Library / Dimensional analysis and similarity / DIM-12
Dimensional analysis and similarity
Cauchy number
Ratio of inertial stress to the fluid’s bulk modulus, which is the square of the Mach number when the bulk modulus is the isentropic one.
- Ref
- DIM-12
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Cy
Cy = ρU²/Ks compares the inertial stress ρU² with the isentropic bulk modulus Ks, the stiffness of the fluid against compression. Because the sound speed is c = √(Ks/ρ), Cy = (U/c)² is the square of the Mach number. A small value means the density changes little and the flow can be treated as incompressible. Cy is used here as the symbol to avoid confusion with the capillary number, which is often written Ca.
Ratio of inertial stress to the fluid’s bulk modulus, which is the square of the Mach number when the bulk modulus is the isentropic one. A small value means the fluid can be treated as incompressible.
Calculator
- Sound speed, c = √(Ks/ρ)1,500 m/s
- Mach number, √Cy0.01
- Density change at a stagnation point, about Cy/20.005 %
- Speed at which Cy = 0.09 (Mach 0.3)450 m/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Cy = 1×10−4 corresponds to a Mach number of 0.01, so the density changes by only about 0.005 % at a stagnation point: the fluid can be treated as incompressible.
- The usual limit for ignoring compressibility in a gas is Mach 0.3, a Cauchy number of 0.09, where the density change reaches about 5 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ks Isentropic bulk modulus | 0.1 to 500,000 MPa | 4.5e-7 | 2.25 | 2.25 | +10% | −9.09 % |
| U Characteristic speed | 0.1 to 2,000 m/s | 4.444e-9 | 1.778 | 1.778 | +10% | +21 % |
| ρ Fluid density | 0.1 to 2,000 kg/m³ | 1e-8 | 2e-4 | 2e-4 | +10% | +10 % |
Sweep of Characteristic speed
| Characteristic speed (m/s) | Cauchy number |
|---|---|
| 0.1 | 4.444e-9 |
| 0.3448 | 5.285e-8 |
| 1.189 | 6.285e-7 |
| 4.101 | 7.475e-6 |
| 14.14 | 8.889e-5 |
| 48.77 | 0.001057 |
| 168.2 | 0.01257 |
| 580 | 0.1495 |
| 2,000 | 1.778 |
Sweep one variable
Variables and how each one contributes
Cauchy number
What the equation gives you. Everything else feeds it.
Fluid density
The fluid density, in kg/m³. Raises the Cauchy number in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 1×10−4 to 1.1×10−4.
Characteristic speed
The characteristic flow speed, in m/s. Raises the Cauchy number with the square of the speed, so speed is the strongest lever. Raising it by 10 % (to 16.5) moves the result by +21 %, from 1×10−4 to 1.21×10−4.
Isentropic bulk modulus
The isentropic bulk modulus, in MPa: about 2250 for water, and γp for an ideal gas (about 0.14 for air at atmospheric pressure). Lowers the Cauchy number in inverse proportion: a stiffer fluid is less compressible. Raising it by 10 % (to 2,475) moves the result by −9.09 %, from 1×10−4 to 9.091×10−5.
Worked example with the values above
- Convert the modulus: Ks = 2,250 MPa = 2.25×109 Pa.
- Inertial stress: ρU² = 1,000 × 15² = 225,000 Pa.
- Cy = 225,000 / 2.25×109 = 1×10−4, a Mach number of 0.01.
Assumptions and limits
- Ks is the isentropic bulk modulus of the fluid at the operating state, so that c = √(Ks/ρ).
- A characteristic speed that represents the strongest flow in the region of interest.
Common mistakes
- Entering the modulus in Pa or GPa instead of MPa.
- Using the isothermal bulk modulus of a gas, which is smaller than the isentropic one by the factor γ.
- Confusing Cy with the capillary number Ca = μU/σ.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (ρ × 1, U × 2, Ks × −1). Speed dominates. The bulk modulus of a liquid also changes with pressure, temperature and any dissolved or entrained gas, which lowers it sharply.
Where it is used
- Checking whether compressibility matters in a gas or liquid flow, or in a water-hammer or sloshing problem.
- Relating the incompressible-flow assumption to the Mach number.
Maths you need
- Bulk modulus: pressure needed to compress a fluid, per fractional volume change
- Mach number: speed relative to the speed of sound