Piston Velocity

When a piston is moving, it creates a shock that moves at a speed greater than that of the piston itself. The unknown data are the piston velocity, the temperature, and, other conditions ahead of the shock. Therefore, no Mach number is given but pieces of information on both sides of the shock. In this case, the calculations for U_s can be obtained from equation (5.24) that relate the shock velocities and Shock Mach number as

$${U_x \over U_y} = { M_{sx} \over M_{sx} - {{U_y}^{'} \over c_x} } = {( k +1) {M_{sx} } ^2 \over 2 + (k -1) {M_{sx}}^{2} }$$ (5.64)

Equation (5.64) is a quadratic equation for M_sx. There are three solutions of which the first one is M_sx=0 and this is immediately disregarded. The other two solutions are

$$M_{sx} = { (k + 1){ U_y }^{'} \pm \sqrt{\left[{U_y }^{'} (1 + k)\right]^2 + 16{c_x}^{2}} \over 4 \; c_x }$$ (5.65)

The negative sign provides a negative value which is disregarded, and the only solution left is

$$M_{sx} = { (k + 1){ U_y }^{'} + \sqrt{\left[{U_y }^{'} (1 + k)\right]^2 + 16{c_x}^{2}} \over 4 \; c_x }$$ (5.66)

or in a dimensionless form

$$M_{sx} = { (k + 1){ M_{yx} }^{'} + \sqrt{\left[{M_{yx} }^{'} (1 + k)\right]^2 + 16} \over 4 }$$ (5.67)

Where the ``strange'' Mach number is M_sx′ = U_x′ /C_x. The limit of the equation when c_x⇒ ∞ leads to

$$M_{sx} = {(k + 1){ M_{yx} }^{'} \over 4 }$$ (5.68)

As one additional ``strange'' it can be seen that the shock is close to the piston when the gas ahead of the piston is very hot. This phenomenon occurs in many industrial applications, such as the internal combustion engines and die casting. Some use equation (5.68) to explain the next Shock-Choke phenomenon.