1. 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

2. 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
Isentropic (no shocks)
Seek rates of change along physical waves
Continuity
Momentum
Irrotational

3. ABREVIATED DERIVATION
Geometrical Statement
See Figure 9.6(a)
Continuity + Momentum
Characteristic Equations
Introduction of m+ and m–
For 2D, r → ∞
q+w = constant on characteristic m+ = I+
q-w = constant on characteristic m– = I–

4. NOMENCLATURE u n m– q+m Left Running Mach Wave m q m Reference Axis
Right Running
Mach Wave m+ n
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–

5. 2-D HALF NOZZLE: HOMEWORK #5, QUESTION 4
Table 1: Method of Characteristics Results for 2-D Shaped Nozzle
Point
I+ (q+w)
I- (q-w)
q(º)
w(º) M
m(º)
m+q
m-q
1.1 1 2 3 4 5 6 7 8 9 10

6. METHOD OF CHARACTERISTICS EXAMPLE

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

8. AIAA 2nd Responsive Space Conference 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

9. LINEAR AEROSPIKE ENGINE (X-33)

10. AEROSPIKE NOZZLE PERFORMANCE ENHANCEMENT