1.1 Compressible Flows

Supersonic flows are fundamentally compressible. At first glance, one might define compressible flows as those with variable density. While partially true, this definition is incomplete, not all flows with density variations are truly compressible in the aerodynamic sense.

Thermodynamic Perspective¶

To understand compressibility, we turn to thermodynamics. The specific volume $v$ is expressed as a function of temperature $T$ and pressure $P$:

Differentiating this expression yields:

Dividing both sides by $v$, we obtain the fractional change in specific volume:

This introduces two key thermodynamic properties:

• Isothermal compressibility $\kappa_T$, which quantifies the change in specific volume due to pressure at constant temperature:

  $$\kappa_T = -\frac{1}{v} \left( \frac{\partial v}{\partial P} \right)_T$$

  (4)

• Coefficient of volume expansion $\beta$, which quantifies the change in specific volume due to temperature at constant pressure:

  $$\beta = \frac{1}{v} \left( \frac{\partial v}{\partial T} \right)_P$$

  (5)

Equation (3) shows that changes in specific volume (or density) result from both pressure and temperature variations. In high-speed flows, where density changes are primarily driven by pressure, the flow exhibits significant compressibility effects. Conversely, in flows like natural convection, where pressure is nearly constant and density changes stem from temperature variations, the flow is often treated as incompressible, despite a non-zero $\beta$ in (5).

In high-speed aerodynamics, compressibility is defined by the dominance of pressure-induced density changes, not merely the presence of density variation.