Partially Open Valve

The previous case is a special case of the moving shock. The general case is when one gas flows into another gas with a given velocity. The only limitation is that the ``downstream' gas velocity is higher than the ``upstream'' gas velocity as shown in Figure (5.17).

Figure: A shock moves into a moving medium as a result of a sudden and complete open valve.

The relationship between the different Mach numbers on the ``upstream'' side is

$$M_x = M_{sx} - {M_x}^{'}$$ (5.76)

The relationship between the different Mach on the ``downstream'' side is

$$M_y = M_{sy} - {M_y}^{'}$$ (5.77)

An additional parameter has be supplied to solve the problem. A common problem is to find the moving shock velocity when the velocity ``downstream'' or the pressure is suddenly increased. It has to be mentioned that the temperature ``downstream'' is unknown (the flow of the gas with the higher velocity). The procedure for the calculations can be done by the following algorithm:

(a)

Assume that ${M_x} = {M_x}^{'} + 1.$

(b)

Calculate the Mach number M_y by utilizing the tables or Potto-GDC.

(c)

Calculate the ``downstream'' shock Mach number $M_{sy} = M_y + {M_y}^{'}$

(d)

Utilizing

$$M_x = \sqrt{T_y \over T_x} \left( M_{sy} \right) - {M_x}^{'}$$

calculate the new ``improved'' M_x

(e)

Check the new and improved M_x against the old one. If it is satisfactory, stop or return to stage (b).

Figure 5.16: The results of the partial opening of the valve.

Earlier, it was shown that the shock choking phenomenon occurs when the flow is running into a still medium. This phenomenon also occurs in the case where a faster flow is running into a slower fluid. The mathematics is cumbersome but results show that the shock choking phenomenon is still there (the Mach number is limited, not the actual flow). Figure (5.16) exhibits some ``downstream'' Mach numbers for various static Mach numbers, M<sub>y</sub>′, and for various static ``upstream'' Mach numbers, M_x′. The figure demonstrates that the maximum can also occurs in the vicinity of the previous value (see following question/example).