Cover Page for Precalculations – Individual Portion
Compressible Flow in a Converging-Diverging Nozzle
Prepared by Professor J. M. Cimbala, Penn State University
Latest revision: 11 January 2012
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________________________________________
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Section number:
ME 325._____
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Precalculations
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Precalculations
If we neglect frictional losses and assume purely one-dimensional flow, then flow quantities such as velocity, Mach
number, pressure, etc. change only in the streamwise direction. If there are no shock waves in the flow and all frictional and
heat transfer effects are neglected, the flow can be considered isentropic. For isentropic flow of an ideal gas, the relationship
between local Mach number and area ratio is
k 1
1  2  (k  1)Ma 2  2( k 1)



(k  1)
A* Ma 

A
(1)
where k is the ratio of specific heats, k= Cp/Cv. For air, k = 1.4, and Equation (1) reduces to
A
A*

1  2  0.4Ma 2 


Ma 
2.4

3.0
(2)
The static pressure (normalized by stagnation pressure P0) at each streamwise location is also known as a function of
Mach number for isentropic flow:
k
P  1
 ( k 1)
 1  (k  1)Ma 2 
P0  2

(3)
3.5
P 
 1  0.2Ma 2 


P0
(4)
which for air reduces to
The Appendix of your fluid mechanics textbook contains tables for isentropic flow of air. These tables were generated
with Equations (2) and (4), along with other similar equations for temperature and density ratio.
(10)
1.
Using either these equations or the tables, calculate the local Mach number and pressure ratio as a function of
streamwise location x, measured from the beginning of the contraction. Complete Table 1, where the area ratio
A/A* and x location are given for each pressure tap. Note that in the diverging part of the nozzle, there are two
possible solutions  one subsonic and one supersonic. Be sure to fill in the numbers for both cases. Three
significant digits are sufficient.
Table 1.
Mach number and pressure ratio as a function of pressure tap location, assuming steady, isentropic, onedimensional, compressible flow of an ideal gas (asterisks indicate an undefined quantity).
Pressure tap
number
Description of
location
Tap location
x (mm)
A/A*
Subsonic
Ma
Subsonic
P/P0
Supersonic
Ma
Supersonic
P/P0
2
beginning of
converging section
0.0
***
0.0
1.00
***
***
3
converging section
3.18
1.763
***
***
4
converging section
6.35
1.172
***
***
5
throat
9.53
1.000
1.00
0.528
6
diverging section
15.88
1.182
7
diverging section
22.23
1.364
8
diverging section
28.58
1.546
9
diverging section
34.93
1.727
***
exit plane
44.45
2.000
1.00
0.528
The pressure ratio across a shock wave can also be found for air from tables in the Appendix of your fluid
mechanics textbook. In our wind tunnel, the pressure regulator keeps stagnation pressure P0 at approximately 100
psia. (pounds force per square inch absolute).
(5)
2.
For our experimental setup, if P0 = 100 psia, calculate the back pressure Pb (not the throat pressure) required to
just choke the flow (curve C in Figure 1). Show your calculations and result below:
Answer: To choke the flow, Pb must be less than
(5)
3.
psia.
Again assuming P0 = 100 psia, calculate the back pressure required to achieve supersonic flow in the entire
diverging portion of the nozzle, but with a shock at the exit plane (curve F in Figure 1). Show your calculations
and results below:
Answer: Pb must be less than
psia.
(4)
4.
Our wind tunnel discharges into the atmosphere, so the minimum obtainable back pressure is approximately
atmospheric pressure, 14.7 psia. Do we have enough pressure ratio to achieve supersonic flow in the entire
diverging portion of the nozzle? Justify your answer in the space below.
(4)
5.
Finally, calculate the back pressure required to achieve “design” conditions (curve H in Figure 1).
Answer: Pb must equal
(2)
6.
psia for design conditions.
Do we have enough pressure ratio to achieve design conditions in our facility? Explain.
Experimental Objectives
a.
b.
c.
d.
To visualize a compressible flow field, including shock waves, with the schlieren optical technique.
To measure mass flow rate as a function of back pressure ratio, and to observe the phenomenon of choking.
To examine the effect of choking on propagation of acoustic noise.
To measure the pressure distribution in a converging-diverging nozzle for various back pressure ratios, with and
without shock waves, and to compare experimental results with the published results.