Mathematical modeling of rarefied gas mixtures

Gupta, Vinay Kumar; Torrilhon, Manuel; Struchtrup, Henning

doi:urn:nbn:de:hbz:82-rwth-2015-052546

Mathematical modeling of rarefied gas mixtures = Mathematische Modellierung verdünnter Gasmischungen = Mathematical modeling of rarefied gas-mixtures

Gupta, Vinay Kumar

2015

Verantwortlichkeitsangabevorgelegt von Vinay Kumar Gupta

ImpressumAachen : Publikationsserver der RWTH Aachen University 2015

UmfangXII, 149 S. : Ill., graph. Darst.

Aachen, Techn. Hochsch., Diss., 2015

Genehmigende Fakultät
Fak01

Hauptberichter/Gutachter
Torrilhon, Manuel (Thesis advisor) ; Struchtrup, Henning (Thesis advisor)

Tag der mündlichen Prüfung/Habilitation
2015-09-24

Online
URN: urn:nbn:de:hbz:82-rwth-2015-052546
URL: https://publications.rwth-aachen.de/record/538381/files/538381.pdf
URL: https://publications.rwth-aachen.de/record/538381/files/538381.pdf?subformat=pdfa

Einrichtungen

Inhaltliche Beschreibung (Schlagwörter)
Mathematik (frei) ; Boltzmann equation (frei) ; kinetic theory (frei) ; gas mixtures (frei) ; linear stability (frei) ; heated cavity (frei) ; lid-driven cavity (frei) ; regularized moment equations (frei)

Thematische Einordnung (Klassifikation)
DDC: 510

Kurzfassung
In dieser Arbeit wird ein mathematisches Framework für Grads Gleichungen höherer Ordnung für verdünnte Gasmischungen entwickelt. Die vollen nichtlinearen 13-Momenten- und 26-Momentengleichungen werden für Gasmischungen bestehend aus N einatomigen Gasen hergeleitet. Die Boltzmann'schen Kollisionsintegrale, welche Grads Momentensystemen zugehörig sind, werden berechnet und für das Maxwell'sche und für das Interaktionspotential harter Kugel präsentiert. Die zugehörigen Randbedingungen werden durch ein Maxwell-Akkommodationsmodell bestimmt, welches von einatomigen Gasen auf Gasmischungen erweitert wurde. Die Momentengleichungen und Randbedingungen werden dann auf den Fall binärer Gasmischungen eingeschränkt und es wird durch eine lineare Stabilitätsanalyse gezeigt, dass Grads 2×13-Momenten- und 2×26-Momentengleichungen für binäre Gasmischungen linear stabil für die beiden erwähnten Interaktionspotentiale sind. Im letzten Teil werden Benchmark-Probleme der Fluiddynamik in einer und zwei Dimensionen für die hergeleiteten Gleichungen untersucht. Zum Abschluss werden die regularisierten 17-Momentengleichungen (R17) für binäre Gasmischungen mit Maxwell-Interaktionspotential hergeleitet.

A mathematical framework based on the higher order Grad's moment equations for studying processes in rarefied gas-mixtures is developed. The fully non-linear Grad's N×13-moment (N×G13) and Grad's N×26-moment (N×G26) equations for a gaseous mixture comprised of N monatomic-inert-ideal gases are derived. The strategy for computing the production terms (or Boltzmann collision integrals) associated with the moment equations for gaseous mixtures interacting with any interaction potential is presented. Employing this strategy, the explicit expressions of the non-linear production terms associated with the N×G13 and N×G26 equations are computed and presented for Maxwell as well as hard-sphere interaction potentials.The boundary conditions complementing the N×G13 and N×G26 equations are derived by extending the Maxwell's accommodation model for a single gas to a gaseous mixture. The moment equations and the boundary conditions are then restricted to binary gas mixtures and the linear stability analysis is performed to conclude that the Grad's 2×13-moment (2×G13) and Grad's 2×26-moment (2×G26) equations for a binary gas-mixture are linearly stable for both Maxwell as well as hard-sphere interaction potentials.Next, the 2×G13 and 2×G26 equations, specialized to Maxwell and hard-sphere interaction potentials, along with the boundary conditions are exploited to study some benchmark problems of fluid mechanics in simple geometries. The heat transfer in a binary gas-mixture confined between two infinite plates having different temperatures is analyzed with all four types of moment systems (2×G13 and 2×G26 equations, both with Maxwell and hard-sphere interaction potentials) and the results are compared with those existing in the literature. Furthermore, a one-dimensional problem of binary gas-mixture having one component infinitely diluted is solved analytically with all four types of moment systems in order to study the flow of the diluted component in the mixture.The numerical method based on finite differences for solving the aforementioned moment systems is demonstrated and employed to various problems in order to study the convergence of the numerical method. The convergence is analyzed for the above one-dimensional problems as well as for the standard two-dimensional problems of bottom-heated and lid-driven square cavities. The preliminary results on heat transfer for the latter two problems are also presented. However, the detailed comparison of the results with those obtained with highly accurate (particle based) methods is left for the future.Moreover, for model reduction, the Grad's moment equations for a binary gas-mixture with Maxwell interaction potential are regularized by employing the order of magnitude method and the regularized 17-moment (R17) equations for a binary gas-mixture with Maxwell interaction potential are derived; the R17 equations are third order accurate in the Knudsen number. The linear stability of R17 equations is analyzed and it is concluded empirically that the R17 equations are linearly stable for binary gas-mixtures with small and moderate mass differences while unstable for those with large mass differences.

OpenAccess:
PDF PDF (PDFA)
(additional files)