Fully Turbulent Wakes at Low Reynolds Numbers: the Case of the Thin Flat Plate

This paper demonstrates through direct numerical simulation and experimental comparison that the wake flow behind a thin two-dimensional flat plate becomes fully turbulent at a relatively low Reynolds number of 400, exhibiting statistical and spectral characteristics indistinguishable from higher-Reynolds-number turbulent wakes, a transition path that differs significantly from that of canonical circular or square cylinders.

The Gist

Imagine you are holding a thin, flat piece of cardboard (like a playing card) in a strong wind. As the wind hits the card, it creates a messy, churning trail of air behind it called a "wake." For a long time, scientists believed that for this wake to become truly chaotic and "turbulent," the wind would have to be blowing very fast, or the card would have to be a specific shape like a round pipe or a square block.

This paper tells a different story. The researchers discovered that if you use a thin, flat plate, the air behind it becomes fully chaotic and turbulent at a much lower wind speed than anyone expected. In fact, it happens at a speed where, for other shapes, the air is still relatively calm and orderly.

Here is a breakdown of their findings using simple analogies:

1. The "Turbulence Threshold" Surprise

Think of turbulence like a crowded dance floor.

• The Old Belief (Round Cylinders): If you have a round pole in the wind, the air behind it starts as a calm, rhythmic dance (swaying back and forth). It takes a lot of energy (high speed) before the dancers start bumping into each other, spinning wildly, and creating a chaotic mess (turbulence). This transition happens slowly over a wide range of speeds.
• The New Discovery (Thin Flat Plate): The researchers found that for a thin flat plate, the "dance floor" goes from calm to a wild mosh pit almost instantly. Even at a relatively low wind speed (Reynolds number 400), the air behind the plate is already fully chaotic. It doesn't go through the slow, rhythmic stages that round poles do. It jumps straight to the party.

2. How They Proved It

To be sure they weren't just imagining things, the team acted like detectives comparing crime scenes.

• The Simulation (The Virtual Lab): They used supercomputers to simulate the wind hitting the plate at low speeds (Re 150 and Re 400).
• The Real-World Test (The Wind Tunnel): They also looked at real experiments where the wind was blowing much faster (Re 12,500 and Re 19,700).
• The Match: When they compared the slow-speed computer simulation (Re 400) with the high-speed real-world experiments, the patterns matched perfectly. The "fingerprints" of the turbulence—how the air moved, how much energy it had, and how it swirled—were identical.
• The Control Group: When they looked at the simulation at an even lower speed (Re 150), the patterns were totally different. It was still in the "calm" phase, not yet chaotic. This proved that the transition to chaos happens somewhere between 150 and 400.

3. The "Fingerprint" of Turbulence

How do you know if a flow is truly turbulent? The paper looks for specific "signs of life" in the data:

• The Energy Spectrum (The Sound of the Wind): In a calm flow, the energy is concentrated in a few specific notes (like a flute playing a single tone). In a turbulent flow, it sounds like white noise or static, with energy spread across a huge range of frequencies. The researchers found that at Re 400, the "sound" of the wind behind the plate was already full of this chaotic static, just like in the high-speed experiments.
• The "Intermittency" (The Occasional Scream): In a truly turbulent flow, the air doesn't just swirl gently; it has sudden, intense bursts of speed and rotation. The researchers found these "screams" in the data at Re 400, but they were absent at Re 150.

4. Why Is This Different?

The paper suggests the reason for this sudden jump is the shape of the object.

• Round/Square Objects: When wind hits a round or square object, the back part of the object acts like a shield, stabilizing the air flow behind it. It takes a lot of energy to break that stability.
• The Thin Plate: Because the plate is so thin, there is no "back" to shield the air. The pressure fluctuations (the push and pull of the air) are directly connected to the swirling vortices right from the start. It's like trying to balance a pencil on its tip versus balancing a bowling ball; the pencil (the thin plate) is inherently unstable and tips over into chaos much faster.

The Bottom Line

This paper changes our understanding of how air flows around flat objects. It proves that thin flat plates create fully turbulent wakes at surprisingly low speeds, much lower than round or square objects. The transition to chaos is not a slow, gradual process for these shapes; it is a sudden, fundamental shift that happens very early in the speed range.

The researchers did not discuss how this applies to building bridges, designing cars, or medical devices. They strictly focused on proving that this phenomenon exists and how the physics of the air flow differs from what we previously thought.

Technical Summary

Technical Summary: Fully Turbulent Wakes at Low Reynolds Numbers: the Case of the Thin Flat Plate

Problem Statement
The numerical simulation of the nominally two-dimensional (2D) flow past a thin flat plate normal to a uniform stream has historically proven challenging, with computational fluid dynamics (CFD) studies yielding conflicting results regarding wake dynamics at low Reynolds numbers ($Re < 1000$). While experimental data suggests that integral parameters (such as drag coefficients and Strouhal numbers) depend weakly on $Re$ in the 3D regime, simulations have reported vastly different wake behaviors. A central question remains: does the wake of a thin flat plate transition to a fully turbulent state immediately following the onset of 3D instabilities, similar to canonical cylinder wakes, or does it follow a distinct, multi-stage transition path? The authors investigate whether the wake past a thin flat plate is already fully turbulent at a relatively low Reynolds number of $Re = 400$, a regime where canonical circular and square cylinder wakes typically exhibit strong $Re$-dependence and non-turbulent fluctuation characteristics.

Methodology
The study employs a multi-faceted approach combining Direct Numerical Simulations (DNS) and experimental validation:

1. Numerical Simulations: Two independent DNS codes were utilized to ensure robustness:

   • Spectral/hp Element Method (SEM): Implemented in Nektar++, used for the primary high-accuracy DNS at $Re = 400$ (DNS400) and $Re = 150$ (DNS150).
   • Finite Volume Method (FVM): Implemented in OpenFOAM, used for mesh convergence studies and cross-validation (DNS400V).
   • The computational domain was truncated and subjected to no-slip conditions on the plate, free-stream inlet conditions, and specific outlet conditions proposed by Dong et al. Periodicity was enforced in the spanwise direction.
   • Mesh convergence was rigorously tested, ensuring the resolution of the Kolmogorov length scale ($\eta$) with $\Delta x/\eta$ values comparable to or better than existing literature for similar geometries.

2. Experimental Data: Two experimental datasets were acquired using time-resolved stereoscopic Particle Image Velocimetry (PIV) in a wind tunnel at the University of Calgary:

   • EX12K: $Re = 12,500$
   • EX20K: $Re = 19,700$
     These datasets represent fully turbulent regimes and serve as the benchmark for the low-$Re$ simulations.

3. Analysis: The study compares global integral quantities, mean velocity fields, Reynolds stresses, turbulent kinetic energy (TKE) budgets, energy spectra, and statistical properties of velocity gradients (specifically non-Gaussianity) across the different $Re$ cases.

Key Contributions and Results

• Early Onset of Turbulence: The primary finding is that the wake of a thin flat plate becomes fully turbulent at $Re = 400$. This is significantly lower than the transition observed in canonical circular ($Re \approx 10^5$) or square ($Re \approx 6000$) cylinder wakes.
• Quantitative Agreement with High-$Re$ Experiments: The DNS results at $Re = 400$ (DNS400) show remarkable agreement with experimental data at $Re = 12,500$ and $19,700$ (EX12K, EX20K) regarding:
  • Global Quantities: Mean drag coefficient ($C_D$) and Strouhal number ($St$) are slow-varying functions of $Re$ for $Re \ge 400$.
  • Flow Structure: Mean streamlines, velocity profiles, and Reynolds stress distributions ($u'^2, v'^2, w'^2$) are statistically indistinguishable between $Re=400$ and the high-$Re$ experiments within measurement uncertainty.
  • Energy Budgets: The spatial distribution of TKE production and advection terms in the $k$-transport equation are consistent across $Re=400$ and the high-$Re$ cases.
• Turbulent Signatures at Low $Re$: The $Re=400$ case exhibits definitive characteristics of turbulence absent in the $Re=150$ case:
  • Energy Spectra: The fluctuating velocity components display broadband spectra with an inertial range scaling approximately as $f^{-5/3}$ (Kolmogorov's law), indicative of an energy cascade. In contrast, the $Re=150$ spectra show energy transfers between only a few modes.
  • Non-Gaussianity: Probability Density Functions (PDFs) of fluctuating strain ($\Sigma$) and rotation ($\Omega$) rates exhibit heavy tails, signifying intermittency and rare high-intensity events characteristic of turbulence. The $Re=150$ case displays Gaussian distributions.
• Distinct Transition Path: The transition at $Re=150$ involves 3D instabilities but lacks the coherent, fully turbulent structure seen at $Re=400$. The $Re=150$ wake features weaker backflow, slower wake recovery, and non-monotonic evolution of the wake half-width, differing fundamentally from the behavior of canonical cylinders where 3D instabilities (Modes A, B, C) evolve over a wide $Re$ range before full turbulence.

Significance and Claims
The paper argues that the path to transition to turbulence for a thin flat plate is fundamentally different from that of canonical cylinders. While cylinder wakes undergo a prolonged sequence of instabilities (Mode A, B, C) over a wide Reynolds number range before reaching a fully turbulent state, the thin flat plate wake transitions to a fully turbulent state almost immediately after the onset of 3D instabilities (between $Re=150$ and $Re=400$).

The authors suggest that this difference may be attributed to the geometry's lack of a streamwise afterbody. For cylinders, the afterbody stabilizes the flow and decouples pressure fluctuations from the separation point. In contrast, for thin flat plates, pressure fluctuations are directly coupled to the vortex-formation region, potentially driving an earlier and more rapid transition to turbulence. These findings help reconcile discrepancies in previous literature regarding the sensitivity of thin plate wakes to aspect ratios and Reynolds numbers, establishing that the "fully turbulent" regime for this geometry begins at a much lower $Re$ than previously assumed for canonical bluff bodies.