# Imaging Flow Velocimetry with Laser Mie Scattering

## Abstract

**:**

## 1. Introduction

## 2. Measurement Approaches

#### 2.1. Application of Mie Scattering

^{3}with air at normal pressure and room temperature as fluid, about $2\times {10}^{13}$ fluid molecules are present according to the ideal gas law. Hence, the advantage of the higher scattering cross-section of seeding particles is partially equalized by the low number of scatterers. As a result, an increase of the scattering power of 1000 remains for the considered example. Another beneficial aspect of using seeding particles is the reduced light extinction, because the seeding is usually applied locally. Furthermore, the angular-dependent scattering and polarization effects have to be taken into account. For instance, Mie scattering has in general a stronger forward scattering than sidewards and backward scattering, which means a reduced light extinction in comparison to Rayleigh scattering. As a result, the potential of Mie scattering approaches for imaging flow measurements with acceptable uncertainty also at high measurement rates is illustrated, in particular with respect to the necessary distribution of the available light energy over space and time.

#### 2.2. Scattered Light Evaluation

- Doppler principles,
- Time-of-flight principles.

**Doppler principles**are based on the optical Doppler effect that occurs for light scattering at a moving object. In this fashion, the frequency shift (Doppler frequency) of the light scattered on a single particle (or multiple particles) is measured, and the Doppler frequency depends on the particle velocity. The relation between the particle velocity ${\overrightarrow{v}}_{\mathrm{p}}$ and the Doppler frequency ${f}_{\mathrm{D}}$ reads: [29,30]

**time-of-flight principles**are based on the kinematic velocity definition, which reads for one velocity component:

## 3. Developed Measurement Techniques and Their Imaging Evolution

#### 3.1. Fundamentals for the Evaluation of the Measurement Techniques

- The flow velocity is a vector quantity. Therefore, the number of measured velocity components is an important property. The abbreviated form 1c, 2c or 3c means that one, two or three components are measured, respectively.
- In order to characterize the spatial behavior of the flow velocity, the number of resolved space dimensions is an important property. Measurements are for instance pointwise, along a line, planar or volumetric, which is indicated by the abbreviated forms 0d, 1d, 2d or 3d, respectively. Characterizing the measurement along each space dimension is possible with the following parameters (cf. Figure 5a):
- -
- spatial resolution,
- -
- spatial distance between the adjacent measurements,
- -
- number of measurements along the respective space dimension or size of the measurement volume in the respective space dimension.

- In addition, the temporal behavior of the measured flow velocity is characterized with the parameters
- -
- temporal resolution,
- -
- temporal distance between the sequent measurements or measurement rate,
- -
- number of measurements along the time dimension or measurement duration.

The term measurement rate requires temporally-equidistant measurements. Otherwise, a mean measurement rate can be specified. The introduced terms are explained in Figure 5b. Note that the temporal resolution does not necessarily equal (or is smaller than) the reciprocal value of the measurement rate. Both quantities are independent. - Each velocity value over space and time can finally be characterized with a measurement uncertainty by applying the international guide to the expression of uncertainty in measurements (GUM) [38,39]. According to the GUM, the measurement uncertainty is a parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand.
- Due to possible cross-sensitivities (for instance with respect to the temperature, vibrations or electromagnetic fields) or other impairments of the measurements (for instance limited optical access), it is important if the measurements concern non-reactive or reactive flows (flames), if the measurements are performed in a laboratory or in an industrial environment, if the measurement object is a simplified model or the real measurement object. For this reason, the surrounding and boundary conditions have to be described.

#### 3.2. Measurement Techniques Using the Doppler Principle

#### 3.2.1. Amplitude-Based Signal Evaluation

#### 3.2.2. Frequency-Based Signal Evaluation

#### 3.3. Measurement Techniques Using the Time-Of-Flight Principle

#### 3.3.1. Time Measurement

#### 3.3.2. Space Measurement

## 4. Fundamental Measurement Limits

#### 4.1. Seeding

#### 4.1.1. Influence on the Flow

^{13}/m

^{3}, which are hardly achievable [138,139], the density change is below 1%. Since the particle concentration is typically several orders of magnitude lower, the influence of the seeding on the air flow density is usually negligibly small. The change in viscosity and the general change of the flow due to the presence of two phases is not known here. However, the low volume percentage of the seeding leads to the assumption that it is negligible as well. The volume percentage is shown in Figure 6b over the particle concentration. It is always below 0.001%. As a result, the retroaction of the seeding is usually negligible compared with other error sources.

#### 4.1.2. Flow Sampling Phenomenon

#### 4.2. Particle Motion

#### 4.2.1. Flow Following Behavior

#### 4.2.2. Brownian Motion

#### 4.2.3. Illumination Effects

#### 4.3. Photon Shot Noise

## 5. Application Examples, Challenges and Perspectives

## 6. Conclusions

## Conflicts of Interest

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**Figure 1.**Calculated scattering cross-section over the particle radius normalized by the wavelength for a spherical particle made of diethylhexyl sebacate (DEHS) with a refractive index of 1.45 at 650 nm.

**Figure 2.**Measurement arrangement of Doppler principles illustrated for the light scattering on a single particle (measurement of one velocity component, acceleration neglected).

**Figure 3.**Measurement arrangement of time-of-flight principles illustrated for the position measurement of a scattering particle (measurement of one velocity component, acceleration neglected).

**Figure 5.**Illustration of the terms (

**a**) spatial resolution, spatial distance of the measurements and size of the measurement volume along one space dimension and (

**b**) temporal resolution, measurement rate and measurement duration.

**Figure 6.**(

**a**) Change of the mean density of the fluid due to the seeding and (

**b**) volume percentage of the seeding over the particle concentration.

**Figure 7.**Mean free distance of the seeding particles divided by the particle diameter ${d}_{\mathrm{p}}$ over the seeding particle concentration for two different particle diameters.

**Figure 8.**Deviation of (

**a**) the amplitude and (

**b**) the phase of a cosinusoidal excitation due to the particle-following behavior of the particles listed in Table 1.

**Figure 9.**Minimal achievable measurement uncertainty due to photon shot noise over the flow velocity v for an average photon rate per particle of ${\dot{N}}_{\mathrm{photon}}=2.7\times {10}^{8}{\mathrm{s}}^{-1}$ (

**a**) for a single particle (that passes the measurement volume with the respective dimension $w=100\mathsf{\mu}\mathrm{m}$) and (

**b**) for multiple particles (that occur during the measurement time $T=0.1\mathrm{s}$). The formulas are from Table 2 and ${c}_{1}=0.9$, ${c}_{2}={c}_{3}={c}_{4}=0.1$, $M=1$, $\lambda =532\mathrm{n}\mathrm{m}$, ${\tau}^{\prime}=$ 2 GHz${}^{-1}$.

**Table 1.**Material, density ${\rho}_{\mathrm{p}}$ and diameter ${d}_{\mathrm{p}}$ of two typical seeding particles.

Material | ${\mathit{\rho}}_{\mathbf{p}}$ (kg/m${}^{3}$) | ${\mathit{d}}_{\mathbf{p}}$ ($\mathsf{\mu}$m) |
---|---|---|

diethylhexyl sebacate (DEHS) | 912 | 1 |

titanium dioxide (TiO${}_{2}$) | 3900 | 0.4 |

**Table 2.**Square root of the Cramér–Rao bound due to photon shot noise for Doppler and time-of-flight measurement techniques, which is the minimal achievable measurement uncertainty $u\left(v\right)$ for a single particle and $u\left(\overline{v}\right)$ for multiple particles, respectively. DGV, Doppler global velocimetry; LDA, laser Doppler anemometer; L2F, laser-2-focus anemometry; PTV, particle tracking velocimetry.

$\mathit{u}\left(\mathit{v}\right)$ for a Single Particle | $\mathit{u}\left(\overline{\mathit{v}}\right)$ for Multiple Particles | ||
---|---|---|---|

DGV | ${c}_{1}\xb7{\displaystyle \frac{{\left|v\right|}^{1/2}\xb7{\displaystyle \frac{\lambda}{|{\tau}^{\prime}|}}}{\sqrt{{\dot{N}}_{\mathrm{photon}}w}}}$ | ${c}_{1}\xb7{\displaystyle \frac{1}{\sqrt{M}}}\xb7{\displaystyle \frac{{\displaystyle \frac{\lambda}{|{\tau}^{\prime}|}}}{\sqrt{{\dot{N}}_{\mathrm{photon}}T}}}$ | ${c}_{1}={\displaystyle \frac{\sqrt{\tau +{\tau}^{2}}}{|\overrightarrow{o}-\overrightarrow{i}|/\sqrt{2}}}$ |

LDA | ${c}_{2}\xb7{\displaystyle \frac{{\left|v\right|}^{3/2}}{\sqrt{{\dot{N}}_{\mathrm{photon}}w}}}$ | ${c}_{2}\xb7{\displaystyle \frac{\left|v\right|}{\sqrt{{\dot{N}}_{\mathrm{photon}}T}}}$ | ${c}_{2}=\sqrt{{\displaystyle \frac{3}{\pi}}}\xb7{\displaystyle \frac{d}{w}}$ |

L2F | ${c}_{3}\xb7{\displaystyle \frac{{\left|v\right|}^{3/2}}{\sqrt{{\dot{N}}_{\mathrm{photon}}w}}}$ | ${c}_{3}\xb7{\displaystyle \frac{\left|v\right|}{\sqrt{{\dot{N}}_{\mathrm{photon}}T}}}$ | ${c}_{3}={\displaystyle \frac{\tilde{b}}{w}}$ |

PTV | ${c}_{4}\xb7{\displaystyle \frac{{\left|v\right|}^{3/2}}{\sqrt{{\dot{N}}_{\mathrm{photon}}w}}}$ | ${c}_{4}\xb7{\displaystyle \frac{\left|v\right|}{\sqrt{{\dot{N}}_{\mathrm{photon}}T}}}$ | ${c}_{4}={\displaystyle \frac{0.94}{{w}_{\mathrm{px}}}}$ |

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Fischer, A. Imaging Flow Velocimetry with Laser Mie Scattering. *Appl. Sci.* **2017**, *7*, 1298.
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Fischer, Andreas. 2017. "Imaging Flow Velocimetry with Laser Mie Scattering" *Applied Sciences* 7, no. 12: 1298.
https://doi.org/10.3390/app7121298