Page 121 - 48Fundamentals of Compressible Fluid Mechanics

5.3. THE MOVING SHOCKS 83
5.3 The Moving Shocks

In some situations, the shock wave
flow
isn’t stationary. This kind of situation direction
is arisen in many industrial applica- !
tions. For example, when a valve is
4
suddenly closed and a shock is prop-
agating upstream. On the other ex- c.v.
treme when suddenly valve is opened
or a membrane is ruptured a shock Stationary Coordinates
occurs and propagates downstream
(the opposite direction of the previous
case). In some industrial applications
a liquid (metal) is pushed in two rapid ?@4A+B6@DCFE G
stages to a cavity through a pipe sys- 02143+56187:9 ;
tem. This liquid (metal) is pushing gas HJI.K
(mostly) air which creates two shock )+*-,.)(/ "$#&%(' <>=
stages. As a general rule, the shock
can move downstream or upstream.
c.v.
The last situation is the most general
case which this section will be dealing Moving Coordinates
with. There is further more general-
Fig. 5.5: Comparison between stationary shock
ized cases where the moving shock is created which include a change in the phys-
and moving shock in ducts
ical properties which this book will not be dealing (at this stage). The reluctance to
deal with the most general case because it is rather specialized and complicated
beyond what even early graduate students work. In these changes (for examples,
opening value and closing valve on the other side) create situations where different
shocks are moving in the tube. In case where two shocks collide into one shock
which can move upstream or downstream is the general case. As specific exam-
ple, which is common in die casting process, After the first shock moving a second
shock is created in which its velocity is dictated by the upstream and downstream
velocity.
In cases when the shock velocity can be approximated as a constant (the
majority of the cases) or as a nearly constant, the previous analysis equations,
and the tools developed in this chapter can be employed. In these cases, the
problem can be reduced to the previously studied shock, i.e., to the stationary case
when the coordinate are attached to shock front. In such case, the steady state is
obtained in the moving control value. It has to be mentioned that the direction of the
shock alone doesn’t determine the “upstream” side of the shock. The determining
factor is the relative velocity of the flow to the shock.
For this analysis, the coordinates move with the shock. Here, the prime
’ will be denoting the values of the static coordinates. Note that this notation is
contrary to the conversion notation in the literature. The reason the deviation from

4 Later on the dimensional analysis what is suddenly