MAE 4262 ROCKETS AND MISSION ANALYSIS Method of

MAE 4262: ROCKETS AND MISSION ANALYSIS Method of Characteristics Summary and Rocket Nozzle Examples Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

METHOD OF CHARACTERISTICS • • Universal way to examine supersonic flow Name comes from looking at characteristic lines (Mach waves) propagating downstream Detailed derivation in Section 9. 3 of Compressible Fluid Dynamics, by Philip Hill Use intrinsic coordinates, see Figure 9. 6(a) shown below Continuity Momentum Irrotational • • Isentropic (no shocks) Seek rates of change along physical waves

ABREVIATED DERIVATION Geometrical Statement See Figure 9. 6(a) Continuity + Momentum Characteristic Equations Introduction of m+ and m– For 2 D, r → ∞ q+w = constant on characteristic m+ = I+ q-w = constant on characteristic m– = I–

NOMENCLATURE Le M ft R ac u h nn W in av g e m– m q n q+m Right R un Mach W ning ave u m Reference Axis q-m m+ q: Angle between u and Reference Axis m: Angle between u and ± Mach Waves n: reference normal vector, perpendicular to u (or streamline) q+w = constant on characteristic m+ = I+ q-w = constant on characteristic m– = I–

2 -D HALF NOZZLE: HOMEWORK #5, QUESTION 4 Table 1: Method of Characteristics Results for 2 -D Shaped Nozzle I+ (q+w) I- (q-w) q(º) w(º) M Point 0 1. 1 1 2 3 4 5 6 7 8 9 10 3 m(º) m+q m-q

METHOD OF CHARACTERISTICS EXAMPLE

SPIKE EXAMPLE Ideal Aerospike Contour Centerline Spike Nozzle a ho Pc O Cowl he Me=3 Pa qo L

AIAA 2 nd Responsive Space Conference 2004 -7003 A STATUS REPORT ON THE DEVELOPMENT OF A NANOSAT LAUNCH VEHICLE AND ASSOCIATED LAUNCH VEHICLE TECHNOLOGIES by: John M. Garvey and Eric Besnard Figure 6: Static Fire Test of Aerospike Nozzle. From Ignition to Continuous Operation

LINEAR AEROSPIKE ENGINE (X-33) http: //www. msfc. nasa. gov/news/photos/1998/photos 98 -204. htm http: //www. nasaexplores. com/show 2_articlea. php? id=01 -001 http: //observe. arc. nasa. gov/nasa/ootw/1997/ootw_971001/x 33. html

AEROSPIKE NOZZLE PERFORMANCE ENHANCEMENT