Some Remarks on the Equations of Burnett and Grad

Struchtrup, Henning

doi:10.1007/978-1-4613-0017-5_17

Cited by 13 publications

(17 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…14,19 Furthermore, due to their hyperbolic nature, the well-known Grad's 13-moment (G13) equations for a single gas obtained via Grad's method of moments manifest non-physical sub-shocks for flows with Mach numbers above 1.65 14,20 and do not capture Knudsen boundary layers. 21,22 Nevertheless, by considering more moments, Knudsen boundary layers can be captured 21,23 and smooth shock structure can be obtained for higher Mach numbers. 20 In order to surmount the deficiencies inherent to both Chapman-Enskog expansion method and Grad's method of moments, Struchtrup and Torrilhon 24 -for single gases-introduced a new method, often referred to as the regularized moment method, which regularizes the original G13 equations for a single gas by means of a Chapman-Enskog expansion of Grad's 26-moment (G26) equations around a pseudo-equilibrium and leads to the regularized 13-moment (R13) equations.…”

Section: Introductionmentioning

confidence: 99%

Regularized moment equations for binary gas mixtures: Derivation and linear analysis

2016

Self Cite

View full text Add to dashboard Cite

The applicability of the order of magnitude method [H. Struchtrup, “Stable transport equations for rarefied gases at high orders in the Knudsen number,” Phys. Fluids 16, 3921–3934 (2004)] is extended to binary gas mixtures in order to derive various sets of equations—having minimum number of moments at a given order of accuracy in the Knudsen number—for binary mixtures of monatomic-inert-ideal gases interacting with the Maxwell interaction potential. For simplicity, the equations are derived in the linear regime up to third order accuracy in the Knudsen number. At zeroth order, the method produces the Euler equations; at first order, it results into the Fick, Navier–Stokes, and Fourier equations; at second order, it yields a set of 17 moment equations; and at third order, it leads to the regularized 17-moment equations. The transport coefficients in the Fick, Navier–Stokes, and Fourier equations obtained through order of magnitude method are compared with those obtained through the classical Chapman–Enskog expansion method. It is established that the different temperatures of different constituents do not play a role up to second order accurate theories in the Knudsen number, whereas they do contribute to third order accurate theory in the Knudsen number. Furthermore, it is found empirically that the zeroth, first, and second order accurate equations are linearly stable for all binary gas mixtures; however, although the third order accurate regularized 17-moment equations are linearly stable for most of the mixtures, they are linearly unstable for mixtures having extreme difference in molecular masses

show abstract

Section: Introductionmentioning

confidence: 99%

Regularized moment equations for binary gas mixtures: Derivation and linear analysis

2016

Self Cite

View full text Add to dashboard Cite

show abstract

“…Higher order ChapmanEnskog expansions yield the Burnett 15 and super-Burnett 16 equations, which are well known to be unstable in time dependent processes 17,18 and to give unphysical solutions in steady state. 19 Grad's 13 moment equations, or systems with more moments, are stable but exhibit unphysical sub-shocks for high speed flows. 20 Moreover, the Grad equations are not linked to the Knudsen number; hence, it is difficult to know a priori which set of moments should be considered for a given process.…”

Section: Introductionmentioning

confidence: 99%

Evaporation boundary conditions for the R13 equations of rarefied gas dynamics

et al. 2017

Self Cite

View full text Add to dashboard Cite

The regularized 13 moment (R13) equations are a macroscopic model for the description of rarefied gas flows in the transition regime. The equations have been shown to give meaningful results for Knudsen numbers up to about 0.5. Here, their range of applicability is extended by deriving and testing boundary conditions for evaporating and condensing interfaces. The macroscopic interface conditions are derived from the microscopic interface conditions of kinetic theory. Tests include evaporation into a half-space and evaporation/condensation of a vapor between two liquid surfaces of different temperatures. Comparison indicates that overall the R13 equations agree better with microscopic solutions than classical hydrodynamics

show abstract

“…The 13 moment equations do not describe Knudsen boundary layers [29,30,24], increasing the number of moments allows to compute these [31,24,32].…”

Section: Grad Moment Methodsmentioning

confidence: 99%

“…In [30] it was shown that this iteration method is equivalent to the CE expansion of the moment equations. In the original CE method one first expands, and then integrates the resulting distribution function to compute its moments.…”

Section: Combining the Chapman-enskog And Grad Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Model Reduction in Kinetic Theory

Struchtrup

Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena

View full text Add to dashboard Cite

Summary. Methods to derive macroscopic transport equations for rarefied gases from the Boltzmann equations are presented. Featured methods include the Chapman-Enskog expansion, Grad's moment method, and the author's order of magnitude method. The resulting macroscopic equations are compared and discussed by means of simple problems, including linear stability, shock wave structures, and Couette flow.

show abstract

Some Remarks on the Equations of Burnett and Grad

Cited by 13 publications

References 22 publications

Regularized moment equations for binary gas mixtures: Derivation and linear analysis

Regularized moment equations for binary gas mixtures: Derivation and linear analysis

Evaporation boundary conditions for the R13 equations of rarefied gas dynamics

Model Reduction in Kinetic Theory

Contact Info

Product

Resources

About