Exact Solutions to the Three-Dimensional Navier–Stokes Equations Using the Extended Beltrami Method
Bibliographic record
Abstract
Abstract In a 1966 publication, Chi-Yi Wang used the streamfunction in concert with the vorticity equations to develop a methodology for obtaining exact solutions to the incompressible Navier–Stokes equations, now known as the extended Beltrami method. In Wang's approach, the vorticity is represented by the sum of a linear function of the streamfunction and an assumed auxiliary function, such that the vorticity equation can be reduced to a quasi-linear partial differential equation, and exact solutions are obtainable for many choices of the auxiliary function. In the present work, a natural extension of Wang's formulation to three-dimensional flows in arbitrary orthogonal curvilinear coordinates has been derived, wherein two auxiliary functions are formed at the outset, with the caveat that the pressure and velocity components may vary in two spatial dimensions. As is the case with two-dimensional extended Beltrami flows, exact solutions are only obtainable when the forms of the auxiliary functions are “simple enough” to render the governing equations solvable. To demonstrate the solutions which may be obtained using the extended formulation, the well-known Kovasznay flow is generalized to a three-dimensional flow. A unique solution in plane polar coordinates is found. An extension to the solution to Burgers vortex has been derived and discussed in the context of existing literature. Finally, a new 3D swirling flow solution which is the angular analogue to Kovasznay flow has been developed.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".