有限Reynolds number对turbulence自相似交叉统计和分形测量的影响
Effect of finite Reynolds number on self-similar crossing statistics and fractal measurements in turbulence
摘要：
随机模拟用于创建基于湍流零交叉的合成一维电报近似（TA）信号，其中交叉之间的间隔由具有指数$α$的幂律概率分布支配。确定模拟TA信号统计的幂律指数，即计盒分形维数$D_1$、能谱指数$β_{TA}$和非线性指数$μ_{TA}$。对于幅度无变化的二进制TA信号，参数线性相关为$D_1 = 2 - β_{TA} = 1 - μ_{TA}$。如果交叉间隔分布具有代表有限雷诺数流动的有限幂律区域（即惯性子区），则关系不变。然而，有限分布产生的统计量不是真正的标度不变的，并且扭曲了统计指数和$α$之间的线性关系。该行为是由于从生存函数或互补累积分布中明显可见的有限尺寸效应，对于有限雷诺数，其仅近似自相似，并且具有不同于$α$的有效指数。有效指数的表达式恢复了$α$和TA统计之间的预期关系。研究结果表明，如何有限的雷诺数可以影响指标的自相似性，分形，和单点测量观察到的不稳定性。
英文摘要：
Stochastic simulations are used to create synthetic one-dimensional telegraph approximation (TA) signals based on turbulent zero crossings, where the interval between crossings is governed by a power law probability distribution with exponent $α$. The power law exponent is determined for statistics of simulated TA signals, namely the box-counting fractal dimension $D_1$, energy spectrum exponent $β_{TA}$, and an intermittency exponent $μ_{TA}$. For the binary TA signal with no variability in amplitude, the parameters are related linearly as $D_1 = 2 - β_{TA} = 1 - μ_{TA}$. The relations are unchanged if the crossing interval distribution has a finite power law region (i.e. inertial subrange) representing a flow with finite Reynolds number. However, the finite distribution yields statistics that are not truly scale-invariant, and distorts the linear relation between the statistic exponents and $α$. The behavior is due to finite-size effects apparent from the survival function, or the complementary cumulative distribution, which for finite Reynolds number is only approximately self-similar and has an effective exponent differing from $α$. An expression presented for the effective exponent recovers the expected relations between $α$ and the TA statistics. The findings demonstrate how a finite Reynolds number can affect indicators of self-similarity, fractality, and intermittency observed from single-point measurements.
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