Solution of the Falkner_Skan equation to determine reverse flow.pdf

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1、SolutionoftheFalkner–SkanequationtodeterminereverseflowL.QuartapelleandA.Scandroglio18May20061.Contents1Introduction91.1Prandtlboundarylayerequations..................101.2Streamfunctionformulation....................132Falkner–Skanproblem132.1Achangeofvariables........................132

2、.2Self-similarequation........................152.3Reductiontoafirst-ordersystem..................192.4Boundaryandintegralconditionsforthefirst-ordersystem....203Sixth-orderLinearMulti-PointMethod213.1Constructionofthesixth-orderaccuratescheme..........213.2Linearsystemwithblockborderedqua

3、dridiagonalmatrix.....263.3Directsolutionalgorithmfortheblocklinearsystem.......283.4Newtonmethodfortheblocknonlinearsystem..........314NumericalsolutionoftheFalkner–Skanproblem334.1TheinitialguessforNewtonmethod................334.2Thesupplementaryvorticitycondition...............375Con

4、tinuationmethod395.1ThecontinuationmethodappliedtotheFalkner–Skanproblem..395.2Reverseflowsolutions.......................415.3Discussionofsomeknownresults.................456Conclusion46..Two,ofcoursetherearetwo.ItseemsperfectlynaturalnowÐÐDeath&Co.,14November1962SylviaPlath..AbstractThi

5、sworkisconcernedwithsteady,two-dimensional,incompressiblevis-cousflowswithintheapproximationofPrandtlboundarylayertheoryforthecaseofasemi-infiniteplate.Theanalysisoftheequationsgoverninglaminarflowofthiskindisbasedontheconceptofsimilaritysolutionsandleadstoasingleordinarydifferentialequatio

6、nknownasFalkner–Skanequation.Itssolutionsarecomputedbymeansofanewnumericalmethodwhichreducesthisthirdorderequationtoafirst-ordersystem.TheproposedschemeisaLinearMulti-PointMethodofsixth-orderaccuracysuggestedbyPaoloLu-chini,implementedhereforthefirsttime.Thisoptimalschemediscretizesthefirst

7、-ordersystemsupplementedbytwo-pointboundaryconditionsbymeansofablockborderedquadridiagonalmatrix.Thediscretization“cir-cumvents”thefirstDahlquist'sstabilitybarrierthankstothepresenceofonefullrowintheborderedsystemofalgebraicequationswhichallowstheexactrespectofthefundament