The Evolution of Acoustic Radiation by an Ensemble of Vortex Rings in Air

The evolution of acoustic radiation emitted by an ensemble of vortex rings in air is studied on the basis of nonstationary Navier–Stokes equations. We use the expansions of required functions into a power series of the initial vorticity which is a small value. The Navier–Stokes equation system reduces to a parabolic system with constant coefficients for the higher derivatives. The problem is posed as follows. The vorticity is defined inside the toroid at t = 0. The other parameters of the gas are assumed to be constant throughout the space at the initial instant of time. The solution is expressed in terms of multiple integrals, which are calculated using Korobov grids. The density oscillations were investigated. The results show that the frequency spectrum depends on time; high-frequency oscillations are observed at small times and low-frequency oscillations then occur. At the same time, the amplitude of high-frequency oscillations decreases in comparison with low-frequency oscillations. Thus, a transition of energy from the high-frequency spectrum to the lowfrequency spectrum occurs. These results can be useful for modeling decaying grid turbulence.