4262: Rockets and Mission Analysis
Homework #5
Assigned: November 6, 2014
Due: November 20, 2014
Question 1:
In class we developed an expression for the divergence loss of a conical nozzle is developed and
shown to be:

1  cos 
2
Consider a linear (as opposed to axisymmetric) rocket nozzle, with a wedge-shaped exit profile,
as shown below. Assuming all properties are constant on the cylindrical surface (shown by a
dashed line), calculate the divergence loss factor , due to the finite angle .
R


Figure 1: Linear Rocket Nozzle Schematic
Plot the divergence loss for a conical and linear nozzle for divergence angles from 0 to 45
degrees. For how larger of a range of half angles would you believe these results? What physical
phenomenon may begin to occur as the half angle is continually increased?
1
Question 2:
The schematic below shows an ideally expanded 2-D spike 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.
Centerline
Spike Nozzle

ho
Pc
O
he
Me=3
Pa
L
o
Cowl
Figure 1: 2-D Spike Nozzle Schematic
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 Prandtl-Meyer flow. All of the
expansion characteristics pass through the point O. The flow is choked at Mach 1 at the throat
location, where the height is ho.
a. Based on the above statement and schematic, calculate o.
b. Using continuity, calculate the ratio he/ho
c. Calculate the angle  made by the last expansion characteristic with the final flow
direction.
d. Calculate the length ratio L/ho
e. Calculate the matching pressure ratio Pc/Pa.
2
Question 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 it is 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 (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 iterate on this nozzle
design? If you had limited resources (time, computers, money) what would be your next steps to
iterating on this nozzle design?
3
Question 4:
To practice actually computing nozzle contours using the Method of Characteristics, this
problem will ask you to begin a design of a 2-D half-nozzle. The nozzle is to be designed to
expand a =1.25 ideal gas from a uniform M=1.1 condition to a uniform M=3 condition. The
grid to be used will be unrealistically coarse (only 4 expansion Mach lines and their centerline
reflections – but if you can do this, you can as many as required for the accuracy desired). The
configuration to be analyzed is shown below:
7w
4w
9w
11
0
M=3
4
M=1.1
7
3
2
1
9
6
5
8
10
The spacing of the m+ characteristics from the point 0 is arbitrary; we will select these lines by
equally subdividing the increment in the I+ invariant between lines 0-1 and 0-4-10. To make the
execution of this calculation more clear, use the following steps:
1) Identify the regions that a uniform flow, simple flow and complex flow exist. For each of
these regions list which:
a. characteristic lines are straight or not
b. invariants I+ and I- are constant or not.
2) What is the value of o and e, which represent the flow direction angle at the inlet and
exit, respectively?
3) Using the Prandtl-Meyer expansion function, calculate the P-M angles at the inlet and the
exit.
4) Fill in the following table with as much information as you can for I+, I-,  and :
4
Point
0
1
2
3
4
5
6
7
8
9
10
Table 1: Method of Characteristics Results for 2-D Shaped Nozzle
I+ (+)
I- (-)
(º)
(º)

(º)
+
-
1.1
1.1
3
5) Now make a second pass at the table to calculate M and , and subsequently the angles of
the two characteristics through each point, which are + (for m-) and - (for m+).
Note: Here the +/- symmetry is unfortunately broken. If you are in a time crunch, write
down explicitly how you would find M and .
Note: The characteristics are straight in simple region 0-1-4 and in the simple region 4w4-10-11. The flow angle (hence the wall angle) at 4w is the same as at 4, and similarly
(7w)=(7), (9w)=(9), and (11)=(10).
6) Comment on these results, noting that the calculation is actually decoupled from the
geometry. Specifically, what have we said about the geometry of the nozzle so far and
what haven’t we specified about the geometry so far. Comment on the universality of the
table developed above and identify what the next steps are to completing an actual nozzle
design in (x,y) Cartesian coordinates.
7) The remaining task, and certainly the most tedious, is to place each grid point on the (x,y)
Cartesian map so that the nozzle may be fabricated. Point 0 is located at (x,y)=(0,1) in
any units you wish. Calculate the (x,y) coordinate pairs for all other points, and plot this
nozzle to scale. Comment on this calculation, in terms of overall nozzle length and the
required Ae/A* to generate a Mach 3 exit flow. We only used 4 characteristic lines, how
would you characterized the performance of this method? Plot a similar nozzle using the
curve fit methods used on the previous homework assignment and overlay the two
nozzles on top of each other. How do they compare?
5
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


 1
K
 1
  K arctan 
 M  1
6
6.1
2
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?
7