[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-201786-105":59,"doc-detail-201786-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","algorithm-and-code-development-for-unsteady-three-dimensional-navier-stokes-equations-research-summary","Algorithm and Code Development for Unsteady Three-Dimensional Navier-Stokes Equations - Research Summary","","Research develops an improved numerical algorithm and computer code to solve the unsteady three-dimensional Euler and Navier-Stokes equations for aeroelastic applications. The work updates NASA Ames’s aeroelasticity code ENSAERO by adopting a streamwise upwind algorithm with central-differencing replaced where higher accuracy is required. The method is extended for moving grids using a finite-volume conservative framework and incorporates an implicit extension for computational efficiency. Validation uses unsteady test cases over oscillating wings.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/algorithm-and-code-development-for-unsteady-three-dimensional-navier-stokes-equations-research-summary/201786/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/algorithm-and-code-development-for-unsteady-three-dimensional-navier-stokes-equations-research-summary/201786.png","ImageObject",300,407,{"name":92,"@type":93},"Asher","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-10","2026-09-04",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",16,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What problem does the research address for aeroelastic computations?","Question",{"text":113,"@type":114},"It addresses the need for a sophisticated algorithm and computer code to simulate unsteady flow phenomena in three dimensions, enabling accurate prediction of unsteady pressure amplitude and phase for aeroelastic applications.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the new approach improve upon the previous ENSAERO central-differencing method?",{"text":118,"@type":114},"The research replaces central differencing with a streamwise upwind algorithm to produce more accurate solutions for the Euler and Navier-Stokes equations, with steady-state computations confirming higher resolution.",{"name":120,"@type":111,"acceptedAnswer":121},"How is the algorithm adapted to handle moving grids?",{"text":122,"@type":114},"The algorithm is extended using the finite-volume concept so it remains conservative for any motion of the coordinate system, enabling implementation 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\n(AA一15“03-)AL.  !i9M ANuC491-.503vLn  T门：VT.AJY THD:“-0I*LNSI0*ALNAVF:9一5     L0AT'NS annuol²eort,!  \nApr.,  -n三r.17/1 (“CAT Inst.Unc15170CSCL 2008B/4  00151  \nJune 1991  \nNCC2-605  \nMCAT Institute3933 Blue Gum DriveSan Jose,CA 95127  \nAlgorithm and Code Developmentfor Unsteady Three-DimensionalNavier-Stokes Equations  \nDr.Shigeru Obayashi  \nJune 1991  \nNCC2-605  \nMCAT Institute3933 Blue Gum DriveSan Jose,CA 95127  \nAlgorithm and Code Development forUnsteady Three-DimensionalNavier-Stokes Equations  \nShigeru Obayashi  \n## Introduction\n\nResearch in aeroelasticity with computational methods,including the nu-merical prediction of futter boundaries,requires a sophisticated algorithmand computer code to simulate unsteady flow phenomena in three dimen-sions.The purpose of this research is to develop an improved algorithm andcomputer code for solving the unsteady three-dimensional Euler and Navier-Stokes equations.These equations accurately describe the flow phenomenafor aeroelastic applications.  \nAn algorithm that uses central differencing has being used in the com-puter code,ENSAERO,for aeroelasticity at the NASA Ames Research Cen-ter.However,modern upwind algorithms can produce more accurate solu-tions for the Euler and Navier-Stokes equations.Among upwind algorithms,a streamwise upwind algorithm has recently been developed by Obayashiand Goorjian(see Appendix A).Most multidimensional upwind algorithmsare frst constructed in one dimension and then extended to multiple di-mensions by applying the one-dimensional procedure in each coordinate di-rection.On the other hand,the present method follows the flow physicsmore closely than the coordinate upwind methods.The steady-state com-putations confirmed the higher resolution of the present algorithm over thecentral-difference method as well as over other upwind methods.  \nThe objective of this research is to update Ames's aeroelasticity code,ENSAERO,by using the improved streamwise upwind algorithm.This re-port summarizes briefly the work performed during the period,April 1,1989  \nthrough March 30,1991.Additional details on the various aspects of thestudy are given in Appendices.  \n## Research Efforts\n\nThe following specific objectives have been performed:  \n·The algorithm was extended to handle the moving grid system.Thefinite-volume concept is essential to extend the algorithm.The result-ing algorithm is conservative for any motion of the coordinate system.  \n·Two extensions to an implicit method were considered.The implicitextension that makes the algorithm computationally efficient is imple-mented into ENSAERO.  \n● The new fow solver has been validated through the solution of testproblems.The test cases include three-dimensional problems with fixedand moving grids.  \n·Finally,the new fow solver has been implemented into NASA Ames'saeroelasticity code,ENSAERO.  \nResults  \nIn this work,only the first-order time-accurate methods are considered be-cause of computational efficiency.However,time accuracy is an essentialrequirement for aeroelastic computations.Numerical schemes used for flowcalculations in aeroelasticity must guarantee the correct calculation of am-plitude and phase of unsteady pressures.In order to verify the time accuracyof the present code,unsteady flows over various wings undergoing prescribedoscillatory and ramp motions have been computed.  \nF-5 Wing  \nThe first test case shown here is an unsteady viscous flow over the F-5 wingwhich has an aspect ratio of 2.98,a taper ratio of 0.31 and a leading edgesweep angle of 31.92°.The test case is chosen at M=0.9,Re=9×10⁶  \nand the angle of attack of 0°.Since the airfoil section is supercritical,thefow is shock-free on the most of the wing surface at this fow condition.Inthe unsteady case,the wing is pitching with a pitch amplitude of 0.11°ata reduced frequency of 0.55 as illustrated in Fig.1.The experiment","cbCaikMF7xvZRd1Y","https://ap.wps.com/l/cbCaikMF7xvZRd1Y","pdf",2839174,"English","# Introduction\n# Research Efforts\n# Results\n## F-5 Wing\n## Clipped Delta Wing","[{\"question\":\"What problem does the research address for aeroelastic computations?\",\"answer\":\"It addresses the need for a sophisticated algorithm and computer code to simulate unsteady flow phenomena in three dimensions, enabling accurate prediction of unsteady pressure amplitude and phase for aeroelastic applications.\"},{\"question\":\"How does the new approach improve upon the previous ENSAERO central-differencing method?\",\"answer\":\"The research replaces central differencing with a streamwise upwind algorithm to produce more accurate solutions for the Euler and Navier-Stokes equations, with steady-state computations confirming higher resolution.\"},{\"question\":\"How is the algorithm adapted to handle moving grids?\",\"answer\":\"The algorithm is extended using the finite-volume concept so it remains conservative for any motion of the coordinate system, enabling implementation within the aeroelasticity code ENSAERO.\"}]","Algorithm and Code Development for Unsteady Three-Dimensional Navier-Stokes Equations - Research Summary | PDF",202]