Nonlinear time series analysis and clustering for jet axis identification in vertical turbulent heated jets

In the present work we approach the hydrodynamic problem of discriminating the state of the turbulent fluid region as a function of the distance from the axis of a turbulent jet axis. More specifically, we analyzed temperature fluctuations in vertical turbulent heated jets where temperature time series were recorded along a horizontal line through the jet axis. We employed data from different sets of experiments with various initial conditions out of circular and elliptical shaped nozzles in order to identify time series taken at the jet axis, and discriminate them from those taken near the boundary with ambient fluid using nonconventional hydrodynamics methods. For each temperature time series measured at a different distance from jet axis, we estimated mainly nonlinear measures such as mutual information combined with descriptive statistics measures, as well as some linear and nonlinear dynamic detectors such as Hurst exponent, detrended fluctuation analysis, and Hjorth parameters. The results obtained in all cases have shown that the proposed methodology allows us to distinguish the flow regime around the jet axis and identify the time series corresponding to the jet axis in agreement with the conventional statistical hydrodynamic method. Furthermore, in order to reject the null hypothesis that the time series originate from a stochastic process, we applied the surrogate data method.

Figure 1

Schematic diagram of turbulent jet flow.

Figure 2

Perspective view of the experimental setup.

Figure 3

Time series at various positions.

Figure 4

Shadowgraph view of the experimental setup. The green areas refer to the boundary region, while the blue and red areas refer to the jet axis region and inner regions, respectively. The red arrows indicate the limits of the measurement zone.

Figure 5

Mean, median (left vertical axis), and variance (right vertical axis) of the time series as a function of measurement positions for the various cases of jets.

Figure 6

Hurst exponent of the time series as a function of measurement positions with R/S method and exponent with the detrended fluctuation analysis (DFA).

Figure 7

Hjorth parameters mobility and complexity of the time series as a function of measurement positions.

Figure 8

(a),(b) Mutual information and cumulative mutual information of the time series as a function of measurement positions.

Figure 9

Dendrogram based on the measures obtained from the time series. The sketch above is a schematic representation of the location of the various regions in the measurement setup (FJA: far from jet axis region; CJA: close to jet axis region; JA: jet axis region).

Figure 10

Normalized excess (above ambient) and rms temperature by the maximum center line time-averaged temperature is plotted versus r/z, along with exponential fit to the time-averaged normalized temperature.

Figure 11

Mean, median (left vertical axis), and variance (right vertical axis) of the time series as a function of measurement positions for the various cases of jets (see Table 3). (a) Second case, (b) third case, (c) fourth case, and (d) fifth case.

Figure 12

Hurst exponent of the time series as a function of measurement positions with R/S method and with the detrended fluctuation analysis (DFA) for the various cases of jets (see Table 3). (a) Second case, (b) third case, (c) fourth case, and (d) fifth case.

Figure 13

Hjorth parameters of the time series mobility and complexity as a function of measurement positions for the various cases of jets (see Table 3). (a) Second case, (b) third case, (c) fourth case, and (d) fifth case.

Figure 14

Mutual information of the time series as a function of measurement positions for the various cases of jets (see Table 3). (a) Second case, (b) third case, (c) fourth case, and (d) fifth case.

Figure 15

Cumulative mutual information of the time series as a function of measurement positions for the various cases of jets (see Table 3). (a) Second case, (b) third case, (c) fourth case, and (d) fifth case.

Figure 16

Dendrogram based on the measures obtained from the time series of (a) the second case, (b) third case, (c) fourth case, and (d) fifth case studies. The sketch above is a schematic representation of the location of the various regions in the measurement setup (FJA: far from jet axis region; CJA: close to jet axis region; JA: jet axis region).