1. INTRODUCTION

Despite its apparent simplicity, determining the quantitative patterns of the multiphase liquid flow in small-sized gaps is a complicated task. High-speed digital photography techniques have already been applied to study cavitation flows by Reinke et al. (2023) [9] and Skripkin et al. (2022) [10]. Based on the data obtained, the patterns of motion of cavitation bubbles and cavities have been revealed. Besides, the frequency characteristics of the flow have been evaluated.

A common technique for measuring liquid and gas flow velocities is non-contact flow diagnostics methods such as laser Doppler anemometry (LDA). The first prototype of a modern LDA system in single-beam mode was introduced by Yeh and Cummins in 1964 [13], and by Vom Stein in 1969 [12] for two-beam mode. When two coherent laser beams intersect, a measuring volume is formed. Tracer particles, crossing the measuring volume, re-emit modulated light in accordance with the dark and light interference bands, which allows, knowing the distances between the bands, calculating the normal component of the tracer velocity (Feng et al. 2011 [2], Pedersen et al. 2003 [7], Kabardin et al. 2021 [3]). The work of Rastello et al. (2022) [8] proved the effectiveness of using LDA for various flow velocities of a liquid with a free turbulent boundary in a narrow channel. Zhang et al. (2022) [15] investigated the two-phase bubble flow using the LDV method. He differentiated the LDV signal from bubbles and tracer particles and demonstrated that the signal amplitude of millimeter bubbles is about an order of magnitude greater than that of tracers or microbubbles. The effective application of the LDA method for three-dimensional turbulent flows is shown by Yaacob et al. (2022) [14], Klimov et al. (2018) [4], Zhang et al. (2020) [17].

In order to study the characteristics of pressure pulsations that occur when a spiral cavitating vortex core appears in a Francis turbine, Deng et al. (2022) [1] applied the LDA method. The fundamental frequency of velocity pulsations was shown to be close to that of pressure pulsations and become higher as the cavitation coefficient increased. The laser Doppler anemometry system was used to measure velocity pulsation signals in various areas in the centrifugal pump (Zhang et al. 2023) [16]. Zhang et al. 2023 [16] conducted studies at various volumetric fluid flow rates. Using time and frequency analysis, the velocity spectrum was determined. The velocity spectrum was characterized by discrete components corresponding to the frequency of passage of the blades and their higher harmonics. Khan et al. 2023 [5] experimentally studied a jet generated in a Pelton turbine using a laser Doppler anemometry system. The intensity of turbulence and the instantaneous velocity of the jet were analyzed at various points along the jet stream. A high-speed wake-up zone appeared in the center of the jet to decrease in the axial direction of the jet. The jet velocity profile obtained using the LDV method almost completely coincided with the analytical one. The flow characteristics near the S-shaped hydrofoil in a 100 mm wide cavitation tunnel were studied in more detail (Liu et al. 2021 [6]). Two characteristic frequencies of the cavity separation were detected using the LDA system. The determination of quantitative characteristics of cavitation flows using the LDA method has now become one of the basic tools. However, there is, in fact, no work on analyzing the characteristics of the cavitation flow velocity in slit channels using the LDA method. Therefore, in this paper we have focused on adapting the LDA method to study fluid flows in slit channels.

2. EXPERIMENTAL FACILITIES

To measure the flow velocity in a slit channel by laser Doppler anemometry, an experimental stand with a closed hydrodynamic circuit is used (Fig. 1). The slit gap is a channel with a length \(l = 276\) mm, a width \(W = 120\) mm, and a height \(h = 1.2\) mm. A hydrofoil is located inside the slit channel, with its chord, \(C\), being 70 mm, the angle of attack equaling to \(\alpha =21^{\circ}\), and the span coinciding with the width of the slit gap. The hydrofoil is made according to a fourth-degree polynomial and corresponds to the NACA0012 hydrofoil. A more detailed description of the experimental stand may be found in the work of Skripkin et al. 2023 [11].

Fig. 1
figure 1

A scheme of the working area with a hydrofoil and a flow diagnostic system.

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In the course of experimental studies, the superficial flow velocity varied and amounted to 14.16, 16.21, and 18.67 m/s. The Reynolds number calculated from the superficial velocity, \(U_{0}\), the characteristic size of the bluff body, \(C\), and the kinematic viscosity, \(v\), Re = \(C\cdot U_{0}/v\) is \(\sim 10^{6}\).

3. MEASURING TECHNIQUE

A two-component laser Doppler anemometry (LDA) system was used in the research (Fig. 1). It was designed on the basis of a Mitsubishi ML1013R semiconductor laser with a wavelength of 684 nm (red color range) and a power of 70 MW. During experimental measurements, the anemometer was mounted on an automatic coordinate-movement device, with the minimum positioning step of 0.1 mm. The LDA employs a two-frequency differential optical circuit based on a Bragg cell with a frequency shift of 80 MHz, the focal length of the lens of 500 mm, and the size of the measuring area in water of \(0.1\times 0.1\times 0.5\) mm. Polyamide particles of neutral buoyancy and light scattering particles of natural origin were used as tracers in experiments. The LDA allows alternately measuring the longitudinal and transverse components of the velocity vector when moving the positioning device with a 0.1 mm step. According to the LDA Instrument Certificate, the accuracy of average velocity measurement is 0.5%.

To measure the longitudinal component of the flow velocity in the slit channel, the measuring equipment was configured according to the following algorithm. The anemometer was positioned on a coordinate device at a distance of approximately 450 mm from the measuring area. On the way of the laser beam there was a channel wall made of polymethylmethacrylate with a thickness of 20 mm. Realizing micro-displacements of the LDA closer and further over the slit channel height, the channel boundaries were determined by the characteristic drop in the velocity value. The central area of the channel was selected for correct measurement of values. The measuring volume was positioned across the channel and made up a third of the channel height.

Fig. 2
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A diagram of a slit channel with a grid of coordinates of the measured points.

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The grid of coordinates of the measured points was constructed taking into account the location of the bluff body in the slit channel, to be finer in the area of gradient flows, and coarser in the area of uniform flows. The zero coordinate was the area of the leading edge of the hydrofoil (Fig. 2).

When the installation was operating at high flow rates, the amplitude of vibrations of the walls of the slit channel began to reach large values. When measuring the flow velocity values, the channel walls fell into the measuring volume, generating noise effects with an absolute value of up to 2 m/s. Such data were successfully filtered using the threshold method, since the main flow velocity was about 9–11 m/s.

During measurements in multiphase flows, small bubbles of the cavity also act as tracer particles. Such bubbles, in addition to moving with the flow, have their own rate of ascent due to the buoyancy forces. Their ascent rate depends on the bubble diameter and in our problem does not exceed 0.1 m/s. Thus, the bubble ascent rate is correlated as 1/100 to the liquid flow velocity in the slit channel, which is a small value being within the measurement error of the method.

Each point in the flow velocity profiles is obtained with averaging 1000 measurements.

4. RESULTS

This section presents experimental results of measuring profiles of the longitudinal component of the average flow velocity obtained by laser Doppler anemometry in a slit channel. The leading edge of the hydrofoil is selected as the origin point.

In Fig. 3, the profile of the longitudinal component of the average velocity is obtained in section \(x = 10\) mm. The area of the profile rupture corresponds to the area of the hydrofoil location in the slit gap. Above the profile, the flow velocity is at a maximum of 15 m/s, and under the profile it is 9 m/s, which corresponds to real values. A similar trend is also observed for the velocity profile obtained in the cross-section \(x=20\) mm (Fig. 4).

In the central region of the hydrofoil, a number of points in the boundary layer are obtained by laser Doppler anemometry (Figs. 5 and 6) in the area of the occurrence of the second phase in the fluid flow. It is noteworthy that in the vicinity of the hydrofoil boundary, the flow velocity takes the opposite sign relative to the main flow of the liquid. Such a change is completely correct. A so-called return flow occurs near the hydrofoil wall to be realized most of the time.

Fig. 3
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The profile of the longitudinal component of the average flow velocity obtained in section \(x=10\) mm.

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Fig. 4
figure 4

The profile of the longitudinal component of the average flow velocity obtained in section \(x = 20\) mm.

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Fig. 5
figure 5

The profile of the longitudinal component of the average flow velocity obtained in section \(x = 30\) mm.

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Fig. 6
figure 6

The profile of the longitudinal component of the average flow velocity obtained in section \(x = 40\) mm.

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Fig. 7
figure 7

The profile of the longitudinal component of the average flow velocity obtained in section \(x = 50\) mm.

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Fig. 8
figure 8

The profile of the longitudinal component of the average flow velocity obtained in section \(x = 60\) mm.

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The thickness of the boundary layer increases significantly with increasing distance from the leading edge of the hydrofoil (Figs. 7 and 8). In the region of the trailing edge, it is about 30 mm. The flow velocity in this area is much less than the velocity of the main flow, which, being completely physical, confirms the correctness of the data obtained by laser Doppler anemometry for a slit with a hydrofoil located in it.

5. CONCLUSIONS

The results of our work prove the applicability of the laser Doppler anemometry method to the study of cavitating hydrofoils in slit channels of a characteristic size of 1 mm A technique for obtaining flow velocity profiles near hydrofoil with a measurement error not exceeding 0.5% has been elaborated.

From the measurement results, profiles of the longitudinal components of the average flow velocity in various sections of the slit channel have been obtained. Measurements using the LDA method are shown to reflect the real flow pattern in the vicinity of the hydrofoil, and serve to obtain velocity values in the near-wall region of a bluff body when the second phase occurs in the flow.

FUNDING

The research was funded by the Russian Science Foundation (project no. 19-79-10217).

CONFLICT OF INTEREST

The authors of this work declare that they have no conflicts of interest.