Turbulent mixing of a hydrogen jet in crossflow: direct numerical simulation and model assessment

This study utilizes direct numerical simulation (DNS) to evaluate large eddy simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) models for hydrogen jet mixing in crossflow, revealing that while LES accurately predicts both flow and mixing, RANS significantly under-predicts mixing due to an overestimated turbulent Schmidt number and the invalidity of assuming isotropic turbulent diffusivity.

The Gist

Imagine you are trying to mix a cup of tea. You drop a sugar cube in, and you stir it with a spoon. If you stir gently, the sugar stays in a clump at the bottom. If you stir violently, the sugar dissolves quickly and spreads evenly throughout the cup.

This paper is about a very high-tech version of that mixing problem, but instead of sugar and tea, scientists are mixing Hydrogen gas (the fuel) with Air (the oxygen) inside an engine. Specifically, they are looking at a "Jet in Crossflow" (JICF) scenario: imagine a powerful hose spraying hydrogen into a strong wind tunnel of air. The goal is to mix them perfectly so the engine runs clean and efficiently.

Here is the breakdown of what the researchers did, using simple analogies:

1. The Three "Mixing Chefs"

To figure out how to mix this hydrogen best, the team used three different "cooking methods" (computer simulations) to predict what happens:

• DNS (Direct Numerical Simulation): This is the Master Chef. It simulates every single tiny swirl and eddy of the gas. It's like watching the mixing process in ultra-high-definition slow motion, frame by frame, down to the smallest molecule. It is incredibly accurate but takes a supercomputer years to cook a single meal.
• LES (Large Eddy Simulation): This is the Sous Chef. It ignores the tiniest, most annoying swirls and focuses on the big, important movements. It's a great balance of speed and accuracy.
• RANS (Reynolds-Averaged Navier-Stokes): This is the Fast-Food Worker. It uses a simple formula to guess the average result. It doesn't see the swirls; it just guesses the "average" mix. It is very fast and cheap, which is why car companies use it for designing engines.

2. The Big Discovery: The Fast-Food Worker is Wrong

The researchers ran all three simulations on a hydrogen jet and compared the results.

• The Verdict: The Master Chef (DNS) and the Sous Chef (LES) agreed perfectly. They both showed that the hydrogen mixes quickly and spreads out beautifully.
• The Problem: The Fast-Food Worker (RANS) failed miserably. It predicted that the hydrogen would stay in a tight, unmixed clump, like a sugar cube that never dissolved. It thought the mixing was much slower than it actually is.

3. Why Did the Fast-Food Worker Fail?

The team investigated why the RANS model was so bad at predicting the mix. They found two main reasons, which they explain using the concept of "The Mixing Scaler":

Reason A: The Wrong "Mixing Speed" (Turbulent Viscosity)
Imagine the turbulence (the swirling chaos) is the engine that stirs the pot. The RANS model underestimated how strong the engine was. It thought the stirring was weak, so the hydrogen didn't move much.

Reason B: The Wrong "Recipe Ratio" (Schmidt Number)
This is the trickiest part. In fluid dynamics, there is a number called the Schmidt Number that tells you how fast a gas mixes compared to how fast it flows.

• The RANS model used a standard, "textbook" recipe ratio (0.78).
• The real data (from the Master Chef) showed that for Hydrogen, the ratio is actually much lower (around 0.4).
• The Analogy: It's like the RANS model was trying to mix honey (thick, slow) when it was actually mixing water (thin, fast). Because it used the wrong "recipe," it calculated that the hydrogen would mix very slowly.

4. The "Direction" Problem (Anisotropy)

There was a second, more subtle error. The RANS model assumed that mixing happens the same way in all directions (like a ball expanding evenly in all directions).

• The Reality: In this specific engine setup, the mixing is lopsided. The hydrogen spreads differently sideways than it does up or down.
• The Analogy: Imagine trying to push a crowd of people through a door. The RANS model assumed they would spread out in a perfect circle. In reality, they are being pushed in a specific direction by the wind, creating a long, stretched-out shape. The RANS model's "perfect circle" assumption was completely wrong for this situation.

5. Why Does This Matter?

Hydrogen engines are the future of clean transportation (trucks, trains, etc.). To build them, engineers need computer models that work fast (like RANS) but are also accurate.

Currently, the models used by car companies are underestimating how well hydrogen mixes. This means they might design engines that are too big or inefficient because they think the fuel won't mix well enough.

The Takeaway:
This paper provides a "Gold Standard" dataset (the Master Chef's recipe) that proves the current fast models are flawed. It tells engineers: "Stop using the old textbook numbers. The hydrogen mixes faster and in a more lopsided way than you think. You need to update your formulas to account for this."

By fixing these formulas, we can design better, cleaner, and more efficient hydrogen engines for the future.

Technical Summary

1. Problem Statement

The study addresses the critical challenge of modeling the turbulent mixing of a hydrogen (H₂) jet injected into an air crossflow (JICF), a configuration representative of Port Fuel Injection (PFI) systems in heavy-duty hydrogen-fueled internal combustion engines (ICEs).

• Context: Efficient H₂-air premixing is vital for engine performance and emissions. While Computational Fluid Dynamics (CFD) is the primary tool for design, standard industry models (RANS) often struggle to accurately predict mixing in complex JICF scenarios.
• Specific Challenges:
  • Density Ratio: Unlike previous studies using unity density ratios, H₂ is much lighter than air, creating a low density ratio that significantly alters flow dynamics.
  • Variable Properties: Physical properties (density, viscosity, diffusivity) vary with local H₂ concentration.
  • Model Limitations: The standard Gradient Diffusion Hypothesis (GDH) used in RANS assumes isotropic turbulent diffusivity and a constant turbulent Schmidt number ($Sc_t$). Previous studies suggest these assumptions fail in JICF, often requiring unphysical tuning of $Sc_t$ (e.g., lowering it to 0.2–0.3) to match experiments.
• Objective: To generate high-fidelity Direct Numerical Simulation (DNS) data for a realistic H₂ JICF configuration and use it to rigorously assess the accuracy of Large Eddy Simulation (LES) and Reynolds-Averaged Navier-Stokes (RANS) models, specifically focusing on the turbulent species flux models.

2. Methodology

The authors employed a multi-fidelity approach comparing three simulation methods against a specific geometry mimicking a heavy-duty H₂ engine intake.

• Physical Configuration:
  • A round H₂ jet (diameter $D=19$ mm) inclined at $60^\circ$ injects into a rectangular air channel.
  • Conditions: $T=325$ K, $P=4.3$ bar.
  • Reynolds numbers: $Re_{jet} = 4,600$, $Re_{crossflow} = 22,100$.
• Direct Numerical Simulation (DNS):
  • Solver: Nek5000 (spectral element method).
  • Resolution: ~93 million grid points (7th-order polynomials) with $y^+ < 1$ at walls.
  • Role: Served as the "ground truth" benchmark.
• Large Eddy Simulation (LES):
  • Solver: CONVERGE CFD (finite-volume).
  • Model: Dynamic Structure model.
  • Resolution: ~8.9 million cells (Level 4 mesh, min size 0.25 mm, $y^+ \approx 5$).
• Reynolds-Averaged Navier-Stokes (RANS):
  • Solver: CONVERGE CFD.
  • Model: RNG $k-\epsilon$ turbulence model.
  • Mixing Model: Standard Gradient Diffusion Hypothesis (GDH) with a default turbulent Schmidt number ($Sc_t = 0.78$).
  • Resolution: ~1.2 million cells (Level 3 mesh, min size 0.5 mm, $y^+ \approx 10$).
• Analysis Strategy:
  • Statistical data (mean velocity, Reynolds stresses, species mass fractions) were collected after flow stabilization.
  • Turbulent transport properties ($D_t$, $\nu_t$, $Sct$) were extracted directly from DNS data using a "least-squares" optimization approach to compare against RANS assumptions.

3. Key Contributions

1. Unique DNS Dataset: Provided the first high-resolution, fully-resolved DNS dataset specifically for a hydrogen jet in crossflow with realistic density ratios and variable transport properties, filling a gap in the literature where most prior studies used unity density ratios.
2. Comprehensive Model Assessment: Systematically evaluated LES and RANS against DNS for both flow dynamics and scalar mixing, moving beyond simple visual comparisons to quantitative analysis of turbulent fluxes.
3. Root Cause Analysis of RANS Failure: Identified specific physical and modeling reasons for RANS under-prediction of mixing, moving beyond the general observation that "RANS fails" to pinpoint why.
4. Validation of Anisotropy: Demonstrated that the assumption of isotropic turbulent diffusivity in standard RANS models is invalid for H₂ JICF configurations.

4. Key Results

A. Flow Field and Reynolds Stresses

• LES: Showed excellent agreement with DNS for both mean velocity fields and Reynolds stress components ($u'u'$, $v'v'$, etc.). It successfully captured large-scale turbulent structures.
• RANS:
  • Predicted mean velocity fields relatively well (qualitatively).
  • Significantly under-predicted all Reynolds stress components. This indicates a failure to capture the intensity of turbulence fluctuations.

B. Hydrogen Mixing Prediction

• LES: Achieved excellent agreement with DNS regarding H₂ mass fraction distribution, capturing the "kidney-shaped" vortex structure and downstream diffusion accurately.
• RANS:
  • Significantly under-predicted mixing.
  • Predicted a distinct separation between the primary jet and secondary branches, creating a low-H₂ region on the leeward side that did not exist in DNS/LES.
  • Showed higher peak H₂ concentrations and smaller mixing clouds at the outlet compared to DNS.

C. Analysis of Turbulent Transport Properties (The "Why")

By extracting properties from DNS, the authors identified two main causes for RANS failure:

1. Under-prediction of Turbulent Viscosity ($\nu_t$): RANS significantly underestimated $\nu_t$ compared to DNS. Since turbulent diffusivity ($D_t$) is often linked to $\nu_t$, this directly reduced the predicted mixing rate.
2. Over-prediction of Turbulent Schmidt Number ($Sct$):
   • The default RANS $Sct$ (0.78) was much higher than the effective $Sct$ derived from DNS (which peaked around 0.4–0.5).
   • Since $D_t = \nu_t / Sct$, a high $Sct$ combined with a low $\nu_t$ resulted in a turbulent diffusivity ($D_t$) in RANS that was far too small compared to DNS.

D. Anisotropy of Turbulent Diffusivity

• Isotropy Assumption Invalid: The study analyzed the components of $Sct$($S_{ct,x}, S_{ct,y}, S_{ct,z}$). Results showed $S_{ct,x}$ was near zero, while $S_{ct,y}$ varied dramatically (often $>1$).
• Misalignment Angle: The angle ($\theta$) between the actual species flux (from DNS) and the flux predicted by the GDH model was large (peaking around $37^\circ$, with regions up to $90^\circ$). This confirms that turbulent species flux is not aligned with the mean concentration gradient, invalidating the standard GDH assumption for this flow.
• Correction Limits: Even adjusting $Sct$ to a "best-fit" value (0.44) only partially improved RANS predictions, confirming that a scalar correction is insufficient to fix the fundamental anisotropy issue.

5. Significance and Implications

• Model Development: The study provides strong evidence that improving RANS predictions for H₂ engines requires more than just tuning the Schmidt number. Future models must account for:
  • More accurate prediction of turbulent viscosity ($\nu_t$).
  • Anisotropic turbulent diffusivity (e.g., using Generalized Gradient Diffusion Hypothesis - GGDH, or Higher Order models).
• Engineering Practice: The findings explain why industry-standard RANS models often fail to predict H₂ mixing accurately, leading to suboptimal injector designs. The provided DNS dataset serves as a critical benchmark for developing next-generation turbulence and mixing models.
• Hydrogen Economy: As the industry transitions to hydrogen-fueled heavy-duty engines, this research offers the necessary physical insights to design efficient, low-emission combustion systems by improving the fidelity of the CFD tools used in their development.