[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82566-en":3,"doc-seo-82566-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82566,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Visualizing Lagrangian Heat Transport Paths and Density Structures in Unsteady Heat Transfer","Convective heat transfer is commonly visualized using Eulerian scalar temperature fields, which limit understanding of the underlying transport mechanisms. A Lagrangian viewpoint, analogous to mass transport along fluid trajectories, can expose coherent structures and routes hidden to Eulerian analysis. Heat transport is inherently non-conservative and aperiodic, restricting existing visualization approaches. This work introduces a particle-based method using time-reparameterized spacetime thermal transport to accumulate path contributions and reveal finite-time attracting/repelling density structures and coherent transport paths.","Visualizing Lagrangian Heat Transport Paths and Density Structures in  \nUnsteady Heat Transfer  \nBesm Osman* Andrei Jalba† Michel Speetjens‡ Anna Vilanova§  \nEindhoven University of Technology  \narXiv :2607 .00909v1 [physics .flu-dyn] 1 Jul 2026  \nFigure 1: Visualization of thermal transport structures using our proposed method. Left: finite-time density structures with transport paths revealed by particle path accumulation. Right: the 3D spacetime domain along which particles are integrated.  \nABSTRACT  \nConvective heat transfer is traditionally visualized from a Eulerian perspective using scalar temperature fields, offering limited insight into the underlying transport mechanisms. A Lagrangian view, analogous to mass transport along fluid paths, can reveal coherent structures and transport routes invisible from a Eulerian view of temperature. However, heat transport is aperiodic and non-conservative, hampering the application of fluid mixing and transport visualization techniques, developed primarily for timeperiodic, conservative transport. We present a particle-based visualization technique that addresses these challenges by advecting massless particles along a time-reparameterized spacetime formulation of thermal transport, accumulating path contributions to reveal coherent transport routes and finite-time attracting and repelling structures that conventional methods cannot show.  \nIndex Terms: Scientific visualization, Flow visualization, Lagrangian methods, Coherent structures.  \n1 INTRODUCTION  \nHeat transfer in fluid flows (i.e., convective heat transfer) is relevant for a wide range of practical applications including the conventional processing industry, pharmaceutical devices, thermal-storage units for sustainable energy systems, and high-precision equipment. Convective heat transfer is traditionally visualized as scalar maps of temperature fields and investigated with heat transfer coefficients  \n* e-mail: [b.osman@tue.nl](b.osman@tue.nl)[ ](b.osman@tue.nl)†[e-mail: A.C.Jalba@tue.nl](e-mail: A.C.Jalba@tue.nl)  \n‡[e-mail: M.F.M.Speetjens@tue.nl](e-mail: M.F.M.Speetjens@tue.nl)  \n§[e-mail: A.Vilanova@tue.nl](e-mail: A.Vilanova@tue.nl)  \ncharacterizing the heat transfer rate at fluid-solid interfaces [5, 1] . However, this view offers only limited insight into the actual physical transport mechanisms: temperature fields reveal only the thermal distribution at a particular instant, not where heat is moving to or where it originated, and heat-transfer coefficients provide no insight into thermal transport pathways. Meaningful visualization of convective heat transfer would instead reveal the thermal paths themselves, the coherency of transport along them, and the structures that emerge from this transport.  \nSpeetjens proposed an alternative Lagrangian approach to convective heat transfer [10] that relies on the notion that convective heat transfer fundamentally is the transport of thermal energy by a net heat flux encompassing the combined effect of fluid motion and molecular diffusion. Convective heat transfer thus becomes the transport of thermal energy along certain paths (“thermal paths”), in a similar way as fluid motion is the transport of fluid parcels by the flow along fluid paths. This mass-transfer analogy admits visualization of convective heat transfer by the topology of such thermal paths using well-established Lagrangian methods and concepts from flow visualization and analysis of mixing by the topology of fluid paths [9] . However, two essential differences between fluid and thermal transport render the utilization of existing Lagrangian methods for thermal transport non-trivial. Namely, heat transport is (i) non-conservative (i.e., net heat flux is divergent), and (ii) aperiodic, whereas fluid transport is conservative and generally periodic in time. Poincar maps require periodicity that thermal transport does not have. Lagrangian coherent structure techniques [3] can capture transport structures in ap","cbCaimGG08XsVV0y","https://ap.wps.com/l/cbCaimGG08XsVV0y","pdf",11346948,1,5,"English","en",105,"# Abstract\n# Introduction\n# Background: Lagrangian Thermal Formalism","[{\"question\":\"Why is the Eulerian temperature-field perspective limited for convective heat transfer visualization?\",\"answer\":\"Temperature maps show thermal distribution at an instant but do not indicate where heat is moving or originating, and heat-transfer coefficients do not expose thermal transport pathways.\"},{\"question\":\"What makes applying standard Lagrangian transport visualization techniques challenging for heat transfer?\",\"answer\":\"Thermal transport is non-conservative (net heat flux is divergent) and aperiodic, so assumptions like periodicity required by some methods do not hold.\"},{\"question\":\"How does the proposed visualization technique reveal coherent thermal transport routes?\",\"answer\":\"It advects massless particles in a time-reparameterized spacetime formulation of thermal transport, then accumulates particle path contributions on an accumulation grid using a Gaussian kernel to generate coherent path structures and finite-time density evolution.\"}]",1784181564,13,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"visualizing-lagrangian-heat-transport-paths-and-density-structures-in-unsteady-heat-transfer","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/visualizing-lagrangian-heat-transport-paths-and-density-structures-in-unsteady-heat-transfer/82566/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why is the Eulerian temperature-field perspective limited for convective heat transfer visualization?","Question",{"text":75,"@type":76},"Temperature maps show thermal distribution at an instant but do not indicate where heat is moving or originating, and heat-transfer coefficients do not expose thermal transport pathways.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What makes applying standard Lagrangian transport visualization techniques challenging for heat transfer?",{"text":80,"@type":76},"Thermal transport is non-conservative (net heat flux is divergent) and aperiodic, so assumptions like periodicity required by some methods do not hold.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed visualization technique reveal coherent thermal transport routes?",{"text":84,"@type":76},"It advects massless particles in a time-reparameterized spacetime formulation of thermal transport, then accumulates particle path contributions on an accumulation grid using a 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