ME 322
Spring 2012
GAS DYNAMICS
Homework 1:Solutions
B1.1. A van der Waals gas satisfies the thermal equation of state
( p + ) (1 ) = RT
2
where and are constants.
(a) Determine the general form of the caloric equation of state for e = e ( T , ) .
(b) Obt
ME 322
Spring 2012
GAS DYNAMICS
Homework 8: Solutions
8B.1 A shock wave propagates into a polyatomic gas with an effective (constant) specific
heat ratio = 1.2 and a gas constant R = 225 J/kg K .The gas ahead of the shock is
moving in the same direction a
ME 322: Gas Dynamics
Class notes
1. Introduction
1.1 Background and outline
Gas dynamics is concerned with compressible flow. Applications include subsonic, transonic,
supersonic, and hypersonic airfoils and wings, rocket nozzles, supersonic wind tunnel d
12. Combustion waves
12.1: Introduction
Neglecting diffusive effects, one-dimensional steady combustion phenomena are
equivalent to flows with heat addition (Rayleigh lines, see chapter 3 of Anderson and
Class notes 3). A schematic diagram is shown in fig
6. Two-dimensional steady supersonic flows: expansion waves
6.1. Natural co-ordinates
For two-dimensional steady supersonic flows, it is often convenient to introduce natural
co-ordinates ( s, n ) with s measured along a streamline and n measured normal t
4. One-dimensional adiabatic flow with friction
Duct flows with wall friction are of significance in long pipe lines, narrow diameter exhausts,
etc. In this section it is assumed that the pipe is well insulated without internal heat generation so
that the
8: Propagating Shock Waves
vp
W
v2 = v p
v1
Figure 8.1. Travelling shock wave
Consider a piston driven shock wave travelling at a fixed speed W in a tube (see Fig. 8.1).
The gas speed ahead of the shock wave, relative to the tube, is v1 and the gas speed
3. One dimensional flow with heat addition
Duct flows with heat release are characteristic of many combustion phenomena. Generally, heat
addition/subtraction is associated with exothermic/endothermic reactions that occur in nonequilibrium gas flows and ap
5. Oblique shock waves
5.1 Mach waves
Consider a weak disturbance propagating at a supersonic speed U . The disturbance omits
signals that travel at the local sound speed a (see Figure 5.1). Clearly the angle that the wave
front makes relative to the axis
9. Unsteady Isentropic Flows Centered Waves
t
particle path
piston path
x = v pt
expansion
fan
wave front
x = a4t
region 4
x
Figure 9.1 Centered expansion wave generated by piston withdrawal
If a piston is impulsively withdrawn in a tube (see Fig. 9.1), a
10: Shock tubes
10.1. Introduction
A conventional shock tube configuration is sketched below in Fig. 10.1. Initially, the
high-pressure driver gas is separated from the low-pressure driven gas by a suitable
diaphragm.
High-pressure
driver section
p = p4,
2. One dimensional flow: sound speed and shock waves
2.1 Isentropic flow
For one dimensional unsteady adiabatic non-diffusive flow the governing equations are:
Continuity
D
u
+
= 0,
Dt
x
(2.1)
Momentum
Du
p
+ 1
= 0,
Dt
x
(2.2)
D
p
e + 1 p + 12 u 2 ) = 1
,
ME 322
Spring 2012
GAS DYNAMICS
Homework 3: Solutions
3B.1 Air enters a subsonic combustor of constant diameter 20cm at a Mach number M 1 = 0.5 .
The entry stagnation pressure is p01 = 1.2 MPa and the entry stagnation temperature T01 = 350 K .
(a) What is
ME 322
Spring 2012
GAS DYNAMICS
Homework 2:Solutions
2.B.1. What is the maximum air speed that can be attained for adiabatic flow from a reservoir at
an initial temperature of 500 K ?
Solution
If adiabatic
1
RT + V 2 =
RT0
1
2
1
or
V=
2 R
(T0 T ) .
1
A
ME 322
Spring 2012
GAS DYNAMICS
Homework 4: solutions
4.B.1 An insulated duct of diameter 10 cm has an average friction coefficient f = 0.005. Air
enters the duct at high speed with a static pressure p1 = 50 kPa, and an inlet temperature T1 = 400
K. The m
ME 322
April 2012
Review Material: Final exam
A list of topics for which you are responsible is given below. Copies of the Final
Examinations for 2010 and 2011 are attached. The exam is open Class notes (see Course
Site) and open book (Anderson). Old home
ME 322
Spring 2012
GAS DYNAMICS
Homework 5: solution
5B.1. A plane shock wave impinges on a straight wall at an angle 1 = 30 . If the upstream flow
properties are M1 = 3.0, p1 = 101 kPa , and T1 = 295 K, calculate the pressure, temperature, Mach
number, s
ME 322
Spring 2012
GAS DYNAMICS
Homework 9: Solutions
9.B1. In a conventional shock tube, helium is employed as the driver gas and argon as the
driven gas. The driven gas temperature T1 = 300 K . Determine accurate plots of the
driving pressure ratio p4 p
ME 322
Spring 2012
GAS DYNAMICS
Homework 11: Solution
11B.1 (i) A Chapman-Jouget detonation propagates through a combustible mixture with a
heat release q = 3MJ/kg . The upstream gas temperature T1 = 295 K , and the upstream
pressure p1 = 101 kPa . Assume
M.E. 322
Spring 2012
GAS DYNAMICS
Homework 6: solution
6B.1 A convergent-divergent nozzle is attached to an air-filled reservoir chamber in
which the stagnation pressure p0 = 500 kPa and the stagnation temperature T0 = 300 K .
The nozzle throat diameter D
M.E. 322
Spring 2012
GAS DYNAMICS
Homework 7
7.B.1 A supersonic wind tunnel and diffuser combination has a fixed geometry with a
first throat area At1 = 0.0025 m2, a working section area AWS = 0.00875 m2, and a second
throat area At 2 = .0068 m2 .
(a) For
ME 322
Spring 2012
GAS DYNAMICS
Homework 10
10.B1. In a conventional shock tube, helium is employed as the driver gas and argon as
the driven gas. For tailoring, determine the required temperature ratio T4 T1 as a function
of the tailored shock Mach numbe
7. Supersonic nozzle flows, diffusers and wind tunnels
7.1 Quasi-one-dimensional flow
A(x)
u
p
T
x
Figure 7.1. Schematic picture of nozzle flow
Assume that the nozzle cross-sectional area A ( x ) varies slowly, that there is no heat addition
and that diff