Coherent structures in Newtonian and viscoelastic turbulent planar jets

This study employs spatio-temporal Koopman decomposition to reveal that while viscoelastic and Newtonian planar jets share similar global coherent structures, elasticity-driven near-field streaks and stretched polymer filaments uniquely sustain elastic turbulence in the potential core of viscoelastic jets even at low Reynolds numbers.

The Gist

Imagine you are watching two different kinds of water fountains.

The first fountain is a standard, "normal" water jet (what scientists call a Newtonian jet). It shoots out fast and immediately starts churning into a chaotic, foamy mess. This happens because the water is moving so fast that tiny swirls and eddies form instantly, breaking the smooth stream apart.

The second fountain is a "special" jet (a viscoelastic jet). It looks like water, but it has a tiny amount of long, stretchy polymer chains mixed in—like adding a drop of very thin slime. Surprisingly, even though this second fountain is moving much slower than the first one, it doesn't stay smooth. Instead, it suddenly starts churning and becoming turbulent, just like the fast one.

The big mystery the authors of this paper wanted to solve is: How does the slow, "slimy" fountain get so chaotic without moving fast?

The Detective Work: Breaking the Flow into "Snapshots"

To figure this out, the researchers used a mathematical tool called HODMD. Think of this like a super-smart camera that doesn't just take a picture of the water; it takes thousands of pictures and then uses a computer to break the motion down into its most important "building blocks" or patterns.

They wanted to find the coherent structures. Imagine a chaotic crowd of people running. Even though it looks messy, if you look closely, you might see a few distinct groups: a line of people marching in step, a group waving their arms in a circle, or a few people running in a straight line. These organized groups are the "coherent structures." The researchers wanted to see what these groups looked like in both fountains.

The Two Different Worlds

1. The Fast, Normal Fountain (Newtonian)
In the fast fountain, the chaos starts with big, rolling waves (like the ripples you see when you throw a stone in a pond). These waves grow and break apart, creating a mix of big swirls and tiny, fast-moving bubbles. The "building blocks" of this chaos are mostly big, rolling waves that happen far away from the nozzle.

2. The Slow, "Slimy" Fountain (Viscoelastic)
In the slow fountain, the story is very different.

• The Surprise: Right at the very beginning, near the nozzle, the flow doesn't form big rolling waves. Instead, it forms long, thin streaks.
• The Analogy: Imagine a calm river where, suddenly, long, thin ribbons of water start stretching out parallel to the flow, like long strands of spaghetti floating in a stream.
• The Trigger: These "spaghetti strands" (streaks) are caused by the stretchy polymers. As they stretch, they create high-pressure zones that pull the fluid apart. This stretching creates a "tug-of-war" that eventually snaps the smooth flow into chaos.

The "Rubber Band" Effect

The paper explains that in the slow fountain, the polymers act like rubber bands.

1. The flow creates these long, thin streaks.
2. The rubber bands (polymers) get stretched tight between these streaks.
3. The tension gets so high that the rubber bands snap back, violently shaking the water and creating turbulence.

This is unique because usually, you need high speed (inertia) to make water turbulent. Here, the "elasticity" (the stretchiness) does all the work, even though the water is moving slowly.

What About the "Slime" Itself?

The researchers also looked at the polymers themselves, not just the water.

• They found that the polymers stretch into long filaments right where the water streaks are.
• They also saw a different pattern called "center-mode" structures. Imagine the water in the middle of the jet forming a shape that looks like an arrowhead or a narwhal's tusk. These shapes appear in the middle of the flow and help sustain the chaos.

The Big Conclusion

The main takeaway is that the "slimy" fountain becomes turbulent in a completely different way than the normal one.

• Normal Fountain: Chaos comes from big, fast rolling waves breaking apart.
• Slimy Fountain: Chaos starts with long, thin streaks near the beginning. These streaks stretch the polymers like rubber bands, which then snap and trigger the turbulence.

The researchers emphasize that this process is three-dimensional. If you only looked at the fountain from the side (a 2D view), you would miss the long, thin streaks entirely and wouldn't understand how the turbulence starts. The "spaghetti strands" are the secret key that turns a slow, smooth stream into a chaotic mess.

Technical Summary

1. Problem Statement

The study investigates the dynamics of viscoelastic turbulent planar jets, specifically focusing on how the addition of long-chain polymers alters flow behavior compared to Newtonian jets.

• Context: While Newtonian turbulence is well-studied, viscoelastic turbulence (driven by elastic instabilities rather than inertial ones) remains less explored, particularly in free shear flows like jets.
• The Gap: It is unclear how elastic turbulence is triggered and sustained in planar jets at low Reynolds numbers ($Re$) where inertial turbulence would normally be laminar. Specifically, the role of coherent structures (organized flow patterns) in this transition and the interaction between polymer elasticity and flow instability in the near-field region are not fully understood.
• Objective: To compare the global and near-field coherent structures of a high-$Re$ Newtonian jet with a low-$Re$ viscoelastic jet (elastic turbulence regime) to elucidate the mechanisms driving elastic turbulence.

2. Methodology

The authors employed a combination of Direct Numerical Simulations (DNS) and advanced data-driven modal decomposition.

• Numerical Simulations:

  • Solver: In-house solver Fujin using the Oldroyd-B model for viscoelasticity.
  • Cases:
    1. Newtonian Jet: High Reynolds number ($Re = 6750$) to ensure fully developed turbulence.
    2. Viscoelastic Jet: Low Reynolds number ($Re = 20$) with high Deborah number ($De = 100$) and elasticity number $E > 1$. In this regime, turbulence is sustained solely by elasticity (inertial effects are negligible), and the equivalent Newtonian jet would be laminar.
  • Domain: 3D Cartesian grids with non-reflective outflow and periodic spanwise boundaries. The viscoelastic domain was significantly larger to accommodate the slower development of the turbulent region.

• Data-Driven Analysis:

  • Technique: Spatio-Temporal Koopman Decomposition (STKD), an extension of Higher Order Dynamic Mode Decomposition (HODMD).
  • Process:
    1. Temporal Decomposition: HODMD extracts dominant temporal modes (frequencies and growth rates) from the flow data.
    2. Spatial Decomposition: Each temporal mode is further decomposed in the spanwise direction to identify traveling waves and standing waves.
  • Advantage: This method allows for the extraction of coherent structures from highly nonlinear, broadband data without requiring linearization or prior knowledge of the governing equations.

3. Key Contributions

1. Application of STKD to Viscoelastic Jets: First application of spatio-temporal Koopman decomposition to characterize coherent structures in low-$Re$ viscoelastic planar jets.
2. Identification of Near-Field Streaks: Discovery of elasticity-driven, streamwise-elongated streaks in the potential core of the viscoelastic jet, which are absent in the Newtonian case at comparable near-field locations.
3. Mechanism of Elastic Turbulence: Proposing a specific mechanism where near-field streaks generate localized high-shear regions that stretch polymers, leading to elastic instabilities that trigger and sustain turbulence.
4. 3D Nature of Elastic Turbulence: Demonstrating that elastic turbulence in planar jets is inherently three-dimensional, driven by spanwise-inhomogeneous streaks, challenging the notion that it is purely a 2D phenomenon.

4. Key Results

A. Global Flow Structures

• Similarities: Both jets exhibit low-frequency, streamwise-elongated structures (streaks) and high-frequency, spanwise-coherent wave packets in the far-field.
• Differences:
  • Newtonian: Low-frequency streaks are dominant, extending globally and penetrating deep into the jet core. High-frequency modes show clear symmetric (varicose) Kelvin-Helmholtz (K-H) rollers.
  • Viscoelastic: Low-frequency streaks are less dominant in the far-field due to disruption by large-scale flapping motions. High-frequency modes resemble Orr wave packets more than K-H rollers.

B. Near-Field Dynamics (The Critical Difference)

• Newtonian: The near-field is characterized by the growth of K-H rollers (varicose instability) starting around $x/h \approx 5$.
• Viscoelastic:
  • Elasticity-Driven Streaks: Instead of K-H rollers, the flow immediately develops short, thin, streamwise-elongated streaks along the shear layers of the potential core ($x/h < 20$).
  • Interaction: These streaks are not standing structures; they travel downstream and interact with high-frequency wave packets (both shear-layer and jet-column modes).
  • Breakdown: The interaction between the streaks and the wave packets causes the streaks to break down, modifying the bulk flow and triggering the transition to turbulence.

C. Polymer Field Dynamics

• Polymer Filaments: The analysis of the conformation tensor reveals that the near-field streaks stretch polymers into streamwise filaments. This confirms that the high shear between adjacent streaks is the primary driver of polymer stretching.
• Center-Mode Structures: At higher frequencies, the polymer field exhibits "center-mode" structures (sheet-like formations merging at the jet centerline), similar to those seen in wall-bounded elastic turbulence.
• Amplification: The polymer field amplifies the velocity modes by an order of magnitude, particularly at higher frequencies, indicating a strong feedback loop where velocity fluctuations drive polymer stretching, which in turn sustains the turbulence.

5. Significance and Conclusions

• Trigger Mechanism: The study establishes that in low-$Re$ viscoelastic jets, near-field streaks are the primary trigger for elastic turbulence. They create localized high-shear zones that stretch polymers, generating elastic stresses that destabilize the flow.
• Three-Dimensionality: The research highlights that elastic turbulence in planar jets is a 3D phenomenon. Neglecting the spanwise-inhomogeneous streaks (as might happen in 2D simulations) would miss the critical mechanism driving the transition.
• Engineering Implications: Understanding these coherent structures is vital for controlling mixing, drag reduction, and noise generation in industrial applications involving polymer-laden flows. The findings suggest that manipulating the near-field streaks could be a strategy for controlling the onset of elastic turbulence.

In summary, the paper provides a comprehensive physical picture of how elastic turbulence emerges in planar jets, identifying a distinct pathway driven by the interplay between near-field streaks, polymer stretching, and wave packet interactions, distinct from the inertial mechanisms of Newtonian turbulence.