Author
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Bailey, Brian |
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Submitted to: Journal of Fluid Mechanics
Publication Type: Peer Reviewed Journal Publication Acceptance Date: 6/14/2016 Publication Date: 7/12/2016 Citation: Bailey, B.N. 2016. Numerical considerations for Lagrangian stochastic dispersion models: Eliminating rogue trajectories, and the importance of numerical accuracy. Journal of Fluid Mechanics. doi: 10.1007/s10546-016-0181-6. Interpretive Summary: Models that simulate how particles move in fluids (e.g., air, water) are used to study problems throughout the sciences. Such examples include the wind blowing particles that leave a smoke stack, or airborne pathogens spreading disease. In a commonly used modeling framework, unphysical results often occur, where the computed speed of particles can become so large that they would effectively launch into outer space. This research explores this problem, and formulates a solution that ensures such unphysical behavior does not develop. This solution will improve the accuracy of particle models across a wide range of applications, and allow them to be applied to study more complex problems than before. Technical Abstract: When Lagrangian stochastic models for turbulent dispersion are applied to complex flows, some type of ad hoc intervention is almost always necessary to eliminate unphysical behavior in the numerical solution. This paper discusses numerical considerations when solving the Langevin-based particle velocity evolution equation. Unphysically large or ‘rogue’ velocities are caused when the numerical integration scheme becomes unstable. By calculating the total derivative in the production term along particle trajectories rather than using the Eulerian definition, it was possible to formulate a fully implicit integration scheme that is always stable. Additionally, when the generalized anisotropic model was used, it was critical that the velocity covariance tensor used to drive the model be realizable, otherwise unphysical behavior became problematic regardless of the integration scheme. A method is presented to ensure realizability, and thus eliminate such behavior. It was also found that the numerical accuracy of the integration scheme determined the degree to which the second law of thermodynamics or ‘well-mixed condition’ was satisfied. Perhaps more importantly, it also determined the degree to which modeled Eulerian particle velocity statistics matched the specified Eulerian distributions. Several test cases are presented to illustrate these points, both in the context of ensemble averaged (RANS) and large-eddy simulation (LES) models. It is recommended that all models be verified by not only checking the well-mixed condition, but also by checking that computed Eulerian statistics match the specified Eulerian statistics. The source code and input data for all of the test cases are provided in the manuscript’s online material. |
