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= ± u 0 and ρρ ⎛⎞ +∇=⇒ + ω ×+∇= ⎜⎟ ⎝⎠ u ru r u r ± 2 11 po p0 2 u uu u Use intrinsic co-ordinates Eq. of motion along s : r 1 o ∂∂ + = up u ss ρ (1) Eq. of motion along n: r 22 12 == u Rn n Now 1 ∂ϑ Rs 2 ∂ϑ =− u sn o ϑ + = u u ns (2) ( g o o d n o p i n i t ) (can also get this from o u ) Continuity: () 2o π δ= urn s 16.512, Rocket Propulsion Lecture 8 Prof. M. Martinez-Sanchez Page 1 of 20

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1111 o ∂∂∂ δ ++ + = ur n susrs ns ρ () 1 sin sin ∂∂ δϑ = ϑ r ss sr ϑ n ∂δ ∂ϑ ⎛⎞ = −δ δ ⎜⎟ ⎝⎠ n ds n s sn 11 s i n u ϑ +− = u n r ϑ Homentropic: == dp 22 s p c d c 16.512, Rocket Propulsion Lecture 8 Prof. M. Martinez-Sanchez Page 2 of 20
s o ρ ∂∂ ϑ +− = 2 ϑ 1 s i n c u sus n r 1p and, from s eq. of motion, =− u u ss so you can eliminate p s : 2 1s ϑ −+ = uu u r c i n ϑ 2 2 1 ⎛⎞ i n ∂ϑ ϑ + = ⎜⎟ ⎝⎠ us n c r M 2 -1 2 i n ϑ += Mu nr ϑ (4) Introduce the Mach angle 11 2 sin tan 1 −− == M M μ 2 1 tan 1 = M Then (2) 2 1 tan o −∂ ∂ ϑ + = un s And (4) 2 1t a n tan −∂ ϑ ϑ n r s i n 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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t h e n tan o tan sin tan ∂ω ∂ϑ += ∂∂ ∂ω ∂ϑ ϑ ns sn r 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 ⎛⎞ + ϑ = + ϑ = ⎜⎟ ⎝⎠ ϑ ϑ ϑ−ω ϑ−ω =− r r r r μμ sin sin +− ϑ In (s, n) coordinates, 11 cos cos ⎫⎧ == ⎬⎨ ⎩⎭ rr so cos sin 1 cos sin 1 + + = −= = r ± r ± m m ( m + , m - are lengths along c haracteristics) inclined inclined + ϑ+ ϑ− m m sin sin sin sin + ϑ ⎧⎫ ϑ+ω =+ ⎪⎪ ⎨⎬ ϑ r m r m or 2-D, r →∞ ϑ+ω = const . along m + I + (inclined F ) = const . along m - I - (inclined ) 16.512, Rocket Propulsion Lecture 8 Prof. M. Martinez-Sanchez Page 4 of 20
2-D Simple Regions Consider a uniform region; the flow from it enters some disturbed region, like a wall turning. One of the m families originate in the uniform region (m + in example) and carries a constant invariant, e.g. oo ϑ+ω=ϑ +ω everywhere downstream. Along one of the other ch nce aracteristics (m - here), we carry a constant (which varies from ch. To ch. of that family); we can evaluate it at the wall, for insta ww −ω=ϑ −ω along each m - line and ϑ ( ) woow w o so 2 ω o =ϑ +ω − then w l o n g e a c h m ϑ=ϑ +ω +ϑ −ω = ϑ ϑ=ϑ ω=ϑ+ω−ϑ−ω → ω=ϑ+ω−ϑ =ω o

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## This note was uploaded on 11/07/2011 for the course AERO 16.512 taught by Professor Manuelmartinez-sanchez during the Fall '05 term at MIT.

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lecture_4_5 - 16.512, Rocket Propulsion Prof. Manuel...

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