The fluid becomes compressible when it is subjected to a pressure field causing them to flow, i.e., the fluid will be compressed or be expanded to some extent because of the pressure acting on them. The time rate of change of velocity of the fluid elements in a given pressure gradient is a function of the fluid density, whereas the degree of compression is determined by the isentropic bulk modulus of compression. The term compressible flows are defined as the variable density flow. The variations in fluid density for compressible flow require attention to density and other fluid property relationships. The fluid equation of state, often unimportant for incompressible flows, is vital in the analysis of compressible fluids.
The sound waves are the infinitesimal pressure disturbances, and the speed at which these waves propagate in a medium is known the speed of sound or acoustic speed. Further, the term compressible flow reflects the variation in density due to pressure change from one point to another in the flow field. The change in density with respect to pressure has strong effects on the wave propagation.
In turbomachines, the speed of the rotor should be in the range of 270–450 \(\mathrm {ms^{1}}\), to avoid the excessive stresses generated due to rotation. The studies also reveal that the loss in efficiency mounts rapidly when the rotor speed approaches the sonic velocity. Thus, for air compressors the limiting design factor on rotational speed may be either stress or compressibility considerations. In hydrogen compressors, the fluid compressibility will never be a factor, whereas compressibility is a major design factor for the compressor working with Freon22 as fluid.
The speed of sound is a property that varies from point to point and if there exists a large difference in the speeds between the body and the compressible fluid surrounding it, the compressibility of the fluid medium influences the flow around the body. Thus, both the inertial forces and elastic forces due to fluid compressibility should be accounted in the analysis. The ratio of inertial force to elastic force is a nondimensional parameter, called the Mach number.
von Karman proposed three rules of supersonic flows which are applicable for small disturbances. These rules, however, can be to large disturbances but for qualitatively purposes only.

Rule of Forbidden Signals: The effect of pressure changes produced by a body, moving at a speed faster than the sound, cannot felt upstream of the body.

Zone of Action and Zone of Silence: A stationary point source in a supersonic stream produces effects only on the points that lie on or inside the Mach cone, extending downstream from the point source.

Rule of Concentrated Action: The proximity of circles representing the various flow situations is a measure of the intensity of the pressure disturbance at each point in the flow field.
The flow regimes can be classified based on the value of the Mach number.

For \(\mathrm {\text {0}<M<\text {1}}\), the flow is termed as subsonic. In a subsonic field, the presence of small disturbance, traveling with acoustic speed, will be felt throughout the flow domain. Thus, the subsonic flows are essentially “prewarned” to the disturbance.

For \(\mathrm {\text {0.8}<M<\text {1.2}}\), the flow is termed as transonic flow.

For \(\mathrm {M=\text {1}}\), the flow is called sonic flow.

For \(\mathrm {M>\text {1}}\), the flow is called supersonic flow. Since the flow speed is above the speed of sound, they are no more “prewarned”.

For \(\mathrm {M>\text {5}}\), the flow is called hypersonic flow.
If the rate of change of fluid properties normal to the streamline direction is negligible as compared to the rate of change along the streamlines, the flow can be assumed to be onedimensional. For flow in ducts, this means that all the fluid properties can be assumed to be uniform over any cross section of the duct. These properties which define the state of a system are called static properties, and the properties at a state which is achieved by decelerating the flow to rest through an isentropic means (i.e., reversible and adiabatic process) are known as stagnation properties.
For the compressible flows, changes in enthalpy and the kinetic energy are much larger than that in elevation. Thus, between any two points, “1” and “2”, along a streamline the specific static enthalpy
\(\mathrm {\left( h\right) }\) and fluid velocity
\(\mathrm {\left( v\right) }\) are related by
$$\begin{aligned} \mathrm {h_{1}+\frac{v_{1}^{2}}{2}}&\, \mathrm {=h_{2}+\frac{v_{2}^{2}}{2}} \end{aligned}$$
For isentropic flow of a perfect gas,
$$\begin{aligned} \mathrm {\frac{p_{0}}{p}}&\, \mathrm {=} \mathrm {\left( \frac{T_{0}}{T}\right) ^{\frac{\gamma }{\gamma 1}}=\left( \frac{\rho _{0}}{\rho }\right) ^{\gamma }}\\ \mathrm {\frac{T_{0}}{T}}&\, \mathrm {=} \mathrm {1+\frac{\gamma 1}{2}M^{2}}\\ \mathrm {\frac{p_{0}}{p}}&\, \mathrm {=} \mathrm {\left[ 1+\frac{\gamma 1}{2}M^{2}\right] ^{\frac{\gamma }{\gamma 1}}}\\ \mathrm {\frac{\rho _{0}}{\rho }}&\, \mathrm {=} \mathrm {\left[ 1+\frac{\gamma 1}{2}M^{2}\right] ^{\frac{1}{\gamma 1}}} \end{aligned}$$
The parameter
\(\mathrm {M^{*}}\) is defined as the ratio of the local velocity to the velocity of sound at the choked state
\(\mathrm {\left( M=\text {1}\right) }\). It is expressed as
$$\begin{aligned} \mathrm {M^{*2}}&\, \mathrm {=\frac{\left[ \frac{\left( \gamma +1\right) }{2}M^{2}\right] }{\left[ 1+\frac{\left( \gamma 1\right) }{2}M^{2}\right] }} \end{aligned}$$
The mass flow rate through a streamtube of crosssectional area
\(\mathrm {A}\) is given by
$$\begin{aligned} \mathrm {\frac{\mathring{m}}{A}}&\, \mathrm {=p_{0}M\sqrt{\frac{\gamma }{RT_{0}}}\left( 1+\frac{\gamma 1}{2}M^{2}\right) ^{\frac{\gamma +1}{2\left( \gamma 1\right) }}} \end{aligned}$$
The maximum mass flow rate per unit area is given by
$$ \mathrm {\mathrm {\left( \frac{\mathring{m}}{A}\right) _{max}}=\frac{\mathring{m}}{A^{*}}=p_{0}\sqrt{\frac{\gamma }{RT_{0}}}\left( \frac{\gamma +1}{2}\right) ^{\frac{\gamma +1}{2\left( \gamma 1\right) }}} $$
i.e., for a given stagnation conditions, the maximum mass flow rate per unit area is directly proportional to
\(\mathrm {\frac{p_{0}}{\sqrt{T_{0}}}}\).
The variation of flow area
\(\mathrm {A}\) through the nozzle relative to the throat area
\(\mathrm {A^{*}}\) for the same mass flow rate and stagnation properties of a perfect gas is
$$\begin{aligned} \mathrm {\frac{A}{A^{*}}}&\, \mathrm {=\frac{1}{M^{2}}\left[ \frac{2}{\left( \gamma +1\right) }\left( 1+\frac{\left( \gamma 1\right) }{2}M^{2}\right) \right] ^{\frac{\left( \gamma +1\right) }{\left( \gamma 1\right) }}} \end{aligned}$$
This is known as area–Mach number relation.
The three reference speeds for studying the compressible flows are
\(\mathrm {v_{\text {max}}}\) corresponding to a given stagnation state, the speed of sound at the stagnation temperature
\(\mathrm {a_{0}}\), and the critical speed
\(\mathrm {v^{*}}\). They are given as
$$\begin{aligned} \mathrm {v_{\text {max}}}&\, \mathrm {=\sqrt{\frac{2\gamma RT_{0}}{\gamma 1}}}\\ \mathrm {a_{0}}&\, \mathrm {=\sqrt{\gamma RT_{0}}} \\ \mathrm {v^{*}}&\mathrm {=\left[ \frac{2\gamma }{\gamma +1}RT_{0}\right] ^{\nicefrac {1}{2}}} \end{aligned}$$
When the speed of sound is plotted as a function of the speed of the flow for an adiabatic flow of a gas, it results an ellipse known as adiabatic flow ellipse. It is given by the following relation:
$$\begin{aligned} \mathrm {\frac{v^{2}}{v_{\text {max}}^{2}}+\frac{a^{2}}{a_{0}^{2}}}&\, \mathrm {=1} \end{aligned}$$