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).
( ) ( ) ( ) ( ) (.) 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, ± ±
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 deflected @, 9 AX tn γ - γ THEORETICAL (4.44) TbleA.5 () 9 O 9.449 3.45 O
( ) ( ) ± ± 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, < >
defelction through ch wve µ sin dx dx dx chrcteristic chrcteristic chrcteristic ch wve ngle tn µ µ ( ± µ ) (.48)
ETHOD OF CHARACTERISTICS Eqution Summry υ @ 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)
cnceltion mx Expnsion region Left nd right wves nd - Curved chrcteristics reflection Strightening region Left wves only only Stright chrcteristics
LONG NOZZLE rbitrry expnsion section length left nd right expnsion wves mx
P- compression wve P- expnsion wve REFLECTION CANCELATION
α µ α µ -
INIU LENGTH NOZZLE no reflections υ
ETHOD OF CHARTACTERISTICS Initil ACH Line grid ch lines re chrcteristics nd - constnt on ch lines υ ( ) design mx 3 υ @, First Chrcteristic Vlue (.)
b c ( ) c b design c c c c (.) @ c (.) @, mx mx mx mx mx mx mx
ETHOD OF CHARACTERISTICS Chrcteristic Vlue Clcultion υ on center line deflection is (.) (.) initil sets the initil ch wve fn
Point υ υ α α ( ) ( ) CALCULATION SPREAD SHEET υ @ nozzle design mx divide into initil ch wve fn µ mx υ υ ( ) ( ) µ µ
DESIGN ACH NuBER.4 AXIU 8.377 NUBER OF WAVES 7 Exmple., pge 4 INITIAL.375 3 Pt.75.375.375.4 74. 73.75 74.475 6.75 3.375 3.375.9 57. 53.85 6.575 3.75 6.375 6.375.3 49.8 43.45 56.75 4 8.75 9.375 9.375.4 45. 35.85 54.575 5 4.75.375.375.5 4. 8.75 53.475 6 3.75 5.375 5.375.6 38..75 53.475 7 36.75 8.375 8.375.7 35.6 7.5 53.975 8 36.75 8.375 8.375.7 35.6 7.5 53.975 9 6.75-6.75 6.75.3 49.3 49.3 49.3.75-6.75 3 9.75.53 44.4 4.4 47.4 8.75-6.75 6.75.63 37.8 3.8 43.8 4.75-6.75 9 5.75.73 37.3 8.3 46.3 3 3.75-6.75 8.75.73 35.3 3.3 47.3 4 36.75-6.75 5.75.84 3.9 7.9 47.9 5 36.75-6.75 5.75.84 3.9 7.9 47.9 υ ) / ) / ( ( µ α α
6.75 -.75.75.53 4.8 4.8 4.8 ( ) / ( ) / 7 8.75 -.75 3 5.75.63 37.8 34.8 4.8 8 4.75 -.75 6 8.75.73 35.3 9.3 4.3 9 3.75 -.75 9.75.84 3.9 3.9 4.9 36.75 -.75 4.75.94 3 9 43 36.75 -.75 4.75.94 3 9 43 8.75-8.75 8.75.73 35.3 35.3 35.3 ( ) / ( ) / 3 4.75-8.75 3.75.84 3.9 9.9 35.9 4 3.75-8.75 6 4.75.94 3 5 37 5 36.75-8.75 9 7.75.5 9.. 38. 6 36.75-8.75 9 7.75.5 9.. 38. 7 4.75-4.75 4.75.94 3 3 3 ( ) / ( ) /. 8 3.75-4.75 3 7.75.5 9. 6. 3. 9 36.75-4.75 6 3.75.6 7.6.6 33.6 3 36.75-4.75 6 3.75.6 7.6.6 33.6 3 3.75-3.75 3.75.6 7.6 7.6 7.6 ( ) / ( ) / 3 36.75-3.75 3 33.75.8 6 3 9 33 36.75-3.75 3 33.75.8 6 3 9 34 36.75-36.75 36.75.4 4.6 4.6 4.6 35 36.75-36.75 36.75.4 34.6 34.6 34.6
.84.3.84.73.7.63.53.6
ETYHOD OF CHARACTERISTICS Contour Construction α i υ α i α i α i