© B. Chen et al., Published by EDP Sciences 2025

1 Introduction

The periodic variations in mass and velocity of oil within a pipeline, caused by changes in system pressure over time, are referred to as pulsating flow. This phenomenon occurs alongside external pressure pulsations introduced by the system, and its internal dynamics are influenced by the time-dependent changes in the wall boundary layer due to the viscosity of the oil. Shear wave excitation generates coherent structures in the oil from the wall, leading to the development of hairpin or horseshoe vortices in the viscous sublayer. Consequently, large-scale vortices form in the main flow, which eventually break down into smaller vortices until they collapse, resulting in the observed pulsation of the oil [1–3]. He et al. [4] investigated the transient velocity characteristics of pulsating pipe flow, focusing on how mean velocity and pulsation amplitude influence the radial distribution and phase behavior of the root-mean-square values of axial mean velocity, radial velocity, and axial pulsating velocity across different frequency ranges. Papadopoulos et al. [5] simulated low-Reynolds-number, high-frequency sinusoidal pulsating flow in straight pipes, discovering that the fluid's pulsation amplitude is nearly sinusoidal, with frequency being the primary factor influencing fluid characteristics. They noted that turbulence pulsation at harmonic frequencies does not occur at higher frequencies.

Currently, technologies like LDA and Particle Image Velocimetry (PIV) are commonly employed in both domestic and international studies of pulsating flow to indirectly assess pulsating characteristics [6,7]. However, these methods have frequency band limitations, making it challenging to thoroughly and accurately analyze the characteristics of oil pulsating flow. With advancements in sensor and signal acquisition technologies, it is now feasible to directly measure and extract the characteristics of oil pulsation signals in pipelines. Oil pulsating flow is inherently unsteady and destabilized by external influences-pump operation modes, pipeline geometry, and connection types-that imprint a tangled spectrum of signal components on any measurement. Therefore, effectively extracting and analyzing pulsating flow signals is a critical challenge that requires urgent attention.

The Short-Time Fourier Transform (STFT) is effective for analyzing stationary signals, but it lacks physical relevance when examining unsteady, unstable pulsating flows. In contrast, the wavelet transform offers simultaneous localization of signals in both time and frequency domains, making it applicable across various fields. Unlike the Fourier transform, the wavelet transform enhances local properties of data in both domains, allowing for the detection of abrupt changes in time series and analysis of their primary change periods, as well as their impacts over different intervals. Daubechies [8] has noted that the wavelet transform is particularly well-suited for investigating periodic changes in time series data. For instance, Zhao Zhongnan [9] utilized a sym4 wavelet basis to analyze frequency domain fluctuations in two-phase flow pressure differences, revealing significant changes in primary frequency signals with varying flow patterns. Liu et al. [10] employed the Morlet wavelet function to study pressure signals in oil with different particle concentrations, exploring the relationship between pulsating flow pressure signals and particle concentration changes. Gao et al. [11] analyzed engine combustion pressure pulsations using wavelet methods, extracting key information from the energy spectrum of decomposed layers. Wang et al. [12] applied wavelet analysis to the multi-scale decomposition of PIV velocity data, examining how surfactants affect turbulence characteristics in multi-scale channels. Farge et al. [13] used adaptive wavelet transforms to investigate turbulence and coherent structures based on numerical simulations.

Given the multi-scale, quasi-periodic character of pulsating oil flow, the present study applies wavelet analysis to interrogate pressure signals. The paper is organized as follows: Section 2 details the experimental rig and the wavelet-based extraction technique; Section 3 analyzes the modulus, modulus square, and variance of the wavelet coefficients post-decomposition, establishing the relationship between oil pressure signals and their vibration characteristics. Finally, Section 4 presents the conclusions.

2 Experimental device and method

2.1 Experimental device

Figure 1 presents the schematic of the oil-pulsating-flow test rig, comprising the test section and the data-acquisition system. The square test section (0.04 m × 0.04 m × 0.5 m) is fabricated with transparent glass on its top and front faces for optical access. A Cartesian coordinate system is defined with the origin at the lower-left corner: x is the streamwise direction, y the spanwise, and z the wall-normal direction.

The measurement chain consists of YP-01S pressure transducers (−0.1–2.0 MPa, 0.2 % FS, 0–5 V), temperature and flow transmitters, plus accelerometers for vibration monitoring. Signals are digitized with an NI PXI card and analyzed in LabVIEW, capturing real-time pressure transients at the test-section inlet and outlet. The flow transmitter, located at the outlet of the pipeline, is a lwgb-6 model with an accuracy of 1%, which measures the oil flow. The temperature transmitter, HSLHW-SX, has an accuracy of 0.25% FS and is used to monitor temperature changes during oil operation. Additionally, a vibration transmitter, model LC0123, is installed at the inlet of the experimental pipeline, with a precision of 0.08%, to detect vibrations in the testing pipeline. Real-time pressure, flow-rate, temperature, and vibration data are acquired by an NI-PXI board and streamed to LabVIEW, where the oil's dynamic pressure, acceleration, flow, and temperature signals are filtered, denoised, and analyzed.

The working fluid is transparent, colorless 25# transformer oil. A fresh batch is first filtered through neutral paper into a reservoir, then fed into the loop by a peristaltic pump at an initial velocity of V₀ = 0.0362 m s⁻¹. After passing through a hydrodynamic-development length, the oil enters the measurement section and finally returns to the reservoir, completing the closed circuit.

Fig. 1 The schematic diagram of the experimental device of oil pulsation flow.

2.2 Experimental methods

Using LabVIEW, we built a virtual instrument whose front panel is split into two main zones: a configuration pane and a real-time display.

• The configuration pane sets the NI-DAQmx task—connection topology, sampling rate, sample count, and acquisition mode (see Tab. 1 for exact values).

• The display pane plots time histories of inlet pressure, outlet pressure, flow rate, temperature, and pipeline-oil vibration (acceleration).

Below the plots, push-buttons let the operator apply digital filters, save waveform data, or capture the current screen image in a single click.

The program block diagram connects the front panel's input and output programs through structures like cycles and sequences, facilitating communication between the front panel controls. During the experiment, oil flows through the pipeline for 10 min, and once the system's oil index stabilizes, data sampling for each parameter begins.

To extract the oil's true pressure-pulsation signature, the acquisition chain first low-pass-filters each raw pressure trace. The inlet and outlet pressures are then differenced to yield the dynamic pressure drop along the test section. This residual, however, still contains the pipe's structural vibration; it is therefore synchronously corrected by subtracting the acceleration transducer signal. The resulting waveform represents the clean pulsating-pressure component of the oil alone. All subsequent wavelet results are derived from this corrected waveform, forming a reproducible, experiment-analysis pipeline that differs from the common practice of using simple differential pressure alone. During the entire run, oil flow rate and temperature are held fixed; their signals remain stable and are logged only for reference.

2.3 The wavelet analysis method

Although wavelet analysis is widely used for non-stationary signals, its effectiveness hinges on matching the wavelet basis to the signal's features, since each basis produces distinct outcomes. Common wavelet functions include the Daubechies wavelet, developed by renowned wavelet analysis expert Ingrid Daubechies, with db1 representing the Haar wavelet. Other wavelets lack clear expressions. The Mexican wavelet, which is the second derivative of the Gaussian smoothing function, is frequently used in visual information processing and edge detection. The Morlet wavelet, a complex-valued wavelet, can extract amplitude and phase information from the analyzed time process or signal; it does not have a scale function or orthogonality and is often used in the analysis of geophysical processes and fluid turbulence [14].

The pressure signal of oil pulsating flow is characterized as a non-stationary sequence with features such as quasi-periodicity, trends, randomness, and variability. It exhibits a multi-layer, multi-scale structure with local time-varying properties. Consequently, the Morlet wavelet is chosen for analyzing this pressure signal [10].

To suppress the boundary effects at both ends of the record, the first 1000 points of the oil pulsating-flow dataset are chosen for data-extension padding. A MATLAB program is created to symmetrically extend the data at both ends, adding 12 data points before and after the pressure signal. The pressure signal is then decomposed into six layers using the Morlet wavelet, yielding wavelet coefficients for each layer. These coefficients are truncated to match the sampled-data length, yielding the wavelet-analysis coefficients of the oil pulsating-flow pressure signal.

3 Results and analysis

From the six-layer Morlet wavelet decomposition, 64 groups of wavelet coefficients are obtained. The modulus, modulus squared, and variance of these coefficients represent the energy density distribution, energy spectrum, and pulsating energy decomposition of the oil pulsating flow at various scales. To analyze the multi-period variation characteristics of the oil pulsating flow over time and its trends at different scales, the modulus, modulus squared, and variance of the wavelet coefficients post-decomposition are examined.

3.1 The distribution of the real part of the wavelet coefficients along time on different scales

Figure 2 presents the real component of the Morlet-wavelet coefficients for the oil pulsating-flow signal across time and scale, revealing its inherent periodicity. A pronounced cyclic pattern is evident in the 35–64 scale band. Across the full time series, 33 maxima and 32 minima are identified; the largest positive coefficient is 0.14 and the most negative is −0.13, yielding an overall amplitude of 0.27. In the time series range of 287–612, the difference is smaller, varying from 0.032 to −0.001, with an amplitude of 0.031, indicating a relatively weak periodic variation trend in this segment of the time series.

As the wavelet scale shrinks, the real-part coefficient span tightens, signaling fewer regions of pronounced periodicity. With finer scales, frequency resolution drops while temporal resolution rises, allowing abrupt pressure-amplitude jumps in the pulsating flow to be pinpointed at precise instants. Conversely, at larger decomposition scales, as illustrated in Figure 2, the primary period of the pulsating flow is evident, while smaller scales, such as the second and third periods, begin to interact, making them less distinct than the first primary period.

Fig. 2 The distribution of the real part of the wavelet coefficient along time.

3.2 The distribution of the modulus of wavelet coefficients at different scales along time

Figure 3 presents the magnitude of the Morlet-wavelet coefficients across scales for the decomposed oil pulsating-flow signal. It shows that regions with high modulus values align with areas of significant difference between the maximum and minimum values of the small wave coefficients over time, as seen in Figure 2. This suggests that the modulus of the wavelet coefficients can represent the energy density distribution of the oil pulsating flow across different decomposition scales; larger modulus values indicate a more pronounced periodicity in the pulsating flow.

Wavelet analysis of the oil pulsating flow shows pronounced periodic signatures at scales 40–50, 20–30, and 10–15, where the modulus of the wavelet coefficients peaks. At finer scales, the maximum modulus declines monotonically along the time axis, progressively eroding the clarity of these periodic patterns.

Fig. 3 The distribution of the modulus of the wavelet coefficients along time.

3.3 The distribution of the modulus square of wavelet coefficients at different scales along time

Figure 4 presents the squared magnitude of the Morlet-wavelet coefficients for the oil pulsating flow, plotted across time and scale.

The squared magnitude of the wavelet coefficients maps the energy landscape of the oil-pulsation pressure signal in the time–scale plane. Energy peaks sharply at scales 40–50, exposing the dominant periodic component of the pulsation. Elsewhere, the energy appears as sparse, intermittent patches, confirming that periodicity waxes and wanes along the record. These scale-localized bursts and varying cycle strengths underscore the signal's intricate multi-scale, multi-level structure.

Fig. 4 The distribution of the modulus square of the wavelet coefficients along time at different scales.

3.4 The distribution of the variance of wavelet coefficients along the scale

Figure 5 presents the variance map of Morlet-wavelet coefficients for the oil pulsating flow across scales. The variance profile reveals how the pressure-signal energy is apportioned among decomposition scales, pinpointing the dominant scale and the pulsation's principal period.

Figure 5 shows the variance of the wavelet coefficients across scales; five distinct maxima appear at scales 6, 11, 20, 27, and 43. Scale 43 carries the largest variance, pinpointing the dominant periodic component of the pressure pulsation. The subsequent sub-dominant periods follow in descending order: 27, 20, 11, and finally 6. Furthermore, as shown in Table 2, the variance amplitude for the first main period is 8.6 × 10⁻³, which is 1.69, 1.71, 1.91, and 2.32 times greater than the amplitudes of the second to fifth main periods, with the differences between the second and fifth variances being relatively minor. This suggests that the pressure signal's vibration in the oil pulsating flow is primarily influenced by the first main period, along with interactions between this primary period and the others.

The variance analysis of wavelet coefficients allows for the determination of the distribution of wavelet coefficients for each primary period throughout the time series, as illustrated in Figure 6. Figures 6a–6e represent the distribution of wavelet coefficients for the first through fifth primary periods of the pressure signal vibrations from the oil pulsating flow over time.

Figure 6 reveals pronounced modulation in the oil pulsation pressure trace: the vibration frequency drops as the dominant period lengthens. Specifically, the trace in Figure 6a oscillates at 32.26 Hz—almost identical to the 30 Hz fundamental linked to the oil pump's 1800 rpm rotation. In Figure 6b, the frequency is 56.6973 Hz, approximately double that of the first primary period (the spindle speed frequency). The frequency in Figure 6c is 78.2031 Hz, which is 2.4 times that of the first primary period, and its wavelet coefficient distribution resembles that in Figure 6b, indicating that the wavelet decomposition scale for this primary period is influenced by noise pollution in the signal.

Figures 6d and 6e show the wavelet coefficient distributions for the fourth and fifth primary periods, with corresponding pressure signal frequencies of 130.0127 Hz and 264.9131 Hz, respectively, which are roughly four and eight times the frequency of the first primary period. This is related to the wavelet decomposition scale ranges for each primary period. Additionally, Figure 2 indicates that the wavelet decomposition scales for the first to fifth primary periods fall within a scale range that exhibits a clear periodic trend. For example, the decomposition scale for the first primary period is 43, which aligns with the scale range of 35–64 that shows significant periodicity in Figure 2. Similarly, the decomposition scales for the second through fifth primary periods correspond to the scale ranges of 20–35, 10–20, and 0–10, all of which demonstrate clear periodicity in Figure 2.

Furthermore, from Figures 6a–6e, it can be observed that as the decomposition scale for the primary period increases, the vibration frequency of the oil pulsating-flow pressure signal decreases exponentially, approximately in a 2:1 relationship as shown in Table 2. This trend aligns with the scale ranges exhibiting clear periodicity corresponding to the decomposition scales of each primary period. In summary, the scale ranges (6, 11, 20, and 43) with distinct periodicity related to the decomposition scales of the primary periods generally increase in a manner consistent with exponentially decreasing, approximately following a 2:1 ratio as shown in Table 2.

Fig. 5 The distribution of the variance of the wavelet coefficient at different scales.

Fig. 6 The distribution of the wavelet coefficients of each main period along time.

4 Conclusions

The pressure signal in oil pulsating flow, which is characterized by its multi-scale quasi-periodic nature, exhibits complex multi-layered and multi-scale vibrations. To analyze this signal, the Morlet wavelet function is utilized to decompose it into six distinct layers, resulting in corresponding wavelet coefficients. Subsequently, an in-depth analysis of the pressure signal's energy density, energy spectrum, and vibration energy distribution over time is performed using the real part of the wavelet coefficients, their modulus, the squares of the modulus values, and variance. The distribution of the real part of the wavelet coefficients indicates the periodic variations in the oil pulsating flow, with clear periodicity observed in the scale range of 35 to 64. Throughout the entire time series, there are 16 peaks and troughs in the wavelet coefficients, and the area exhibiting significant periodic changes diminishes as the wavelet decomposition scale decreases. The modulus of the wavelet coefficients reflects the energy density distribution of the oil pulsating flow over time at various decomposition scales; larger modulus values indicate more pronounced periodicity in the pulsating flow. The most intense vibration energy in the oil pulsating-flow is found within the decomposition scale range of 40–50, and the flow's temporal distribution is marked by locality and discontinuity. The pressure signal from the oil pulsating-flow shows five main periods and unique modulation features; as the decomposition scale linked to the primary period rises, the vibration frequency of the pressure signal exponentially decreasing, approximately following a 2:1 ratio. The derived five-period fingerprint and the synchronous correction protocol can be implemented on standard condition-monitoring systems, extending the impact beyond academic fluid dynamics. Crucially, this study demonstrates—within the low-speed square-duct—that the Morlet wavelet retains its ability to resolve coherent pulsation packets even when viscous sublayer effects dominate. These results offer important insights into the pressure signal of oil pulsating-flow, which is essential for comprehending oil pollution mechanisms and developing control strategies.

Funding

This study was funded by National Natural Science Foundation of China (51375516).

Conflicts of interest

Authors disclose that we have no a financial relationship with the organization that sponsored the research. We should also state that we have full control of all primary data and that agree to allow the journal to review their data if requested.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Author contribution statement

Liu Ge: Conceptualization, Methodology, Software, Formal analysis, Investigation, Data curation, Writing − original draft, Visualization. Chen Bin: Validation, Writing − review & editing, Supervision, Funding acquisition. Chen Xi: Validation, Resources, Data curation.

References

All Tables

All Figures

	Fig. 1 The schematic diagram of the experimental device of oil pulsation flow.
In the text

	Fig. 2 The distribution of the real part of the wavelet coefficient along time.
In the text

	Fig. 3 The distribution of the modulus of the wavelet coefficients along time.
In the text

	Fig. 4 The distribution of the modulus square of the wavelet coefficients along time at different scales.
In the text

	Fig. 5 The distribution of the variance of the wavelet coefficient at different scales.
In the text

	Fig. 6 The distribution of the wavelet coefficients of each main period along time.
In the text