The Reynolds number is a dimensionless quantity used to compare inertial effects with viscous effects in a fluid flow.
It is one of the most important tools for predicting whether a flow is smooth and layered or irregular and turbulent.
1. Reynolds number with dynamic viscosity
The common form is
\[
\begin{aligned}
Re &= \frac{\rho v L}{\eta}.
\end{aligned}
\]
Here:
- \(\rho\) is fluid density.
- \(v\) is a representative flow speed.
- \(L\) is the characteristic length.
- \(\eta\) is dynamic viscosity.
2. Reynolds number with kinematic viscosity
Kinematic viscosity is
\[
\begin{aligned}
\nu &= \frac{\eta}{\rho}.
\end{aligned}
\]
Substituting this into the Reynolds-number formula gives
\[
\begin{aligned}
Re &= \frac{vL}{\nu}.
\end{aligned}
\]
3. Physical meaning
The Reynolds number compares two effects:
\[
\begin{aligned}
Re
&\sim
\frac{\text{inertial effects}}{\text{viscous effects}}.
\end{aligned}
\]
When \(Re\) is small, viscosity dominates and the flow tends to be smooth and orderly.
When \(Re\) is large, inertia dominates and the flow is more likely to become irregular or turbulent.
4. Pipe-flow classification
For internal flow in a circular pipe, a common classroom guide is:
| Reynolds number range |
Flow regime |
Typical behavior |
| \(Re<2300\) |
Laminar |
Smooth layers, little mixing, stable streamlines |
| \(2300\le Re\le4000\) |
Transitional |
Can switch between smooth and disturbed behavior |
| \(Re>4000\) |
Turbulent |
Irregular motion, eddies, strong mixing |
5. Characteristic length
The characteristic length \(L\) depends on the physical situation.
| Situation |
Typical characteristic length |
| Pipe flow |
Pipe diameter or hydraulic diameter |
| Flow around a sphere or cylinder |
Object diameter |
| Flat plate boundary layer |
Distance from the leading edge |
| Custom geometry |
The length scale most relevant to the flow structure |
6. External flow around objects
For flow around objects, such as a sphere or cylinder, the transition behavior is different from pipe flow.
A very small value such as
\[
\begin{aligned}
Re &\ll 1
\end{aligned}
\]
indicates creeping flow, where viscous effects dominate strongly.
This is the range where Stokes’ law can be useful.
At larger Reynolds numbers, wake formation, vortex shedding, and turbulence can appear.
The exact thresholds depend on shape, roughness, and flow disturbances.
7. Flat plate boundary layer
For flow over a flat plate, the Reynolds number is often based on distance \(x\) from the leading edge:
\[
\begin{aligned}
Re_x &= \frac{\rho vx}{\eta}.
\end{aligned}
\]
A common ideal estimate is that transition may begin around
\[
\begin{aligned}
Re_x &\approx 5\times10^5.
\end{aligned}
\]
However, surface roughness, free-stream turbulence, pressure gradient, and edge shape can move the transition point.
8. Solving for speed
If a target Reynolds number is known, solve for speed:
\[
\begin{aligned}
v &= \frac{Re\,\eta}{\rho L}.
\end{aligned}
\]
With kinematic viscosity:
\[
\begin{aligned}
v &= \frac{Re\,\nu}{L}.
\end{aligned}
\]
9. Solving for characteristic length
If the needed size scale is unknown:
\[
\begin{aligned}
L &= \frac{Re\,\eta}{\rho v}.
\end{aligned}
\]
Or, using kinematic viscosity:
\[
\begin{aligned}
L &= \frac{Re\,\nu}{v}.
\end{aligned}
\]
10. Solving for viscosity
If \(Re\), speed, length, and density are known:
\[
\begin{aligned}
\eta &= \frac{\rho vL}{Re}.
\end{aligned}
\]
The corresponding kinematic viscosity is
\[
\begin{aligned}
\nu &= \frac{vL}{Re}.
\end{aligned}
\]
11. Worked example
Water flows at \(2\ \mathrm{m/s}\) through a pipe of diameter \(5\ \mathrm{cm}\).
Use
\[
\begin{aligned}
\rho &= 1000\ \mathrm{kg/m^3},\\
\eta &= 1.0\times10^{-3}\ \mathrm{Pa\,s},\\
v &= 2\ \mathrm{m/s},\\
L &= 0.05\ \mathrm{m}.
\end{aligned}
\]
Then
\[
\begin{aligned}
Re
&= \frac{\rho vL}{\eta}\\
&= \frac{(1000)(2)(0.05)}{1.0\times10^{-3}}\\
&= 100000.
\end{aligned}
\]
For pipe flow, \(Re=100000\) is far above \(4000\), so the flow is classified as turbulent.
12. Formula summary
| Goal |
Formula |
Use |
| Reynolds number |
\(Re=\rho vL/\eta\) |
Use density and dynamic viscosity |
| Reynolds number |
\(Re=vL/\nu\) |
Use kinematic viscosity |
| Kinematic viscosity |
\(\nu=\eta/\rho\) |
Convert dynamic viscosity to kinematic viscosity |
| Speed for target Re |
\(v=Re\,\eta/(\rho L)\) |
Find speed needed to reach a regime boundary |
| Length for target Re |
\(L=Re\,\eta/(\rho v)\) |
Find required diameter or length scale |
| Viscosity for target Re |
\(\eta=\rho vL/Re\) |
Find viscosity that gives a desired Reynolds number |
13. Assumptions and cautions
- Reynolds number is a guide, not a complete flow solution.
- Critical values depend on geometry and disturbances.
- Pipe-flow thresholds should not be blindly applied to external flow.
- Surface roughness, inlet disturbances, bends, and vibration can trigger turbulence earlier.
- For non-Newtonian fluids, the simple formula may require an apparent viscosity.
- For compressible high-speed gases, Mach number and density changes may also matter.
Key idea: low Reynolds number means viscosity dominates; high Reynolds number means inertia dominates and turbulence becomes likely.
