# Turbulence

A flow is characterized as *laminar* or *turbulent* depending on its [Reynolds
number](https://en.wikipedia.org/wiki/Reynolds_number)

$$
\mathrm{Re} = \frac{uL}{\nu},
$$

where $u$ and $L$ correspond to a *characteristic* velocity and length scale,
respectively, and $\nu$ is the kinematic viscosity of the fluid.  The
dimensionless Reynolds number is the ratio of inertial forces versus viscous
forces.  High Reynolds number flows are dominated by inertial forces for which
the flow is turbulent; whereas low Reynolds number flows are dominated by
viscous forces in which case the flow is laminar (sheet-like).  

In *"A First Course in Turbulence"*, Hendrik Tennekes and John  L. Lumley
describe large Reynolds number flows as follows:

> **Large Reynolds numbers** Turbulent flows always occur at high Reynolds
> numbers. Turbulence often originates as an instability of laminar flows if the
> Reynolds number becomes too large. The instabilities are related to the
> interaction of viscous terms and nonlinear inertia terms in the equations of
> motion. This interaction is very complex: the mathematics of nonlinear partial
> differential equations has not been developed to a point where general
> solutions can be given. Randomness and nonlinearity combine to make the
> equations of turbulence nearly intractable; turbulence theory suffers from the
> absence of sufficiently powerful mathematical methods. This lack of tools
> makes all theoretical approaches to problems in turbulence trial-and-error
> affairs. Nonlinear concepts and mathematical tools have to be developed along
> the way; one cannot rely on the equations alone to obtain answers to problems.
> This situation makes turbulence research both frustrating and challenging: it
> is one of the principal unsolved problems in physics today.

Turbulence is chaotic and contains vortical structures ranging over a wide
spectrum of length scales.  Vortices in a flow field are described by the
*vorticity* $\bm{\omega}$ which can be computed from the velocity field $\bm{u}$
by taking its curl

$$
\bm{\omega} = \nabla\times\bm{u}.
$$

Vorticity can be computed and visualized in tools like
[ParaView](https://www.paraview.org/) by selecting the gradient filter and
passing the velocity field as input.  Alternatively, the ParaView Python module
allows to script this filter instead of using the GUI with a code similar to:

```python
import paraview.simple as pv

reader = pv.OpenDataFile('some_data.bin')
vort = pv.Gradient(
    ComputeGradient=0,
    ComputeVorticity=1,
    Input=reader,
    ScalarArray='velocity',
    VorticityArrayName='vorticity'
)
```

Note that the script must be executed either with `pvpython` or `pvbatch`.
Computing the vorticity magnitude $\lvert\bm{\omega}\rvert$ for an example
laminar flow at low Reynolds number results in the example visualization shown
below. Vorticity magnitude is high (yellow color) at no-slip walls and vorticity
decays quickly downstream of solid obstacles due to dominating viscous forces.
The fluid in this flow can be thought of as something similar to honey.

![](./figs/laminar.png)

In a similar way, the vorticity magnitude for a turbulent flow at high Reynolds
number is shown in the next image.  Here the flow is chaotic as can be seen by
the many "swirls".  Viscous forces are much less pronounced for this type of
flow and thus turbulent eddies advect further downstream from the solid
obstacles before they die out.  Fluids behaving this way for typical
applications include air or water, for example.

![](./figs/turbulent.png)