 This slide is amongst the last slides for external flow. We have a flow across a cylinder and this is a topic of research in itself. So, Phd after Phd is obtained on this. We are just going to focus on the heat transfer characteristics or Nusselt number behavior as a function of theta that is theta is measured from the stagnation point. What is the stagnation point? Isentropically. Isentropically brought to rest. So, what does it mean? So, when a fluid is brought to rest isentropically then that essentially we will call as a stagnation condition. So, static versus stagnation that is the major difference one people say will have the dynamic component u square by 2 in this that u square by 2 will be equal to 0. So, what happens is as the flow flows cross over the cylinder the Reynolds for a given Reynolds number if you are seeing especially for so called laminar flow whatever be the case the heat transfer coefficient is going to decrease from this stagnation point and then it comes to a minimum value. This minimum value is roughly at about theta equal to 80 degrees where separation point in laminar flow that is called as a separation point in laminar flow. What is flow separation? What is flow separation? Adverse and favorable there are two words. So, adverse pressure gradient versus favorable pressure gradient favorable means it is going to aid in the direction of aid the flow. So, what is the pressure gradient for a flow to occur? Should it be positive or negative? Positive or negative? Dp by dx should be negative. Now, what will happen at the point of separation or as the flow progresses the Dp by dx is slowly going to come to a value which is equal to 0 and after that this Dp by dx is going to become positive. Why should that happen? No that is fine, but why should the pressure gradient progressively come to 0 first? Your velocity increases at the cost of which of the equations that we derived tell you that Bernoulli or he showed one form u du by dx is equal to Dp by dx that equation that is the balance essentially. So, as one of the size is going to increase the other is going to be affected it will come to 0. Further downstream this quantity will start to become positive and when it is becoming positive what is happening to the flow when Dp by dx is becoming positive the fluid will have to do what? So, I am trying to enter a compartment of a Bombay local train I have 1000s of people trying to come out I have to put in that extra effort and I may succeed in getting into the train or I may fall back into the platform. So, when this I am able to overcome the resistance offered by the people I enter the compartment. So, for small values of positive pressure gradient the fluid probably is able to be attached when this Dp by dx starts to increase and increase and increase what happens like instead of 5 people coming out 50 people are coming out of the platform I am just not able to enter I design myself to be waiting for the next train. What happens to the flow? Flow does not have a choice it is just taken off from the solid surface. So, the reverse flow which is happening is because of the positive pressure here positive pressure gradient is not good that is why it is called as the adverse pressure gradient. The good or favorable pressure gradient for the flow is Dp by dx is less than 0 Dp by dx is less than 0 becomes equal to 0 at that point all of us would have drawn in our exams velocity distribution would be they will show a vertical line or tangential or something like that we would draw and then afterwards we will draw a reverse flow and the velocity profile will be drawn. So, the reverse flow or separation will happen because of the adverse pressure gradient 5 people are trying to get down from the train you are trying to enter the compartment you do not have to do much you know can you can push them across and get it only 10 seconds the train is going to stop. So, you will somehow get in with less effort 50 people are trying to get down you have to fight across for 50 people and there is a good chance that you might not be able to enter the train. So, precisely what is happening the fluid is going to do the momentum of the fluid is going to keep it in motion in that direction when the obstruction from the pressure the adverse pressure gradient becomes larger than what it is able to match it will have to go in the backward direction. So, it is a common sensical because it is not a the magnitude is of one of the sides is becoming larger than the other one. So, other one is going to win the fluid will have to flow back in the opposite direction that is why there is a reverse flow. In fact, if you people would have drawn this also several of you give this right. So, I will have like this and then at d p by d x equal to 0 I will draw this and at d p by I will draw this this is d p by d x equal to 0 d p by d x greater than 0 d p by d x less than 0 I think we all seen this. So, this is the point where that correct my equation prove if I am wrong u d u by d x is equal to 1 by rho minus 1 by rho d p by d x this is the equation right what is the left hand side of this equation it is essentially the momentum associated with the fluid correct. So, this momentum associated with the fluid is balanced by the pressure gradient correct when d p by d x is negative in a flow which is the normal good flow it is called as a favorable because it is favorable for the flow to occur. So, this value is normally negative that means the value is positive u d u by d x is greater than 0 when this becomes 0 d p by d x becomes 0 u d u by d x is equal to 0 this is where you get d u by d x is equal to 0. Now, when d p by d x becomes positive u d u by d x becomes negative what does this mean u is not negative or d u by d x is not negative u d u by d x becomes less than 0 why not u? Why not u? u also becomes negative right. So, the flow reverses here in this if you see here the flow is going to reverse you should have a negative velocity correct negative velocity. So, this is the primary reason for separation to occur that means the momentum of the fluid is incapable of overcoming the pressure gradient when the pressure gradient was favorable the momentum it was not a problem once it starts to become adverse it can overcome for some time afterwards it will have to give up and it will have to resign itself to separate from the solid surface. So, this occurs at about 80 degrees for laminar flow Nusselt number increases with increase increasing theta as a result of good mixing in separated flow region that is why you see a spike there if you see the plot here this is where separation is occurred then you see a spike because of the mixing at the region of separation and then it just continues to go up like this. Now, if I go for turbulent flow which is the other Reynolds number there will be two minima one is primarily sharp increase at about 90 degrees is due to transition from laminar to turbulent flow second thing separation occurs further downstairs here that is near about 140 135 why should separation be delayed at higher Reynolds number why should separation be delayed I said 80 degrees for laminar flow and then I am showing the same thing at about 135 yeah madam. So, what so what what does it do you are right velocity profile is fuller so what does it mean look at look at this equation what does this equation tell you in turbulent flow the fluid will possess lesser or greater amount of momentum greater amount of momentum. So, it has more strength to be able to withstand any adverse pressure gradient geometry is the same only thing is velocity has changed the flow has become turbulent when the flow has become because the flow has become turbulent inherently it has a greater amount of momentum associated. So, as a wrestler can enter even with 50 people trying to come out of the train you and I cannot enter. So, that is the we are laminar flow the wrestler is a turbulent flow you can just muscle his way through. So, when I when the fluid carries larger amount of momentum it is able to you know fight the adverse pressure gradient to a greater theta value afterwards finally it has to be. So, the separation is delayed at a point you have to take yeah these curves are important of course, two things to note separation comes in fluid mechanics all of us would have studied but students will not have understood why separation is delayed in turbulent flow and after the separation the mixing in the wake region what is the wake region what is the wake region it is said here you know muscle number increases with increasing theta as a result of intense mixing sorry sorry sorry as a result of intense mixing in the wake region see here and again here also. What is this wake region? So, at after the flow is separated there is a good mixing in the layer in the region there and that is called as a wake region. So, there because of good mixing heat transfer as a result. So, whenever there is a good mixing the a beard turbulence or due to separation you are going to have an increasing heat transfer coefficient. Because that is the only velocity I know I have no idea I have no clue whatsoever about the fluctuating component that is that is the reason always we try to represent everything in terms of average that is the mean flow. Yeah as we see the Reynolds averaging equation we are having the mixing in different planes like u u bar v bar correct minus rho u prime v prime bar yeah again these turbulence models for example k epsilon attempt is made to represent that eddy viscosity is again like defining eddy viscosity into del u bar by del y only I am imposing I am trying to make it look like tau t equal to minus mu del u by del y tau laminar equal to minus mu del u by del y tau turbulent equal to eddy viscosity epsilon m into del u bar by del y point is because the easiest thing I will be knowing is the mean flow I can have no idea about the turbulent flow. So, essentially it should be average it will be the average velocity. To answer your question even little bit more basic when I am doing an experiment or as a student what velocities do I know? Average. Only the average velocity we do not even bother do we bother when we do when we tell the when you are designing a pipe for a house or for carrying water from a pump to the upper tank velocity at the center should be so much at the very very sure we just say average velocity. So, this is just because of our limitation to measure the velocity, but it should have certain kind of certain potential which will affect this definitely if we are able to measure there is nothing like problem is we are incapable of doing that yes sir it is a limitation. So, bunch of correlations are there for various geometries and I am not going to go through this we can take there is a problem for cross flow across a cylinder and Churchill Chu correlation which was presented there. Right time to take your question. Whose question? This question about that film temperature. So, here we take the properties at the film temperature see there are correlations based on the wall temperature also and the fluid temperature also, but the properties within the boundary where should I take the properties actually within the boundary layer, but it is quite difficult to take the property variation within the boundary layer, because I need to know the temperature variation within the boundary layer. Here again I know only two things what is that I know T s, T s and T infinity. So, it is the again the engineering necessity which drives me to take the properties at film temperature. Few people have given correlations in papers only on fluid temperature only on wall temperature, but it is kind of customary I use this word very consciously it is customary to use these properties at film temperature at film temperature to have the pumping power requirement, but I had asked you the question whether it is equally important when we are working with the external flows because as we see. You had a question yeah how will I decide whether see now let us take an example let us say I have a heated plate I have a heated plate now I need to cool that there is some limitation for me for some reason I have a metallurgical limit that my temperature of the plate cannot go above let us say this plate this is my plate and now I will ask you a question come on decide me the fan capacity or a blower capacity so that I can cool this such that my temperature of this plate is not going to exceed in any case 100 degree Celsius how will you make sure heat flux is fixed for some reason heat flux is fixed how do you do that how do you do that can I put any fan according to you it doesn't matter any fan I can put but how do I put how do I decide the fan capacity yes I need to know the velocity which I have to pump at the inlet so that will decide my Reynolds number that will from that Reynolds number I can go back and calculate my heat transfer coefficients from the heat transfer coefficients I know the heat flux I will calculate back calculate my wall temperatures if my velocity what I chose will yielded my wall temperatures lower than 100 I am safe otherwise I will have to increase my Reynolds number I have not yet reached in natural so your question will again get postponed to natural connection this is when I am solving it by force you can okay naturally we will handle when we get naturally naturally we will handle