Dynamic three-dimensional measurement and analysis of rotating polygons

Rotating polygons are geometric patterns formed spontaneously by the free surface of a rotating fluid driven by a rotating disk at the bottom of a partially filled cylindrical container. Most previous studies mainly predict the number of polygon apexes as a function of disk rotation speed and initial fluid depth. We propose to experimentally characterize this three-dimensional instability by means of highly accurate measurements of the free surface height using Fourier transform profilometry. For a given initial height of 26 mm in a cylindrical cavity of radius 140 mm, these measurements enabled us to not only revisit quantitatively the previous analysis in the fully developed regime but also extend the analysis to the spin-up regime (starting from rest). This latter regime has not been explored. In addition, we describe the mixed-mode regime at the boundary between two clearly established polygons. Thanks to precise space-time resolved surface height measurements, we propose a decomposition of each azimuthal mode that provides valuable information on the growth and dynamics of rotating polygons and can, therefore, serve as a basis for future numerical simulations and theoretical models.