Sound generation by head-on and oblique collisions of two vortex rings

Abstract

Sound generation by head-on and oblique collisions of two vortex rings with equal strength is investigated by a direct numerical simulation. The three-dimensional, unsteady, compressible Navier-Stokes equations are solved by a finite difference method, not only for a near-vortical field but also for a far-acoustic field. The theory of vortex sound is also applied to investigate the detailed relation between the vortex ring motion and the acoustic wave mode. Generation of two sound pressure waves, which are called the first and second pulses, is observed in the numerical results. The theoretical results show that, for the head-on collision, the generations of the first and second pulses are due to the variations of vorticity moment, associated with a decrease in the translational velocity of the vortex ring near the collision plane and its stretching motion, respectively. On the other hand, for the oblique collision, in addition to such generation mechanisms, the reconnection of vortex lines also affects the generation of the second pulse. The reconnection of vortex lines occurs more rapidly for a smaller collision angle, and therefore its contribution to the second pulse becomes large. In fact, in the oblique collision at a right angle, the generation of the second pulse is closely related to the reconnection of vortex lines, which can be seen by analyzing both the near and far fields. Furthermore, for a smaller collision angle, the reconnection of vortex lines leads to more rapid variation of the vorticity, which enhances asymmetric features of the second pulse exhibited by the octupole mode ͑third-order wave mode͒.

Sound generation by a vortex ring collision with a wall

“…In this case, reconnection of vortex lines occurs between the real and image vortex rings on the outer circumference of the rings. 8 For the no-slip case, the vorticity fields on the symmetric plane ͑x 1 =0͒ are presented in Fig. 14.…”

“…For the free-slip case, the grid spacings are prescribed to be the same as those in the study of Nakashima,8 in which the reliability of the grid system has been confirmed for collision of two vortex rings; the grid spacings are fixed to be ⌬x v = 0.05, ⌬x s = 1, and ⌬x b = 5, and the time step is ⌬t = 0.02. For the no-slip case, a typical example of the distributions of grid spacing is plotted against x i ͑i =1,2,3͒ in Fig.…”

“…8 The threedimensional, unsteady, compressible Navier-Stokes equations are solved by a finite difference method. An eighthorder-accurate compact Padé scheme proposed by Lele 27 is used for spatial derivatives, together with the fourth-orderaccurate Runge-Kutta scheme for a time integration.…”

“…[3][4][5][6][7][8] Inoue et al 7 investigated the acoustic sound in an axisymmetric flow field generated by head-on collision of two coaxial vortex rings. By directly solving the axisymmetric compressible Navier-Stokes equations, they successfully captured the generation and propagation processes of sound pressure waves.…”

“…7,8,26 In the DNS of CAA, the compressible Navier-Stokes equations are solved directly over the whole domain from a near-vortical flow field to a far-acoustic field, usually by using highly accurate schemes in both space and time in order to precisely capture the far-field sound pressure. One of the advantages of DNS is its capability of clarifying both the generation and propagation processes of sound pressure waves without restrictions imposed in the approximate methods as mentioned above.…”