Shock-Choke Phenomenon

Assuming that the gas velocity is supersonic (in stationary coordinates) before the shock moves, what is the maximum velocity that can be reached before this model fails? In other words, is there a point where the moving shock is fast enough to reduce the ``upstream'' relative Mach number below the speed of sound? This is the point where regardless of the pressure difference is, the shock Mach number cannot be increased.

Figure: The maximum of ``downstream'' Mach number as a function of the specific heat, $k$ .

This shock-choking phenomenon is somewhat similar to the choking phenomenon that was discussed earlier in a nozzle flow and in other pipe flow models (later chapters). The difference is that the actual velocity has no limit. It must be noted that in the previous case of suddenly and completely closing of valve results in no limit (at least from the model point of view). To explain this phenomenon, look at the normal shock. Consider when the ``upstream'' Mach approaches infinity, $M_x=M_{sx}\rightarrow \infty$ , and the downstream Mach number, according to equation (5.38), is approaching to $(k-1)/2k$ . One can view this as the source of the shock-choking phenomenon. These limits determine the maximum velocity after the shock since $U_{max} = c_yM_y$ . From the upstream side, the Mach number is

$$M_{x}= M_{sx} = \cancelto {\infty} {\sqrt{T_y \over T_x}} \left(k-1 \over 2k \right)$$ (5.69)

Thus, the Mach number is approaching infinity because of the temperature ratio but the velocity is finite.

To understand this limit, consider that the maximum Mach number is obtained when the pressure ratio is approaching infinity ${P_y \over P_x} \rightarrow \infty$ . By applying equation (5.23) to this situation the following is obtained:

$$M_{sx} = \sqrt{ {k+1 \over 2k} \left( {P_x \over P_y} - 1 \right) + 1 }$$ (5.70)

and the mass conservation leads to

$$U_y \rho_y = U_s \rho_x$$

$$\left( U_s - {U_y}^{'} \right) \rho_y = U_s \rho_x$$

$${{M_y}^{'}} = \sqrt{T_y \over T_x} \left( 1 - { \rho_x \over \rho_y } \right) M_{sx}$$ (5.71)

Substituting equations (5.26) and (5.25) into equation (5.71) results in

$${M_y}^{'} = {1 \over k} \left( 1 - {P_y \over P_x} \right) \sqrt{... ..._x} \right) \over \left( k+1 \over k-1\right) +\left( {P_y \over P_x} \right)}}$$ (5.72)

When the pressure ratio is approaching infinity (extremely strong pressure ratio), the results is

$${M_y}^{'} = \sqrt{2 \over k (k -1)}$$ (5.73)

What happens when a gas with a Mach number larger than the maximum Mach number possible is flowing in the tube? Obviously, the semi steady state described by the moving shock cannot be sustained. A similar phenomenon to the choking in the nozzle and later in an internal pipe flow is obtained. The Mach number is reduced to the maximum value very rapidly. The reduction occurs by an increase of temperature after the shock or a stationary shock occurs as it will be shown in chapters on internal flow.

$\rule[-0.1in]{0.pt}{0.3 in}\mathbf{k}$	$\mathbf{{M_x}}$	$\mathbf{M_y}$	$\mathbf{{M_y}^{'}}$	$\mathbf{{T_y} \over {T_x} }$
1.30	1073.25	0.33968	2.2645	169842.29
1.40	985.85	0.37797	1.8898	188982.96
1.50	922.23	0.40825	1.6330	204124.86
1.60	873.09	0.43301	1.4434	216507.05
1.70	833.61	0.45374	1.2964	226871.99
1.80	801.02	0.47141	1.1785	235702.93
1.90	773.54	0.48667	1.0815	243332.79
2.00	750.00	0.50000	1.00000	250000.64
2.10	729.56	0.51177	0.93048	255883.78
2.20	711.62	0.52223	0.87039	261117.09
2.30	695.74	0.53161	0.81786	265805.36
2.40	681.56	0.54006	0.77151	270031.44
2.50	668.81	0.54772	0.73029	273861.85

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Table of maximum values of the shock-choking phenomenon.

The mass flow rate when the pressure ratio is approaching infinity, $\infty$ , is

$${\dot{m} \over A} = {U_y}^{'} \rho_y = {M_y}^{'} c_y \rho_y = {M_y}^{'} \overbrace{\sqrt{kRT_y}}^{c_y} \overbrace{P_y \over R T_y}^{\rho_y}$$

$$= {{M_y}^{'} \sqrt{k} P_y \over \sqrt{RT_y}}$$ (5.74)

Equation (5.74) and equation (5.25) can be transferred for large pressure ratios into

$${\dot{m} \over A} \sim \sqrt{T_y} {P_x \over T_x} {k-1 \over k+1}$$ (5.75)

Since the right hand side of equation (5.75) is constant, with the exception of $\sqrt{T_y}$ the mass flow rate is approaching infinity when the pressure ratio is approaching infinity. Thus, the shock-choke phenomenon means that the Mach number is only limited in stationary coordinates but the actual flow rate isn't.