Modified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> D Laplacian for smooth pressure reconstruction based on time-resolved velocimetry (1): analysis and numerics
Bibliographic record
Abstract
Abstract We analyze a smooth pressure solver based on the ‘modified Poisson equation’: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>ξ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>∂</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∂</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> is the pressure field, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="bold-italic">u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> is the velocity field measured by time-resolved image velocimetry, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>ξ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> is a tunable parameter to control the solver’s diffusive behaviour in time. This modified Poisson equation aims at obtaining smooth pressure fields from potentially noisy image velocimetry measurements, and is a part of the current four-dimensional (4D) pressure solver (implemented in, for example, DaVis 10.2) by LaVision. This work focuses on investigating three aspects of the ‘modified Poisson equation’: smoothing effect, error propagation, and drift in time. We first provide rigorous analysis and validate that this solver can sufficiently smooth the computed pressure field by setting a large enough <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>ξ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . However, a large value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>ξ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> may cause large errors in the reconstructed pressure fields. Then we introduce an upper bound on the error in the reconstructed pressure fields to quantify the error propagation dynamics. Finally, we discuss the potential drift due to the partitioning in time, which is an optional strategy used in LaVision’s current 4D pressure solver to reduce computational costs. Our analysis and validation not only show that careful choice of the parameters (e.g. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>ξ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> ) is needed for smooth and accurate pressure field reconstruction but provide theoretical guidelines for parameter tuning when similar pressure solvers are used for time-resolved image velocimetry data.
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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 | 0.006 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.002 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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; both teacher heads agree on what is shown here.
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".