Space–time formation of very-large-scale motions in turbulent pipe flow

Abstract

We examine the origin of very-large-scale motions (VLSMs) in fully developed turbulent pipe flow at friction Reynolds number, $\mathit{Re}_{\unicode[STIX]{x1D70F}}=934$, using data from a direct numerical simulation. The VLSMs and the packet-like large-scale motions (LSMs) found in this study are very similar to those found in earlier studies. Three-dimensional time-evolving instantaneous fields show that one component of the process leading to the large streamwise length of VLSMs is the concatenation of adjacent streamwise LSMs caused by the continuous elongation of LSMs due to the strain component of the mean shear. Spatial organization patterns of the VLSMs and LSMs and their properties are studied by separating auto-correlation of the streamwise velocity fluctuations into the components of the VLSM and the LSM defined by low-pass/high-pass filtering in the streamwise direction. The structures of the two-point spatial correlations of the streamwise velocity component of the VLSMs and the LSMs in the streamwise-azimuthal plane are characterized by multiple maxima and complex patterns that beg explanation in terms of patterned coherent arrangements of the LSMs. Using proper orthogonal decomposition (POD), it is found that the X-shape correlation pattern of the VLSMs results from the superposition of very long helically inclined structures and streamwise-aligned structures. Further explanation of the patterns in the correlations of the VLSMs and LSMs is provided through the study of synthetically constructed arrangements of simple hairpin packet models of the LSM. Head-to-tail alignment of the model packets along streamwise and helical directions suggested by the eigenvalues of the POD creates a pair of long roll-cells centred above the logarithmic layer, and bracketing the LSMs. These roll-cells are pure kinematic consequences of the induction within the LSM packets, but they may also serve to organize smaller packets.