fluid_mechanics

586 Chapter 11 ■ Compressible Flow
The changes in fluid
properties across
a sound wave are
very small compared
to their local values.
in Fig. 11.1a. The speed of the weak pressure pulse is considered constant and in one direction
only; thus, our control volume is inertial.
For an observer moving with this control volume 1Fig. 11.1b2, it appears as if fluid is entering
the control volume through surface area A with speed c at pressure p and density r and leaving
the control volume through surface area A with speed c dV, pressure p dp, and density
r dr. When the continuity equation 1Eq. 5.162 is applied to the flow through this control volume,
the result is
or
(11.26)
(11.27)
Since 1dr21dV2 is much smaller than the other terms in Eq. 11.27, we drop it from further consideration
and keep
(11.28)
The linear momentum equation 1Eq. 5.292 can also be applied to the flow through the control volume
of Fig. 11.1b. The result is
(11.29)
Note that any frictional forces are considered as being negligibly small. We again neglect higher
order terms [such as 1dV2 2 compared to c dV, for example] and combine Eqs. 11.26 and 11.29
to get
or
rAc 1r dr2A1c dV2
rc rc r dV c dr 1dr21dV2
r dV c dr
crcA 1c dV21r dr21c dV2A pA 1 p dp2A
crcA 1c dV2rAc dpA
rdV dp c
(11.30)
From Eqs. 11.28 1continuity2 and 11.30 1linear momentum2 we obtain
or
c 2 dp
dr
c B
dp
dr
(11.31)
This expression for the speed of sound results from application of the conservation of mass and
conservation of linear momentum principles to the flow through the control volume of Fig. 11.1b.
These principles were similarly used in Section 10.2.1 to obtain an expression for the speed of surface
waves traveling on the surface of fluid in a channel.
The conservation of energy principle can also be applied to the flow through the control volume
of Fig. 11.1b. If the energy equation 1Eq. 5.1032 is used for the flow through this control volume,
the result is
dp
r d aV2 b g dz d1loss2
2
or, neglecting 1dV2 2 compared to c dV, we obtain
(11.32)
For gas flow we can consider g dz as being negligibly small in comparison to the other terms in the
equation. Also, if we assume that the flow is frictionless, then d1loss2 0 and Eq. 11.32 becomes
dp 1c dV22
c2
r 2 2 0
r dV dp c
(11.33)