Precise measurement of fluid flow rate attracts a lot of attention due to the applicable demands in oil industry so that the flow rate of oil/water can be monitored in real time to determine the volume of the exploration oil^[1]. Traditional flowmeters mainly include volumetric flowmeters, turbine flowmeters, differential pressure flowmeters, ultrasonic flowmeters, electromagnetic flowmeters, and so on^[2,3]. The main drawback of those flowmeters is that their output signal is electrical, and they have a low signal to noise ratio because of the transmission of the sensor signal to a recording device a long distance away. Recently, the development trend of flowmeters is high integration, high accuracy, and miniaturization^[4]. Flow measurement based on fiber optic components is a more promising and preferred solution because of the advantages that fiber optic recording devices have over their electric analogs. Those advantages are that optical fibers have an almost boundless extent and great flexibility, are insensitive to electromagnetic disturbances, can withstand high temperatures, are resistant to the action of corrosive media, and are explosion- and fire-safe^[5]. Some researchers have proposed optical flowmeter sensors based on fiber Bragg gratings (FBGs). Qiao et al.^[6] proposed the target type flowmeter based on FBGs, the relationship between the central wavelength shift of the FBG and the flux is derived, and the experimental results verify the proposed sensor that can measure the flux range from 200 to 1200cm3/s. Zhao et al.^[7] proposed a microprobe-type optical fiber flowmeter based on the differential FBG measurement method. A couple of FBGs that were stuck on the inner wall were used to measure the strain of the cantilever. The flow rate can be obtained by monitoring the difference of the two shifted Bragg wavelengths. The experiment results showed that the resolution of the accuracy was 3.6% in the region of 0–22.5m3/h.

Higher accuracy flow measurement cannot be achieved because of the strain resolution limit of the FBG, so the FBG-based flow sensor has been improved. Cashdollar et al.^[8] presented FBG flow sensors powered by in-fiber light, and the self-heated FBG sensor was used to measure gas flow with a sensitivity of 0.35m/s. Caldas et al.^[9] demonstrated the possibility of heating a silver-coated FBG structure with a high-power laser diode when the cladding region is excited through a long period grating with the same wavelength resonance as the laser emission and a flow speed resolution of 0.08m/s is achieved using this new configuration. Liu et al.^[10] presented a miniature fluidic flow sensor based on a short FBG inscribed in a single mode fiber and heated by Co²⁺-doped multimode fibers. A small flow rate of 0.005m/s or 0.002m/s can be distinguished for a particular phase of water or oil, respectively, at a certain laser power.

The flow sensors described above are intrusive, so with the target hindering the fluid flow, this type of flowmeter would cause some pressure loss in actual application. Another significant drawback of intrusive flowmeters is that their components are damaged by corrosion from aggressive media.

In this Letter, the non-intrusive flow measurement method based on a distributed feedback fiber laser (DFB-FL) is proposed to solve the above problem. A DFB-FL has s higher strain resolution (10−14Hz−1/2 level^[11]) than an FBG as the sensing element of the flow sensor. The flow sensor based on a DFB-FL is sensitive enough to detect the flow inside the pipe even though it is installed on the outer wall of the pipe.

The shift of DFB-FL wavelength Δλ with strain ε can be expressed as^[12]Δλλ=(1−Pe)ε,(1)where λ is the wavelength of the DFB-FL and Pe is the photoelastic coefficient of the fiber.

The pipe can be regarded as a cylinder, and the circumferential strain ε of the outer wall of the cylinder due to the internal pressure P is defined as^[13]ε=Δrr=2Pr12E(r22−r12),(2)where Δr is the radial displacement, r1 is the inner diameter of the cylinder, r2 is the outer diameter of the cylinder, r is the radius of the cylinder (r1≤r≤r2), P is the internal pressure of the cylinder, and E is the elastic modulus.

The shift wavelength of the DFB-FL Δλ with internal pressure P can be obtained from Eq. (1) and Eq. (2): Δλ=λ(1−Pe)2Pr12E(r22−r12).(3)

According to Pittard’s research^[14], when fluid molecules reach the pipe wall, more than 90% of the kinetic energy is converted into the form of pressure, which means that pressure is the main form of energy transfer between the fluid and the pipe wall. Consider a turbulent flow through a horizontal pipe of circular cross section. The velocities of the fluid can be expressed in terms of a time average (u¯, v¯) and a fluctuation (u′, v′), as shown by^[15]$$\begin{gathered} u=u¯+u′, \\ v=v¯+v′, \end{gathered}$$(4)where u is the velocity in the direction of the primary pipe axis and v is the velocity perpendicular to the pipe axis.

According to Prashun’s research^[16] on round tube turbulence, the pressure fluctuation P′ is proportional to the fluctuation of the fluid velocity: P′∝u′v′¯∝(u′)2¯.(5)

Equation (3) can be described as Δλ′=λ(1−Pe)2P′r12E(r22−r12)∝(u′)2¯(6)

Equation (6) shows that the dynamic change of the DFB-FL wavelength shift is proportional to the fluctuation of the fluid velocity.

The flow sensor based on the DFB-FL is designed and tested according to the above principle. Figure 1 shows the schematic diagram of the flow sensor. The rigid L-beam mass is connected to the bracket through the diaphragm and can be rotated around the diaphragm. The DFB-FL with the length of 40 mm is bonded between the mass and the bracket as a sensitive component. When fluid generates the vibration signal through the pipeline, the mass rotates around the diaphragm due to inertia, which changes the axial strain of the DFB-FL and causes the central wavelength of the DFB-FL to change.

Figure 2 shows a system diagram of the flow measurement based on the DFB-FL. Pump light with a wavelength of 980 nm set to 90 mW enters the DFB-FL-based flow sensor, and the wavelength change of the DFB-FL Δλ is converted into a phase change Δϕ by using a Michelson interferometer. The phase change can be expressed as^[17–19]Δϕ=2πnsλ2Δλ,(7)where n is the fiber refractive index, s is the difference in length between the two arms of the interferometer, and s=5m.

Equations (6) and (7) show that the information of the vibration signal induced by internal flow of pipeline can be restored by the external flow sensor based on DFB-FL.

The interferometer output signal I can be expressed as^[20]I=A+Bcos[Ccos(ω0t)+Δϕ],(8)where A is the average optical power of the interferometer output signal, B=κA, κ≤1, and κ is the interference fringe visibility. Ccos(ω0t) is the phase generated carrier and Δϕ is the phase change induced by the tested signal.

The interference signal from the systems is demodulated via phase demodulation [such as the typical phase-generated carrier (PGC) demodulation algorithm^[21] if the intensities follow Eq. (8)].

The flow sensor based on the DFB-FL is fixed on the BK 4808 standard vibration test platform. The acceleration value of the vibrating platform remains consistent and the frequency value is increased from 10 Hz to 500 Hz. The frequency response curve of the flow sensor is shown in Fig. 3. It can be seen from Fig. 3 that the frequency response of the flow sensor is relatively flat. Finally, the frequency value of the vibrating platform is fixed at 310 Hz and the acceleration value is gradually increased. The linear response curve of the flow sensor is shown in Fig. 4. Figure 4 shows that the response of the flow sensor has a high linearity and the R² value of the fitting curve reaches 99.4%. It can be seen from Figs. 3 and 4 that the designed flow sensor’s performance parameters meet the requirements of flow measurement.

The DFB-FL-based flow sensor is mounted on the outer wall of the steel pipes to measure the flow rate. The best mounting direction of the DFB-FL sensor is parallel to the pipe cylinder axis. The flow experiment setup is shown in Fig. 5. The two tanks filled with water are connected by three steel pipes with inner diameters of 68 mm. A water pump installed in the middle of the bottom steel pipe circulates the liquid in the experiment setup, and the valves are installed in the side steel pipes to control the flow rate. The electronic flow sensor is used for flow calibration.

The flow from 0.6m3/h to 25.5m3/h is modulated in the pipe by controlling the valve. The DFB-FL-based flow sensor measures the flow, and the time domain signal based on the DFB-FL measured by the demodulation instrument is shown in Fig. 6. In Fig. 6, the x axis represents the measurement time, the y- axis is the flow, and the z axis represents the demodulation in radians. There are nine curves corresponding to the nine different flow values in Fig. 6. From the preliminary analysis in the time domain, the variation of the waveform amplitude under different flows is not seen.

The signal frequency domain analysis is further carried out in order to find the regularity. The frequency domain signal based on the DFB-FL is shown Fig. 7. In Fig. 7, the x axis is the frequency signal, the y axis is the flow, and the z axis represents the fast Fourier transform (FFT) amplitude. There are nine curves corresponding to the nine different flow values in Fig. 7. From the preliminary analysis of the data in Fig. 7, the FFT amplitude in the range of 200 Hz to 400 Hz under different flows has a certain regularity. So, the different frequency intervals are set in the range of 0 to 2000 Hz for each curve, and the energy values in different frequency intervals are calculated. The calculated energy values in the range of 200 Hz to 400 Hz under different flows and flow rates are shown in Table 1. The relation curves between the flow and energy are fitted by a quadratic curve in Fig. 8. In Fig. 8, the x axis is the calculated energy values in the range of 200 Hz to 400 Hz, and the y axis is different flow values. The R² value of the fitting curve reaches 99.1%. 0.6m3/h of the minimum flow is achieved in the experiment, and the minimum flow rate is 0.046 m/s. The above measurement indicators are accomplished well for flow measurement.

Some factors (such as the structural design of the DFB-FL and the material and diameter of the pipe) may affect the selection of the effective frequency range. Therefore, in the actual application process, it is necessary to calibrate the on-site flow rate, which will cause trouble in the actual application. Sensor structure design and demodulation algorithms need to be further improved to solve current problems.

In conclusion, a new non-intrusive flow measurement method based on the DFB-FL is proposed. The new method includes a new flow turbulence test principle based on the DFB-FL and energy analysis method with different frequency intervals. The response curve is relatively flat in the frequency range of 10 Hz to 500 Hz, and the response of the flow sensor has a high linearity. The flow from 0.6m3/h to 25.5m3/h is accurately measured and a minimum flow rate of 0.046 m/s is achieved.