Implicit flux-split Euler schemes for unsteady aerodynamic analysis involving unstructured dynamic meshes

and 12, however,usea spatialdiscretization basedon central differencingwith explicit artificial dissipation,and use a Improved algorithms for the solution of the timetemporal discretization involving explicit time-marching dependent Euler equations are presented for unsteady basedon a multi-stageRunge-Kuttatime integration. The aerodynamicanalysisinvolvingunstructuredynamicmeshes, explicitartificialdissipationusedin such schemestendsto The improvements havebeendevelopedrecentlyto the spatial smearshockwavesoverseveralgridcellsand requiresthe and temporaldiscretizations used by unstructuredgridflow tuningof free parametersthatscalethe dissipation.Also,the solvers. The spatial discretizationinvolvesa flux-split explicitRunge-Kuttatime-integration has a step size that is approachwhichis naturallydissipativeand capturesshock limitedby the Courant-Fredricks-Lewy (CFL) conditionto wavessharplywith at mostone grid pointwithinthe shock verysmallvalues. Consequently, thousands(andoccasionally structure. The temporaldiscretizationinvolvesan implicit tensof thousands) of timestepsare requiredto obtainsteadytime-integrationscheme using a Gauss-Setdelrelaxation statesolutions, andthousandsof stepsper cycleof motionare procedurewhichis computationally efficientfor eithersteady requiredforunsteadysolutions.Therefore, the purposeof the or unsteadyflow problems. For example,very large time paperis to reporton improvements thathavebeendeveloped stepsmaybe usedfor rapidconvergence to steadystate,and recentlyto the spatialand temporaldiscretizationsof the the stepsize for unsteadycasesmaybe selectedfortemporal unstructuredgrid flow solverswhichresolvethe numerical accuracyrather than for numericalstability. Steady and issues describedabove. The spatial discretizationnow unsteadyflowresultsare presentedforthe NACA0012airfoil involvesa so-calledflux-splitapproach,whichis similarto to demonstrateapplicationsof the new Eulersolvers. The discretizations presentedin Refs. 10, 13, and 14 basedon unsteady results were obtained for the airfoil pitching either the flux-vectorsplitting(FVS) of van Leer15 or the harmonicallyabout the quarter chord. The resulting flux-differencesplitting(FDS) of Roe.16 These flux-split instantaneouspressuredistributionsand lift and moment discretlzationsaccount for the local wave-propagation coefficientsduring a cycle of motioncomparewell with characteristicsof the flow and they captureshockwaves experimental data. The paperpresentsa descriptionof the sharply with at most one grid point within the shock Eulersolversalongwithresultsandcomparisons whichassess structure. A furtheradvantageis that these discretizations the capability, are naturallydissipativeand consequentlydo not require additionalartificialdissipationterms or the adjustmentof free parametersto controlthe dissipation.Furthermore, the Introduction temporaldiscretizationhasbeenchangedto an implicitimeintegration scheme involving a Gauss-Seidel relaxation Considerable progresshas beenmadeover the past two proceduresimilarto discretizations presentedin Refs. 17 and decadeson developingcomputational f uid dynamics(CFD) 18. This relaxationschemeis unconditionally stableandthus methodsfor aerodynamic analysis.l,2RecentworkinCFD has allowsthe selectionof the stepsize basedon the temporal focusedprimarilyon developingalgorithmsforthe solutionof accuracydictatedby the problembeingconsidered, ratherthan the Euler and Navier-Stokesequations. For unsteady on the numericalstabilityof the algorithm. Consequently, aerodynamic andaeroelasticanalysis, thesemethodsgenerally very largetime stepsmaybe usedfor rapidconvergenceto requirethat the meshmoveto conformto the instantaneous steadystate,andan appropriate stepsizemaybe selectedfor positionof the movingordeformingbody underconsideration, unsteadycases, independentof numericalstabilityissues. Manyof themethodsthatarecurrently beingdevelopedassume Steadyandunsteadyresultsare presentedfor the NACA0012 that the mesh movesrigidlyor that the meshshearsas the airfoil to demonstrateapplicationsof the new Eulersolvers. body deforms. These assumptionsconsequentlylimit the The unsteady flow results were obtained for the airfoil applicabilityof the proceduresto rigid-bodymotionsor pitchingharmonicallyaboutthe quarterchord. The paper small-amplitudedeformations. Furthermore,these methods presentsa descriptionof the Eulersolversalongwithresults of solutiontypicallyassumethatthecomputational gridhas an andcomparisons whichassessthecapabilily. underlying geometrical structure. As an alternative, • algorithmshave beendevelopedrecentlywhichmakeuseof unstructuredgrids. 3-12 In twodimensionsthesegridsare ._;I[LP, __J_ typicallymadeup of triangles,andin threedimensionsthey consistof an assemblageof tetrahedra.Theunstructured grid Inthepresentstudy, the flowis assumedto begovernedby methodshavedistinct advantagesoverstructured gridmethods the two-dimensional time-dependent Euler equationswhich in that they can easily treat the mostcomplexof geometric maybe writtenin integral formas configurationsas well as flow conditions,and that the unstructured gridcan be movedto treatrealisticmotionsand "_tJ'J" Qdxdy +fa(Fdy -Gdx )= 0 (1) structural deformationsof these configurations. 10"12 n wherethe vectorof conservedvariablesQ andthe convective The resultspresentedby the authorin Refs. 11 and 12 fluxesF andG aregivenby demonstratedthat (1) the methods producesolutionsof comparableaccuracyto resultsobtainedusingstructuredgrid methodology can easilyanalyzecomplexaircraftgeometries undergoingstructural deformation. 12 Themethodsof Refs.11 Q = (2a) *ResearchScientist, UnsteadyAerodynamics Branch, _P:)