# Imaging Flow Velocimetry with Laser Mie Scattering
## 4 Fundamental Measurement Limits
### Particle Motion
#### 4.2.1 Flow Following Behavior
Note that an alternative and well-known figure of merit for the measurement deviation is the slip [26]:  
\[s=\frac{v-v_{\mathrm{P}}}{v}=-\frac{\Delta v}{v}. \tag{9}\]  
As a result, the measurement deviation \(\Delta v\) depends on the characteristic time constant and the acceleration of the particle, which includes spatial and temporal changes of the flow velocity. Since both quantities are usually not available during the flow measurement, the measurement deviation is a fundamental measurement limit. The limit is calculated as an example, assuming a cosinusoidal variation of the flow velocity with the frequency \(f\), the amplitude \(v_{\mathrm{a}}\) and the phase \(\varphi_{\mathrm{a}}\). Solving the equation of motion Equation (7) of the particle and inserting the solution into Equation (8) leads to the measurement deviation:  
\[\Delta v=v_{\mathrm{a}}\frac{f}{\sqrt{f^{2}+f_{\mathrm{g}}^{2}}}\cos\left(2 \pi ft+\varphi_{\mathrm{a}}-\arg\left(1+j\frac{f}{f_{\mathrm{g}}}\right)- \frac{\pi}{2}\right), \tag{10}\]  
with \(j\) as imaginary unit and \(f_{\mathrm{g}}=1/(2\pi\tau_{\mathrm{P}})\) as the \(3\,\mathrm{dB}\) limit frequency of the transfer function:  
\[\frac{V_{\mathrm{p}}(j\omega)}{\overline{V}(j\omega)}=\frac{1}{1+j\frac{f}{f_ {\mathrm{g}}}},\qquad\mathrm{with}\;\;\omega=2\pi f. \tag{11}\]  
For a harmonic oscillation of the flow velocity with the frequency \(f\), the measurement deviation of a single velocity value is in the interval \(\pm v_{\mathrm{a}}\frac{f}{\sqrt{f^{2}+f_{\mathrm{g}}^{2}}}\). Without knowledge of \(t\), \(f\), \(f_{\mathrm{g}}\) and \(\varphi_{\mathrm{a}}\), this represents a fundamental measurement limit. For the particles listed in Table 1 in air flows at room temperature, the \(3\,\mathrm{dB}\) limit frequency is \(f_{\mathrm{g}}=46.8\,\mathrm{kHz}\) and \(f_{\mathrm{g}}=69.2\,\mathrm{kHz}\), respectively. Considering \(f=10\,\mathrm{kHz}\) for instance, the maximal deviation \(|\Delta v|\) then amounts to \(0.21\times v_{\mathrm{a}}\) and \(0.14\times v_{\mathrm{a}}\). It is usually the case that the amplitude \(v_{\mathrm{a}}\) and/or the phase \(\varphi_{\mathrm{a}}\) of the velocity oscillation is of interest from a time series measurement of the flow velocity. Taking the particle-following behavior into account, the amplitude and phase deviations according to the Equation (11) read for a frequency \(f\):  
\[\Delta v_{\text{a}}(f) =v_{\text{a}}\cdot\left(\frac{1}{\sqrt{1+\frac{f^{2}}{f_{\text{g}} ^{2}}}}-1\right) \tag{12}\] \[\Delta\varphi_{\text{a}}(f) =-\arg\left(1+j\frac{f}{f_{\text{g}}}\right). \tag{13}\]  
For the particles listed in Table 1, the resulting amplitude and phase deviation with respect to the frequency is shown in \[-\frac{3}{2}d_{\mathrm{P}}^{2}\sqrt{\pi\rho\eta}\int\limits_{t_{0 }}^{t}\frac{\mathrm{d}(\vec{v}_{\mathrm{p}}-\vec{v})}{\mathrm{d}t^{\prime}} \frac{1}{\sqrt{t-t^{\prime}}}\mathrm{d}t^{\prime}+\vec{F}.\]0 Comparing the amplitudes of the flow and particle velocity oscillation for instance, the relative measurement deviation for \(f=10\,\text{kHz}\) is \(-2.2\%\) and \(-1\%\), respectively. If the oscillation frequency \(f\) and the characteristic time constant \(\tau_{\text{p}}\) or the limit frequency \(f_{\text{g}}\) of the particle is known, a correction of the measurement deviation is possible in principle. However, this is typically not the case. Hence, it is shown for the example of a harmonic flow oscillation that the particle-following behavior leads to a fundamental measurement limit.