Optimization of Second-Order Transport Models for Transition-Continuum Flows

Jonathan F. MacArt; Mikolaj Kryger

REVIEW 3 major objections 5 minor 42 references

Optimization of Second-Order Transport Models for Transition-Continuum Flows

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A posteriori-optimized second-order transport coefficients bring continuum shock predictions close to Boltzmann-DSMC data across Mach 1.1–10, including heat flux and stress excluded from training.

desk verdict Useful adjoint-based calibration of the Paolucci–Paolucci closure with credible out-of-sample shock results, but the stress/heat-flux 'validation' is not independent and the paper overclaims what it proves. read the letter →

arxiv 2411.13515 v2 pith:SDDNVDC6 submitted 2024-11-20 physics.flu-dyn physics.comp-ph

Mikolaj Kryger , Jonathan F. MacArt This is my paper · ORCID

classification physics.flu-dynphysics.comp-ph PACS 47.45.-n47.40.Nm

keywords second-ordercontinuumtransportmodelPaolucci–Paolucciconstitutivetheoryadjoint-basedoptimizationtransition-continuumflowviscousshockstructuredirectsimulationMonteCarloBoltzmannequationhypersonicnonequilibriumflows

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

This paper asks whether the second-order continuum transport theory of Paolucci and Paolucci, whose extra coefficients cannot be measured directly, can be made accurate for transition-continuum shocks by calibrating them against Boltzmann-equation data. The authors show that adjoint-based, PDE-constrained optimization of the eight free coefficients, minimizing error in density, velocity, and temperature, yields argon shock profiles that track DSMC data from Mach 1.1 to 10 far better than first-order Navier–Stokes or the original second-order model. A biased minibatch weighting that up-weights low-Mach training cases fixes the lower-Mach accuracy that unbiased training misses. The optimized models also reproduce viscous stress and heat flux, quantities excluded from the objective, which the authors read as evidence that the fitted closures capture the actual nonequilibrium transport. If correct, the work provides a route to accurate continuum descriptions of hypersonic transition-continuum flows at much lower cost than particle simulation.

What carries the argument

The load-bearing object is the one-dimensional second-order constitutive closure obtained from the Paolucci–Paolucci entropy-restricted theory, in which heat flux and stress contain products of gradients: $q = -(k + k_\star \partial u/\partial x)\,\partial T/\partial x - k_{\star\star}(\partial u/\partial x)(\partial \rho/\partial x)$, with an analogous expression for the stress. The unknown coefficient functions are collapsed into power-law forms such as $\mu_* = \rho_*^{\beta_1} T_*^{\gamma_1}$ and $k_\star = k_{\star 0}\,\rho_*^{\beta_4} T_*^{\gamma_4}$, leaving eight free parameters. The optimization uses the discrete adjoint of the steady Navier–Stokes operator: adjoint variables $\hat{V}$ satisfy $(\partial F/\partial V)^\top \hat{V} = -(\partial J/\partial V)^\top$, which gives the parameter gradient $\nabla_\theta J = \hat{V}^\top \partial F/\partial \theta$ without forming $\partial V/\partial \theta$. A biased minibatch average over Mach 2, 5, and 8 training cases up-weights low-Mach gradients, and early stopping at the first local loss minimum limits overfitting.

What would settle it

Pick a 1D argon shock at the same Mach numbers but a freestream Knudsen number two to five times larger or smaller than the training case, solve it with the optimized parameters, and compare density, heat flux, and stress against DSMC. If the untrained moments drift well off the DSMC data while first-order Navier–Stokes does not, the power-law closure form, not the calibration, is the limiting error; a second test is to extract $k_\star$ and $k_{\star\star}$ locally from DSMC fields and check whether any power-law fit over $\rho$ and $T$ leaves structured residuals.

Watch

Extended reading notes

Core claim

The central claim is that the Paolucci–Paolucci second-order constitutive closure, with its higher-order thermal-conductivity and viscosity coefficients represented by eight-parameter power laws, contains enough freedom to describe nonequilibrium argon shocks once those parameters are chosen by solving the flow equations inside an optimization loop. When the objective is the L2 mismatch of density, velocity, and temperature against DSMC moments, the optimized parameters reduce density error at high Mach numbers relative to both first-order Navier–Stokes and the unoptimized PP18 model, and the bias-weighted models do so at low Mach numbers as well. Because the viscous stress and heat flux were not in the loss but still agree with the Boltzmann moments, the paper argues the improved shock structure is not merely curve-fitting but is consistent with the nonequilibrium physics. Out-of-sample testing across Mach 1.1 to 10 shows the a posteriori-trained models remain stable and accurate, unlike a priori-trained parameters.

Load-bearing premise

The argument rests on the assumption that the true higher-order transport coefficients in these argon shocks are well represented by power laws in density and temperature, with the first-order viscosity exponents fixed and bulk viscosity zero; if the true coefficients have a different functional form, no adjustment of the fitted parameters can recover the nonequilibrium physics.

Editorial extensions

If this is right

Adjoint-optimized second-order models reduce L2 density error relative to first-order Navier–Stokes and unoptimized PP18 at high Mach numbers, and bias weighting extends the gains to low Mach numbers.
Predicted viscous stress and heat flux, which are not in the objective function, closely match DSMC moments, indicating that the calibrated closure is consistent with the nonequilibrium transport physics.
Inverse shock thickness predictions for Mach numbers above 4 become more accurate than both first-order Navier–Stokes and PP18, while the biased models match PP18 at low Mach within experimental uncertainty.
A priori-trained parameters can converge on training data yet produce unstable and inaccurate out-of-sample shock solutions, demonstrating that the PDE constraint is essential in this regime.
The optimized parameter trends (reduced $k_{\star 0}$ and $k_{\star\star 0}$, near-zero $\beta_4$, $\gamma_4$, $\gamma_5$) suggest a simplified higher-order closure may be sufficient for shocks at this Knudsen number.

Reading between the lines

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

Directly testable extension: extract $k_\star$ and $k_{\star\star}$ locally from DSMC fields using the constitutive equations and fit them over density and temperature; structured residuals would show the power-law closure itself, not the calibration, is the limiting error.
The same adjoint pipeline should transfer to two- and three-dimensional flows if the adjoint system is solved with sparse iterative linear algebra instead of dense elimination; the authors state this expectation but do not demonstrate it.
The optimized $\beta_4$, $\gamma_4$, and $\gamma_5$ values near zero suggest these higher-order conductivity exponents could be fixed or dropped for shocks at the present freestream Knudsen number, though different flows or Knudsen regimes may require nonzero values.
Replacing the power-law ansatz with a neural-network or tabulated closure trained by the same objective could relax the main closure assumption and likely improve extrapolation, at the cost of a much larger parameter space.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

You should know two things about this paper. First, it is a genuinely useful application of adjoint-based a posteriori calibration to the Paolucci–Paolucci second-order closure: the authors fit eight higher-order transport coefficients in a PDE-constrained loop, compare against a priori training, and show real out-of-sample gains in density profiles and shock thickness at Mach numbers not used in training. Second, the Section 4.2 stress/heat-flux validation is not the independent check it claims to be. In a steady 1D shock, once you have matched ρ, u, and T, the viscous stress and heat flux are fixed by conservation. So matching τ and q is a numerical consistency check, not evidence that the closure captures the nonequilibrium physics. The stress-test note is correct.

What is actually new: the parameterization of the second-order coefficients, the biased minibatch weighting that keeps low-Mach cases from being ignored, and the demonstration that a priori-trained models become unstable out of sample. The adjoint equations (22–25) are standard and correctly derived. The out-of-sample tests at M=9 for models trained on {2,5,8} are the strongest part of the evidence, and the improvements at M>4 are convincing.

Soft spots, in proportion. The abstract and conclusion say the results are “consistent with the nonequilibrium flow physics,” but the stress/heat-flux comparison cannot support that; the claim should be removed or severely qualified. The paper treats DSMC targets as exact: no statistical error bars appear anywhere, and the fitted parameters are reported as point values with no sense of uncertainty. This matters because the loss landscape may have multiple minima and the optimization starts from the PP18 parameters. No code or data is shipped, which makes reproduction harder than it should be. The power-law closure is a genuine limitation, though the authors acknowledge it openly. There is also a minor internal inconsistency in Section 4: the text says the optimized higher-order coefficients are “more sensitive to density gradients” while the density exponents mostly decrease in magnitude toward zero.

Who is this for? Anyone working on transition-continuum modeling or PDE-constrained closure calibration will get value from the method and the comparison with a priori training. The paper deserves a serious referee; the central method and the out-of-sample results are solid enough to warrant revision rather than rejection.

Recommendation: send it to peer review, but instruct the authors to address the independence issue in the stress/heat-flux validation and to provide uncertainty estimates or justify their absence.

Referee Report

3 major / 5 minor

Summary. The paper proposes an adjoint-based, a posteriori optimization framework for calibrating the higher-order transport coefficients in the Paolucci-Paolucci second-order continuum model, using one-dimensional argon shock waves as test cases. Target data are moments of DSMC/Boltzmann solutions, and the objective function is an L2 error in primitive variables (density, velocity, temperature). Parameters are optimized for individual Mach numbers and for minibatch training over M=2,5,8 with different bias weightings, then evaluated over M=1.1-10. The authors report that the optimized models improve density profiles, inverse shock thickness, shock asymmetry, and temperature-density separation relative to first-order Navier-Stokes and the unoptimized PP18 model. They also compare predicted viscous stress and heat flux, which are not in the loss, to DSMC moments and interpret the agreement as evidence that the optimized models capture the underlying nonequilibrium physics.

Significance. The methodological contribution is useful: the adjoint derivation (Eqs. 22-25) is standard and correct, the discrete treatment via algorithmic differentiation is appropriate, and the biased minibatch scheme is a sensible remedy for loss-scale imbalance across Mach numbers. The comparison with a priori optimization is instructive and clearly demonstrates the value of PDE-embedded training. If the empirical improvements are robust, the approach provides a practical route for calibrating higher-order continuum closures when direct transport-coefficient data are unavailable. However, the paper's central claim of physical validity is not supported as strongly as stated. In a steady one-dimensional shock, the viscous stress and heat flux are algebraically determined by the primitive variables through the conservation equations, so the Section 4.2 stress/heat-flux comparison is not an independent test of closure physics. The acknowledged power-law closure assumption (Eq. 10) then becomes the main unresolved limitation.

major comments (3)

[Section 2.3, Eq. (10)] The viscous stress and heat-flux validation is not independent of the primitive-variable loss. In a steady one-dimensional shock, integrating the momentum and energy conservation equations with the same upstream boundary conditions gives τ(x) = ρu^2 + p - (ρ∞u∞^2 + p∞) and q(x) = ρ∞u∞E∞ + p∞u∞ - (ρEu + pu - τu). Thus, once ρ, u, and T are specified, τ and q are fixed by conservation alone. Since the objective function (20) includes all three primitive variables, a good match of V to the DSMC target automatically produces a good match of τ and q to the DSMC moments, regardless of whether the constitutive closure is physically correct. The close agreement in Figure 8 therefore tests numerical consistency of the forward solution, not closure adequacy. The statement in Section 4.2 that 'the optimized models capture the true physics of these nonequilibrium shocks' is an overclaim; the stress/heat-flux comparison should be reframed as a consistency check, and the conclusions revised accordingly.
[Section 3, Eqs. (13)-(15)] The assumed power-law closure for the higher-order transport coefficients, with fixed first-order exponents β1 = 0.5 and γ1 = 0.72 and bulk-viscosity parameter κ0 = 0, is ad hoc, as the authors acknowledge ('this is neither a unique closure nor necessarily globally optimal'). Because the Section 4.2 stress/heat-flux comparison is not an independent physical test, the claim that the optimized parameters 'correctly capture the nonequilibrium physics' rests entirely on this closure assumption. If the true coefficients violate the power-law form, no parameter optimization can recover the correct physics, and the observed profile improvements would reflect effective calibration rather than physical closure. Please either add a genuinely independent validation (e.g., a different Knudsen number, a differently shaped flow, or a direct comparison of coefficient functions against kinetic-theory data) or temper the conclusions to present the optimized parameters as effective, flow-calibrated coefficients.
[Section 3] The DSMC target data are treated as exact throughout the optimization and evaluation, but no statistical error bars, particle-number convergence criteria, or uncertainty estimates are reported for the moment data, nor for the optimized parameter sets in Table 1. Consequently, the error differences in Figure 6 are not accompanied by any confidence intervals, and it is unclear whether the reported improvements over PP18 are statistically significant at the lower Mach numbers where differences are small. Please report DSMC statistical uncertainty at least for representative cases and use it to assess the robustness of the comparisons and the optimized parameter values.

minor comments (5)

[Section 2.4] The specific gas constant for argon is given as R = 208.12 kJ/(kg·K); this should be J/(kg·K) (or 0.20812 kJ/(kg·K)). As written, the value is three orders of magnitude too large and would affect the equation of state if taken literally.
[Section 3.1.3 and Section 3.1.1] There are minor typographical issues: in Section 3.1.3, 'Thea priori-trained' is missing a space, and in the heading of Section 3.1.1, 'Loss F unction' contains an extra space.
[Figure 2] The caption of Figure 2(b) refers to 'a posteriori simulations' before the a posteriori method has been introduced in Section 3.2; this appears to mean forward simulations using the a priori-optimized parameters and should be reworded for clarity.
[Section 4.1] The aggregate claims about improved accuracy across the full Mach range should be qualified by the fact that M=2, 5, and 8 are in-sample training cases for the M258 models, while M=1.1, 3, 4, 6, 7, 9, and 10 are out-of-sample. The improvements in Figure 6(a) include training points, so the paper should distinguish in-sample, interpolated, and extrapolated conditions when summarizing the results.
[Section 3.1.1, Eq. (16)] The normalization terms 'max qe · qe' and 'max τe : τe' in the a priori loss are not defined; please specify whether the maximum is taken over the spatial domain and, if so, over the target fields.

Circularity Check

1 steps flagged · score 6.0 of 10

Section 4.2 stress/heat-flux validation is forced by conservation once ρ,u,T are fitted; core density-error improvements remain independent.

fitted input called prediction [Section 4.2 (Viscous Stress and Heat Flux), around Eq. (20) and Fig. 8]
"Unlike for a priori training (Section 3.1), the a posteriori training process does not include the viscous stress and heat flux in its loss function (20), yet the optimized models still predict these quantities. The similarity of the modeled viscous stress and heat flux to those obtained from the Boltzmann distribution function can be used to assess the physical realism of the optimized second-order models. ... The results confirm that the optimized models capture the true physics of these nonequilibrium shocks."

In a steady 1D shock, the x-momentum and energy conservation equations imply τ = ρu^2 + p − (ρ∞u∞^2 + p∞) and q = (ρ∞u∞E∞ + p∞u∞) − (ρEu + pu − τu) once the upstream boundary conditions are fixed. The objective (20) fits V = (ρ,u,T) directly to the DSMC target, so any optimized solution that reproduces V must reproduce the DSMC τ and q purely through conservation, independent of the constitutive closure. The stress/heat-flux comparison therefore tests consistency of the fitted primitive-variable field, not whether the closure captures nonequilibrium physics; the claim that these quantities are 'not included in the objective function' is misleading because τ and q are algebraic functions of the fitted variables and the boundary data.

full rationale

The adjoint-based parameter optimization itself is not circular: the eight optimized parameters are fitted to DSMC-derived primitive variables, and the paper evaluates holdout Mach numbers (e.g., M∞=9) before claiming generalizability; the a priori baseline comparison and experimental shock-thickness data provide additional independent checks. The self-citation to Nair et al. [25] is not load-bearing because that prior work validates DSMC targets against independent experimental data. The one significant circular element is the Section 4.2 'validation': in a steady 1D shock, the x-momentum and energy equations make τ and q algebraic functions of ρ, u, T and the upstream state, so optimizing J(V) = (1/2)∫(V−Ve)² dx against DSMC forces post-optimization τ and q to match DSMC regardless of whether the second-order closure is physically correct. Thus the claim that the close match 'indicates that the learned models correctly capture the nonequilibrium physics' is not supported by that comparison; it tests numerical consistency only. The paper's density-profile improvements, shock-thickness trends, and out-of-sample behavior remain as empirical evidence, so the circularity is partial rather than total. The authors also explicitly acknowledge that the power-law closure 'is neither a unique closure nor necessarily globally optimal,' which is an honest limitation, not a circular step.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The central results rest on the assumed validity of the PP18 second-order constitutive closure, the specific power-law parameterization of the transport coefficients, and the use of DSMC moment data as ground truth. The model parameters are fitted to the DSMC primitive variables; no independent first-principles derivation of the transport coefficients is provided.

free parameters (9)

k0_star = 0.380 to 0.594 across optimized models (Table 1)
Reference second-order thermal conductivity coefficient; optimized a posteriori against DSMC primitive variables.
k0_starstar = 1.076 to 3.728 (Table 1)
Reference second-order conductivity for the density-gradient term; optimized.
beta3 = 0.595 to 0.897
Density exponent of first-order thermal conductivity; optimized.
gamma3 = 0.941 to 1.005
Temperature exponent of first-order thermal conductivity; optimized.
beta4 = -0.280 to 0.165
Density exponent of second-order conductivity k*_star; optimized.
gamma4 = -0.019 to 0.282
Temperature exponent of second-order conductivity k*_star; optimized.
beta5 = -0.813 to 0.341
Density exponent of second-order conductivity k**_star; optimized.
gamma5 = -0.197 to 0.245
Temperature exponent of second-order conductivity k**_star; optimized.
minibatch_biases = chi = [1,1,1], [3,1,1], [7,2,1]
Hand-chosen gradient weights per Mach number; influence final parameters and model accuracy.

assumptions (6)

domain assumption The PP18 second-order constitutive functionals (Equations 5-6, 8-9) are a valid closure for nonequilibrium gas dynamics.
Assumed from Paolucci and Paolucci (2018); the paper optimizes only the parameters, not the closure form.
ad hoc to paper The higher-order transport coefficients follow the power-law forms in Equation (10).
The authors state the power-law representation is 'neither a unique closure nor necessarily globally optimal'; the choice bounds the expressiveness of the model.
domain assumption DSMC solutions of the Boltzmann equation provide exact target data for the flow and for validation moments.
Used as ground truth in Equations (13)-(15); DSMC statistical error is not quantified.
domain assumption Fixed first-order viscosity exponents beta1=0.5 and gamma1=0.72 from Macrossan and Lilley are valid for argon over T=300-8000 K.
Taken from experimental power-law fits; not re-optimized, so the model relies on their accuracy outside the fit range.
domain assumption Bulk viscosity kappa0=0 for monatomic argon (Chapman-Enskog expansion).
Standard for monatomic gases; justifies removing beta2 and gamma2 from optimization.
domain assumption The 1D steady normal shock at fixed freestream Knudsen number is a representative test for transition-continuum closure calibration.
The conclusions on coefficient values are drawn from this single flow class; the authors note the parameters may not transfer to other flows or Knudsen numbers.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Optimization of Second-Order Transport Models for Transition-Continuum Flows." pith.science (2026). https://pith.science/paper/SDDNVDC6

@misc{pith2026241113515,
  author       = {Pith},
  title        = {Pith review of: Optimization of Second-Order Transport Models for Transition-Continuum Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SDDNVDC6}},
  note         = {Machine review of arXiv:2411.13515}
}

abstract

Modeling transition-continuum hypersonic flows poses significant challenges due to thermodynamic nonequilibrium and the associated breakdown of the continuum assumption. Standard continuum models such as the Navier-Stokes equations are inaccurate for these flows, and molecular models can be inefficient due to the large number of computational particles required at moderately high densities. We explore computational modeling of transition-continuum flows using a second-order constitutive theory that provides closures for the terms representing the molecular transport of momentum and energy. We optimize the second-order model parameters for one-dimensional viscous shocks using an adjoint-based optimization method, with the objective function comprising the primitive flow variables. Target data is obtained from moments of distribution functions obtained by solving the Boltzmann equation. We compare results using optimized second-order models, the unoptimized second-order model, and the first-order Navier-Stokes model for Mach numbers $M\in[1.1,10]$ and observe improvements to the shock profiles and shock thickness calculations. We validate the optimized models by comparing the predicted viscous stress and heat flux, which are not included in the objective function, to those obtained by integrating the distribution function. The close match to these moments indicates that the satisfactory performance of the optimized second-order models is consistent with the nonequilibrium flow physics.

Figures

Figures reproduced from arXiv: 2411.13515 by the authors.

**Figure 2.** (a) In-sample heat flux and viscous stress magnitudes for first-order, PP18 second-order, and [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Model training hierarchy for a posteriori optimization [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Graphical depiction of initial (PP18) and [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: Density, velocity and temperature profiles for [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: Density error (in kg/m3 ) (a) and the inverse shock thickness (b) for first-order Navier–Stokes, second-order PP18, and a posteriori-optimized second-order models across the range of tested Mach numbers. first-order solution due to its inaccurate heat flux. All optimiz…

**Figure 7.** Figure 7: Shock asymmetry quotient (left) and temperature–density separation distance (right) for [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: Heat flux (top) and viscous stress (bottom) magnitude for [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

[1]

J. D. Anderson, Modern Compressible Flow With Historical Perspective 3rd Ed., McGraw Hill, 2003

work page 2003
[2]

Cercignani, R

C. Cercignani, R. Illner, M. Pulvirenti, Particle Simulation of the Boltzmann Equation, Springer New York, New York, NY, 1994, pp. 286–311

work page 1994
[3]

T. E. Schwartzentruber, I. D. Boyd, Progress and future prospects for particle-based simulation of hypersonic flow, Progress in Aerospace Sciences 72 (2015) 66–79

work page 2015
[4]

I. D. Boyd, T. E. Schwartzentruber, Direct Simulation Monte Carlo, Cambridge Aerospace Series, Cambridge University Press, 2017, p. 183–251

work page 2017
[5]

Grad, On the kinetic theory of rarefied gases, Communications on Pure and Applied Mathematics 2 (4) (1949) 331–407

H. Grad, On the kinetic theory of rarefied gases, Communications on Pure and Applied Mathematics 2 (4) (1949) 331–407

work page 1949
[6]

Grad, The profile of a steady plane shock wave, Communications on Pure and Applied Mathematics 5 (3) (1952) 257–300

H. Grad, The profile of a steady plane shock wave, Communications on Pure and Applied Mathematics 5 (3) (1952) 257–300

work page 1952
[7]

M¨ uller, T

I. M¨ uller, T. Ruggeri, Rational Extended Thermodynamics, Vol. 37, Springer-Verlag, New York, 1998

work page 1998
[8]

Torrilhon, Modeling nonequilibrium gas flow based on moment equations, Annual Review of Fluid Mechanics 48 (1) (2016) 429–458

M. Torrilhon, Modeling nonequilibrium gas flow based on moment equations, Annual Review of Fluid Mechanics 48 (1) (2016) 429–458. 18

work page 2016

Show all 42 references

[9]

B. C. Eu, A modified moment method and irreversible thermodynamics, The Journal of Chemical Physics 73 (6) (1980) 2958–2969

1980
[10]

Al-Ghoul, B

M. Al-Ghoul, B. C. Eu, Generalized hydrodynamics and shock waves, Physical Review E 56 (3) (1997) 2981

1997
[11]

R. S. Myong, Thermodynamically consistent hydrodynamic computational models for high-knudsen- number gas flows, Physics of Fluids 11 (9) (1999) 2788–2802

1999
[12]

Jiang, W

Z. Jiang, W. Zhao, Z. Yuan, W. Chen, R. S. Myong, Computation of hypersonic flows over flying configurations using a nonlinear constitutive model, AIAA Journal 57 (12) (2019) 5252–5268

2019
[13]

Vincenti, C

W. Vincenti, C. Kruger, Introduction to Physical Gas Dynamics, Wiley, 1965

1965
[14]

Bellomo, P

N. Bellomo, P. LeTallec, B. Perthame, The solution of the nonlinear Boltzmann equation: A survey of analytic and computational methods, Computers & Mathematics with Applications 30 (7) (1995) 21–30

1995
[15]

R. K. Agarwal, K.-Y. Yun, R. Balakrishnan, Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime, Physics of Fluids 13 (10) (2001) 3061–3085

2001
[16]

C. J. Lee, Unique determination of solutions to the Burnett equations, AIAA Journal 32 (5) (1994) 985–990

1994
[17]

Comeaux, D

K. Comeaux, D. Chapman, R. MacCormack, An analysis of the Burnett equations based on the second law of thermodynamics, in: 33rd AIAA Aerospace Sciences Meeting and Exhibit, 1995, p. 415

1995
[18]

R. S. Jadhav, A. Gavasane, A. Agrawal, Improved theory for shock waves using the oburnett equations, Journal of Fluid Mechanics 929 (2021) A37

2021
[19]

Paolucci, C

S. Paolucci, C. Paolucci, A second-order continuum theory of fluids, Journal of Fluid Mechanics 846 (2018) 686–710

2018
[20]

Paolucci, Second-order constitutive theory of fluids, Continuum Mechanics and Thermodynamics 34 (1) (2022) 185–215

S. Paolucci, Second-order constitutive theory of fluids, Continuum Mechanics and Thermodynamics 34 (1) (2022) 185–215

2022
[21]

A. S. Rana, V. K. Gupta, H. Struchtrup, Coupled constitutive relations: a second law based higher-order closure for hydrodynamics, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474 (2018) 20180323

2018
[22]

Brenner, Kinematics of volume transport, Physica A: Statistical Mechanics and its Applications 349 (1) (2005) 11–59

H. Brenner, Kinematics of volume transport, Physica A: Statistical Mechanics and its Applications 349 (1) (2005) 11–59

2005
[23]

C. J. Greenshields, J. M. Reese, The structure of shock waves as a test of Brenner’s modifications to the Navier–Stokes equations, Journal of Fluid Mechanics 580 (2007) 407–429

2007
[24]

Sirignano, J

J. Sirignano, J. F. MacArt, J. B. Freund, DPM: A deep learning PDE augmentation method with application to large-eddy simulation, Journal of Computational Physics 423 (2020) 109811

2020
[25]

A. S. Nair, J. Sirignano, M. Panesi, J. F. MacArt, Deep Learning Closure of the Navier–Stokes Equations for Transition-Continuum Flows, AIAA Journal 61 (12) (2023) 5484–5497

2023
[26]

Li, Deep reinforcement learning: An overview, arXiv preprint (2017)

Y. Li, Deep reinforcement learning: An overview, arXiv preprint (2017)

2017
[27]

Rabault, M

J. Rabault, M. Kuchta, A. Jensen, U. R´ eglade, N. Cerardi, Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control, Journal of Fluid Mechanics 865 (2019) 281–302. 19

2019
[28]

Garnier, J

P. Garnier, J. Viquerat, J. Rabault, A. Larcher, A. Kuhnle, E. Hachem, A review on deep reinforcement learning for fluid mechanics, Computers & Fluids 225 (2021) 104973

2021
[29]

X. Liu, J. F. MacArt, Adjoint-based machine learning for active flow control, Physical Review Fluids 9 (1) (2024) 13901

2024
[30]

Duraisamy, Perspectives on machine learning-augmented Reynolds-averaged and large eddy simula- tion models of turbulence, Physical Review Fluids 6 (5) (2021) 050504

K. Duraisamy, Perspectives on machine learning-augmented Reynolds-averaged and large eddy simula- tion models of turbulence, Physical Review Fluids 6 (5) (2021) 050504

2021
[31]

J. F. MacArt, J. Sirignano, J. B. Freund, Embedded training of neural-network subgrid-scale turbulence models, Physical Review Fluids 6 (2021) 050502

2021
[32]

Sirignano, J

J. Sirignano, J. F. MacArt, Deep learning closure models for large-eddy simulation of flows around bluff bodies, Journal of Fluid Mechanics 966 (2023) A26

2023
[33]

Sirignano, J

J. Sirignano, J. F. MacArt, K. Spiliopoulos, PDE-constrained models with neural network terms: Op- timization and global convergence, Journal of Computational Physics 481 (2023) 112016

2023
[34]

Alsmeyer, Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam, Journal of Fluid Mechanics 74 (3) (1976) 497–513

H. Alsmeyer, Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam, Journal of Fluid Mechanics 74 (3) (1976) 497–513

1976
[35]

M. N. Macrossan, C. R. Lilley, Viscosity of argon at temperatures greater than 2000 K from measured shock thickness, Physics of Fluids 15 (11) (2003) 3452–3457

2003
[36]

Linzer, D

M. Linzer, D. F. Hornig, Structure of Shock Fronts in Argon and Nitrogen, The Physics of Fluids 6 (12) (1963) 1661–1668

1963
[37]

Camac, Argon and nitrogen shock thicknesses, Proceedings of the Fourth International Symposium on Rarefied Gas Dynamics 1 (1965) 240

M. Camac, Argon and nitrogen shock thicknesses, Proceedings of the Fourth International Symposium on Rarefied Gas Dynamics 1 (1965) 240

1965
[38]

Schmidt, Electron beam density measurements in shock waves in argon, Journal of Fluid Mechanics 39 (2) (1969) 361–373

B. Schmidt, Electron beam density measurements in shock waves in argon, Journal of Fluid Mechanics 39 (2) (1969) 361–373

1969
[39]

Garen, R

W. Garen, R. Synofzik, A. Frohn, Shock tube for generating weak shock waves, AIAA Journal 12 (8) (1974) 1132–1134

1974
[40]

P. L. Roe, Characteristic-based schemes for the euler equations, Annual Review of Fluid Mechanics 18 (1986) 337–365

1986
[41]

Paszke, S

A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, S. Chintala, PyTorch: An Imperative Style, High-Perform...

2019
[42]

O. A. Montesinos L´ opez, A. Montesinos L´ opez, J. Crossa, Overfitting, Model Tuning, and Evaluation of Prediction Performance, Springer, 2022, pp. 109–139. 20

2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

J. D. Anderson, Modern Compressible Flow With Historical Perspective 3rd Ed., McGraw Hill, 2003

work page 2003

[2] [2]

Cercignani, R

C. Cercignani, R. Illner, M. Pulvirenti, Particle Simulation of the Boltzmann Equation, Springer New York, New York, NY, 1994, pp. 286–311

work page 1994

[3] [3]

T. E. Schwartzentruber, I. D. Boyd, Progress and future prospects for particle-based simulation of hypersonic flow, Progress in Aerospace Sciences 72 (2015) 66–79

work page 2015

[4] [4]

I. D. Boyd, T. E. Schwartzentruber, Direct Simulation Monte Carlo, Cambridge Aerospace Series, Cambridge University Press, 2017, p. 183–251

work page 2017

[5] [5]

Grad, On the kinetic theory of rarefied gases, Communications on Pure and Applied Mathematics 2 (4) (1949) 331–407

H. Grad, On the kinetic theory of rarefied gases, Communications on Pure and Applied Mathematics 2 (4) (1949) 331–407

work page 1949

[6] [6]

Grad, The profile of a steady plane shock wave, Communications on Pure and Applied Mathematics 5 (3) (1952) 257–300

H. Grad, The profile of a steady plane shock wave, Communications on Pure and Applied Mathematics 5 (3) (1952) 257–300

work page 1952

[7] [7]

M¨ uller, T

I. M¨ uller, T. Ruggeri, Rational Extended Thermodynamics, Vol. 37, Springer-Verlag, New York, 1998

work page 1998

[8] [8]

Torrilhon, Modeling nonequilibrium gas flow based on moment equations, Annual Review of Fluid Mechanics 48 (1) (2016) 429–458

M. Torrilhon, Modeling nonequilibrium gas flow based on moment equations, Annual Review of Fluid Mechanics 48 (1) (2016) 429–458. 18

work page 2016