[CFD] Conservative, Advective & Material Derivative forms of the Navier-Stokes Equations

A basic overview of the material derivative, conservative and advective forms of the Navier-Stokes equations.
Time stamps
0:00 Introduction
0:00 Introduction
0:57 Example Problem
7:56 Material Derivative
10:21 Navier-Stokes derivation
15:36 Surface and volume forces
19:36 Conservative and advective form
22:59 Derivation
27:39 Equivalence of forms
28:54 Other transport equations
31:40 Outro
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Disclaimer
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The methods, algorithms, equations, formulae, diagrams and explanations in this talk are for educational and demonstrative purposes only. They should never be used to analyse, design, accredit or validate real scientific / engineering / mathematical structures and flow systems. For such applications, appropriate trained, qualified and accredited (SQEP) engineers / scientists should be consulted along with the appropriate documentation, procedures and engineering standards. Furthermore, the information contained within this talk has not been verified, peer reviewed or checked in any way and is likely to contain several errors. It is therefore not appropriate to use this talk itself (or any of the algorithms, equations, formulae, diagrams and explanations contained within this talk) as an academic or technical reference. The reader should consult the original references and follow the verification and validation processes adopted by your company / institution when carrying out engineering calculations and analyses. Fluid Mechanics 101 and Dr. Aidan Wimshurst are not accountable or liable in any form for the use or misuse of the information contained in this talk beyond the specific educational and demonstrative purposes for which it was intended.

[@swap_pat29]

The best explanation of material derrivative so far

[@abrahakahsay]

Dr. Aidan, when I want to understand some CFD concepts, you are the first to come to mind. Thanks for making it easy.

[@lih3391]

Best explanation I've seen so far as a high school student. Thank you!

[@TahsinEngin-Akademi]

Excellent explanation of Material Derivative and conservative & non-conservative forms of transport equations. You are doing a great job. Please make a video on using of tensor notation in Fluid Dynamics.

[@josemonteiro5988]

As someone recently graduated i want to thank you for your videos because help me a lot during the making of my thesis!!

[@fluidmechanics101]

Well done on your graduation. Glad I could help 😄

[@kevint1160]

Dr. Adian. I would like to apologize first but where were you when i was doing my thesis 8 years ago. Great job. You rekindle my passion for Fluid Dynamics.

[@abdofast5]

Absolutely amazing explanation!! This is the first time i really understand this subject throughly. Please, keep sharing more knowledge you have got all my support. Bravo ❤

[@sergniko]

This was a useful talk

[@hungnguyenthanh8833]

I always look forward to your lessons that appearing everyday. Hope to see you more 😀😀😀

[@TheGamingHungary]

Excellent video, thank you Aiden!

[@alisadeghi4698]

Crystal Clear, Thanks!

[@anderssonbenitoherrera4026]

(The best explanation of material derrivative so far)^n

[@Rafael-es3go]

This is very well explained, your videos are amazing, thanks!!

[@Rafael-es3go]

just 1 thing: audio is available only in 1 speaker around 18:00

[@fluidmechanics101]

Noted. Thank you

[@malothrajkumar541]

Hi Dr. Adian, Thank you very much for your high quality educational videos on CFD, you’re one of my favourites!! and you inspired me in many ways. A small typo correction on one of the slide at 29:02. “ Remember to start with the *Conservative* form instead it is written Convective. Thank you again, I have taken all of your courses on Udemy as well❤.

[@abdelkaderalkelanee1187]

Thanks a lot

[@makavelilcf]

Great lecture Aidan! a small correction: conservative or divergent form is a form of N-S equation allowing to apply integral conservation laws using Gauss-Green theorem for example. It doesn't concern time term since that's a pure volumetric term, density comes inside just due to the introduction of the continuity equation into the advective form of N-S.

[@fluidmechanics101]

Great correction. Thank you!

[@mohammaderfankarami1744]

Hi dear, Thank you for your useful data. I really appreciate you. In the following, can I ask you to share some videos about spars matrix? and how does fluent solve spars matrix?

[@nourtarek9984]

Hello Aidan, I've purchased your courses on Udemy and wanted to know if they are any different than the ones on your website .. Thank you for your great content you inspire me ^^

[@fluidmechanics101]

They are exactly the same 👍 I have them on my website and on udemy because sometimes people have problems with payment processors

[@rza_ramezanii]

Dear Aiden, I sincerely appreciate your invaluable materials and your knack for simplifying complex concepts. As I endeavor to develop online courses focused on Python and its applications in mechanical engineering, I'm eager to learn about the tools you utilized for course creation. Could you kindly share the video editing software you employed? Additionally, which software did you use for crafting engaging figures and slides? Lastly, I'm curious if your website was built using WordPress. Thank you for your guidance. Warm regards, Reza

[@fluidmechanics101]

Figures: inkscape Slides: Beamer latex Video editing: shotcut All free software

[@rza_ramezanii]

@@fluidmechanics101 Thank you immensely. Your guidance will be remembered and cherished as I continue on my journey.

[@56rockit]

Sorry, @3:19 the explanation and the diagram is not quite correct: the faster we move the mesurement point from left to richt the diagram implies that the temp will rise. This is not the case. When moved faster the end tempererature will be reached earlier i.e. the rate of temp change will be steeper.

[@hcphysicist3921]

For meshing for thesis which should i learn icem or ansa??

[@fluidmechanics101]

Whichever one your school/university/institution has access to 😃 if you have both, it will depend on what problem you are trying to solve. They are both better at different things (Personally I prefer ANSA)

[@user-dj8vo9df3g]

Hi. I have an issue. I practically use CFD during my study, but because i use codes mostly written by other people i don't quite undertand all the details of methods implemented there. Don't have hierarchy of existing methods i CFD neither. For example i know there are FVM, FDM and FEM. Are there any other methods what are their MAIN features, advantages/disadvantages. What the difference betwen pressure/density based solvers. Next, as i work with code based on FVM i know some basic info about its methods and features i know such words as tvd an weno (i don't quite understand is that features of specific schemes or their "families"). Also there are different approaches to obtain convective fluxes such as flux/vector splitting shemes (f.e. roe, hll family riemann solvers, ausm family solvers, are there essential alternatives). Reconstruction methods for solvers such as MUSCL (are there any other, can it be used not only in tvd?). Such a big zoo of methods an approaches just to solve euler equations. And i don't feel i have understanding of the relations between them. I mean something like table or tree of hierarchy with short commentaries near its' components. I undertand there are your videos that touch some of these questions but they are specific to some particular problem. It would be just great if there were lecture on basic information about "everything that currently exists in CFD" just to build foundation for everything else. p.s. Sorry if I ask some childish questions. But when one start working practicaly and didn't have proper learning before it is so hard to figure everything out without good teacher.

[@kuijaye]

But why is the last form called the conservative form?

[@fluidmechanics101]

Essentially, the mass flux is calculated and stored on the cell face. So that way you can guarantee that the 'mass flow rate * the quantity' leaving one cell is equal and opposite the amount entering the neighbouring cell. None of the quantity is 'lost' numerically, so we say that it is 'conservative'

[@fluidmechanics101]

There is a good explanation in 'Notes on Computational Fluid Dynamics General Principles' with a good diagram to show this numerically

[@sameerverma6949]

Sir can you please make video on Eulerian wall film model of ansys fluent..

[@danielodeliro2412]

Thank u sir I have a question but its not related to your lecture Can you please explain concept of Static pressure Dynamic pressure Total pressure Mechanical pressure Thermodynamic pressure Stagnation pressure Mean pressure Average pressure And which one of pressure is in NS equation?

[@josemonteiro5988]

if i am not mistaken NS use static pressure, even if the input in the inlet is total pressure.

[@fluidmechanics101]

I would check out a general fluid dynamics textbook. You should be able to find the differences in there.

[@danielodeliro2412]

.

[@nicoskal85]

ⅆT/ⅆt=u ⋅(∇T) would you consider this form to be correct as well? Would you rather show that the whole dot product operates upon T? The idea behind the pool explanation is just brilliant - thank you for that!

[@fluidmechanics101]

Yes, this form is fine as well. As long as nabla dot U = 0, then they are equivalent. You only really need to care about the difference when you have a numerical method, because nabla dot U only equals zero at convergence. So before convergence, you might need to account for the difference between the two forms.