 So good morning yesterday we started looking at fundamentals of how natural convective heat transfer occurs and also we derived the governing equations and the non-dimensional form of the governing equations we find that the more important term coming in the buoyancy force is the non-dimensional group which is Grashof by re square so this is important non-dimensional parameter which is governing basically the strength of natural convection with respect to the forced convection so the Reynolds number that we defined you should remember when we non-dimensionalize that governing equations was such that we used re is equal to ur x l by mu so in this case the reference velocity ur from the order of magnitude analysis we have to also express this as do you remember what it is G ? you all – T 8 x L correct so essentially the velocity here what we are referring to is nothing but a velocity scale arising out of the buoyancy force okay now there could also be a case where you can have a bulk motion well along with the natural convection you can also have a bulk motion in that case you have to modify the definition of Reynolds number to a conventional definition where we define it based on the free stream velocity okay so however still the same non-dimensional combination will be there only that the Reynolds number in that case will be defined based on the bulk velocity so this is a important non-dimensional number therefore governing the strength of the natural convection to the force convection so this is the ratio of your buoyancy force to the inertial force now so you can have considering therefore two scenarios where you can have the force convection acting vertically upward or going downward okay so we call this case where the buoyancy and the direction of the bulk motion are in acting together so this is called buoyancy aided natural convection or buoyancy aided you can say forced convection okay so these are two different scenarios in mixed convection okay so one is a buoyancy aided convection the other is a buoyancy opposed convection where the buoyancy is in this case is trying to act upward because the wall temperature is greater than the free stream temperature however the bulk motion is acting downward so in fact depending on the strength of the forced convection the boundary layer growth could be either way so if your force convection dominant then the boundary layer will actually start growing from top to bottom in this case okay so this is decided by the ratio of Grashof number 2 re square so if this strength is around one then both of them will be equally dominant if this is very small then Reynolds number will be the dominant one and therefore you might actually end up having a boundary layer growing in this way so depending on this if you draw the velocity profile so unlike the pure natural convection case you will have a finite value of velocity profile at the edge of the boundary layer and this should approach your U infinity right so you do not have actually a minima at this point it will be a finite value depending on what the U infinity is and this will vary so if your U infinity is considerably large you can actually end up with actually going for a gradual profile and then which will approach your conventional boundary layer velocity profile okay without going through the maxima and then approaching U infinity depends on the strength of the forced convection in this case nevertheless now you can use the same set of governing equations that we derived yesterday and we can find the solution for the mixed convection case also however the one of the more simpler approaches in a empirical manner will be to find the Nusselt number independently for forced convection and then natural convection and then simply blend it blend these two values together depending on whether it is a buoyancy assisted case or a buoyancy a post case. So for the case of buoyancy aided one you see that the boundary layer growth is much faster and it is thicker than the case of buoyancy a post okay so therefore the when you look at the sign it can be either plus or minus depending on you have a buoyancy assisted or a buoyancy a post case so now the value of M here depends on the kind of configuration so I will just list out the value of that index M for a few cases for a few configurations for example if you have a vertical plate and you have an so this index M will be something like 3.5 okay so this is usually between 3 and 4 okay so this value is generally taken to be 3.5 and if you have a for example a horizontal cylinder also this value if depending on the condition boundary condition if it is a constant wall temperature this is again taken to be 3.5 the same horizontal cylinder with constant wall flux is taken to be 4 and similarly for a sphere also with a uniform surface temperature this value is taken to be 3.5 okay so depending on the configurations you can use values of M appropriately and blend the values from the independent results so usually it is between 3 and 4 okay so now what we will do is that so I just also will give a representation of how the Nusselt numbers might probably look in the buoyancy assisted case so when we plot Nusselt number as a function of RE for a fixed value of Grashof number so that is I maintain say the Grashof number to be something like 10.5 and vary only the Reynolds number okay so I can go anyway from therefore natural convection regime all the way to forced convection regime depending on the value of Reynolds number that I vary okay so for very low values of Reynolds number so then what happens this will be in the natural convection regime so typically you end up with value for natural convection now this will be your natural convection value so you see what happens at low Reynolds numbers since it is dominated by buoyancy irrespective of what the Reynolds number is the Grashof number is also a constant so the Nusselt number will remain a constant it will be governed by the value of Grashof number that you use now when you again go to very high value of Reynolds numbers so then the buoyancy will be insignificant compared to the inertial force and therefore Nusselt number will be purely dictated by the forced convection in forced convection case Nusselt number is directly proportional to Reynolds number and therefore it progressively increases in the case of forced convection so if you plot forced convection case you will have something like this so this is your forced convection case so therefore the combined convection will have to transition from the natural to the force like this so this will be your combined or mixed convection so this is the assisted case so initially when you talk about lower Reynolds number so you have only natural convection which is governing the value of Nusselt number very high Reynolds number it is only the forced convection which is governing it and in between where the ratio of Gr by Re square is of the order of 1 so you have both of them both the forced convection as well as the natural convection so your Nusselt number values will be transitioning smoothly from the natural to the forced convection is that okay so this is a kind of more realistic case in many applications you can actually have both the forced and natural convection to be equally significant and we cannot therefore neglect the effect of mixed convection in those cases so now what we will do is first we will take up the pure natural convection case where we do not have any bulk motion and try to look at some solutions to the fluid flow and heat transfer problem so what are the different ways of solving it just like you have your external force convective boundary layer for which we have the flashes and pole house and solution we can approach by using similarity methods so similarity solutions are the exact solutions that we are going to do and once again this can be done for either a constant wall temperature boundary condition or a constant heat flux boundary condition so the constant wall temperature boundary condition case was originally attempted by pole house and along with the external forced convection boundary layer we also started looking at the case of natural convection and he derived the similarity equation for this case also but this was later on solved for wide range of Prandtl numbers by another person called Ostrich okay originally during pole house and time there were numerical methods were very few so he could not find a very general solution for the Nusselt number for different Prandtl number so only for fixed Prandtl number he was able to get the solution but nevertheless that similarity equation was solved using numerical methods later on by Ostrich and he generalized to a different range of Prandtl numbers and the other solution is for the constant heat flux boundary condition and this was done by two people sparrow and great so this is extension to the basic pole house and solution to constant heat flux boundary condition and apart from this you also have the approximate methods so one is the use of similarity solution the other is the approximate technique which is not as rigorous but nevertheless gives you very useful correlation which is close to the exact solution so the approximate method has been derived by a person called square okay so using the momentum and energy integral technique especially for the constant wall temperature boundary condition okay later on people extended the solution to also constant heat flux boundary condition so like this similar to the external force convection you have similarity and approximate solutions also for the natural convection problem so we will start with the similarity solution with constant wall temperature boundary condition okay and later on we will move to the modification required in the similarity variable for the constant heat flux and then we will go to the approximate methods okay so the similarity solution was originally developed by pole house and similarity equation as such so we will look at the solution first before we go to the other extensions okay now just the same with blushes equations you have to first start with defining similarity variable similar transformation from XY coordinate to coordinate which is called the similarity variable coordinate so we do not know what is it but let us say that ? will be a function of y by some ? right so we will go on the same lines as blushes did okay naturally because they we are talking about boundary layer growth and therefore the similarity variable should be able to map the boundary layer thickness at different x locations and when you plot y by ? this should be a similarity variable which should collapse all the velocity profiles okay the same starting point as the blushes solution but in this case we do not know what the order of ? should be what was the similarity variable in the case of blushes equation the case of blushes solution the similarity variable was assumed to be y by x rex to the power half because ? the order of magnitude of ? was found out to be x by square root of Reynolds number local Reynolds number okay so now we have to find a similar transformation in the case of natural convection that we have to replace Reynolds number with our Grashof number okay so yesterday we have seen that the equivalence between Reynolds and Grashof number so according to this non-dimensional number we have Grashof number to the power half is of the order of Reynolds number correct so therefore if you want to use Reynolds number to the power half this will be Grashof number to the power one-fourth so what Polhausen did was to simply so it is re raised to one by two this is going a square yeah that is correct so in that case we may yeah I mean this is a generic representation yeah but you can you can plot it exactly you know I have just given an idea here the shape of the curve could be different okay so what you are suggesting is it should go like this right okay you know at very low values of Reynolds number here so your Nusselt number with a force convection will be very less so this will be insignificant okay so you are saying that this value should be shifted up here no but that is what I am saying for very low values of Reynolds number if you plot the value of Nusselt number from the natural convection so this will be much larger than the force convection value right so this becomes dominant only after the Reynolds number becomes sufficiently large this is a generic representation okay I am not plotting any values here so you can actually calculate it for fixing Grashof number say 10 power 5 you will find that this value will be different for 10 power 6 this will again change okay so but this is a generic representation saying that very low value of Reynolds numbers okay your force convection contribution is small okay if you therefore put it into that empirical correlation so it will be mostly governed by the natural convection value and once you cross a threshold Reynolds number then the force convection becomes dominant so then that will be decided only by the force convection value so in between these two is where you use this empirical correlation and blend both the values of forced and natural convection so that blending will give you a smooth transition all right so so let us come back to the solution so when we replace therefore Polhausen looked at the same similarity variable and therefore he replaced Reynolds number with Grashof number so in this case therefore what happens you have Grashof number to the power 1 by 4 okay so this is of the order so what he did you can also use this find a similarity equation solve it so it will not change the solution but nevertheless to be precise he used Grashof number to the by 4 the whole raise to the power 1 by 4 is anyway constant they shouldn't affect the final solution for say skin friction coefficient or nusselt number because everything will get adjusted in the end okay so even if you do not do this you might get different constants in the similarity equation but then finally when you calculate the CF and nusselt number they will get scaled proportionally okay by this constant right so he used this as the similarity variable many of the textbooks in fact show that this is your exact similarity variable there are few textbooks which omit a constant and they go ahead and solve they get slightly different constants in the similarity equation but finally the non dimensional numbers all will be the same okay and in fact the Blasch's equation we did not use any of these constants so now the next step is to find the stream function okay so when we solve the governing equations we have to solve in terms of the stream function because then the continuity equation becomes redundant okay so therefore what is the appropriate transformation for transforming the stream function which is a function of x y to another function which is only a function of the similarity variable eta okay so that is the next step that you have to find so how do we do that so once again we write down the equations for relating stream function and velocity field so u is equal to d psi by dy and v is equal to minus d psi by dx so let us take this particular relation and we can integrate it so therefore psi will be nothing but integral u dy and let us also assume that if you find the right transformation variable then u by say u reference will be a function of only this variable ? right that is the purpose of finding the similarity variable so that finally when you plot u by u reference so it will not vary as x and y but it will all collapse as a function of ? right so therefore we can substitute for u into this so you have u reference which is taken out into G of ? so now we have to transform the variable y in terms of ? so we will write this as dy by d ? into d ? correct and dy by d ? can be expressed from the similarity variable here okay now if you do that I will just give you what will be the expression for ? please check that it comes out to be u reference times x by rash of number to the power 1 by 4 into 4 raise to the power 1 by 4 into this integral G of ? d ? will be nothing but another function of ? which will say f of ? and this dy by d ? is nothing but x by x into 4 raise to the power 1 by 4 by rash of number to the power 1 by 4 okay and integral G of ? d ? is nothing but another function f of ? so therefore this is your transformation from of the flow field from x y coordinate to a ? coordinate this is your transformation from x y coordinate to ? coordinate through the similarity variable so this is your similarity variable and this is the corresponding transformation of the flow field so now that you have u reference we also know that u reference can be related to G ? ? t okay so in fact pole house and gave you also introduce the constant 4 into this it does not matter if you do not introduce any of the constants here or here also finally it will not matter all these are scaled up or down accordingly so the final non dimensional numbers will not change but let us do it exactly the same way he did it so if you substitute for u reference here and therefore combine the terms you get 4 into rash of by 4 power 1 by 4 into ? times f of ? okay so what we are doing is we are writing G ? into T wall – T infinity in terms of rash of number okay since we know the definition of rash of number as G ? into T wall – T infinity so this is a local rash of number we will define it with the local coordinate so I am just substituting for G ? into T wall – T infinity as rash of number into new square by x cube into this and already I have a rash of number so I just combine the terms finally I will get this as my relation between ? and f okay so therefore I find the transformation now we can go ahead proceed start calculating you then V du by dx du by dy all the derivatives and then we can finally plug them into the governing equation so if this were the right similarity variable so we should be able to transform this PDE into similarity equation which is only a function of ? so that shows that we have whatever we have assumed for ? is correct so if you have some terms in x or y that means we have still not found the right similarity variable so let us do that the next step is therefore to find you which is d ? by dy and how do you find this now ? is a function of y and we have to convert this in terms of ? that means we can write this as d ? by d ? into d ? by dy okay the ? is a function of y through ? so now we will write d ? by d ? into similarity variable is a function of ? so this is a partial derivative d ? by dy using the chain rule of differentiation okay so same way we can also calculate the V velocity how do we calculate V velocity now V velocity is nothing but minus d ? by dx now if you look at this ? is directly a function of x correct through Grashof number and also function of x through ? so therefore in this case we can write this as d ? by dx this is your normal derivative with respect to x plus it is a function of x through ? so therefore this is d ? by d ? into partial derivative of ? with respect to x so this is exactly the same way we did the blushes solution so can you calculate and tell me what the velocity is you and we are so when you write d ? by d ? in this case the other terms are all constants only f is a function of ? okay so essentially you have 4 by into d ? by dy which is nothing but 1 by 4 4 into 1 by x into Grashof number raise to the power 1 by 4 okay so this gives my U to be 2 into Grashof number raise to power 1 by 2 into ? by x into df by d ? okay so we have 4 raise to the power 1 by 4 into 4 raise to the power 1 by 4 okay so this becomes therefore 2 4 by 2 right so this is 2 and we have Grashof number raise to the power 1 by 4 into so 1 by 4 plus 1 by 4 so we have 1 by 2 okay and then we have ? by x into df by d ? similarly you can find V also I will what I will do is I will write down the final expression here you can it will take some time about 5 to 10 minutes to simplify and do this you can check this a little bit later so it comes out to be minus 3 by 4 raise to the power 1 by 4 Grashof number by 4 U by ? minus okay so all you have to do is find d ? by dx treating other things as constant and differentiating with respect to x then the second term you have df by d ? into d ? by dx okay so put together you have function of Grashof number x ? and y okay so now using knowing therefore U and V we can calculate the derivatives okay so we will see how the derivatives are calculated therefore the first step is du by dx okay now how do you calculate du by dx now you should tell me so again U is a function of x directly and also through ? okay so we can write this as therefore du by dx plus du by du by d ? into d ? by dx right okay so du by dx so you have to differentiate by keeping the other terms constant only as a function of x only with respect to x you should differentiate du by dx the other is du by d ? so in that case the other terms are all constant so you have only d by d ? of df by d ? so you will have a d ? by d ? square term into d ? by dx right so if you again substitute for d ? by dx and so on so this should come out to be so when you write du by dx that df by d ? will be a constant term so that will just come as it is and from the second term you will have a d square f by d ? square okay so you have to come one from du by dx and the other from du by d ? into d ? by dx that is this term where yeah so here when you differentiate you only treat the other terms as constant right yeah that is what so you you have Grashof number to the power half by x so this is actually x power 3 by 2 minus 1 okay so that is what 1 by 2 so we have therefore 1 by 2 x power minus 1 by 2 right so this is what you are writing here again you are splitting that as therefore Grashof number to the power half by x square so here you have x power 3 by 2 minus 2 that is what minus 1 by 2 right okay so I we are finally putting wherever possible in terms of Grashof number again you do not want to carry that g ? ? t together so you are rewriting them in terms of Grashof number again right so the next step is to calculate the gradient with respect to y so how do you calculate du by dy so you is a function of y only through ? you look at the expression for you right we do not have any other term of y sticking there so therefore we can write this as du by d ? into d ? by dy so now you should be able to tell me in terms of Grashof number what this is this is much simpler term so you have Grashof number raise to power 1 by 4 into Grashof number raise to the power 1 by 2 1 by 2 1 by 4 so what will be you should have 3 by 4 3 by 4 and you have nu by x square into you have d square f by ? square okay so we can also find out the second derivative of velocity with respect to y so d square u by dy square so this is again the same way you can write this as what you have du by dy you already have du by dy so this will be a function of y only through ? so you can write this as d by d ? into du by dy into d ? by dy so that comes out to be Grashof number into nu by x cube into d cube f by d ? cube okay please check once again so we have all this is constant you have to differentiate with respect to ? d cube f by d ? cube into d ? by dy so d ? by dy will again give Grx power 1 by 4 by x into 4 raise to the power 1 by 4 okay so now that we have all the terms what I ask you to do is please substitute into the x momentum equation okay so our momentum equation for this case okay so we can write the last term in terms of ? we can now define for the constant wall temperature case T minus T infinity by T wall minus T infinity as ? and therefore we can write this as G ? into T wall minus T infinity into ? okay so you can go ahead and plug for you du by dx v du by dy d square u by dy square and G ? into T wall minus T infinity you can write this in terms of Grashof number again right so what this will be in terms of Grashof number so your Grashof number once again G ? T wall minus T infinity into x cube by nu square so this will be simply Grashof number x by x cube into nu square ? so please plug in the other terms you will get some terms cancel some terms will cancel away and tell me what will be the final equation will you be able to transform the PDE into a OD function of ? okay as you keep doing I will also write down the energy equation so the energy equation can also we can use the differentiation by parts here so you can write this as you can assume now ? to be only a function of ? now let us use the variable ? of ? here okay so when you plot this non dimensional temperature also as a function of the similarity variable they should all collapse same way like your u by u reference okay in that case then ? will be only a function of ? so we can split this as d ? by d ? into d ? by dx and so on so I will give you the final similarity equation here it comes out to be d cube f by d ? cube plus 3 times f into d square f by d ? square minus twice df by d ? whole square plus ? equal to zero so finally we have successfully transformed PDE into OD as a function of only ? and now you see we also have the ? term here so please note that finally when we are solving this equation by shooting method we have to make sure we solve both the momentum and the energy equations simultaneous okay so this is where the coupling comes from so you can compare this to the Blasius equation so there we had d cube f by d ? cube plus we had f by 2 into d square f by d ? square the other terms were not there so compared to that we have now additional terms so to complete it now we can also transform the energy equation as a function of only ? by substituting for u v d ? by dx d ? by dy and so on okay so I will again give you the final similarity equation for energy d square ? by d ? square plus so you get d square ? by d ? square plus 3 times Prandtl number where Prandtl number is the ratio of momentum to thermal diffusion so in the case of Paul Haussens solution for flat plate external force convection there was PR into f by 2 so now we have modified this okay so tomorrow we will stop here you can also go home and verify whether you are getting the final equations so tomorrow we will apply the boundary conditions and then look at the application of shooting method for solving this equation then how the velocity and temperature profiles look after solution based on that we will develop the expression for Nusselt number all right thank you.