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Stochastic simulations are used to create synthetic one-dimensional telegraphapproximation (TA) signals based on turbulent zero crossings, where theinterval between crossings is governed by a power law probability distributionwith exponent $\alpha$. The power law exponent is determined for statistics ofsimulated TA signals, namely the box-counting fractal dimension $D_1$, energyspectrum exponent $\beta_{TA}$, and an intermittency exponent $\mu_{TA}$. Forthe binary TA signal with no variability in amplitude, the parameters arerelated linearly as $D_1 = 2 - \beta_{TA} = 1 - \mu_{TA}$. The relations areunchanged 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 trulyscale-invariant, and distorts the linear relation between the statisticexponents and $\alpha$. The behavior is due to finite-size effects apparentfrom the survival function, or the complementary cumulative distribution, whichfor finite Reynolds number is only approximately self-similar and has aneffective exponent differing from $\alpha$. An expression presented for theeffective exponent recovers the expected relations between $\alpha$ and the TAstatistics. The findings demonstrate how a finite Reynolds number can affectindicators of self-similarity, fractality, and intermittency observed fromsingle-point measurements.

Effect of finite Reynolds number on self-similar crossing statistics and fractal measurements in turbulence