A Measuring Method for Three-dimensional Turbulent Velocities Based on Vector Decomposition and Synthesis

2.1 Principle of velocity detection

The 3C tripod structure is designed to adapt vector characteristics of flow, so as to be able to sense each of the velocity vectors by a silicon piezoresistance beam as shown in Figure 1. The flow sensing ball is 3 mm in diameter and the core of the ball is the meeting point of the three vector support bars. The other end of each tripod bar is spliced to one end of the strain sensing silicon piezoresistance beam that measures the vector pressure of the bar in the direction of the flow. The other ends of the silicon beam of each of the tripod bars are spliced to the ceramic support base by low temperature glass. Once the ball is exposed to flow, the velocity of the flow creates strains on the sensing parts which converted the strains into electrical output signals by the Wheatstone bridge of the signal conditioning unit. The signal conditioning unit (as shown in Figure 7) is a multi-vector processing apparatus, in which the measured 3C vectors can be synthesized to the flow turbulence velocities’ directions corresponding to the vectors which represent the turbulence velocities of the flow, and they are longitudinal velocity, vertical velocity, horizontal velocity, denoted as $u$, $v$, $w$.

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Figure 1. 3C tripod sensing structure

Since the silicon piezoresistance beam is made of highly rugged material and the force created by the flow in flume is quite small, the deformation in the silicon piezoresistance beams could go undetected in analysis. The changes in resistivity of the piezoresistance are dominated by strain caused by the front ball and the tripod bars. The detecting 3C structure is a vibration free system and the beam curve angle is small in order to keep the transducer shell profile size as small as possible. The relationship between the force of the external flow and the angle could be taken as a linear relationship [15] .The vibrating differential equation of the system can be described as:  

$m \frac{d^{2} x}{d t^{2}}+c \frac{d x}{d t}+k x=F(t)$    (1)

Let $k / m=\omega_{p}^{2}, c / m=2 D$ , and Formula 1 can be rewritten as:

$\frac{d^{2} x}{d t^{2}}+2 D \frac{d x}{d t}+\omega_{p}^{2} x=F(\mathrm{t})$      (2)

Where $m$ is the mass of the beam and the relative structure materials, $c$is the resistivity coefficient and $k$ is the elastic coefficient, $\omega_{p}$ is the systematic free vibrating radian frequency, $D$ is the attenuation coefficient.

The turbulence flow is not periodic, but within a relatively long period of time, and the average turbulence velocities in both vertical and horizontal direction of a flume could be thought of as zero. In a given time period, the portion of turbulence velocity is made of multi-harmonic waves and is considered as the excitation of the system, described as follows:

$F(\mathrm{t})=\sum_{i=0}^{n} F_{i} \sin \left(\omega_{i} t+\varphi_{i}\right)$     (3)

Where n is the highest frequency in the turbulence flow, $F_{i}$ is the amplitude of the force, $\varphi_{i}$ is the phase angle, and $\omega_{i}$ is the flow radian frequency acting on the transducer. Normally $\omega_{i}$ is less than 100 Hz in the lab’s fluid flow flume where the average velocity of sediment content fluid flow is less than 3m/s, meanwhile, and n decreases as the sediment content increases. As seen in Formulas 2 and 3, the general solution equation of the system could be to say:

$\begin{aligned} x(t)=& e^{-D t}\left(C_{1} e^{t \sqrt{D^{2}-\omega_{p}^{2}}}+C_{2} e^{-t \sqrt{D^{2}-\omega_{p}^{2}}}\right) +\sum_{i=1}^{n} X_{i} \sin \left(\omega_{i} t+\phi_{i}\right) \end{aligned}$    (4)

In the above equation, the first part is a free vibration function of the system, which is a fast diminishing one. The second part represents the forced vibration of the system when the sensor is excited by the flow. In a linear system, it is thought of as corresponding to the acting force. With a suitable silica gal filler damper in the 3C structure tube, the resonant vibration of the sensor should be higher than 500 Hz, as indicated by the pulse jet calibration test, which matches the frequency response demand of the turbulence measurement.

2.2 Principle of vector decomposition and synthesis

The force analysis in three-dimensional space is shown in Figure 2. The angle between each support bar and the centricity of the tripod is $\theta$. In the preliminary design, $\theta$ is 10° to fit the size demand of the transducer. The angle between two of any of the three fixed support bars is 120°. The angle between any of the three support bars and the silicon piezoresistance beams is 90° to keep a good linearity signal conditioning of the Wheatstone bridge, as shown in Figure 3.

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Figure 2. Force analysis of 3C structure for velocity detection

Let $\vec{f}$denote the force (F) acting on the target bar. Let$f_{1}$, $f_{2}$ and $f_{3}$ denote the magnitude of the forces acting on the support bars. The direction of the forces is along the support bars. According to the decomposition and synthesis principle of vectors, the force of each support bar can be decomposed in x, y, z directions as follows.

(1) Bar 1’s branch force

$f_{x}^{1}=f_{1} \sin \theta, f_{y}^{1}=0, f_{z}^{1}=f_{1} \cos \theta$      (5)

(2) Bar 2’s branch force

$f_{x}^{2}=-f_{2} \sin \theta \cos 60^{\circ}$ ,

$f_{y}^{2}=f_{2} \sin \theta \sin 60^{\circ}$ ,

$f_{z}^{2}=f_{2} \cos \theta$        (6)

(3) Bar 3’s branch force

$f_{x}^{3}=-f_{3} \sin \theta \cos 60^{\circ}$ ,

$f_{y}^{3}=-f_{3} \sin \theta \sin 60^{\circ}$ ,

$f_{z}^{3}=f_{3} \cos \theta$        (7)

Set: $H=\left[f_{x}, f_{y}, f_{z}\right]$ ,

$A=\left[f_{1}, f_{2}, f_{3}\right]$ ,

$B=\left(\begin{array}{ccc}\sin \theta & 0 & \cos \theta \\ -\sin \theta \cos 60^{\circ} & \sin \theta \sin 60^{\circ} & \cos \theta \\ -\sin \theta \cos 60^{\circ} & -\sin \theta \sin 60^{\circ} & \cos \theta\end{array}\right)$ , 

Then, $\vec{f}$ along x, y, z directions’ force $f_{x}$ , $f_{y}$ ,  $f_{z}$ can be calculated as follows.

$H=A^{*} B$   (8)

According to the Bernoulli Equation of fluid mechanics, in the flow of the lab’s flume (the flume is 33 m long, 0.5m width, 0.5m height, flow is controlled by valves between 3cm/s-3m/s), kinetic energy ($f$) of the fluid is proportional to the square of velocity ($v^{2}$) and fluid density ($\rho$),  which can be simply expressed as:

$f \propto \rho|\vec{v}|^{2}$     (9)

According to the above analysis, the instant fluid velocity could be expressed as:

$|\vec{v}|=Q_{m} \sqrt[4]{f_{x}^{2}+f_{y}^{2}+f_{z}^{2}}$    (10)

In equations 9 and 10,  $f=\vec{f} |=\sqrt{f_{x}^{2}+f_{y}^{2}+f_{z}^{2}}$ , $Q_{m}$ can be obtained by a calibration experiment of the transducer [13], where $\vec{v}$ and $\vec{f}$ have the same vector direction. Therefore, the longitudinal velocity, vertical velocity and horizontal velocity ($u$, $v$, $w$) can be expressed respectively as follows.

$u=v_{z}=|\vec{v}| \frac{f_{z}}{\sqrt{f_{x}^{2}+f_{y}^{2}+f_{z}^{2}}}$      (11) 

$v=v_{x}=|\vec{v}| \frac{f_{x}}{\sqrt{f_{x}^{2}+f_{y}^{2}+f_{z}^{2}}}$      (12)

$w=v_{y}=|\vec{v}| \frac{f_{y}}{\sqrt{f_{x}^{2}+f_{y}^{2}+f_{z}^{2}}}$      (13)

With a series of measurements of the velocities in a time interval, the series data could be described as $\{u\}$ , $\{v\}$ , $\{w\}$ . The velocities of each direction and respective turbulence velocities can be calculated by above analysis.

2.3 Principle of strain measurement by silicon piezoresistance beam

The 3C structure with three silicon piezoresistance beams as the sensing element to detect the three different directions velocity vectors is the critical element of the transducer. The various strains on the 3C structure are converted to electrical output signals by Wheatstone bridges which are integrated into the silicon beams with the electronic signal processing unit [19-20]. The sketch map of the piezoresistances on a silicon beams is shown in Figure 3. The detection circuit of the three-dimensional velocity by each Wheatstone bridge is shown in Figure 4. With a suitable exciting constant current source, relative differential signal conditioning and filter circuits, the flow velocity can be measured.

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Figure 3. Arrangement of piezoresistances on silicon beam

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Figure 4. Wheatstone bridge circuits to detect force

According to the measurement principle of Wheatstone bridges, once the silicon beam detects the stress $f_{i}$ (that is $f_{1}$ , or $f_{2}$ , or $f_{e}$ ) from the support bar in one direction, the piezoresistances R₂, R₄ will increase, while R₁, R₃ will decrease. The larger the stress acting on the beam, the larger the output $U_{0}$ of the Wheatstone bridge. The relationship between the output voltage $U_{0}$ and the stress $f_{i}$ can be expressed as:

$U_{o}=\frac{4 \mu R_{2}^{0} R_{4}^{0}}{R_{1}+R_{2}+R_{3}+R_{4}} I_{a} \bullet f_{i}$      (14)

or

$U_{O}=\frac{4 \mu R_{1}^{0} R_{3}^{0}}{R_{1}+R_{2}+R_{3}+R_{4}} I_{a} \bullet f_{i}$     (15)

In Formulas 14 and 15, $I_{a}$ is the value of the exciting constant current source, $R_{1}^{0}$ , $R_{2}^{0}$ , $R_{3}^{0}$ , $R_{4}^{0}$ is the initial value of each silicon piezoresistance,  $\mu$ is a constant corresponding to the bridge sensitivity to the applied force or the flow velocity. Wherein, $\mu=K \frac{\sigma_{i}}{E}$ ( $\sigma_{i}=\frac{f_{i}}{S}$ , $\sigma_{i}$ is strain force of silicon beam, S is the force area of silicon piezoresistanes, E is the elastic modulus, K is a strain sensitivity coefficient of silicon piezoresistanes) .

In order to improve the measurement sensitivity of the silicon piezoresistive effect, a finite element method (FEM) is used to analyze the stress distribution of the force and deformation of the elastic cantilever [21-22]. The finite element model is established by ANSYS software, in which N-type silicon is chosen as the support base, elastic modulus E take 130Gp, poison ratio is 0.278, parameters of beams are 450$\mu m$*150$\mu m$*50$\mu m$. The strain force ($\sigma_{i}$) distribution on the cantilever silicon beam under the action of corresponding stress force is shown in Figure 5. Owing to the difference of  the strain distribution on each beam, combined with the measurement characteristic of the Wheatstone bridge, the measurement accuracy of each direction can be distinguished and improved .

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Figure 5. Stress distribution on the three silicon beam

4.1 Validation of forces’ decomposition and synthesis

According to the design of the transducer structure, the force will be automatically decomposed into three components along support bars when the flow sense ball is under impact of flow

fluid pressure. In order to verify the feasibility of the- 3C structural design and the principle of forces decomposition and synthesis, numerical analysis is given by the static stress analysis function module of Solidworks software. Examples provided are under assumptions that the transducer’s sensing ball (diameter of 3 mm) is placed in fluid flow, which is fully impacted by the measured flow. According to the Bernoulli Equation of fluid mechanics, the applying forces are set at a

range from 0.05N to 0.5N, corresponding to the flow velocity from 0.5m/s to 3 m/s [24]. The forces ($f_{1}$, $f_{2}$, $f_{3}$) along three support beams are obtained through different placement angles ($\beta$) and different applying forces ($F$). In this experiment they are $\beta$=$15^{\circ}$ , $30^{\circ}$,  $45^{\circ}$, $60^{\circ}$, $90^{\circ}$. The specific data is shown in Table 1. $f_{x}$ , $f_{y}$ , $f_{z}$  are the components of acting force $f$ in $x$, $y$ , $z$ directions which is shown in Figure 2. They are calculated based on the principle of decomposition and synthesis and the data is in Table 1. Thus $v_{x}$ , $v_{y}$ , $v_{z}$  can be obtained using the same method. The calculation results are curve fitting with the Matlab software, exactly as shown in Figure 8-(a), (b), (c), (d), (e).

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(a)

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(b)

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(c)

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(d)

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(e)

Figure 8. The fitting curve $f_{z}$ and $\mathcal{V}_{z}$ , $f_{x}$ and $\mathcal{V}_{x}$ , $f_{y}$ and $\mathcal{V}_{y}$ in different placement angles 

It can be seen in the above results, when $\beta$ = $15^{\circ}$ , $30^{\circ}$ , $45^{\circ}$ , $60^{\circ}$ , $90^{\circ}$ , $f_{z}$ and $v_{z}$ , $f_{x}$ and $v_{x}$  can be compared with $y=\xi x^{2}(\xi>0)$ curve fitting. when $\beta$ = $15^{\circ}$ , $30^{\circ}$ , $45^{\circ}$, $f_{y}$ and $v_{y}$ can be compared with $y=-\xi x^{2}(\xi>0)$ curve fitting. But when $\beta$ is $60^{\circ}$ or $90^{\circ}$ , $f_{y}$ and $v_{y}$ can be compared with$y=\xi x^{2}(\xi>0)$ curve fitting . This shows that when the $\beta$ < $45^{\circ}$ , the $f_{y}$ form the reaction force, the direction is opposite in $y$ direction, and on the other hand, $f_{y}$ and $v_{y}$  in the same direction.

Through the above analysis it can be speculated that $f_{x}$ , $f_{y}$ , $f_{z}$ and the corresponding $v_{x}$ , $v_{y}$ , $v_{z}$ can be fitted with $|y|=\xi x^{2}(\xi>0)$ curve when the transducer is placed at any angle. At the same time, as seen in Figure 9, the result verifies that the force ($F$) of fluid is proportional to the square of velocity ($v^{2}$). Proportional relations reflect a consistent linear relationship of the transducer, which is one of the most important characteristics of the transducer.

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Figure 9. The relation F and V form the static stress analysis

Because $v_{x}$ , $v_{y}$ , $v_{z}$ are related to longitudinal velocity, vertical velocity, horizontal velocity ($u$ , $v$ , $w$),  the relationship between $f_{x}$ , $f_{y}$ , $f_{z}$and the corresponding $v_{x}$ , $v_{y}$ , $v_{z}$at different placement angles can be simulated in three-dimensional space as shown in Figure 10 (a), (b), (c) .

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(a)X direction

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(b)Y direction

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(c)Z direction

Figure 10. The relationship between $f_{x}$ , $f_{y}$ , $f_{z}$and the corresponding $v_{x}$ , $v_{y}$ , $v_{z}$ at different placement angles in three-dimensional space

4.2 Characteristic analysis of transducer

In this paper, the numerical simulation method is used for testing the relation between the force (F) and the velocity (V) by CFD. The numerical result is simulated by commercial code ANSYS Fluent 12.1. The standard k-ε model is adopted during simulation [25], and 2D axial-symmetric modeling is used in the simulating strategy. The calculating domain is shown in Figure 11 (a). The boundary condition of inlet and outlet, other than the wall, is velocity inlet and pressure outlet respectively. The mesh around the sensor is shown in Figure 11 (b) and the grid number is 50, 000.

Because the front ball is the only part of the transducer exposed to the flow, the knife-shaped shell with a sharp front was designed to reduce the impact of the transducer on the original flow. The vortex comparison between the single ball and the transducer in flow is shown in Figure 12. The vortex is created behind the ball when a single ball is derived by the flow. Because of the transducer shell’s shape, as seen in Figure 6, the front ball cannot sense the vortex at the tail of the 3C structure.

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Figure 11. CFD model of the sensor

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Figure 12. Vortex comparison between single ball and

transducer in flow

The relation between F and V is shown in Figure 13 as calculated by Fluent software. The relation curve between F and V can be fitted with $y=\xi x^{2}(\xi>0)$ curve, which verifies that the force ($F$) of fluid is proportional to the square of velocity ($v^{2}$).

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Figure 13. Relation between F and V from the CFD

According to formula 10, assuming $Q_{m}$ =1, the V-F curve restults in that shown in Figure 14. The relationship between V-F could be formulated as $y=\xi x^{2}(\xi>0)$. The analysis using Matlab, where $Q_{m}$ =12.656, the relation curve between F-V from static stress analysis can match the relation curve between F-V from CFD perfectly as shown in Figure 14.

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Figure 14. $Q_{m}$ theoretical calibration analysis

4.3 Result of measuring turbulent velocities

The transducer’s performance is studied in a flume, shown in Figure 15, which simulates a flume environment in The State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, where the flume is 33m long, 0.5m width, 0.5m height. In the experiment, water depth varies from 5 to 30cm, flow velocity is between 3cm/s-3.5 m/s. the width of flow section is 1.2 m and the length of flow in 12m. With the depth of flow section being 0.16 m for example, five testing points are selected and their corresponding feature values are measured, results of which are shown in Table 2. The velocities data of $v_{x}$, $v_{y}$ and $v_{z}$ (in 0.6 h) is collected in 30 second periods as shown in Figure 16.

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Figure 15. Measuring environment in laboratory flume

This experiment is carried out in transparent water by acquiring current velocities in a certain period of time continuously [26], then average of velocities is obtained. The whole experiment process, including data acquisition, data processing and data display, is completed by computers and the corresponding client app.

Table 2. Feature value of velocity measurement

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Figure 16. The velocities data of $v_{x}$ , $v_{y}$ and $v_{z}$ (0.6 h) in 30 seconds

As is shown above, the method is based on the elastic component whose natural frequency of vibration is above 100Hz and can measure three-dimensional turbulent velocity at the same testing point in a period of time. In the experiment, it is found that the flow turbulence can be well detected at a water depth of 0.1m when current average velocity is below 1.5m/s.