Common Dimensionless Numbers

There are certain Pi Terms (Dimensionless Numbers) that appear so frequently within Fluid Mechanics that they have been given special names. Some of these are Reynolds Number, Froude Number, Euler Number, Cauchy Number, Mach Number, Strouhal Number, and Weber Number.

Reynolds Number

Named after Osborne Reynolds, Reynolds Number is one of the most common dimensionless numbers in Fluid Mechanics. In order to calculate Reynolds Number you will need to take the ratio of the inertial forces over the viscous forces. To calculate Reynolds Number you will need the following variables; length, $l$ (or diameter), velocity, $v$ , fluid density, $ρ$ , and fluid viscosity, $μ$ . As result, to calculate Reynolds number you will use the following equation.

(Eq 1) $R e = \frac{ρ v l}{μ}$

If Reynolds Number is extremely small than the viscous forces are dominant. In turn this will cause creeping flow. On the other hand if Reynolds Number is exceptionally large than the viscous forces can be considered negligible.

Froude Number

Next, let’s take a look at Froude number. Named after William Froude, its purpose is to help analyze free surface flows. In order for Froude number to do this it will take the ratio of the inertial forces over the gravitational forces. As a result, to calculate Froude number you will need to use the gravitational constant $g$ which is $9.81 \frac{m}{s^{2}}$ or $32.2 \frac{f t}{s^{2}}$ . Gravitational forces are so important due to their heavy influence on free surface flow. In turn, to calculate Froude Number you will need to use the following equation.

(Eq 2) $F r = \frac{v}{\sqrt{g l}}$

Finally, one use of Froude Number is to study water flow around a ship.

Euler Number

Named after Leonard Euler, Euler Number is the ratio of pressure forces, $p$ , over the inertial forces. In turn, this dimensionless number is used when pressure differences are of interest. This will result in the following equation.

(Eq 3) $E u = \frac{p}{ρ v^{2}}$

In addition, the pressure difference, $Δ p$ may be of interest. As a result, Euler Number will be $E u = Δ p / ρ v^{2}$ . Finally, Euler Number is sometimes referred to as the cavitation number since a modification of it $(p_r-p_v)/\frac{1}{2}ρv^2 is used when analyzing cavitation.

Cauchy Number & Mach Number

Both the Cauchy Number, named after Augustin Louis de Cauchy, and the Mach Number, named after Ernst Mach, are used when the compressibility of the fluid is important. Both take the ratio of inertial forces over the compressibility force. However, the form each equation takes is different. First, to find Cauchy number you will use the following equation.

(Eq 4) $C a = \frac{ρ v^{2}}{E_{v}}$

$E_{v}$ = bulk modulus of elasticity

Next, the following equation represents the Mach Number.

(Eq 5) $M a = \frac{v}{c}$

$c = \sqrt{E_{v} / ρ}$ which is the speed of sound

Out of these two dimensionless number, Mach Number is the most common. When the Mach number is less than 0.3 the inertial forces caused by fluid motion are significantly small. As a result, there will be no noticeable effects on the fluid’s density.

Strouhal Number

Named after Vincenze Strouhal, Strouhal number relates unsteady flows to a characteristic frequency of oscillation $ω$ . In order to do this the ratio of inertial local forces over inertial convective forces will be taken. In turn, the following equation will result.

(Eq 6) $S t = \frac{ω l}{v}$

In turn, Strouhal Number has the capability to represent the flow of fluid over an immersed solid object like a wire. This is because as the fluid flows over the wire a regular pattern of vortices known as a Karmen vortex trail will form. As a result, an oscillating flow with a discrete frequency will form.

Weber Number

Named after Moritz Weber, Weber Number analyzes interface between two fluids. As a result, it is especially important when surface tension, $σ$ , is important. It is found by taking the ratio of inertial forces over surface tension forces.

(Eq 7) $W e = \frac{ρ v^{2} l}{σ}$

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