Effiziente Simulation von nichtlinearen Strömungen in verdünnten Gasen
Mathematik
Zusammenfassung der Projektergebnisse
Computational fluid dynamics (CFD) is typically based on the Navier-Stokes equations for the velocity field combined with the energy balance and Fourier’s law if the temperature field is of interest. However, these classical models are valid only for processes close to thermal equilibrium. For situations in rarefied gases or in microscopic settings classical CFD is known to produce physically wrong results, so that they can not be used in these cases. Over the past years new continuum models have been developed on the bases of kinetic gas theory which extend the range of applicability of conventional fluid dynamics, like moment approximations of the Boltzmann equation. Given the vision of extending modern day CFD to enable it for the demanding computation of rarefied gas flows, the results of this project yield a relevant and important perspective on moment approximations of the Boltzmann equation. Nonlinear processes remain difficult to compute with reduced models like moment equations, because of the gap between the instabilities of perturbation-based approaches and the computational inefficiencies of highly nonlinear models. This project adds results to both approaches. The entropic quadature approach improves the nonlinear maximum entropy closure, while another paper shows how the linear structure of moment equations can be used to compute weakly nonlinear processes. The results also show that further methodological research is needed to bridge this gap. Efficient and effectively reduced models like moment equations can be used for optimization and inverse problems in rarefied gases which requires a large number of successive forward computation. As an example this project investigated the Crookes radiometer with a range of different geometries in three space dimensions – a setup that would have been prohibitively expensive when based on the Boltzmann equation directly. The radiometer allows a study using linear, steady-state moment equations, like the regularized 13-moment system and single 3D simulation runs are a matter of minutes. In the future the usage of moment approximations requires adaptivity along two directions. Larger moment systems and nonlinear closures are possible to construct, but their complexity increases dramatically. Hence, the number of moments must be increased very carefully only locally in space and time where absolutely needed. Similarly, linear and perturbation-based methods must be used as much as possible. Switching the model to more moments or nonlinear closures locally, requires an understanding of the properties of the model hierarchies and, ultimately, reliable error estimates, possibly empirically and augmented by machine learning techniques.
Projektbezogene Publikationen (Auswahl)
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Entropic quadrature for moment approximations of the Boltzmann-BGK equation. Journal of Computational Physics, 401, 108992.
Böhmer, Niclas & Torrilhon, Manuel
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Structured derivation of moment equations and stable boundary conditions with an introduction to symmetric, trace-free tensors. Kinetic and Related Models, 16(3), 458-494.
Bünger, Jonas; Christhuraj, Edilbert; Hanke, Andrea & Torrilhon, Manuel
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Generic moment systems and FEM-based numerical solutions for the Boltzmann equation. AIP Conference Proceedings, 3050(2024), 070001.
Christhuraj, Edilbert & Torrilhon, Manuel
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GitHub Repository: 3-Dimensional FEM Implementation for the R13-Equations with Application to Crookes Radiometer, Source Code (2024)
E. Christhuraj
