混沌时间序列的Lyapunov指数和噪声水平估计及其在湍流射流中的应用
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摘要
湍流射流是气流床气化炉等射流反应器的基本流动形态。本文提出了从混沌时间序列估计Lyapunov指数和噪声水平的方法,并利用提出的方法研究了圆湍流射流和平面湍流射流速度时间序列的性质,发现了两种射流的一些相似规律。具体内容可归纳如下：
1.提出了一种可以从含有噪声的混沌时间序列中计算Lyapunov指数谱的新方法。该方法利用随机噪声和潜在动力系统相互独立的性质,提出了一种可以消除噪声影响的平均方法。通过这种方法可以从含有噪声的数据中得到潜在动力系统的映射方程,从而可以估计出潜在动力系统的Lyapunov指数谱。对Logistic映射、Henon映射、广义Henon映射和Lorenz系统生成的时间序列进行了仿真计算,结果表明,当噪声水平分别小于15%、20%、10%和7%时,该方法对这四个混沌系统的计算结果都是可靠的。另外,提出的方法对白噪声的分布类型不敏感,而且随着时间序列长度的增加,计算结果变得更加精确。
2.提出了一种可以从含有噪声的混沌时间序列中同时估计最大Lyapunov(?)指数和噪声水平的新方法。该方法首先研究了噪声对嵌入相空间中两点距离的影响,然后根据最大Lyapunov指数在不同维嵌入相空间中不变的性质,提出了从时间序列同时估计最大Lyapunov指数和噪声水平的方法。仿真计算结果表明,当噪声水平小于10%时,该方法对时间序列的最大Lyapunov指数和噪声水平的估计都是可靠的,而且对白噪声和色噪声均有效。另外,结合非线性降噪方法对该方法作了改进,发现改进的方法可用于含30%噪声的混沌时间序列。
3.用热线风速仪采集了圆射流和平面射流的速度时间序列,并用提出的混沌时间序列分析方法计算了两种射流的最大Lyapunov指数和其中包含的随机噪声。结果表明,两种射流的最大Lyapunov指数均随射流出口雷诺数的增加而增加,但出口雷诺数相同时,平面射流的最大Lyapunov指数比圆射流的大。在射流近场区,两种射流的最大Lyapunov (?)旨数沿流动方向都是先增加后减小。两种射流的随机噪声均随射流出口雷诺数和离开喷嘴出口的距离的增加而增加。对相同的出口雷诺数和相同的无因次位置,平面射流的随机噪声比圆射流的大。结果还发现,两种射流的最大Lyapunov与湍流积分时间尺度的-1次方成正比,与量纲分析结果一致,并且比例系数相等；另外,两种射流中的随机噪声与湍流的Kolmogorov速度尺度有相同的线性关系。由此可见,湍流可分解为确定性的混沌和随机噪声两部分,其中湍流中大涡的运动遵循混沌运动规律,而湍流中小涡的随机运动构成了湍流的随机噪声。
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.
1.提出了一种可以从含有噪声的混沌时间序列中计算Lyapunov指数谱的新方法。该方法利用随机噪声和潜在动力系统相互独立的性质,提出了一种可以消除噪声影响的平均方法。通过这种方法可以从含有噪声的数据中得到潜在动力系统的映射方程,从而可以估计出潜在动力系统的Lyapunov指数谱。对Logistic映射、Henon映射、广义Henon映射和Lorenz系统生成的时间序列进行了仿真计算,结果表明,当噪声水平分别小于15%、20%、10%和7%时,该方法对这四个混沌系统的计算结果都是可靠的。另外,提出的方法对白噪声的分布类型不敏感,而且随着时间序列长度的增加,计算结果变得更加精确。
2.提出了一种可以从含有噪声的混沌时间序列中同时估计最大Lyapunov(?)指数和噪声水平的新方法。该方法首先研究了噪声对嵌入相空间中两点距离的影响,然后根据最大Lyapunov指数在不同维嵌入相空间中不变的性质,提出了从时间序列同时估计最大Lyapunov指数和噪声水平的方法。仿真计算结果表明,当噪声水平小于10%时,该方法对时间序列的最大Lyapunov指数和噪声水平的估计都是可靠的,而且对白噪声和色噪声均有效。另外,结合非线性降噪方法对该方法作了改进,发现改进的方法可用于含30%噪声的混沌时间序列。
3.用热线风速仪采集了圆射流和平面射流的速度时间序列,并用提出的混沌时间序列分析方法计算了两种射流的最大Lyapunov指数和其中包含的随机噪声。结果表明,两种射流的最大Lyapunov指数均随射流出口雷诺数的增加而增加,但出口雷诺数相同时,平面射流的最大Lyapunov指数比圆射流的大。在射流近场区,两种射流的最大Lyapunov (?)旨数沿流动方向都是先增加后减小。两种射流的随机噪声均随射流出口雷诺数和离开喷嘴出口的距离的增加而增加。对相同的出口雷诺数和相同的无因次位置,平面射流的随机噪声比圆射流的大。结果还发现,两种射流的最大Lyapunov与湍流积分时间尺度的-1次方成正比,与量纲分析结果一致,并且比例系数相等；另外,两种射流中的随机噪声与湍流的Kolmogorov速度尺度有相同的线性关系。由此可见,湍流可分解为确定性的混沌和随机噪声两部分,其中湍流中大涡的运动遵循混沌运动规律,而湍流中小涡的随机运动构成了湍流的随机噪声。
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.
引文
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