Reynolds number

The Reynolds number characterises the relative importance of inertial and viscous forces in a flow. It is important in determining the state of the [[flow]], whether it is [[laminar]] or [[turbulent]]. At high Reynolds numbers flows generally tend to be [[turbulent]], which was first recognized by [[Osborne Reynolds]] in his famous [[pipe flow experiments]]. Consider the [[Navier-Stokes equations|momentum equation]] which is given below

:<math>
\frac{\partial}{\partial t}\left( \rho u_i \right) +
\frac{\partial}{\partial x_j}
\left[ \rho u_i u_j + p \delta_{ij} \right] =  \frac{\partial}{\partial x_j} \tau_{ij}
</math>

The terms on the right are the [[inertial forces]] and those on the left correspond to [[viscous forces]]. If <math>U</math>, <math>L</math>, <math>\rho</math> and <math>\mu</math> are the reference values for [[velocity]], [[length]], [[density]] and [[dynamic viscosity]], then

inertial force ~ <math>\frac{\rho U^2}{L}</math>

viscous force ~ <math>\frac{\mu U}{L^2}</math>

Their ratio is the Reynolds number, usually denoted as <math>Re</math>

:<math>
Re = \frac{\mbox{inertial force}}{\mbox{viscous force}} = \frac{\rho U L}{\mu}
</math>

In terms of the kinematic viscosity

:<math>
\nu = \frac{\mu}{\rho}
</math>

the Reynolds number is given by

:<math>
Re = \frac{U L}{\nu}
</math>

==Reynolds number as a ratio of time scales==

Consider an impulsively started flat plate moving in its own plane with [[velocity]] <math>U</math>. Due to the [[no-slip condition]] on the plate a [[boundary layer]] gradually develops on the plate. At time <math>t</math>, the thickness of the [[boundary layer]] is of the order of <math>\sqrt{\nu t}</math>  (see Batchelor(1967), section 4.3). Let <math>L</math> be the [[characteristic length scale]]. The time taken for [[viscous]] and [[convective]] effects to travel a distance <math>L</math> is

:<math>
T_{v} = \frac{L^2}{\nu}
</math>

and

:<math>
T_{c} = \frac{L}{U}
</math>

The ratio of viscous to convective time scales is

:<math>
\frac{ T_{v} }{ T_{c} } = \frac{(L^2/\nu)}{(L/U)} = \frac{UL}{\nu} = Re
</math>

Thus the Reynolds number is a measure of the viscous and convective time scales. A large Reynolds number means that viscous effects propagate slowly into the [[fluid]]. This is the reason why boundary layers are thin in high Reynolds number flows because the fluid is being convected along the [[flow]] direction at a much faster rate than the spreading of the [[boundary layer]], which is normal to the [[flow]] direction.

==References==
*{{reference-book | author=Batchelor, G K | year=1967 | title=An Introduction to Fluid Dynamics | rest=Cambridge University Press}}

*{{reference-book |author=Rott, N | year=1990 | title=Note on the history of the Reynolds number| rest =Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 1–11}}

[[Category: Dimensionless parameters]]