Euler Number

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 – p_s)/ρv², where p₀ is the characteristic pressure and p_s 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 = c_p/c_v (c_p is the specific heat at constant pressure, and c_v 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.

单词	euler number
释义	Euler Number Euler Number 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 – p_s)/ρv², where p₀ is the characteristic pressure and p_s 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 = c_p/c_v (c_p is the specific heat at constant pressure, and c_v 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.
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