The use of moment-closure methods to predict continuum and moderately rarefied flow offers
many modelling and numerical advantages over traditional methods. The maximum-entropy
family of moment closures offers models described by hyperbolic systems of balance
laws. In particular, the twenty-one moment model of the maximum-entropy hierarchy offers a
hyperbolic treatment of viscous flows exhibiting heat transfer. This twenty-one moment
model has the ability to provide accurate solutions where the Navier-Stokes equations lose
physical validity due to the solution being too far from local equilibrium. Furthermore,
its first-order hyperbolic nature offers the potential for improved numerical accuracy as
well as a decreased sensitivity to mesh quality. Unfortunately, higher-order
maximum-entropy closures cannot be expressed in closed form. The only known affordable
option is to propose approximations. Previous approximations to the fourteen-moment
maximum-entropy model have been proposed [McDonald and Torrilhon,
2014]. Although this fourteen-moment model also predicts viscous flow with heat
transfer, the necessary moments to close the system renders it more difficult to
approximate accurately than the twenty-one moment model. The proposed approximation for
the fourteen-moment model also has realizable states for which hyperbolicity is lost.
Unfortunately, the velocity distribution function associated with the twenty-one moment
model is an exponential of a fourth-order polynomial. Such a function cannot be integrated
in closed form, resulting in closing fluxes that can only be obtained through complex
numerical methods. The goal of this work is to present a new approximation to the closing
fluxes that respect the maximum-entropy philosophy as closely as possible. Preliminary
results show that a proposed approximation is able to provide shock predictions that are
in good agreement with the Boltzmann equation and surpassing the prediction of the
Navier-Stokes equations. Furthermore, Couette flow results as well as lid-driven cavity
flows are computed using a novel approach to Knudsen layer boundary conditions. A
dispersion analysis as well as an investigation of the hyperbolicity of the model is also
shown. The Couette flow results are compared against Navier-Stokes and the free-molecular
analytical solutions for a varying Knudsen number, for which the twenty-one moment model
show good agreement over the domain. The shock-tube problem is also computed for different
Knudsen numbers. The results are compared with the one obtained by directly solving the BGK
equation. Finally, the lid-driven cavity flow computed with the twenty-one moment model
shows good agreement with the direct simulation Monte-Carlo (DSMC) solution.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/45657 |
| Date | 23 November 2023 |
| Creators | Giroux, Fabien |
| Contributors | McDonald, James Gerald |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
| Rights | Attribution-NonCommercial-NoDerivatives 4.0 International, http://creativecommons.org/licenses/by-nc-nd/4.0/ |
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