# Imaging Flow Velocimetry with Laser Mie Scattering
## 4 Fundamental Measurement Limits
### Seeding
#### 4.1.2 Flow Sampling Phenomenon  
Since not the flow velocity, but the seeding particle velocity is actually measured, the flow field is sampled in space. The sampling is not equidistant due to the random position of the seeding particles. Hence, the determined particle velocities are a random sample of the flow velocity field to be measured. As a result of the flow sampling, the complete spatiotemporal flow behavior cannot be reconstructed from the velocities of the particles in general, even if the particle positions and the measurement time stamps are exactly known. Therefore, it is desirable to increase the sample size, which is possible by increasing the seeding particle concentration and by decreasing the temporal and spatial resolution (spatiotemporal averaging). Increasing the seeding particle concentration is only possible to a certain extent, because this deteriorates the optical access due to the increased light extinction. In addition, except for DGV, single particles or at least particle patterns need to be resolved in the measurement volume. Assuming a (locally) temporally-constant flow, the temporal resolution can be reduced by increasing the averaging time. One typical example is comprised of LDA measurements in stationary boundary layer flows [140]. Assuming a (locally) spatially-constant flow, the spatial resolution can be reduced by increasing the spatial averaging. Many flows exist, which are neither temporally nor spatially constant. An example is an eddy with an almost fixed structure during the measurement, which moves due to a non-zero mean flow velocity [141]. Due to the translational motion of the flow velocity field of the eddy, a temporal averaging also means a spatial averaging according to [PERSON]'s hypothesis, and vice versa [142]. As a result, the resolution of flow structures can be limited by spatial or temporal averaging. Actually, measurement systems always have a limited spatial and temporal resolution. For this reason and due to the flow sampling phenomenon, the measured mean particle velocity in general exhibits random fluctuations and is not identical to the true mean flow velocity, if spatial (eddies, gradients) or temporal (turbulence) fluctuations of the flow velocity occur. Space, time and velocity follow an uncertainty principle, which leads to a fundamental limit of measurability. The finding is illustrated as an example. A stationary flow is considered with a velocity gradient in the measurement volume. The dimension of the measurement volume is the finite spatial resolution. The resulting measurement uncertainty is described by [PERSON] et al. in [143; 144; 145]. [PERSON] et al. derived the relation between the spatial resolution, the temporal resolution and the resulting standard deviation \(\sigma_{v}\) of the measured flow velocity in the measurement volume [139]. Assuming a uniform distribution of the seeding particles in space, neglecting the particle transit time through the measurement volume in comparison with the temporal resolution \(\Delta t\) and neglecting the flow velocity variations in the measurement volume in comparison with the mean flow velocity \(\bar{v}\), the relation reads:  
\[\sigma_{v}\sim\frac{G}{\sqrt{c_{\rm p}|\bar{v}|\Delta t}}\sqrt{\frac{\Delta x }{\Delta y}}. \tag{5}\]  
The symbol \(G\) denotes the absolute value of the flow velocity gradient, \(c_{\rm p}\) is the particle concentration and \(\Delta x\), \(\Delta y\) are the respective spatial resolution. Both axes are perpendicular to the flow direction, and \(x\) is oriented parallel and \(y\) perpendicular to the direction of the flow velocity gradient. The proportionality constant of Equation (5) results from the light intensity distribution of the illumination in the measurement volume and can be determined with a simulation [139]. Hence, an uncertainty in space (finite spatial resolution) and a spatial flow velocity gradient leads to a flow velocity uncertainty.