{"id":"https://openalex.org/W3174694876","doi":"https://doi.org/10.3390/e23060782","title":"Variable-Order Fractional Models for Wall-Bounded Turbulent Flows","display_name":"Variable-Order Fractional Models for Wall-Bounded Turbulent Flows","publication_year":2021,"publication_date":"2021-06-20","ids":{"openalex":"https://openalex.org/W3174694876","doi":"https://doi.org/10.3390/e23060782","mag":"3174694876","pmid":"https://pubmed.ncbi.nlm.nih.gov/34202955"},"language":"en","primary_location":{"id":"doi:10.3390/e23060782","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23060782","pdf_url":"https://www.mdpi.com/1099-4300/23/6/782/pdf?version=1648636403","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/23/6/782/pdf?version=1648636403","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5046508214","display_name":"Fangying Song","orcid":"https://orcid.org/0000-0002-1493-3819"},"institutions":[{"id":"https://openalex.org/I80947539","display_name":"Fuzhou University","ror":"https://ror.org/011xvna82","country_code":"CN","type":"education","lineage":["https://openalex.org/I80947539"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fangying Song","raw_affiliation_strings":["College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China","institution_ids":["https://openalex.org/I80947539"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5009658255","display_name":"George Em Karniadakis","orcid":"https://orcid.org/0000-0002-9713-7120"},"institutions":[{"id":"https://openalex.org/I27804330","display_name":"Brown University","ror":"https://ror.org/05gq02987","country_code":"US","type":"education","lineage":["https://openalex.org/I27804330"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"George Em Karniadakis","raw_affiliation_strings":["Division of Applied Mathematics, School of Engineering, Brown University, Providence, RI 02912, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Division of Applied Mathematics, School of Engineering, Brown University, Providence, RI 02912, USA","institution_ids":["https://openalex.org/I27804330"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5009658255"],"corresponding_institution_ids":["https://openalex.org/I27804330"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.8901,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.69603524,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"23","issue":"6","first_page":"782","last_page":"782"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10104","display_name":"Nanofluid Flow and Heat Transfer","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/turbulence","display_name":"Turbulence","score":0.7362539768218994},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5774564743041992},{"id":"https://openalex.org/keywords/reynolds-number","display_name":"Reynolds number","score":0.5656511783599854},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5311381220817566},{"id":"https://openalex.org/keywords/direct-numerical-simulation","display_name":"Direct numerical simulation","score":0.5303353071212769},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.46913549304008484},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.4628877341747284},{"id":"https://openalex.org/keywords/open-channel-flow","display_name":"Open-channel flow","score":0.4609001874923706},{"id":"https://openalex.org/keywords/quantum-nonlocality","display_name":"Quantum nonlocality","score":0.4494037330150604},{"id":"https://openalex.org/keywords/pipe-flow","display_name":"Pipe flow","score":0.4324130415916443},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.42893218994140625},{"id":"https://openalex.org/keywords/reynolds-stress","display_name":"Reynolds stress","score":0.42020758986473083},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.41534528136253357},{"id":"https://openalex.org/keywords/mechanics","display_name":"Mechanics","score":0.3397901654243469},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.29391348361968994}],"concepts":[{"id":"https://openalex.org/C196558001","wikidata":"https://www.wikidata.org/wiki/Q190132","display_name":"Turbulence","level":2,"score":0.7362539768218994},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5774564743041992},{"id":"https://openalex.org/C182748727","wikidata":"https://www.wikidata.org/wiki/Q178932","display_name":"Reynolds number","level":3,"score":0.5656511783599854},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5311381220817566},{"id":"https://openalex.org/C18533594","wikidata":"https://www.wikidata.org/wiki/Q1227572","display_name":"Direct numerical simulation","level":4,"score":0.5303353071212769},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.46913549304008484},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4628877341747284},{"id":"https://openalex.org/C180925781","wikidata":"https://www.wikidata.org/wiki/Q514096","display_name":"Open-channel flow","level":3,"score":0.4609001874923706},{"id":"https://openalex.org/C138622882","wikidata":"https://www.wikidata.org/wiki/Q7269079","display_name":"Quantum nonlocality","level":4,"score":0.4494037330150604},{"id":"https://openalex.org/C204561356","wikidata":"https://www.wikidata.org/wiki/Q1148018","display_name":"Pipe flow","level":3,"score":0.4324130415916443},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.42893218994140625},{"id":"https://openalex.org/C147196274","wikidata":"https://www.wikidata.org/wiki/Q7319685","display_name":"Reynolds stress","level":3,"score":0.42020758986473083},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.41534528136253357},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.3397901654243469},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.29391348361968994},{"id":"https://openalex.org/C121040770","wikidata":"https://www.wikidata.org/wiki/Q215675","display_name":"Quantum entanglement","level":3,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e23060782","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23060782","pdf_url":"https://www.mdpi.com/1099-4300/23/6/782/pdf?version=1648636403","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:34202955","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/34202955","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:4dceed1b38a8487fa96aca1db6f003de","is_oa":true,"landing_page_url":"https://doaj.org/article/4dceed1b38a8487fa96aca1db6f003de","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 23, Iss 6, p 782 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/23/6/782/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e23060782","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy; Volume 23; Issue 6; Pages: 782","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:8234867","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/8234867","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e23060782","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23060782","pdf_url":"https://www.mdpi.com/1099-4300/23/6/782/pdf?version=1648636403","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3174694876.pdf","grobid_xml":"https://content.openalex.org/works/W3174694876.grobid-xml"},"referenced_works_count":36,"referenced_works":["https://openalex.org/W323536346","https://openalex.org/W1580865944","https://openalex.org/W1977046327","https://openalex.org/W1989266909","https://openalex.org/W2004872575","https://openalex.org/W2038286209","https://openalex.org/W2048950352","https://openalex.org/W2067530013","https://openalex.org/W2092721527","https://openalex.org/W2093820071","https://openalex.org/W2119104548","https://openalex.org/W2128337144","https://openalex.org/W2140030715","https://openalex.org/W2155155203","https://openalex.org/W2165076033","https://openalex.org/W2165742943","https://openalex.org/W2167500235","https://openalex.org/W2168479071","https://openalex.org/W2171651455","https://openalex.org/W2287209854","https://openalex.org/W2292897919","https://openalex.org/W2534431640","https://openalex.org/W2709969531","https://openalex.org/W2740817067","https://openalex.org/W2748985244","https://openalex.org/W2792089488","https://openalex.org/W2802768264","https://openalex.org/W2889205466","https://openalex.org/W2918335093","https://openalex.org/W2982546823","https://openalex.org/W2990449391","https://openalex.org/W2999451223","https://openalex.org/W3021996758","https://openalex.org/W6684171834","https://openalex.org/W6685041794","https://openalex.org/W6843829459"],"related_works":["https://openalex.org/W1986812219","https://openalex.org/W2497626802","https://openalex.org/W2106451319","https://openalex.org/W2152145107","https://openalex.org/W2110223908","https://openalex.org/W4389790645","https://openalex.org/W1497066214","https://openalex.org/W2205547892","https://openalex.org/W2318043555","https://openalex.org/W2123619231"],"abstract_inverted_index":{"Modeling":[0],"of":[1,44,71,85,167,179],"wall-bounded":[2,33],"turbulent":[3,168,180,201,291],"flows":[4],"is":[5,207,241],"still":[6],"an":[7],"open":[8],"problem":[9],"in":[10,17,32,41,165,229,243],"classical":[11],"physics,":[12],"with":[13,80,89],"relatively":[14],"slow":[15],"progress":[16],"the":[18,23,29,60,66,69,72,77,90,93,137,153,162,171,176,186,230,237,244,259,266,274,285,297,301],"last":[19],"few":[20],"decades":[21],"beyond":[22],"log":[24],"law,":[25],"which":[26,240],"only":[27],"describes":[28],"intermediate":[30],"region":[31],"turbulence,":[34],"i.e.,":[35,117],"30-50":[36],"y+":[37],"to":[38,58,68,135,225,248,284],"0.1-0.2":[39],"R+":[40],"a":[42,50,81,104,194,210,253],"pipe":[43,123],"radius":[45],"R.":[46],"Here,":[47],"we":[48,75,96,142,189,263,289],"propose":[49,190],"fundamentally":[51],"new":[52,234],"approach":[53],"based":[54],"on":[55],"fractional":[56,83,101,197,214,267],"calculus":[57],"model":[59,199,235,290],"entire":[61],"mean":[62],"velocity":[63],"profile":[64],"from":[65,92,130,170,185],"wall":[67,172],"centerline":[70],"pipe.":[73],"Specifically,":[74],"represent":[76],"Reynolds":[78,109],"stresses":[79],"non-local":[82],"derivative":[84],"variable-order":[86,138],"that":[87,98,182,220],"decays":[88],"distance":[91],"wall.":[94,187],"Surprisingly,":[95],"find":[97],"this":[99,221],"variable":[100,196,213],"order":[102],"has":[103],"universal":[105,302],"form":[106],"for":[107,112,200,257,272],"all":[108],"numbers":[110],"and":[111,122,140,149,276,294],"three":[113],"different":[114],"flow":[115],"types,":[116],"channel":[118],"flow,":[119,121],"Couette":[120],"flow.":[124],"We":[125,251],"first":[126],"use":[127,252],"existing":[128],"databases":[129],"direct":[131],"numerical":[132,217],"simulations":[133],"(DNSs)":[134],"lean":[136],"function":[139],"subsequently":[141],"test":[143],"it":[144],"against":[145],"other":[146],"DNS":[147],"data":[148],"experimental":[150],"measurements,":[151],"including":[152,193],"Princeton":[154],"superpipe":[155],"experiments.":[156],"Taken":[157],"together,":[158],"our":[159],"findings":[160],"reveal":[161],"continuous":[163],"change":[164],"rate":[166],"diffusion":[169],"as":[173,175],"well":[174],"strong":[177],"nonlocality":[178],"interactions":[181],"intensify":[183],"away":[184],"Moreover,":[188],"alternative":[191],"formulations,":[192],"divergence":[195],"(two-sided)":[198],"flows.":[202],"The":[203,216],"total":[204],"shear":[205],"stress":[206],"represented":[208],"by":[209],"two-sided":[211],"symmetric":[212],"derivative.":[215],"results":[218],"show":[219],"formulation":[222],"can":[223],"lead":[224],"smooth":[226],"fractional-order":[227],"profiles":[228],"whole":[231],"domain.":[232],"This":[233],"improves":[236],"one-sided":[238],"model,":[239],"considered":[242],"half":[245],"domain":[246],"(wall":[247],"centerline)":[249],"only.":[250],"finite":[254],"difference":[255],"method":[256],"solving":[258,273],"inverse":[260,275],"problem,":[261],"but":[262],"also":[264],"introduce":[265],"physics-informed":[268],"neural":[269],"network":[270],"(fPINN)":[271],"forward":[277],"problems":[278],"much":[279],"more":[280],"efficiently.":[281],"In":[282],"addition":[283],"aforementioned":[286],"fully-developed":[287],"flows,":[288],"boundary":[292],"layers":[293],"discuss":[295],"how":[296],"streamwise":[298],"variation":[299],"affects":[300],"curve.":[303]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}