A discussion of hyperbolicity in CATHENA 4: Virtual Mass and phase-to-interface pressure differences
Bibliographic record
Abstract
Abstract It is well known that the one-dimensional equations of motion for two-phase flow are non-hyperbolic. Non-hyperbolicity can lead to numerical instabilities, destroying the solution. However, researchers in the last few decades were able to show that inclusion of virtual mass and/or phase-to-interface pressure differences in the momentum equations successfully render the equations of motion hyperbolic. In the present paper, the effect of including virtual mass and phase-to-interface pressure terms in the momentum equations on the hyperbolicity of the two-phase model in the CATHENA 4 code is discussed. The study is motivated by the fact that the inclusion of either model has been shown in the open literature to lead to a hyperbolic system separately. However, no known study exists that examine hyperbolicity in the presence of both these terms in the momentum equations. In this work, both terms are considered in the model equations simultaneously and their implications on the hyperbolicity of the two-phase model are discussed. Specifically, it is shown that in the case of mixed flow, there is a distinct region of non-hyperbolicity that developers need to be aware of when their equations include both the virtual mass and the phase-to-interface terms. Selecting the coefficients of phase-to-interface pressure difference terms properly ensures that the equations are hyperbolic for a wide range of conditions.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".