Unified scaling and shape laws for turbulent premixed methane and hydrogen jet flames

This study establishes a unified scaling framework incorporating flame speed and shape factors that successfully describes the turbulent premixed jet flames of both hydrogen and methane across a wide range of operating conditions, demonstrating that despite hydrogen's distinct thermodiffusive effects, both fuels follow consistent turbulent scaling laws when analyzed within this common model.

The Gist

Imagine you are trying to predict how a campfire will behave when a strong wind blows through it. Now, imagine you have two different types of wood: one is a standard, slow-burning oak log (representing Methane, the main component of natural gas), and the other is a bundle of super-fast, highly reactive kindling (representing Hydrogen).

For years, scientists had a great "rulebook" for predicting how the oak log fire would dance in the wind. But when they tried to use that same rulebook for the hydrogen kindling, it didn't quite work. Hydrogen is tricky; it diffuses (spreads out) much faster than heat, creating little cellular patterns and making the flame behave differently.

This paper is like a massive experiment where researchers built a new, universal rulebook that works for both types of fire, even though they behave so differently.

The Big Experiment: The Wind Tunnel Kitchen

The researchers set up a giant, high-tech kitchen burner in a lab. They shot jets of mixed fuel and air into a room, creating turbulent flames. They didn't just do this once; they did it hundreds of times, changing:

• How fast the wind blew (Reynolds number): From a gentle breeze to a hurricane.
• The size of the nozzle (the hole the fire comes out of): Small, medium, and large.
• The fuel: Pure Methane vs. Pure Hydrogen.

To "see" the invisible flames, they used a special camera that takes pictures of the OH chemiluminescence*. Think of this as a night-vision camera that only sees the "soul" of the fire—the glowing parts where the chemical reaction is actually happening. They took 700 snapshots for every single test to get a perfect average picture of the flame's shape.

The Two Magic Numbers

The core discovery of this paper is that they found a way to describe both fires using the same mathematical formula, but with two "magic numbers" (constants) that change depending on the fuel.

Think of the formula as a recipe for a cake. The ingredients (turbulence, speed, size) are the same for both cakes, but the flavor depends on two secret spices:

1. The "Speed Spice" (Factor $\alpha$):

   • This measures how much the wind makes the fire burn faster.
   • Methane: It's a mild spice. The wind makes it burn a little faster, but not wildly.
   • Hydrogen: It's a super-hot, spicy pepper! Because hydrogen is so reactive and diffuses quickly, the wind makes it burn much faster than methane does. The researchers found that Hydrogen needs a "speed factor" about 8 times larger than Methane to fit the same equation.

2. The "Shape Spice" (Factor $\gamma$):

   • This measures the overall shape of the fire. Does it look like a sharp cone? A long cylinder? A blob?
   • As the fire gets longer, it naturally changes shape. The researchers found that both fuels follow the exact same "shape-shifting" rule (a power law), but Hydrogen's shape is just slightly more compact (like a tighter, fatter cylinder) compared to Methane's.

The "Universal Translator"

Before this study, scientists had to use different, complicated math for Hydrogen because it was so different from Methane. They had to guess and tweak the numbers for every new experiment.

This paper says: "Stop guessing. We found the universal translator."

By adjusting just those two "spices" (the speed factor and the shape factor), the same mathematical model can accurately predict:

• How fast the fire will burn.
• How long the flame will be.
• What shape it will take.

Why Does This Matter?

Hydrogen is the future of clean energy. We want to burn it in power plants and cars to stop climate change. But because Hydrogen behaves so differently from the natural gas we use today, it's hard to design engines that are safe and efficient.

If you try to build a Hydrogen engine using the old "Methane rulebook," the engine might explode or fail. This new "Universal Rulebook" gives engineers a reliable way to design these new engines without having to test every single scenario from scratch. It bridges the gap between the old world of natural gas and the new world of hydrogen.

In a Nutshell

The researchers took two very different types of fire (Methane and Hydrogen), threw them into a wind tunnel, and discovered that they actually follow the same underlying dance steps. They just need a slightly different "tempo" (speed factor) and a slightly different "posture" (shape factor) to fit the music. This discovery provides a simple, powerful tool to help us safely harness the power of hydrogen for a cleaner future.

Technical Summary

1. Problem Statement

Turbulent premixed flame scaling laws are traditionally derived for fuels with a Lewis number ($Le$) close to unity, such as methane ($CH_4$). However, hydrogen ($H_2$) has a low Lewis number ($Le < 1$), which induces significant thermodiffusive instabilities (TDI) and preferential diffusion. These effects alter local burning rates and flame structures, creating uncertainty regarding whether classical scaling laws (developed for $Le \approx 1$) remain valid for hydrogen.

While extensive studies exist on lean premixed $H_2$ flames, most focus on specific configurations or fuel compositions. There is a lack of systematic comparison between $H_2$ and $CH_4$ across a wide range of turbulence intensities to determine if:

1. TDI-induced modifications follow a systematic trend.
2. Classical scaling laws break down fundamentally, or if they can be extended with fuel-specific corrections.

2. Methodology

The study employs a systematic experimental approach comparing turbulent premixed $CH_4$/air and $H_2$/air jet flames.

• Experimental Setup:

  • Burner: A round jet burner with a central fully premixed jet, stabilized by a concentric laminar pilot flame and an outer air coflow.
  • Fuels: $CH_4$ at stoichiometric conditions ($\phi = 1.0$) and $H_2$ at lean conditions ($\phi = 0.4$).
  • Parameters: Experiments covered Reynolds numbers ($Re_j$) from 5,000 to 60,000 and effective Karlovitz numbers ($Ka^*_{eff}$) from 3 to 368. Nozzle diameters ($d_j$) of 6, 9, and 12 mm were used.
  • Diagnostics: Spatially resolved OH chemiluminescence imaging* (700 frames per case) was used to capture flame structure. A multi-Otsu thresholding method segmented the flame to determine flame length ($h_f$) and mean flame surface area ($A_0$).

• Theoretical Framework:
  The authors propose a unified scaling framework based on Damköhler's small-scale limit (SSL) for turbulent burning velocity, introducing two fuel-dependent, constant prefactors:

  1. Flame Speed Factor ($\alpha$): Quantifies the enhancement of local burning rates due to fuel-specific effects (differential diffusion).
  2. Shape Factor ($\gamma$): Describes the scaling of the mean flame geometry (transition from conical to columnar).

  The model relates normalized turbulent burning velocity ($s_T/u_j$) and flame length ($h_f/d_j$) to these factors, assuming the flame operates in the small-scale regime.

3. Key Contributions

• Unified Scaling Framework: The study demonstrates that despite the distinct physical mechanisms of $H_2$ (thermodiffusive instability) and $CH_4$ (unity Lewis number), both fuels obey the same underlying dynamic scaling laws when corrected by specific constant prefactors ($\alpha$ and $\gamma$).
• Systematic Comparison: It provides the first comprehensive dataset comparing $H_2$ and $CH_4$ across a wide range of $Re$ and $Ka$, isolating the effects of fuel properties from turbulence intensity.
• Generalized Correlations: The paper derives generalized correlations for turbulent burning velocity and flame length that account for differential diffusion without requiring case-specific tuning, offering a pathway for validating RANS/LES models for low-carbon fuels.

4. Key Results

A. Turbulent Burning Velocity

• Scaling Behavior: Both fuels follow the small-scale limit scaling law ($s_T \propto Re^{1/2}$).
• Flame Speed Factor ($\alpha$): A significant difference was found in the magnitude of $\alpha$:
  • Hydrogen: $\alpha_{H2} \approx 0.28$
  • Methane: $\alpha_{CH4} \approx 0.036$
  • Interpretation: The order-of-magnitude difference in $\alpha$ reflects the enhanced burning rates in $H_2$ driven by thermodiffusive effects. However, once this factor is applied, the scaling is consistent across all tested conditions.
• Model Accuracy: The model reproduces experimental data well. Minor discrepancies (approx. 20% overestimation of $s_T$) occur for $H_2$ at very high Karlovitz numbers ($Ka^*_{eff} > 160$), suggesting a potential suppression of thermodiffusive effects in extreme turbulence that the current model does not fully capture.

B. Flame Shape and Geometry

• Geometric Evolution: As $Re_j$ increases, flames transition from conical to columnar (cylindrical) structures.
• Shape Factor ($\gamma$): The mean flame geometry follows a power-law relationship with a fuel-dependent constant:
  • Hydrogen: $\gamma_{H2} = 1.15$
  • Methane: $\gamma_{CH4} = 0.98$
  • Interpretation: $H_2$ flames are slightly more compact than $CH_4$ flames, but both follow the same effective power-law slope (-0.2) regarding the relationship between flame length and shape factor.
• Flame Length Model: By combining the burning velocity model and the shape factor, a unified model for flame length ($h_f$) was derived. It shows good agreement for shorter flames but deviates for very long $CH_4$ flames (where the transition to a cylindrical shape is faster than predicted) and high-$Ka$ $H_2$ flames.

5. Significance and Implications

• Validation of Scaling Laws: The study confirms that classical turbulent combustion scaling laws are not fundamentally broken by low Lewis number effects; rather, they require adjustment via physically based, fuel-specific constants.
• Model Development: The derived correlations ($\alpha$ and $\gamma$) provide a benchmark for validating and calibrating turbulence-chemistry interaction models (RANS/LES) for hydrogen and hydrogen-blended fuels without arbitrary tuning.
• Low-Carbon Transition: By establishing the limits of regime validity and providing a unified framework, this work supports the development of combustion technologies for hydrogen, a key energy carrier for decarbonization.
• Future Directions: The authors plan to extend this framework to $CH_2/H_2$ blends and investigate the robustness of these parameters across different equivalence ratios.

In summary, the paper successfully bridges the gap between unity-Lewis-number and low-Lewis-number combustion physics, proving that a unified model can describe turbulent premixed flames across vastly different fuel types through the introduction of two invariant, fuel-dependent prefactors.