## 2 Materials and Methods  
### The UVP with Doppler Pulse Repetition Technique  
The UVP method can obtain instantaneous velocity profile of fluid using ultrasonic waves by means of an ultrasound reflector. Figure 1 displays the UVP principle, consisting of ultrasound transmission, an echo signal, and velocity profile obtained. An ultrasonic pulse is emitted repeatedly from the transducer along the measurement line, and the echo reflected from the surface of the reflector such as from a particle is derived from the same transducer. The particles dispersed in the fluid according to the main flow. In the case of water, the density of the particle is nearly equal (0.98 g/cm\({}^{3}\)). Therefore, the velocity of a particle is assumed to be equal to the velocity of water at that position. The echo signals reflected from moving particles contain the Doppler signal. The Doppler frequency \(f_{Di}\) directly relates to the velocity of a moving particle at the position in measuring line. Consequently, the velocity of the particle \(V(x)\) can be obtained as follows in Equation (1). The velocity profile of the fluid along the channel diameter can be computed if tracer particles are sufficiently dispersed.  
\[V(x)=\frac{cf_{Di}}{2f_{0}sin\theta} \tag{1}\]  
where \(f_{0}\) is the basic frequency of ultrasound, \(\theta\) is the incident angle, \(x\) is a position in the horizontal axis, and \(i\) is a position in the measuring line.  
The Doppler pulse repetition technique is used to extract the Doppler signal from the echo signal, to estimate the Doppler frequency and calculate fluid velocity, respectively. The Doppler signal can be  
Figure 1: Ultrasonic Velocity Profiler (UVP) measurement configuration, an echo signal and velocity profile reconstruction.  
demodulated from echo signals reflected from a moving particle using ultrasonic emission repetition as shown in Figure 2. Figure 3 shows a block diagram of the UVP system based on the Doppler pulse repetition technique. An ultrasonic pulser/receiver, operated in the pulse-echo mode, emits an ultrasonic pulse through an ultrasonic transducer. The echo signal \(e(t)\) is then reflected and sent to the pulser/receiver via the same transducer [23]. It is expressed as:  
\[e(t)=A_{n,i}\cos 2\pi\bigg{(}f_{0}(t-t_{n,i})+n\frac{f_{D,i}}{f_{PRF}}\bigg{)} \tag{2}\]  
where \(t_{n}\) represents the delay time of the echo at \(n^{th}\) pulse, \(A\) is the amplitude, and \(f_{PRF}\) is the pulse repetition frequency.  
In the quadrature demodulation section, the echo signals are multiplied by the cosine and sine components, and a finite impulse response filter is then used as a low-pass filter to eliminate the carrier wave component or basic frequency. Therefore, the Doppler signal \(D(n)\) can be extracted from the echo signals as modeled in Equation (3)-(6) The Doppler frequency is then estimated in the frequency estimation section by autocorrelation or Fast Fourier Transform [23; 24]. Finally, the computed Doppler frequency is plugged into Equation (1) to demonstrate fluid velocity.  
Figure 3: Block diagram of the UVP system based on the Doppler pulse repetition technique.  
Figure 2: Demodulation of the Doppler signal.  
Continuous signal  
\[D_{t}\left(t\right)=\left\{2e\left(t\right)\left(\cos\left(2\pi f_{0}t\right)+j \sin\left(2\pi f_{0}t\right)\right)\right\}_{LOPLPASS} \tag{3}\]  
\[D_{t}\left(t\right)=\sum A_{n,j}\cos 2\pi\left(\frac{f_{D,t}n}{f_{PRF}}-f_{0}t_{n,j }\right)-j\sum A_{n,j}\sin 2\pi\left(\frac{f_{D,t}n}{f_{PRF}}-f_{0}t_{n,j}\right) \tag{4}\]  
Discrete signal  
\[D_{t}\left(n\right)=X_{t,j}\left(n\right)+jX_{Q,j}\left(n\right) \tag{5}\]  
\[D_{t}\left(n\right)=A_{n,j}\cos\left(\frac{2\pi f_{D,t}}{f_{PRF}}-\phi_{t} \right)-jA_{n,j}\sin\left(\frac{2\pi f_{D,t}}{f_{PRF}}-\phi_{t}\right) \tag{6}\]  
where \(\phi\) is the initial phase of the in-phase signals \(X_{I}\) and the quadrature-phase signals \(X_{Q}\).