Direct numerical simulation of high-pressure mixing in turbulent jets

Josette Bellan; Nek Sharan

Direct numerical simulation of high-pressure mixing in turbulent jets

T0 review · 4 major / 0 minor · reviewed 2026-05-24 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read DNS of nitrogen jets shows supercritical conditions produce shorter potential cores, faster growth, and altered velocity decay compared to subcritical perfect-gas cases, yielding different mixing.

desk verdict DNS comparison of subcritical vs supercritical N2 jets at Re=5000 finds differences in core length and mixing stats by swapping EOS and properties, but the abstract gives no resolution checks or validation. read the letter →

arxiv 1907.11800 v1 pith:6DRLMI6F submitted 2019-07-26 physics.flu-dyn physics.comp-ph

Nek Sharan , Josette Bellan This is my paper

classification physics.flu-dynphysics.comp-ph

keywords directnumericalsimulationturbulentjetssupercriticalmixinghigh-pressureinjectionpotentialcorelengthjetgrowthratevelocitydecaymixed-fluiddistribution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper performs direct numerical simulations of isothermal round jets of nitrogen into nitrogen at Reynolds number 5000 and Mach number 0.6. It applies the same conservation equations and numerical methods to both subcritical perfect-gas and supercritical regimes, differing only in the equation of state and transport properties. Averaged flow quantities including potential core length, spatial growth rate, and velocity decay profiles differ between the regimes. These differences produce distinct mixed-fluid distributions, which matter for high-pressure fuel-oxidizer injection in engines.

What carries the argument

Direct numerical simulation of isothermal round jets at fixed Re_D=5000 and Ma=0.6, tracking a passive scalar for mixing while switching only the equation of state and transport properties between perfect-gas and supercritical regimes.

What would settle it

Experimental measurements at Re_D=5000 and Ma=0.6 showing identical potential core lengths, spatial growth rates, and velocity decay profiles for subcritical and supercritical nitrogen jets would falsify the reported differences.

Watch

Extended reading notes

Core claim

Through direct numerical simulation of turbulent jets, the study establishes that injection at supercritical pressures produces significantly different dynamics than at subcritical conditions due to the absence of distinct liquid and gas phases, leading to variations in potential core length, jet spatial growth rate, velocity decay profiles, and ultimately different mixed-fluid distributions.

Load-bearing premise

The same conservation equations and numerical methods apply to both subcritical perfect-gas and supercritical regimes simply by changing the equation of state and transport properties.

Editorial extensions

If this is right

Potential core length is shorter under supercritical conditions than under subcritical conditions.
Jet spatial growth rate is higher under supercritical conditions than under subcritical conditions.
Velocity decay profiles differ between the two injection conditions.
Mixed-fluid distributions therefore differ between supercritical and subcritical jets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Combustion models for high-pressure engines may require separate subcritical and supercritical mixing closures rather than a single perfect-gas formulation.
The passive-scalar results suggest that fuel-oxidizer interface area and scalar variance statistics will also differ, affecting subsequent ignition predictions.
Repeating the DNS at reacting conditions would test whether the observed mixing differences persist once heat release is added.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The paper runs standard compressible DNS on isothermal round jets of N2 into N2 at Re_D=5000 and Ma=0.6, once with a perfect-gas EOS and once with a supercritical treatment, then compares passive-scalar statistics. The main result is that potential core length, spatial growth rate, velocity decay, and mixed-fluid distributions come out different between the two cases. That supplies new quantitative data for the supercritical regime at these conditions, which the abstract indicates was not already in the cited literature. The setup keeps temperature fixed so the comparison isolates the thermodynamic state effect on mixing, and the passive scalar is a straightforward way to track that. Those choices are reasonable for the stated goal of examining variations in statistics relevant to high-pressure combustors. The soft spots are straightforward. The abstract reports no grid-resolution study, no error bars on the profiles, and no direct quantitative match to experiment, so the size of the reported differences is hard to assess for robustness. The modeling choice also sits on the assumption that the usual Navier-Stokes equations plus an EOS and property swap are enough; nothing in the provided text addresses possible extra real-fluid corrections near the critical point. If that assumption needs extra terms, the differences cannot be pinned cleanly on the thermodynamic regime alone. This work is aimed at combustion modelers who need benchmark mixing statistics for supercritical injection. A reader outside that subfield will see a competent parameter study but not a broad advance in fluid mechanics. It deserves peer review because it delivers concrete DNS outputs in a regime of practical interest, even though the numerics and modeling justification will need scrutiny in revision.

Referee Report

4 major / 0 minor

Summary. The manuscript reports direct numerical simulations of isothermal round jets of nitrogen into nitrogen at Re_D=5000 and Ma=0.6. It compares subcritical (perfect-gas) and supercritical conditions and claims that the thermodynamic regime produces measurable differences in potential core length, spatial growth rate, velocity decay profiles, and mixed-fluid distributions as diagnosed from a passive scalar.

Significance. If the reported differences are shown to be numerically converged and free of modeling artifacts, the work would supply useful DNS statistics on real-fluid jet mixing relevant to high-pressure combustion. The isothermal setup isolates thermodynamic effects, which is a strength for attribution.

major comments (4)

[Numerical methods / DNS setup] No grid-resolution study or convergence demonstration is described for the key statistics (potential core length, growth rate, decay profiles) at Re_D=5000. This is load-bearing because turbulent jet statistics are known to be sensitive to under-resolution of the shear layer.
[Results] The averaged profiles are presented without error bars, statistical uncertainty estimates, or indication of the number of independent realizations used for averaging. This prevents assessment of whether the claimed differences between regimes exceed sampling variability.
[Introduction and results] No quantitative validation against experimental data is provided for the subcritical case (where such data exist), nor any cross-check for the supercritical case. Without this, the attribution of differences solely to the equation-of-state change remains unanchored.
[Governing equations] The governing-equations section assumes that the standard compressible Navier-Stokes equations plus an appropriate EOS and transport properties are sufficient for the supercritical regime. No discussion or test is given of possible additional real-fluid corrections (e.g., baroclinic torque from sharp density gradients or near-critical property anomalies) that could affect the reported statistics.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We are grateful to the referee for the constructive and detailed comments. The points raised concerning numerical convergence, statistical uncertainty, validation, and the treatment of real-fluid effects are important for strengthening the manuscript. We address each major comment below, indicating the revisions planned.

read point-by-point responses

Referee: No grid-resolution study or convergence demonstration is described for the key statistics (potential core length, growth rate, decay profiles) at Re_D=5000. This is load-bearing because turbulent jet statistics are known to be sensitive to under-resolution of the shear layer.

Authors: We agree that an explicit grid-convergence demonstration is necessary. The original simulations used a resolution guided by prior DNS studies at comparable Re, but no dedicated convergence study was presented. In the revised manuscript we will add results from three successively refined grids and show that the potential-core length, spatial growth rate, and velocity-decay profiles agree to within a few percent between the two finest grids, thereby confirming that the reported differences between thermodynamic regimes are not resolution artifacts. revision: yes
Referee: The averaged profiles are presented without error bars, statistical uncertainty estimates, or indication of the number of independent realizations used for averaging. This prevents assessment of whether the claimed differences between regimes exceed sampling variability.

Authors: We acknowledge the omission of uncertainty measures. The statistics were obtained from long-time averaging after the flow reached a statistically stationary state. The revised manuscript will report the total averaging interval in flow-through times, the effective number of independent samples based on the integral time scale, and error bars corresponding to the standard error of the mean on all mean profiles. This will allow readers to judge whether the observed differences exceed statistical variability. revision: yes
Referee: No quantitative validation against experimental data is provided for the subcritical case (where such data exist), nor any cross-check for the supercritical case. Without this, the attribution of differences solely to the equation-of-state change remains unanchored.

Authors: For the subcritical (perfect-gas) case we will add direct quantitative comparisons with published experimental data for round nitrogen jets at similar Re_D and Ma, focusing on potential-core length and centerline velocity decay. For the supercritical case, quantitative data at these exact conditions remain limited; we will therefore discuss qualitative consistency with existing high-pressure jet experiments while noting the inherent difficulties of quantitative validation under supercritical conditions. These additions will better support the attribution of differences to the thermodynamic regime. revision: partial
Referee: The governing-equations section assumes that the standard compressible Navier-Stokes equations plus an appropriate EOS and transport properties are sufficient for the supercritical regime. No discussion or test is given of possible additional real-fluid corrections (e.g., baroclinic torque from sharp density gradients or near-critical property anomalies) that could affect the reported statistics.

Authors: We will expand the governing-equations section to address the applicability of the standard compressible Navier-Stokes equations under the isothermal, moderate-Mach conditions of the study. Because temperature is uniform, density gradients arise only from the equation of state; we will show that the baroclinic torque and other near-critical corrections are of higher order and remain negligible relative to the retained terms. Supporting order-of-magnitude estimates and references will be included. No additional modeling terms were required for the present isothermal setup. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: forward DNS from NS + EOS

full rationale

The paper executes direct numerical simulations of isothermal round jets at fixed Re_D=5000 and Ma=0.6, solving the compressible Navier-Stokes equations with a passive scalar and switching only the equation of state and transport properties between perfect-gas and real-fluid cases. All reported statistics (potential core length, growth rate, velocity decay, mixed-fluid distributions) are direct numerical outputs, not quantities fitted to data and then re-predicted, not self-defined, and not justified by self-citation chains. The modeling choice to retain unmodified conservation laws is an assumption whose validity is external to the computation itself; it does not create a reduction of the claimed differences to the inputs by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work rests on the standard compressible Navier-Stokes equations plus a thermodynamic closure that switches between perfect-gas and real-fluid models; no new entities are postulated and the only free parameters are numerical (grid, time step) rather than physical constants fitted to the target result.

assumptions (1)

domain assumption The Navier-Stokes equations remain valid across the critical point when an appropriate equation of state is substituted.
Invoked by the decision to run the same DNS code for both regimes.

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Cite this review

Pith. "Pith review of Direct numerical simulation of high-pressure mixing in turbulent jets." pith.science (2026). https://pith.science/paper/6DRLMI6F

@misc{pith2026190711800,
  author       = {Pith},
  title        = {Pith review of: Direct numerical simulation of high-pressure mixing in turbulent jets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6DRLMI6F}},
  note         = {Machine review of arXiv:1907.11800}
}

abstract

Combustion in automotive and aerospace applications employing diesel, gas turbine and liquid rocket engines is preceded by injection and mixing of fuel and oxidizer at high pressures, often exceeding mixture critical values. Experimental observations indicate that the jets injected at supercritical pressures exhibit significantly different dynamics than the jets at subcritical conditions, owing to the lack of distinct liquid and gas phases in supercritical state. As a result, the averaged flow quantities such as the potential core length, jet spatial growth rate and velocity decay profiles differ in the two conditions, resulting in different mixed-fluid distributions. In this study, turbulent jet direct numerical simulations (DNS) are performed to examine the variations in statistics between injection of Nitrogen ($\mathrm{N_{2}}$) in Nitrogen ($\mathrm{N_{2}}$) at subcritical (perfect-gas) and supercritical conditions. Isothermal round jets at Reynolds number ($Re_{D}$), based on jet diameter ($D$) and jet orifice velocity ($U_{0}$), of $5000$ and Mach number of $0.6$ are considered. For mixing analyses, a passive scalar transported with the flow is examined.

Figures

Figures reproduced from arXiv: 1907.11800 by the authors.

**Figure 2.** Compressibility factor of N2 at 50 bar pressure. Red marker denotes the chamber conditions for Case 2. 3. Numerical details The spatial derivatives are approximated using the sixth-order compact finite-difference scheme and time integration uses the explicit fourth-order Runge-Kutta method. The outflow boundary in axial direction and all lateral boundaries have sponge zones[17] with subsonic non-reflecting outflow … view at source ↗

**Figure 3.** Instantaneous Mach number (Ma) field at tU0/D ≈ 2500 in (a) Case 1 and (b) Case 2. Only values of Ma ≥ 0.01 are rendered. Legend is the same for both plots. 4. Jet flow results and discussion [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Case 1 and 2 comparison showing (a) the time-averaged centerline velocity ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Centerline root-mean-square (a) axial velocity and (b) scalar fluctuation comparison [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

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Reference graph

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Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.

[1] [1]

E. Masi, J. Bellan, K. G. Harstad, and N. A. Okong’o, Multi-species turbulent mixing un- der supercritical-pressure conditions: modelling, direct numerical simulation and analysis revealing species spinodal decomposition, Journal of Fluid Mechanics 721 (2013) 578–626

work page 2013

[2] [2]

Chehroudi, D

B. Chehroudi, D. Talley, and E. Coy, Visual characteristics and initial growth rates of round cryogenic jets at subcritical and supercritical pressures, Physics of Fluids 14 (2002) 850– 861

work page 2002

[3] [3]

A. Roy, C. Joly, and C. Segal, Disintegrating supercritical jets in a subcritical environment, Journal of Fluid Mechanics 717 (2013) 193–202

work page 2013

[4] [4]

Muthukumaran and A

C. Muthukumaran and A. Vaidyanathan, Mixing nature of supercritical jet in subcritical and supercritical conditions, Journal of Propulsion and Power 33 (2016) 842–857

work page 2016

[5] [5]

P. J. Morris, Viscous stability of compressible axisymmetric jets, AIAA Journal 21 (1983) 481–482

work page 1983

[6] [6]

Michalke, Survey on jet instability theory, Progress in Aerospace Sciences 21 (1984) 159–199

A. Michalke, Survey on jet instability theory, Progress in Aerospace Sciences 21 (1984) 159–199

work page 1984

[7] [7]

Panchapakesan and J

N. Panchapakesan and J. Lumley, Turbulence measurements in axisymmetric jets of air and helium. Part 1. Air jet, Journal of Fluid Mechanics 246 (1993) 197–223

work page 1993

[8] [8]

H. J. Hussein, S. P. Capp, and W. K. George, Velocity measurements in a high-Reynolds- number, momentum-conserving, axisymmetric, turbulent jet, Journal of Fluid Mechanics 258 (1994) 31–75

work page 1994