What is the Cauchy Number?
The Cauchy number (Ca) is a dimensionless quantity used in fluid mechanics to characterize the compressibility effects in a flow. It is named after the French mathematician Augustin-Louis Cauchy (1789–1857), who made profound contributions to continuum mechanics.
Mathematical definition:
Ca = ρ v² / K
where ρ = density (kg/m³), v = flow velocity (m/s), K = bulk modulus of elasticity (Pa).
Theoretical Background & Derivation
The Cauchy number arises naturally when non‑dimensionalizing the compressible Navier‑Stokes equations. By scaling the momentum equation with a reference velocity U, density ρ₀, and bulk modulus K, the coefficient in front of the pressure gradient term becomes 1/Ca. Thus Ca governs the importance of pressure work due to compressibility.
For an ideal gas undergoing an isentropic process, the bulk modulus K = γ p, where γ is the specific heat ratio and p is the pressure. Using the ideal gas law p = ρ R T and the definition of the speed of sound c = √(γ p/ρ), one obtains the simple relation:
Ca = M²
where M = v/c is the Mach number. This demonstrates that for gases, the Cauchy number is exactly the square of the Mach number.
Note: In older literature, Ca is sometimes defined as ρ v² / (γ p) or with other factors, but the modern consensus (per ISO 80000‑11) is the definition used here.
Physical Interpretation & Flow Regimes
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Ca ≪ 1 (e.g., Ca < 0.1): Compressibility forces dominate. The fluid behaves as nearly incompressible; density changes are negligible. Typical for liquids at moderate speeds and low-speed gas flows.
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Ca ≈ 1 (0.1 < Ca < 10): Inertial and compressibility forces are comparable. Compressibility begins to influence pressure distribution and may cause mild wave effects.
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Ca ≫ 1 (Ca > 10): Inertial forces dominate; compressibility effects are strong. Shock waves, expansion fans, and large density variations appear. Common in high-speed aerodynamics (M > 1).
Engineering Significance & Applications
The Cauchy number is crucial in several fields:
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Aerospace engineering: Assessing compressibility effects on airfoils, predicting wave drag, and designing supersonic inlets.
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Hydraulic transients (water hammer): In pipelines, the wave speed depends on the bulk modulus of the fluid; Cauchy number helps estimate pressure surge magnitudes.
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Underwater acoustics: Sound propagation in water relates to K and ρ; Ca appears in the reflection/transmission coefficients at interfaces.
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High‑pressure systems: In hydraulic presses or fuel injection, fluid compressibility affects performance and stability.
Case Study: Water Hammer in a Penstock
Consider a 1000 m long water pipeline (ρ = 1000 kg/m³, K = 2.2 GPa) with an initial velocity of 2 m/s. If a valve closes rapidly, the pressure rise Δp ≈ ρ v c, where c = √(K/ρ) ≈ 1483 m/s. Here Ca = ρv²/K = (1000×4)/(2.2e9) ≈ 1.8×10⁻⁶ ≪ 1, confirming incompressible behaviour during steady flow. However, during the transient, the compressibility of water determines the wave speed and the magnitude of the water hammer. Thus even a tiny Ca can be dynamically important.
Relationship with Other Dimensionless Numbers
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Mach number M: Ca = M² for ideal gases. For liquids, no direct relation exists because sound speed depends on K and ρ (c = √(K/ρ)).
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Euler number Eu = p/(ρv²): Eu ≈ 1/Ca when pressure scaling is based on bulk modulus.
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Compressibility factor Z: In real gases, Ca can be corrected using Z.
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Strouhal number St: In unsteady flows, Ca interacts with St to determine the importance of compressible unsteady effects.
Typical Values of Bulk Modulus & Density
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Fluid
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Bulk Modulus K (Pa)
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Density ρ (kg/m³)
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Speed of sound c (m/s)
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Air (1 atm, 20°C)
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1.42 × 10⁵
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1.205
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343
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Water (fresh)
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2.15 × 10⁹
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998
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1481
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Seawater
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2.34 × 10⁹
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1025
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1510
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Ethanol
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1.06 × 10⁹
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789
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1150
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Glycerin
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4.35 × 10⁹
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1260
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1900
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Hydraulic oil
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1.5–2.0 × 10⁹
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850–900
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1300–1500
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Common Misconceptions & Pitfalls
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Confusing Cauchy number with Mach number: They are only equivalent for ideal gases; for liquids, Ca can be small even at high M if defined with gas sound speed – always use the correct bulk modulus.
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Assuming bulk modulus is constant: K varies with pressure and temperature, especially for gases and near phase change.
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Using Ca alone for unsteady flows: In transient problems, the time scale also matters; consider the Cauchy number in conjunction with the Strouhal number.
Historical Note
Augustin-Louis Cauchy originally introduced the concept of “modulus of compression” in his 1827 work on elasticity. The dimensionless number bearing his name was later adopted in fluid mechanics to quantify compressibility effects. Early aerodynamicists like Prandtl and Glauert used the equivalent Mach number, but Cauchy’s formulation remains fundamental in continuum mechanics.
Frequently Asked Questions
Mach number (M = v/c) compares flow velocity to the speed of sound. Cauchy number (Ca = ρv²/K) compares inertial forces to compressibility forces. For ideal gases, they are related by Ca = M². Mach number is more common in gas dynamics, while Cauchy number is used for any fluid (including liquids) where bulk modulus is known.
Cauchy number is directly applicable to both liquids and gases, whereas Mach number relies on the speed of sound, which may not be easily defined for complex fluids. In hydrodynamics (e.g., water hammer), Cauchy number is used to assess compressibility effects without needing sound speed.
Temperature influences density and bulk modulus. For gases, density decreases with temperature (at constant pressure), and bulk modulus (γp) may remain constant if pressure is constant. For liquids, bulk modulus generally decreases slightly with increasing temperature, while density also decreases. Therefore Ca can change significantly with temperature.
Yes, if velocity is very high (e.g., water jet at several hundred m/s) or bulk modulus is low (e.g., gas‑liquid mixtures). However, for pure water under normal conditions, Ca remains far below 1 because K is huge (~2 GPa). Achieving Ca > 1 in water would require velocities > 1500 m/s, which is impractical.
For an ideal gas with constant γ, yes: Ca = ρv²/(γp) = v²/(γRT) = v²/c² = M². For real gases, the bulk modulus deviates from γp, so Ca may differ slightly. Always use the actual K for the specific gas condition.
For water flows, Ca is usually 10⁻⁶ to 10⁻³. For air at subsonic speeds, Ca ranges from 0 to 1 (M<1). For supersonic aircraft, Ca can exceed 1 up to 10 (M≈3). In hypervelocity flows (re‑entry), Ca can reach 100 or more.
References: White, F.M. "Fluid Mechanics" 8th ed.; Kundu, P.K. "Fluid Mechanics" 6th ed.; ISO 80000-11:2019. Values from NIST Chemistry WebBook.