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# Euler number (physics)

The Euler number is a dimensionless number used in fluid flow calculations. It expresses the relationship between a local pressure drop over e.g. a restriction and the kinetic energy per volume, and is used to characterize losses in the flow.

It is defined as

$\mathit{Eu}=\frac{p(upstream)-p(downstream)}{\frac{1}{2}\rho V^2}$

where

• ρ is the density of the fluid.
• p(upstream) is the upstream pressure.
• p(downstream) is the downstream pressure.
• V is a characteristic velocity of the flow.

Somewhat the same structure, but with a different meaning is the Cavitation number:

The Cavitation number is a dimensionless number used in flow calculations. It expresses the relationship between the difference of a local absolute pressure from the vapor pressure and the kinetic energy per volume, and is used to characterize the potential of the flow to cavitate.

It is defined as

$\mathit{Ca}=\frac{p-p_v}{\frac{1}{2}\rho V^2}$

where

• ρ is the density of the fluid.
• p is the local pressure.
• pv is the vapor pressure of the fluid.
• V is a characteristic velocity of the flow.