Kinematics of the stationary helical vortex mode in Taylor–Couette–Poiseuille Flow

Abstract

We reconstruct a kinematically admissible (volume-preserving) three-dimensional velocity field corresponding to the stationary helical vortex (SHV) mode which is observed in the Taylor–Couette–Poiseuille (TCP) system with a ratio of inner to outer cylinder radii of 0.5 and a length to annulus gap ratio of 16, starting from experimental data obtained via magnetic resonance imaging (MRI) for $\Rey \,{=}\, 11.14$ and $\Ta^{1/2} \,{=}\, 170$ in water. The goal of the present work is to provide a complete kinematic representation of a strongly nonlinear duct flow that is of importance in the fields of mixing and segregation, as well as in the study of the kinematic structure of three-dimensional flows. By a judicious choice of a set of global basis functions that exploit the helical symmetry of SHV, an analytical approximation of the streamfunction is obtained despite the coarse MRI data and the non-uniform distribution of measurement error. This approximation is given in terms of a truncated series of smooth functions that converges weakly in L$_2$, and the reconstruction method is directly applicable to three-dimensional incompressible flows that possess a continuous volume-preserving symmetry. The SHV flow structure consists of a pair of asymmetric counter-rotating helical cells in a double helix structure, foliated with invariant helically symmetric surfaces containing fibre-like fluid particle orbits wrapped around the inner cylinder. Imposing general topological constraints, juxtaposing SHV with neighbouring hydrodynamic modes such as the propagating Taylor vortex flow and direct numerical simulation help corroborate the validity of the reconstruction of the SHV flow field. The kinematically admissible flow field obeys the Navier–Stokes equations with 10% accuracy, which is consistent with experimental error, and has the same flow portrait as the numerically computed flow. Global analysis of the SHV mode indicates that it corresponds to a minimum in dissipation and mixing in comparison with a wide class of perturbed neighbouring modes; hence it is a candidate for the study of particle segregation. To our knowledge, the present study reports the first synthesis of a physically realizable complex open flow that can be represented by an integrable Hamiltonian system starting from point-wise experimental data and using solely kinematic constraints.