Lifting line theory [Aerodynamics #16]

In this lecture, we derive Prandt's famous Lifting Line Theory. Essentially, this theory models a finite-wing + wing tip vortices as a horseshoe vortex, where the wing is replaced by a vortex and the strength distribution along the vortex represents the wing's lift distribution. We go over the example for an elliptic distribution of vortex strength and why that distribution is particularly important to Aerodynamics.
Free downloadable notes (PDF with white background) can be found at my website: sites.udel.edu...

[@windsurfingphd]

I am a retired computer engineering professor, and have been building and flying model aircraft since my childhood from kit plans. But, now I have time in my hands, I started learning the fundamentals of aerodynamics to be able to design my own model aircraft, and understand the concepts of AoA, lift and drag etc tied to a design. This is by far the best lecture I have seen on lifting line theory period. I have seen others in KZbin, and read articles and book chapters, but this is the best :-)) Can you please prepare another video (if you have time, of course), and give us a good example of how to design an entire aircraft design, including the design goals, airfoil selection, moments, angle of wing incidence, tail design etc because I believe this example will connect all the dots of your very articulate lectures. Thanks again.

[@tom_winguill]

I vote for this request. Please do a lecture series on aircraft design professor. It will really make this channel complete.

[@prof.vanburen]

Thank you for the kind words! I am certainly looking to expand the content of the channel, and will definitely consider something more geared towards flight mechanics/aircraft design. (I guess, first, I would have to learn how to design an aircraft!)

[@rolandotillit2867]

Fantastic explanation for the logic and how the logic determines the math. You sir, truly understand because of the clarity in your explanation.

[@prof.vanburen]

thank you so much!

[@setsototitipana8433]

Loving the series with every video😍 but sometimes it's hard to see the numbers and some parameters used in equations😩 if only the notes were given in the description to make it easy to follow along. But the explanation in general is 👌👌 I am loving it

[@prof.vanburen]

Great suggestion! I've linked my website where the notes can be downloaded in each video description.

[@osamaarefi6770]

Thnx prof you saved my day❤

[@prof.vanburen]

Glad to help!

[@flth1040]

There is an error in integral calculation in 15:28. Bounds of integral must be from pi to 0 otherwise lift force will be negative.

[@prof.vanburen]

Thanks for clarifying!

[@shrishkumar6018]

Thank you so much sir!

[@prof.vanburen]

No problem!

[@jonasdondi698]

Thank you Sir for this. I have a question at 16:16. Where did the minus sign go for the final equation for induced angle of attack?

[@prof.vanburen]

Good eye Jonas! I should have dropped the minus sign slightly above this on the gamma term at around 15:26.

[@romboom2405]

Good morning sir ! Thanks for this amazing lecture Regarding the discrete solution of LLT do you know why control/collocation point are chosen to be at 3/4 chord. The reason is usually to ensure flow tangency and recover the 2D incompressible airfoil slope but I never found any mathematical answer about the second one. Thanks for your answer

[@prof.vanburen]

Hi! Sorry for the late reply. Can you expand on your question regarding the 3/4 chord? I am not sure I'm following.

[@stasiekpiekarski]

Why in Biot-Savart Law (4:00 ) you divide by 4 not by 2? There is: v=Gamma/(4*pi*h) not v=Gamma/(2*pi*h)? Is it an error or I'm missing something? 2*pi*h is the circle length and multiplied by velocity gives circulation. Two my knowledge 4 in the denominator appears im the following form: dv=Gamma r x dl /(4*pi*r**2)

[@Mi-nr3wp]

For a semi infinite (0=>infinity), the induced velocity is half that of the infinite conditions (-infinity=>infinity)

[@prof.vanburen]

Sorry I was late to this and thanks to @Mi for covering it!

[@stasiekpiekarski]

@@prof.vanburen Thank you both for the answer! Great lecture!

[@davidbeckschulte8246]

6:40 why would gama(z) be an ellipse? love your content by the way, thanks for reading (sorry for my bad English)

[@prof.vanburen]

Hi and thanks for the kind words! At this point in the video, nothing forces gama(z) to be an ellipse. When you add up the contribution of an infinite number of horseshoe vortices along the lifting line, you consider the induced velocity at a point from all the neighbors, which results in a curve distribution of gama. However, assuming elliptical gama distributions is common, and that leads to elliptical lift distributions which then leads to the elliptical planform wing.

[@davidbeckschulte8246]

@@prof.vanburen , thank you for the reply, hope you have a great day

[@gloxe1282]

Thank you so much for the videos Sir! this is very helpful. Do you have any recommended reference for me to see an example for the detailed explanation for the process of using Lifting Line Theory with a General Lift Distribution (in a tapered wing if possible). This would be very helpful. Again, Thank you soo much for your videos sir! This is very helpful.

[@prof.vanburen]

You're welcome and glad you enjoy them! A good place to start would be Anderson's Fundamentals of Aerodynamics book, it's really stellar.

[@gloxe1282]

@@prof.vanburen Thank you for the answer, Sir! I have another question, if i want to use LLT for a tapered wing with the steps that you showed in 19:26, do i have to calculate the Fourier Series for the specific wing shape? or the steps applies to all planform wing shape (the first step particularly, where we assume with an elliptical wing distribution)? (which means i can use the elliptical distribution as a start despite the planform wing difference), but from here i dont have a clear view on how it will end up to be the distribution of the tapered wing instead of the elliptical one. I am confused at this. Thank you so much for your hard work!

[@mickolesmana5899]

And generally, the performance is hindered by reality. Some mad purple titan: "Reality is often disappointing"

[@prof.vanburen]

When the tail perfectly balances the main wing moment...some big dude: "Perfectly balanced, as all things should be"

[@davidbeckschulte8246]

could you do a video on sweet wings and the "Mittelefekt"?

[@prof.vanburen]

Oh! To be honest, I had to look up specifically Mittelefekt, which I understand is a term regarding the loss of lift near the root due to spanwise flow on swept wings? I certainly can consider a video on three-dimensional flows over finite-span wings after I finish up my Fluid Mechanics series (it's a bit more of an advanced aerodynamics topic). The Airfoils and wings video has some of the flow control devices used to mitigate these effects, but you're right in that I don't ever get into three-dimensional flow details.

[@davidbeckschulte8246]

@@prof.vanburen , i don't know if the mittelefekt emerges due to the span wise flow, but as far as I understand it (not very far) it does describe the loss of lifting force near the Roth. thanks again for answering, have a good day :)

[@SaraKhan-uz1vc]

somebody make a pdf of this

[@prof.vanburen]

I did! You can download all the videos as PDF notes at my website: sites.udel.edu/vanburen/education

[@SaraKhan-uz1vc]

@@prof.vanburen oh great thank you!!! one question is it bad if i do the lifting line theory on a swept wing mini aircraft?

[@prof.vanburen]

@@SaraKhan-uz1vc It depends on how mini the aircraft is and the foil geometry. Embedded in lifting line theory is thin airfoil theory. Low Reynolds number foils tend to be thicker, and that moves further from being considered a "thin airfoil". Also, when viscosity becomes more important then we can no longer assume inviscid. I think it could be okay for a first estimate, but I would be a bit cautious until I knew more about the flyer itself.

[@SaraKhan-uz1vc]

@@prof.vanburen airfoil is the NACA 4415 with the Reynolds number of 500000. The aircraft is a flying wing the weight limit is 16 OZ!

[@prof.vanburen]

@@SaraKhan-uz1vc The NACA 4412 airfoil at that Reynolds certainly follows thin-airfoil-theory behavior (specifically the lift-curve slope of 2*pi). airfoiltools.com/airfoil/details?airfoil=naca4412-il. I think lifting line theory is a good start to estimating performance!

[@dickdekker1734]

For a project we are working on calculating the influence of wingtips on induced drag. However, we were wondering if the lifting line theory can also be used for a wing which is not flat, but instead has winglets.

[@prof.vanburen]

Hmm, good question. I'm not sure it would appropriately capture the end effects as it will change things slightly (generally for the better with regards to induced drag). Potentially you might find measurements on induced drag of winglet systems in literature?

[@dickdekker1734]

@@prof.vanburen It was hard to find an appropriate way to calculate the induced drag of winglets. What we eventually found is that by predicting the downwash profile of a wing (by for instance using computer programs like OpenVSP or XFLR5). An expression can be found for the circulation, resulting in an estimation of the induced drag (as explained in the video). We are however not sure if these estimates come close to the real world. Moreover, we found that using the method of restricted variations that the optimal induced drag is obtained when the downwash is proportional to the local cosine of the dihedral angle. So when having a winglet at for example 90 degrees dihedral, the net sidewash should equal 0. This means that the winglet should not produce thrust, however the winglet causes a upwash on the main wing, reducing the induced drag of the main wing. This way, the winglet mainly helps the main wing perform better, instead of producing thrust itself. Some more considerations can be found in our Aerodynamics Assignment for the University of Twente: dickdekker.jouwweb.nl/induced-drag-of-winglets