Abstract
We analyze a turbulent flow field generated by a fractal grid, with respect to spatial scale and different downstream positions. 2- and N-point statistics are used for the analysis. 2-point statistics are done by a spectrogram, which shows the spectral energy density in scale r and in distance to the grid x. The loglog-derivative in scale of the spectrogram is calculated and illustrates different scaling regions of the energy cascade. A complete characterization of the turbulent cascade is done by N-point statistic in terms of its stochastic process evolving in scale. This analysis is done in scale r at three characteristic downstream positions. The results of 2- and N-point statistic are interpreted and compared with each other, which provide a deeper understanding of the fractal grid wake.
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Notes
- 1.
Calculation of energy spectral density is done by the fft-function of Matlab 2012.
- 2.
L is calculated by integrating the autocorrelation function R(r), cf. Batchelor (1953). Partly autocorrelation functions do not monotonous decrease, until first zero crossing. In such a case, the lower part of autocorrelation function gets extrapolated by exponential function, \(f=a\cdot e^{-b\cdot r}\). f is fitted on R(r), the fit range is between the inflection point \(\partial _{rr}R(r)=0\), (e.g. \(R(r)\approx 0.7\)) and the point where the slope for the first time vanish \(\partial _{r}R(r)=0\), (e.g., \(R(r)\approx 0.1\)). Typically, such a procedure leads to smaller integral length scale compared to the standard procedure of Batchelor (1953).
- 3.
The Taylor length \(\lambda \) is estimated by the procedure proposed by Aronson and Löfdahl (1993).
- 4.
A single realization \(u(\cdot )\) is also named as a single path or trajectory of an increment, which means that the trajectory \(u=(U(x+r)-U(x))/\sigma _\infty \) is considered, where r changes from the upper border of the inertial range (\({\sim } L\)) to the lower border ( \({\sim } \lambda \)), whereas x is fixed.
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Reinke, N., Fuchs, A., Hölling, M., Peinke, J. (2016). Stochastic Analysis of a Fractal Grid Wake. In: Sakai, Y., Vassilicos, C. (eds) Fractal Flow Design: How to Design Bespoke Turbulence and Why. CISM International Centre for Mechanical Sciences, vol 568. Springer, Cham. https://doi.org/10.1007/978-3-319-33310-6_6
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