Bifurcation in compressible moving fluids and suppressing atmospheric turbulence of aircraft-flow amplifies sound

Quasi-accumulation solutions for acoustic waves in a compressible moving fluid are obtained by applying the Lagrange parameter variation method to solve the Lighthill equation. The results demonstrate that nonlinear interactions lead first to period-doubling, followed by odd multiple half-period bifurcations, with all-order sub-harmonics subsequently generated. The amplitudes of these sub-harmonics depend not only on the acoustic Mach number but also on the Mach number of the flow. The latter result indicates that the acoustic wave has been amplified by the momentum of the flow. Furthermore, the relationship between the amplification gain of sub-harmonics and flow velocity is a polynomial function of flow-sound Mach number ratio M/m. If the kinetic energy gained through momentum amplification exceeds the energy loss due to the acoustic attenuation, a chain-reaction of the period-doubling followed by the odd multiple half-period bifurcation can be sustained. As the order number of the approximation increases, the number of degrees of freedom in the flow increases infinitely and the leading terms of the amplitudes for the generated sub-harmonics which are proportional to approach to infinity, whereand , k and are Mach number for flow and sound, the wave-number and absorption coefficient, respectively. The obtained results also indicate that the appropriate control parameter for transitioning from bifurcation to chaos should be instead of Reynolds number. This paper also demonstrates that in the moving fluid sound waves can be amplified through nonlinear interactions, particularly, the first sub-harmonic generates, thereby explaining the experimental results we discovered decades ago. Finally, a potential strategy for suppressing aircraft-induced atmospheric turbulence is proposed based on the present theoretical findings.