Equations for Steady 1D Isentropic Flow

The equations used to describe steady 1D isentropic flow are derived from conservation of mass, momentum, and energy, as well as an equation of state (typically the ideal gas law).

These equations are typically described as ratios between the local static properties (p, T, \rho) and their stagnation property as a function of Mach number and the ratio of specific heats, \gamma. Recall that Mach number is the ratio between the velocity and the speed of sound, a.

These ratios are given here:

Temperature: T_o/T = \left(1+\frac{\gamma -1}{2} M^2\right)

Pressure: P_o/P = \left(1+\frac{\gamma -1}{2} M^2\right)^{\frac{\gamma}{\gamma-1}}

Density: \rho_o/\rho = \left(1+\frac{\gamma -1}{2} M^2\right)^{\frac{1}{\gamma-1}}

In addition to the relationships between static and stagnation properties, 1D nozzle flow offers an equation regarding the choked cross-sectional flow area (recall that the flow is choked when M=1.)

A/A^* = \frac{1}{M}\left(\left(\frac{2}{\gamma+1}\right)\left(1+\frac{\gamma -1}{2} M^2\right)\right)^{\frac{\gamma+1}{2\left(\gamma-1\right)}}

Some excellent references for these equations are:

  • Gas Dynamics Vol. I – Zucrow and Hoffman – 1976
  • Gas Dynamics – John and Keith – 2nd Ed. – 2006