Cauchy number#

Named after: Augustin-Louis Cauchy (1789-1857).

$$\text{Cy} \stackrel{\text{def}}{=} \frac{\rho U^{2}}{K} \sim \frac{\text{inertial stress}}{\text{elastic stress}}$$

Description#

Measures inertial stress relative to elastic or compressibility stress. It indicates when deformation, pressure waves, or material elasticity affect dynamic similarity.

Quantities#

Name	Symbol	SI units	Dimension
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
bulk modulus	$K$	$\mathrm{Pa}$	$\text L^{-1}\,\text M\,\text T^{-2}$

Regimes#

Dynamic similarity

Range	Regime	Description
0 – 1	stress dominated	Reference stresses are large compared with inertial stresses. Deformation or pressure response controls the motion.
1 – ∞	inertia dominated	Inertial stresses exceed the reference stress scale. Dynamic deformation and wave effects become significant.

Name	Symbol	SI units	Dimension
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)
bulk modulus	\(K\)	\(\mathrm{Pa}\)	\(\text L^{-1}\,\text M\,\text T^{-2}\)