 The last topic that we're going to look at in this lecture considers boundary layers with pressure gradient. Now there are different types of pressure gradients that can exist external to the boundary layer and we can have zero pressure gradient which is quite often what is studied for flat plate boundary layer flow but you can also have conditions where there is adverse pressure gradient where the pressure is increasing as the flow moves into it or favorable pressure gradients where the pressure is getting lower and lower and the flow is consequently moving more easily because the pressure is driving it into the flow. So these are the different conditions that we'll be looking at. So what we're going to do we're going to begin looking at the x momentum equation and we'll consider it for the different conditions one where we have adverse so it's difficult for the flow to move into an adverse pressure gradient and favorable it'd be easier for the flow to move in because the pressure gradient would be driving it. So let's take a look at the x momentum equation. So here we have the x momentum equation the terms on the left hand side these are the inertia force terms and then this here is the pressure force term and that's related to the pressure gradient and then finally we have the viscous force terms and so within the fluid element all of these are in balance with one another and what we're going to do we're going to consider the case of looking very close to the wall because that's where the interesting things happen when you have a boundary layer with an adverse pressure gradient. So I mentioned that we're going to look at what's going on very close to the wall and what happens is a fluid particle moving right along the wall has to overcome whatever the pressure gradient might be and so the fluid particle could either be accelerated if we have a favorable pressure gradient or if we have an adverse pressure gradient it's actually being decelerated so in the case of an adverse pressure gradient because that's the one that becomes quite interesting to look at what happens is the fluid itself starts to lose momentum and it starts to slow down it's decelerating and you can actually get to the point where it goes to zero velocity and negative velocity and at that point we achieve what is called separated flow and so the boundary layer separates it's a very bad thing for an airfoil for example because that could lead to stall and and and then loss of lift but for other types of flows flows over cylinders we have separated flow it happens quite often well all the time and bluff body flows so but what we're going to do we're going to look at some of the logic behind what is happening and we're going to use boundary layer equations and boundary layer theory in order to study this and understand it a little better so what we're going to do we're going to consider the case on the wall y equals zero and we know with the no slip no flow condition u and v are equal to zero and I'm also going to say that the u prime v prime term which comes out of the turbulence and the turbulent stress tensor will assume that to be zero as well which it would be the fluctuations on the wall would have to be zero and consequently that term would be zero and looking back to the equation that we used a couple of segments ago when we were looking at the shear stress within the boundary layer you'll recall that we came up with a formulation enabling us to equate the pressure gradient to what was happening in the external flow that's because the boundary layer cannot sustain a pressure gradient normal to the wall and we can also equate that to the shear that is occurring along the wall and so that can be rewritten and so we get these equations here and these apply whether the flow is laminar or turbulent so what we're going to do we're going to look at these relationships and specifically what we will be looking at is we will be examining the first derivative of velocity with respect to y as well as the second derivative of velocity with respect to y and we will be doing this as well as looking at the velocity profile as a function of y and looking at how it changes under different conditions of external pressure gradient so let's take a look at that okay so looking at the equations that we just wrote out for the shear stress for an adverse pressure gradient where we have dp with respect to x greater than zero if we look back at our equations we have a relationship here between dp and dx partial or the derivative of pressure with respect to x and on the left hand side we have the second derivative of velocity with respect to y and so dp by dx is greater than zero as it would be for the the case of an adverse pressure gradient we know that the second derivative of the velocity of you with respect to y has to be positive as well but as we move out towards the outer flow region i'll show you with some plots here in a moment it it needs to become negative at some point and and and so the implication of that is whenever you have the second derivative of a function and if it goes through zero that is an indication of an inflection point and consequently we can make the statement so what that tells us is that when we're dealing with adverse pressure gradients there will be an inflection point at some point at some location above the wall within the flow field and and where that is has implications for what is going on within the valangular so what we're now going to do is we're going to take a look at a number of different velocity profiles for different types of pressure gradients and we'll start off with favorable and then we'll go to zero pressure gradient so what we have here is a case of a favorable pressure gradient where dp by dx is less than zero and this would be a case where we definitely do not have separation and the point of inflection if there is an inflection point it would be within the wall okay so we're now going to move on to the case where the pressure gradient is equal to zero so moving on to the case where we have the scenario of a zero pressure gradient in this case there will be a point of inflection there is no separation so the boundary layer remains attached and the point of inflection is at the wall and we can see that here by looking at the second derivative term we have a second derivative value of zero and consequently that's an indication that we have an inflection point but in this case the inflection point is right at the wall so we've looked at the case of a favorable a zero gradient now let's look at a weak adverse pressure gradient so in this case what is now happening is we have a point of inflection that is above the wall and we can see that the zero crossing with our second derivative is now also above the wall and that is the point where the first derivative of of u with respect to y is at a maximum and and it the second derivative would then be zero at that location and and consequently this is a case where we have no separation but the point of inflection is in the flow now moving on we're going to get to a the next pressure gradient dp by dx which is greater than zero and it's what we will call a critical adverse pressure gradient because what is going to happen the condition right along the wall is going to get to a certain criteria where the shear stress will go to zero and that becomes very significant so i've also drawn out our first and second derivatives of the velocity but the main thing is we can see that at this condition the slope or du by dy at the wall has now gone to zero and and so if we compare it to the previous one we were not at zero we had a finite value so tau wall did not equal to zero however now we have a condition where tau wall is equal to zero because of that and that is what we referred to as being a critical adverse pressure gradient and and that is the case of zero slope at the wall and that would lead to the onset of separated flow or separation and so when the boundary layer gets to this state it is no longer attached and the the the flow enters into potentially unless it can sometimes reattach quite often it does not reattach and then you get full blown separated flow downstream so in the case of an airfoil this would be very very detrimental you would not want to have this very far up far up along the airfoil because that would lead to separated flow install as i mentioned in turbo machinery it can occur and you get rotating stall all kinds of bad effects so the following on let's take a look at an excessive adverse pressure gradient that'll be the last case that we look at so this is a case where dp by dx is greater than zero it is an excessive adverse pressure gradient and this is also a separated flow region and you can even see that we have backflow and consequently the tau wall is less than zero so we have a negative slope at the wall so that is the evolution of the pressure gradient let's go all the way back here so we started with this case here that was no separation things were nice and clean then we went to the point where we have the inflection point right along at the wall and that leads to zero change in slope and that's why this value here is at zero and then we move along and we have the inflection point up above the wall but we still have some shear stress along the wall it's not zero and then as we move along and we get to what we call the critical adverse pressure gradient this is where you have zero shear at the wall and consequently your boundary layer at that point will separate and and then as you go further on you even get backflow where the flow along the wall is not able to overcome the change in pressure and consequently it starts flowing in the opposite direction and that could then lead to massive separation so if you're looking at something like an airfoil and you have the flow so there we have our stagnation point dividing streamline we can have flow coming over the top and we have flow coming over the bottom but what would happen in the boundary layer here you would then get the boundary layer it's starting to form from the front front front and then you get to your separation point and and you get a big massive separation region a lot of recirculation going on in there that would be stall so that that's very bad in the case of an airfoil in the case of a golf ball that's actually maybe not a bad thing because we put dimples in the golf ball and the separation point if there are no dimples in the golf ball it's going to be up here and if you have a laminar boundary layer the separation point will be up on the front but when you put dimples in the golf ball it actually generates a turbulent boundary layer and causes a separation point to move further downstream removing or reducing the drag the form drag due to the wake but in any event the separation point would be right in here and that is one thing that I should say that the golf ball idea brought this to mind and that is the fact that laminar flow is more susceptible to separation than turbulent flow and I should say boundary layer flow so if you have a turbulent boundary layer it would be less susceptible to separation than if you have a laminar boundary layer and the reason is because when you have turbulence you have a lot of large-scale activity and taking place along the wall and consequently the flow right along the wall has more momentum than it would if you just have a laminar flow and that's also why we put dimples in golf balls we put the dimples there in order to cause the flow the boundary layer to transition from laminar to turbulent and the the dimples are basically nucleation sites for instabilities which then lead to transition to the boundary layer and and what that does that causes the separation point to move back further and back here and that causes the drag coefficient to decrease when you have the dimples and the turbulent boundary layer on the golf ball so that's a final point about the boundary layer and how it relates if you have laminar or turbulent and the separation point so that is the boundary layer under adverse or different pressure gradient conditions usually when we study the flat plate it's always studied usually under the zero pressure gradient condition where you have dp by dx is equal to zero you'll sometimes see that assumption being made for flat plate boundary layer flow but just be aware that that is not always the case and when you have different pressure gradients it can have quite significant impact upon the formation of the boundary layer and it can lead to separated flow which really changes the flow regime significantly as you can tell with the separated flow on this airfoil or what's going on behind the golf ball so that is boundary layer flows we've looked at laminar we looked at turbulent we looked at von karmann integral we looked at blazius's solution in the last lecture what we'll be doing is looking at external flows on body so we'll be looking at lift and drag characteristics which we're kind of getting into it here when we're talking about what's going on in a golf ball but anyways that's kind of a transition from laminar or from the boundary layer flows into flows around other objects and bodies which is what we're going to be doing in the last lecture