Relationship between streamwise and azimuthal length scales in a turbulent pipe flow

Abstract

The statistical relationships among the turbulence structures of the streamwise velocity fluctuations along the streamwise and azimuthal directions in a turbulent pipe flow were examined using direct numerical simulation data at Reτ = 3008. Two-point correlations of the streamwise velocity fluctuations showed a linear relationship between the streamwise and azimuthal length scales (lx and lθ), where lθ/lx = 0.07 along the wall-normal distance, indicating the long coherent structures called very-large-scale motions (VLSMs). The one-dimensional pre-multiplied energy spectra of the streamwise velocity fluctuations showed that the streamwise and the azimuthal wavelengths (λx and λθ) grew linearly along the wall-normal distance, λx/y = 20 and λθ/y = 7, respectively. The ratio between the two linear relationships was determined to be λθ/λx = 0.35, indicative of large-scale motions (LSMs). The energetic modes obtained from a proper orthogonal decomposition (POD) analysis using the translational invariance method showed that the averaged helical angles of the wall mode (ix < iθ; β < 0.1 rad, where ix and iθ are the streamwise and azimuthal mode numbers and β is the helical angle) and lift mode (ix ≥ iθ; β ≥ 0.1 rad) were related to lθ/lx = 0.07 (VLSMs) and λθ/λx ≈ 0.35 (LSMs), respectively. The superposition of the energetic POD modes showed the superimposed X-shaped patterns. The helical angle of the wall mode in the near-wall region was similar to that in the outer region, implying the existence of the VLSMs in the entire wall-normal distance. The LSMs showed more inclined X-shaped patterns. The LSMs were concatenated with the azimuthal offsets to form meandering VLSMs. Most of the VLSMs and LSMs in the near-wall region inclined smaller and larger than 10° (0.17 rad), respectively. In the core region, VLSMs were distributed more helically along the azimuthal direction due to the space limitations of the pipe geometry.