# Turbulence Pseudospectral Code

I am writing a 2D pseudo spectral code for turbulence in a box with 1024 grid points with 3/2 aliasing scheme in the vorticity/stream function formulation. the vortices tends to appear very slowly and energy is decaying faster than expected at a Reynolds number of 5*10^4. The faster decay in energy leads to very slow vortices (which is not the case) I am not sure where the problem is. Any help in this would be useful. I have tried different initial conditions like random velocity and gaussian white noise but nothing seems to work.

2D turbulence has an inverse energy cascade which feeds the formation of large-scale vortices. Starting with a random field will produce some structures that are rapidly dissipated, and then you settle into normal evolution.

That said, it's not clear from your question what your initial conditions are, nore really what you expect to see.

I'm not sure I understand the specifics of your code. That aside, turbulence is not random and, assuming you're working on an incompressible flow, the velocity field that you produce using random generators is not likely to satisfy the continuity equation and will be rapidly killed by your numerical scheme. This could explain why you're seeing higher than expected turbulence decay rates.

You could try to generate a turbulent velocity field "naturally" by tripping a boundary layer flow in a periodic computational domain for example. Otherwise, I believe there are some standard benchmark fields for turbulent decay with a very specific initial conditions but I don't know where to find those.

Standard initialization of homogeneous, isotropic turbulence is through the Passot-Pouquet spectrum$^{[1]}$ or the Von Karman-Pao spectrum$^{[2]}$. Both initializations are done in wavenumber space and hence easier to implement with a spectral/pseudo-spectral code.

[1] Passot, Thierry, and Annick Pouquet. "Numerical simulation of compressible homogeneous flows in the turbulent regime." Journal of Fluid Mechanics 181 (1987): 441-466.

[2] https://turbulence.utah.edu/media/karman.html. Also Pao, Yih‐Ho. "Structure of turbulent velocity and scalar fields at large wavenumbers." The Physics of Fluids 8.6 (1965): 1063-1075. for the derivation of the form.