Maximum Entropy Reconstruction for Gas Dynamics

Citation

Summy, Dustin Phillip (2017) Maximum Entropy Reconstruction for Gas Dynamics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9GT5K7W. https://resolver.caltech.edu/CaltechTHESIS:05262017-215132894

Abstract

We present a method for selecting a unique and natural probability distribution function (PDF) which satisfies a given number of known moments and apply it for use in the closure of moment-based schemes for approximately solving the Boltzmann equation in gas dynamics.

The method used for determining the PDF is the Maximum Entropy Reconstruction (MER) procedure, which determines the PDF with maximum entropy which satisfies a given set of constraining moments. For the five-moment truncated Hamburger moment problem in one dimension, the MER takes the form of the exponential of a quartic polynomial. This implies a bimodal structure which gives rise to a small-amplitude packet of PDF-density sitting quite far from the mean. This is referred to as the Itinerant Moment Packet (IMP). It is shown by asymptotic analysis that the IMP gives rise to a solution that, in the space of constraining moments, is singular along a line emanating from, but not including, the point representing thermodynamic equilibrium. We use this analysis of the IMP to develop a numerical regularization of the MER, creating a procedure we call the Hybrid MER (HMER). Compared with the MER, the HMER is a significant improvement in terms of robustness and efficiency while preserving accuracy in its prediction of other important distribution features, such as higher order moments.

We apply the one-dimensional HMER to close a fourth order moment system derived from the Boltzmann equation by using a specific set of moment constraints which allow the full, three-dimensional velocity PDF to be treated as a product of three independent, one-dimensional PDFs. From this system, we extract solutions to the problem of spatially homogeneous relaxation and find excellent agreement with a standard method of solution. We further apply this method to the problem of computing the profile within a normal shock wave, and find that solutions exist only within a finite shock Mach number interval. We examine the structure of this solution and find that it has interesting behavior connected to the singularity of the MER and the IMP. Comparison is made to standard solution methods. It is determined that the use of the MER in gas dynamics remains uncertain and possible avenues for further progress are discussed.

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