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A Best Practices Report on CFD Education in the Undergraduate Curriculum

JeffD.Eldredge1,InancSenocak2,PaulDawson3,JamesCanino4,WilliamW.Liou5,RayLeBeau6,DarrenL.Hitt7,MarkusP.Rumpfkeil8,andRussellM.Cummings9

 

SummaryThis report, written by a working group of the AIAA Fluid Dynamics TechnicalCommittee, is intended toguide thedevelopmentofcomputational fluiddynamics(CFD) instructional content in undergraduate aerospace and mechanicalengineeringcurricula. Thereportaddressesagrowingneedfornewengineerstobecome‘intelligentusers’ofCFD:thatis,tobeabletoobtainasolutionofaflow,andtocriticallyassessthequalityoftheresult. ThereportdistillstheconceptsofCFDinto curricular elements, and establishes reasonable expected outcomes forundergraduate‐levelinstructionoftheseconcepts.ItthenprovidesnumerouscasestudiesofexistingCFDcourses,presentedinahierarchyofvarious‘profiles’—fromCFDlighttoCFDheavy—forinclusionincourseswithlecture,laboratoryordesignformats. Specific needs of mechanical engineering programs are also discussed.Hardware, software and textbook resources are briefly reviewed.

1 AssociateProfessor,Mechanical&AerospaceEngineering,UniversityofCalifornia,LosAngeles2 AssistantProfessor,MechanicalandBiomedicalEngineering,BoiseStateUniversity3 Professor,MechanicalandBiomedicalEngineering,BoiseStateUniversity4 AssistantProfessor,MechanicalandAerospaceEngineering,TrineUniversity5 Professor,MechanicalandAeronauticalEngineering,WesternMichiganUniversity6 AssistantProfessor,AerospaceandMechanicalEngineering,ParksCollege,SaintLouisUniversity7 AssociateProfessorofMechanicalEngineering&ProgramHead,SchoolofEngineering,UniversityofVermont,Burlington,VT.8 AssistantProfessorofMechanicalandAerospaceEngineering,UniversityofDayton,OH.9 Professor,DepartmentofAeronautics,U.S.AirForceAcademy

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I. INTRODUCTION AND MOTIVATION  3

II. CLASSIFICATION OF CORE CFD CONCEPTS  5

III. STRATEGIES FOR INCORPORATING CFD INTO UNDERGRADUATE CURRICULUM  12

A. CFDlight 13

B. CFDmoderate 13

C. CFDheavy 17

D. CFDinLaboratoryCourses 25

E. CFDinDesignCourses 26

IV. CFD‐RELATED CONCEPTS IN MECHANICAL ENGINEERING  29

V. CFD INSTRUCTIONAL RESOURCES  31

A. Hardwareresources 31

B. Open‐sourceversuscommercialsoftware 32

C. Textbooks 34

VI. CONCLUSIONS  36

VII. REFERENCES  38 

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I. Introduction and Motivation

Computationalfluiddynamics,orCFD,hasmaturedsignificantlyinthepasttwentyyears,tothepointthatitisusedascommonlyas—orinlieuof—physicaltestingin aerospace andmechanical engineering. This usepermeates all aspects of real‐worldengineering:researchanddevelopment,productdesign,processengineering,andtestcorrelation.TheuseofCFDinthedesignprocessisgrowingmoreprevalenteachyear.TheuseofCFDintheproductdesignsequenceiswellillustratedbythetestimony of Michael Garrett of Boeing before the U.S. Senate Committee onCommerce, Science, and Transportation in 2006. Mr. Garrett stated, “In 1980,Boeing tested 77wings inwind tunnels to arrive at the final configuration of the767.Just25yearslater,webuiltandtested11wingsforthe787—areductionofover 80%.”[1] In a recent survey of 40 employers of aerospace engineeringgraduateswhose organizations regularly use CFD— that is, stakeholders in fluiddynamicseducation—alloftheseusesofCFDwerewellrepresented.Mostofthesestakeholders were members of industry, with the remaining group made up ofacademicresearchersandsomemembersofgovernmentlabs.

However,undergraduateinstructioninfluiddynamicsisstillbasedsubstantiallyona traditional chalkboard‐styleparadigm, coupledwith laboratoryexperiences thatfocus almost entirelyonphysical experiments. Computational exercises areoftenlimited to plotting of analytical expressions or numerical solution of an ordinarydifferentialequation(forexample,theBlasiusboundarylayersolution).Thus,thereisanapparentlackofparityinthebalanceofexperiment,theoryandcomputationbetweentheundergraduatecurriculumandthepost‐graduateworld.

OnecouldcontendthatCFDismerelyatool,anddoesnotbelonginthecurriculumanymorethananexperimentaldiagnostictool.Butwhenintroducedthoughtfully,CFDhasthepotentialtoilluminatefoundationalconceptsinfluiddynamicsinwaysthataresimplyimpossibleusingtraditionalpedagogicalapproaches. Forexample,in a typical presentation of boundary layer theory, the instructor would usuallyproceed to adverse pressure gradients and on to flow separation. But this lastconcept, although critically important in aerodynamics, lacks a general predictivetheorythatisadequateforpedagogy,sotheinstructortypicallypresentsThwaites’methodasasemi‐empiricalsubstitute.ButthereislittleevidenceofthecontinueduseofThwaites’methodinpractice,andaCFDsolutionofalaminarseparatedflowcanshedmuchmorelightonthisconcept.

Theseobservationsserveasagenuineimpetustorevisit,withfreshperspective,thefluiddynamicscurriculuminundergraduateaerospaceandmechanicalengineeringprograms. To be clear, the education mission itself has not changed. It is asimportantas ever foranaerospaceormechanical engineeringgraduate tohaveafirmgraspof fundamental fluiddynamics concepts, and so thebasic constraint inanycurriculumreformmustbe“First,donoharm”.However,thenatureofhowthisfundamentalknowledgeisappliedinpost‐graduateworkhaschanged:apracticingengineer is now much more likely to need to assess if an attractive‐looking

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streamline plot generated by a commercial CFD code of flow around a wing isphysically sensible. This requires coupling one’s core understanding of fluiddynamics with a working knowledge of the computational methodology itself, atleasttotheextentnecessarytounderstandhowtheCFDsolutionmaystray.Ideally,amodernengineeringeducationshouldprovidestudentswiththetoolstocomputeaCFDsolutionofaseparatedflow,andthenmakeaquickcheckthattheseparationpointisconsistentwithanappropriatecorrelation.

Perhaps the best guide in developing a modern computationally orientedcurriculum can be found by addressing the question, “What do students need toknow about CFD to pursue careers in engineering?” The aforementioned surveyasked respondents to rate the importance of various CFD‐related skills in theiremployees,onascaleof0to5,where5meantveryimportant.Unsurprisingly,thehighest‐rated skill was “Understanding of flow phenomena” (4.6). This skill wasfollowed by “Post processing” (4.1) and “Verification & Validation” (4.1), then by“Gridgeneration”(3.7),“Compressibleflowmethodology”(3.4),andseveralothersratedstill lower. (Interestingly, “Programming”wasonly ratedas3.1.) Thus, thehighest premiumwas placed on setting up a problem correctly and assessing thesuitabilityoftheachievedsolution.

In assessing the extent towhich their new employeeswere preparedwith theseskills, theresponseswereasfollows(againbetween0and5,with5meaningveryprepared): “Understanding of flow phenomena” (2.6), “Post processing” (2.5),“Verification & Validation” (1.9), and “Grid generation” (2.0), and “Compressibleflowmethodology” (1.9). Thus, the perception of preparednesswas quite low ineveryskill,butparticularlyinverificationofsolutions.

Clearly,muchofthisworkingknowledgeofCFDcanbe(andcurrentlyis)gainedbyon‐the‐job training. Some of the respondents of the survey commented on thisexplicitly, stating that they do not expect their new employees with Bachelor’sdegrees to have familiarity with CFD, except to be able to assess whether their“converged”solution is really thesolutionof theproblemtheyare trying tosolve.Butnevertheless,forsuchavaluableandpervasivetoolasCFD,itwouldbeprudenttogivestudents their firstexperience in theclassroomor instructional lab,wherethefocusisoneducationratherthantraining.

This report has been assembled by aworking group of the AIAA Fluid DynamicsTechnicalCommittee[2]. It is intended to illustrate, throughexamplesandexistingcase studies, how CFD can be thoughtfully incorporated into the undergraduatecurriculum. Weseek toaddress theoverarchingquestionsofwhat constitutesan‘intelligent user’ of CFD and how the curriculum can be designed to prepare astudenttobecomesuchauser,whilepreserving(orenhancing)thecoremissionofanundergraduateprogram:toprovideinstructioninfundamentalflowphysics.Aswewillshow,thecurriculumcanincorporateCFDtovariousdegreesandinvariousforms, providing comparabledegreesof preparation inCFDapplication. Wenotethat, although this reporthasbeenwrittenunder thebannerofAIAA, it hasbeen

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writtenwithboth aerospace andmechanical engineering curricula inmind, andaseparatesectionexplicitlyaddressesthedistinctneedsofthelatterdiscipline.

Inthenextsection,wewillpresentaclassificationofcoresubjectsrelatedtoCFD,withtheobjectiveofestablishingahierarchyofthesesubjectsfromwhichtodrawinconstructingcoursesormoduleswithincourses.Inthefollowingsection,wewillpresent a progression of different approaches to introduce CFD content into anaerospace (or mechanical) engineering curriculum, with existing examples thathighlightsomelessonslearned.Wethenbrieflydiscusssometopicsthataremoretypicallyalignedwithmechanicalengineeringinstruction,butwhichareundertheumbrellaofengineeringCFDandthereforewarrantinclusion.Thereportconcludeswith a discussion on the resources for CFD instruction, including hardware,softwareandtextbooks.

Finally,wenotethatthedevelopmentofthisreportfollowsaspecialsessiononCFDin Undergraduate Education, held at the AIAA Aerospace Sciences Meeting inNashville in January 2012, in which several notable examples of courses andcurriculumreformonthesubjectofCFDwerepresented. Thepresentations fromthis meeting can be found in reference [2]. In addition, several articles on thesubject are slated to appear in upcoming issues of the International Journal ofAerodynamics.

II. Classification of Core CFD Concepts In this section,ourobjective is to itemize thecore toolsandconceptsofCFD,andassesstheirplacementinanundergraduatecurriculum.Weareparticularlyfocusedhere on prioritization of the topics, and in particular, in establishing the baselineknowledge of an intelligent user of CFD. Our objective is to distill each of thesetopicstoasufficientdegreethattheycanbeorganized,inthefollowingsection,intovariouslevelsofCFDcurriculum.

ComputerProgrammingBeyond introducing thesyntaxofaprogramming languageof choice (e.g.Fortran,C++, Java,Matlab,orPythonetc.),computerprogramming,most importantly, isanessential skill to develop computational thinking in students. Computerprogrammingalso formally introducesstudents to thecomputerhardwareand itsoperation. One could argue that programming is not an essential skill for CFDeducation on the basis that commercial CFD packages have graphical userinterfaces, but almost all commercial software provides an interface for users totailorthesoftwarefortheirapplication(e.g.userdefinedfunctions(UDF)inANSYSFluent). A practicing engineer may also need to write computer programs tomanipulate experimental and/or computational data. Additionally, someundergraduatestudentscontinuetheirstudiesatthegraduatelevelwheretheymayneedtodevelopcomputationaltoolsintheirresearch.Forthesereasons,computerprogramming is a fundamental skill in aerospace or mechanical engineeringeducation.

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Pre‐processingandgridgenerationThis constitutes the first, and arguably, the most important step in any CFDapplication. It isalsotheoneforwhichstudentslikelyhavetheleastappreciationforthechallengesinvolved,andwouldtendtobreezepastwithoutproperguidance.The interlinked choices for the computational domain, governing equations,boundary and initial conditions, specification of physical parameters, desiredaccuracy and solution time, and solution parameters of interest, all requirecompromises that students seldom appreciatewhen first introduced to the topic.Furthermore, the preparation of the geometry and the grid generation itself aresubtle topics that require an intuition for the expected solution, as well as thecapabilitiesandlimitationsoftheunderlyingnumericalmethodology.Studentsareoftensurprisedtofindthatthereisnoimmediate“Solve”buttontopressoncetheyhaveimportedageometryintoacommercialCFDtool.

Thus, because of its inevitable and challenging role in any CFD project, this topicposesthegreatestchallengeforinstruction.ThisisparticularlytrueinacontextinwhichCFDisdesiredasatoolforachievingacertainobjective—suchasinadesignclass or project team— without providing a foundational experience in a priorcourse. In any context, the topic is best introduced by patientlywalking throughseveralexamplesbeforeleavingthestudentstomakethechoicesthemselves.Itisimportantthatstudentslearntocarefullyobservetheconsequencesoftheirchoicesatthisstage,andtosystematicallydeterminehowtomakeappropriatechoices.

ValidationandverificationVerification and validation (V&V) are the primary means to assess accuracy andreliability of computational simulations, a task that is absolutely mandatory inscience and engineering. Indeed, aptitude for this particular step is likely theclearestindicatorofanintelligentuserofCFD.InlearningV&V,astudentlearnstoavoidblindfaithinoutputfromaCFDcode,butrather,toinspectCFDresultswithacritical eye. Much of the discussion in this subsection is based on the excellentSandia report “Verification and Validation in Computational Fluid Dynamics”[3],which the interested reader is highly encouraged to read for more in depthinformation. There is also a long‐standing AIAA Committee on Standards forComputationalFluidDynamicsthathasthoughtfullylaidoutguidelinesforV&V[4].

Verificationandvalidationareoftenconfused,butitisimportanttodistinguishtheirroles. The fundamental strategyofverification is the identification,quantificationand reductionoferrors in the computationalmodel and its solution (onecanask,“Am I solving theequationscorrectly?”). The fundamentalobjectiveofvalidation,on theotherhand, is toassesshowaccurately thecomputational results comparewith the experimental data (one can ask, “Am I solving the right equations?”).Stateddifferently,verificationisprimarilyamathematicsissue,whereasvalidationisprimarilyaphysicsissue.

There is a natural but unfortunate inclination to give these tasks very shorttreatment,bothinpracticeaswellasinCFDinstruction.Particularlyintheuseofa

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commercialcode,verificationistoooftenentirelyneglected(“thesoftwarepackageI amusing iswell verified, thus Idonothave todo it”) and the typical validationprocedure inCFD(andother fields)usuallyonly involvesgraphicalcomparisonofcomputational results and experimental data. If the computational results“generallyagree”withtheexperimentaldata,thecomputationalresultsaredeclared“validated”. However,with a graphical comparison, one does not commonly seequantification of the numerical error or quantification of computationaluncertainties due to missing initial conditions, boundary conditions, or modelingparameters. Also,anestimateofexperimentaluncertainty isnot typicallyquoted,andinmostcasesitisnotevenavailable.

Because of the time and patience required, the road to instill V&V standards intoCFDusers is longanddifficult. However, it isessential to instillacommitment tothispractice,particularlyasrelianceonCFD(andother)softwarewillonlyincreasein the future. Thus, V&V forms an essential component of any level of CFDinstruction,andshouldbepresentedasinseparablefromtheotherCFDtasks.

Instructionofverificationassessmentshouldfocusontheuseofbenchmarks.Itiscritical for students to learn to distinguish between exact and approximatesolutions, and to recognize that even a highly accurate numerical solution can beregardedasastandardagainstwhichtomeasure.Studentsshouldbecomefamiliarwith some important examples of such benchmarks. Verification necessarilyinvokesadiscussiononmajorsourcesoferror: insufficientconvergenceofspatialand temporal discretization, insufficient convergence of iterative procedures,computer round‐off, andcomputerprogrammingerrors. Thequestions thatmustbecomesecondnaturetostudentsare:“Doesthediscretesolutionconvergetotheexact solutionas thegrid is refined?”and “What is thediscretizationerror that isactually observed for real calculations on finite grids?” In addressing thesequestions, studentsshouldbecomeadeptatcriticallyassessing thepre‐processingsteps,notablythegridconstruction,andtheireffectonthesourcesoferror. TheyshouldalsolearntousestandarderrormetricsandtechniquessuchasRichardsonextrapolation.

Validationassessment,withitsfocusonthesuitabilityoftheunderlyingmodelforthe physics, necessarily requires comparison between sufficiently accuratecomputational results and high‐quality experimental data. The latter are to beunderstood as a more faithful reflection of reality. Consequently, in this task,studentsmustlearntobecognizantofandestimatetheuncertaintyofexperimentaldata, and to faithfully represent the experimental setup in the numerical model.Because of the infeasibility and impracticality of conducting true validationexperimentsforessentiallyallcomplexengineeringsystems,theusualmethodistouse a building‐block approach, wherein the complex system of interest isprogressivelydividedintosimplersubsystemsthatareallvalidatedseparately.

In short, V&V constitutes an essential practice for even casual users of CFD. ItshouldthereforeberegardedasessentialcontentinanylevelofCFDinstruction.

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NumericalmethodologiesforCFDThisisclearlyabroadtopicwithawiderangeofsubtopics,andthisbreadthaloneposesachallenge for fitting itadequately intoanundergraduate‐levelCFDcourse.Furthermore,manyofthetopicsrequireafullcadreofmathematicalprerequisites—linearalgebra,andordinaryandpartialdifferentialequations—someofwhichmaynot be adequately treated in the core engineering curriculum. This requiresonetoberealisticaboutwhatisandisnotpracticalindesigninganundergraduateCFDcourse.Alistoftopicsisasfollows:

ClassificationandcharacteristicsofmodelPDEs Finitedifferenceapproximations

o Basicschemeso Truncationerrorandorderofaccuracyo Schemeconstructiono Fouriererroranalysis

Finitevolumeapproximationso Cell‐andface‐centeredquantitieso First‐andsecond‐orderschemes

SolutioncharacteristicsofsystemsoflinearODEs Timemarchingmethods

o CanonicalmethodsandapplicationtoODEsystemso ApplicationtomodelPDEso Erroranalysiso Stabilityanalysiso Treatmentofnumericalstiffness

Solutionoflinearsystemsofequationso Directmethodso Relaxationmethods

Methodsforcompressibleflowo Centralschemeso Fluxsplittingandupwindingo Artificialdissipationandfluxlimiterso Numericalandphysicalboundaryconditions

Methodsforincompressibleflowo Collocatedandstaggeredgridso Pressure‐correctionmethods

Inordertoestablishanappropriatelevelofpedagogicaltreatmentofthistopicinacertain course, it is perhaps most effective to establish the overall high‐levelobjectiveandthenworkbackwardtodistillthenecessaryprerequisitematerial.Webelieve,aswestressthroughoutthisreport, thattheminimalobjectivemustbetodevelopastudent intoan intelligentuserofCFD. Thismeansthatastudentmustlearn enough about numerical methodologies to be able to make a meaningfulassessmentof, forexample, theexpected levelofnumericaldissipation inaresult,thereasonsforacertainsolutiontofailtoconverge,orthenumericaloriginof(and

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anappropriatecurefor)spuriousnoise.Theseneedsaccompanymostofthehigh‐leveltopicsdescribedinthissection,i.e.pre‐processingandgridgeneration,aswellasverificationandvalidation.Inparticular,solutionverificationreliesonhavinganadequateunderstandingof themajorsourcesoferror innumericalapproximationofthegoverningequations,andtherefore,basicknowledgeofspatialandtemporaldiscretizationschemesandtheirassociatedtruncationerror.

Acompacttreatmentofnumericalmethodologies—say,withinadesigncoursethatlacks a CFD prerequisite — can often be facilitated by judiciously chosendocumentationandtutorialsassociatedwithcommercialsoftware.Forexample,ifthe objective is to provide students with minimal understanding of themethodologies required to obtain a subsonic steady‐state solution around acomplex geometry with a commercial CFD code, then it might be appropriate todesign the syllabus by referencing the code’s underlying low‐speed flowmethodologyandchoosingappropriatetopicsfromthelistabove.

Turbulencemodeling,RANSandlarge‐eddysimulationTurbulence modeling poses a difficult paradox in undergraduate instruction.Reynolds‐averagedNavier‐Stokes(RANS)solvershaveanearlyuniversalpresencein industry, and nearly all engineering‐oriented applications of CFD invoketurbulencemodels intheirsolutions. However,oneisgenerallychallengedtofindenough time to adequately treat this subject in an undergraduate course, givingpreferenceinsteadtothemoreelementarytopicsonthislist.Thus,itiseasytofindCFDuserswhoaresimplyignorantoftheconsequencesofchoosingbetweenRANS,large‐eddy simulation (LES) and hybrid RANS‐LES, or a k‐epsilon versus Spalart–Allmarasmodel.Anditisnotsurprisingtoseestudentsexpectingachaotic,randomlookingsolutionfroma“turbulencemodel”.

Itisnaturaltoclassifythistopicastooadvancedforundergraduateinstructionandsimplyomititfromthecurriculum.Butasmentionedabove,mostengineeringCFDsimulationsrequireaturbulencemodelinthesolution.ItisalsoreasonabletoviewthetopicofturbulencemodelingasanopportunitytoteachthetopicsofturbulenceandhighReynoldsnumberflowphysics,whichgenerallyreceivecursorytreatmentatbestinanundergraduate‐levelfluiddynamicscourse.Oneostensiblereasonforthiscursorytreatmentisthatitisdifficulttovisualizesomeoftheabstractconcepts—statisticalversusinstantaneousorpoint‐wiseproperties,eddyviscosity,etc.Webelieve, if taughtwith reference to a CFD code and its implementation of variousturbulencemodels, that thissubjectcanbe illuminatedmoreeffectively. Studentsshould be aware of the fact that turbulence remains one of the great unsolvedproblemsofclassicalphysics,andasengineerswedevelopmodelsforturbulenceinordertomoveforwardasproblem‐solvers.

Thus,weofferherealistofexpectedoutcomesforundergraduate‐levelinstructioninturbulencemodeling:

1. Familiarity with time averaging of the governing equations, associatedturbulentstresses,andthe“turbulenceclosure”problem.

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2. Recognition of some basic differences between and appropriate uses ofcommonlyusedturbulencemodelsandwallfunctions.

3. Familiaritywiththelawofthewallandnon‐dimensionalwallunits,andtheiruse in setting appropriate grid spacing in conjunction with turbulencemodelsandwallfunctions.

4. AwarenessofcommontroublespotsforRANSsimulations.

It should be emphasized that none of these outcomes requires much, if any,prerequisitematerialfromtheotherCFDconceptsdescribedearlierinthissection,asidefromabasictreatmentofspatialdiscretization.Indeed,theseoutcomescouldbenaturally incorporatedintotheexistingoutcomesofan intermediate‐level fluiddynamicsclass,intheCFDLightorModerateprofilesdescribedinthenextsection.For example, sample results of wall stress from resolved and under‐resolvedturbulent channel or pipe flow simulations could be used to illuminate why it isimportanttorespecttheunderlyingassumptionsinturbulencemodeling.

Large‐, detached‐eddy simulation (DES), hybrid RANS‐LES approaches are moreadvancedthanRANS,andarelikelyonlyfeasibleforinclusioninaseniorlevelCFDcourse.Asetofmoreadvancedexpectedoutcomesassociatedwiththeseconceptsis,forexample,

1. Familiaritywithfiltering(versustimeaveraging)ofthegoverningequations,andstatisticalpropertiesofaflowfield.

2. Understanding of the objectives, requirements and expectations of LESversusthoseofRANS(anddirectnumericalsimulation,DNS).

3. Ability to approximate integral and Kolmogorov length scales, and to usethemtodetermineappropriateLESgridspacing.

4. Familiaritywiththecharacteristicsofbasicsub‐grid‐scalemodels.

5. FamiliaritywiththebasicstrategyofhybridRANS‐LESandDES.

Perhaps, themost importantmessage to ingrain in a student is that adopting theLES approach does not readily mean better results than the RANS approach. Inparticular,arandomandchaoticlookingflowfielddoesnotnecessarilymeanagoodsimulation, and that onemust be aware of statisticalmeasures to gauge success.Because of their relatively higher expense, large‐eddy simulations are likelyimpracticalforcourseprojects.Instead,onemightrelyoninterrogatingadatabaseof “good” and “bad” pre‐computed results in order to investigate turbulencestatisticsanddevelopawarenessofbestpractices.

Finally,itisusefultonotethattheTurbulenceModelBenchmarkingWorkingGroupoftheAIAAFluidDynamicsTechnicalCommitteehas, forseveralyears,workedtoassemble and report an impressive number of benchmark solutions on which toverifyandvalidate turbulencemodels (both forRANSandLES). Thesehavebeen

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welldocumentedforgeneraluse,[5]andtheproblemshaveasignificantamountofeducationalvalue.

PanelandvortexlatticemethodsThesenumericaltoolshavebeenapartofengineeringanalysisandeducationlongerthan CFD, and remain valuable tools in both contexts. In an aerodynamicscurriculum, they provide a studentwith an opportunity to solve for the essentialfeatures of a meaningful aerodynamics problem, with minimal computationalresources.Theintroductionofthemethodsfollowsnaturallyfromthepresentationofclassicalinviscidflowsolutions,andadiscussionofthecouplingoftheseinviscidsolverswithsolutions fromboundary layer theory isphysicallyenlightening. Thenumberofchoicesrequiredforgettingareasonablesolutionwithanexistingopen‐sourcecodeisrelativelysmallcomparedwithafullCFDapplication.Thus,therearefewobstaclestogettingstudentstothepointwheretheycanuseittocompleteclassassignmentsordesignprojects.Thedevelopmentofarudimentarypanelorvortexlatticesolverfromscratchalsoconstitutesavaluablecomputingproject.

Thedisadvantageisthatthetoolshavelimitedapplicability,andthereforecannotbeused to directly illustrate phenomena such as flow separation and transition toturbulence (though they canbeused to present the origins of thesephenomena).For the same reasons, their use in industry is also in decline. Nonetheless, theyremainessentialpedagogicaltools.

SoftwareengineeringThis topic is generic to scientific computing, andmore commonly associatedwithcomputer science than aerospace or mechanical engineering. However, itnonetheless warrants consideration in this discussion because of the inevitableimportantrolethattechnicalcomputingplays(andwillcontinuetoplayevenmore)in all engineering disciplines. Software engineering in this context includes:learning to work in a Unix‐like environment, code compilation, revision control,debugging,codeperformanceprofiling,documentation,datavisualization,anduseofthird‐partylibraries.Italsoincludesdiscussionondatastructuresandlanguageabstractionrelevantto,e.g.,gridgeneration,linearsolvers,etc.

Dr Phil Colella recently organized a course on this topic at UC Berkeley thataddressed the needs of graduate students and advanced undergraduate studentsacross a wide swath of disciplines — from computer science to mechanicalengineeringtobusiness.(SeeColella’spresentationintherecentspecialsessionatreference [2].) What was remarkable for the course was the minimal set ofprerequisites for the course,necessary forattractingabroadenrollment. Indeed,theonlyessentialprerequisitesareprogramming,andpossibly,linearalgebra.Suchacourseisnecessarilyproject‐oriented,andworksmosteffectivelywhenstudentsareabletodefinetheirprojects.Thus,itwouldbeusefulifcoordinatedwithsomeapplication‐orientedprojectclass.

Computerarchitectureandparallelcomputing

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CFDsimulationscanbedemandingandexpensive intermsofcomputinganddatastorage requirements. A majority of industrial fluids engineering applicationsinvolvehighReynoldsnumberturbulentflowsinacomplexgeometrysetting.High‐fidelityCFDsimulationofsuchengineeringproblemstypicallyrequiresfinemeshes.Equallyimportant,itisnotuncommonthattime‐dependentcomputations(e.g.,LESorURANS)areusedinthesetypesofsimulations. IntermsofundergraduateCFDeducation,whereCFDisintroducedinanacceleratedfashionwithtwodimensionalfundamental laminar flow calculations that finish rather quickly on a standarddesktop computer, students can develop a misconception of the computationalrequirements for CFD simulations in general. Therefore, an intelligent CFD usershouldbeabletoestimatethespatialandtemporalresolutionneededtoproduceanacceptable simulation, and be able to identify and distinguish the availablecomputing resources. In other words, an intelligent CFD user should be able toestimate computational turnaround time, memory and file storage requirementsbeforeadoptingorsuggestingCFDanalysisinsolvinganengineeringproblem.Thisskill requires a basic understanding of hardware and software for parallelcomputing.

Overthelastdecade,computerhardwareandsoftwarehasradicallychangedtothepoint where workstations with 64 cores of central processing units (CPU)augmented with multiple graphics processing units (GPU) as accelerators arebecoming common in the engineering workplace. Thanks to industry‐universitypartnerships, high‐performance computing (HPC) platforms aremore available toengineers than before. Most commercial and open‐source CFD models can takeadvantageofparallelcomputing.Equallyimportant,parallelvisualizationsoftware(e.g., Paraview and VisIT) are is freely available for rendering large data sets. Alargemajorityofopen‐sourcesoftwareisbasedontheLinuxoperatingsystemandadoptsMPIforparallelprogramming.NewsimulationsoftwareisbeingdevelopedtotakeadvantageofGPUstoacceleratesimulation.Thisnewsoftwareisbasedonnew languagessuchasCUDAorOpenCLandoften interleavedwithotherparallelprogramming languages (e.g., MPI, OpenMP, Pthreads) to afford large‐scalecomputations onmultipleGPUs. Therefore, undergraduate CFD education shouldconvey the concept of “large‐scale computations” and inform students aboutsoftwareandhardwareplatformsusedinHPC.Studentsshouldbeableto identifyshared memory and distributed memory systems and parallel programmingenvironments that are needed to perform large‐scale computations and ways toprocess large data sets that are too large for a workstation for visualization.Currently,studentsareexposedtotheseskillsetsatthegraduatelevel,butthanksto the open‐source movement, students can actually acquire these skills at theundergraduatelevelinCFDrelatedcourses.

III. Strategies for Incorporating CFD into Undergraduate Curriculum

In the previous section, an effort was made to present the list of CFD‐relatedconcepts and tools hierarchically. This effort was in anticipation of the current

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section, inwhichahierarchyofCFDcurriculumprofiles—from lowtohigh— isdescribed.Inthisfashion,themodularCFDelementscannaturallybeidentifiedforinclusioninanappropriatesetting. Forexample,inthelowestprofilesetting,pre‐computed CFD solutions are to be used to illuminate fundamental fluid dynamicphenomena,andthereforeonlysomeelementsoftheconcepthierarchyareinvoked(inthiscase,post‐processing).

The objective in this section is to present a number of examples, drawingwherepossibleonexistingcurricula. Thisallowsustopresenttheaccompanyingsyllabi,some brief descriptions of sample homework problems and projects, and lessonslearnedonthedevelopmentandcontinuingevolutionoftheseexistingcourses.

A. CFD light ThisrepresentsthelowestprofileinclusionofCFDinanundergraduatecurriculum.In such a case, an instructor of an introductory or intermediate fluid mechanicscourse uses pre‐computed CFD simulations in order to reinforce fundamentalconcepts. With the availability of free high‐quality visualization software (e.g.,ParaviewandVisIT),thisapproachisrelativelystraightforwardtointroduceintoanexistingcurriculum,providedarepositoryofsolutionshasalreadybeengenerated.Studentswouldbeable todownload theCFDsolution filesanduse themtostudyfluid flow inandaroundvariousgeometries. Thisapproachwouldhelpeliminatesoftware cost, and also introducemore realistic engineering simulations into theundergraduatecurriculumwhileavoidinglongsimulationtimesandresourcesandinstruction needed to conduct such simulations. In addition to in‐classdemonstrationsusingtheCFDsolutions,homeworkproblemscanbedevelopedto:estimatethenondimensionalparametersinpre‐computedsimulations,analyzeflowfields and structures to identify a “good” CFD simulation, distinguish inviscid vs.viscoussimulations,andverifytheprinciplesofmassandmomentumconservation,tonameafew.

B. CFD moderate AtthislevelofCFDinstruction,CFDmodulesareusedtoaugmentcoursematerial,by substituting the module into an existing lecture‐form fluid mechanics oraerodynamics course, or into the accompanying lab unit of the course. The CFDcontent is heavier than in the CFD light scenario, in that here, the students areexpected tocompute thesolutionsbeforeanalyzingthem. As theexamplesbelowwillsuggest,thisapproachoftenmakesuseoffreeorin‐houseinviscidaerodynamicsolvers (i.e.panelandvortex latticecodes) inorder tocompute flowsandpredictforcesonliftingsurfaces.

Example1:UniversityofDayton,MechanicalandAerospaceEngineering

Coursename:FundamentalsofAerodynamics Lectureformat,seniorundergraduate(required)/graduatelevel Prerequisites:Programming(Matlab),PartialDifferentialEquations,

IntroductoryFluidMechanics

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ThiscourseassignsseveralCFD‐relatedhomeworkassignments:

1. Students are given a source panel code written in Matlab using complexarithmetic.Theyareaskedtorunthecodefordifferentnumbersofpanelsfora non‐lifting cylinder case and compare the results to the analytic solutionderived in class. In addition, theyhave to comment each lineof the sourcecode(abouttwenty).Finally,theyalsohavetomodifythecodetobeabletorunanon‐liftingellipsecase.

Learning outcomes: Realizing that a source panel code with a sufficientnumber of panels can approximate a known analytic solution, and thenfurtherlearningthatthecode’srealpowerisinanalyzingmorecomplicatedgeometries.Bycommentingthesourcecodestudentsdemonstratethattheyunderstandhowtheunderlyingmathematicalequationsforthesourcepanelapproachare implemented;bychanging thecylinder toanellipse,studentslearnthatitisrelativelyeasytochangegeometries.

2. Students are asked to modify the source panel code from the previousproblemsuchthattheyobtainavortexpanelcode.Theythenrunthevortexpanelcodeforaliftingcylindercasewithgiventotalcirculation.

Learning outcomes: Realizing that source and vortex panel codes are verycloselyrelated;learningtoresolvetheoverconstrainednessoftheproblem.

3. StudentsareaskedtowriteasolverforthePrandtlliftinglineequationusingfour terms in theexpansion toobtain the liftdistributionofa tapered, thinwing with no aerodynamic or geometric twist and symmetric airfoil crosssections at a certain angle of attack. They then have to compute the liftcoefficient,thespanwiseefficiencyfactor,andtheinduceddragcoefficient.

Learning outcomes: Appreciating that the Prandtl lifting line equation isapplicable toverygeneralwingdesigns; realizing that theyhave topickanappropriatenumberof span‐wise locations togenerate sufficientequationstoobtainasolvablelinearsystem.

Example2:SaintLouisUniversity,AerospaceandMechanicalEngineering

Coursename:(Multiplecourses)

Projectsandassignmentsrelatedtocomputationalfluiddynamicsareinterspersedthrough several lecture courses, as part of an overall effort by the department toencouragetheuseofcomputersandcomputationaltechniquesintheundergraduatecurriculum.TheprimarycoursesareIntroductiontoAeronauticsandAstronautics,takenfallsemesterofthesecondyearbyaerospacemajors;FluidDynamics,takenin the spring semesterof the secondyearbyaerospacemajors and the fall of thethird year by mechanical and civil engineering majors; Aerodynamics, taken byaerospacemajorsinthespringofthethirdyear;andIntroductiontoHeatTransfer,takenbybothaerospaceandmechanicalengineeringmajorsintheirsenioryear.Itisintendedthatthecollectiveeffectoftheseprojectsthroughoutthecurriculumwill

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establish the basic usefulness of numerical techniques for solving fluid‐relatedproblems as well as introducing basic techniques and issues related tocomputationalfluiddynamicsandrelatedcomputationalsciences.

Course:IntroductiontoAeronauticsandAstronautics

Lectureformat,sophomoreaerospaceundergraduates,required Prerequisites:CalculusI,PhysicsI

In Intro to Aeronautics and Astronautics, aerospace students are introduced toXFOILandXFLR5.Atthisstage,thegoalistousethesecodestoexaminetheeffectsofairfoilandwingdesignonlift,drag,andmomentatdifferentanglesofattackandReynoldsnumbers.Studentsinvestigatetheeffectsofsweep,aspectratio,thickness,camber, and general airfoil construction on thewing performance. Students alsousethiscode inthe finalclassproject, inwhichtheobjective istodesign,analyze,build,andflyasmallglideraircraft.

Course:FluidDynamics

Lectureformat,sophomoreaerospace,juniormechanicalandcivilundergraduates,required Prerequisites:Statics,CalculusIII

XFOIL and XFLR5 are also used in Fluid Dynamics, although with a greaterconsideration of the role of the boundary layer. In addition, two other projectsintroduce the use of computational techniques. One is the numerical solution ofhydrostatic forces on a curved surface. Students are required to construct aspreadsheetorMatlabcodethatcanintegratethenetforcegeneratedbyaseriesofhydrostatic fluid columns along with numerically determining the center ofpressure for this force.Whileessentiallyanexercise innumerical integration, thisprojectillustrateshowafluidmechanicsproblemcanbetranslatedintoadiscrete,incremental approximation and the role of numerical resolution, as students arerequiredtocomputetheresultforanincreasingnumberofcolumnsuntilsufficientaccuracy is achieved. The second assignment has the students develop and use asimplepotentialflowsolver.ThiscodeistypicallyconstructedinMatlab,andmustallow for the introduction of multiple simple flows by the user. From thisinformation, the code generates the potential flow field. Depending on theinstructor, students also may have an assignment requiring the use of thecommercialcodepackageSC/Tetra.

Course:Aerodynamics Lectureformat,junioraerospaceundergraduates,required Prerequisites:FluidDynamics,AdvancedMathematicsforEngineers

InAerodynamics,XFOILandXFLR5areusedforevaluatingtheeffectsofplanformshapeandsweep.(Aerospacestudentshavealsostartedtousethesecodesaspartof senior design and for various small aircraft (SAE, AIAA DBF) competitions,although this usage has also required a greater discussion of the practicallimitations of these codes.) This course also typically has a project requiring the

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development of panel methods or a horseshoe vortex/lifting‐line/vortex‐latticemethod.Afterbeingintroducedtothebasictechnique,includingsomediscussionofhowtoformulatethetechniqueinacode,studentsdevelopacodefromthegroundupandapplyittoaparticularproblemdesignedinparttoprovidevalidationofthesimulationaccuracy.Theprecisenatureoftheprojectvariesbyinstructorandfromclass to class, but they sharemany commonalities with the University of DaytonAerodynamicsprojectsdescribedpreviously.

Course:IntroductiontoHeatTransfer

Lectureformat,aerospaceandmechanicalseniors,required Prerequisites:Statics,FluidDynamics,ScientificProgramming

Here,thecomputationalassignmentistodevelopatwo‐dimensionaliterativesolverfor theheat conductionequationusing finitedifference techniquesona relativelysimplegeometry.Whilenotfluiddynamics,theprojectintroducesdiscretizationofa differential equation along with concepts like grid resolution and boundaryconditions.Again,theprecisenatureoftheprojectvarieswiththeinstructor.

Example3:BoiseStateUniversity,MechanicalEngineering

Coursename:(Multiplecourses)

CFDcontentisintroducedintoseveralcoursesinthecurriculum,startingwiththelab component of the required junior‐level fluid mechanics class, followed by atechnical elective on aerodynamics, and culminating in a technical elective onparallelscientificcomputing(discussedintheCFDHeavysectionbelow).

Course:FluidMechanicsLaboratory

Labformat,juniorundergraduates Prerequisites:co‐requisitewithfluidmechanicslecturecourse

ThegoaloftheCFD‐relatedcontentinthislaboratory‐formcourseistodemonstratethe use of modern simulation software in engineering, reinforce theoreticalconcepts, and develop student interest in upper level courses where CFD andscientificcomputingarecoveredatamoderate level. Forthesepurposes,asinglelaboratorysessionisdevotedtoCFDinwhichstudentsperformthesimulationofabasiclaminarincompressibleflowproblemusingtheeducationalversionofANSYSFluent. A teaching assistant walks the students through the tutorial example.Simulationsof flowoveracircularcylinderatseveralReynoldsnumbers lessthan100havebeenusedtoreinforcethefollowingconceptsinfluidmechanics.

Flowvisualizationthroughcontourplots,streamlinesandvectorstoreinforceconceptsofpressure‐velocitycoupling,flowseparationandstagnation

Dimensionalanalysisandsimilitudethroughsimulationsetup FlowinstabilitybyrealizingvonKarmanvortexsheddinginthesimulations 

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Course:Aerodynamics/IntroductiontoCFD

Lectureformat,seniorundergraduates(elective)/graduatelevel Prerequisites:FluidMechanics

In this course, Anderson’sFundamentalsofAerodynamics and theANSYS FLUENTacademicsoftwarepackageareadoptedascoursematerials. IntheCFDportionofthe course, which covers about 50% of the course, students (a) computationallysolvetheLaplaceEquationforapotentialflowapplication,usingEXCEL,(b)executeat least five basic ANYSYS FLUENT tutorials, (c) perform a physical wind tunnelexperiment to determine the lift on an airfoil and then simulate that experimentusingFLUENT,(d)learnaboutthefinitedifference,finitevolumeandfiniteelementmethods, (e) learn about various options in FLUENT such as explicit and implicitformulation,meshing options, velocity‐pressure coupling, underrrelaxation, near‐wall treatments, and turbulence theory andRANS turbulencemodeling, (e) learnaboutboundarylayertreatmentsandtheRunge‐KuttamethodinsolvingtheBlasiusEquation, and (f) perform an individual computational project, including both areportandapresentation.Undergraduatesmaychoose2‐Dprojectswhilegraduatestudentschoose3‐Dprojects,ifthereisenoughresolutionfortheirproject,ormoredetailed2‐Dprojects.

SomeCFDclassprojects in thepasthave includedflowaroundvarioustrucksandcars and vehicle parts, flow around projectiles in barrels, flow around bicyclehelmets,flowoveraheatedmetalcatalyst,plugflow,bloodflow,hydraulicchannelflow, flow in hydraulic jumps and waves, flow over dimples on projectiles, flowaroundspinningballs,flowaroundbuildings,andflowaroundwindturbineblades.

Someof the students in the classparticipate inMEclubactivities and compete inregionalandnationalcontests.Forinstance,somestudentsintheAerodesignClubselectCFDprojectsthatmayimprovetheperformanceoftheirremotelycontrolledaircraftduringcompetitions.TheyhaveusedFluenttoimprovetheirairfoilandtailsection designs and they have been successful in regional and internationalcompetitions. Other students in the class have used CFD as part of their seniordesignormaster’sthesisprojects.StudentsfindprojectworkandthecompletionofindividualCFDprojectsasthebestandmostrewardingpartoftheirclass.

C. CFD heavy Inthisformat,CFDistaughtasastand‐alonecourse,andthereforemoreemphasisisplacedonteachingfundamentalconceptsofCFD.Suchacourseisoftenelective,thoughthereisanotableexampleofarequiredcourseattheUSAirForceAcademy,aswillbedescribedbelow.

Example1:U.S.AirForceAcademy Coursename:ComputationalAerodynamics Lecture/projectbased,junior‐levelundergraduate,required Prerequisites:IntroductoryFluidMechanics

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TheU.S.AirForceAcademycreatedthiscoursenearlyadecadeago,andthecoursehasbeencontinuouslyevolvingsincethen.Thecoursetopicsarelargelypresentedtothestudents“justintime”asprojectsandassignmentsrequireagivenknowledgeset.Coursetopicsinclude:

Potentialflowreview Panelmethodsreview Vortexlatticemethods IntroductiontoCFDconcepts Computersystemoverview Vectoralgebrareview Governingequationsoffluidmotionreview Levelsofgoverningequations(assumptionsandsimplifications) Finitedifferencing/finitevolume/finiteelementapproaches Truncationerror Stability,consistency,convergence,andCFLnumber Stabilityanalysis Modifiedequation ClassificationofPDEs Algorithmtypes(explicit,implicit) Steadyandunsteadyflow Timeintegration Boundaryconditions(farfield,solidsurfaces,matchingPDEtypes) Gridgeneration(structured,unstructured,Cartesian,overset,topology,

transformedcoordinates) Thesolutionprocess(initialconditions,convergence,stability/robustness) Gridindependence Timeaccuracy Turbulencemodels(RANS,LES,DES),modeltypes(eddyviscosity,stress

transport) Flowvisualization Parallelcomputing Optimizationapproaches

Thecourseistaughtin‘ActivityMode’,withmostworkconsistingofprojects.Theseprojectsinclude:

Finitedifferencingandorderofaccuracy Waveequationanalysis Heatequationanalysis NACAairfoilnumericalsimulationatvariouspre‐stall(steadyflow)anglesof

attack NACAairfoilpredictionatapost‐stall(unsteadyflow)angleofattack Predictionofairfoilaerodynamicswithpanelmethod Predictionofairplanestabilitywithavortexlatticeprogram

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Specifically,studentslearnedthefollowingaerodynamicconceptswhileperformingtheirprojects:

Boundaryconditionsandtheirrelationshiptoflowtype(viscousorinviscid) Boundarylayerthickness,growth,andvelocityprofiles Importanceofunderstandingboundary‐layertheorywhiledoingCFD(sub‐

layertypesandthicknesses,pressuregradients,etc.) Stagnationpointsandstagnationstreamlines Flowseparationandreattachment Laminarseparationbubbles Airfoil/wingstall Airfoilpressuregradientsasafunctionofangleofattack Airfoilsurfaceandoff‐surfacepressures,circulation,andtheresultingliftand

dragvariationswithangleofattack Therelationshipbetweenpressuregradientsandflowseparation Pressureandskinfrictiondrag Unsteadyvortexshedding Theimpactofwing‐tipvortices CompressibilityeffectsatsubsonicMachnumbers

In addition, the following aerodynamic theories and concepts are taught to thestudentswhiletheyareperformingtheirprojects(orarelearnedinapriorcourse):

NACAairfoildesignationsanddata Potentialflowtheory Kutta‐Joukowskitheorem Thinairfoiltheory Lifting‐linetheory

So,whilemany facultymembersmighthavebelieved theyweregivingupagreatdealbyteachingcomputationalaerodynamicstotheirstudents in lieuofthemoretraditional syllabus,we found thatmany of the same conceptswere still covered,butindifferent(andproject‐based)ways.

Example2:UniversityofDayton,MechanicalandAerospaceEngineering

Coursename:FundamentalsofCFD Lecture/projectbased,seniorundergraduate(elective)/graduatelevel Prerequisites:Programming(Matlab),PartialDifferentialEquations,

IntroductoryFluidMechanics

The main idea behind the course design is to give students an exposure to theapplicationaswellascodedevelopmentaspectsofCFD.ItalsocontainstutorialsontheuseofcommercialCFDsoftware(ANSYSFluent).Theoutlineofthecourseisasfollows:

1.IntroductionandBackground

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1.1ConservationLawsandModelEquations2.SpatialDiscretization

2.1Finite‐DifferenceApproximations2.2GridConvergenceStudies2.3Finite‐VolumeMethods

3.SolvingLinearSystems3.1DirectMethods3.2IterativeMethods3.3Preconditioning

4.Time‐MarchingMethods4.1SomepopularTime‐MarchingMethods4.2ImplementationofImplicitMethods4.3Stability

5.Turbulence5.1DirectNumericalSimulations5.2BasicsofTurbulenceModeling5.3SomepopularTurbulenceModels5.4LargeEddySimulations

The following projects (given as homework assignments every two weeks) arecovered:

1. Students run a quasi‐1d Euler solver (written in Matlab) provided by theinstructor for sub‐ and transonic converging‐diverging nozzles andwrite acomprehensivereport.

Learning outcomes: Being able to write a comprehensive CFD reportincludingdescriptionofgoverningequations,convergencehistories,plotsofvariables of interest, and grid convergence studies; realizing that even"simple"problemsalreadyrequirearelativelylargeamountofsourcecode;being able to identify the four phases of CFD analysis in the source code(problem specification and geometry preparation, selection of governingequations and boundary conditions, selection of gridding strategy andnumericalmethod,assessmentandinterpretationofresults).

2. Studentswrite a potential flow solver in a programming language of theirchoice for a non‐lifting cylinder using second‐order finite‐differenceapproximationsandsuccessiveover‐relaxation(SOR).Theyhavetomakeasimple O‐grid, apply Dirichlet boundary conditions, implement the SORupdateformulaforLaplace'sequationinpolarcoordinates,post‐processthesolutiontocalculatevelocitiesandpressures,comparetheCpdistributiononthe cylinder surface to the analytical solution, and perform a gridconvergencestudytoprovethattheywroteasecond‐orderaccuratecode.

Learningoutcomes:Developingstudent'sabilitytowritecomputersoftware;being able to implement the four phases of CFD analysis (problem

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specificationandgeometrypreparation,selectionofgoverningequationsandboundary conditions, selection of gridding strategy and numericalmethod,assessment and interpretation of results) for a relatively simple linearproblem.

3. AfteratutorialsessionstudentsuseANSYSFluenttosolvetheinviscidflowaroundacylinderproblem(samegeometry,grid,andfree‐streamconditionsas in the secondproject) andcompare the resultswith theirpotential flowsolver results. Afterward they run a low‐Reynolds number steady laminarcase(Re=40)tocontrast.TheywriteacomprehensiveCFDreportaboutbothefforts includingagrid convergence studyof the totaldragon the cylinderforthelaminarcase.

Learning outcomes: Learning how to use a commercial CFD softwareproduct; further developing the capability of students to writecomprehensive CFD reports; realizing that for viscous flows one needs toresolve the boundary layer in order to obtain reasonable (grid converged)valuesforthedrag.

4. Students investigate the flow over a backward‐facing step with ANSYSFluent. First the Reynolds number based on step height is 50 (i.e. laminarflow) and studentsperforma grid convergence study for the reattachmentpointanduseFluent'spathlinecalculationsforflowvisualization.ThentheReynoldsnumberischangedto5100(i.e.turbulentflow)andstudentsmakeagoodqualitymeshwithappropriatey+values.Thenstudentsareaskedtokeepthemeshthesameandtouseafewdifferentturbulencemodelstosolvethisproblem inorder to compare theirperformance. As abenchmark, thereattachmentlocationcalculatedusingdirectnumericalsimulation(DNS)isgiven. Also, studentswrite once again a comprehensive CFD report aboutbothefforts.

Learning outcomes: Realizing that turbulencemodels are a source of highuncertainty;gainingknowledgeabout turbulencemodel fundamentalssuchas turbulence intensity, y+ values, boundary conditions, number of addedequations,etc.

5. Studentswritea solver for theone‐dimensional linear convectionequationfor a simple Gaussian pulse with periodic boundary conditions. Second‐order finite‐differences in space, and explicit and implicit Euler, aswell asMcCormack time‐marchingmethods are to be implemented. Students areasked to change the CFL number and comment on the observed behaviorbasedonlinearstabilitytheory.Theyalsoperformgridconvergencestudiestoobtaintheglobalorderofaccuracyoftheimplementedmethods.

Learning outcomes: Further developing student's capability to writecomputersoftware;beingable toobservethepracticalstabilitybehaviorof

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time‐marching methods as well as the interplay of spatial and temporaldiscretizationerrors.

6. Finalproject:Thestudentshavethelastfourweeksofthesemestertoworkontheirfinalprojects(everyotherlecture,theremaininglecturesareusedtodiscussturbulencemodelinginmoredepth).Theyhavetheoptiontoeitherproposetheirownproject(needstobeapprovedbyinstructor)ortheycaneitherrunsomegenericmulti‐elementairfoiltestcasesorAhmedtrailingcarconfigurations.AcomprehensiveCFDreportaboutthefinalprojecthastobehandedin.

Someexamplesofpaststudentsprojects:blowing/suctiononairfoils,writingamultigridsolverforLaplaceequation,analysisofasimpleheatexchanger,parallelizationofLaplacesolverusingOpenMPandMPI,investigationofvonKarman vortex shedding behind cylinder, shock tube problems,turbomachinery,etc.

Learningoutcomes:Beingabletopursueopen‐endedprojects;beingabletodefinefeasibleprojects;furtherdevelopingthecapabilityofstudentstowritecomprehensiveCFDreports.

Example3:SaintLouisUniversity,AerospaceandMechanicalEngineering

Coursename:IntroductiontoComputationalFluidDynamics Lecture/projectbased,seniorundergraduate(elective)/graduatelevel Prerequisites:FluidDynamics(undergraduate)orequivalent,Scientific

Programming(undergraduate)orequivalent.

This course serves as a technical elective for undergraduates students and agraduatecourse,withbothMechanicalandAerospacemajors(aswellasthestrayPhysics major) having taken the class. To date, 65% of the students have beenundergraduates. Both graduates and undergraduates have common lectures andassignments,althoughonsomeassignmentsadditionalrequirementsareplacedonthegraduatestudents(eitherintheformofadditionalworkorrequiringsoloworkwhiletheundergraduatesmayworkinsmallgroups).Onsomeassignmentsandthefinalprojects,groupsareformedcombininggraduateandundergraduatestudents,whichhasprovenanexcellenteducationalopportunityfortheundergraduates.Theclasshasbeentaughtbothasaonehour lecture, threedayperweekanda1.5 totwohourlecture,oneortwodaysperweek.

Thecourseencompassesthreeparts:

1. IntroductiontobasictheoryandconceptsassociatedwithCFD

IntroducesCFDasadisciplineanddescribesbasicconceptsfromdiscretizationto boundary conditions, along with characterizing basic approaches to CFD.Throughmostof thissection, theemphasis ison finitedifferenceschemesandapplication to theheat equationas a linearanalogue to thenon‐linearNavier‐Stokesequations.Conceptssuchasbasicdiscreteapproximationofderivatives

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and differential equations, upwind versus central methods, explicit versusimplicit methods, numerical stability, multistep and predictor‐correctormethods, orderof truncationaccuracy, thederivationof schemesusingTaylorexpansions,convergence,andnumericaldiffusionandothercommonnumericalerrors are presented, discussed, and then reinforced through a series ofassignments.Theseassignmentsincludederivationofschemes,analysisoferrorand stability criterion, andwriting one‐ and two‐dimensional solvers for heatequation problems using different techniques presented in the class. Forexample,overtwoassignmentsstudentsarerequiredtodevelopasolvertofirstsolveasteady‐statetwo‐dimensionalfinwithconvectiveboundariesandthentomodify this solver to address unsteady problems. The steady‐state case isanalyzedovermultiplegriddensities toexamine theeffectsof grid resolution.Theunsteadysolverisdevelopedtousetwodifferenttypesoftimeintegration(typicallyalow‐orderRunge‐Kutta/predictor‐correctorschemeandthenafully‐implicit first order approach), and comparisons are made of the convergencerateandthenumericalstabilityof the twotechniques. For this firstsectionofthe class, students typically perform computational work in Matlab or Excel,althoughtheyarewelcometouseamoreformalprogramminglanguageshouldtheychoose.

2. Introductiontocommercialsoftware

Thesecondsectiontargetsuseofcommercialsoftware.Studentscompletetwoprojectsusing theSC/Tetrapackage.Classesat thisstageareacombinationoftraditional lecture and computer laboratory sessions. After an introduction toturbulence modeling focuses on RANS techniques, students complete twoprojects,onesimulatingflowoveraflatplateandoneforflowovercylinder.Pre‐processing, includinggridconstruction,processing,andpost‐processingareallcomplete by the students. Because of limitations on available compute powerand software licenses, the simulations are steady‐state and not fully resolved;however, the basic effects of grid and domain resolution and differentturbulence models can still be illustrated in these exercises. Results arecompared to established experimental and/or computational data todemonstratetheneedforvalidationevenforcommercialsoftware.Studentsalsolearn aspects of good grid construction, effects of relaxation and convergencecriteria,andchallengesofworkingwithunstructuredgrids.Anoveralloutcomeisforstudentstorealizethatpre‐packagedcodesshouldnotbeusednaively,butrather,requiretestingandvalidationinthesamemannerasself‐writtencodesdo inorder toensureaccurate results.Once theseassignmentsare completed,students are then required to develop independent small group projectsincluding some form of grid resolution and validation studies. This project ispresentedinbothawrittenreportandoralpresentation.

3. Introductiontocodeconstruction.

Simultaneouswiththeirprojectwork,thefinalpartoftheclassintroducesbasicCFD techniques for finitevolumemethods.Abasic formulationof a staggered‐grid CFD code is presented for 2D incompressible, steady, laminar flow,

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includingmultipleadvectionschemes.Conceptspreviouslyintroducedusingtheheat equation are now revisited in the context of the coupled Navier‐Stokesequations. Approaches for solving multiple equation systems are presented,focusing primarily on iterative schemes. As a final assignment, students areasked todevelopaone‐dimensional, steady, incompressible, constantviscosityNavier‐Stokes solver using the techniques presented in class. A concludinglectureortwoisusedtodiscusshowthesetechniquesareextendedtounsteadyflows,differentmeansofsolvinglinearalgebramatrices,andparallelprocessing.

Theintendedoveralloutcomeofthiscourseistoprovideareasonablegroundinginand appreciation of CFD. The course is intended to provide an educationalexperienceforboththosewhowouldliketouseCFDsolelyasatoolandthosewhowouldultimatelyliketodevelopCFDtechniquesandcodes.

Example4:UniversityofVermont,MechanicalEngineering● Coursename:ComputationalFluidsEngineering● Lecture/projectbased,seniorundergraduate(elective)/graduatelevel● Prerequisites:FluidMechanics

ThiscourseisdesignedasanintroductorycourseinthemethodsandapplicationsofCFD, and uses the ANSYS FLUENT CFD package exclusively as a learning vehicle.The philosophy behind the course is to use the FLUENT software to enable thestudents to perform meaningful engineering calculations without the need ofprogramming; in this way, the student may focus their learning efforts on theunderlyingprinciplesandtheoryofCFDanditsproperapplicationtoobtaincorrectresults. The class features two hours of lecture per week that covers topics innumericaltechniquesinconjunctionwithatwo‐hourCFDlabthatdemonstratestheimplementationofthosetechniques.Thecoursecontentbeginswithfinite‐volumeanalyses of steady and unsteady heat conduction. The next step is convection‐diffusion problems with prescribed velocity fields. The solution of theincompressible,laminarNavier‐Stokesequationsisthenexamined;thisisfollowedbyanintroductionofRANS‐basedturbulencemodels. Theremainingtimeduringthesemesterisallocatedtotopicsofparticular interesttotheclass; thismayvaryfromnon‐Newtonianflowstocompressibleflowdependingontheclass.

Example5:BoiseStateUniversity,MechanicalEngineering● Coursename:ParallelScientificComputing● Lecture/projectbased,seniorundergraduate(elective)/graduatelevel● Prerequisites:DifferentialEquations,Programming

The second upper level technical elective course at Boise State University is thenewlyintroducedParallelScientificComputing.Thecourseisdesignedtointroduceengineeringstudentstothefieldofsupercomputingusingprototypeproblemssuchas2Dconductionheat transferand/orvibratingmembrane. Theemphasis in thecourse is optimization of scientific programs for high performance computing,parallel computing with Message Passing Interface (MPI) and programming the

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massivelyparallelgraphicsprocessingunitswithCUDA.Studentslearnthemodernhardware and software for parallel computing. Additionally, students learnnumericalmethodstosolveordinaryandpartialdifferentialequations,explicitandimplicit formulation, concept of numerical accuracy and stability, boundaryconditions,andbasic iterativesolutionmethods. Thereare threeclassprojects inwhichstudentsarerequiredtosubmitcomprehensivewrittenreportsalongwithaworking computer program. In these projects students apply their knowledge ofhigh performance and parallel computing to solve prototype 2D PDEs on NSFsupercomputers, andassessparallelperformance throughspeedupandscalabilityanalysis. For everyproject students are required todemonstrate formal orderofnumericalaccuracyusingexactsolutionsoftheprototypeproblem.8‐9homeworkproblems on numerical methods with small computer programming tasks areassignedtoreinforceconceptsthatarenotcoveredbytheclassprojects.

D. CFD in Laboratory Courses As discussed in the introduction, traditional fluid mechanics laboratory coursesprovideanaturalopportunityforstudentstoperformnumericalexperimentswithCFDinadditiontophysicalexperiments.Aparticularlyrewardingapproachinthisform allows students to use CFD and physical investigation of the sameconfiguration,inordertoperformadirectcomparisonoftheirresultsandreflectonsourcesofdiscrepancies.

Example1:TrineUniversity,MechanicalandAerospaceEngineering● Coursename:Aerodynamics● Lectureformatwithafewlaboratoryexperiences,seniorundergraduate

(elective)● Prerequisites:FluidMechanics

Thereweretwoobjectivesofthislaboratoryexperience.Thefirstobjectivewastointroduce the students to detached shocks. The second objective was to exposestudents to different levels of numerical modeling for supersonic flow. To theseends,thestudentsusedasupersonicwindtunneltomeasuretheliftanddragforcesonasmall‐scalemodelrocket.Halfofthestudentsintheclassmodeledtherocketusingthe1997versionofMissileDATCOM.Theotherhalfoftheclassmodeledtherocket using ANSYS Fluent. The instructor provided significant assistance to thestudentstaskedwiththeCFDsimulation. Forallpracticalpurposesthe instructorperformedthegridgeneration,thesetupofthecaseinFluent,andtheinitialpost‐processing of the results. Students performed further post‐processing. Thelaboratoryrequired3hoursofclasstime:1hourfortheexperimentand2hoursforthecomputationalmodeling.Studentsspentadditionaltimeoutsidetheclassroomperformingthecomparisonbetweenthevariousresultsandwritingalabreport.

Onreflection,studentsseemedtoenjoythelabbutlearnedlittlewithregardtothenumericalmethods. However, therewassignificantvalue inthecomparisonsthatthe students made between the wind tunnel results and the numerical results.

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ParticularlyinterestingwasthecomparisonoftheactualschlierenimagetotheonecomputedbyFluent.

Example2:SaintLouisUniversity,AerospaceandMechanicalEngineering● Coursename:FluidDynamicsLab● Laboratoryformat,sophomoreaerospace,juniormechanicalandcivil

undergraduates,required● Prerequisites:FluidDynamics(co‐requisite)

All aerospace, mechanical, and civil engineering majors are required to take thisone‐credit fluidlaboratorycourse. Thecoursemeetsonceperweek,andstudentscomplete a total of ten lab assignments over that time, rotating in small groupsthroughthe labequipment.Onestandardexperiment isexperimentallymeasuringthepressuredistribution alongaClark‐Y airfoilmodel in awind tunnel, and thenusing this data to compute lift and drag. To enhance this experience, a separatelaboratoryexperimentbasedonXFOILhasbeencreated. In this lab, thestudentsuseXFOILtorecreatetheexperimentalflowsatthefivestudiedanglesofattackandthe approximate Reynolds number. Students also simulate the airfoil at a higherReynoldsnumber for comparisonwith standardexperimentaldata sets. StudentsaretoexaminethepredictionofseparationandtransitioninXFOILandrelatethatto the computational and experimental lift, drag, and pressure distributions. Forfurther comparison, a second set of airfoil data is provided and the same XFOILcomputationsareconductedinordertoillustratehowchangesinairfoildesigncanresultindifferentaerodynamics.Thestudentresultsarepresentedinawrittenlabreport as per the other experiments in the class. The overall objective is tointroducestudentstothecomplementaryrolesofCFDandexperimentwithinfluiddynamicsandtheroleofvalidationwithinCFD.

E. CFD in Design Courses Perhaps themost desirable context for CFD in the engineering curriculum is in acapstone design course. In no other course do students so regularly requestinformalguidancefrominstructorsinusingCFDinthedesignprocess.Indeed,ifadesigncourseistoserveasarepresentativetemplateforteam‐orientedindustrialdesign,itisonlyfittingthatstudentshaveaccesstoasetofdesigntoolsthatreflectsthesetusedinpractice.

The challenge,of course, is that it is generally impractical to includeanythingbutsuperficial CFD instruction into a course whose content is focused on designparadigms.Theidealscenario,then,isoneinwhichthestudentshavealreadyhadmeaningfulexperienceswithCFDinapreviousproject‐basedcourse.ThisscenarioisreflectedintheexampleoftheU.S.AirForceAcademy,describedbelow.

Example1:U.S.AirForceAcademyPriortothecreationofthecomputationalaerodynamicscourse(describedaboveintheCFDHeavysection),therewerenorequiredelementsofCFDorpotentialflowinthedesigncourse.Fluiddynamicsanddesignfacultyworkedcloselyover

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the past few years to ensure that the elements learned in the computationalaerodynamicscoursewereusedinthedesigncourse.Therealimpactofthishasgrown substantially, especially in the past two years, as students developed a“toolbelt”ofmethods including:XFOIL(airfoilpanelmethod),Tornado(vortexlattice code), CEASIOM (capable of doing Euler predictions, as well as vortexlatticeandDATCOMestimates),andKESTREL(Navier‐Stokessolver).Theabilityto use a wide variety ofmethods has greatly impacted the design course, andcould be one of the most important results of creating the computationalaerodynamicscourse.

An alternative, lower‐profile format is to develop an ad hoc Independent Studycourse(orcomponentwithinacourse)forthosestudentsthatseektoapplyCFDintheircapstonedesignproject(orcompetitionteamproject).Studentsmightfollowa predefined but self‐paced syllabus, with only informal guidance from a facultymember. This approach is generally driven by a desire to simply use CFDcommercialsoftwareasatoolforaspecifictaskintheprojectratherthantoacquirean in‐depthknowledgeof the topic. In these cases, a level of familiaritywith theoperationofthesoftwarebecomesthefocusoftheCFDeducation.Whenproperlymotivated and challenged, students may find this learn‐by‐doing approach to bemoreappealingthanatraditionallectureform.

One can reasonably ask whether this independent study format constitutes ameaningfuleducationalexperience,orissimplytraining.Undoubtedly,somehighlymotivatedstudentswill gatherasmuch fromthis formatas froma formal course.However,itismorelikelythatthemostthatonecanrealisticallyexpectfromsuchaformatisabetterappreciationofthestrengthsandcapabilitiesofCFD.ExamplesofthisatTrineUniversityandWesternMichiganUniversityaredescribedbelow.

Example2:TrineUniversity,MechanicalandAerospaceEngineeringBasedon thegrowing importanceofCFD in thedesignprocess, itwasdeemednecessary by the faculty at Trine University to support those students whowantedtoperformCFDsimulationsaspartoftheircapstonedesignexperiences.During the 2011–2012 school year, one student from a group that planned toparticipateintheASMEHumanPoweredVehicleChallengewantedtouseCFDtodesignashellthatminimizedthedragofthegroup’svehicle. Amentor‐menteerelationship was established between a faculty member and the interestedstudent.Inthiscontextnoformalinstruction,suchasaclassroomlectures,wasgiven. Instead the studentwas coached through each step of the CFD processusingANSYSWorkbench.

Forexample,thestudentfirstgeneratedagridusingthedefaultsettinginANSYSMeshing. Once this initial grid was generated the instructor discussed theimportance of resolving the relevant fluidmechanicswith specific attention tothe boundary layer with the student. The student then investigated variousmethods for refining the grid. Once the refinements were performed theinstructorindicatedthatthestudentneededtodeterminethequalityofthegrid

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before proceeding to attempt a solution. The instructor explained theimportance of cell aspect ratio and skewness and the impact on convergence.Thisdiscussionwasdoneataverytoplevelandwithoutreferencetodifferentialequations or numerical schemes. This type of coaching continued through thesetupofFluentaswell.Partiallybecauseoftheinformalnatureofthiscoaching,thisprocesstookseveralmonthsbeforethefirstCFDsolutionwascompletedtoasatisfactory level. During this time the other students in his group produced amodel thatwas tested in a subsonicwind tunnel. The CFD solutionwas thencomparedtothewindtunnelresultsasaninformalvalidationoftheCFDresults.At this point the students utilized CFD to determine the drag on four differentshelldesigns. Theshellwiththe lowestdragcoefficientascomputedbyFluentwasselectedforthefinalvehicledesign.

Onthequestionofwhetherthisinformalstudyservedaslearningortraining,itisthe instructor’s contention that learning took place. The instructor witnessedthisstudentassistingotherstudentsinaCFDclasswiththeirfinalprojects.Thisassistance indicated that he had learned some important lessons regarding theappropriate use of CFD. Perhaps more important than what he learned, thestudentwasinspiredtoobtainandreadtwobooksonCFDsothathecouldlearnmoreabouttheactualmethodsemployed.Theinterestofthestudenttofurtherexplore CFD on his own was an unexpected but welcome outcome of thisexperience.

Example3:WesternMichiganUniversity,Mechanical&AeronauticalEngineeringTherearetwoexamplesofhands‐onCFDexperiencesindesignatWMU.Thefirstis an independent study course, and the second is within a two‐part capstonedesignproject.

Course:IndependentStudyAnintensive,three‐credit‐hourindependentstudycourseisofferedtointerestedstudents.ItbeginswithbackgroundreadingonCFDingeneral,andmeshingandconvergence in particular. This is followed by hands‐on operation of thesoftware vendor’s tutorial cases that most closely resemble the target case.Students are highly encouraged to explore and observe results of changes ofrelevantparametersandoptions. Facultyguidance insimplifyingthegeometrywhilecapturingtheessentialfeaturesofaflowcanbeimportant.

OneofthestudiesinvolvesthedevelopmentofaCFDmodelforasmallturbineengine combustor. The turbine engine is used in the lab section of theAeropropulsioncourseinAeronauticalEngineeringatWMUandforresearch.Asimulationmodelwith combustion included, developed in ANSYS FLUENT, hasbeensuccessfullydevelopedinpreviousofferingsofthecourse.

Course:SeniorDesignProject

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CFD is an important part of this course, which consists of two consecutivesemester‐long parts. In nearly all cases where the use of CFD is involved,experimentaltesting—eitheronabenchtoporinawindtunnel—isalsoused,andtheresultsofthetwoapproachesarecompared.Asaresult,groupsnormallyhave three members whose main responsibilities will be the design,manufacturing,andCFD, respectively. Allmembersare required to supportallefforts to create a fully integrated project between testing and CFD. Theinstruction for theCFD specialist is very similar to thatused in conducting theindependentstudyclassdescribedabove. Becauseofthefactthatexperimentaldataareavailable,theCFDmodelerismotivatedtodeveloprefinedandaccuratesimulations.SoftwareusedinthepastincludeFLUENT,COMSOL,andSC/Tetra.AnexampleprojectisthedesignoftheexternalbodysurfaceofanFSAEracecarthat competes in the annual SAE design and road competitions. A number ofprojectsconcerndragreductiondevicesforclass‐8trucks.

IV. CFD-Related Concepts in Mechanical Engineering It isappropriatetoaddresstheneedsofmechanicalengineeringinthisdocument,since many universities have joint programs in mechanical and aerospaceengineering,andthetopicsareoftenofinteresttoboth.Here,webrieflycommentontwotopics—heattransferandmultiphaseflows—traditionallyincludedinthemechanical engineering curriculum that are associatedwith a distinct set of CFDconcepts,andthereforemeritseparatediscussion.

ComputationalheattransferComputationalheattransfer,withinthecontextofanintroductoryCFDclass,canbeincorporatedinessentiallytwodifferentmodes:stationaryproblemsinvolvingheatconductionand/orthermalradiation;andcoupledheattransferandfluidflow.Thenecessaryprerequisitesforthiscomponentwouldbeanintroductorycourseinfluidmechanicsaswellasagoodknowledgeofthermodynamics.Anadditionalcourseinheat transfer,common inmostmechanicalengineeringcurricula, isalsobeneficialespeciallyinconnectionwiththermalradiation.

Theinherentvalueofheatconductionisthatthegoverningequationsarelinear,andthus,itiswellsuitedasanintroductorylearningvehicleforfinite‐volume(orfinite‐difference) analysis and associated numerical techniques for linear systems ofalgebraic equations. For cases of steady standalone heat conduction in solids,classicsolutionsofLaplace’sequationcanberecreatednumericallyandrigorouslycompared with analytical solutions. Doing so can allow students to betterappreciateconceptsofspatialgridrefinementandvalidationwithinthesettingofarelativelysimpleproblem.Theintroductionofunsteadyheatconductiontermsmaysimilarly be exploited to introduce concepts of numerical stability and timediscretization. With the establishment of these course concepts, the topicalcoverageofconjugateheattransferatalaterstageisanaturalextensionoftheflowand conduction analysis that links thermal interaction between a flow and itssurroundingsolidgeometry.Finally,itisworthwhilenotingthatitisoftenpossible

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to leverage commercial CFD software to perform these calculations, although notoriginally designed for this purpose. This provides an option for the instructor touseCFDsoftwarethroughoutthecourseiftheysochoose.

The solution of steady, or unsteady, convection‐diffusion problemswith a knownflowfieldrepresentsthenextlevelofsophisticationwhilemaintainingalinearsetofgoverning equations. Here the focus can be upon the numerical treatment ofconvective versus diffusive terms and the associated consequences. In particular,discretizationeffectssuchas instabilityorartificialdiffusioncanbeexamined inaverysimplesetting.Directcomparisonswithanalyticalsolutionsareagainavailableinmanycases.

Thenumerical simulationofnatural convectionphenomena canoften represent ausefulillustrationofthetwo‐waycouplingofthethermalandvelocityfields.Insuchflows, the coupling occurs via a buoyancy term in the momentum equation asopposed to fully thermodynamic coupling in high‐speed compressible flows.Becausethevelocitygradientsarelowinthesetypesofproblems,griddevelopmentisstraightforwardandtherequiredspatial resolution is typicallymodest.Assuch,the computations themselves are well suited for introductory classes. ClassicproblemsinRayleigh‐Benardflowarenaturalchoicesforcourseprojects.

The modeling of thermal radiation heat transfer is a complex topic owing to itsinherentnonlinearnatureaswellasitsdependenceongeometricconfiguration.Italso represents a specialized regime of heat transfer that is of significance underconditions of very high temperatures and/or under highly rarefied conditionswhere convection or diffusion modes of transport are negligible. One potentialutility of radiation coverage would be in a topical module that leverages thecapabilities of commercial software to handle the calculation of radiative viewfactors for realistic surface geometries. For students with prior experience inradiation heat transfer, comparisons with radiative circuit modeling are alsopossible for simplified scenarios. If desired, instructors could also introduce inparallel a discussion involving the calculation of theoretical view factors usingMonteCarlomethodsforlineofsight.

NumericaltechniquesformultiphaseflowThis topic introduces the CFD concept of simulating a system that consists ofmultiple phases. In practical terms, this topic must be limited in scope for anintroductoryCFDcoursegiventheadvancedlevelof thetopic. Unlessthestudenthas prior experience in multiphase flows resulting from an advanced electivecourse,theinstructorshouldbepreparedtoallocateasufficientamountoftimetoprovidea summaryofkey issues. Anaturaldivision insuchcontent is theuseofEulerianmixturemodelsandLagrangianapproaches.

For Eulerian models, the theory of averaging can be presented to develop theaveraged equations of motion for a multiphase mixture based on the volumefractionsofitsconstituents. Thisisappropriateforexaminingflowscenariossuchasbubblyflowsorsuspensionswherethedetailsoftheindividualparticlesarenot

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important,butrathertheiraggregateeffectsare. Theresultingequationscanthenbe solved in amanner usingmethods developed for flows of single fluids. Thus,withappropriatepresentationof theunderlying theory, it ispossible to introducethis into an introductory CFD class. Appropriate model problems include thesimulationofbubblyflowsinpipes,forwhichanextensiveempiricaldatabaseexiststhatcanbeusedforcomparisonandvalidation.

InLagrangianmodelingonetreatsthephasesseparatelyandtrackinterfaces.This,ingeneral,representsamuchmorechallengingcomputationalproblemandislikelybetter suited for an advanced computational class. Nonetheless, a discussion ofcommon methods such as the volume‐of‐fluid (VOF) and level‐set could beincorporatedintoanintroductoryCFDclassprovidedthisisdonecarefully.Whileapresentation of the underlying theory of these approaches can be effectivelydelivered at the introductory level, the numerical implementation represents anundertakingthatmaybebeyondthiscourselevelunlessacommercialCFDcodeisutilized.Tothisend,theVOFmethodisavailablewithintheFLUENTCFDpackage,and the level‐set method is available within COMSOL Multiphysics CFD Module.Appropriatemodelproblems foran introductory coverageofLagrangianmethodscouldincludetheclassicproblemofarisingbubbleinaninfinitemediumatvaryingReynoldsnumber,forwhichtheoreticalandempiricaldescriptionsareavailableforcomparison.

V. CFD Instructional Resources The previous sections have outlined a variety of ways in which CFD can beintegrated into the undergraduate curriculum. The objective of this section is toprovide some guidance regarding the resources that are required for successfulimplementation of the above ideas. This requires a consideration of both thehardware and software resources for the hands‐on computing activities in thecurriculum, as well as a selection of an appropriate textbook for guiding theinstructionofCFDmethodologies.

A. Hardware resources Firstand foremost,auniversitymusthaveaccess tocomputational resources thatare adequate for the desired level of CFD instruction. For the lower‐profileexamples detailed above, it is likely that a student’s personal computer or theworkstationsinaninstructionalcomputinglabwillbeadequate. However,forthecourses involving large‐scale projects or instruction in high‐performancecomputing, a larger resource is necessary. In the example above of theComputationalAerodynamicscourseattheU.S.AirForceAcademy,studentsweregivenaccessto144coresofalocalcomputationalclustertoensurethatthestudentsdidnotgetinthewayofeachotherorotherresearchersbyhavingtocompeteforcomputationaltime.

Dedication of a large number of cores for education may be feasible at largeresearch‐orienteduniversities,butmaynotbepossibleatsmalleruniversitiesandcolleges. Instructors at such institutions might consider using computational

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resources in the Extreme Science and Engineering Discovery Environment(XSEDE)[6]. XSEDEisaNSF‐fundedconsortiumofsupercomputersthatisavailabletoresearchersandeducators.Aneducatorcanrequestaneducationallocationofupto200,000CPU‐hours. Oneoftheauthorshasutilizedaneducationallocationandthe entire process, from request to approval, took less than two weeks.Furthermore, theapplication itself took less than30minutes tocomplete. XSEDEsystemsareagoodwaytointroducesupercomputingtostudents.

B. Open-source versus commercial software Once an instructor has obtained the necessary computers for the students toperformtheirCFDanalyses,theymustselectthesoftwarethatwillbeusedbythestudents.ThisisbothanimportantanddifficultdecisionthatmustbemadewhenintroducingCFDintoacurriculum.Initially,itmustbedeterminedifopensourceorcommercial software will be used. Both forms have their advantages anddrawbacks.

Two advantages of open source software are price and the ability for students toexamine the source code. In particular, by utilizing the XSEDE system and opensource software, a university could implement ahigh‐performanceCFDeducationcomponent intotheircurriculumwith littletonocost. Furthermore, theabilityofstudentstoexaminethesourcecodewouldallowthemtomorefullyunderstandtheimplementationof thevarious schemesand themethodbywhichdata ishandledwithinaCFDcode.Theadvantagesofopensourcesoftware,however,docomeatacost. Most — though not all — open source codes are based on a Unix/Linuxoperating system, with which relatively few undergraduates are familiar. Thus,there is slightly higher overhead, measured in time units, in initiating hands‐onprogresswithopensourcecodes.

CommercialCFDcodes,on theotherhand,placeapremiumon theuser interface.TheGUIsareusuallywellthoughtoutandtheiruseisintuitive,especiallytothosestudents who have worked with computers all of their lives. Furthermore,commercial CFD codes usually contain excellent documentation. Thisdocumentationmighthavetutorialsandtrainingmaterialsavailablethatwillallowthestudentstoquicklyutilizethesoftware.However,selectingacommercialcodemeansaddedcosts foranacademicsite licensewithanadequatenumberofseatsandagenerallackoftransparencyintheimplementationofthenumericalschemes.There aremany commercial software packages for grid generation, solution, andvisualization.A trip through theASMExpositionwill expose the reader to severalcompaniesthathavesoftwarethatperformsallorsomeofthestepsoutlinedabove.As foropen source codes,CFDOnline lists around30different softwarepackagesandprovideslinkstohundredsmore.[7]Abrieflistofsomeoftheseoptionsisgivenbelowinalphabeticalorder.SomeofthecommercialCFDpackagesincludeameshgenerator and/or visualization software. We wish to emphasize that we do notendorseanyspecificproduct.

CommercialSoftware

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MeshGeneration: cubit(csimsoft) GridPro(ProgramDevelopmentCompany) Pointwise/Gridgen(Pointwise,Inc.)

Engineeringleveltoolsforaerodynamics AVIDsoftwaresuite(AVID,LLC) LinAir(DesktopAeronautics,Inc.) VLAERO+(AnalyticalMethods)

CFDSoftware ANSYSFluent/ANSYSCFX(ANSYS) CFD++(MetacompTechnologies) COMSOLMultiphysics(COMSOL) CRUNCH/other(CombustionResearchandFlowTechnologies,Inc) Cart3D(DesktopAeronautics,Inc.) GASP(Aerosoft,Inc.) STAR‐CCM+(CD‐adapco) SC/Tetra(Cradle)

Visualizationsoftware EnSight(ComputationalEngineeringInternational) Fieldview(IntelligentLight) Tecplot(Tecplot,Inc.)

OpenSourceMeshingSoftware Salome SolidMesh

EngineeringLevelAerodynamicssoftware Ceasiom DigitalDatcom XFLR5 XFOIL

CFDSoftwareSomesoftwareisfreetoobtain,butrestrictedinsomeway.Forexample,NASAdevelopedsoftwareissometimerestrictedtoUSpersonsonly. OpenFoam Overflow(restricted) SU2 USM3D(restricted) WIND‐US(restricted)

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Visualizationsoftware Paraview VisIT

C. Textbooks A variety of different textbooks on CFD methodology have been published. Theselectionofthetextbook,likethehardwareandsoftware,isdependentonthescopeof the class, and there isno single text that canbedescribedas “one size fits all”.Rather than review every textbook, the objective here is to highlight some of thestrengths and weaknesses of some representative examples, thereby establishingsomequalitiesthatcanbeusedtocriticallyinspectothertextbooks.

ComputationalFluidDynamics:APracticalApproachJ.Tu,G.Yeoh,C.LiuButterworth‐Heinemann.2007.

Asanintroductorytextattheundergraduatelevelthisisaverygoodbook.ItgivesaniceoverviewandintroductionofCFDsinceitexplainsmajorconceptsverywellusingeasytounderstandexamples.Theearlychaptersprovideaconciseoverviewof the entire CFD implementation to a flow problem. This overview begins byidentifyingandsummarizingtheessentialcomponentsofgridgeneration,problemdefinition, numerical solution and post‐processing. The text then moves on toexamine, in somewhat finer detail, the selection of flow physics and concepts ofnumerical discretization of the governing equations. This is followed by adiscussionofassessingconvergenceandstabilityissues.

Because of its high‐level treatment, this is a great book for someonewho has noexperiencewithCFDandjustwantstogetageneralideaaboutwhatCFDisandhowitworks. On theotherhand, this isnotasuitablebook if theaim is towriteCFDcode or to begin CFD research. Themathematics is oversimplified and does notprovide enough details to gain an in depth understanding. For that, a moreadvancedtextwouldberequired.Furthermore,evenforinstructorswhobasetheirCFD course on a GUI‐driven commercial CFD code (e.g. FLUENT) there areinsufficienttechnicaldetailstoallowthestudenttobeaknowledgeableuserofthesoftware. Ifitwerepossibletobundlethistextwithasetofnotesorusethisasasupplementtoamorequantitativetext,thismightbeausefulcombination.

FundamentalsofComputationalFluidDynamicsH.Lomax,T.H.Pulliam,D.W.ZinggSpringer.2011.

ThisbookpresentssomefundamentalsofCFDbasedonstraightforwardmathematicaltools,givingfastaccesstotheveryinterestinggeneralbehaviorsofPDEfamilies.Thebookisverywellwrittenandeasytounderstandifonehasalreadytakenclassesonpartialdifferentialequationsaswellasnumericalmethods.ThecentralthemeofthebookisaclearandsystematicmethodologytogofromPDEtoODEtofullydiscretizedequations.Thus,itstartswithanintroductionofbasic

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concepts,introducesfinitedifferenceandfinitevolumediscretizations(thoughtheemphasisisonfinitedifferencetechniques),andcontinueswith(linear)stabilitytheoryoftimemarchingmethodsandthesolutionoflinearsystemsusingiterativemethods.Somemoreadvancedtopicssuchasmultigrid,fluxsplitting,andupwinddifferencingstencilsarealsopresentedinthesameclearsetting.

AnIntroductiontoComputationalFluidDynamics:TheFiniteVolumeMethodH.VersteegandW.MalalasekeraPrentice‐Hall.2ndedition,2007.

Thishasproventobeaveryusefultext,andisquitereminiscentofthenow‐classictextbook Numerical Heat Transfer and Fluid Flow by S. Patankar (1980). TheintroductorysectiontoCFDconceptsandapplicationsisratherterseandlacking;amore substantial introduction would be a worthwhile addition. The text firstdevelopstheappropriateconservationlaws,followedbyaself‐containedtreatmentof turbulencemodeling. This separate treatmentof turbulencemodeling isusefulfor an introductory course if the instructorwishes to focus on laminar flows anddefer theadditionalcomplicationsof turbulencetoa later topic.The finitevolumemethod is then developed for steady diffusion and convection‐diffusion problemswith known velocity fields. Pressure‐velocity coupling, and associated classicalalgorithms, is next introduced in order to solve for an unknown flow. Finally,unsteady effects and time discretization are considered. (It should be noted thatthisbookcoversonlysegregatedpressure‐basedsolvers,sothataninstructorwhowishes to cover density‐based schemes or coupled solverswould need additionalsourcesfortheirstudents.)

Theattractivenessofthistextliesinitssuitabilityforincorporationinacoursethatusescommercialsoftwareoroneinwhichstudentswritetheirowncode.Itisnotedthattheapplicabilityofthistexttoinstructionbasedonbothcommercialsoftwareandself‐writtencodeisakeyreasonforitsuseintheSaintLouisUniversitycoursedescribed previously. For use with commercial codes, the depth of coverage isappropriate to allow the students tobecome intelligentCFDusers. In the caseofself‐written codes, the text also provides a self‐contained section addressing thenumericalsolutionofthediscretizedequations.Thetextillustratesbasictechniquesused inFiniteVolumemethodologies common to commercial software,which thestudentscanthenusetoguidetheirowncodeconstructioninthefinalpartof thecourse. The depth of coverage is such that the bulk of this text can indeed becoveredinthespanofaone‐semestercourse.

Aparticularstrengthofthisbookisintheuseofexampleproblemstoillustratethevariousnumericalschemes.Whenteachingundergraduates,theseexampleshelptohighlighttheissueswithimplementationoftheschemes.

ComputationalAerodynamics:AModernEngineeringApproachR.Cummings,B.Mason,S.Morton,D.McDanielCambridgeUniversityPress.2013.

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Thistextbook,whichistobepublishedbyCambridgeUniversityPressin2013,isadirectoutgrowthofthefrustratingprocessattheU.S.AirForceAcademyinfindingasuitable undergraduate textbook for the required course on ComputationalAerodynamics (CA) described above. The target audience is juniors in aerospaceengineeringwhowant(orneed)tolearnCA/CFDinthebroadcontextoflearningtodo computational investigations, while also learning engineering methods andaerodynamics.Inaddition,itisbelievedthatengineerswhoneedtoapplyCA/CFDmethods, butwho have no CA/CFD background,will also find the book valuable.RayCosnerofTheBoeingCompany(theSeniorTechnicalFellowforCFDatBoeing)strongly encouraged the development of this book precisely because of the greatneedforitintheaerospaceindustry.

The book does not assume that the students have had a course in aerodynamicsprior to using the book, but it does assume a skill set that includes IntroductoryFluidMechanics. The typical studentusing thisbookwouldstillbe learningbasicengineering and aerodynamics, but they would not have well‐developed skills incomputationalproblemsolving.

Thecontentofthebookincludes:

● Abriefhistoryofcomputationalaerodynamicsandcomputers(whyandhowCA/CFDisused)

● Engineeringproblemsolving,withemphasisonusingthecomputer,butinthebroadcontextofexperimental,analytical,andengineeringmethods

● Anintroductiontoaerodynamicconcepts● ReviewofthegoverningequationsusedinCA● “Classical”linearcomputationalaerodynamicsmethods● ThecentralideaofCFD–thenumericalsolutionofPDEs● Geometryandgrids● Viscosityandturbulencemodels● TheartofCFD,includingrulesofthumb,overallapproachtosimulationof

aerodynamics,flowvisualization,gridgeneration,convergence,andgridstudies

● ProjectsillustratingbothCA/CFDandaerodynamics.Thebookwillprovidesoftwareaccesssothatanyonewithinanacademicorindustrialenvironmentwouldbeabletoaccomplishthevariousprojectswithreadilyavailablecomputerresources.

Itisimportanttostressthatthisisabookdesignedtoteachaerodynamicsthroughthe use of modern computational tools, and is not a book for CFD algorithmdevelopers.

VI. Conclusions This report has presented an assessment of CFD instruction in the context of theundergraduate aerospace ormechanical engineering curriculum. For the reasonsarticulated in the introduction, computationally oriented concepts should beintrinsic to a modern engineering curriculum, but the manner in which these

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conceptsareincludedrequirescare.Here,wehavetriedtodistilltheconceptsintobasiccategories,fromwhichonecandrawinordertodesignorreformatacourse.Wehavepresented several possibleprofiles forCFD content—asdemonstrationunits or projects within a lecture‐style class, as a course in its own right, or asminimalpreparationasatoolforalaborcapstonedesigncourse.

Throughout,weemphasizethattheoverarchingobjectiveofCFDinstructionshouldbe todevelopa student into an intelligentuserof computational tools for solvingengineering problems. It can be easy to blur the line between instruction andtrainingwhenthetargetis,inmanycases,atoolforengineeringdesign.However,wemaintainthattrueinstruction—thegoalofanyfour‐yearengineeringprogram—must,aboveallelse,providestudentswithfoundationalknowledge,andinfuseinthemthecapacitytothinkcriticallyandtoregardresultswithaskepticaleye.TheprincipaloutcomeofCFDinstructionisnotthebulletpointonthe“Skills”sectionofthecurriculumvitae,buttheabilitytoassessthatflowphysics,atthedesiredleveloffidelity,areappropriatelyaccountedforinCFDsoftwareoutput.

These aims — establishing foundational knowledge and a capability for criticalthinking — should guide any curriculum reform, from a small tweak of a fluidmechanicscoursetoacompleteoverhaulofthethermalsciencescurriculum. It ishopedthatthisreportcanservearoleinassistinganysuchreform.

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VII. References

[1]“TestimonyofMichaelGarrett,Director,AirplanePerformance,BoeingCommercialAirplanes”July,19,2006.[http://www.nscee.edu/NSCEE_hpc_docs/BoeingTestimony.pdf]

[2]CFDEducationinUndergraduateCurriculumWorkingGroup[https://info.aiaa.org/tac/ASG/FDTC/DG/CFD_Ed.aspx]

[3]W.L.OberkampfandT.G.Trucano.“VerificationandValidationinComputationalFluidDynamics”,SandiaReport2002‐0529,2002.

[4]R.R.Cosner,W.L.Oberkampf,C.L.Rumsey,C.R.RahaimandT.I.‐P.Shih,“AIAACommitteeonStandardsforComputationalFluidDynamics:StatusandPlans,”AIAAPaper2006‐0889,44thAIAAAerospaceSciencesMeeting&Exhibit,Reno,NV,2006.

[5]http://turbmodels.larc.nasa.gov/index.html

[6]https://www.xsede.org/

[7]http://www.cfd‐online.com/Links/soft.html