Dynamical Eigenfunction Decomposition of Turbulent Pipe Flow

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Abstract
The results of an analysis of turbulent pipe flow based on a Karhunen-Lo`eve decomposition are presented. The turbulent flow is generated by a direct numerical simulation of the Navier-Stokes equations using a spectral element algorithm at a Reynolds number Re_\tau=150. This simulation yields a set of basis functions that captures 90% of the energy after 2,453 modes. The eigenfunctions are categorised into two classes and six subclasses based on their wavenumber and coherent vorticity structure. Of the total energy, 81% is in the propagating class, characterised by constant phase speeds; the remaining energy is found in the non propagating subclasses, the shear and roll modes. The four subclasses of the propagating modes are the wall, lift, asymmetric, and ring modes. The wall modes display coherent vorticity structures near the wall, the lift modes display coherent vorticity structures that lift away from the wall, the asymmetric modes break the symmetry about the axis, and the ring modes display rings of coherent vorticity. Together, the propagating modes form a wave packet, as found from a circular normal speed locus. The energy transfer mechanism in the flow is a four step process. The process begins with energy being transferred from mean flow to the shear modes, then to the roll modes. Energy is then transfer ed from the roll modes to the wall modes, and then eventually to the lift modes. The ring and asymmetric modes act as catalysts that aid in this four step energy transfer. Physically, this mechanism shows how the energy in the flow starts at the wall and then propagates into the outer layer.

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