Laminar-to-Turbulent Transition of Yield-Stress Fluids in Pipe and Channel Flows

Imagine you are trying to push a giant, thick blob of toothpaste through a long, narrow tube. Now, imagine that toothpaste has a superpower: it acts like a solid block until you push hard enough to break it, and then it suddenly starts flowing like a liquid. This is what scientists call a "yield-stress fluid."

This paper is a high-tech, computer-based experiment that asks a simple but tricky question: What happens when you push this "smart toothpaste" really, really fast? Does it stay smooth, or does it start to swirl and chaos (turbulence) like water in a storm?

Here is the story of their discovery, broken down into everyday concepts:

1. The "Solid Core" vs. The "Slippery Walls"

In normal water, if you speed it up, the whole stream gets chaotic at once. But with these special fluids (like the toothpaste or thick paint mentioned in the paper), something weird happens.

The Plug: Because the fluid is so stubborn, the stuff in the very middle of the pipe refuses to move. It stays stuck together like a solid plug.
The Shear: Only the stuff touching the walls gets squeezed and forced to move. It's like a group of people trying to walk through a hallway; the people in the middle stand still, while only the people hugging the walls are forced to shuffle sideways.

2. The "Tipping Point" (Transition)

The researchers used super-computers to simulate pushing this fluid faster and faster. They were looking for the exact moment the "solid plug" in the middle breaks apart and the whole pipe turns into a chaotic mess (turbulence).

They found that the fluid has a secret rule:

The chaos can only happen if the "push" from the moving fluid is stronger than the "stickiness" holding the plug together.

Think of it like trying to break a piece of hard candy. If you tap it gently (low speed), it stays solid. If you hit it hard enough (high speed), it shatters. The computer simulations showed that the "plug" in the middle of the pipe only shatters when the force of the flow overpowers the fluid's natural desire to stay solid.

3. The Three Zones of Flow

The study mapped out three distinct stages, like a video game level progression:

Level 1: The Calm (Laminar): The fluid moves slowly. The middle is a giant, unmoving solid block. The walls are just sliding past it. It's very orderly.
Level 2: The Wobble (Transitional): You speed up. The solid block starts to crack. Small swirls of chaos appear near the walls, but the middle is still holding on tight. It's a tug-of-war between the fluid's desire to stay solid and the force pushing it.
Level 3: The Storm (Turbulent): You push really hard. The "plug" completely dissolves. The whole pipe is now a swirling, chaotic mess, just like water in a fast river.

4. Why Does This Matter?

You might wonder, "Who cares about toothpaste in a pipe?"
Actually, this stuff is everywhere!

Paint: You don't want your paint to run down the wall like water, but you want it to spread smoothly when you brush it.
Blood: Our blood has properties that act a bit like this.
Slurries: Mining companies pump thick mud and rock mixtures through pipes. If they don't understand when the flow turns chaotic, the pipes can clog or burst.

5. The "Virtual Lab"

The scientists didn't just guess; they built a digital twin of the experiment.

They created a virtual pipe and a virtual channel (like a flat, wide gutter).
They filled it with a digital version of "Carbopol" (a common gel used in labs that acts like toothpaste).
They ran millions of calculations to watch how the "plug" formed, how it broke, and how the turbulence started.
They even compared their computer results to real-world experiments done by other scientists, and the numbers matched perfectly.

The Big Takeaway

The most important thing they learned is that turbulence doesn't just happen because you go fast. In these special fluids, turbulence only happens when the force of the flow is strong enough to physically break the "solid" core in the middle.

It's like realizing that a crowd of people won't start running in panic until the person at the front pushes hard enough to knock over the first person, who then knocks over the next, until the whole line is moving chaotically.

This research gives engineers a better "rulebook" for designing pumps and pipes for everything from industrial manufacturing to medical devices, ensuring they don't get clogged or break under pressure.

Here is a detailed technical summary of the paper "Laminar-to-Turbulent Transition of Yield-Stress Fluids in Pipe and Channel Flows" presented at the ASME 2026 Fluids Engineering Division Summer Meeting.

1. Problem Statement

The transition from laminar to turbulent flow in non-Newtonian fluids, specifically Herschel–Bulkley (HB) yield-stress fluids, remains a complex challenge in fluid dynamics. Unlike Newtonian fluids, HB fluids (e.g., Carbopol, slurries, paints) exhibit a yield stress ( $\tau_y$ ) that creates an unyielded "plug" region in the flow core where shear rates are zero.

The Gap: While experimental studies exist, there is a lack of Direct Numerical Simulations (DNS) that resolve the complete transition sequence (laminar $\to$ transitional $\to$ turbulent) for HB fluids.
The Challenge: Existing models often struggle to capture the interplay between the yield-stress-driven plug, its breakdown due to inertial forces, and the subsequent emergence of near-wall turbulence. Furthermore, a unified understanding of these mechanisms across both pipe and rectangular channel geometries has been limited.

2. Methodology

The authors employed high-fidelity Direct Numerical Simulations (DNS) to investigate pressure-driven flows of HB fluids.

Rheological Model:
- The fluid is modeled using the Herschel–Bulkley constitutive relation: $\tau = \tau_y + k\dot{\gamma}^n$ for $\tau > \tau_y$ .
- To handle numerical stiffness at low shear rates, a bi-viscosity regularization was applied, capping viscosity below a cut-off strain rate ( $\dot{\gamma}_0$ ) to simulate the unyielded plug without singularities.
Governing Equations:
- Incompressible Navier-Stokes equations were solved with a shear-dependent viscosity.
- The nonlinear convective term was implemented in a skew-symmetric form for numerical stability.
- A constant reference viscosity was split off to allow implicit time-stepping for the Laplacian term, enabling stable time steps near the CFL limit.
Dimensionless Parameters:
- Generalized Reynolds Number ( $Re_G$ ): Defined based on wall viscosity ( $\mu_w$ ) to account for shear-thinning and yield stress effects.
- Metzner–Reed ( $Re_{MR}$ ): Used for comparison with experimental correlations.
- Friction Reynolds Number ( $Re_\tau$ ): Used for inner scaling and near-wall normalization.
Computational Domains:
- Pipe Flow: Circular cross-section ( $R=25.4$ mm), length $L=4\pi D$ . Mesh: $\approx 4 \times 10^6$ control volumes using an O-grid topology.
- Channel Flow: Rectangular domain ( $L_x = 0.2$ m). Mesh: $128 \times 129 \times 128 $points ($ \approx 4.2$ million cells).
- Resolution: Near-wall resolution satisfied $y^+ < 1$ .
Validation: Pipe flow results were validated against experimental Laser Doppler Velocimetry (LDV) data for 0.1% Carbopol solutions (Guzel et al., 2009).

3. Key Contributions

First Unified DNS of HB Transition: This work provides the first DNS to resolve the entire laminar-to-turbulent transition sequence for HB fluids in both pipe and channel configurations.
Mechanism Elucidation: It explicitly identifies the physical mechanism of transition: turbulence only sustains when local Reynolds stresses exceed the local yield stress, allowing the plug core to deform and break down.
Regime Classification: The study establishes clear regime boundaries for HB pipe flow based on $Re_G$ , aligning with experimental trends for Carbopol.
Geometry Comparison: It demonstrates that the fundamental physics of plug formation, breakdown, and wall-dominated turbulence are consistent across pipe and planar channel geometries.

4. Key Results

A. Transition Regimes in Pipe Flow

The simulations identified three distinct regimes based on the generalized Reynolds number ( $Re_G$ ):

Laminar ( $Re_G < 1735$ ): Characterized by a strong, stable central plug and negligible turbulence.
Transitional ($1735 < Re_G < 2920$): A narrow window where turbulence intensity rises sharply. The plug boundary begins to erode, and flow exhibits intermittent turbulence and spatial asymmetry.
Fully Turbulent ( $Re_G > 2920$ ): The plug breaks down significantly; turbulence becomes established throughout the cross-section, though the core remains weaker than the near-wall region.

B. Turbulence Statistics and Validation

Turbulence Intensity ( $I$ ): The DNS reproduced the experimental trend where turbulence intensity peaks in the transitional regime at off-center locations ( $r/R \approx \pm 0.75$ ) rather than at the centerline.
Asymmetry: A slight asymmetry in turbulence intensity was observed in the transitional range ( $r/R = +0.75$ vs. $-0.75$ ), consistent with experimental observations of biased states in yield-stress flows.
Velocity Profiles: As $Re_G$ increased, the velocity profile evolved from a "blunted" plug shape to a fuller, turbulent profile with steeper near-wall gradients.

C. Flow Structures (Q-Criterion)

Low $Re_G$ (378): Vortical structures were sparse, large, and organized, confined primarily to the sheared region near the wall. The core remained unyielded.
High $Re_G$ (4488): The flow exhibited dense, small-scale vortical activity throughout the pipe, indicating fully developed turbulence.
Plug Breakdown: The Q-criterion visualizations clearly showed the erosion of the plug boundary as inertia overcame the yield stress.

D. Channel Flow

Channel simulations confirmed the same physics observed in pipes:

Formation of a central plug at low $Re_G$ .
Growth of near-wall turbulence structures (visualized via Q-criterion).
Progressive strengthening of RMS velocity fluctuations ( $u'_{rms}$ ) and turbulent kinetic energy peaks with increasing $Re_G$ .
Strong influence of shear-thinning ( $n < 1$ ) and yield stress on the transition threshold.

5. Significance

Fundamental Physics: The study resolves the long-standing question of how yield stress suppresses and delays transition, confirming that the "plug" acts as a stabilizing mechanism that must be locally overcome by turbulent Reynolds stresses for transition to occur.
Engineering Applications: The findings are critical for designing systems involving high-viscosity non-Newtonian fluids, such as:
- Slurry Transport: Predicting pressure drops and pump requirements.
- Industrial Processing: Optimizing mixing and flow control for paints, polymers, and food products.
- Heat Transfer: Understanding how yield stress affects heat pipe performance and refrigerant transport.
Future Work: The authors plan to validate these DNS predictions experimentally using a 3.6 m acrylic channel with Particle Image Velocimetry (PIV) to further quantify geometry-dependent transition thresholds.

In summary, this paper provides a rigorous computational framework that bridges the gap between theoretical rheology and experimental observations, offering a unified view of how yield-stress fluids transition from laminar plug flow to wall-dominated turbulence.