fluid_mechanics

592 Chapter 11 ■ Compressible Flow
Mach cone
Aircraft moving with velocity
V and Mach number Ma
z
α
x = Vt
F I G U R E E11.4a
observer, the “sound” of the aircraft is perceived. The angle a in
Fig. E11.4 is related to the elevation of the plane, z, and the ground
distance, x,by
a tan 1 z x
Also, assuming negligible change of Mach number with elevation,
we can use Eq. 11.39 to relate Mach number to the angle a. Thus,
Combining Eqs. 1 and 2 we obtain
Ma
tan1
1000
Vt
Ma 1
sin a
1
sin 3tan 1 11000Vt24
The speed of the aircraft can be related to the Mach number with
V 1Ma2c
(4)
where c is the speed of sound. From Table B.4, c 343.3 ms.
Using Ma 1.5, we get from Eqs. 3 and 4
1
1.5
sin e tan 1 1000 m
c
11.521343.3 ms2t df
(1)
(2)
(3)
F I G U R E E11.4b NASA
Schlieren photograph of shock waves from a
T-38 aircraft at Mach 1.1, 13,000 feet.
or
t 2.17 s
(Ans)
COMMENT By repeating the calculations for various values
of Mach number, Ma, the results shown in Fig. E11.4c are obtained.
Note that for subsonic flight (Ma 1) there is no delay
since the sound travels faster than the aircraft. You can hear a subsonic
aircraft approaching.
t, s
3
2.5
2
1.5
1
0.5
F I G U R E E11.4c
(1.5, 2.17 s)
0
0 0.5 1 1.5 2
Ma
2.5 3 3.5 4
11.4 Isentropic Flow of an Ideal Gas
An important class
of isentropic flow
involves no heat
transfer and zero
friction.
In this section, we consider in further detail the steady, one-dimensional, isentropic flow of an ideal
gas with constant specific heat values 1c p and c v 2. Because the flow is steady throughout, shaft work
cannot be involved. Also, as explained earlier, the one-dimensionality of flows we discuss in this
chapter implies velocity and fluid property changes in the streamwise direction only. We consider
flows through finite control volumes with uniformly distributed velocities and fluid properties at
each section of flow. Much of what we develop can also apply to the flow of a fluid particle along
its pathline.
Isentropic flow involves constant entropy and was discussed earlier in Section 11.1, where
we learned that adiabatic and frictionless 1reversible2 flow is one form of isentropic flow. Some
ideal gas relationships for isentropic flows were developed in Section 11.1. An isentropic flow is
not achievable with actual fluids because of friction. Nonetheless, the study of isentropic flow
trends is useful because it helps us to gain an understanding of actual compressible flow phenomena