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

This study utilizes direct numerical simulations and theoretical analysis to demonstrate that realistic no-slip end-wall boundary conditions in Taylor-Couette flow induce multiple stable states with pronounced hysteresis, distinct transition sequences, and altered angular momentum transport compared to periodic conditions.

The Gist

Imagine a giant, transparent jar filled with honey. Inside this jar, there is a smaller, spinning rod in the center, and the jar itself is stationary. This setup is called Taylor–Couette flow. It's a classic physics experiment used to understand how fluids move, swirl, and eventually turn chaotic (turbulent).

For a long time, scientists studied this by pretending the jar was infinitely tall, like a straw that goes up and down forever. They ignored the top and bottom lids. But in the real world, jars have lids! This paper by Kriening and his team asks: What happens when we actually put lids on the jar?

Here is the story of their discovery, explained simply:

1. The "Infinite Straw" vs. The "Real Jar"

Imagine you are trying to spin a rope. If the rope is infinite, the spin travels smoothly forever. But if the rope is tied to the floor and the ceiling, the ends get stuck.

In fluid physics, the "ends" are the top and bottom lids of the jar. Because the fluid sticks to these lids (a rule called "no-slip"), it creates a traffic jam of swirling motion right near the top and bottom. The researchers found that ignoring these lids is like trying to understand a traffic jam by only looking at the middle of the highway and ignoring the on-ramps and off-ramps. The lids change everything.

2. The "Ghost Traffic" (Axial Transport)

The team realized that when the fluid hits the top and bottom lids, it doesn't just stop; it gets pushed sideways, creating a hidden "ghost traffic" of momentum moving up and down.

They updated the old math formulas to account for this ghost traffic. Think of it like updating a GPS map. The old map only showed the main highway (radial flow). The new map adds the side streets and exits (axial flow) caused by the lids. This new math matched real-world experiments perfectly, whereas the old math was off.

3. The Magic of "Multiple Personalities" (Multiple Stable States)

This is the most exciting part. Usually, if you spin a jar at a specific speed, you expect the fluid to settle into one specific pattern, like a calm, organized dance.

But the researchers discovered that with realistic lids, the fluid has multiple personalities.

• The Scenario: Imagine you have a jar spinning at a speed of "500 mph."
• The Surprise: If you start the jar gently, the fluid might settle into a pattern with 18 giant swirls (like 18 large donuts stacked on top of each other).
• The Twist: If you start the jar with a little "kick" (a tiny perturbation) at that exact same speed, it might settle into a pattern with 24 smaller swirls.

Both patterns are stable. They are both happy to stay there forever. It's like a ball sitting in a valley. If you push it slightly, it rolls into a different valley that is just as deep. The fluid has "memory" of how it started. This is called hysteresis.

4. The "Merge and Split" Dance

The researchers watched these swirls (called "rolls") over time.

• Sometimes, if the fluid is in a "wrong" state for that speed, two neighboring swirls will slowly drift together and merge into one giant swirl, like two soap bubbles combining.
• This happens very slowly, like watching paint dry, but it changes the whole system.
• They found that at very high speeds, the fluid gets so turbulent that it actually calms down the chaos, allowing these organized swirls to survive again. It's like a chaotic crowd suddenly organizing into a marching band when the music gets loud enough.

5. Why Does This Matter?

You might ask, "Who cares about spinning honey?"

• For Science: It proves that even in a simple, predictable system, the outcome isn't always unique. The history of how you started the system matters. It's a lesson in complexity: small changes in the beginning can lead to totally different, stable endings.
• For Industry: Many machines (like hard drives, chemical mixers, or heat exchangers) use rotating fluids. If engineers understand that they can "choose" a specific flow pattern by how they start the machine, they could design systems that mix chemicals faster or transfer heat more efficiently.

The Bottom Line

This paper is like finding a secret door in a familiar room. By finally paying attention to the top and bottom of the jar (the lids), the scientists discovered that the fluid inside is much more complex, flexible, and full of surprises than we thought. They showed us that nature doesn't always pick just one path; sometimes, it has several paths it's happy to walk down, depending on how you nudge it.

Technical Summary

1. Problem Statement

Taylor–Couette (TC) flow, the motion of fluid between two concentric rotating cylinders, is a canonical model for studying hydrodynamic stability and turbulence. Historically, many numerical studies have employed periodic boundary conditions (PBC) in the axial direction to simplify simulations and avoid end-wall effects. However, real-world experimental setups and industrial applications utilize no-slip boundary conditions at the top and bottom lids.

The authors identify a critical gap: PBCs fail to capture the complex dynamics induced by Ekman layers at the end walls, which significantly alter the flow structure, angular momentum transport, and the stability landscape. Specifically, the existence of multiple stable states (hysteresis) and the coexistence of different flow patterns (e.g., different numbers of Taylor vortex rolls) at identical Reynolds numbers ($Re$) under realistic boundary conditions remain poorly understood. The paper aims to resolve these discrepancies by performing Direct Numerical Simulations (DNS) with fully realistic no-slip boundaries and extending the theoretical framework for angular momentum transport.

2. Methodology

The study employs a combination of high-fidelity Direct Numerical Simulations (DNS) and theoretical analysis.

• Numerical Setup:

  • Code: Simulations were conducted using the in-house code goldfish, utilizing a fourth-order finite-volume discretization on staggered grids and a third-order Runge–Kutta time-marching scheme.
  • Geometry: Two aspect ratios ($\Gamma = H/d$) were investigated: $\Gamma = 11$ and $\Gamma = 30$, with a radius ratio $\eta = r_i/r_o \approx 0.909$ (inner cylinder rotating, outer stationary).
  • Boundary Conditions: No-slip conditions were applied to all surfaces (inner cylinder, outer cylinder, top, and bottom lids).
  • Singularity Handling: A significant numerical challenge arises at the junction of the rotating inner cylinder and stationary lids, creating a velocity discontinuity. The authors introduced a sigmoid smoothing function ($\psi$) near the lids to transition the azimuthal velocity smoothly, ensuring numerical stability while minimizing non-physical second derivatives.
  • Initialization: To explore multiple stable states, the authors developed a specific initialization strategy. Instead of starting from zero velocity (which biases the system toward asymmetric Taylor vortex flow), they initialized simulations with analytically constructed Taylor vortex fields containing specific numbers of rolls ($n$) and added controlled white noise perturbations to trigger transitions.

• Theoretical Extension:

  • The authors extended the classical angular-momentum-flux framework (Eckhardt–Grossmann–Lohse model) to account for axial transport.
  • They derived a modified total angular momentum flux ($J_\omega$) that includes a radial term and an axial correction term arising from the $z$-dependence of the azimuthal velocity induced by Ekman layers. This ensures the flux is radially constant in statistically steady states, unlike the periodic approximation.

3. Key Contributions

1. Extended Angular Momentum Theory: The derivation of a modified angular momentum flux equation (Eq. 2.13) that explicitly includes axial transport terms due to no-slip end walls. This corrects the radial non-constancy found in periodic approximations and improves agreement with experimental data.
2. Numerical Validation of Realistic BCs: Successful DNS validation against experimental data from Martínez-Arias et al. (2014) and Ramesh et al. (2019), demonstrating that realistic boundary conditions are essential for capturing the correct onset of instability and transport scaling.
3. Discovery of Multiple Stable States: Systematic mapping of the parameter space ($Re, n$) revealing that multiple distinct flow states (with different roll numbers $n$) can coexist at the same Reynolds number. The final state depends heavily on initial conditions, leading to pronounced hysteresis loops.
4. Transition Mechanisms: Identification of the physical mechanisms driving transitions between states, including roll merging events and the role of modal energy budgets (specifically the interplay between axisymmetric and non-axisymmetric modes).

4. Key Results

• Impact of Boundary Conditions:

  • Realistic no-slip conditions lead to asymmetric roll configurations that are stable and exhibit significantly higher angular momentum flux compared to symmetric states.
  • The smoothing function introduces a constant offset in $Re$ and $J_\omega$, which can be systematically corrected to match experiments.

• Multiple Stable States & Hysteresis:

  • In the range $400 \le Re \le 3000$, the system exhibits multistability. For a given $Re$, the flow can settle into different stable states depending on the initial number of rolls ($n$).
  • Hysteresis: The system shows strong hysteresis; increasing $Re$ from a low value yields a different sequence of states than decreasing $Re$ from a high value.
  • Phase Space Dynamics:
    • Low $Re$ (< 900): Large accessible phase space with multiple coexisting stable states.
    • Intermediate $Re$ (900–2000): The system converges to a single stable state (strong convergence), often involving roll merging.
    • High $Re$ (> 2500): A "regrowth" of the accessible phase space occurs, where multiple roll configurations regain stability even in turbulent regimes.

• Transition Dynamics:

  • Transitions between states (e.g., from 26 rolls to 20 rolls) occur via roll merging. Space-time diagrams show that neighboring rolls approach axially, their interface weakens, and they coalesce into a larger vortex.
  • Energy Budget Analysis: During transitions, non-axisymmetric modes ($m > 0$) exhibit transient amplification. The axisymmetric mode ($m=0$) remains dominant but shows a temporary reduction in energy fraction prior to a transition, suggesting it acts as an energy reservoir for configurational changes.
  • Flow Evolution Sequence: As $Re$ increases, the flow evolves: Taylor Vortex Flow (TVF) $\to$ Wavy Vortex Flow (WVF) $\to$ Modulated/Chaotic WVF $\to$ Turbulent WVF $\to$ Axisymmetric Turbulent TVF.

• Transport Scaling:

  • The angular momentum flux efficiency increases with the number of rolls.
  • Unlike 2D Rayleigh–Bénard convection, the scaling exponent of the flux depends on the specific roll configuration (aspect ratio of the rolls), highlighting the 3D nature of TC flow with end walls.

5. Significance

• Fundamental Physics: The study challenges the notion of a unique turbulent state for fixed control parameters in wall-bounded flows. It demonstrates that history dependence (initial conditions) is a fundamental feature of TC flow with realistic boundaries, creating a complex attractor landscape.
• Engineering Applications: The existence of multiple stable states with different transport efficiencies offers new avenues for flow control. By selecting specific initial conditions or perturbations, one could potentially optimize industrial mixing or heat transfer processes by targeting states with higher angular momentum flux.
• Methodological Benchmark: The work provides a robust framework for simulating wall-bounded turbulent flows, emphasizing the necessity of resolving end-wall effects and the limitations of periodic boundary approximations in predicting transport properties and stability.

In conclusion, Kriening et al. provide a comprehensive picture of how realistic boundary conditions reshape the stability landscape of Taylor–Couette flow, revealing a rich phenomenology of multistability, hysteresis, and complex transition dynamics that are invisible in idealized periodic models.