6. Gas dynamics. Ideal gases Speed of infinitesimal disturbances in still gas
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1 6. Gs dynmics Dr. Gergely Krisóf De. of Fluid echnics, BE Februry, 009. Seed of infiniesiml disurbnces in sill gs dv d, dv d, Coninuiy: ( dv)( ) dv omenum r r heorem: ( ( dv) ) d q m dv d dv llievi heorem d In sel In wer In ir ~5000 m/s ~500 m/s ~340 m/s Equion of se: Idel gses R We lso ssume h he secific hes re consn. Inernl energy: u cv Enhly: h u c Ru Secific gs consn: R c cv ; 834 R ir 87 9 J kg K Rio of secific hes: c cv eg. for ll diomic gses:.4
2 he seed of sound in idel gses We ssume isenroic comression, which is very fs nd he effec of he fricion is negligible, hus: cons. ln ln ln d 0 ( cons. ) d R R Eg. for ir: 0 C: 33 m/s 0 C: 343 m/s Nonliner wve rogion Wh if we genere noher smll disurbnce? dv v dv v > becuse: - he second wve roges in gs flow of dv velociy. - he second wve roges in gs flow hving higher seed of sound:. he second wve will cch u o he firs wve. comression wve is seeening, nd finlly i becomes shock wve: Shock wves reed s disconinuiy (finie jum) of he se vribles (,, nd ). Proges fser hn he smll disurbnces. (Only shock wves cn do so.) Exnsion wves behve in he oosie wy: Decelerion of suersonic flows re generlly cused by shock wves. I is dissiive rocess. (Cuses hed losses.)
3 nlogy Wves breking in shllow wer nlogy Hydrulic jum in sink Resonnce in closed ie φ Pie lengh: 6.05 m Dimeer: 36 mm Pison dislcemen: 50 cm 3. [P] Hz we mesured: Crnk ngle φ [rd] 3
4 Progion of smll disurbnces in subsonic nd in suersonic flow Posiions of n objec hving velociy v ime insns 0,-,- nd -3 seconds nd lso showing he wve frons sred in hose insns: v0 v< subsonic v v> suersonic licion Schlieren imge of gun fire [h:// ch cone µ v ch number: v ch ngle: µ rc sin rc sin v 4
![below: [n lbum of fluid moion] Shericl rojecile o](/docs-images/94/119476776/images/5-1.jpg "he soluion nlogy Cerenkov rdiion Vrible")
5 Problem #6. Esime he ch number on he bsis of he shdowgrm below: [n lbum of fluid moion] Shericl rojecile o he soluion nlogy Cerenkov rdiion Vrible cross-secion chnnel () Coninuiy: Euler equion: Definiion of : d dv 0 v d v dv d gs flow, v, nd deend only on x x dv d v v d 3 dv d dv v v dv d v ( ) 5
6 Vrible cross-secion chnnel () ( ) dv d v ccelerion Decelerion Subsonic < Convergen Divergen Suersonic > Divergen Convergen If hen d0: he re hs n exreme vlue (minimum). gs flow < > Lvl nozzle ex subsonic flow rns-sonic flow wih norml shock suersonic flow x Under-exnded: Suersonic jes Over-exnded: [n lbum of Fluid oion, 68] 6
7 Energy equion () W Q v v r r r r ( u ) dv ( u ) v d Q W v d V V For sedy se: v r r ( h ) v d Q W Denoing he mss weighed verge of he sgnion (ol) enhly in crosssecions nd by h, nd h,, i reds: ( h h ) q Q W,, m Energy equion () hin srem ube he srem ube cn be regrded s moving wll. We ly he energy equion for sedy flow under he following ssumions: -he srem ube is hermlly isoled (Q0); -he sher sress is 0 over he srem ube (W0). We obin: h, h, Isenroic flow () I. lw of hermodynmics: for n idel gs: ds ( ) ds du d cvd cvd R for isenroic flow: d c v R R c cv cv cv d ( ) 7
8 Isenroic flow () d d ( ) d d d d ( ) d d ( ) Isenroic flow (3) Reference ses 0 mx 0 Isenroic flow (5) We cn exress emerure s funcion of : v h h v c c R c v ( ) c 8
9 9 Isenroic flow (6) Locl ressure nd densiy cn be exressed in erms of he ch number hrough he isenroic relions: he criicl rios (for he se of ): For.4: ( ) f Isenroic flow (8) v q m ss flow-re: q m ( ) q m m q Isenroic flow (9) he inverse of he bove funcion lso gives he ch number for given /.
10 Problem #6.3 ou ) Wh is he oimum ou / rio of he nozzle of rocke hruser designed for ner ground fligh, if he chmber ressure 0 br, nd.3. Plese, use he gs bles! b) Clcule he mss flow-re for 300 K, R46 J/kg-K nd ou 0 cm! c) Plese, clcule he hrus! o he soluion hrus funcion he momenum heorem for vrible cross-secion sedy chnnel flow reds: F ro ( v ) ( v ) ( ) 0 F ro F.3.5 F F F v F ( v ) v known funcions of. E.g: Norml shock wves () v v 4 unknowns. We cn elimine one by using: R Coninuiy: omenum low: Energy equion:,,,, v v v ( v ) ( v ) sedy flow is observed! v v c v c v 0
11 Norml shock wves () ch number ws he key o isenroic flows we should ry o solve his roblem for ( ). v... / R ( R )... v v... v ( )... v c... R v... c... Norml shock wves (3) () (b) (c) ( R ) /... ( )... R... b - c 0.5 ( ) ( ) I is qudric formul for We cn rrnge i ino he olynomil form: (...) (...) (...) 0 4 Norml shock wves (4) his brnch belongs o n exnsion shock. Is i vlid?
12 Norml shock wves (5) Pressure rio: ( ) f (b) emerure rio: ( ) g (c) ( ) h Norml shock wves (6) / / / / Problem #6.4 here is srong sionry norml shock in divergen chnnel he cross-secion chrcerized by w. in ou w.4 in in kp 00 K in 70 3 ou in / w in / ) Clcule he ch number he oule ( ou )! b) Plese, deermine he oule ressure ( ou )! o he soluion
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