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lecture_4_5

# lecture_4_5 - 16.512 Rocket Propulsion Prof Manuel...

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16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 4-5: Nozzle Design: Method of Characteristics The Method of Characteristics (Ideal Gas) (Ref. Phillip Thompson Compressive Fluid Dynamics , McGraw Hill, 1972, Ch. 9) 2-D or axisymmetric Homentropic as well as isentropic o ∇ × = ur u plus ( ) 0 ρ = ur ± u 0 and ρ ρ + = ⇒ ∇ + ω × + = ur ur ur ur ± 2 1 1 p o p 0 2 u u u u Use intrinsic co-ordinates Eq. of motion along s : r 1 o + = u p u s s ρ (1) Eq. of motion along n : r 2 2 1 2 = = − u p u R n n ρ Now 1 ∂ϑ R s 2 ∂ϑ = − u u u s n o ∂ϑ + = u u n s (2) (good no p in it) (can also get this from o ∇ × = ur u ) Continuity: ( ) 2 o π δ = u r n s ρ 16.512, Rocket Propulsion Lecture 8 Prof. M. Martinez-Sanchez Page 1 of 20

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1 1 1 1 o ∂δ + + + = u r n s u s r s n s ρ ρ ( ) ( ) 1 sin sin δ ϑ = ϑ r s s s r ϑ n ∂δ ∂ϑ = δ δ n ds n s s n 1 1 sin u ∂ϑ + = − u s s n r ρ ρ ϑ Homentropic: ρ ρ = = dp 2 2 s p c d c 16.512, Rocket Propulsion Lecture 8 Prof. M. Martinez-Sanchez Page 2 of 20
so ρ ∂ϑ + = − 2 ϑ 1 sin c u s u s n r 1 p and, from s eq. of motion, ρ = − 1 p u u s s so you can eliminate p s : 2 1 s ∂ϑ + = − u u u s u s n r c in ϑ 2 2 1 s 1 in ∂ϑ ϑ + + = u u u s n c r M 2 -1 2 1 sin ∂ϑ + = M u u s n r ϑ (4) Introduce the Mach angle 1 1 2 1 1 sin tan 1 = = M M μ 2 1 tan 1 = M μ Then (2) 2 1 tan o ∂ϑ + = M u u n s μ And (4) 2 1 tan tan ∂ϑ ϑ + = M u u s n r μ μ sin Introduce the Prandtl-Meyer function ( ) M ω by 2 d d (to be integrated later) 1 ω = u M u 16.512, Rocket Propulsion Lecture 8 Prof. M. Martinez-Sanchez Page 3 of 20

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then tan o tan sin tan ∂ω ∂ϑ + = ∂ω ∂ϑ ϑ + = n s s n r μ μ μ add and subtract to obtain the “Characteristics form” (single differential operator per equation) ( ) ( ) ( ) ( ) tan sin sin sin tan cos sin tan sin sin sin tan cos sin ϑ + ω + ϑ = + ω + ϑ = ϑ ϑ ϑ − ω ϑ − ω = − s n r s n r s n r s n r μ μ μ μ μ μ μ μ μ μ sin sin + ϑ In (s, n) coordinates, 1 1 cos cos = = r r μ μ μ μ so cos sin 1 cos sin 1 + + + = ∇ = = ∇ = r ± r ± s n m s n m μ μ μ μ (m + , m - are lengths along c haracteristics) inclined inclined + ϑ + ϑ − m m μ μ ( ) ( ) sin sin sin sin + ϑ ϑ + ω = + ϑ ϑ − ω = − r m r m μ μ or 2-D, r → ∞ ϑ + ω = const . along m + I +
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lecture_4_5 - 16.512 Rocket Propulsion Prof Manuel...

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