Reynolds decomposition and Reynolds Averaged Navier-Stokes (RANS) [Fluid Mechanics #11]

In this video, we take a deep dive into the Reynolds Averaged Navier-Stokes equations (a.k.a. RANS). In practice, RANS is a valuable tool to have in your toolbelt when approaching real-world flow scenarios.
To get these equations, we need to apply Reynolds Decomposition, separating a time-varying signal into the mean and fluctuating components, to the Conservation Equations and then average them. We will explore the complexities of Reynolds Decomposition and Averaging, specifically with non-linear terms, and the rules we can apply to averages of mean and fluctuating quantities.
Then, we fully derive the RANS equations in their entirety, and show how we can use it to study turbulent flows in a way we couldn't when considering the traditional Navier-Stokes equations. An important feature that makes turbulence different from laminar flows are the Reynolds Stresses, which are analogous to viscous forcing on a macroscopic scale.
Free downloadable notes (PDF with white background) can be found at my website: sites.udel.edu...

[@danieldusentrieb2912]

I'm currently writing my bachelors thesis about fluid dynamics and your videos are incredibly helpful for catching up on the basics :D Thank you!

[@prof.vanburen]

Good luck with your thesis! What topic in fluids are you covering?

[@VascoBreitenfeld]

It's very impressive how this lecture breaks down a very complex concept, this helped me very much!

[@prof.vanburen]

Thanks and I'm glad you liked it!

[@erik_jan09]

thank you for very clear explanations, sending this to all my engeneering friends!!

[@prof.vanburen]

Glad you like it and thanks for sharing!

[@aidakiadsguyz4026]

Great and clear breakdown and explanation of the subject! Thanks a lot ^^

[@prof.vanburen]

No problem and thanks!

[@devins9402]

Very clear! Very good quality! Wish it can inspire more people for the science and engineering!

[@prof.vanburen]

Thanks Devin!

[@karanpannu4106]

Thank You sir for the great explanation. It was totally helpful not only in the view of academics but also in terms of the logical conceptualization background.

[@prof.vanburen]

Thank you so much!

[@Moonlight-gg3nf]

Thank you so much for a great effort in explain.

[@prof.vanburen]

Most welcome!

[@arashasadiabadi6013]

Thank you for your video!

[@prof.vanburen]

No problem!

[@unforgotable1]

Wow! Thank you for the great lecture.

[@prof.vanburen]

Thank you!!

[@GauravKumar-ue7nz]

So Good Lecture. Thank you.

[@prof.vanburen]

Aw thanks!

[@binbai2472]

Excellent lecture!

[@prof.vanburen]

Thank you!

[@eugenesadie7707]

Awesome video, thank you! For the X-momentum in the flow between plates example - can you do all the eliminations (steady, fd, 2d, etc) before you decompose and average to have less terms to compute?

[@prof.vanburen]

Hmm, good question. I haven't done it out the reverse way because I think you might lose important terms if you simplify at the beginning, but I'd have to check. Regardless, if you start with the RANS equations and eliminate from there, isn't it the same amount of terms to compute anyway?

[@vaibhavbandal8325]

Amazingly explained.
Can you please tell me what is the average of product of one averaged quantity and its gradient.
i.e. average of (u.bar)*(derivative of u.bar)?

[@prof.vanburen]

Thanks Vaibhav! I am not sure I completely understand your question, but would like to help. Can you expand a bit?

[@samueldaramola8303]

Very nice

[@prof.vanburen]

Thanks!

[@samueldaramola8303]

@@prof.vanburen could you show how to derive a distribution for the shear stress over the depth of a steady, uniform, 2D river flow that accounts for wind shear blowing over the top of the water from the RANS equations?

[@shahvaizkhan6279]

Can you please apply RANS to P = rho RT and show that
P/Pavg = T/Tavg + g/rhoavg

[@shahvaizkhan6279]

Or if you can guide me? I am unable to comprehend it according to your method.

[@prof.vanburen]

Once I get through the end of the semester let me circle back to this! It actually inspired me to give a RANS of P=rho R T question on my final quiz

[@shahvaizkhan6279]

@@prof.vanburen I figured it out!

[@bluemantle8269]

I don't understand tis part 16:55

[@prof.vanburen]

Hi! Which part at 16:55, can you be more specific?

[@corinnesmith829]

I'm also confused on the same section... why are we doing the product rule? I'm not sure where did the second derivative of u came from

[@corinnesmith829]

AHH figured it out, this lecture was extremely helpful THANK YOU!!!! :D