I know that spectral-methods which are weighted-residual methods can be applied to solve for the incompressible N-S equations. In particular, they are applied to Direct Numerical Simulations (DNS) for studying turbulence. However, I'm still unclear how the continuity condition is enforced in such cases, and what trial functions are chosen.

  • $\begingroup$ Have you tried looking for papers describing the codes that the people in the area use? There is material detailing the implementation of several of those codes. $\endgroup$ – Hydro Guy Oct 1 '15 at 23:47
  • $\begingroup$ I did, and with some shame I have to admit I'am still left with the same questions as I did when I started. I have just started using the nek5000 code, which has an extensive documentation that I have perused. However... $\endgroup$ – Amitvikram Dutta Oct 1 '15 at 23:49
  • $\begingroup$ I still don't use nek myself, a post-doc in my group is using it though, and liking very much. I have some papers on spectral-type codes for CFD, I can mail them to you if you wish, I just have to find where I left them $\endgroup$ – Hydro Guy Oct 1 '15 at 23:51
  • $\begingroup$ Sure, any additional material is appreciated. Unfortunately, nek uses fortran which I had looked at in disdain at the beginning of my programming education (ah the arrogant punk I was). As a result its slow going. If you could send the papers to my e-mail address I'd be very grateful. $\endgroup$ – Amitvikram Dutta Oct 1 '15 at 23:54
  • $\begingroup$ I'm look on my things and if I don't send it today, by the end of the weekend it will be sent. You are a C/C++ programmer? I had a similar learning history, but when I tried to learn fortran, I did only the bare minimum to be able to read fortran code from my previous advisors, personally I prefer C way more for number crunching, though there are some things in C that are frankly outdated in anno 2015. $\endgroup$ – Hydro Guy Oct 2 '15 at 13:14

For incompressible spectral DNS, often you'll see reference to John Kim, Parviz Moin and Robert Moser (1987): Turbulence statistics in fully developed channel flow at low Reynolds number, Journal of Fluid Mechanics, 177, pp 133-166, which includes details on how continuity is enforced. You might also find the book Spectral Methods in Fluid Dynamics by Canuto et al., Springer 1988 of interest.


Frequently some sort of velocity-pressure splitting is employed which gives a step in the method where the velocity and pressure are coupled directly.

  • $\begingroup$ I have the impression that the mass continuity equation if 'assumed' satisfied if one have $\rho=0$ and also $\nabla \cdot \vec v =0$, thus something like velocity-pressure spliting would only serve as to assure divergence-free velocity, but it doesn't explicitly adress the conservation of mass, does it? $\endgroup$ – Hydro Guy Oct 4 '15 at 3:28
  • $\begingroup$ @HydroGuy, I'm no expert in these methods. Just working from memory. Listen to Rhys. The Chorin splitting assumes incompressiblity. I think you could reformulate it for non-constant density, but not zero. $\endgroup$ – Bill Barth Oct 4 '15 at 3:39
  • $\begingroup$ I typed it wrong, I ment $\rho=$ constant, not $\rho=0$, it's late night and I'm tired, I'll correct the typo. Edit: I'm not fiding the link to edit the previous answer, thus just consider that it's $\rho$=constant. What I meant is that Choring splitting assumes incompressibility, but many people also assume $\rho$= constant as a direct consequence of incompressibility. I might be a bit ignorant, but I haven't seen any clear direct demonstration of $\nabla \cdot \vec v = 0$ implies $\rho$ independant of space and time. $\endgroup$ – Hydro Guy Oct 4 '15 at 3:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.