Cauchy Number formula |
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\( Ca \;=\; \dfrac{ v^2 \cdot \rho }{ B }\) (Cauchy Number) \( v \;=\; \sqrt{ \dfrac{ Ca \cdot B }{ \rho } }\) \( \rho \;=\; \dfrac{ Ca \cdot B }{ v^2 }\) \( B \;=\; \dfrac{ v^2 \cdot \rho }{ Ca }\) |
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| Symbol | English | Metric |
| \( Ca \) = Cauchy Number | \(dimensionless\) | \(dimensionless\) |
| \( v \) = Fluid Velocity | \(ft\;/\;sec\) | \(m\;/\;s\) |
| \( \rho \) (Greek symbol rho) = Fliud Density | \(lbm\;/\;ft^3\) | \(kg\;/\;m^3\) |
| \( B \) = Bulk Modulus of Elasticity | \(lbm\;/\;in^2\) | \(Pa\) |
Cauchy number, abbreviated as Ca, a dimensionless number, is the ratio of inertial force to compressibility force in a flow. When the compressibility is important the elastic forces must be considered along with inertial forces. The Cauchy number provides a measure of the importance of viscous effects compared to inertial effects in a fluid flow. It helps determine the behavior of the flow, such as whether it is dominated by viscous forces (low Cauchy number) or inertial forces (high Cauchy number).
It’s commonly used in problems involving high speed flows (like aerodynamics or hydrodynamics) where compressibility becomes relevant, such as in the study of shock waves or the interaction of fluids with deformable structures. For incompressible fluids, the Cauchy Number is less relevant since the bulk modulus approaches infinity.
Cauchy Number Interpretation
- High Cauchy Number (Ca >> 1) - Inertial forces dominate over elastic forces. This typically occurs in high speed flows where the fluid’s momentum is significant compared to its compressibility or the stiffness of the medium. For example, this might be seen in supersonic flows or when a fluid impacts a relatively flexible structure. Compressibility effects may still matter, but the flow’s kinetic energy drives the behavior.
- Low Cauchy Number (Ca << 1) - Elastic forces dominate over inertial forces. This suggests the fluid or structure resists deformation strongly relative to the momentum of the flow. It’s common in situations with highly incompressible fluids (where K is very large) or slow flows (where v is small). Here, elastic properties like stiffness or compressibility play a bigger role in the system’s response.
- Cauchy Number (Ca ≈ 1) - Inertial and elastic forces are of comparable magnitude, and both effects need to be considered equally. This transitional regime might occur in specific engineering scenarios, like the vibration of structures in a moderately compressible flow.
