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Reynold Number
Equation
The key criterion for determining whether a medium is laminar or turbulent is the Reynolds number, which compares the energy associated with inertia to that associated with viscosity. The former depends on the liquid density ($\rho_w$), maximum Speed ($v_{max}$), and the typical Dimension of the System ($R$), while the latter depends on the viscosity ($\eta$), defining it as:
 $ Re =\displaystyle\frac{ \rho R v }{ \eta }$ |
The inertia of a medium can be understood as proportional to the density of kinetic energy, given by:
$\displaystyle\frac{\rho_w}{2}v^2$
where the liquid density ($\rho_w$) and the mean Speed of Fluid ($v$) are.
If we consider the viscose force ($F_v$) as:
$F_v=S\eta\displaystyle\frac{v}{R}$
where the section or Area ($S$), the viscosity ($\eta$), the mean Speed of Fluid ($v$), and the typical Dimension of the System ($R$) are properties of the medium.
Let's recall that energy equals the viscose force ($F_v$) multiplied by the distance traveled ($l$). The density of energy lost due to viscosity will be equal to the force multiplied by the distance divided by the volume $S l$:
$\displaystyle\frac{F_vl}{Sl}=S\eta\displaystyle\frac{v}{R}\displaystyle\frac{l}{Sl}=\eta\displaystyle\frac{v}{R}$
Therefore, the relationship between the density of kinetic energy and the density of viscous energy is equal to a dimensionless number known as the number of Reynold ($Re$). If the number of Reynold ($Re$) is several orders of magnitude greater than one, inertia dominates over viscous force and the flow becomes turbulent. On the other hand, if the number of Reynold ($Re$) is small, viscous force dominates and the flow is laminar.
| $ Re =\displaystyle\frac{ \rho R v }{ \eta }$ |
The original article in which Osborne Reynolds introduces the number bearing his name is:
"An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels," authored by Osborne Reynolds, published in Philosophical Transactions of the Royal Society of London, Vol. 174, pp. 935-982 (1883).
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