On multiple stable states in Taylor-Couette flow with realistic end-wall boundary conditions

TL;DR

This work shows that realistic no-slip end-wall boundary conditions profoundly alter Taylor-Couette flow dynamics by enabling axial angular-momentum transport via Ekman layers and by supporting multiple long-lived roll configurations at fixed $Re$. The authors extend the classical angular-momentum-flux framework to include axial flux $J_z$ and derive a radially conserved total flux $J^\omega$, improving agreement with experiments and boundary-layer predictions. Through DNS with careful lid-smoothing and initialization strategies, they demonstrate strong multistability and hysteresis across a broad range of $Re$ and $Ta$, and they reveal transition mechanisms via modal-energy budgets and space-time analyses of roll merging. The findings have implications for transport optimization and flow-control in wall-bounded systems, illustrating how end-wall physics shapes stability landscapes and transition pathways in closed shear flows.

Abstract

We investigate Taylor-Couette flow with realistic no-slip boundary conditions at all surfaces through direct numerical simulations (DNS) and theoretical analysis. Imposing physically consistent end-wall conditions at the top and bottom lids significantly alters the flow dynamics compared to that for periodic boundary conditions. We extend the classical angular-momentum-flux framework to account for axial transport, which leads to a significantly improved agreement with the Eckhardt-Grossmann-Lohse model (Eckhardt et al. 2007). A systematic exploration of the parameter space $(Re, n) $ uncovers multiple long-lived states with different roll number $n$ configurations at identical Reynolds numbers $Re$, giving rise to pronounced hysteresis loops occurring under realistic boundary conditions. Our DNS for no-slip axial end caps reveal a sequence of structural transitions: as the inner-cylinder Reynolds number increases, the flow evolves from Taylor vortex flow through chaotic wavy vortex flow and turbulent wavy vortex flow to an axisymmetric turbulent Taylor vortex flow. Using modal energy budgets we identify transition mechanisms and quantify how the accessible phase-space volume and associated roll-specific angular momentum flux depend on control parameters and the specific flow state. Our findings demonstrate the impact of realistic boundary conditions on the dynamics in Taylor-Couette flow, and how they change the stability landscape of multiple states. The coexistence of distinct flow patterns and their stability analysis offers promising insights into transition dynamics between laminar and turbulent regimes in closed sheared flows.

Figures (15)

• Figure 1: Example of a TC flow as obtained in the DNS for $\hbox{Re} = 5600$, $\varGamma = 2\pi$, $\eta = 5/7$ as in the setup of ostillaOptimalTaylorCouette2013, but with solid top and bottom lids, illustrated by isosurfaces of the axial velocity $u_z.$ Only half of the computational domain is shown.
• Figure 2: Angular momentum flux development for different smoothing ranges ($\epsilon=2\%$, 3%, 5%, 8% and 10%) in the range of $50 \leq \hbox{Re} \leq 700$ for $\varGamma=30$ and $\eta=0.909$.
• Figure 3: Instantaneous snapshots of axial velocity within the Taylor--Couette system in central vertical cross-sections for $\varGamma = 30, \eta = 0.909$ and different $\hbox{Re}$ (a) $\hbox{Re} = 1500$ and (b) $\hbox{Re} = 4500$, initialized with $\mathbf{u}_{\text{initial}}=\mathbf{0}$. Presented is a TVF state on the left and a WVF state on the right.
• Figure 4: Angular momentum flux development for moderate $\hbox{Re}$ values in the setup of rameshSuspensionTaylorCouette2019 for $\varGamma=11$, $\eta=0.914$. The blue region highlights the onset of convection in the experiments, which agrees perfectly with our simulation result ($\blacklozenge$). The earlier transition in comparison to the periodic assumption is due to perturbations by Ekman vortices (see figure \ref{['fig:flowfields_ramesh']}).
• Figure 5: Central vertical cross-sections of the instantaneous vertical velocity for two different Reynolds numbers $(a,c)\,\,\hbox{Re}=90$ and $(b,d)\,\,\hbox{Re}=120$ in the setup of rameshSuspensionTaylorCouette2019 for $\varGamma=11$ and $\eta=0.914$. The colour coding resolves the strength in axial velocity in $(\textit{a},\textit{b})$, where blue (red) denotes negative (positive) velocites. Smooth contiguous regions indicate areas below the filter threshold of $10^{-6}$. In $(\textit{c},\textit{d})$ the amplitude of the axial velocity is shown on a logarithmic scale ranging from lower (blue) to higher (red) velocities to enhance the contrast in roll-configuration visibility. $(\textit{a},\textit{c})$ At $\hbox{Re}=90$, convection rolls form initially from the upper and lower boundaries and penetrate into the centre of the domain. Due to the limited angular frequency of the inner wall, the roll formation cannot yet cover the full domain. $(\textit{b},\textit{d})$ At $\hbox{Re}=120$, the external driving is strong enough to enable a domain-wide convection roll development.