#1.1 Flow equations and turbulence scales

A fluid is defined as a medium that can continuously deform or move –i.e. flow– and, unlike a solid, has a zero shear modulus. Its motion is described under the continuum hypothesis, meaning that instead of considering individual particles of given mass and momentum, infinitesimal elements of density \rho, velocity \boldsymbol{u} and pressure p are introduced. Their behavior is described by the Navier-Stokes equations, the first one being the continuity equation: \frac{\partial \rho}{\partial t} + \nabla (\rho \boldsymbol{u}) = 0, which, considering an incompressible fluid, simplifies to \nabla \boldsymbol{u} = 0. The second one, the momentum equation, is Newton’s second law applied to the infinitesimal element, with \nu the fluid’s kinematic viscosity: \frac{{D} \boldsymbol{u}}{{D} t} = -\frac{1}{\rho} \nabla p + \nu \nabla^2 \boldsymbol{u} + \textrm{other forces}. Each fluid particle is accelerated by the pressure gradient force, directed toward lower pressure zones, while also being “pushed” by viscosity forces –other fluid particles. By considering the inertial term {D} \boldsymbol{u}/{D} t = \partial \boldsymbol u /\partial t + (\boldsymbol u \cdot \nabla) \boldsymbol u, steady flow conditions leave the spatial derivative as the dominant part, thus {D} \boldsymbol{u}/{D} t \sim (\boldsymbol u \cdot \nabla) \boldsymbol u. By introducing velocity and length scales |\boldsymbol{u}| \sim u_0 and \nabla \sim 1/\ell_0 at which the external forces occur, the acceleration induced by the inertial force is on the order of u_0^2/\ell_0 and the viscosity-induced acceleration is of order \nu u_0/\ell_0^2. Consequently, the ratio between the inertial and viscous forces gives the Reynolds number \textrm{Re} = \frac{u_0\ell_0}{\nu}.

This dimensionless number expresses the balance between inertial and viscosity forces, and its magnitude characterizes the flow regime. For low Reynolds, viscosity dominates and the momentum equation can be linearized, which results in a predictable behavior, with little sensitivity to perturbations: laminar flow. For higher and higher Reynolds, turbulence progressively appears as the inertial forces take over, resulting in a chaotic, unpredictable regime.The original pipe flow experiment conducted by Reynolds (1883)² showed the flow to be laminar under 2300 Re and fully turbulent above 4000 Re. However, these threshold values vary with experimental conditions.

Richardson (1922)⁴ showed that energy mechanically dissipates into smaller and smaller coherent structures, up until a point where molecular viscosity thermally dissipates the kinetic energy. As a consequence, turbulent flows are characterized by eddies distributed over wide ranges of time and length scales. For given macroscopic velocity and length scales u_0 and \ell_0, the Kolmogorov microscales characterize the very smallest eddies with length \eta, velocity u_\eta and time constant \tau_\eta (Kolmogorov 1941)⁵: \left\{ \begin{aligned} \eta & \sim \ell_0\text{Re}^{-3/4},\\ u_\eta & \sim u_0\text{Re}^{-1/4},\\ \tau_\eta & \sim \frac{\ell_0}{u_0} \text{Re}^{-1/2}. \end{aligned} \right.

The more turbulent a flow is at a given macroscale, the smaller and shorter the Kolmogorov microscales are. As an example, an airliner with a wing chord of 10 m at cruise altitude and speed operates at a Reynolds number of approximately 2 × 10⁸. This translates to spacetime Kolmogorov microscales of \eta = 5 µm and \tau_\eta = 3 µs. Figure 1.1 shows the order of magnitude of the spatiotemporal Kolmogorov microscales for various real applications. The airliner example falls into the aircraft part of the diagram (shown in red), which present the most challenging flow configurations to measure.

Anemometry is the action of measuring one or several quantities in the flow, like the velocity, pressure, density or even temperature fields. Due to the multi-scale property of turbulent flows, by far the most common in real world applications, the measurement volumes and timescales should be as small as possible, ideally on the order of the Kolmogorov microscales. Although information about the macroscale properties of flows can still be obtained with some time or space averaging, intensive research had been made to resolve down these measurements over the past century, accompanying the engineering breakthroughs in modern aviation. A state of the art of the flow measurement techniques is presented below, emphasizing miniaturized sensor technologies.