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Flow
Reynolds number and how much random noise is added to the ordered vortex array.
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Model checks
The science
Taylor-Green vortex
The 2D Taylor-Green vortex is the flow u = sin x cos y e^(-2 nu t), v = -cos x sin y e^(-2 nu t) on a periodic box of side 2 pi: an array of counter-rotating vortices. It is an exact solution of the incompressible Navier-Stokes equations, and its kinetic energy decays as E(t) = E0 e^(-4 nu t) with E0 = 1/4. The Reynolds number is Re = 1/nu (velocity scale 1, length scale 1).
The solver
The page integrates the vorticity equation w_t + u.grad(w) = nu laplacian(w) with a pseudo-spectral method on a 64 x 64 grid (FFT, 2/3-rule dealiasing, classical RK4), with the stream function from laplacian(psi) = -w. Energy and enstrophy obey dE/dt = -2 nu Z, with Z = (1/2) mean(w^2).
2D versus 3D
In two dimensions the ordered vortex array does not break into turbulence: nearby like-signed vortices can merge and energy moves to large scales (the inverse cascade). The famous breakdown of the Taylor-Green vortex into small-scale turbulence needs vortex stretching, which exists only in 3D; the 3D companion shows that. Here, add a perturbation to see 2D vortex interaction.
Simplifications
Periodic box, 64 x 64 grid (so Re above a few thousand is under-resolved), no forcing, no walls, constant viscosity. The random perturbation is band-limited noise.
What to try
- With no perturbation, compare the measured energy with the exact exponential at several Re.
- Add 30% perturbation at Re = 1000: vortices of the same sign merge and the energy spectrum shifts to larger scales.
- Lower Re to 20: viscosity smooths everything and the flow decays within a few time units.
- Turn on tracers to see how fluid parcels are carried around the vortex cells.