Estimating the skin-friction coefficient in drag-reduced pipe flows of a flexible polymer
Bibliographic record
Abstract
In turbulent pipe flows, drag-reducing polymers are commonly used to reduce skin-friction drag; however, predicting this reduction in industry applications, such as crude oil pipelines, remains challenging. The skin-friction coefficient ( $C_f$ ) of polymer drag-reduced turbulent pipe flows can be related to three dimensionless parameters: the solvent Reynolds number ( $Re_s$ ), the Weissenberg number ( $Wi$ ) and the ratio of solvent viscosity ( $\eta _s$ ) to zero-shear-rate viscosity ( $\eta _0$ ), denoted as $\beta$ . The function that relates these four dimensionless numbers was determined using experiments of various pipe diameters ( $D$ ), flow velocities ( $U$ ) and drag-reducing polyacrylamide solutions. The experiments included measurements of streamwise pressure drop ( $\Delta P$ ) for determining $C_f$ , and measurements of shear viscosity ( $\eta$ ) and elastic relaxation time ( $\lambda$ ). This experimental campaign involved 156 flow conditions, each characterised by distinct values for $C_f$ , $Re_s$ , $Wi$ and $\beta$ . Experimental results demonstrated good agreement with the relationship: $C_f^{-1/2} = \widehat {A}\log _{10}(Re_sC_f^{1/2})+\widehat {B}$ , where $\widehat {A} = 27.6(Wi \beta )^{0.346}$ and $\widehat {B} = 122/15-58.9(Wi \beta )^{0.346}$ . Based on this relationship, onset and maximum drag reduction are predicted to occur when $Wi \beta$ equals $3.76 \times 10^{-3}$ and
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 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 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".