#Introduction
The Navier-Stokes Navier-Stokes continuity equation:
\nabla \boldsymbol{u} = 0,
momentum equation:
\frac{{D} \boldsymbol{u}}{{D} t} + \frac{1}{\rho} \nabla p = 0,
where:
\boldsymbol{u} the velocity
field,
\rho the density field,
t the time.
equation remains one –among five others– of the
yet unsolved millennium problems (Fefferman 2006)1.
Anyone who brings the slightest hint of evidence towards the
existence and uniqueness of a solution can claim a
one-million-dollar prize. This highlights mankind’s limitations
when it comes to understanding, predicting and controlling fluid
flows, especially when high speeds and complex geometries are
involved. Turbulence is an emerging property of such
flows, which is commonly observed in the everyday life: a
waterfall, a gust of wind, the smoke from a chimney. Turbulence
is seemingly random and unpredictable, and being involved in so
many application fields –airplanes, cars, pipes, engines–,
considerable efforts have been made since Reynolds
(1883)2 first
discovered it, to better appreciate its nature.
Stokes equation gives turbulence its chaotic and multi-scale nature, (de Wit et al. 2024)3 making it seemingly random and unpredictable.
The chaotic nature of turbulence makes it challenging to simulate: the state of a turbulent flow will be highly sensitive to slight uncertainties in the initial conditions,and numerical rounding in the computer simulations can yield widely diverging outcomes. As a consequence, predicting and controlling turbulence has become a whole research field: Computational Fluid Dynamics (CFD).
- 3D: VSAERO, PMARC, CMARC
- 2D: PROFILE, XFOIL
However, these methods involve pre- and post-validations, usually performed using real-flow measurements in wind tunnels or flight tests. Flow measurement is thus key to initiate and validate numerical simulations, in applications ranging from weather forecasting to Space Shuttle re-entry dynamics.
The second challenge imposed by turbulence is its
scale-invariance property (Richardson 1922)4.
described turbulence as composed of eddiesAn eddy is roughly defined as
coherent motion over a region of a certain size \ell popeTurbulentFlows200?.
of varying sizes. Large eddies which scale \ell_0 is similar to the flow scale
L are unstable and break up,
thus transferring their energy to smaller eddies, which also
break up into smaller and smaller eddies. Kolmogorov
(1941)5 showed the
scalar self-similarity in the energy cascade, and determined the
smallest eddy size \eta \approx
\ell_0\cdot\text{Re}^{-3/4} at which point, the energy
dissipation happens thermally instead of mechanically.
…
In conclusion, flow measurements should be as spatio-temporally resolved as much as possible to resolve Kolmogorov microscales, as well as non-intrusive so that it does not interfere with the flow itself –even the slightest interference could completely change the turbulent state of the. In practice, it means that sensor sizes should be smaller than the spatial microscale, and that their bandwidth should cover its temporal microscale. Figure 1.1 shows the order of magnitudes encountered in several industrial applications. Aeronautics is the most demanding one.
The challenge of measuring turbulence has been introduced above, …