Triple decomposition is a powerful analytical method for a deep understanding of the flow structure by extracting the mean value, organized coherent motion, and stochastic part from a fluctuating quantity. Here, we perform the triple decomposition of the spatial two-dimensional data, especially pressure-sensitive paint (PSP) data, since the PSP method is widely used to measure the pressure distribution on a surface in wind tunnel testing. However, the PSP data measuring near atmospheric pressure contain significant noise, and this makes it difficult to conduct the decomposition. To construct phase-averaged data representing an organized coherent motion, we propose a relatively simple method based on a multi-dimensional scaling plot of the cosine similarity between each PSP datum. Then, the stochastic part is extracted by selecting phase-averaged data with an appropriate phase angle based on the similarity between the measurement and phase-averaged data, and the PSP data are successfully decomposed. Moreover, we consider sparse optimal sensor positions, in which the data are effectively represented, based on the stochastic part as a data-driven approach. The optimal sensor positions are determined as a combinatorial optimization problem and estimated using Fujitsu computing as a service digital annealer. We reconstruct the pressure distribution from the pressure data at the optimal sensor positions using the mean value, organized coherent motion, and stochastic part obtained from the triple decomposition. The root mean square error between the pressure measured by a pressure transducer and the reconstructed pressure obtained by the proposed method is small, even when the number of modes and sensor points is small. The application of PSP measurement is expected to expand further, and the framework for calculating triple decomposition and sparse representation based on the decomposition will be useful for detailed flow analysis.

## I. INTRODUCTION

^{1}In the triple decomposition, a fluctuating quantity $fx,t$ at position vector $x$ and time $t$ is decomposed as

^{1,2}For example, a raw signal is divided into multiple phase angle ranges based on a reference signal, and the signals in each phase angle range are averaged to construct a phase-averaged signal.

^{2}However, reference signals are generally not always available. As a method that does not require a reference signal, an optimal frequency-domain filter deduced from power-spectral estimates is used to extract $f\u0303x,\varphi t$.

^{3}These methods are useful for data measured by a point measurement technique such as a hot-wire anemometer and a laser Doppler velocimeter (LDV). The proper orthogonal decomposition (POD) is used to extract a coherent structure for the analysis of spatial two-dimensional (2D) data measured by techniques such as particle image velocimetry (PIV)

^{4}and Schlieren method.

^{5}The extracted coherent structure reconstructed by truncated POD modes was similar to that obtained by the phase-averaged fields.

^{4}For example, the averaged pressure and acceleration structures are estimated from PIV data based on this method.

^{6}It has also been applied to the analysis of the coherent velocity fluctuation of swirling turbulent jets.

^{7}The triple decomposition based on the optimal mode decomposition (OMD)

^{8}is proposed to decompose a flow field consisting of large and small coherent structures.

^{9}The OMD method is a generalization of the dynamic mode decomposition (DMD) method

^{10}because the OMD method optimizes a subspace basis, whereas the DMD method uses the POD modes as a subspace basis. This method is applied to the analysis of the effect of free-stream turbulence on the near-field growth of the near wake of a cylinder.

^{11}

^{12–15}has been applied to various fields of aerodynamics such as aerospace engineering,

^{16–20}fluid machinery,

^{21–25}automotive engineering,

^{26–28}and micro-engineering.

^{29–31}The PSP method enables the measurement of the pressure on the surface to which the PSP coating is applied. The method utilizes oxygen quenching of photoluminescence, and the pressure is measured by the variation of the emission intensity of the PSP coating containing luminescent dye. Since the variation in the emission intensity is small for a small pressure variation, the detection of the small pressure variation is difficult due to the relatively large camera noise compared with the variation of the emission intensity. Therefore, the PSP data contain significant noise. For PSP data, by explicitly describing the measurement noise $nx,t$ at the position vector $x$ and time $t$, we can decompose pressure $px,t$ as

^{32–37}These methods enable the selection of the modes to reconstruct the pressure distribution with reduced noise. For example, the empirical selection of POD modes with small noise

^{32}or mode selection by sparse modeling have been proposed.

^{35,37}Although these denoising methods effectively remove the noise, to the best of our knowledge, there are no reports on the application of POP and DMD to the triple decomposition for PSP measurement data. This is because even the first few POD/DMD modes with large energy contributions contain the noise;

^{32,35,36}thus, it is difficult to separate a large-scale structure from the noise. Moreover, data representing a large flow structure reconstructed from these modes do not always coincide with phase-averaged data. As a phase-average method for the PSP method, the conditional image sampling (CIS) method is proposed;

^{38}the CIS method requires the acquisition of a reference signal. In our previous study, we proposed a phase-average method leveraging a time-series clustering method without using a reference signal.

^{39}

In this study, we conduct the triple decomposition for noisy PSP data. We estimate $p\u0303x,\varphi t$ using a simple phase-average method based on our previously proposed method.^{39} Then, the stochastic part $p\u2032x,t$ is extracted, and the noise $nx,t$ is removed using the POD analysis and sparse modeling. After conducting the triple decomposition, we estimate optimal sensor positions to effectively represent the distribution using Fujitsu computing as a service digital annealer.^{40,41} Sparse sensing is realized at optimal sensor positions, which are determined to be the solution to the optimal sensor placement problem.^{27,42–45} As a demonstration, we apply the proposed method to the PSP data measuring the pressure distribution of the Kármán vortex in our previous studies.^{46,47} The time-series pressure data are triple-decomposed and reconstructed as noise-suppressed data. The reconstructed pressure data are compared with those measured by a pressure transducer. This triple decomposition without using a reference signal is useful for analyzing flow field data with significant noise, such as the PSP data.

## II. PROPOSED METHOD FOR TRIPLE DECOMPOSITION

The pressure distribution $px,t$ measured by the PSP method can be decomposed into Eq. (3). First, the mean distribution for time is calculated. Second, the phase-averaged distribution is extracted as the coherent motion. Third, the stochastic part is obtained by calculating $px,t\u2212p\xafx\u2212p\u0303x,\varphi t$. Finally, the noise of the stochastic part data is reduced. The details of the proposed method are provided in this section.

### A. Calculation of mean distribution for time

### B. Estimation of coherent motion

^{39}The brief explanation of the method is provided in this paper. The cosine similarities between each data point are calculated, and the cosine similarity $cos\u2009\theta i,\u2009j$ is defined as

### C. Estimation of the stochastic part

*et al.*,

^{9}which focuses on flow velocity and not pressure, $p\u0303x,\varphi tn$ is estimated by the minimization problem as follows:

^{48–50}However, the estimation of $\alpha tn$ for all pixels of all measured images results in high computational cost. In our previous studies,

^{37,51}we proposed an estimation method for the pressure distribution $px,tn$ based on the selected optimal sensor points determined as the optimal sensor placement problem. Here, we propose the optimal points for detecting the stochastic part $p\u2032tn$. The same algorithm using Fujitsu computing as a service digital annealer

^{51}can be applied, except that the pressure distribution is replaced by the stochastic part $p\u2032tn$. We briefly introduce the algorithm for determining optimal points. More details are provided in our previous study.

^{51}The idea of the method is that the large variation of the POD mode at an optimal point well represents the characteristic of the distribution, and points with similar variation of the mode do not need to be selected. We consider the row vector $u\u0302i$ of $U\u0302\Sigma \u0302$, and $u\u0302i$ represents the variation of the mode at point $i$. The weight $wu\u0302i,u\u0302j$ between points $i$ and $j$ is defined as

^{37}

## III. RESULTS AND DISCUSSION

### A. Triple mode decomposition for PSP data

We applied the proposed method to the PSP data obtained in our previous study.^{46,47} The data were the pressure distribution induced by the Kármán vortex behind a square cylinder. The Reynolds number was $1.1\xd7105$. The time interval of $\Delta t$ was $1.0\u2009ms$. Before applying the proposed method, the PSP distributions were processed by a $5\xd75$ smoothing spatial filter. A typical example of the processed data are shown in Fig. 1. As shown in the figure, the noise was still large for further analysis.

Next, we calculated the cosine similarity for the 21 800 data points by Eq. (6). Since the data contained large amounts of noise, a truncated-SVD of rank five was used to reduce the noise. It is noted that these truncated-SVD data were only used for the calculation of the cosine similarity not for the following phase-averaging calculation. The MDS plot is shown in Fig. 2. The data on the circle with a radius of $1/2$ shown in the red dashed line represents a periodic phenomenon; thus, we obtained the phase-averaged data by dividing the data on the circle in 12 subsets ( $M=12$) and averaging each subset. Here, the number of the divisions per period $M=12$ was determined because the amount of data were not enough to reduce the noise by averaging for some phase range for larger $M$. The phase-averaged data are shown in Fig. 3, where the phase angle $\varphi m$ ( $m=1,\u20092,\u2009\u2026,\u200912$) is defined in Eq. (7). In this figure, the phase-averaged distributions of $p\xafx+p\u0303M=12x,\varphi m$ are presented for ease of visibility. The noise was significantly reduced by averaging.

^{52}The amplitudes for the first and third POD modes are shown in Fig. 5. The discrete amplitudes for phase angle were interpolated by following function:

^{51}Then, the stochastic part $p\u2032x,tn$ was extracted from $p\u0302x,tn$ by Eq. (22) as shown in Fig. 10. Finally, the noise reduced pressure distribution was obtained by calculating the sum $p\xafx+p\u0303x,\varphi tn+p\u2032x,tn$ as shown in Fig. 11, and the pressure distributions were successfully reconstructed by the method. The twin vortices aligned just behind both ends of the square cylinder as shown at $t6$ in Fig. 11 were sometimes observed. In such a case, the similarity was small for any $p\u0303Lx,\varphi l$ and the amplitude of $p\u0303Lx,\varphi l$ was also small. This correlation between the similarity and amplitude was clearly observed as shown in Fig. 12. This indicates that the contribution of $p\u0303Lx,\varphi l$ is small for the phenomena that deviate from the periodic coherent motion.

### B. Sparse sensing at optimal sensor points

As a data-driven approach, the pressure distribution can be reconstructed from the pressure data at the optimal sensor points estimated by the POD modes of $p\u0302x,tn$ based on triple decomposition. The mean value $p\xafx$, organized coherent motion $p\u0303Lx,\varphi l$, and optimal sensor positions $S$ were given/pre-calculated as a prior knowledge. The reconstruction of the pressure distribution was conducted by the following method. First, we calculated the cosine similarities between the pressure data at optimal sensor positions and $p\u0303LSx,\varphi l$ ( $l=1,\u20092,\u2009\u2026,\u2009L$) and selected $p\u0303Lx,\varphi lmax$ at the phase $\varphi lmax$ that gives the maximum similarity. Second, $p\u0302x,tn$ was calculated using $p\xafx$ and $p\u0303Lx,\varphi lmax$. Third, the SVD of $p\u0302x,tn$ was calculated and the amplitudes of the POD modes were determined following Eq. (22) to extract $p\u2032x,tn$ using Eq. (19) from the noisy data of $p\u0302x,tn$. Finally, the pressure distributions were obtained by $p\xafx+p\u0303Lx,\varphi lmax+p\u2032x,tn$.

The root mean square error (RMSE) between the pressure data measured by a pressure transducer through a pressure tap and the reconstructed data are shown in Fig. 13. The position of the pressure tap is shown in Fig. 13(a). For comparison, the RMSE results obtained by the method based on the previous study^{37,51} were also shown. The pressure distributions were reconstructed from the POD modes calculated from the observed data $px,t$, and the amplitudes of these POD modes were determined by LASSO at the optimal sensor positions that were determined by these POD modes in the previous method. That is, the difference between these methods is whether the sensor position is determined based on the POD modes of $p\u0302x,tn$. The horizontal axis in Fig. 13(b) represents the number of POD modes used to reconstruct the pressure distribution, and this was varied by varying $\lambda $ in LASSO. The RMSE was calculated for $128$ time steps, and the standard deviation of the number of modes used for each time step is shown in the figure. The results for the optimal sensor points of 25 and 60 are shown. The RMSE values were small for a small number of optimal sensor points and POD modes in the proposed method. This result indicates that triple decomposition enables a sparser representation.

## IV. CONCLUSIONS

We conducted the triple decomposition of noisy pressure-sensitive paint (PSP) data that are decomposed to the mean value, organized coherent motion, and stochastic part. We proposed a relatively simple method based on the multi-dimensional scaling (MDS) plot of the cosine similarity to construct phase-averaged data representing an organized coherent motion for periodic flows. Since the constructed phase-averaged data are typically discrete with respect to the phase angle direction, we produced the data with a sufficient phase angle step size by interpolating the data between the phase angles. Then, the stochastic part can be extracted by selecting phase-averaged data with an appropriate phase angle based on the similarity between the measurement data and phase-averaged data. The noise of the stochastic part was suppressed by superposing the proper orthogonal decomposition (POD) modes, whose amplitudes were determined by the least absolute shrinkage and selection operator (LASSO) at optimal sensor positions. We used Fujitsu computing as a service digital annealer for the estimation of optimal sensor positions based on the POD modes of the stochastic part. In this manner, the triple decomposition of the noisy PSP data was performed. The application of PSP measurements is expected to expand further, and triple decomposition will be very useful.

The pressure distribution was successfully reconstructed from the pressures at the optimal sensor positions using the phase-averaged data and the POD modes of the stochastic part. The reconstructed pressures agreed well with those measured by a pressure transducer, even when the number of sensors was small compared with that in the previous study. As a result, a sparser representation was achieved using the triple decomposition.

## ACKNOWLEDGMENTS

A part of this work was supported by the Collaborative Research Project (J23I041 and J24I064) and the Low Turbulence Wind Tunnel Facility of the Advance Flow Experimental Research Center at the Institute of Fluid Science, Tohoku University. The authors express their gratitude to Dr. Yasuhumi Konishi, Mr. Hiroyuki Okuizumi, and Mr. Yuya Yamazaki for their assistance during the wind tunnel testing. We also gratefully appreciate Tayca Corporation for providing the titanium dioxide.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors declare the following competing interest: Takahiro Kashikawa and Koichi Kimura are employees of Fujitsu Ltd. All other authors declare no competing interest.

### Author Contributions

**Koyo Kubota:** Conceptualization (supporting); Data curation (equal); Formal analysis (equal); Investigation (lead); Methodology (supporting); Software (lead); Validation (equal); Visualization (equal). **Makoto Takagi:** Investigation (supporting); Methodology (supporting); Software (supporting). **Tsubasa Ikami:** Data curation (supporting); Investigation (supporting); Resources (equal); Validation (supporting); Writing – review & editing (supporting). **Yasuhiro Egami:** Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Resources (equal); Writing – review & editing (supporting). **Hiroki Nagai:** Investigation (supporting); Resources (equal); Writing – review & editing (supporting). **Takahiro Kashikawa:** Methodology (supporting); Software (supporting); Validation (supporting). **Koichi Kimura:** Methodology (supporting); Software (supporting); Validation (supporting). **Yu Matsuda:** Conceptualization (lead); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (supporting); Methodology (lead); Project administration (lead); Resources (equal); Software (supporting); Supervision (lead); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (lead).

## DATA AVAILABILITY

The part of the flow measurement dataset is available in Zenodo at https://doi.org/10.5281/zenodo.10215642 (Ref. 47).

## REFERENCES

*Pressure and Temperature Sensitive Paints*

*In-Flight Application of Pressure Sensitive Paint*

*The Elements of Statistical Learning: Data Mining, Inference, and Prediction*

*Statistical Learning with Sparsity: The Lasso and Generalizations*