Cauchy Number
Dimensionless ratio of inertial forces to elastic forces. Values well below 1 indicate incompressible flow.
Ca = ρv² / Bₛ
Density from Cauchy Number
Solves for the fluid density given the Cauchy number, bulk modulus, and flow velocity.
ρ = Ca × Bₛ / v²
Velocity from Cauchy Number
Determines the flow velocity from the Cauchy number, bulk modulus, and fluid density.
v = √(Ca × Bₛ / ρ)
Bulk Modulus from Cauchy Number
Solves for the bulk modulus given density, velocity, and Cauchy number.
Bₛ = ρv² / Ca
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. For an ideal gas the Cauchy number equals the square of the Mach number.
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.
- Identify the known variables: ρ = 1,000 kg/m³, v = 15 m/s, Bₛ = 2.2 × 10⁹ Pa
- Write the Cauchy number formula: Ca = ρv² / Bₛ
- Square the velocity: v² = 15² = 225 m²/s²
- Calculate the numerator: ρ × v² = 1,000 × 225 = 225,000
- Divide by bulk modulus: Ca = 225,000 / 2,200,000,000 = 0.000102
- Interpret: Because Ca ≪ 1, compressibility effects are negligible and the water flow can be modeled as incompressible
Because Ca ≪ 1, compressibility effects are negligible.
When to Use Each Variable
- Solve for Cauchy Number — when you know the fluid density, flow velocity, and bulk modulus, e.g., checking whether compressibility effects matter in a pipe flow simulation.
- Solve for Density — when you have a target Cauchy number, bulk modulus, and velocity, e.g., selecting a fluid for a compressibility experiment.
- Solve for Velocity — when you know the Cauchy number, bulk modulus, and density, e.g., finding the maximum flow speed that keeps Ca below a threshold.
- Solve for Bulk Modulus — when you know the Cauchy number, density, and velocity, e.g., back-calculating fluid stiffness from measured flow conditions.
Key Concepts
The Cauchy number is a dimensionless ratio of inertial forces to elastic (compressional) forces in a fluid flow. When Ca is much less than 1, compressibility effects are negligible and the flow can be modeled as incompressible. For ideal gases, the Cauchy number equals the square of the Mach number, making it a direct indicator of compressibility regime.
Applications
- Aerospace engineering: determining whether airflow around a wing requires compressible flow equations
- Pipeline design: verifying that water-hammer pressure waves are negligible in liquid pipelines
- Underwater acoustics: assessing compressibility effects in sonar propagation models
- Hydraulic machinery: checking whether cavitation or compressibility affects pump performance
Common Mistakes
- Using gauge pressure instead of absolute bulk modulus — the bulk modulus is an absolute material property, not a pressure reading
- Forgetting to square the velocity — the Cauchy number depends on v², so doubling velocity quadruples Ca
- Confusing Cauchy number with Mach number — Ca = M² only for ideal gases, not for liquids
Frequently Asked Questions
What is the physical meaning of the Cauchy number in fluid dynamics?
The Cauchy number measures the ratio of inertial forces to elastic (compressional) forces in a flowing fluid. It tells engineers whether a fluid's compressibility matters: when Ca is much less than 1, the fluid behaves as if it were incompressible, and simpler equations apply. When Ca approaches or exceeds 1, density changes become significant and compressible flow equations are required.
How does the Cauchy number relate to the Mach number?
For an ideal gas, the Cauchy number equals the Mach number squared (Ca = M²). This is because the speed of sound in an ideal gas is c = √(Bₛ/ρ), so Ca = ρv²/Bₛ = v²/c² = M². For liquids, the Cauchy number still applies but the Mach number relationship requires using the actual speed of sound in the liquid.
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. Values well below 1 indicate the fluid can be treated as incompressible.
When can you assume incompressible flow?
Flows with Ca < 0.1 (or M < 0.3) can be modeled as incompressible with less than about 5% density variation. This covers nearly all liquid flows and low-speed gas flows.
What is the bulk modulus of common fluids?
Water has a bulk modulus of about 2.2 GPa. Engine oil is around 1.5 GPa. Air at sea level has an effective isentropic bulk modulus of only about 142 kPa. The huge difference explains why liquids are nearly incompressible while gases compress easily.
Why does doubling velocity quadruple the Cauchy number?
The Cauchy number depends on velocity squared (Ca = ρv²/Bₛ). Doubling v means v² increases by a factor of 4, so Ca also quadruples. This is why compressibility effects grow rapidly with speed — a flow at 600 m/s has 4× the Cauchy number of a flow at 300 m/s.
Is the Cauchy number used outside of fluid mechanics?
Yes. The Cauchy number also appears in solid mechanics and structural dynamics, where it compares dynamic loading (inertial stress) to material stiffness. In that context, Ca indicates whether a structure will deform significantly under dynamic forces or remain effectively rigid.
Cauchy Number Formula
The Cauchy number is a dimensionless ratio of inertial forces to elastic (compressibility) forces in a fluid:
Ca = ρv² / Bₛ
Where:
- Ca — Cauchy number (dimensionless)
- ρ — fluid density, measured in kg/m³
- v — flow velocity, measured in m/s
- Bₛ — isentropic bulk modulus of the fluid, measured in pascals (Pa)
When Ca ≪ 1, the fluid is effectively incompressible. For an ideal gas, Ca = M² (Mach number squared), making the Cauchy number a direct measure of compressibility regime.
Worked Examples
Aerodynamics
Does a subsonic aircraft need compressibility corrections?
An aircraft cruises at 250 m/s at altitude where air density is 0.41 kg/m³ and the effective bulk modulus (isentropic) is about 57,000 Pa. Calculate the Cauchy number.
- Ca = ρv² / Bₛ = 0.41 × 62,500 / 57,000
- Ca = 25,625 / 57,000
- Ca = 0.4496
Ca = 0.45 (≈ M = 0.67) means compressibility effects are noticeable. Aerodynamicists apply Prandtl-Glauert corrections above Ca ≈ 0.09 (M = 0.3).
Underwater Acoustics
Is seawater flow in a sonar dome compressible?
Seawater (ρ = 1,025 kg/m³, Bₛ = 2.34 × 10⁹ Pa) flows past a submarine sonar dome at 15 m/s. What is the Cauchy number?
- Ca = ρv² / Bₛ = 1,025 × 225 / 2,340,000,000
- Ca = 230,625 / 2,340,000,000
- Ca = 9.86 × 10⁻⁵
Ca ≪ 1, confirming the flow is effectively incompressible. Sonar models can safely use the incompressible Navier-Stokes equations for the flow field around the dome.
Shock Wave Analysis
What bulk modulus would make a flow compressible at 50 m/s?
A researcher wants to study compressibility effects at Ca = 0.5 with a fluid of density 800 kg/m³ flowing at 50 m/s. What bulk modulus is needed?
- Bₛ = ρv² / Ca = 800 × 2,500 / 0.5
- Bₛ = 2,000,000 / 0.5
- Bₛ = 4,000,000 Pa (4 MPa)
A bulk modulus of 4 MPa is very low compared to liquids (water is 2.2 GPa), so the researcher would need a highly compressible fluid or gas at elevated pressure.
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.
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