Moving Normal Shocks -1 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Moving Normal Shocks So far,

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Moving Normal Shocks -1 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Moving Normal Shocks So far, considered changes across shock wave for the case of the shock not moving –observer “sitting” on the shock, moving with shock What happens to properties if we consider the shock to be moving –observer not moving at same speed as shock v1p11T1v1p11T1 v2p22T2v2p22T2 1 2

Moving Normal Shocks -2 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Coordinate Transformation –Galilean transform –switch directions (+) and add shock speed, v s Now shock problem looks same as stationary (steady) problem that we already solved vivi vgvg vsvs + transform + v 1 =v s -v i v 2 =v s -v g v s -v s =0 First, convert moving shock to stationary shock

Moving Normal Shocks -3 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Shock Property Ratios Static properties –property you would measure if moving with flow –so, unaffected by transformation –e.g., still use with M 1 =v 1 /a 1 ; a 1 2 =  RT 1 ; T 1 =T i ; etc. vivi vgvg vsvs transform v 1 =v s -v i v 2 =v s -v g Stagnation properties –depend on velocity; not same after transform –find using static properties and M i, M g

Moving Normal Shocks -4 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Example: Known Shock Speed T i =300 K p i = 1 atm v s =520 m/s Given: Normal shock moving at 520 m/s into still air (300 K, 1 atm) Find: 1.Temperature behind shock (after shock passes) 2.Velocity of gas behind shock (in “lab” reference frame) 3.Stagnation temperature before and after shock (in lab ref. frame) Assume: Air TPG/CPG with  =1.4 v g T g T og T oi

Moving Normal Shocks -5 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Solution: Known Shock Speed Analysis: Transform to stationary shock T i =300 K p i = 1 atm v s =520 m/s T g T og T oi vgvg transform v 1 =v s v 2 =v s -v g T2T2 –find M 1 in stationary frame –M 2 from B.1 or (VII.11)

Moving Normal Shocks -6 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Solution: Known Shock Speed Analysis (con’t): T i =300 K p i = 1 atm v s =520 m/s T g T og T oi vgvg transform v 1 =v s v 2 =v s -v g T2T2 –T 2 from B.1 or (VII.9) –v g from B.1 or (VII.8) –T oi, T og T o not same for moving shock

Moving Normal Shocks -7 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Example: Known Postshock Pressure Given: Supersonic projectile (or equivalently piston) pushing gas ahead in tube filled with initially still air Find: 1.shock speed (v s ) (lab reference frame) 2.projectile speed (v p ) (lab reference frame) Assume: Air TPG/CPG with  =1.4 v p T 1 =294 K p 1 = kPa v i =0 p 2 =608 kPa vg=vg= vsvs T2T2 –p 2 measured –leading shock produced

Moving Normal Shocks -8 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Solution: Known Postshock Pressure Analysis: Transform to stationary shock –use p information B.1/ (VII.12) M’s + B.1 or (VII.9) in lab reference frame, supersonic behind shock in shock’s ref. frame, subsonic behind shock transform v 1 =v s v 2 =v s -v p v g = v p vsvs T2T2 T 1 =294 K,

Moving Normal Shocks -9 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Example: Postshock Speed Known Given: Normal shock moving into still gas (at T 1 ) produces known gas speed (v g ) behind shock Find: –Expression for shock speed v s in terms of v g Assume: –TPG/CPG vgvg T1T1 vsvs v i =0

Moving Normal Shocks -10 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 (VII.18) another normal shock relation Solution: Postshock Speed Known Analysis: In stationary ref. frame –velocity ratio v 2 =v s -v g vsvs T1T1 (VII.19)

Moving Normal Shocks -11 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Numerical Example: Known v g Given: Piston impulsively set into motion at 400 m/s in 25cm 2 tube filled with initially still 290K, 100 kPa Find: 1.Mach number of shock, M s (relative to unshocked gas) Assume: Air TPG/CPG with  =1.4, no friction on piston v p vg=vg= MsMs p2p2 –leading shock produced T 1 =290 K p 1 =100 kPa v i =0 v p = 400 m/s FpFp 2.Force, F p, required to keep piston moving

Moving Normal Shocks -12 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Solution: Known v g Analysis: Transform to stationary shock –use (VII.19) transform v 1 =v s v 2 =v s -v p v g = v p vsvs p2p2 T 1 =290 K p 1 =100 kPa 235 lb f to move 2.2 inch diam. piston w/o friction B.1 or (VII.12) M 1 =M s ( v i =0)