Acoustic radiation emitted by three-dimensional (3D) vortex rings in air has been investigated on the basis of the unsteady Navier-Stokes equations. Power series expansions of the unknown functions with respect to the initial vorticity which is supposed to be small are used. In such a manner the system of the Navier-Stokes equations is reduced to a parabolic system with constant coefficients at high derivatives. The initial value problem is as follows. The vorticity is defined inside a toroid at t = 0. Other gas parameters are assumed to be constant throughout the whole space at t = 0. The solution is expressed by multiple integrals which are evaluated with the aid of the Korobov grids. Density oscillations are analyzed. The results show that the frequency band depends on the initial size of the vortex ring and its helicity. The presented data may be applied to the study of a flow in a wake region behind an aerodynamic body.
Acoustic radiation by a Hill vortex has been studied on the basis of the Navier – Stokes equations. We used the expansion of unknown functions into power series of the initial vorticity which was assumed to be small. The Navier-Stokes equations were reduced to a parabolic system with constant coefficients for higher derivatives. The problem is set in such a manner. The vorticity is defined inside a sphere at t = 0. The initial density and the temperature are assumed to be constant throughout the whole space. The solution is represented as a power series whose coefficients are multiple integrals. They have been calculated with the aid of Korobov grids. The density oscillations have been investigated. The frequency spectra were studied at various distances from the vortex. The results may be useful for modelling decaying grid turbulence.
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