Influence of the dispersive and dissipative scales<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>β</mml:mi></mml:math>on the energy spectrum of the Navier-Stokes<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:math>equations
Bibliographic record
Abstract
Lundgren's vortex model for the intermittent fine structure of high-Reynolds-number turbulence is applied to the Navier-Stokes $\ensuremath{\alpha}\ensuremath{\beta}$ equations and specialized to the Navier-Stokes $\ensuremath{\alpha}$ equations. The Navier-Stokes $\ensuremath{\alpha}\ensuremath{\beta}$ equations involve dispersive and dissipative length scales $\ensuremath{\alpha}$ and $\ensuremath{\beta}$, respectively. Setting $\ensuremath{\beta}$ equal to $\ensuremath{\alpha}$ reduces the Navier-Stokes $\ensuremath{\alpha}\ensuremath{\beta}$ equations to the Navier-Stokes $\ensuremath{\alpha}$ equations. For the Navier-Stokes $\ensuremath{\alpha}$ equations, the energy spectrum is found to obey Kolmogorov's $\ensuremath{-}5∕3$ law in a range of wave numbers identical to that determined by Lundgren for the Navier-Stokes equations. For the Navier-Stokes $\ensuremath{\alpha}\ensuremath{\beta}$ equations, Kolmogorov's $\ensuremath{-}5∕3$ law is also recovered. However, granted that $\ensuremath{\beta}<\ensuremath{\alpha}$, the range of wave numbers for which this law holds is extended by a factor of $\ensuremath{\alpha}∕\ensuremath{\beta}$. This suggests that simulations based on the Navier-Stokes $\ensuremath{\alpha}\ensuremath{\beta}$ equations may have the potential to resolve features smaller than those obtainable using the Navier-Stokes $\ensuremath{\alpha}$ equations.
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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 | 0.000 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.015 | 0.001 |
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".