uv 2 c v a N D u a 2 (1

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Kerrie McCoy
5 years ago
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1 3 simultneous liner equtions in ( Φ xy u Φ Φ xx xy ) Φ dx dx xx u ( ) dx du u uv ( ) - dx du uv Φ c Φ xy dv dv xy v ( ( Φ v v yy Φ ) ) Φ yy N D du from (.6) dv Φ from (.5) from (.7) xy is indeterminte on chrcteristic of When both N nd D re, (.8) Φ is indeterminte. N of xy D defines the chrcteristic the solution, defines properties long the chrctersitic. the solution. - nd f(x, y).

2 ( ) ( ) ( ) ( ) (.) chrcteristic long C υ() (.) chrcteristic long C υ() TbleA.5 (4.44) tn γ γ tn γ γ υ - eyer Function the Prndtl, υ d V dv d u v u uv - ) v ( ) u ( dv du )dxdv v ( du ) u ( numertor N, ± ±

3 Subsititying into υ d υ ( ) ( 4.35) d d γ Prndtl eyer Function Tble A.5 ( ) υ( ) γ γ ( ) tn ( ) tn ( ) (), ngle υ γ γ γ d γ d d, (4.45) dv V the Prndtl eyer Function is the mximum through supersonic flow cn be 9 AX tn γ - γ THEORETICAL (4.44) TbleA.5 () 9 O O

4 ( ) ( ) ± ± chrcteristic chrcteristic u uv dx (.) u v u uv dx ) v ( dx uv dx ) u ( )(dx) v ( dx uv ) u ( D,deno min tor eqution. differentil prtil ellipticl chrcteristics two imginry, eqution. differentil prbolic prtil chrcteristic through ech point one rel, eqution. differenti hyperbolic prtil chrcteristics rel, < >

5 defelction through ch wve µ sin dx dx dx chrcteristic chrcteristic chrcteristic ch wve ngle tn µ µ ( ± µ ) (.48)

6 ETHOD OF CHARACTERISTICS Eqution Summry nozzle design mx P Expnsion υ i ACH - µ sin Chrcteristic Vlues υ i Lines υ constnt long υ re (.) (.) constnt long Chrcteristics chrcteristic chrcteristic combining, ( ) (.4) υ ( ) (.5) Angle- Chrcteristic nd initil flow, α dx - tn dx α µ α µ - chrcteristic for chrcteristic for tn(µ ± ) (µ ± ) Chrcteristic Chrcteristic (.4)

7 cnceltion mx Expnsion region Left nd right wves nd - Curved chrcteristics reflection Strightening region Left wves only only Stright chrcteristics

8 LONG NOZZLE rbitrry expnsion section length left nd right expnsion wves mx

9 P- compression wve P- expnsion wve REFLECTION CANCELATION

10 α µ α µ -

11 INIU LENGTH NOZZLE no reflections υ

12 ETHOD OF CHARTACTERISTICS Initil ACH Line grid ch lines re chrcteristics nd - constnt on ch lines υ ( ) design mx 3 First Chrcteristic Vlue (.)

13 b c ( ) c b design c c c c c mx mx mx mx mx mx mx

14 ETHOD OF CHARACTERISTICS Chrcteristic Vlue Clcultion υ on center line deflection is (.) (.) initil sets the initil ch wve fn

15 Point υ υ α α ( ) ( ) CALCULATION SPREAD SHEET nozzle design mx divide into initil ch wve fn µ mx υ υ ( ) ( ) µ µ

16 DESIGN ACH NuBER.4 AXIU NUBER OF WAVES 7 Exmple., pge 4 INITIAL Pt υ ) / ) / ( ( µ α α

18 ( ) / ( ) / ( ) / ( ) / ( ) / ( ) / ( ) / ( ) /

21 ETYHOD OF CHARACTERISTICS Contour Construction α i υ α i α i α i

MASS+MOMENTUM u a. v x. u x. v a. u y. v y. v x ORGANIZE SO AS TO SOLVE FOR THE VELOCITY

MASS+MOMENTUM u a. v x. u x. v a. u y. v y. v x ORGANIZE SO AS TO SOLVE FOR THE VELOCITY The Method of Chrcteristics D Isentropic Compressible Flow IRROT. v x u y MASS+MOMENTUM u u x v v y uv ( 1 ) (1 ) v x ORGANIZE SO AS TO SOLVE FOR THE VELOCITY DERIVATIVES Prtil Differentil Eqns.? TO MATRIX