Abstract
In order to understand the fundamental measurement capabilities of different flow velocity measurement principles based on Mie scattering, a fundamental equation of how to calculate the shot noise limit for a respective signal model is derived. The derivation is based on the well-known rules of uncertainty propagation and yields the Cramér–Rao bound without the necessity to calculate the Fisher information. The derived equation is next applied to compare the shot noise limit for Doppler and time-of-flight principles including laser Doppler anemometry (LDA), planar Doppler velocimetry (PDV), laser-two-focus velocimetry (L2F), particle tracking velocimetry (PTV) and particle image velocimetry (PIV). The comparison is performed for an identical mean laser power, while two cases are studied in detail: measuring on a single seeding particle as well as measuring on multiple seeding particles and averaging. LDA, L2F and PTV/PIV obey a similar shot noise limit. For the case of a measurement on multiple seeding particles, the minimal achievable measurement uncertainty is directly proportional to the absolute value of the measured velocity component and inversely proportional to the spatial resolution. The respective shot noise limit for PDV is almost independent of the measured flow velocity component and the spatial resolution. Since PDV is sensitive with respect to a different flow velocity component depending on the observation direction, a comparison with the other principles is only reasonable to a certain extent. However, all shot noise limits in case of measuring on multiple seeding particles show the expected inverse proportionality to the square root of the total number of detected photons and thus also to the square root of the measurement time. Considering a comparable spatiotemporal resolution, an identical mean light power and typical measurement configurations, the PDV shot noise limit is the largest. As a final result, it is derived that each measurement principle obeys an uncertainty principle between position and the respective component of the wave vector, which is in agreement with Heisenberg’s uncertainty principle. Therefore, a common basis is provided to assess the fundamental measurement capabilities of Doppler and time-of-flight measurement systems on the basis of what is possible within the quantum mechanical constraints.Graphic abstract
Highlights
Different Doppler and time-of-flight principles for flow velocity field measurements exist, which are in the following represented by laser Doppler anemometry (LDA) as Doppler principle using light mixing, planar Doppler velocimetry (PDV) as Doppler principle using light filtering, Laser-2-Focus Anemometry (L2F) as time-of-flight principle based on time measurements and particle tracking velocimetry (PTV)/particle image velocimetry (PIV) as time-of-flight principle based on position measurements
The minimal achievable measurement uncertainty for LDA, L2F and PTV is similar while it is comparably high for PDV because the scattered light and the information obtained is attenuated by the light filtering
A key difference between the different measurement principles is that PDV measures a mixture of in-plane and out-ofplane velocity component, LDA and L2F measure the in-plane velocity component transverse to optical axis of the illumination and PTV/PIV allows to measure both inplane velocity components
Summary
When confronted with a flow measurement task, finding the proper choice from the huge variety of different measurement principles is difficult. Doppler principles evaluate the frequency change of the scattered light from a particle that is caused by a nonzero flow velocity relative to the measurement system. Introduced as a point measurement device, enhanced LDA techniques enabling profile (Czarske 2001) and field measurements (Coupland 2000; Voigt et al 2008; Meier and Rösgen 2012) are known today Another Doppler technique is to separately measure the frequency of the illuminated light and the observed scattered light, and evaluating the frequency difference in a second step. Different Doppler and time-of-flight principles for flow velocity field measurements exist, which are in the following represented by LDA as Doppler principle using light mixing, PDV as Doppler principle using light filtering, L2F as time-of-flight principle based on time measurements and PTV/PIV as time-of-flight principle based on position measurements.
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