Structural similarity in radial correlations and spectra of longitudinal velocity fluctuations in pipe flow

Abstract

The paper describes correlation measurements in both broad and narrow frequency bands of the longitudinal velocity fluctuations in fully developed pipe flow at four positions for a reference probe whilst a second probe was traversed radially from deep in the sublayer to a position near the axis with both longitudinal and transverse separations zero (Δx = Δz = 0). Such measurements require that both the Covariant (Co) and Quadrature (Quad) correlations be determined for each of the 15 frequencies used to constrain the wave component λx.The new data demonstrate that low frequency, large scale turbulence fluctuations extend over the majority of the radial region and that these components are highly correlated. By using a similarity variable kxy, along with a normalized wall distance y/y REF, both correlation functions, i.e. the Co and the Quad components, are shown to collapse. The physical significance of this is discussed.The broad-band data do not collapse because of the large range of wave sizes. However, the present experiment does show that strong radial correlations exist even when one probe is at y+ = 1. This conflicts with the earlier data of Favre, but agrees with the more recent work of Comte-Bellot. There is a significant amount of turbulent energy in frequencies less than 16 Hz (ω+ = 0·008) for turbulent flows of about 105 Reynolds number.The spectral function ωΦ(ω) is also presented for a range of y+ values. Using this form for the power spectral density, along with the stochastic wave modelling and similarity arguments of this paper, it is shown how a consistent explanation for the behaviour of these spectra is obtained. In addition some preliminary results from cross-spectral analyses are presented and suggestions made as to their physical significance.

The large-scale dynamics of near-wall turbulence

“…All the modes in the upper-right-hand corner of the spectral plane, which corresponds to the spectral production tail, have correlation heights which are larger than y + = 30, and are therefore essentially coherent over the full integration height. That result is consistent with those of Bullock, Cooper & Abernathy (1978), who computed covariances for velocity signals filtered in different frequency bands. In a pipe at Re τ = 2600 they found that the correlation coefficient of u between y + = 50 and all the y < y was larger than 0.6 when λ + x & 1000, but that it vanished in the sublayer for the shorter wavelengths λ + x < 250.…”

An experimental investigation of wall pressure fluctuations beneath non-equilibrium turbulent flows

“…The double peak in iu(w) near the wall is also reported by others; fcr example, the double peak is in the data of Perry and Abell (1975) and in the data of Bullock et al (1978). Bullock et al (1978) At high enough Reynolds numbers, the separation between the low and high wavenumber regions will he large and the high wavenumber region will be isotý'opic and independent of the character of the production process which can be highly anisotropic.…”

“…However, very near the wall'in the buffer region, whore the mean velocity gradient is large, the spectrum in the inertial wavenumber range reflects the influence of the production process, namely the mean velocity gradient (dU/dy), and that the spectrum in this range scales as Sk-1 . It is interesting to noto that Bullock et al (1978) also obtained a k-1 dependence to their near wall data (200<y+<500) which they attributed as being due to the predominance of the lower frequency peak (u wave) that was present in the spectral data. Nevertheless, the frequency of the velocity spectra can be used to gain insight to the turbulence structure of the flow field.…”

“…Bradshaw (1971) suggested that data presentation in this format might be advantageous since it displays the relative contribution each frequency range makes to the overall mean-square value since w4P(w)d(logw)=•w()dw. Samuel and Joubert (1974), as well as Bullock et al (1978), presented velocity spectra in this format to emphasize the differences between measurements at different wall locations. Strickland and Simpson (1975) state that the peak in the first-moment spectral density of the streamwise velocity fluctuations occur due to a statistically periodic turbulence phenomena which they argue is the "bursting" phenomena that occurs near the wall.…”

“…Bullock et al (1978) At high enough Reynolds numbers, the separation between the low and high wavenumber regions will he large and the high wavenumber region will be isotý'opic and independent of the character of the production process which can be highly anisotropic.…”