a b s t r a c tA two-step derivative-free iterative algorithm is presented for solving nonlinear equations. Error analysis shows that the algorithm is fourth-order with efficiency index equal to 1.5874. A lot of numerical results show that the algorithm is effective and is preferable to some existing derivative-free methods in terms of computation cost.

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The acoustic streaming motion in a compressible air-filled two-dimensional cylindrical resonator driving by a piston is simulated by using the gas-kinetic scheme, and the effects of acoustic field intensity on the formation process of flow structure as well as streaming pattern are investigated numerically for the practical applications of high-intensity acoustic devices. Therefore, five cases with different excitation amplitudes are considered in simulation ranging from the linear to highly nonlinear regions. The validation of the developed model is verified by comparing the numerical results of streaming velocities with the theoretical ones for the linear case. The wave form of pressure and velocity and the transient flow field structure as well as the resulting streaming pattern are found to be strongly correlated to the excitation amplitude. Be observed for the linear case is a sine wave and a uniform of quasi-one-dimensional flow field as well as classical Rayleigh streaming. Periodic shock waves and strongly two-dimensional flow fields as well as the irregular acoustic streaming fields have been observed for the nonlinear cases. And some new physical phenomena have also been revealed for the highly nonlinear case, and these facts are that the wave form will deviate from typical “shock wave”-type toward a more distorted type, and at the same time, the two-dimensional transient flow fields are characterized by the changing flow direction and circulatory flow pattern, which are all correlated to a turbulent streaming with various scales of irregular and unsteady vortices throughout the resonator. Detailed explanations are provided for these nonlinear phenomena, and a critical Reynolds number for the transition to turbulence is also numerically obtained which is consistent with the experiment result from the previous publication. The present work has demonstrated that the gas-kinetic scheme is capable of resolving large Reynolds number of nonlinear acoustic problems with no any restriction on the nonlinearity level.

Nonlinear standing waves and acoustic streaming in an axial-symmetrical resonator with exponentially varying cross-sectional area were studied. A two-dimensional gas-kinetic Bhatnagar-Gross-Krook scheme based on the non-structure triangular grid was established to simulate nonlinear acoustic oscillations in the resonator. Details of the transient and steady flow fields and streaming were developed. The effects of winding index of the exponential-shape resonator, the displacement amplitude of the acoustic piston on the streaming, and the vortex pattern were analyzed. The results demonstrate that the acoustic streaming pattern in such resonators is different from the typical Rayleigh flow in a constant cross-sectional area resonator. No obvious shock wave appeared inside the exponential-shape resonator. The comparison reveals that with increasing the displacement amplitude of the acoustic piston energy dissipation is accompanied by vortex break-up from a first-level to a second-level transition, and even into turbulent flow. This research demonstrates that the exponential-shape resonator, especially that with a winding index of 2.2 exhibits better acoustic features and suppression effects on shock-wave, acoustic streaming, and the vortex.

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