Estimates Of Lyapunov Exponents And Noise Level From Chaotic Time Series And Their Application In Turbulent Jets

Turbulent jets are the basic flow patterns in entrained-bed gasifier and other jet reactors. In this paper, the methods for estimating the Lyapunov exponents and noise level from chaotic time series were proposed, and the velocity time series of round turbulent jets and plane turbulent jets with the proposed method were studied and then some similar laws of the two kinds of jets were found. The main contents and results are summarized as follows:1. A novel method for estimating the Lyapunov spectrum from a noisy chaotic time series is presented. An averaging method is proposed to cope with this noise based on the property of mutual independence between the random noise and the underlying dynamical system. The mappings equations of the underlying deterministic system can be obtained from the noisy data via the method, and then the Lyapunov spectrum of the underlying deterministic system can be estimated. We demonstrate the performance of our algorithm for the time series of Logistic map, Henon map, the generalize Henon map and Lorenz system. It is found that the proposed method provides a reasonable estimate of Lyapunov spectrum for these four systems when the noise level is less than15%,20%,10%and7%, respectively. Furthermore, our method is not sensitive to the distribution types of the white noise, and the results of our method become more accurate as the length of the time series increasing.2. A novel method for estimating simultaneously the largest Lyapunov exponent and noise level from a noisy chaotic time series is presented. The influence of noise on the distance of two points in an embedding phase space is researched, and then our algorithm is proposed based on the invariant of the largest Lyapunov exponent in different dimensional embedding phase spaces. With numerical simulation, we find that the proposed method provides a reasonable estimate of the largest Lyapunov exponent and noise level when the noise level is less than10%of the signal content, and the method is useful not only for the wihte noise but also for the color noise. Furthermore, combining the nonlinear noise reduction, an improved methd is proposed and it can be used for the chaotic time series with30%noise.3. The velocity time series of round jets and plane jets are acquired with the hot-wire anemometer. The largest Lyapunov exponents and the random noise of two kinds of jets are computed with the proposed method of chaotic time series. The results show that the largest Lyapunov exponents of two kinds of jets increase with the exit Reynolds numbers of jets, but the largest Lyapunov exponents of the plane jets are larger than that of round jets at the same exit Reynolds number. In the near field of jet, the largest Lyapunov exponents of two kinds of jets increase first and then decrease along the flow direction. The random noise of the two kinds of jets increase both as the exit Reynolds number and the distance away from the nozzle exit. For the same exit Reynolds number and the same dimensionless position, the random noise of plane jets is larger than that of round jets. The results also show that the largest Lyapunov exponents of two kinds of jet are in direct proportion to the reciprocal of the integral time scale of turbulence and the proportionality coefficients are equal, which is in accordance with the results of the dimensional analysis. In addition, the random noise of the two kinds of jets has the same linear relation with the Kolmogorov velocity scales of turbulence. Consequently, the turbulence is composed of the deterministic chaos and the random noise, where the motion of large eddies in the turbulence follows the rule of chaotic motion and the random noise arises from the random motion of small eddies in the turbulence.