 All right in the last lecture what we did is we took a look at the flat plate boundary layer and so we're looking at external flows force convection and what we're going to do now is we're going to move on to external flows over bluff bodies. We're going to begin by looking at flow across cylinders. So looking at force convection across cylinders this is important to engineers for a number of different reasons. One of the most common applications within heat exchangers, cross flow heat exchangers, are where you would have a tube bundle and flow coming across multiple cylinders which we'll be taking a look at in a later lecture. But essentially what we have are a number of cylinders so the flow over cylinders is of interest there. Another place could be mass flow sensors inside of automobiles for measuring the mass flow rate of air coming in. And a third application could be in the measurements of velocity for either experimental or HVAC applications heating ventilation and air conditioning. And these are quite often referred to as being hot wire anemometry or hot wire anemometers. And so if you're looking at HVAC applications you're measuring the velocity at a low frequency and if you're doing experiments you could be measuring it at a high frequency and so the difference would be your cylinder would be much smaller in the case of an experiment. So anyway those are three different applications of areas where we would have an interest in connective heat transfer over a cylinder. So let's begin by taking a look at what the flow field itself looks like. And so what we're going to do here we're going to take a look at flow visualization and there you can see on the top normal speed on the bottom I'm going to slow it down so that becomes slow motion and you can see the red points there denote separation points. That's where the boundary layer is separating and then downstream of there on this cylinder we have what we refer to as being a very broad wake with a strong recirculation zone. And so this has a fairly large implication on to the connective heat transfer that will be taking place around the cylinder because we only have attached flow on the front of the cylinder. And then once we hit the separation point we have what we call separated flow as you can see with the strong recirculation zones that exist. And so what we're going to do let's begin by taking a look at what the flow field itself looks like on a cylinder and then that will help us understand some of the physics in terms of the heat transfer. Okay so what I've drawn here is the flow over a cylinder up until a certain point and what we have to begin with let's see we have the free stream out here on the left and through Bernoulli's equation we know that the total pressure is going to equal the static pressure plus the dynamic pressure which is one half rho u infinity squared. And what I've drawn on the cylinder I've drawn pressure taps at locations one two and three and these are static pressure taps. So they're measuring the static pressure in the boundary layer coming over the cylinder and if we were to have an inviscid flow which no flow would ever be that way but if we had a flow without viscosity what we would find is that the velocity on the top of the cylinder would equal two times u times the free stream twice the free stream. Now in reality what happens is the boundary layer forms and the velocity does not behave in that way as we'll see momentarily. But with this what we could do with these static pressure taps we can then say pressure wall at one so that would be static pressure wall at location one is the total pressure we calculated the total pressure out here minus one half rho u one squared. Now I've drawn this right at the front and that would be a oops where did my cylinder go there we go that would be what we call a stagnation point and so really the velocity there would be zero but anyways let's just say it's u one for now p wall location two again would be p naught minus one half rho u two squared and then p wall at location three again the total pressure minus one half rho u three squared. So what's happening here the velocity is going from a very low velocity at the front and the flow is accelerating as it comes around the cylinder and consequently what is happening is the static pressure is going lower and lower and that that's a region that we referred to as being a favorable pressure gradient because as you're flowing along the pressure is getting lower and lower so the pressure is essentially pushing the fluid and causing it to accelerate so if we were to look at the velocity at these different locations we would have u one is equal to zero because that was what we call a stagnation point and then u two is greater than u one obviously because the flow is starting to move and accelerate around and then u three is greater than u two and this represents the flow accelerating around the body but from the flow visualization that we just saw we saw a case where we have this broad weight and and then a very large recirculation zone and that does play a very big impact or part in what is happening with the flow around the cylinder so let's take a look at what is going on with the pressure distribution around a cylinder and so bear with me I'm going to sketch out the pressure distribution here won't be the most accurate thing but anyway so it'll give us an idea as to what is happening okay so what we have that's my best effort of being able to show the pressure distribution and not at all pretty but anyways what's going on first of all we have the inviscid curve now inviscid that's what that assumes that there is no separation on the cylinder you can calculate this using potential flow theory and and that would be something that you'd take in a fluid mechanics course but basically a model the cylinder is a doublet with a free stream and and with that you can then come up with the shape and the pressure distribution around the cylinder and what we find from that is we have a functional form for the pressure distribution in reality however what is happening if we look at our cylinder the flow comes along the stagnation point is at the front everything is going good but it will depend upon the nature of the boundary layer so the boundary layer on the on the cylinder as the flow is coming up and around is a laminar we get here there will be a separation point of its laminar and if it's turbulent the separation point will be further delayed on the backside of the cylinder and that's why these two pressure distributions look different and and what we can see is that the turbulent boundary layer has a very different pressure distribution from that says laminar that should be laminar we should have the R in there the laminar boundary layer so and just by looking at the pressure distributions alone we know that there's something different going on let's take a quick look at a schematic in terms of what this might look like so if we have the laminar boundary layer let me just sketch a cylinder here so this would be the case of a laminar boundary layer and for the laminar boundary layer the separation point is that around theta equals 82 degrees so that's typically where you will find the separation point with the cylinder if you have a laminar boundary layer for me so we have our stagnation point right here and then as the flow is moving up and around we have a boundary layer forming and if that boundary layer is laminar separation will occur at about 82 degrees and it is symmetric so we would have a separation point that either of those locations and what the separation point means that the boundary layer is lifting off of the body and and so from the flow biz you could see we had kind of the structures coming off and then in the downstream we had large-scale vertical structures that they call that the von Karman vortex street but essentially it's a broad wake in the downstream and the implications to heat transfer is this region in the downstream is being impacted by the fact that you have all of this recirculating flow so this here is a separation point and if we're to look at the drag coefficient drag coefficient for a cylinder with the laminar boundary layer it's around 1.2 now it's going to depend upon the Reynolds number you got to be careful because a very very low Reynolds number would be what we call creeping flow and it would be a very different drag coefficient but that would be I don't know I'm guessing a couple thousand up to I don't know what the transition oh here we have critical Reynolds number three times 10 to the 5 so I'll get into that in a moment now let's draw the cylinder again and what I'm going to do here is this case we're going to assume that we have a different Reynolds number and the Reynolds number here would be where we have a turbulent boundary layer so this could occur in a number of different ways obviously if the Reynolds number was higher you can also sometimes trip the boundary layer on a cylinder what you do is you either put a wire or a sandpaper at the front and that can cause the boundary layer to transition but we would have our stagnation point the boundary layer is forming around here but let's say it then goes through transition it becomes turbulent when it goes to turbulent what happens is the separation point the flow is actually able to make it around the top of the cylinder here it is not it separates before it gets to the top but for the turbulent boundary layer it's actually able to make it around the top of the cylinder it's coming around to the backside and at about 120 degrees that's where we find separation for the turbulent boundary layer so theta approximately 120 degrees and with that consequently we have a much narrower weight and and that then consequently has implications on to the convective heat transfer on a cylinder and so here is CD 0.3 would be an approximation this is why golf balls are dimpled golf balls are dimpled in order to cause the boundary layer to transition and through the transitioning process the drag coefficient goes down and golf balls fly much further than if you only had a laminar boundary layer forming on the golf ball so that is the purpose of the dimples and they cause the boundary layer to transition okay so where does that transition take place the critical Reynolds number for a cylinder is about three times ten to the five and so that would be Reynolds number based on that would be Reynolds number based on diameter I would assume yeah it must be okay so three times ten to the five and that is where we go through the transition process and then your separation point moves further downstream you have very different heat transfer characteristics so that has an impact on convective heat transfer and obviously that means that we need relationships that will then be a function of Reynolds number and that's what we'll be seeing in a later segment of this lecture