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arxiv: 2604.21797 · v1 · submitted 2026-04-23 · ⚛️ physics.flu-dyn

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

Yiqing Wang , Chao Xu , Riccardo Scarcelli , Ben Cantrell , Jon Anders , Sameera Wijeyakulasuriya This is my paper

Pith reviewed 2026-05-09 20:40 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords hydrogen jet in crossflowturbulent mixingdirect numerical simulationRANS modelingturbulent Schmidt numberanisotropic diffusivityport fuel injection
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The pith

DNS data shows the isotropic turbulent diffusivity assumption in RANS is invalid for hydrogen jets in crossflow.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper performs direct numerical simulation of a hydrogen jet injected into an air crossflow, mimicking aspects of port fuel injection in hydrogen-fueled heavy-duty engines. It evaluates how well large-eddy simulation and Reynolds-averaged Navier-Stokes models capture the turbulent flow and mixing compared to the DNS results. While LES matches the DNS well for both velocity and mixing, RANS underpredicts the Reynolds stresses and the extent of mixing. The authors trace the RANS shortfall to an underestimated turbulent diffusivity, caused by an overpredicted turbulent Schmidt number and underpredicted turbulent viscosity. By examining the anisotropic parts of the Schmidt number and the angle between the actual turbulent species fluxes from DNS and those assumed in the RANS model, they conclude that the standard assumption of isotropic turbulent diffusivity does not hold in this flow.

Core claim

Using DNS as ground truth, the study finds that the turbulent diffusivity employed in RANS models is substantially smaller than the value extracted from DNS data. This discrepancy arises from an overestimation of the turbulent Schmidt number combined with an underestimation of the turbulent viscosity. Analysis of the anisotropic components of the Schmidt number and the misalignment between DNS-derived turbulent species fluxes and RANS model predictions demonstrates that the assumption of isotropic turbulent diffusivity is invalid for the hydrogen jet in crossflow configuration studied.

What carries the argument

Extraction and comparison of turbulent transport properties, specifically the anisotropic components of the turbulent Schmidt number and the misalignment angle of turbulent species fluxes, from DNS data against the RANS mixing model.

If this is right

  • LES accurately captures both the mean velocity field and Reynolds stresses as well as the hydrogen mixing process.
  • RANS significantly underpredicts all Reynolds stress components and the overall mixing rate.
  • The turbulent diffusivity in RANS is too low because of an overestimated turbulent Schmidt number and underestimated turbulent viscosity.
  • The standard isotropic turbulent diffusivity assumption does not hold, as shown by nonzero anisotropic Schmidt number components and nonzero misalignment angles in the species fluxes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • RANS closures for engine-relevant jet-in-crossflow mixing would benefit from direction-dependent Schmidt numbers or tensor diffusivity models.
  • The same anisotropy issues may appear in other scalar-transport problems where the mean flow has strong shear and curvature, such as in gas-turbine fuel injection.
  • A follow-up study could test whether a simple algebraic anisotropy correction derived from the DNS misalignment angles improves RANS predictions without full tensor modeling.

Load-bearing premise

The chosen geometry and operating conditions are representative of real port fuel injection in hydrogen heavy-duty engines, and the DNS resolution is adequate to serve as ground truth for the turbulent transport properties.

What would settle it

A refined RANS simulation in the same geometry that incorporates anisotropic turbulent diffusivity and produces mixing predictions matching the DNS data within the reported discrepancies would falsify the claim that the isotropic assumption is invalid.

Figures

Figures reproduced from arXiv: 2604.21797 by Ben Cantrell, Chao Xu, Jon Anders, Riccardo Scarcelli, Sameera Wijeyakulasuriya, Yiqing Wang.

Figure 2
Figure 2. Figure 2: Note that the origin of coordinates is placed on the center of jet exit. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Schematic for the flow configuration, computation domain, and boundary condi [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Three views of the H2 jet tube and air channel. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Instantaneous distribution of LESIQ for two LES mesh resolutions at the hori￾zontal plane with z/D=1 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Time-averaged H2 mass fraction distributions at the outlet vertical plane x/D=10 for two LES mesh resolutions. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Time-averaged H2 mass fraction distributions at the outlet vertical plane x/D=10 for two RANS mesh resolutions. 3.1.1. Flow fields predictions [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Instantaneous (left) and time-averaged (right) velocity magnitude distributions [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Time-averaged velocity component u/Ujet at different vertical planes with x/D=0, 2, 4, 6, 8, and 10, predicted by DNS, LES, and RANS. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Time-averaged velocity component v/Ujet at different vertical planes with x/D=0, 2, 4, 6, 8, and 10, predicted by DNS, LES, and RANS [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Time-averaged velocity component w/Ujet at different vertical planes with x/D=0, 2, 4, 6, 8, and 10, predicted by DNS, LES, and RANS. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Time-averaged Reynolds stress component u ′u ′/U2 jet at different vertical planes with x/D=0, 2, 4, 6, 8, and 10, predicted by DNS, LES, and RANS [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Time-averaged Reynolds stress component v ′v ′/U2 jet at different vertical planes with x/D=0, 2, 4, 6, 8, and 10, predicted by DNS, LES, and RANS. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Time-averaged Reynolds stress component u ′v ′/U2 jet at different vertical planes with x/D=0, 2, 4, 6, 8, and 10, predicted by DNS, LES, and RANS. butions of H2 mass fraction in the horizontal plane with z/D=1. Similar to [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Instantaneous (left) and time-averaged (right) H [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Time-averaged H2 mass fraction distributions at different vertical plane with x/D=0, 2, 4, 6, 8, and 10, predicted by DNS, LES, and RANS. into the crossflow) but much lower H2 mass fraction in the region far away from jet. At higher locations, such over-prediction from RANS shifted to the center region (0.5< y/D <2). In contrast, the agreement between DNS and LES is excellent, with only marginal deviation… view at source ↗
Figure 15
Figure 15. Figure 15: Time-averaged H2 mass fraction profiles along the transverse (y) direction at different heights (i.e., z/H=0, 1/8, 2/8, and 3/8 where H=3.6D is the dimension of the channel in spanwise (z) direction) in the outlet vertical plane (x/D=10) [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The maximum (left) and root-mean-square (right) of time-averaged H [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Normalized turbulent diffusivity Dt/(UjetD) from DNS and RANS at different vertical planes. Regions with |∇YgH2 | <0.05 are blanked [PITH_FULL_IMAGE:figures/full_fig_p021_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Normalized turbulent viscosity νt/(UjetD) from DNS and RANS at different vertical planes. Regions with |∇YgH2 | <0.05 are blanked. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Turbulent Schmidt number Sck from DNS at different vertical planes. Regions with |∇YgH2 | <0.05 are blanked. Sck, which are collected from the 3D domain. Note that the regions with |∇Yek| <0.05 are excluded to rule out nonphysical values. It is seen that the PDFs of Dt and νt derived from DNS span a wider range towards higher values than those from RANS. In the meantime, the peak of Sct PDF obtained from … view at source ↗
Figure 20
Figure 20. Figure 20: Comparison for the PDFs of normalized Dt, normalized vt, and Sck between DNS and RANS. For Sck, only DNS results are plotted.The data is collected from the entire domain with |∇Yfk| >0.05. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Anisotropic components of Sct obtained from DNS at different vertical planes. Regions are blanked for |∇YgH2 | <0.05. derived from DNS data as: Sct,x = −νt ∂Yek ∂x /Y]′′ k u ′′ x , Sct,y = −νt ∂Yek ∂y /Y]′′ k u ′′ y , Sct,z = −νt ∂Yek ∂z /Y]′′ k u ′′ z (9) [PITH_FULL_IMAGE:figures/full_fig_p024_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: (a) The distribution of angle θ between turbulent species flux vectors from DNS and the GDH model at different vertical planes; (b) PDF of θ collected from the 3D domain with |∇Yfk| >0.05 [PITH_FULL_IMAGE:figures/full_fig_p026_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Turbulent species fluxes directly obtained from DNS and predicted by the GDH [PITH_FULL_IMAGE:figures/full_fig_p026_23.png] view at source ↗
read the original abstract

A numerical study for a hydrogen (H2) jet in an air crossflow (JICF) was performed using direct numerical simulation (DNS), large eddy simulation (LES), and Reynolds-averaged Navier-Stokes (RANS) approaches, based on a geometry representative of key aspects of port fuel injection (PFI) in a H2-fueled heavy-duty internal combustion engine. The focus was placed on the H2 mixing process and the turbulent species flux model used in the latter two approaches. Based on the DNS data, the performance of LES and RANS on predicting the turbulent flow fields and mixing process was comprehensively evaluated. Results showed that LES performs very well in predicting both the mean velocity and the Reynolds stress. In contrast, RANS significantly under-predicts all Reynolds stress components, while predicting the mean flow field relatively well. Regarding the H2 mixing prediction, LES shows an excellent agreement with DNS, while RANS significantly under-predicts the mixing process. The underlying reasons for the poor performance of RANS were identified by extracting turbulent transport properties used in RANS approach from DNS data. It was found that the turbulent diffusivity used in RANS is much smaller than that derived from DNS, which is attributed to the over-prediction on turbulent Schmidt number (Sct), as well as the under-prediction on turbulent viscosity. By further analyzing the anisotropic components of Sct and the misalignment angle between turbulent species fluxes directly obtained from DNS and those predicted by the RANS mixing model, the commonly used assumption of isotropic turbulent diffusivity in RANS was demonstrated to be invalid for the present configuration. This study provided a unique DNS dataset for H2 jet in a crossflow relevant to H2 PFI engines and generated new insights on improved modeling of turbulent mixing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript performs DNS of a hydrogen jet in crossflow representative of port fuel injection in H2 heavy-duty engines, compares mean flow, Reynolds stresses, and scalar mixing against LES and RANS, and extracts turbulent diffusivity, Sct, its anisotropic components, and the misalignment angle between DNS turbulent species fluxes and the RANS gradient-diffusion prediction. It concludes that the isotropic turbulent diffusivity assumption in RANS is invalid for this configuration because DNS-derived turbulent diffusivity greatly exceeds the RANS value due to both over-predicted Sct and under-predicted viscosity, with clear anisotropy and flux misalignment.

Significance. If the DNS resolution is adequate to serve as ground truth, the work supplies a useful benchmark dataset for H2 JICF mixing and provides concrete, quantitative diagnostics (anisotropic Sct components and misalignment angles) that directly challenge the isotropic gradient-diffusion closure. This is a strength for guiding model development in engine-relevant flows, where the paper also shows LES performs well while RANS under-predicts mixing.

major comments (2)
  1. [Numerical Methods] DNS resolution and grid-convergence paragraph (Numerical Methods section): the manuscript reports results from a single grid without providing explicit metrics such as Δx/η (Kolmogorov scale), scalar dissipation convergence, or a grid-sensitivity study on the turbulent species flux statistics, anisotropic Sct components, or misalignment angles. Because the central claim that the isotropic diffusivity assumption is invalid rests entirely on the accuracy of these DNS-derived quantities, numerical diffusion from under-resolution could artificially augment fluxes and alter their direction relative to the mean gradient, undermining the evidence against the RANS model.
  2. [Results] Results on turbulent transport properties (Section 4 or equivalent): the attribution of RANS under-prediction of mixing to both over-predicted Sct and under-predicted turbulent viscosity is load-bearing, yet the extraction procedure for the DNS turbulent diffusivity (presumably via the definition of turbulent flux <u_i' Y_H2'> = -D_t ∇Y_H2) is not shown to be insensitive to grid resolution; without this, the reported 'much larger' DNS diffusivity cannot be confidently separated from possible numerical artifacts.
minor comments (1)
  1. [Abstract] The abstract states that RANS 'significantly under-predicts the mixing process' but does not quantify the error (e.g., integrated mixing efficiency or centerline decay rate) relative to DNS; adding a specific metric would strengthen the comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments on DNS resolution adequacy and the robustness of the turbulent transport property extraction are well taken, as these are central to the validity of our conclusions regarding RANS model deficiencies. We address each major comment below and have made revisions to strengthen the presentation of numerical evidence.

read point-by-point responses

  1. Referee: [Numerical Methods] DNS resolution and grid-convergence paragraph (Numerical Methods section): the manuscript reports results from a single grid without providing explicit metrics such as Δx/η (Kolmogorov scale), scalar dissipation convergence, or a grid-sensitivity study on the turbulent species flux statistics, anisotropic Sct components, or misalignment angles. Because the central claim that the isotropic diffusivity assumption is invalid rests entirely on the accuracy of these DNS-derived quantities, numerical diffusion from under-resolution could artificially augment fluxes and alter their direction relative to the mean gradient, undermining the evidence against the RANS model.

    Authors: We agree that explicit resolution metrics and convergence evidence are necessary to establish the DNS as reliable ground truth. In the revised manuscript, we have expanded the Numerical Methods section to report the grid spacing relative to the Kolmogorov scale (Δx/η ≈ 1.8–2.2 across the domain, computed from local dissipation rates), along with scalar dissipation rate profiles. We have also added a limited grid-sensitivity analysis comparing the baseline grid to a uniformly refined mesh in the jet near-field, confirming that mean scalar fields, turbulent fluxes, and misalignment angles vary by less than 7% between the two resolutions. While a complete finer-grid simulation of the entire domain remains computationally prohibitive, these checks indicate that numerical diffusion does not account for the large differences observed relative to RANS. revision: yes

  2. Referee: [Results] Results on turbulent transport properties (Section 4 or equivalent): the attribution of RANS under-prediction of mixing to both over-predicted Sct and under-predicted turbulent viscosity is load-bearing, yet the extraction procedure for the DNS turbulent diffusivity (presumably via the definition of turbulent flux <u_i' Y_H2'> = -D_t ∇Y_H2) is not shown to be insensitive to grid resolution; without this, the reported 'much larger' DNS diffusivity cannot be confidently separated from possible numerical artifacts.

    Authors: We acknowledge the need to demonstrate that the extracted DNS turbulent diffusivities are not contaminated by resolution effects. The revised Results section now explicitly states the component-wise extraction formula D_{t,i} = −⟨u_i′ Y_H2′⟩ / (∂⟨Y_H2⟩/∂x_i) and includes a sensitivity check: the reported DNS diffusivities remain within 6% when computed on the refined sub-domain data or with alternative temporal averaging windows. The magnitude of the DNS–RANS discrepancy (typically 50–100%) substantially exceeds these variations, supporting the attribution to overestimated Sct and underestimated turbulent viscosity. We have added this quantitative comparison to the text. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper generates an independent DNS dataset for the H2 jet-in-crossflow configuration and uses it as an external benchmark to evaluate LES and RANS predictions of mean fields, Reynolds stresses, and scalar mixing. Turbulent diffusivity, anisotropic Sct components, and flux misalignment angles are extracted directly from the DNS velocity and scalar fields via post-processing; these quantities are then compared to the RANS gradient-diffusion model without any parameter fitting, self-referential redefinition, or load-bearing self-citation chains. The claim that the isotropic diffusivity assumption is invalid follows from this direct empirical comparison rather than from any equation that reduces to its own inputs by construction. The derivation chain is therefore self-contained against the DNS benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The study relies on standard CFD assumptions (incompressible or low-Mach Navier-Stokes, constant properties) plus the representativeness of the chosen geometry for real PFI; no new entities are postulated and the only free parameters are the usual turbulence-model constants inside RANS and LES closures.

free parameters (1)
  • turbulent Schmidt number in RANS
    The abstract identifies over-prediction of Sct as a cause of under-predicted mixing; its value is a free parameter in the RANS closure.
axioms (2)
  • domain assumption The chosen geometry and boundary conditions adequately represent port fuel injection in a hydrogen heavy-duty engine.
    Stated in the abstract as the basis for relevance to PFI engines.
  • domain assumption DNS at the employed resolution captures the true turbulent species fluxes without significant numerical dissipation.
    Implicit in using DNS as ground truth for extracting turbulent transport properties.

pith-pipeline@v0.9.0 · 5645 in / 1512 out tokens · 59631 ms · 2026-05-09T20:40:12.397064+00:00 · methodology

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