Acoustic Radiation by 3D Vortex Rings in Air

Fedor Shugaev,Oxana Solenaya,Dmitri Cherkasov

doi:10.3390/aerospace2040627

Abstract

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.

Highlights

Vortical structures play an important role in the sound radiation of gaseous flows
The aim of the present investigation is to determine the frequency band of acoustic radiation emitted by 3D vortex rings in air
A new method for solving the equations is set forth. It uses power series expansion of the unknown functions with respect to the initial vorticity which is supposed to be small. By applying this procedure the Navier–Stokes equations are reduced to a parabolic system with constant coefficients

Summary

Introduction

Vortical structures (vortex rings and cylindrical vortices) play an important role in the sound radiation of gaseous flows. These structures convert a fraction of their rotational energy into sound waves [1]. The aim of the present investigation is to determine the frequency band of acoustic radiation emitted by 3D vortex rings in air. A new method for solving the equations is set forth It uses power series expansion of the unknown functions with respect to the initial vorticity which is supposed to be small. By applying this procedure the Navier–Stokes equations are reduced to a parabolic system with constant coefficients. The topic of this paper is the study of the frequency band of acoustic radiation by a 3D solitary vortex ring in air and the evolution of the radiation

Governing Equations

Initial Value Problem

Solution to the Problem

Results and Discussion