978-0131206687 Chapter 4 Part 1

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Problem 1. – A helium flow with a velocity of 2500 m/s and static temperature of 300 K

undergoes a normal shock. Determine the helium velocity and the static and stagnation

temperatures after the wave. Assume the helium to behave as a perfect gas with constant

γ = 5/3 and R = 2077 J/kg·K.

m/s 7067.1456

7162.1

V ,7162.1

V

2

1

2====

ρ

Problem 2. – A normal shock occurs at the inlet to a supersonic diffuser, as shown in

Figure P4.2. Ae/Ai is equal to 3.0. Find Me, pe, and the loss in stagnation pressure (poi –

poe). Repeat for a shock at the exit. Assume γ = 1.4.

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e

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()

kPa2961.1391847.018526.170

p

1ppp

o1

2o

oiboio =−=

⎟

⎠

⎞

⎜

⎝

⎛−=−

Problem 3. – Sketch p versus x for the three cases shown in Figure P4.3. Assume

isentropic flow except for flow across the normal shocks.

M > 1

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59

Figure P4.4

Problem 5. – A supersonic flow at Mach 3.0 and γ = 1.4 is to be slowed down via a

normal shock in a diverging channel. For the conditions shown in Figure P4.5, find p2/p1

and pe/pi.

To = 300 K

po = 200 kPa Athroat = 50 cm2

Aexit = 4 Athroat

Ashock = 2 Athroat

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61

Figure P4.6

9719.0

T

2

2o =

⎠

⎝

Problem 7. – Determine the back pressure necessary for a normal shock to appear at the

exit of a converging-diverging nozzle, as shown in Figure P4.7. Assume γ = 1.4.

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A

p

*

1

2o ==

()()( )

1256.29662.012.2

A

*

2

*

1

t

*

2

From this area ratio we are able to extract the exit Mach number again using the

Newton-Raphson method, therefore, the static to total temperature ratio

decrease 82.9502%

6833.1

2870.0

1100

=

⎟

⎠

⎞

⎜

⎝

⎛−=

Problem 9. – A flow system consists of two converging-diverging nozzles in series (see

Figure P4.9a. If the area ratio (exit to throat) of each nozzle is 3.0 to 1, find the area ratio

A3/Al necessary to produce sonic flow at the second throat, with a shock at A2. Assume

isentropic flow except for the normal shock. Find the percent of loss in stagnation

pressure for this flow. At another operating condition, a shock appears at A3 (Figure

P4.9b). Find the percent of loss of stagnation pressure for this condition.

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A

p

*

2

*

1

2o

% loss in stagnation pressure =

()

%3800.551004462.01100

p

1o

2o1o =−=

⎟

⎠

⎜

⎝

(b) For shock at A3, we have from part (a) A3/A1 = A3/A* = 2.411. Using this area

ratio, we can find the Mach number on the upstream side of the shock, i.e.,

Problem 10. – For the system shown in Figure P4.10, Mi = 2.0, Ai = 20 cm2, throat area =

15 cm2, shock area = 22 cm2, and exit area = 25 cm2. With the working fluid behaving as

a perfect gas with constant γ = 1.3, find the following:

(a) Throat Mach number

(b) Exit Mach number

(c) Ratio of exit static pressure to static pressure at i

Figure P4.10

1 2

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65

7732.1

A

*

1

i=.

Hence,

()

3299.17732.1

20

A

*

1

i

*

1

From which we determine the result of part (a),

22

A

t

s

s=== ; therefore from the Newton-Raphson method

At this Mach number we can compute the total pressure ratio across the shock

*

2

1

1o

2o

A

6502.0

p

Thus,

(c) From the various Mach numbers computed thus far, we may determine the

following pressure ratios and form the string,

1

p

1o

2o

oe

e

e=

⎞

⎛

Problem 11. – A jet plane uses a diverging passage as a diffuser (Figure P4.11). For a

flight Mach number of 1.8, determine the range of back pressures over which a normal

shock will appear in the diffuser. Ambient pressure and temperature are 25 kPa and

220 K. Find the mass flow range handled by the diffuser for the determined back pressure

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range. Also, the inlet and exit area are Ai = 250 cm2, Ae = 500 cm2. Assume isentropic

flow except for the shocks. Take γ = 1.4.

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For a shock at the exit,

1740.0

p

1o

1

e=

⎠

⎝

∞

The diffuser is choked so it passes the same mass flow for the back pressure range,

Problem 12. – Air (γ = 1.4) enters a converging-diverging diffuser with a Mach number

of 2.8, static pressure pi of 100 kPa, and a static temperature of 20°C. For the flow

situation shown in Figure P4.12, find the exit velocity, exit static pressure, and exit

stagnation pressure.

1 2 e

i

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*

1

2o

A

p==

()()

8025.58289.05001.3

25.0

A

*

2

*

1

i

*

2

From this area ratio we can compute the exit Mach number

=

Problem 13. – Write a computer program that will yield values of p2/p1, ρ2/ρ1, T2/T1, and

po2/po1 for a fixed normal shock with a working fluid consisting of a perfect gas with

constant γ = 1.20. Use Mach number increments of 0.05 over the range M = 1.0 to M =

2.5.

1

−

γ

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