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Next: Moving Shock into Stationary Up: The Moving Shocks Previous: Shock or Wave Drag   Index

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 Us. Equations (5.42) to (5.53) can be transformed into simpler equations when Mx is zero and Us is a positive value.

Figure: Comparison between a stationary shock and a moving shock in a stationary medium in ducts.
Image intoStationaryMedium
Image intoStationaryMediumMoving
The ``upstream'' Mach number reads
The ``downstream'' Mach number reads
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

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 Msx, or Msy 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

And substituting equation (5.57) into (5.48) results in
Figure: The moving shock Mach numbers as a result of a sudden and complete stop.
Image shockDnamics
The temperature ratio in equation (5.58) and the rest of the right-hand side show clearly that Msx has four possible solutions (fourth-order polynomial Msx 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 Msx. For very small values of the upstream Mach number, Mx∼ ε equation (5.58) provides that Msx∼1 + 1/2ε and Msy∼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, Mx, is provided herein:

  1. Guess Mx > 1,
  2. Using shock table or use Potto--GDC to calculate temperature ratio and My,
  3. Calculate the Mx = Mx′ - sqrt(Tx/Ty)My
  4. Compare to the calculated Mx to the given Mx. and adjust the new guess Mx > 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 Mx = Mx′ - sqrt(Tx/Tx) My
  4. Compare the new Mx approach the old Mx, if not satisfactory use the new Mx to calculate Mx= 1 + Mx then return to part \eqref{shock:list:mvshock}.

next up previous index
Next: Moving Shock into Stationary Up: The Moving Shocks Previous: Shock or Wave Drag   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21