Shock Result from a Sudden and Complete Stop

The general discussion can be simplified in the extreme case when the shock is moving from a still medium. This situation arises in many cases in the industry, for example, in a sudden and complete closing of a valve. The sudden closing of the valve must result in a zero velocity of the gas. This shock is viewed by some as a reflective shock. The information propagates upstream in which the gas velocity is converted into temperature. In many such cases the steady state is established quite rapidly. In such a case, the shock velocity ``downstream'' is U_s. Equations (5.42) to (5.53) can be transformed into simpler equations when M_x is zero and U_s is a positive value.

Figure: Comparison between a stationary shock and a moving shock in a stationary medium in ducts.

The ``upstream'' Mach number reads

$$M_x = {U_s + U_x \over c_x} = M_{sx} +M_x$$ (5.54)

The ``downstream'' Mach number reads

$$M_y = { \left\vert U_s \right\vert \over c_y } = M_{sy}$$ (5.55)

Again, the shock is moving to the left. In the moving coordinates, the observer (with the shock) sees the flow moving from the left to the right. The flow is moving to the right. The upstream is on the left of the shock. The stagnation temperature increases by

$$T_{0y} - T_{0x} = U_s \left( {T_x \over c_x} {k -1 \over 2} \left... ...right. - \left. { T_y \over c_y} {k -1 \over 2} \left( {M_{sy}} \right) \right)$$ (5.56)

The prominent question in this situation is what will be the shock wave velocity for a given fluid velocity, ${U_x}^{'}$ , and for a given specific heat ratio. The ``upstream'' or the ``downstream'' Mach number is not known even if the pressure and the temperature downstream are given. The difficulty lies in the jump from the stationary coordinates to the moving coordinates. It turns out that it is very useful to use the dimensionless parameter M_sx, or M_sy instead of the velocity because it combines the temperature and the velocity into one parameter.

The relationship between the Mach number on the two sides of the shock are tied through equations (5.54) and (5.55) by

$$\left({M_y}\right)^{2} = { \left({M_x}^{'} +M_{sx}\right)^{2} + {2 \over k -1} \over {2k \over k -1} \left({M_x}^{'} + M_{sx}\right)^{2} - 1 }$$ (5.57)

And substituting equation (5.57) into (5.48) results in

$$M_x = \overbrace{\sqrt{T_x \over T_y}}^{f(M_{sx})} \sqrt{ \left({... ... {2 \over k -1} \over {2k \over k -1} \left({M_x}^{'} + M_{sx}\right)^{2} - 1 }$$ (5.58)

Figure: The moving shock Mach numbers as a result of a sudden and complete stop.

The temperature ratio in equation (5.58) and the rest of the right-hand side show clearly that M_sx has four possible solutions (fourth-order polynomial M_sx has four solutions). Only one real solution is possible. The solution to equation (5.58) can be obtained by several numerical methods. Note, an analytical solution can be obtained for equation (5.58) but it seems utilizing numerical methods is much more simple. The typical method is the ``smart'' guessing of M_sx. For very small values of the upstream Mach number, M_x^′∼ ε equation (5.58) provides that M_sx∼1 + 1/2ε and M_sy∼1 - 1/2ε (the coefficient is only approximated as 0.5) as shown in Figure (5.11). From the same figure it can also be observed that a high velocity can result in a much larger velocity for the reflective shock. For example, a Mach number close to one (1), which can easily be obtained in a Fanno flow, the result is about double the sonic velocity of the reflective shock. Sometimes this phenomenon can have a tremendous significance in industrial applications.

Note that to achieve supersonic velocity (in stationary coordinates) a diverging-converging nozzle is required. Here no such device is needed! Luckily and hopefully, engineers who are dealing with a supersonic flow when installing the nozzle and pipe systems for gaseous mediums understand the importance of the reflective shock wave.

Two numerical methods and the algorithm employed to solve this problem for given, M_x^′, is provided herein:

1. Guess M_x > 1,
2. Using shock table or use Potto--GDC to calculate temperature ratio and M_y,
3. Calculate the M_x = M_x′ - sqrt(T_x/T_y)M_y
4. Compare to the calculated M_x′ to the given M_x′. and adjust the new guess M_x > 1 accordingly.

The second method is ``successive substitutions,'' which has better convergence to the solution initially in most ranges but less effective for higher accuracies.

1. Guess M _x = 1 + M _x′,
2. \label{shock:list:mvshock} using the shock table or use Potto--GDC to calculate the temperature ratio and M _y ,
3. calculate the M_x = M_x′ - sqrt(T_x/T_x) M_y
4. Compare the new M_x approach the old M_x, if not satisfactory use the new M_x′ to calculate M_x= 1 + M_x′ then return to part \eqref{shock:list:mvshock}.