Conservation laws [Aerodynamics #5]

[@arjunmonga1054]

Thanks for this series! I'm using this as a crash course and I really appreciate it.

[@prof.vanburen]

No problem! Hope you're enjoying it

[@AinaweeUAE]

Thank you very much for these lecture series Prof. Van Buren! Really appreciate it and the notes!

[@prof.vanburen]

Aw thanks, glad you like them!

[@matteoliguori2544]

this is really amazing. truly don't mean to be whiny, but red text for equations does make this significantly harder to watch for me, (some weird stuff with dislexia and red text, actually very interesting topic). Since you decided to make this available on KZbin for free and as I know how much work this must have been, I guess you are passionate about making this as accessible as possible, one easy way to impove accessibility would be to just change the colour you use for equations. Never the less, thank you for all of the hard work!

[@prof.vanburen]

Thank you so much for the nice words and helping bring awareness to accessibility! I had not even considered the text color in regards to dyslexia. Do the white background version of the notes help out? This doesn't change the red problem, but maybe red on white is not the same as red on black. See, for example: sites.udel.edu/vanburen/files/2019/08/Lecture-1_How-we-study-aerodynamics.pdf. Would a black/white version of the notes be more helpful (although there would be some categorizing/style benefits lost in this)?

[@evanbw2629]

@@prof.vanburen Thank you for the great lectures! I also have real issues reading text like that, it seems to be related to the high contrast between page and text colour. There are such a variety of ways that people deal with it though; I often use a purple filter when I'm reading black text from a white page. It'd be impossible to cater to everyone with the recordings. However, might I suggest that you make the one note files available to download? Then people could change things like background and text size/ colours to suit their individual needs. It would also allow people to condense the notes, or pull out and group key concepts. Side note: on the website, adding a link to download the whole folder of lecture notes for each subject would be nice.

[@xiangwenyan3772]

Dear Prof., could you please explain more at "7:33 The reason for needing both of these is because of our fixed observation window". Is that compared with the case when we describe the issue with Lagrangian method? Thanks really.

[@prof.vanburen]

Hi! Yes, you at least have it right with respect to my understanding. If you follow the particle, as in solid mechanics, you would only need to account for the acceleration in time. However, when you fix your window and particles pass you by, you can get flow accelerating in space despite the fact that in your window flow is steady (no time-change in velocity). Please see my Material Derivative video from the Fluid Mechanics playlist, this is talked about in a lot more detail!

[@046_MDMAHFUZAHMED]

Professor in 3:25 you said, if fluid is leaving faster than it's coming in then this term is positive and in 3:41 you said, if fluid leaves our cube faster than it comes in this represents a negative change in density over time. So how for the same condition left hand side positive and right-hand side negative?

[@prof.vanburen]

Good question! On the LHS we are seeing the change in mass flow rate. Positive means fluid is leaving our box at a greater rate than entering. On the RHS we have the "mass" of our box, so if more fluid leaves the box than enters (a positive LHS term) then the mass needs to decrease in time (a negative RHS term). I hope this helps.

[@046_MDMAHFUZAHMED]

Thanks a lot Professor Buren. Your comment realy helps me to understand the topic.

[@zknowlton21]

Great video, really helped a lot. Thanks!

[@prof.vanburen]

Glad you liked it!

[@vardhanshah5283]

I believe assuming the flow is inviscid is a less common assumption, or am I missing something? Would love to hear from you professor, massive fan of your teaching style and content!

[@prof.vanburen]

Thanks!! Inviscid is certainly a *big* assumption, but it's surprisingly effective in a ton of aerodynamic applications.

[@x.ruizdealegria]

Good night Prof.! What is the second term of the momentum equation from the left? I have never seen 3 terms in the acceleration part of the equation, only the material derivative term

[@prof.vanburen]

Hey Xabier! Have you seen the full compressible version before? The traditional (and far more popular) form of the Navier-Stokes equations is the incompressible form where the LHS is just the material derivative. Here at NASAs website you can find an alternative version to this compressible form, where there are fewer terms with more variables left inside the derivatives (www.grc.nasa.gov/www/k-12/airplane/nseqs.html).

[@yempadayugesh8195]

Prof. are there simplified versions of the momentum equations which can be memorized for interviews. Like the external forces acting on a fluid element can be summed to the change in the momentum flux

[@prof.vanburen]

Hi! Sorry I'm so late to this. I try and remember the equations through the physical source of each term. However, this doesn't help if you need to be able to recreate the exact equation. For this, people use math symbol shortcuts (like the del operator) to get the equations more manageable, and then you just have to remember what those math shortcuts are and a much smaller version of the main equations.

[@mohammedouallal2]

I wonder why did you add the second term in the left hand side of N-S eq?

[@prof.vanburen]

Hi Mohammed and thanks for all the comments and following the series! Do you mean the rho*u*(derivative group)? These terms are for compressible flows only, they do not appear in the traditional incompressible N-S equations that you might see more frequently.

[@mohammedouallal2]

​Hi Prof. Van Buren, don't mention it. It is always a pleasure to see someone dedicated and sharing! I believe that the compresibility is taken care of in the continuity equation and should be used in the navier stokes in oder to get the final form.

[@tom_winguill]

I guess the shortened form of the Euler equations isn't correct professor. Density is also a variable, so, the derivative of a uvw form must result in three terms. But the result is as if density is a constant.

[@prof.vanburen]

Hey thanks for bringing this up! I've always found the mathematical shorthand a bit confusing, so I'm not surprised I represented it confusingly. I checked with the Anderson text to be sure, but this is how it is represented. Sometimes I make mistakes in converting from mathematical shorthand versions (Anderson) to the expanded differential forms, but I am not sure that's the case here. Note that I find the NASA N-S equations to be most useful to look at as they also don't use the shorthand version www.grc.nasa.gov/www/k-12/airplane/nseqs.html, whereas Anderson's book does. I should note that the shorthand below the boxed equation in 12:26 is only the shorthand version of the x-equation, so technically there would be three total of those shorthand versions. This was not clearly labeled or mentioned, as far as I can tell. My fault! I am not quite sure exactly what you mean by "Density is also a variable, so, the derivative of a uvw form must result in three terms". Maybe the confusing bit is in my expansion of the left-hand terms it is really partially expanded? I am treating rho*u as a variable, and velocity vector V as a variable, and doing the product rule on those. In this case, density and u are still variables, so that could product rule again into a third set, but to be consistent with other forms of the N-S equations this isn't always done. I'm not sure if this helps at all?

[@tom_winguill]

@@prof.vanburen Ok, my bad, you just took rho*u as a single entity so that it will look consistent with NS equations. I just got a bit confused because usually the product rule of differentiation of three variables result in three terms, I was just wondering if you are considering rho as a constant, so that you have only two variables to differentiate, so that you got two terms. Sometimes I make these very silly mistakes. I must have thought it twice before asking you and giving you a trouble. I am very thankful to you for clearing my every stupid doubt. I feel very lucky to find you. Next time I will think it twice before it was posting here. I hope you make more courses on aerospace subjects. You have a lot of potential in teaching. Just bring that potential to kinetic and you can take this channel to greater heights😉.

[@aleksandrekobeshavidze559]

Super Danke!

[@prof.vanburen]

Thanks so much!

[@hrkalita159]

I think your momentum equation is not correct

[@prof.vanburen]

Can you be more specific? What's the correct one?