Shock (fluid dynamics)

Shock in Gas Dynamics Terminology means an abrupt discontinuity in the flow field. It occurs in flows when the local flow speed exceeds the local sound speed. To put it in other words it means when the Mach number of the flow exceeds '1', then alone the concept of Shock makes sense.

Explanation of phenomena

Shock is formed due to coalescing (coming together) of various small pressure pulses. Sounds waves are pressure wave and it is at the speed of the sound wave the disturbances are communicated in the medium. When an object is moving in a flow field the object sends out disturbances which propagate at the speed of sound and adjusts the remaining flow field accordingly. However, if the object itself happens to travel at speed greater than sound, then the disturbances created by the object would not have traveled and communicated to the rest of the flow field and this results in an abrupt change of property, which is termed as shock in gas dynamics terminology.

Shocks are characterized by discontinuities changes in flow properties such as velocity, pressure, temperature etc. Typically shock thickness is of a few mean free paths (of the order of 10 ^-8 m). Shocks are irreversible occurrence (Non-Isentropic) in supersonic flows.

Normal Shock formulas

$\mathbf {T_{02}} =\mathbf {T_{01}}$

$M_{2}=({\frac {{\frac {2}{\gamma -1}}+{M_{1}}^{2}}{{\frac {2\gamma }{\gamma -1}}{M_{1}}^{2}-1}})^{0.5}$

${\frac {p_{2}}{p_{1}}}={\frac {1+\gamma M_{1}^{2}}{1+\gamma M_{2}^{2}}}={\frac {2\gamma }{\gamma +1}}M_{1}^{2}-{\frac {\gamma -1}{\gamma +1}}$

${\frac {T_{2}}{T_{1}}}={\frac {1+{\frac {\gamma -1}{2}}M_{1}^{2}}{1+{\frac {\gamma -1}{2}}M_{2}^{2}}}={\frac {(1+{\frac {\gamma -1}{2}}M_{1}^{2})({\frac {2\gamma }{\gamma -1}}M_{1}^{2}-1)}{\frac {(\gamma +1)^{2}M_{1}^{2}}{2(\gamma -1)}}}$

${\frac {a_{2}}{a_{1}}}={({\frac {T_{2}}{T_{1}}})}^{0.5}$

${\frac {\rho _{2}}{\rho _{1}}}={\frac {p_{2}}{p_{1}}}{\frac {T_{1}}{T_{2}}}$

${\frac {p_{01}}{p_{1}}}=(1+{\frac {\gamma -1}{2}}M_{1}^{2})^{\frac {\gamma }{\gamma -1}}$

${\frac {p_{02}}{p_{2}}}=(1+{\frac {\gamma -1}{2}}M_{2}^{2})^{\frac {\gamma }{\gamma -1}}$

Note :

'1' refers to upstream properties

'2' refers to down stream properties

subscript '0' - refers to Total or stagnation properties

T refers to Temperature

M refers to Mach number

P refers to Density

$\mathbf {\rho }$ refers to Density

$\gamma$ is the ratio of the specific heats