# A statistical method for transforming temporal correlation functions from one-point measurements into longitudinal spatial and spatio-temporal correlation functions

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## Abstract

The transformation of temporal, one-point correlation functions into longitudinal spatial and spatio-temporal correlation functions in turbulent flows using a simple statistical convection model is introduced. To illustrate and verify the procedure, experimental data (one-point and two-point) have been obtained with a laser Doppler system from a turbulent, round, free-air jet.

## 1 Introduction

In the study of turbulence, temporal and spatial correlation functions are fundamental quantities defining characteristic scales of motion. In particular, integral scales and Taylor microscales are directly defined from the correlation function and, under the hypothesis of local isotropy, an estimate of the rate of dissipation per unit mass can be obtained.

Experimentally, measurements of correlation functions have often been performed using single-point measurement techniques, for example with stationary hot-wire probes (e.g., Favre 1965) or laser Doppler anemometers (e.g., Romano et al. 1999). Temporal Eulerian correlation functions can be obtained directly from the measured time series. Two-point or spatial correlation functions can be obtained with an array of multiple probes or, sequentially, with two measurement probes at varied separations. With particle image velocimetry (PIV), the Eulerian spatial correlation can be measured directly from the spatially resolved velocities at given time instances. However, the temporal and spatial resolution of PIV usually lacks the requirements to obtain small scales, for example the Taylor microscale, with sufficient accuracy. High-speed particle tracking (e.g., Ouellette et al. 2006) allows the Lagrangian correlation statistics to be obtained.

*U*, that is, \(u^\prime/U\ll 1, \) thus, spatially fluctuating quantities of the advected fluid along a path line of the fluid can be observed as temporal fluctuations at a given point. The functions investigated in the past span from local derivatives of velocities or of passive scalars like the temperature, where TH reads

*f*, like correlations or structure functions, using

Although TH requires small fluctuations compared with the mean advective velocity, it has not seldom been applied to turbulent flows, where the above condition is no longer met. Here, as expected, it has been observed that TH works well only at small scales, smaller than the typical scale of the flow (Lin 1953), where local homogeneity can be assumed. Corrections due to the fluctuating advective velocity may or may not be necessary (Heskestad 1965; Tennekes 1975; Browne et al. 1983; Hill 1996).

If applied to functions on larger scales, corrections of the TH become necessary, considering the variations of the velocity in a turbulent flow. Cholemari and Arakeri (2006) introduce methods which give a correspondence between the temporal and longitudinal spatial correlations by translating the time lag into separation or vice versa. However, they cannot be used to derive the spatio-temporal correlations, which decrease in amplitude for larger separations. Cenedese et al. (1991) predict the decrease in correlation by transfer functions, which work as a phase shifter from lower to higher frequencies. In He and Zhang (2006) and Zhao and He (2009), an elliptical approximation based on the second-order Taylor-series expansion of the spatio-temporal correlation function is introduced, reproducing well the shift and deformation of the correlation peak. However, the latter two methods describe the effect phenomenologically and require empirical parameters to be adapted to the measured data.

In the present study, an integral method based on the probability density distribution of the fluid velocity is proposed to transform temporal into longitudinal spatial and spatio-temporal correlation functions (Sect. 2). The procedure is validated by comparing results to two-point correlations measured directly with a laser Doppler system from a turbulent, round, free-air jet (Sect. 3).

## 2 Integral time-to-length transform

*R*(0, τ) with the time delay τ and no separation (ξ = 0). The transformation into the longitudinal spatial correlation function with the separation ξ and no time delay is then

*U*yielding

Turbulent flows with fluctuating velocity *u*(*t*) may significantly differ from this "frozen" hypothesis. The idea of the new integral time-to-length transform (ITLT) is that any fluid structure with a specific temporal correlation function, which has been measured at a specific point, can move to another point in flow direction within a certain time. This time depends on the varying fluid velocity *u*. While TH assumes that this velocity is constant, namely the mean velocity *U*, ITLT considers that the velocity and also the time to cover the separation between the two points can vary.

*u*yields a different time of flight θ. For given ξ and

*u*, the time of flight is determined by

*u*does not change within the separation ξ (or within the time of flight θ). If

*u*has a probability density function

*p*

_{u}(

*u*), then (for a given ξ) the time of flight has a probability density of

## 3 Experimental verification

*u*component. The two channels acquire the velocity samples independently (free-running mode) without coincidence. The Reynolds number is 20,000 based on the inner mean velocity and diameter. The laser Doppler data have been taken in the center of the jet at a distance

*x*= 320 mm (40

*d*

_{i}) downstream. The separation of the measurement volumes has been varied between ξ = 0 and ξ = 32 mm (4

*d*

_{i}) with symmetrical shifts of ±ξ/2 with respect to

*x*. For one-point measurements, the appropriate data sets have been selected from the two-point measurements.

Flow specifications

Outer diameter, | 140 mm |

Inner diameter, | 8 mm |

Outer velocity (at nozzle exit) | 0.5 m/s |

Inner velocity (at nozzle exit), | 35.9 m/s |

Outer volume flux | 27.6 m |

Inner volume flux | 6.5 m |

Kinematic viscosity, ν | 14 × 10 |

Reynolds number, \({\rm Re}=\frac{U_0d_{\rm i}}{\nu}\) | 2 × 10 |

Specification of the laser Doppler system

Laser | Ar |

Wavelength | 514.5 and 488 nm |

Optical configuration | Fiber-coupled probe, backscatter |

Frequency shift | Bragg cell, 40 MHz |

Focal length | 310 mm |

Measurement volume | \(400\,\upmu\hbox{m} \times 50\,\upmu\hbox{m}\) |

Processor | IFA750 |

The mean velocity *U* and the RMS velocity \(u^\prime\) are obtained from the measurements as ensemble averages applying transit time weighting (Hösel and Rodi 1977; Fuchs et al. 1994). The mean velocity decays from 7.85 to 7.15 m/s over the measurement region, while the RMS velocity decays from 1.44 to 1.40 m/s. This corresponds approximately to an inverse relation with the distance to some virtual origin lying within the nozzle; the virtual origin of the RMS data being slightly further inside the nozzle than the mean velocity data set. A turbulence intensity \(u^\prime/U\) of about 19 % is found, slightly increasing with the distance from the nozzle. Comparing with Wygnanski and Fiedler (1969), this small value indicates that the present free jet may not be fully developed in the measurement region; this is of no direct consequence for the present study.

*R*(ξ, τ) have been normalized, yielding the correlation coefficient functions ρ(ξ, τ) with

Based on the mean velocity *U* and the RMS velocity \(u^\prime\) obtained from the measured data set, the temporal correlation function is transformed into the longitudinal spatial correlation function using TH [Eqs. (1), (3)] and ITLT [Eqs. (5), (6)]. In the present study, a Gaussian distribution of the velocity *u* is assumed to derive the probability density function of times of flight. Alternatively, the probability density can be derived directly from the measured data.

*d*

_{i}). The shift in time of the maximum correlation corresponds to the mean time of flight to cover the separation. The height of the maximum decreases, which indicates that the shape of fluid structures changes during their passage. Furthermore, the peak width increases, indicating spatial diffusion of the fluid structures. Similar behavior has been shown by Kerhervé et al. (2008) for a turbulent round jet and by Cenedese et al. (1991) and by Chatellier and Fitzpatrick (2005) for other turbulent flow configurations.

TH simply shifts the autocorrelation function to the time of flight given by the separation of the probes divided by the mean velocity. The shape of the obtained spatio-temporal correlation function is the same as the autocorrelation function. Therefore, the transform based on TH is not able to reproduce the degradation of the correlation height or the expansion of the correlation width. However, the accuracy of the predicted time shift by TH is additionally limited due to the fact that the maximum position of the correlation peak may deviate from the separation divided by the mean velocity due to a skewness of the deformed peak. This yields a correlation peak traveling slower than the mean velocity as observed by He and Zhang (2006) and in Zhao and He (2009).

In contrast, ITLT is able to recover the spatio-temporal correlation correctly, including the time shift, the degradation of the correlation height and the expansion of the correlation width. Even the skewness of the peak is reproduced correctly.

Although the spectral transfer functions from the autocorrelation to the modeled spatio-temporal correlation functions are redundant, if the time responses are correct, it is still interesting to verify the correspondence of the obtained transfer functions with the results in Cenedese et al. (1991). Therefore, the experimentally obtained spatio-temporal correlation functions and the pendents obtained by TH and ITLT are Fourier transformed and divided by the Fourier transform of the autocorrelation function.

*d*

_{i}). The result of TH rotates in the complex plane with unit magnitude, corresponding to a simple time shift. Although the experimental data strongly scatter, ITLT obviously reproduces the amplitude decreasing with increasing frequency, yielding similar results as in Cenedese et al. (1991).

Unfortunately, if *u* changes within the separation ξ (or within the time of flight θ) the probability density function of times of flight changes and also the fluid structure passing by changes, yielding an additional degradation of the correlation. This case is a strong limitation of the present transformation method, which requires the temporal correlation in the Lagrangian framework to be much longer than the time of flight for a certain velocity and a given separation of the measurement volumes.

## 4 Longitudinal spatial correlation

In deriving the spatio-temporal correlation function, ITLT is clearly superior to the TH. However, if only the longitudinal spatial correlation function is required at turbulence levels at least up to 25 %, the TH performs as well (Fig. 4).

To estimate the longitudinal spatial correlation function from two-point measurements, only the values at τ = 0 (coincidence) are measured for several separations ξ, while all other time lags of the spatio-temporal correlation are not taken into account. However, on the left tail of the spatio-temporal correlation function, the two transform methods almost coincide (Fig. 5). Significant deviations are visible only at the peak center and on the right tail of the spatio-temporal correlation function. Only for turbulence levels above at least 25 % could we expect deviations on the left tail of the spatio-temporal correlation functions occurring between the TH and ITLT, yielding also differences between the longitudinal spatial correlation functions.

## 5 Conclusion

An integral time-to-length transform method has been introduced. It is capable of reproducing the longitudinal spatial-temporal correlation by considering fluctuations of the varying convective velocity. Therefore, it is able to provide longitudinal spatial and spatio-temporal correlation functions from temporal correlation functions obtained from single-point measurements. In the case of turbulent flows, it is superior to Taylor’s hypothesis of a “frozen” flow. The integral transform method is able to recover the spatio-temporal correlation correctly, including the time shift, the degradation of the correlation height, the expansion of the correlation width and the skewness of the peak.

On the contrary, the transform of a temporal correlation function into a longitudinal spatial correlation function based on Taylor’s hypothesis is possible up to turbulence intensities of at least 25 %, because the systematic errors are small, even if the model of a “frozen” flow is far from reality. However, for turbulence intensities beyond 25 %, differences between the methods may occur also at the time lag τ = 0; hence, differences of the obtained longitudinal spatial correlation function must be expected as well. However, Taylor’s hypothesis is not able to recover the temporal development of fluid structures and, therefore, is not capable of transforming temporal correlation functions into spatio-temporal correlations, which the integral method is able to reproduce reliably.

## Acknowledgments

The help of C. Schneider in performing the laser Doppler measurements at EKT (TU Darmstadt), the fruitful discussions with X. He and the financial support of the Deutsche Forschungsgemeinschaft (SFB568) are gratefully acknowledged.

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