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a similarity criterion used in the study of the motion of liquids or gases.
The Euler number Eu characterizes the relation between the pressure forces acting on an elementary volume of a liquid or gas and the inertia forces. It is defined by the formula
(sometimes 2p/ρv2), where p2 and p1 are the pressures at two characteristic points of the flow (or of a body moving in it), ρv2/2 is the velocity head, ρ is the density of the liquid or gas, and v is the fluid velocity (or the velocity of the body).
In the case of liquid flows with cavitation, the analogous criterion is called the cavitation number κ = 2(p0 – ps)/ρv2, where p0 is the characteristic pressure and ps is the saturation vapor pressure of the liquid.
For compressible gas flows the relation between the Euler number in the form Eu = 2p/ρv2 and two other similarity criteria—the Mach number M and the ratio of the specific heats of the medium γ—is given by the equation Eu = 2/γM2, where γ = cp/cv (cp is the specific heat at constant pressure, and cv is the specific heat at constant volume).
It should be noted that in the American literature two Euler numbers are often used. Euler number 1 equals the friction head divided by two times the velocity head, and Euler number 2 is equal to two times the Fanning friction factor.
The Euler number is named for L. Euler.