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Mathematics for turbulence research and study 

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December 6, 2022, 21:27 
Mathematics for turbulence research and study

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Hello everyone~!
I'm interested in turbulence. So I used to read some books and papers but it is hard to what they mean. The reason that I can't understand is caused by that I'm not good at mathematics. Just to arrange lectures that I've taken, there are vector calculus, linear algebra and engineering mathematics 1. Contents that I've experienced in engineering mathematics 1 are just method for solving differential equations such as below(The reason that I've written below is because each different major handles different parts in engineering mathematics.)  separation of variables  exact / non exact differential equations  homogenous / non homogenous linear differential equation  2nd order linear ordinary differential equation  ordinary differential equation with constant  homogenous Euler Cauchy equation  method of undetermined coefficient  variation of parameter  higher order ordinary differential equation  power series  Laplace transform Uppers are what I've learned until now. And also I'm going to learn engineering mathematics 2 soon. Contents that I'm going to learn in engineering mathematics 2 will be below  Fourier Series / integration / transform  Complex function  Integration of complex function  Taylor Series  Laurant Series  Residue theorem / integration  Conformal mapping But I'm not sure these are enough. When I read turbulence paper, there were lots of things that I can't understand mathematically. For example  Pass filter  conditional correlation coefficient  spectral density function  StokesHelmholtz decomposition  conditional average(expectation)  two point correlation  cross spectral density function  probability density function I think I need to learn mathematics more. So I want to be recommended subjects or lectures that can cover uppers and mathematics for turbulence research/study. I think there can be probability theory and statistics. But I don't know what else there are. And also I want to ask one more thing. When you forumers are stucked by mathematics or formula in paper, do you guys just neglect those things? For me, it is hard to understand meaning of author's intention or physical situation without understanding mathematical formula. And also once I try to study unknown mathematical part, I had to retrace previous part consistently because that part is related to many other parts often. What do you think about it? I want to listen your advice and wisdom. Thank you~! 

December 7, 2022, 07:17 

#2  
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The way it worked for me was, from what I can get of your posts, quite opposite to yours. I didn' really delve into actual research papers before being actually ready for it, which was around the time I started my master thesis. Before that, what I did was massive reading of any relevant textbook in fluid mechanics and numerical methods at large (of course, also according to my financial capabilities, but back then almost everything was below 50 bucks). The main advantage here is that yes, you can skip a topic you seem to not digest, and later find it again in a different book with a different exposition. And it goes over and over (say, around ten books for each topic) until you finally get it. Research papers, in contrast, may be quite unique in their exposition or just plain wrong and you wouldn't have a backup. I am pretty stupid, but I found I have this little capability to put things I don't understand in a mental folder, and seemingly related things I don't understand in a folder close to that folder, etc. And I can do that at a pretty large scale (as there are a lot of things I don't understand at first). At a certain point you just get it because you've read so much about it, probably more than most people. I also found that in order for this to work I had to let go any fear of math I don't understand. Not understanding certain math can be as natural as not understanding someone speaking, probably even bad, a different language. Again, however, this works for textbooks, where a larger picture can still be drawn without figuring out single parts. For a paper it really doesn't make sense, you can just read the abstract for that. 

December 7, 2022, 20:00 

#3  
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I'm going to read textbook often. 

December 8, 2022, 06:18 

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u need mathematical physics background esp statistical mechanics and quantum field theory (group theory, functional analysis and more.) also, u must get familiar with basic experimental techniques and results as well. this is a multidecade effort.


December 8, 2022, 11:56 

#5  
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Filippo Maria Denaro
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You research area in turbulence could be in theory, experiments, numerical simulation. Each one of this area would require more specific studies after you have read the general introduction to turbulent flows. 

December 8, 2022, 20:35 

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I think it is far from fluid mechanics. 

December 8, 2022, 20:41 

#7  
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So after learning RANS modeling, I've read some other paper. But I'm stucked by mathematics. I couldn't understand their formula. So I was trying to study that part of mathematics but it was related to other parts. In my opinion, because I have no fundamental of that subject, I had to trace back to previous part infinitely. This is why I'm trying to find specific mathematic subject for turbulence. What do you do at situation like this? 

December 9, 2022, 05:27 

#8  
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If you stay in the continuous realm where 99% of this things are nowadays studied, you are fine without quantum. However, statistical mechanics is kind of the actual ground for fluid dynamics and something you want to study at a certain point. You can still do this without disturbing quantum theories, but certain topics in statistical mechanics actually need them to get correct results. It really depends from how deep you want to go. I never delved in to it and would never mention this to someone asking like you. 

December 9, 2022, 05:28 

#9  
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see Developments in the theory of turbulence: Leslie, D. C. Also the McComb Books. 

December 9, 2022, 23:59 

#10  
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I've thought one youtube video that treats statistical physics of turbulence (https://www.youtube.com/watch?v=3frL...ndex=1&t=3756s) I think I can refer to this. Thank you~! 

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mathematics, turbulence, turbulence analysis 
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