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How It Works
The Cauchy number compares a fluid's inertial forces to its resistance to compression (bulk modulus). When Ca is much less than 1, the flow behaves as though the fluid is incompressible; when Ca approaches or exceeds 1, compressibility effects become significant and must be included in the analysis.
For an ideal gas the Cauchy number equals the square of the Mach number, so it bridges compressibility analysis and acoustic phenomena. Engineers use it in shock-wave studies, high-speed aerodynamics, and underwater acoustics.
Example Problem
Water (ρ = 1,000 kg/m³) flows at 15 m/s through a pipe. The bulk modulus of water is about 2.2 × 10⁹ Pa. What is the Cauchy number?
- Ca = ρv² / Bₛ = 1,000 × 15² / 2,200,000,000
- Ca = 225,000 / 2,200,000,000 = 0.000102
Because Ca « 1, compressibility effects are negligible and the flow can be treated as incompressible.
Frequently Asked Questions
What is the Cauchy number in fluid mechanics?
The Cauchy number (Ca) is a dimensionless ratio of inertial forces to elastic (compressional) forces in a flow. It equals ρv² / Bₛ, where ρ is density, v is velocity, and Bₛ is the bulk modulus. Values well below 1 indicate the fluid can be treated as incompressible.
How is the Cauchy number related to the Mach number?
For an ideal gas, the Cauchy number equals the Mach number squared (Ca = M²). This makes it a convenient single parameter for assessing compressibility in gas dynamics without separately computing the speed of sound.
When can you assume incompressible flow?
A common engineering rule is that flows with Ca < 0.1 (or equivalently M < 0.3) can be modeled as incompressible with less than about 5% density variation. Water at typical pipe velocities has Ca on the order of 10⁻⁴, so it is almost always incompressible.
What is the bulk modulus of common fluids?
Water has a bulk modulus of about 2.2 GPa at room temperature, making it very resistant to compression. Engine oil is around 1.5 GPa. Air at sea level has an effective isentropic bulk modulus of only about 142 kPa, which is why gas flows become compressible at much lower speeds.
Related Calculators
- Mach Number Calculator — compute the ratio of flow velocity to the speed of sound.
- Euler Number Calculator — relate pressure drop to inertial forces in a flow.
- Weber Number Calculator — compare inertial forces to surface tension forces.
- Reynolds Number Calculator — determine whether a flow is laminar or turbulent.
- Speed Converter — convert between m/s, ft/s, km/h, and other velocity units.