# Pseudospectral method for Rayleigh-Benard with constant temperature gradient

$$\nabla\cdot \mathbf{u} = 0 \\ \frac{\partial \mathbf{u}}{\partial t}+\left(\mathbf{u}\cdot \nabla\right)\mathbf{u} = -\nabla p+\nu\nabla^2\mathbf{u}+\alpha g\theta\mathbf{e}_z\\ \frac{\partial\theta}{\partial t}+ \left(\mathbf{u}\cdot \nabla\right)\theta = \kappa\nabla^2\theta+\mathbf{u}_z\frac{\Delta T}{H}$$

Can I apply Fourier-spectral method for 3D-Rayleigh-Benard convection with a constant temperature gradient in the vertical direction?

I want to take all the directions as periodic.

There are papers which have done this with free-slip boundary condition in the vertical direction using the Fourier method.

But there is a paper that has done it use Lattice-Boltzmann method with triply-periodic boundaries.

• What is your concrete question? Just whether one can use such a method? Why would one not be able to use such a method? – Wolfgang Bangerth Aug 12 '19 at 12:39
• I'll add that it doesn't make sense to make the domain periodic in z-direction, because your temperature profile is not periodic. – Wolfgang Bangerth Aug 12 '19 at 12:39
• @WolfgangBangerth That is exactly my doubt. I have seen papers that use free-slip boundary condition in the z-direction while using Fourier-pseudospectral method. At the same time, there is a different paper that does Rayleigh-Benard simulation in with triply periodic BCs with the Lattice-Boltzmann method. How does having periodic BC in z-direction takes account of the anisotropy, is not explained in the paper and I am confused. Link to the paper: pdfs.semanticscholar.org/c73f/… – user162281 Aug 13 '19 at 9:18
• I want to know if similar is possible in fourier-pseudospectral method. If no, then why it has been possible for LB method? If yes, then can you direct me to any paper or something? – user162281 Aug 13 '19 at 9:48
• I don't know -- I don't actually know anything about pseudospectral methods, I was just trying to understand your question :-) – Wolfgang Bangerth Aug 13 '19 at 20:50