Spatial topologies of nondissipative dynamics or superfluid in a turbulent pipe flow

Abstract

Turbulence is a common phenomenon characterized by its chaotic nature in time and coherent structures in space. A recent study was able to solve the temporal component of turbulent velocity and produce a temporal correlation function analytically by the hypothesis of isentropic motion or superfluid in a viscous fluid [W. Chen, “On Taylor correlation functions in isotropic turbulent flows,” Sci. Rep. 13, 3859 (2023)]. However, the spatial distribution of the turbulent velocity is still unknown. In this study, the spatial topology of a turbulent pipe flow [Jackel et al., “Coherent organizational states in turbulent pipe flow at moderate Reynolds numbers,” Phys. Fluids 35, 045127 (2023)] was investigated with the theory of nondissipative dynamics or superfluid. Ten elementary excitation modes on the boundary of the second law have been identified. The temporal, radial, azimuthal and longitudinal components of the longitudinal velocity have been solved and specified. The spatial topology on the cross section is described by the employment of orthogonal correlation functions. This theory satisfactorily agrees with the experimental data at the azimuthal wavenumber from 2 to 7. In the spatial topology, each azimuthal wavenumber corresponds to one pair of positive and negative velocity torsos along the mean longitudinal flow. Many other spatial topology examples involve the combinations of three basic structures of resonant superfluids, i.e., nodes, antinodes, and saddles. The essential differences between the flow fields of a regular fluid and superfluid are summarized. This work provides spatial solutions and methods to complement the temporal solutions in earlier studies [W. Chen, “On Taylor correlation functions in isotropic turbulent flows,” Sci. Rep. 13, 3859 (2023); W. Chen, “An asymmetric probability density function,” Phys. Fluids 35, 095117 (2023)]. The method and results should advance the understanding of turbulence and coherent states.