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 ^5.4 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 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. As a general rule, the shock can move downstream or upstream. The last situation is the most general case, which this section will be dealing with. There are more genera cases where the moving shock is 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 which different shocks are moving in the tube. The general case is where two shocks collide into one shock and moves upstream or downstream is the general case. A specific example is common in die-casting: after the first shock moves a second shock is created in which its velocity is dictated by the upstream and downstream velocities.

Figure: Comparison between stationary shock and moving shock in ducts

In cases where the shock velocity can be approximated as a constant (in the majority of cases) or as near constant, the previous analysis, equations, and the tools developed in this chapter can be employed. The problem can be reduced to the previously studied shock, i.e., to the stationary case when the coordinates are attached to the shock front. In such a case, the steady state is obtained in the moving control value.

For this analysis, the coordinates move with the shock. Here, the prime ' denote the values of the static coordinates. Note that this notation is contrary to the conventional notation found in the literature. The reason for the deviation is that this choice reduces the programing work (especially for object-oriented programing like C++). An observer moving with the shock will notice that the pressure in the shock is

$${P_x}^{'} = P_x \qquad {P_y}^{'} = P_y$$ (5.42)

The temperature measured by the observer is

$${T_x}^{'} = T_x \qquad {T_y}^{'} = T_y$$ (5.43)

Assuming that the shock is moving to the right, (refer to Figure (5.6)) the velocity measured by the observer is

$$U_x = U_s - {U_x}^{'}$$ (5.44)

Where U_s is the shock velocity which is moving to the right. The ``downstream'' velocity is

$${U_y}^{'} = U_s - U_y$$ (5.45)

The speed of sound on both sides of the shock depends only on the temperature and it is assumed to be constant. The upstream prime Mach number can be defined as

$${M_x}^{'} = {U_s - U_x \over c_x} = { U_s \over c_x} - M_x = M_{sx} - M_x$$ (5.46)

It can be noted that the additional definition was introduced for the shock upstream Mach number, M_sx = U_s /c_x. The downstream prime Mach number can be expressed as

$${M_y}^{'} = {U_s - U_y \over c_y} = { U_s \over c_y} - M_y = M_{sy} - M_y$$ (5.47)

Similar to the previous case, an additional definition was introduced for the shock downstream Mach number, M_sy. The relationship between the two new shock Mach numbers is

$${U_s \over c_x} = {c_y \over c_x} {U_s \over c_y}$$

$$M_{sx} = \sqrt{T_y \over T_x}M_{sy}$$ (5.48)

The ``upstream'' stagnation temperature of the fluid is

$$T_{0x} = T_x \left( 1 + {k-1 \over 2} {M_x}^{2} \right)$$ (5.49)

and the ``upstream'' prime stagnation pressure is

$$P_{0x} = P_x \left( 1 + {k-1 \over 2} {M_x}^{2} \right) ^{k \over k-1}$$ (5.50)

The same can be said for the ``downstream'' side of the shock. The difference between the stagnation temperature is in the moving coordinates

$$T_{0y} - T_{0x} = 0$$ (5.51)

It should be noted that the stagnation temperature (in the stationary coordinates) rises as opposed to the stationary normal shock. The rise in the total temperature is due to the fact that a new material has entered the c.v. at a very high velocity, and is ``converted'' or added into the total temperature,

$$T_{0y} - T_{0x} = T_y \left( 1 + {k -1 \over 2} \left(M_{sy} - {... ...- T_x \left( 1 + {k -1 \over 2} \left(M_{sx} - {M_{x}}^{'}\right)^2 \right)\\$$

$$0 = \overbrace {T_y \left( 1 + {k -1 \over 2} { {M_{y}}^{'} }^2 \... ...}^{{T_{0y}}^{'}} + T_y M_{sy} {k -1 \over 2} \left( {M_{sy}} - 2 M_y \right)$$

$$- \overbrace{T_x \left( 1 + {k -1 \over 2} { {M_{x}}^{'} }^2 \right)}^{{T_{0x}}^{'}} - T_x M_{sx} {k -1 \over 2} \left( {M_{sx}} - 2 M_x \right)$$ (5.52)

and according to equation (5.51) leads to

$${T_{0y}}^{'} - {T_{0x}}^{'} = U_s \left( {T_x \over c_x} {k -1 \o... ...ight) - { T_y \over c_y} {k -1 \over 2} \left( {M_{sy}} - 2 M_y \right) \right)$$ (5.53)

Again, this difference in the moving shock is expected because moving material velocity (kinetic energy) is converted into internal energy. This difference can also be viewed as a result of the unsteady state of the shock.