Building on homework modules 1 and 2, write a spectral code to simulate 2D turbulence in a 1536 × 1536 square periodic box. Start with the 2D Navier-Stokes equations in streamfuncion / vorticity formulation and discretize them spatially using a Fourier spectral method. Use your favorite dealiasing scheme (e.g. 2/3 rule would use 1024×1024 Fourier modes) and a third-order, low storage, semi-implicit Runge-Kutta method to integrate in time. Solve these equations for a Reynolds number
where L is the size of the box, ν is the kinematic viscosity and u0 is the root mean square of the velocity fluctuations at t = 0. Set a random initial condition with energy spectrum
Run this simulation and
- Estimate the transient time that is required for the random initial conditions to transform into a sea of compact vortices.
- Plot the time evolution of the total enstrophy and energy in the box. Which one decaysfaster? Why?
- Plot 2D vorticity fields for the instants of time when the energy is 90%, 80%, 70%,60% and 50% of its initial value. Describe qualitatively the evolution of vortex number and size.