4262: Rockets and Mission Analysis
Homework #6
Assigned: March 16, 2004
Due: March 25, 2004
1. Consider a rocket nozzle designed such that its exit plane Mach number is Me=3.
The chamber pressure is Pc=100 atm, and Cp/Cv=1.25. Calculate the angle  of the
gas flow just after the exit lip when the rocket operates in a 1 atm external pressure
( measured from the wall direction at the exit). Is the rocket ideally-, over-, or
under-expanded? See the figure below for details.

Nozzle Lip
Me=3
Pc=100
=1.25
Figure 1: Nozzle Exit Lip Flow
2. The schematic below shows an ideally expanded 2-D nozzle which expands uniform
sonic flow (with Cp/Cv=1.2) to a uniform Me=3 exit condition. The sonic flow is
directed at an angle o (to be determined) to the final flow direction.
Spike Nozzle

ho
Pc
O
he
Me=3
Pa
L
o
Cowl
Figure 2: 2-D Spike Nozzle Schematic
1
The whole expansion area is a “simple flow region” because it is adjacent to uniform
regions, from which one of the two families of characteristics will propagate a constant
invariant into it. The expansion is centered at the cowl lip (point O), and it is a PrandtlMeyer flow. All of the expansion characteristics pass through O. The flow is chocked at
Mach 1 at the throat location, where the height is ho.
a.
b.
c.
d.
e.
Based on the above statement and schematic, calculate o.
Using continuity, calculate the ratio he/ho
Calculate the angle a made by the last expansion characteristic with the final flow.
Calculate the length ratio L/ho
Calculate the matching pressure ratio Pc/Pa.
3. A detailed calculation is performed using the method of characteristics to size a bell
(shaped) nozzle for a future rocket concept. The throat radius of the rocket (to provide
the necessary mass flow) is 10 cm, and the exit diameter is 2 m, which gives an area
ratio, =100. The method of characteristics solution has yielded an ideal length of 5m
for this nozzle. The designers have determined that this length, although yielding a
perfectly axial flow, will end up weighing too much, and its better to trade some
thrust performance for a reduction in length and weight. The consensus is that the
performance will suitable with a nozzle of length 2 m (instead of the requisite 5m).
Your job is to shape this nozzle. A suggested method is outlined below:

r*
Part 2: Polynomial Fit
y=Ax2+Bx+C
r*
re
Part 1: Circular Arc
r=r*, =45º
Design the nozzle in two parts.
Part (1): For the diverging portion immediately following the throat use a circular arc
with radius equal to the throat radius and terminating at 45 degrees.
Part (2): Fit the end of the circular arc with a polynomial curve. Determine the
coefficients (A, B, and C) of this curve to match with the circular arc and to have zero
slope at the exit of the nozzle.
Plot the nozzle geometry (preferably to scale) and comment on your results. Judging by
the shape, how do you think this nozzle will perform. What are your thoughts on this
method? If you had unlimited resources (time, computers, money) what would be your
next steps to iterating on this nozzle design? If you had limited resources (time,
computers, money) what would be your next steps to iterating on this nozzle design?
2
EXTRA CREDIT 1:
It may be shown that the angle of the Mach line emanating from a point disturbance is
given by:




2
M

1


  arctan 
1
1
Using this relation as well as the expansion of a supersonic flow (with Mach number M
and flow velocity U) around an infinitesimal corner with angle d, show that:
d  M 2  1
2
dU
U
Comment on the physical phenomena associated with dU>0, dU<0, and dU=0 (i.e., what
is happening to the wall and the flow for each of the cases). For this reason, since both
expansions and compression waves may be considered, we can write equation 2 using the
defined variable , as:
d  M 2  1
3
dU
  d
U
Next use the definition of Mach number, U=Ma, and its logarithmic derivative to
eliminate both dU/U and da/a to arrive at an expression solely between turning angle and
Mach number:
M 2 1
d  
1
1
  1M 2
2
4
dM
M
This expression is the basis for the Prandtl-Meyer expansion function. For our interests in
rocket nozzles, the primary application is to expansions, i.e., increasing M and decreasing
. Therefore, for convenience, we may define the Prandtl-Meyer angle (M), which
increases when  decreases and is zero at the sonic point.
d  d
  0; M  1
5.1
5.2
Equation 4 is integrated from the sonic point to any value of M, with the result:
 M 2 1 
  arctan


K


  K arctan 
3
 M  1
2
6.1
K
 1
 1
6.2
The extra credit problem has a fair amount of derivation associated with it. What do you
think is the point of this problem? Once we arrive at equation 6.1, can we better
understand the performance of rocket nozzles? How does this derivation tie in with the
derivation of the governing forms for the method of characteristics?
EXTRA CREDIT 2:
Starting with a general representation of plane streamline flow, derive the characteristic
equations:
 
tan  sin 

  tan       
n 
r
 s
 
tan  sin 

  tan       
n 
r
 s
1.1
1.2
What do these equations imply?
Show that equations 2.1 and 2.2 are the derivatives along the characteristic lines m + and
m-, where the slope o the characteristic lines m+ and m- respectively is given by tan(-)
and tan(+)
d
      sin  sin 

r
dm
d
sin  sin 
     
r
dm 
2.1
2.2
Comment on the results of this derivation. What is the point of deriving all of this? How
does it relate to rocket nozzle design?
4