Numerical study of the precession-driven flow inside a sphere using helical wave decomposition.

This article focuses on the numerical investigation of precession-driven fluid flows inside a rotating sphere using a spectral method based on helical wave decomposition (HWD). The study identifies four flow states—steady, periodic, quasi-periodic, and turbulent—and finds that steady and periodic flows exhibit polarity symmetry in kinetic energy distribution, while quasi-periodic and turbulent flows show polarity asymmetry, with the quasi-periodic case having identical frequency spectra for opposite polarities. At high Reynolds numbers, the helical wave energy spectra follow a λ^(-7/3) scaling, differing from the k^(-2) scaling observed in homogeneous turbulence under precession. The spectral dynamic equation reveals that energy input from boundary motion acts as a body force across all scales, driving a forward cascade of energy from large to small scales, whereas the Coriolis force induces an inverse cascade transferring energy from small to large scales; both polarities receive energy symmetrically, resulting in nearly identical energy spectra. This work demonstrates that HWD provides a comprehensive framework for analyzing wall-bounded turbulent flows in bounded domains, highlighting distinct energy transfer mechanisms compared to classical Fourier-based turbulence analyses.