📂Fluid Mechanics

Reynolds Number: Distinction between Laminar and Turbulent Flow

Definition ¹

Explanation

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We refer to the numerator of the Reynolds number as the inertial force and the denominator as the viscous force. In other words, the Reynolds number expresses the ratio of inertial to viscous forces.

A large inertial force means the fluid possesses substantial energy independent of its intrinsic properties, and a small viscous force means the fluid can move much more freely. In any case, if $\textrm{Re}$ increases, the fluid motion becomes more complex and irregular, approaching the turbulence we envision.

Imagine a water pipe pouring out a thin stream: if you open the valve, $d$ increases and the velocity $v$ becomes faster, so the Reynolds number increases and the flow actually becomes turbulent and gushes out. An extreme example of very high viscosity is slowly flowing lava, in which case it is easy to imagine laminar flow.

Laminar flow

Turbulent flow

Derivation ³

Since this is a definition, a derivation is not strictly necessary, but we can examine how the Reynolds number is obtained from the ratio of inertial force $F_{n}$ to viscous force $F_{t}$. This is not very rigorous, so treat it as a heuristic.

First, consider an infinitesimal cube of fluid with edge length $l$ and mass $m$.

The relation with density is $m = \rho l^{3}$, and because $l$ is sufficiently small, the acceleration $a$ can be expressed in terms of the velocity $u$ as follows. $$ a = {\frac{ du }{ dt }} \approx {\frac{ u }{ l / u }} = {\frac{ u^{ 2 } }{ l }} $$ Accordingly, the inertial force $F_{n}$ can be approximated as: $$ \begin{align*} F_{n} =& ma \\ \approx& \rho l^{3} \cdot {\frac{ u^{ 2 } }{ l }} \\ =& \rho l^{2} u^{2} \end{align*} $$

On the other hand, the viscous force $F_{t}$ is the product of the viscous stress $\tau$ acting on the area $l^{2}$, hence $F_{t} = - \tau l^{2}$.

Newton’s law of viscosity: $$ \tau = \mu \left( \nabla \mathbf{u} + \left( \nabla \mathbf{u} \right)^{T} \right) $$

If Newton’s law of viscosity is taken in a one-dimensional form considering only the $x$ direction, it is $\tau = - \mu du / dy$ and $du / dy \approx u / l$, so $F_{t}$ can also be approximated as follows. $$ \begin{align*} F_{t} =& - \tau l^{2} \\ =& - \left( - \mu {\frac{ du }{ dy }} \right) l^{2} \\ \approx& \mu \frac{u}{l} l^{2} \\ =& - \mu u l \end{align*} $$ The ratio of the two forces $F_{n}$ and $F_{t}$ is then: $$ {\frac{ F_{n} }{ F_{t} }} \approx {\frac{ \rho l^{2} u^{2} }{ \mu u l }} = {\frac{ \rho u l }{ \mu }} = \mathrm{Re} $$