Cauchy number

For isentropic processes, the Cauchy number may be expressed in terms of Mach number. The isentropic bulk modulus ${\displaystyle K_{s}=\gamma p}$ , where ${\displaystyle \gamma }$  is the specific heat capacity ratio and p is the fluid pressure. If the fluid obeys the ideal gas law, we have

${\displaystyle K_{s}=\gamma p=\gamma \rho RT=\,\rho a^{2}}$ ,

where

${\displaystyle a={\sqrt {\gamma RT}}}$  = speed of sound, (SI units: m/s)

R = characteristic gas constant, (SI units: J/(kgK) )

T = temperature, (SI units: K)

Substituting K (K_s) in the equation for Ca yields

${\displaystyle \mathrm {Ca} ={\frac {u^{2}}{a^{2}}}=\mathrm {M} ^{2}}$ .

Thus, the Cauchy number is square of the Mach number for isentropic flow of a perfect gas.

• Massey, B. S.; Ward-Smith, J. (1998). Mechanics of Fluids (7th ed.). Cheltenham: Nelson Thornes. ISBN 0-7487-4043-0.