# How are Spectral Methods applied to CFD? In particular, how is the pressure-velocity coupling implemented?

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.

• 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. Oct 1 '15 at 23:47
• 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... Oct 1 '15 at 23:49
• 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 Oct 1 '15 at 23:51
• 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. Oct 1 '15 at 23:54
• 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advisors, personally I prefer C way more for number crunching, though there are some things in C that are frankly outdated in anno 2015. Oct 2 '15 at 13:14

• 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? Oct 4 '15 at 3:28
• 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. Oct 4 '15 at 3:48