# 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.

Edit: $$\theta$$ here is the temperature fluctuation and not the actual temperature. Knowing that the equations can be solved using periodic boundary conditions.

I'm not sure if the question would be helpful to anyone (since it leaves out some crucial detail).

• 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