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---
title: "Reynolds number"
sort_title: "Reynolds number"
date: 2021-05-04
categories:
- Physics
- Fluid mechanics
- Fluid dynamics
layout: "concept"
---
The [Navier-Stokes equations](/know/concept/navier-stokes-equations/)
are infamously tricky to solve,
so we would like a way to qualitatively predict
the behaviour of a fluid without needing the flow $$\va{v}$$.
Consider the main equation:
$$\begin{aligned}
\pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v}
= - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v}
\end{aligned}$$
In this case, the gravity term $$\va{g}$$
has been absorbed into the pressure term:
$$p \to p\!+\!\rho \Phi$$,
where $$\Phi$$ is the gravitational scalar potential,
i.e. $$\va{g} = - \nabla \Phi$$.
Let us introduce the dimensionless variables $$\va{v}'$$, $$\va{r}'$$, $$t'$$ and $$p'$$,
where $$U$$ and $$L$$ are respectively a characteristic velocity and length
of the system at hand:
$$\begin{aligned}
\va{v} = U \va{v}'
\qquad
\va{r} = L \va{r}'
\qquad
t = \frac{L}{U} t'
\qquad
p = \rho U^2 p'
\end{aligned}$$
In this non-dimenionsalization, the differential operators are scaled as follows:
$$\begin{aligned}
\pdv{}{t}
= \frac{U}{L} \pdv{}{t'}
\qquad \quad
\nabla
= \frac{1}{L} \nabla'
\end{aligned}$$
Putting everything into the main Navier-Stokes equation then yields:
$$\begin{aligned}
\frac{U^2}{L} \pdv{\va{v}}{t'} + \frac{U^2}{L} (\va{v}' \cdot \nabla') \va{v}'
= - \frac{U^2}{L} \nabla' p' + \frac{U \nu}{L^2} \nabla'^2 \va{v}'
\end{aligned}$$
After dividing out $$U^2/L$$,
we arrive at the form of the original equation again:
$$\begin{aligned}
\pdv{\va{v}}{t'} + (\va{v}' \cdot \nabla') \va{v}'
= - \nabla' p' + \frac{\nu}{U L} \nabla'^2 \va{v}'
\end{aligned}$$
The constant factor of the last term
leads to the definition of the **Reynolds number** $$\mathrm{Re}$$:
$$\begin{aligned}
\boxed{
\mathrm{Re}
\equiv \frac{U L}{\nu}
}
\end{aligned}$$
If we choose $$U$$ and $$L$$ appropriately for a given system,
the Reynolds number allows us to predict the general trends.
It can be regarded as the inverse of an "effective [viscosity](/know/concept/viscosity/)":
when $$\mathrm{Re}$$ is large, viscosity only has a minor role,
but when $$\mathrm{Re}$$ is small, it dominates the dynamics.
Another way is thus to see the Reynolds number
as the characteristic ratio between the advective term
(see [material derivative](/know/concept/material-derivative/))
to the viscosity term, since $$\va{v} \sim U$$:
$$\begin{aligned}
\mathrm{Re}
\approx \frac{\big| (\va{v} \cdot \nabla) \va{v} \big|}{\big| \nu \nabla^2 \va{v} \big|}
\approx \frac{U^2 / L}{\nu U / L^2}
= \frac{U L}{\nu}
\end{aligned}$$
In other words, $$\mathrm{Re}$$
describes the relative strength of intertial and viscous forces.
Returning to the dimensionless Navier-Stokes equation:
$$\begin{aligned}
\pdv{\va{v}}{t'} + (\va{v}' \cdot \nabla') \va{v}'
= - \nabla' p' + \frac{1}{\mathrm{Re}} \nabla'^2 \va{v}'
\end{aligned}$$
For large $$\mathrm{Re} \gg 1$$,
we can neglect the latter term,
such that redimensionalizing yields:
$$\begin{aligned}
\pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v}
= - \frac{\nabla p}{\rho}
\end{aligned}$$
Which is simply the main [Euler equation](/know/concept/euler-equations/)
for an ideal fluid, i.e. a fluid without viscosity.
## Stokes flow
A notable case is so-called **Stokes flow** or **creeping flow**,
meaning flow at $$\mathrm{Re} \ll 1$$.
In this limit, the Navier-Stokes equations can be linearized:
since $$\mathrm{Re}$$ is the advective-to-viscous ratio,
$$\mathrm{Re} \ll 1$$ implies that we can ignore the advective term, leaving:
$$\begin{aligned}
\boxed{
\pdv{\va{v}}{t}
= - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v}
}
\end{aligned}$$
This equation is called the **unsteady Stokes equation**.
Usually, however, such flows are assumed to be steady
(i.e. time-invariant), leading to the **steady Stokes equation**,
with $$\eta = \rho \nu$$:
$$\begin{aligned}
\boxed{
\nabla p
= \eta \nabla^2 \va{v}
}
\end{aligned}$$
This equation is much easier to solve than the full Navier-Stokes equation
thanks to being linear,
and has some interesting properties, such as time-reversibility.
## References
1. B. Lautrup,
*Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
CRC Press.
2. R. Fitzpatrick,
[Dimensionless numbers in incompressible flow](https://farside.ph.utexas.edu/teaching/336L/Fluid/node17.html),
University of Texas.
|
