hello and welcome to our next lecture in aerodynamics last time we drove deeper into the world of incompressible and inviscid flows by introducing elementary flow fields the building blocks of aerodynamics four main building blocks the uniform flow the source and sink the doublet and the vortex were combined in various ways to make four more complex flows we covered the flow around a semi-infinite body a ranking oval a stationary cylinder and a rotating cylinder by making flows with streamlined patterns that looked like these bodies we could recreate the flow behavior as if the body existed in our flow today we'll explore the rotating cylinder in more depth in order to introduce the cuttachikowski theorem a powerful lawn aerodynamics that relates a flow's circulation to the lift produced by a body so let's jump in if you remember in the last video we ended with the stationary cylinder and the rotating cylinder let's recreate their streamline patterns here for the stationary cylinder you might notice that the flow is symmetric about the vertical plane meaning there is no drag produced by the flow and it is symmetric in the horizontal plane meaning there is no lift in the rotating cylinder our streamline pattern is again symmetric about the vertical plane so again there's no drag in the flow however across the horizontal plane we have asymmetry this means there is a non-zero lift produced by the body so what's the main difference between these two flows that causes the asymmetry the case of the rotating cylinder is really the case of the stationary cylinder with the addition of the point vortex elemental flow and that point vortex leads to added circulation which leads to lift this is the heart of the kodachakowski theorem an equation that relates the lift per unit span in a flow field to the circulation along with the flow velocity and density so circulation added to a flow indicates lift interestingly this relation was found independently by cutta a german mathematician and jakowski a russian physicist at the turn of the 20th century so they have to share the glory our goal for the majority of this video will be to try and calculate the lift force from the velocity field of a rotating cylinder because that leads us to the kodachikowski theorem through an example from our elemental flows video we derive the velocity field from the stream function so let's write them down again in their entirety remember we're in cylindrical coordinates still so the velocity is in the radial r and as muthal theta directions we want to get the lift force from this velocity field and to do this we're going to need to get the pressure from the velocity field and then from the surface pressure we need to get to the body force we will prefer to work with appropriate non-dimensional coefficients like the lift drag and pressure coefficients instead of dealing with the raw quantities step 1 we'll find pressure from velocity the pressure coefficient is defined similarly to the lift and drag coefficients where the change in pressure relative to the free stream quantity is divided by one-half rho u squared commonly known as the dynamic pressure we assume we know the dynamic pressure but how do we get to delta p from the velocity field well if our flow is incompressible and inviscid which it is we can assume bernoulli holds true for our flow and the bernoulli equation relates a flow's pressure to the velocity so write out the bernoulli equation between two points on a streamline point one we will take to be far from the foil so the free stream quantity is denoted by the subscript infinity will be used and with some rearrangement we can set the difference in pressure delta p equal to a function of the flow density and velocity difference let's plug this back into our definition of the pressure coefficient do some simplification and we find we can define the pressure coefficient from velocity alone now we have a way to take what we know the velocity field and say something about the pressure note we're going to be concerned with the flow along the surface because we want the pressure distribution on that surface for a circle the flow on a surface is defined to be the theta velocity at the radial location of the circle radius where r equals big r on to step two we want to turn the pressure distribution into a force this is something we were exposed to a few videos ago when we discussed aerodynamic forces recall our equations for finding the normal force and axial force of a body from integrals of the pressure and shear distribution on the upper and lower surfaces of that body let's write them down out and in their entirety for us flow is inviscid so all those pesky sheer terms go away these equations are for an arbitrary body like an airfoil but we can take some shortcuts because our shape is a cylinder first we can turn the surface coordinate s into the x and y components notice for a second we have popped back into the cartesian coordinate system but don't get comfortable it doesn't last that long and since our body is on an axis the center line of the cylinder is effectively aligned with the flow velocity which means the lift and drag are equal to the normal and axial forces respectively let's make these simplifications into the above equations and write out what we have so far we've already non-dimensionalized our pressure into the pressure coefficient so we need to express our equations in terms of lift and drag coefficients too in the lift and drag coefficient equations we typically deal with airfoils that have a cord c but here we will replace that c value with the cylinder diameter d which is our important streamwise length scale okay so we've got them in cartesian form but that's no good for our round flow so we'll have to transform back into the cylindrical coordinates apply the transformations for x dx y and d y to theta d theta r and d r and for convenience we'll write out the diameter as two times r the cylinder radius since r shows up elsewhere in our equation and we can cancel it out after the transformation we're back to being uncomfortable in our cylindrical coordinate system these integrals deal separately with the top and bottom surfaces because a lot of aerodynamic bodies are different on the top and bottom however for a cylinder the top and bottom are equal so we can combine our integrals and integrate around the entire cylinder surface from 0 to 2 pi before we put our pressure coefficient into this equation let's try to get it in its final form take what we derived above and plug our velocity field for a rotating cylinder into the pressure coefficient equation we can expand our squared parentheses and get it a bit easier to work with for the upcoming integration and now we're ready to calculate the lift and drag force for our flow finally and just in case we forgot what it looks like let's sketch the flow again here we notice at the beginning of the lecture that because we have symmetry about the vertical axis we expect there is no drag in the flow let's check that take our drag equation derived above and plug in our coefficient of pressure there are only four terms to worry about here in the integral all multiplied by cosine theta conveniently sine and cosine functions are oscillatory about a mean of zero that means many of their products integrate to zero first the integral of cosine theta from zero to two pi is zero which gets rid of two of our terms also the integral of sine squared times cosine is also zero getting rid of our second term and the integral of sine and cosine is 0 because of their anti-correlation so we can get rid of our third term with no terms left we can confidently say that the drag of this flow is analytically zero just like we anticipated note this is only for an incompressible and inviscid assumption now let's move to the lift with a similar approach we said the asymmetry about the horizontal axis means there is non zero lift let's see what this is take the lift coefficient equation and plug in our pressure coefficient again we have four terms in our integral and again we'll use some convenient properties of the integrals of sine functions so that we can get rid of all but one of our terms this is because sine squared does not oscillate about zero therefore it has a non-zero integral let's do out the integral for that term realizing that most of it's a constant and we find that the lift coefficient is equal to the circulation over the cylinder radius and the flow velocity let's turn this back into the lift per unit span and simplify where we can and finally we arrive at an equation relating the lift per unit span to the flow circulation and this is known as the cuttachikowski theorem importantly although we derive this for a rotating cylinder it actually works for flows over bodies with arbitrary shape making it a powerful tool in aerodynamics let's say we have flow over an airfoil far from the foil we know flow is inviscid however near the surface in reality we have this pesky boundary layer region where the viscosity is important let's draw a loop around the foil purposefully enclosing all of the area where we can't assume flow is inviscid meaning that flow outside of our loop isn't visited entirely and calculate the circulation of the loop as long as all of the flow circulation production is accounted for meaning we have enclosed all of our viscous flow we can use the cuttachikowski theorem to get the lift you might be wondering how this works because our airfoil isn't spinning like the cylinder was well the curvature of the airfoil and its angle caused the flow to rotate around the body which adds this circulation however keep in mind i am usually careful to say that circulation indicates lift it does not cause it lift and drag forces are made by pressure and shear stress distributions and unless there's an odd body force that's it for the most part surface stress is the only way a fluid can force an object circulation is merely the footprint of the lift in a way cuttackowski is analyzing the size and depth of the footprint to say something about the pedestrian walking nevertheless this is a hugely popular and powerful concept in aerodynamics and is widely used in practice you can see cuttachikowski in both experiments and simulations say you're working for boeing and you designed a new airfoil but now that you have a new airfoil you have to get its lift characteristics so you make yourself a model you scale it properly and you stick it in a wind tunnel you can get the lift a number of ways you could directly measure it with a force sensor otherwise you could somehow measure the surface pressure distribution and integrate that though i wouldn't recommend it or you could measure the flow field around the foil and calculate the circulation to get you the lift force from the flow field any option works really but a lot of the time you already plan to measure the flow field anyway to assess other flow behaviors in your measurements so getting circulation can be easy that way in simulations this theorem is at the heart of the vortex panel method this simulation is where arbitrary bodies are replaced by sheets of small vortices for the simulation to calculate the lift it looks at the circulation of the vortices in the sheet thereby using the cutter jakowski theorem and that's it let's review we started today by revisiting the stationary and rotating cylinder from our elemental flows lecture and noting that the only difference between them that could cause a lift force is the addition of the vortex which is driven by circulation we took the velocity field of the rotating cylinder with the goal of calculating the lift first we turn the velocity into pressure via the bernoulli equation then we calculated the lift force from that pressure distribution by calculating the lift and drag directly from the velocity field we confirmed our suspicion that the lift force was non-zero and turns out to be directly correlated to the flow circulation this example reveals the katajokowski theorem which works for any flow where the circulation can be calculated in a way that you can capture all of the circulation producing mechanisms and we ended with how you might come across karachikowski as an aerodynamicist i hope you enjoyed the video and thanks for watching