[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125858-en":3,"doc-seo-125858-105":31,"detail-sidebar-cat-0-en-105":93},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},125858,1099523882367,"Hazel","https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc",8,"Research & Report","Non-Unique Machine Learning Mapping in Data-Driven Reynolds Averaged Turbulence Models","Growing interest in machine learning for turbulence modelling has produced many data-driven models, yet training and prediction can suffer from non-unique mapping (NUM), which arises when the same invariant inputs correspond to multiple outputs. Most NUM research focuses on one-dimensional channel flows, even though many ML models are trained using two-dimensional flow data. This study provides the first detailed NUM analysis for two-dimensional flows, using a periodic hills flow and an impinging jet. It quantifies NUM, identifies low-strain/rotation or near-pure-shear regions as dominant contributors, and introduces viscosity ratio as a supplementary input variable to reduce NUM and improve tensor-basis model accuracy.","Non-Unique Machine Learning Mapping in Data-Driven Reynolds Averaged Turbulence Models  \nAnthony Man, Mohammad Jadidi, Amir Keshmiri, Hujun Yin, Yasser Mahmoudi *  \nSchool of Engineering, The University of Manchester, Manchester, M13 9PL, UK  \n*[Corresponding author:](Corresponding author: yasser.mahmoudilarimi@manchester.ac.uk)[ yasser.mahmoudilarimi@manchester.ac.uk](Corresponding author: yasser.mahmoudilarimi@manchester.ac.uk)  \nAbstract  \nRecent growing interest in using machine learning for turbulence modelling has led to many proposed data-driven turbulence models in the literature. However, most of these models have not been developed with overcoming non-unique mapping (NUM) in mind, which is a significant source of training and prediction error. Only NUM caused by one-dimensional channel flow data has been well studied in the literature, despite most data-driven models having been trained on two-dimensional flow data. The present work aims to be the first detailed investigation on NUM caused by two-dimensional flows. A method for quantifying NUM is proposed and demonstrated on data from a flow over periodic hills, and an impinging jet. The former is a wall-bounded separated flow, and the latter is a shear flow containing stagnation and recirculation. This work confirms that data from two-dimensional flows can cause NUM in data-driven turbulence models with the commonly used invariant inputs. This finding was verified with both cases, which contain different flow phenomena, hence showing that NUM is not limited to specific flow physics. Furthermore, the proposed method revealed that regions containing low strain and rotation or near pure shear cause the majority of NUM in both cases – approximately 76% and 89% in the flow over periodic hills and impinging jet, respectively. These results led to viscosity ratio being selected as a supplementary input variable (SIV), demonstrating that SIVs can reduce NUM caused by data from two-dimensional flows and subsequently improve the accuracy of tensor-basis machine learning models for turbulence modelling.  \nKeywords: Turbulence modelling, Machine learning, Reynolds stress, Non-unique mapping, Multivalue problem, Supplementary input variable, Tensor-basis neural networks.  \nNomenclature  \nVariable Meaning Unit  \n􀜾 􀯜􀯝 Anisotropy tensor -  \n􀜤 Impinging jet inlet width m  \n􀜥􀰓 Boussinesq hypothesis parameter -  \n􀝃􀯡 Scalar coefficients ofthe general effective- -viscosity hypothesis  \n􀝀 Distance between two arbitrary scatter points -  \n􀝌 and 􀝍  \n􀜪ℎ Hill height m  \n􀝇 Turbulent kinetic energy m2/s2  \n􀜮􀯫 Domain length in periodic hills case m  \n􀝊􀮼􀯂 Number of conflicting instances in a grid cell -􀝎􀰔 Viscosity ratio (􀝎􀰔 = 􀟥􀯧⁄(100􀟥 + 􀟥􀯧)) -  \n􀡾 Non-dimensional mean rotation rate (􀡾 = -􀝇􀝎􀯜􀯝⁄􀟝)  \n􀝎􀯜􀯝 Mean rotation rate 1/s 􀜴􀝁􀯧 Turbulent Reynolds number (􀜴􀝁􀯧 = 􀝇 2⁄􀟥􀟝) -  \n􀡿 Non-dimensional mean strain rate (􀡿 = -􀝇􀝏􀯜􀯝⁄􀟝)  \n􀝏􀯜􀯝 Mean strain rate 1/s 􀝑 + Nondimensional streamwise velocity (􀝑+ = -  \n􀝑̅1⁄􀝑􀰛)    \n􀝑􀰛 Friction velocity (√􀟬􀯪⁄􀟩) m/s 􀝑′􀯜 Velocity fluctuation in ith direction m/s  \n􀝑̅􀯜 Mean velocity in ith direction m/s  \n􀜷􀯕 Bulk inlet velocity in periodic hills case m/s  \n􀜷􀯜􀯡 Uniform inlet velocity in impinging jet case m/s  \n􀝔􀯜 Distance in ith direction m  \n􀢞􀯧􀯥 Input variables ofthe two-dimensional GEVH -(􀢞􀯧􀯥 ≡ {􀝐􀝎 (􀡿2), 􀝐􀝎 (􀡾2)})  \n􀝕 + Nondimensional distance from the wall (􀝕+ = -  \n􀝔2􀝑􀰛⁄􀟥)  \nSymbol  \n􀟬􀯪 Wall shear stress m2/s2  \n􀟬􀯜􀯝 Reynolds stress m2/s2  \n􀟜􀯜􀯝 Kronecker delta -  \n􀟝 Turbulent kinetic energy dissipation rate m2/s3  \n􀟙 Non-dimensional shear velocity gradient (􀟙 = -(􀝇⁄􀟝)(􀟲􀝑̅1⁄􀟲􀝔2))  \n􀟥 Kinematic molecular viscosity m2/s  \n􀟥􀯧 Kinematic eddy viscosity m2/s  \n􀟩 Density kg/m3  \n􀟚 Steepness factor in periodic hills case -  \nSubscript  \n1 Streamwise direction  \n2 Wall-normal direction  \n3 Spanwise direction  \n􀝌 Arbitrary scatter point 􀝌  \n􀝍 Arbitrary scatter point 􀝍  \n\n|  | Superscript\u003Cbr>̅⬚̃⬚ | Reynolds-averaged\u003Cbr>Min-max normalized quantity |\n| --- | --- | --- |\n|  | Abbreviat","cbCait6t0hgqtR6D","https://ap.wps.com/l/cbCait6t0hgqtR6D","pdf",3952328,5,1,44,"English","en",105,"# Abstract\n# Introduction\n# Nomenclature\n## Variables\n## Symbols and Subscripts\n# Keywords","[{\"question\":\"What problem does NUM address in data-driven turbulence modelling?\",\"answer\":\"Non-unique mapping (NUM) occurs when identical invariant inputs correspond to different target outputs, creating training and prediction errors. The study treats NUM as a significant source of inaccuracy.\"},{\"question\":\"How does the work quantify non-unique mapping for two-dimensional flows?\",\"answer\":\"It proposes a quantification method for NUM and demonstrates it on data from a periodic hills flow and an impinging jet. The analysis identifies where conflicting mappings occur in the flow field.\"},{\"question\":\"Why is viscosity ratio selected as a supplementary input variable, and what effect does it have?\",\"answer\":\"Viscosity ratio is added as a supplementary input variable to reduce NUM caused by two-dimensional flow data. This subsequently improves the accuracy of tensor-basis machine learning models for turbulence modelling.\"}]","Non-Unique Machine Learning Mapping in Data-Driven Reynolds Averaged Turbulence Models | PDF",1785901625,111,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"non-unique-machine-learning-mapping-in-data-driven-reynolds-averaged-turbulence-models","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/non-unique-machine-learning-mapping-in-data-driven-reynolds-averaged-turbulence-models/125858/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-08-23","2026-08-05",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What problem does NUM address in data-driven turbulence modelling?","Question",{"text":77,"@type":78},"Non-unique mapping (NUM) occurs when identical invariant inputs correspond to different target outputs, creating training and prediction errors. The study treats NUM as a significant source of inaccuracy.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does the work quantify non-unique mapping for two-dimensional flows?",{"text":82,"@type":78},"It proposes a quantification method for NUM and demonstrates it on data from a periodic hills flow and an impinging jet. The analysis identifies where conflicting mappings occur in the flow field.",{"name":84,"@type":75,"acceptedAnswer":85},"Why is viscosity ratio selected as a supplementary input variable, and what effect does it have?",{"text":86,"@type":78},"Viscosity ratio is added as a supplementary input variable to reduce NUM caused by two-dimensional flow data. This subsequently improves the accuracy of tensor-basis machine learning models for turbulence modelling.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":20,"slug":139},19,"General","general"]