Instabilities and turbulence#

Common phenomena and important concepts#

Instability is a very general concept: “when a state is unstable”. So it is not surprising that instabilities are everywhere: in flows (what we will study during this course) but also in many other systems. Here are two examples from the fields of solid mechanics and fluid-structure interactions:

A system can be characterized as “turbulent” when structures with a broad range of scales interact non-linearly in a system with several degrees of freedom.

Instabilities and turbulence are very general concepts that can be applied to a lot of systems (populations, ecology, society, climate, etc.).

Flow instabilities and turbulence are subjects with many practical interests (vehicles, machines, biological fluids, geophysical fluids, etc.). They have been widely studied. They are very good models for research on instabilities and turbulence in general.

2013_FordFusion_CFDTopandSide.png
airfoil_turbulence.jpg

Fig. 1 Turbulent wake of vehicles, car and airfoil#

Two types of descriptions:

Mechanisms (instability) and statistics (turbulence).

Reynolds experiment and Reynolds number#

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Fig. 2 Sketch of Reynolds’s dye experiment, taken from his 1883 paper.#

Osborne Reynolds (1842-1912) carried out experiments on the flow in a long tube. A drawing of Reynolds and its experimental setup is shown in Fig. 2. This video of a recent experiment helps to understand what appends.

Water flow through a tube and a small flow volume of ink is injected at the beginning of the tube.

  • For low velocity, the flow is laminar and the filament of ink stays linear along the tube.

  • For larger velocity the flow becomes unstable and there is a transition to turbulence. The filament of ink is perturbed and there is mixing.

More precisely, depending on the velocity and tube radius, the flow can be:

  • laminar (straight not perturbed ink filament),

  • perturbed,

  • turbulent (fast mixing).

This flow exhibits a hysteresis! The instability is said “subcritical”.

When the tube is long enough, the length of the tube is not a relevant parameter and there are only three relevant physical quantities:

  • viscosity of the water \(\nu\) (in m\(^2\)/s),

  • diameter of the tube \(D\) (in m),

  • velocity of the water \(U\) (in m/s).

We see that there are only 2 units and 3 variables so we can only form 1 non-dimensional number in application of the \(\Pi\) theorem:

\[ Re = \frac{U D}{\nu}.\]

This is the Reynolds number. It is one of the most important non-dimensional numbers in fluid dynamics.

An example: the wake behind a cylinder#

Fig. 3 shows four pictures of the flow behind a cylinder. We see that these four states are qualitatively very different, from laminar (regular, steady, symmetrical) to nearly turbulent. Fig. 4 presents a really turbulent case. This video (at \(t = 3\) min) shows a clear example of the vortex shedding behind a cylinder for different velocities (Reynolds numbers).

Let’s recall and introduce very important concepts:

  • For this flow, the control parameter is also the Reynolds number \(Re\).

  • Instabilities. Stable and unstable states.

  • Breaking of symmetry (space and time).

  • Vortex shedding with a characteristic frequency: Strouhal number \(St = fD / U \simeq 0.2\)

  • Transition to turbulence (see Fig. 4).

  • For this configuration there is no hysteresis. This is a “supercritical” instability.

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Fig. 3 Visualization of a flow behind a cylinder for different Reynolds numbers. (a) \(Re = 1.54\); (b) \(Re = 26\); (c) \(Re = 200\); (d) \(Re = 8000\). Taken from GHP.#

_images/turb_wake_cylinder.jpg

Fig. 4 Visualization of the turbulent wake of a cylinder at large Reynolds number.#

Note

Fluid-structure interaction… See this video.

Questions#

  • Write the vectorial version of the Navier-Stokes equations for an incompressible fluid.

  • Give a definition of the Reynolds number as a ratio of two terms of the Navier-Stokes equations.

  • Give the characteristic time of a diffusive process along a length \(L\).

  • Give the characteristic time of a advective process along a length \(L\).

  • What is the Strouhal number? What is its typical value for the wake of a cylinder?