The turbulent von Kármán flow in a cylinder has been studied by helical-wave decomposition (HWD) based on direct numerical simulations. HWD provides a powerful tool for analyzing wall-bounded flows by decomposing the flow field into global orthogonal eigenmodes with different scales. The flow visualizations show that the flow field exhibits significant spatial inhomogeneity and anisotropy, characterized by laminar boundary-layer near the endwalls, turbulent boundary-layers on the sidewall adjacent to the endwalls, jetlike flow in the middle of the sidewall, and a central mixing region. The numerical results indicate that the global helical-wave energy spectra of the fluctuating flow in the whole domain show a scaling range of ${\lambda }^{-5/4}$ with wave number $\lambda$. At the same time, the second-order structure functions in physical space are found to exhibit a 2/3 scaling. The finding suggests that the direct relationship between the energy spectrum and the second-order structure function observed in homogeneous isotropic turbulence (HIT) no longer holds. Furthermore, we investigated the nonlinear inter-scale energy transfer in the fluctuating flows. Within the $-5/4$ scaling range of the energy spectrum, the energy transfer function of fluctuating modes remains nonzero; this range is termed the scaling range to distinguish it from the inertial subrange in HIT where the transfer function is zero. This finding reveals the impact of the moving boundary on the energy cascade dynamics at small scales.

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• Energy cascade
• Three-dimensional turbulence
• Turbulence
• Turbulence simulations