## 1 Introduction
[14] utilized particle velocity measurement and the laser-induced fluorescence method to measure the velocity field and temperature field of the thermal jet, and obtained the vertical variation of the main axis velocity of a thermal jet. [PERSON] et al. [15] studied turbulent distribution using a statistical analysis method, and the results show that the average velocity and turbulence intensity distribution of a single jet and a cold coaxial jet are almost the same. The strong temperature gradient in the hot coaxial jet enhances the mixing, resulting in a steep increase in turbulence intensity distribution near the end of the potential core. Compared with the distribution of a single jet and a cold coaxial jet, the turbulence intensity distribution obviously moves upstream. The second-order statistical characteristics of turbulence can be divided into Euler spatial correlation, Euler time correlation, and Lagrange time correlation [16]. For spatial turbulence, Reynolds stress can be written as the second-order correlations of two velocities in three directions on a spatial point, which belongs to a Euler spatial correlation [17]. [PERSON] et al. [18] used the Improved Instability-Sensitive Reynolds Stress Model (IIS-RSM) method to study the statistical and dynamic flow characteristics in open channels with emerging jet configurations, and obtained the corresponding isocontours of the first- and second-order turbulence statistics on the central vertical plane. [PERSON] et al. [19] used the two-dimensional particle image velocimetry (2D-PIV) technique to study the Reynolds stress distribution and self-similarity of semi-confined impinging jets. The results show that the velocity scale of Reynolds stress decays with the power law, while its length scale increases linearly. Unlike free impinging jets, Reynolds stress does not show self-similarity when using external scaling. The autocorrelation function is a statistic that describes the degree of correlation between random variables at different times [20]. It is usually regarded as Euler time correlation because it usually focuses on the time dependence of signals in a fixed spatial position. [PERSON] et al. [21] used the power spectral density (PSD) function to study the stochastic process characteristics of fluid dynamics, and obtained the autocorrelation function and the autospectrum of fluid dynamics. [PERSON] et al. [22] used the Hilbert transform-based autocorrelation envelope to study jet noise, and the results show that large-scale noise radiation from an F-22A Raptor base deviates from the similar spectrum model. [PERSON] et al. [23] used dual particle image velocimetry (dual-PIV) to study the space-time correlation of two subsonic isothermal circular jets with Mach numbers of 0.6 and 0.9, characterized the space-time correlation scales of the radial and axial velocity components, and obtained the self-similarity of the correlation functions in the shear layer and on the jet axis. [PERSON] [24] studied the order reduction algorithm by generating a smaller system matrix directly from the snapshot, and proposed a method to recover the coherent mode from the ratio of the time cross-correlation of a time step to the autocorrelation of the original field. The square of the time coefficient amplitude (usually the power spectrum in the frequency domain) is more consistent with the Euler time correlation because it usually describes the energy distribution of the signal at different time points (or frequency components). Autospectrum usually refers to the power spectral density (PSD) of the signal. The autospectrum describes the energy distribution of the signal in the frequency domain. It is obtained by the Fourier transform through an autocorrelation function (usually analyzed in the time domain), so it is more in line with the description of the Euler spatial correlation. [PERSON] et al.