Basics of Fluid Mechanics, 2014a

414 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL
U x = M x × c x =3× 331.2 = 993.6[m/sec]
Now the velocity downstream is determined by the inverse ratio of ρ y /ρ x =
U x /U y =3.85714.
P 0y =
(
P0y
U y = 993.6/3.85714 = 257.6[m/sec]
P 0x
)
× P 0x =0.32834 × 55.1[bar] =18.09[bar]
End Solution
When the upstream Mach number becomes very large, the downstream Mach
number (see equation (11.90)) is limited by
M 2 y = 1+ 2
✘✘✘✘✘✿ ∼0
(k−1)M 2 x
2 k
1
M 2 x
k−1 − ✚ ✚❃∼0
= k − 1
2 k
(11.103)
This result is shown in Figure 11.12. The limits of the pressure ratio can be obtained
by looking at equation (11.84) and by utilizing the limit that was obtained in equation
(11.103).
11.5.4 The Moving Shocks
In some situations, the shock wave is not stationary. This kind of situation arises in
many industrial applications. For example, when a valve is suddenly 8 closed and a
shock propagates upstream. On the other extreme, when a valve is suddenly opened
or a membrane is ruptured, a shock occurs and propagates downstream (the opposite
direction of the previous case). In addition to (partially) closing or (partially) opening of
value, the rigid body (not so rigid body) movement creates shocks. In some industrial
applications, a liquid (metal) is pushed in two rapid stages to a cavity through a pipe
system. This liquid (metal) is pushing gas (mostly) air, which creates two shock stages.
The moving shock is observed by daily as hearing sound wave are moving shocks.
As a general rule, the moving shock can move downstream or upstream. The
source of the shock creation, either due to the static device operation like valve operating/closing
or due to moving object, is relevant to analysis but it effects the boundary
conditions. This creation difference while creates the same moving shock it creates
different questions and hence in some situations complicate the calculations. The most
general case which this section will be dealing with is the partially open or close wave.
A brief discussion on the such case (partially close/open but due the moving object)
will be presented. There are more general cases where the moving shocks are created
which include a change in the physical properties, but this book will not deal with them
at this stage. The reluctance to deal with the most general case is due to fact it is
highly specialized and complicated even beyond early graduate students level. In these
changes (of opening a valve and closing a valve on the other side) create situations in
8 It will be explained using dimensional analysis what is suddenly open.