fixed of course if you fix your M you fix your F also okay and also you have to decide which Prandtl number you are calculating so you can tablet this for different values of mantle number so for M is minus 0.0753 okay okay I am just giving you some value of M here for which if you have Prandtl number of 0.7 this value becomes 0.242 okay I am just tabulating all the values here okay so anybody remember now value of M equal to 0 flat plate what should be the value of Prandtl number 0.7 probably I must have yeah I think you can calculate and tell me what should be the value of d ? by d ? for Prandtl number 0.7 okay so for Prandtl number of 1 what should be the value 0.332 why yeah because for the case where your Prandtl number equal to 1 I mean the velocity and the temperature profile should be identical so the curvature d d cube d square f by d ? square should be exactly equal to d ? by d so this should be 0.332 so this value should be 0.332 into Prandtl number power 1 by 3 okay so what should be the value so it will be something like 0.2292 it will be reduced so then this is 0.307 and 0.585 this is 0.730 so on so if you go to M equal to 1 the stagnation flow 0.496 0.523 okay so this has some kind of values I am just giving you this because tomorrow when I ask you to compute using the shooting method you should be able to match with these tabulated values all right so why we are calculating the slope okay so because we finally are interested in the heat transfer coefficient and Nusselt number so therefore you can define your local heat transfer coefficient in the same way the wall it flux divided by T wall minus T infinity okay if you substitute for minus k dT by dy at y equal to 0 and write in terms of d ? by d ? you should get k T wall minus T infinity into so you can write your dT by dy as d ? by d ? at ? equal to 0 into d ? by dy okay which is nothing but square root of U infinity by µx divided by T wall minus T infinity okay so finally if you define your Nusselt number local Nusselt number as Hx by k so that will result in d ? by d ? T equal to 0 into so you have a x here so square root of µ infinity x by µ which is nothing but your if you if you divided if you divide multiply H into x by k so what happens to this particular term Reynolds number square root of Reynolds number okay this is the same as what we did for the flat plate okay there is nothing new here only thing you should now know for which configuration the value of d ? by d ? you have to pick put it there and then you will get the Nusselt number profile for that particular value of M okay and also now it is a function of Prandtl number so you may have to fit a curve as a function of Prandtl number and you should bring the Prandtl number dependence okay now for the stagnation point flow if you are interested in values of Prandtl number about one okay that is something in this particular region right here okay you can in fact fit a curve to the values of d ? by d ? in this Prandtl number regime close to one and you will get a nice curve fit of this particular form which is 0.57 times Prandtl number to the power 0.4 can you can you just check if you can you substitute Prandtl number as 0.8 and check whether you are getting something close to 0.523 okay so this is this is the kind of fit that you can do for M equal to one around Prandtl number close to one okay so you are not saying here Prandtl number is exactly one it has some Prandtl number dependence but I mean the dependence is relatively kind of similar you know you have about 0.5 to 0.57 variation here okay so therefore if you substitute for that so your d ? by d ? is a function of Prandtl number now this becomes 0.57 into rex power half Prandtl number to the power 0.4 so this is the case for M equal to one Prandtl number close to one okay so this is the local variation that you find for the stagnation point flow case of course you are you know the variation with respect to Prandtl number for M equal to 0 pole house and already did that it did the curve fit for different values of Prandtl number small Prandtl numbers intermediate and large and you can use those values okay does it does it make sense okay so that this is a reasonably good fit okay for Prandtl number close to one okay now so we this is one one example to show you for the stagnation for example stagnation flow how we can define the local variation in Nusselt number okay so all this can be also equally verified by you you can calculate the values of the slope compare that for different values of M and Prandtl number with the tabulated values and you can yourself correlate with these values right here okay so now one more thing as we said if you look at the flow past a cylinder okay the flow that we are looking at right here so apart from U infinity suppose you heat this particular surface so you are maintaining this at T wall equal to constant okay so this is a flow apart from the flow you also have a temperature profile okay now this U infinity is a function of X here whereas still it is having some temperature T infinity so you have a velocity boundary layer you have a thermal boundary layer which is simultaneously growing so if you are interested in the stagnation region for the cylinder what is the variation in the Nusselt number okay so now as we as we already shown you can describe that by the stagnation flow M equal to 1 the same correlation will apply for this region as well okay now X is defined in this particular manner okay where this is your R this is your origin okay so now it is it is not so convenient to operate in terms of local X for a cylinder and sphere what is a more convenient characteristic length instead of X diameter okay so usually when you talk about cylinder flow past circular cylinder the characteristic length that is chosen is the radius or diameter of the cylinder okay so therefore we can transform your local X into terms of R similarly in the Reynolds number also and we can define a Reynolds number based on the characteristic length which is the radius of the cylinder okay so if you do the transformation so you already know so how do you do this transformation you already know that your REX is defined based on U infinity X into X by you so this is the definition of your local Reynolds number right now so where you are defining based on your local velocity yeah that is how that is how we have to transform we will see how we will transform it okay so now you can replace this as you can write this as to U infinity you can you can you can write your U infinity in terms of U infinity of X in terms of to U infinity X by R so this is your local velocity profile related to the free stream velocity profile for this stagnation region okay so you can substitute for U infinity in terms of the constant free stream velocity X and R okay if you do that this will be X by R into X by ? okay so now what I can do is I can multiply and divide by R so I can say that this is R and there will be an X square by R square okay so therefore this is how my local Reynolds number is related to now I can define I can define this as my Reynolds number based on the radius of the cylinder okay which is nothing but U infinity R by ? this is now this is the constant free stream velocity okay and the characteristic dimension is the radius so now I have transformed from the local velocity and the local coordinate I have transformed that to constant velocity and a fixed coordinate so the fixed dimension here is R okay and you have a factor X square by R square therefore if you substitute you can you can find that your current NUX is also HX by K right if you substitute for REX from there you can finally write NU in terms of the radius which is nothing but HR by K so you have a 2 square root of 2 which comes out and multiplies with 0.57 that becomes 0.81 and this will be RE R to the power half and your Prandtl number to the power 0.4 okay so this is what you finally get in terms of fixed dimensions are and so this is the expression for the cylinder when you look at the stagnation flow okay so in terms of the cylindrical dimensions okay so you can simply transform from your local coordinate to the cylinders fixed dimensions which is basically the radius okay the same way you can also do that for sphere okay for the case of three-dimensional wedge flows okay now whatever we were discussing so far or two-dimensional wedge flows the same two-dimensional wedge flows can be transformed using what is called as a manganese transformation okay so if you if you happen to go through the boundary layer theory by listing okay so he talks about three-dimensional boundary layers so there one way of deriving the similarity solution for three-dimensional wedge flows is to take the two-dimensional wedge flow solution apply what a particular kind of transformation called manganese transformation and you will get the similarity solution for three-dimensional wedge flows and for that particular case where m equal to 1 for three-dimensional wedge flows that is the stagnation region stagnation flow for three-dimensional wedge flows and that will be similar to the stagnation region flow for sphere in 3d okay like we have equivalent to a 2d wedge flow stagnation region we can correlate that to stagnation region of a cylinder same way a three-dimensional axis symmetric wedge flow stagnation flow can be correlated to the stagnation region of a sphere okay so you can also have a similar relationship for Nusselt number for a sphere from the applying the manganese anyway manganese transformation is beyond the scope of this class so I am just giving you an idea that you can also do that for three-dimensional wedge flows okay so I think with that we have more or less covered the flows with pressure gradient terms any any questions so far whatever we have done yes yes so this this just only just only says the Nusselt number in the stagnation region that is it that is that is only for that okay so it is not really you are not really going varying the X because your variation in the X is actually confined to a small region here the stagnation point so you just say that what is the stagnation point Nusselt number for example okay so based on the free stream Reynolds number and for a particular value of Prandtl number you directly get the Nusselt number in the stagnation region of a cylinder okay you do not once again look at the local variation of the Nusselt number and things like that because the stagnation region is a very small region correct so the next class on Tuesday what we will do for the same wedge problem okay now the wedge problem that we have taken we can add one slightly one complicated boundary condition which is so far we have assumed that your Nusselt exist at the wall which is correct but it is quite possible you can have some small vertical velocities at the wall correct so this is your local coordinate X and Y so therefore you can have your V velocity and U velocity at the wall anyway your U velocity is equal to 0 because it cannot slip tangentially but it is quite possible that you are blowing or you can suck now this is called flows with transpiration okay so this is a typical flow with transpiration so in that case you can have a small value of vertical velocity at the wall it will not be that much but it will be small enough so typical applications are if you are looking at boundary layer separation control okay so typically you can blow some small velocity okay in order to control the separation point or if you have a massive separation you can suck the separation bubble by means of flow suction and therefore you can avoid separation so typically in like airfoils now you can do this kind of flows to control the drag and you know stalling of the airfoil and so on okay so these are these are extension of the same falconers can solution the same solution or same equation the similarity equation what we derived will exist and you hold true for this case also only the boundary condition will now change okay so far we have said at y equal to 0 v equal to 0 but now we will be V has a small component so that has to be included and we can use that as a more generalized solution okay so if you do this solution this will be the most general solution for whatever we have seen till now okay that includes all kinds of configurations also different kinds of boundary conditions for limiting case where v equal to 0 it becomes the solution that we had derived till now so okay we look at this particular case in the next class and with that we will complete the similarity solutions okay.