Enstrophy spectrum in freely decaying two-dimensional self-similar turbulent flow

We consider freely decaying two-dimensional homogeneous and isotropic turbulent motion in a self-similar limit that is achieved at large Reynolds numbers based on time and the mean kinetic energy of the flow provided that initial average enstrophy tends to infinity as fluid viscosity tens to zero. In this case, the enstrophy dissipation rate has a nonzero finite limit. The vorticity correlation function and the spectral enstrophy density are investigated in an inertial range of distances and wave numbers where these functions are free from the influence of viscosity and large-scale flow parameters. It turns out that in freely decaying two-dimensional self-similar turbulence, the inertial range exists in real space but is absent in the space of wave numbers. This means that turbulent eddies of the appropriate size do not contribute to the spectral density and the known $k^{-1}$ law does not hold. The spectral enstrophy density at large wave numbers behaves nonmonotonically: it first decreases faster than in accordance with the $k^{-1}$ law and then, in the dissipation region, has a growth portion and a second maximum. The enstrophy spectral flux at the boundary of the dissipation region is only a fraction of the enstrophy dissipation rate.

Figure 1

Qualitative form of the vorticity correlation function.