Cauchy Number Calculator

What is the Cauchy Number?

The Cauchy number (Ca) is a dimensionless quantity in fluid mechanics that measures the ratio of inertial forces to elastic (compressional) forces in a fluid flow. It is named after the French mathematician Augustin-Louis Cauchy and plays a vital role in determining whether a fluid can be treated as incompressible or if compressibility effects must be accounted for in engineering analysis.

The Cauchy number is mathematically expressed as:

Ca = ρv² / Bₛ

Where Ca is the Cauchy number (dimensionless), ρ is the fluid density in kg/m³, v is the flow velocity in m/s, and Bₛ is the isentropic bulk modulus of the fluid in pascals. When Ca is much less than 1, compressibility effects are negligible and the flow can be modeled as incompressible. When Ca approaches or exceeds 1, density changes become significant and compressible flow equations are required.

Applications of the Cauchy Number

The Cauchy number is widely used in aerospace engineering, high-speed piping systems, water hammer analysis, and any application where fluid compressibility may affect system performance. For ideal gases, the Cauchy number equals the square of the Mach number (Ca = M²), making it a direct indicator of the compressibility regime in gas flows.

Engineers use the Cauchy number to evaluate whether the simpler incompressible flow equations are sufficient for their analysis or whether the more complex compressible flow models must be employed. This decision has significant implications for computational fluid dynamics (CFD) simulations, where treating a flow as incompressible when it is not can lead to inaccurate results.

How to Use This Calculator

This calculator allows you to solve for any of the four variables in the Cauchy number equation:

• Cauchy Number — Calculate Ca when you know density, velocity, and bulk modulus
• Flow Velocity — Determine the velocity from Ca, bulk modulus, and density
• Fluid Density — Find the density given Ca, bulk modulus, and velocity
• Bulk Modulus — Compute the bulk modulus from density, velocity, and Ca

Interpretation of Results

When interpreting the Cauchy number, remember these guidelines:

• Ca < 0.1 — Compressibility effects are negligible; incompressible flow assumptions are valid
• 0.1 < Ca < 1.0 — Transitional regime; compressibility may need to be considered
• Ca > 1.0 — Compressibility effects are significant; use compressible flow equations

For a given fluid, increasing the flow velocity increases the Cauchy number, making compressibility more important. Conversely, fluids with a higher bulk modulus (stiffer fluids) have lower Cauchy numbers at the same velocity, meaning they behave more like incompressible fluids.