Application of Proper Orthogonal Decomposition in Spatiotemporal Characterization and Reduced-Order Modeling of Rotor–Stator Interaction Flow Field

Abstract

The periodic unsteady flow induced by rotor–stator interaction (RSI) is the primary cause of blade forced vibration and fatigue failure. Therefore, analyzing the excitation characteristics of RSI flow fields under multi-parameter conditions is essential for vibration analysis and optimization in fluid–structure interaction. This study derives the Toeplitz structure of the correlation matrix in proper orthogonal decomposition (POD) for strictly periodic flow fields and reveals that the POD spatial modes appear in pairs with a 90° spatial phase difference, which originates from the cosine and sine form of the eigenvectors of the Toeplitz matrix. Taking a 1.5-stage compressor cascade as an example, the POD method is employed to effectively extract the main spatiotemporal characteristics of the RSI flow field, and the spatial symmetry and phase difference of the POD modes are further interpreted from a physical perspective. To address the high computational cost and resource demands arising from large-scale similar cases in multi-parameter excitation optimization and analysis, a reduced-order modeling (ROM) method based on time–space and parameter decoupling is proposed using multi-parameter POD. Spatial bases are extracted through the first-level POD, and a second-level POD is applied to the first-level coefficients to obtain temporal bases and coefficients that are solely parameter-dependent. A radial basis function (RBF) interpolation is used to establish the mapping between parameters and the second-level coefficients, enabling efficient multi-parameter ROM construction. The resulting ROM achieves a relative prediction error of less than 1.4% under typical operating conditions and less than 2.9% near the choke boundary, improving computational efficiency by four orders of magnitude while maintaining accuracy, thereby providing an effective approach for aerodynamic excitation acquisition.

1. Introduction

Gas turbines operate based on the Brayton cycle, where axial compressors play a critical role in pressurizing the incoming air prior to combustion, as their performance directly impacts the efficiency, stability, and reliability of the entire engine system [1]. The growing demand for aero-engines with higher pressure ratios, increased speeds, and improved efficiency has driven the evolution of compressor design, resulting in more complex geometries. This evolution includes higher inlet flow speeds, increased tip speeds, progressively greater stage loads, and more compact designs. However, these advancements also intensify the transient effects between the rotor and stator [2]. Rotor–stator interaction [3], characterized by the periodic unsteady flow induced by the wake of the upstream blade row and the potential flow of the downstream blade row, is the primary cause of forced blade vibrations. In practice, high-cycle fatigue failure of blades caused by RSI vibration has become a key issue affecting the structural integrity and reliability of engines during their operational life cycle. Analyzing the unsteady excitation caused by RSI is essential for the design and optimization of compressors.

In early studies on RSI flow field excitation, Mailach et al. [4] experimentally measured the pressure distributions on both the pressure and suction surfaces of the blade. They analyzed the influence of the upstream blade wake and the downstream blade potential flow on the unsteady aerodynamic forces and performed a spectral analysis of the time-domain history of the blade surface lift. They found that the primary frequency components of unsteady excitation caused by RSI are harmonics of the blade passing frequency. Monk et al. [5] demonstrated the effectiveness of stator asymmetric designs in reducing aerodynamic loads by performing a spectral analysis of unsteady aerodynamic forces in a three-stage compressor. They found that this asymmetric design reduces the amplitude of all harmonics. Li et al. [6] systematically studied the impact of the rotor–stator gap on aerodynamic damping. Through spectral analysis of the unsteady aerodynamic forces on the blade surface, they demonstrated the existence of an optimal gap for flutter stability and forced response stresses. The aforementioned studies primarily obtained local information by analyzing the spectral components of unsteady excitation at specific points in the flow field, as well as macroscopic information through lift and torque. With advancements in compressor design, researchers have developed new methods to characterize unsteady excitation caused by rotor–stator interactions. Fruth et al. [7] calculated the harmonic components of aerodynamic forces using Fourier transformation and defined a normalized “stimulus” metric to quantify the intensity of unsteady excitation, which was used to evaluate the differences in unsteady aerodynamic excitation under varying clocking positions. Mailach et al. [8] introduced dimensionless forms of blade lift and torque to quantitatively analyze the aerodynamic forces and dynamic responses induced by RSI flow fields, thereby revealing the physical mechanisms of blade row interactions. More recently, Akhlaghi et al. [9] performed spectral analysis on unsteady pressure signals in a low-speed axial-flow compressor with vaned-recessed casing treatment. They found that the dominant frequencies align with the blade passing frequency and its harmonics, while additional low-frequency components emerged near stall, indicating intensified unsteady behavior and broader frequency content in the tip region.

As compressor flow-induced vibration issues become increasingly significant, analyzing unsteady flow field excitation based on lift and torque at a single point or on individual blades is no longer sufficient to meet the demands. Therefore, there is an urgent need to develop new methods for evaluating unsteady flow field excitation. Modal decomposition techniques, which are based on feature extraction, provide a novel approach to address this need. Modal decomposition of unsteady flow field excitation enables decoupling in both time and space. The spatial distribution of the excitation can be intuitively derived from the resulting modes, whereas the corresponding coefficients contain the temporal distribution information of the excitation components [10]. Common modal decomposition methods used in the compressor field include proper orthogonal decomposition [11,12], dynamic mode decomposition (DMD) [13,14], and spectral proper orthogonal decomposition (SPOD) [15]. Among these, the POD method is the most widely applied due to its high computational efficiency and strong physical interpretability [16]. Cizmas et al. [17] used the RSI flow field of a single-stage turbine as an example to validate the effectiveness of the POD method in capturing the main excitation features of the RSI flow field. Lei et al. [18] performed a POD analysis on the unsteady flow in the blade wake affected by laminar vortex shedding in the compressor blade row and further investigated the relationship between the blade wake and laminar vortex shedding in the RSI flow field, quantifying the contributions of wake instability, laminar vortex shedding, and separated shear layer flapping to the overall flow loss. Qiao et al. [19] analyzed the POD modes of the T106 turbine blade row and revealed the specific effects of the upstream wake and leakage flow on turbine performance. The study showed that the periodic wake significantly reduces the energy loss within the passage, while the leakage flow has a greater impact in the vicinity of the casing. Wang et al. [20] applied POD to the flow field of rotating stall in an axial compressor, successfully capturing the main excitation features at the blade tip during the stall process, including the pre-stall, stall, and stable stall stages. This approach provides a more direct and accurate explanation of the rotating stall process.

Currently, developed methods for characterizing flow field excitation are all based on high-fidelity unsteady flow fields obtained through computational fluid dynamics (CFD) numerical simulations. However, with the development of variable cycle technology [21], the frequent switching of compressor operating modes has led to a sharp increase in the complexity of flow field calculations. Large-scale computations of similar samples under varying parameters are required, and the computational cost of such calculations is unaffordable for engineering applications [22]. Therefore, the efficient analysis of unsteady flow fields under multi-parameter conditions remains an urgent problem to solve. It is an effective approach to construct reduced-order models by combining feature extraction techniques such as POD with data mining methods like radial basis functions [23], Gaussian regression [24], and physics-informed neural networks [25]. By replacing full-order models with ROMs to obtain sample data, computational costs can be significantly reduced. This method has been widely applied in aerospace [26], biomedical [27], energy systems [28], and other fields, where its efficacy has been well demonstrated.

ROMs can be classified into intrusive and non-intrusive methods based on the reduction process. Intrusive methods rely on direct access to and modification of the high-fidelity numerical model code [29]. These methods reduce computational costs by introducing reduction techniques into the numerical solution process. However, they are complex to develop and less practical for engineering applications. In contrast, non-intrusive methods do not require modification of the original model and instead rely on snapshots of data generated from high-fidelity models for reduced-order modeling [30]. Given the demand for efficiency and compatibility in engineering applications, non-intrusive methods are more suitable due to their broad applicability [31]. Xiao et al. [32] developed a non-intrusive ROM by combining two-level radial basis function (RBF) interpolation with POD modes. Using a multiphase porous media flow example, they demonstrated that the model significantly improved computational efficiency while providing accurate solutions and analyzed the sources of error in the method [33]. Min et al. [34] trained POD mode coefficients using RBF neural networks and achieved rapid flow field prediction by linearly combining the predicted POD mode coefficients at non-sample points with the POD mode bases. They compared this approach with a backpropagation neural network and demonstrated the superior efficiency and accuracy of RBF networks. In the compressor field, Luo et al. [35] used a non-intrusive ROM combined with POD to perform aerodynamic optimization design on the last 4.5 stages of a compressor. By altering the camber and stagger angles along the span, they successfully improved adiabatic efficiency while maintaining the mass flow rate.

ROMs in the compressor field are primarily focused on stability analysis and aerodynamic performance prediction, involving multi-parameter steady flow fields and single-parameter transient flow fields. For multi-parameter transient flow fields, the snapshot matrix composed of numerous time steps results in prohibitively high computational costs. The design philosophy of variable cycle technology in aero-engines poses new challenges for the rapid acquisition of transient flow fields in compressors under multiple parameters. To address this issue, this paper constructs a multi-parameter ROM based on snapshot POD, utilizing snapshot POD to reduce computational load. Through two-stage POD, the temporal, spatial, and parametric information are completely decoupled. The first POD separates the spatial bases and its coefficients, where the coefficients contain both parameter and time information. After processing these coefficients, a second POD is applied to isolate the temporal bases, leaving the coefficients with only parameter information. RBF interpolation is then used to establish the relationship between the parameters and the coefficients from the second POD. For new parameters, the unsteady flow field is predicted by combining the predicted second POD coefficients with the temporal and spatial bases.

There are four more sections in this paper. Section 2 introduces the theoretical foundation, including the derivation of spatial symmetry and phase difference of POD modes in periodic flow fields, and the methodology and workflow for constructing a multi-parameter ROM based on multi-parameter POD (MPOD) combined with RBF interpolation. Section 3 presents the model and numerical computation methods used in this study. Section 4 is devoted to the results and discussion, analyzing the spatiotemporal distribution characteristics of the RSI flow field excitation in a 1.5-stage compressor cascade, and establishing the ROM for the RSI flow field based on three parameters: the angle of the variable inlet guide vane (VIGV), the angle of the variable stator vane (VSV), and the outlet pressure. The accuracy and effectiveness of the MPOD-RBF (MPR) ROM method are also validated. Finally, Section 5 summarizes the conclusions drawn from this study.

2. Theoretical Foundations

This section presents the theoretical foundations adopted in this study. The POD method is first introduced to extract the dominant spatiotemporal features of the unsteady RSI flow field, with emphasis on revealing the mathematical essence of POD modes. Subsequently, a high-efficiency ROM framework is constructed based on a multi-parameter POD approach, enabling the decoupling of spatial, temporal, and parametric couplings. This MPR-ROM method allows for accurate flow field prediction across a wide range of operating conditions. The combination of POD and ROM provides an effective approach for the spatiotemporal characterization of unsteady flow excitations.

2.1. Theory of Spatiotemporal Characterization Analysis for Flow Fields

2.1.1. Snapshot POD

POD is a data-driven dimensionality reduction technique designed to extract dominant spatiotemporal coherent structures, referred to as modes, from high-dimensional complex flow fields. It constructs low-dimensional representations to capture the primary dynamic features of the flow [36]. The core principle of POD lies in identifying an optimal set of orthogonal basis functions that retain the maximum possible flow field energy or variance with the fewest modes. This method is widely applied in fluid mechanics for unsteady flow analysis and excitation feature extraction, particularly in scenarios requiring efficient characterization of flow-induced vibrations and dynamic loading.

Taking a physical quantity at each grid node of the unsteady flow field as an example, the physical time step is defined as $\Delta t$. The physical quantities of all nodes at the i-th discrete time step are recorded as a snapshot ${\mathit{u}}_{\mathit{i}}$. By collecting snapshots from n time steps and subtracting the time-average mean, the fluctuating snapshot matrix is formed as

$$\mathit{U}(x,t)=\left[\begin{array}{cccc}{\mathit{u}}_{\mathbf{1}}& {\mathit{u}}_{\mathbf{2}}& \cdots & {\mathit{u}}_{\mathit{n}}\end{array}\right].$$

(1)

The snapshot matrix $\mathit{U}(x,t)$ contains both spatial and temporal information of the flow field. It is assumed that the spatial information in the matrix can be projected onto a space formed by a set of complete orthonormal bases:

$$\mathbf{\Phi }=\left[\begin{array}{c}{\mathit{\phi }}_{\mathbf{1}}\left(x\right),{\mathit{\phi }}_{\mathbf{2}}\left(x\right),\cdots ,{\mathit{\phi }}_{\mathit{n}}\left(x\right)\end{array}\right].$$

(2)

The fluctuating snapshot matrix is decomposed into a superposition of the product of the spatial bases and their corresponding coefficients:

$$\mathit{U}(x,t)=\sum _{i=1}^n{\mathit{\phi }}_{\mathit{i}}\left(x\right){\mathit{a}}_{\mathit{i}}\left(t\right),$$

(3)

where ${\mathit{\phi }}_{\mathit{i}}\left(x\right)$ represents the spatial basis vector, that is, the i-th POD spatial mode, and ${\mathit{a}}_{\mathit{i}}\left(t\right)$ denotes the mode coefficient corresponding to the i-th POD spatial mode.

The snapshot-based POD extracts modes by solving the eigenvalue problem of the spatial correlation matrix $\mathit{R}$, which is defined as

$$\mathit{R}={\displaystyle \frac{1}{n}}\mathit{U}{\mathit{U}}^{\mathrm{T}},$$

(4)

$$\mathit{R}{\mathit{\phi }}_{\mathit{i}}\left(x\right)={\sigma }_i{\mathit{\phi }}_{\mathit{i}}\left(x\right),$$

(5)

where $N_s$ i the number of grid nodes; the corresponding mode coefficients are expressed as

$${\mathit{a}}_{\mathit{i}}\left(t\right)={{\mathit{\phi }}_{\mathit{i}}}^{\mathrm{T}}\mathit{U}.$$

(6)

The eigenvalue ${\sigma }_i$ represents the contribution of the corresponding spatial mode to the original flow field. The sum of the eigenvalues represents the total energy of the flow field [37]. The larger the eigenvalue, the more prominent the corresponding spatial mode is in the original flow field. This leads to the concept of mode energy proportion to evaluate the impact of truncating spatial modes on the original flow field as

$$E={\displaystyle \frac{{\sum }_{i=1}^N{\sigma }_i}{{\sum }_{i=1}^n{\sigma }_i}},$$

(7)

where N represents the truncated mode order, and E denotes the proportion of truncated mode energy. The closer E is to 1, the higher the degree to which the truncated modes restore the dynamic characteristics of the flow field. This is used to characterize the error caused by mode truncation.

2.1.2. Spatial Symmetry of POD Modes

In rotating machinery flow fields, the flow structures typically exhibit spatial periodicity in the circumferential direction, meaning that highly correlated flow patterns may exist at different spatial locations, such as between different blades. Assuming that the flow field strictly conforms to spatial periodicity,

$$\mathit{U}(x+p,t)=\mathit{U}(x,t),$$

(8)

within a single time snapshot:

$$\mathit{u}(x+p)=\mathit{u}\left(x\right),$$

(9)

where p is the spatial periodic length, expressed as a multiple of the blade pitch.

Expanding the elements of the spatial correlation matrix, we obtain

$$R_{ij}={\displaystyle \frac{1}{N_s}}\sum _{k=1}^n{\mathit{u}}_{\mathbf{k}}\left(x_i\right){\mathit{u}}_{\mathbf{k}}\left(x_j\right),$$

(10)

where $N_s$ is the number of grid nodes, and $R_{ij}$ represents the statistical correlation of the flow field between positions $x_i$ and $x_j$. Due to the spatial periodicity of $u\left(x\right)$, $\mathit{R}$ also exhibits corresponding periodic symmetry:

$$\mathit{R}(x_i+p,x_j+p)=\mathit{R}(x_i,x_j).$$

(11)

Fixing $x_i$, the eigenvalue problem in Equation (5) is transformed as

$$\sum _{j=1}^{N_s}\mathit{R}(x_i+p,x_j+p){\mathit{\phi }}_{\mathit{i}}(x_j+p)={\sigma }_i{\mathit{\phi }}_{\mathit{i}}(x_i+p),$$

(12)

and due to the periodicity of $\mathit{R}$, this equation can be rewritten as

$$\sum _{j=1}^{N_s}\mathit{R}(x_i,x_j){\mathit{\phi }}_{\mathit{i}}(x_j+p)={\sigma }_i{\mathit{\phi }}_{\mathit{i}}(x_i+p),$$

(13)

compared with the original eigenvalue problem:

$$\sum _{j=1}^{N_s}\mathit{R}(x_i,x_j){\mathit{\phi }}_{\mathit{i}}\left(x_j\right)={\sigma }_i{\mathit{\phi }}_{\mathit{i}}\left(x_i\right).$$

(14)

Thus, it can be concluded that if the correlation matrix $\mathit{R}(x_i,x_j)$ exhibits periodicity, the corresponding eigenvectors should satisfy the following form:

$${\phi }_i(x+p)=c{\phi }_i\left(x\right),$$

(15)

where c is an arbitrary constant.

Furthermore, according to Equations (4) and (11), the correlation matrix $\mathit{R}$ is a symmetric and positive semi-definite matrix with translational invariance in space. That is, it satisfies

$$R(x_i,x_j)=R\left(\right|x_i-x_j\left|\right).$$

(16)

This indicates that the elements of $\mathit{R}$ depend only on the relative distance between $x_i$ and $x_j$, rather than their absolute positions. As a result, the correlation matrix $\mathit{R}$ possesses the structure of a symmetric Toeplitz matrix. For such matrices, the eigenvectors typically take the form of exponential functions:

$${\phi }_i\left(x\right)=e^{i{k}_{i}x},$$

(17)

where $k_i$ is the Fourier wavenumber. The complex exponential function can be expanded as

$$e^{i{k}_{i}x}=cos\left(k_{i}x\right)+isin\left(k_{i}x\right).$$

(18)

Therefore, the real-valued POD modes used in this study can be expressed as a linear combination of orthogonal trigonometric functions:

$${\phi }_i\left(x\right)=cos\left(k_{i}x\right),{\phi }_{i+1}\left(x\right)=sin\left(k_{i}x\right).$$

(19)

This reveals that POD modes in periodic flow fields often appear in pairs, with a phase difference of 90° in space.

Based on this, the constant c in Equation (15) can be further determined:

$$c=\pm 1.$$

(20)

To further determine the value of c, we incorporate physical considerations and assume that the total circumferential length L of the compressor is an integer multiple of the fundamental periodic length:

$$L=Np,$$

(21)

where N is the number of periods. After a shift of N full periods, corresponding to a full circumferential rotation, the following relation should be satisfied:

$${\phi }_i(x+Np)=c^N{\phi }_i\left(x\right).$$

(22)

Given that a compressor is a closed structure, both the flow field and the modal structures must return to their original states after one full circumferential cycle. Thus, the following condition must hold:

$${\phi }_i(x+Np)={\phi }_i\left(x\right),$$

(23)

which leads to

$$c^N=1.$$

(24)

Here, N can be either an odd or even integer. However, when N is odd and $c=-1$, Equation (24) is clearly violated since the mode would reverse its sign after one full rotation, contradicting the assumption of full circumferential periodicity. Therefore, the only valid solution for c is

$$c=1.$$

(25)

Equation (15) can thus be rewritten as

$${\mathit{\phi }}_{\mathit{i}}(x+p)={\mathit{\phi }}_{\mathit{i}}\left(x\right),$$

(26)

This indicates that the POD modes inherit the spatial periodicity of the snapshot matrix constructed from flow fields with inherent spatial periodic structures.

To further derive the specific form of the Fourier wavenumber $k_i$, substituting Equation (17) into (26) yields

$$e^{i{k}_i(x+p)}=e^{i{k}_{i}x},$$

(27)

which implies

$$e^{i{k}_{i}p}=1.$$

(28)

Therefore,

$$k_{i}p=2\pi i,i\in \mathbb{Z},$$

(29)

and the wavenumber can be expressed as

$$k_i={\displaystyle \frac{2\pi i}{p}}.$$

(30)

Similarly, when constructing the correlation matrix in Equation (4) as a time correlation matrix, the symmetry and phase difference in the time domain can also be derived. This imposes specific requirements on the form of the snapshot matrix. For constructing a snapshot matrix of a rotating machinery flow field, which exhibits periodicity in both time and space, a complete snapshot matrix must contain at least one full period of both temporal and spatial data.

2.2. Theory of MPR-ROM

2.2.1. Multi-Parameter POD

The physical quantity ${\mathit{u}}_{\mathit{i}}^{\mathbf{j}}$ of all nodes at the i-th discrete time step of the j-th sample is recorded as a snapshot. The multi-parameter fluctuating snapshot matrix of m samples and n time steps is given as

$$\mathit{U}(x,t,\alpha )=\left[{\mathit{u}}_1^1\cdots {\mathit{u}}_n^1,\cdots ,{\mathit{u}}_1^m\cdots {\mathit{u}}_n^m\right]=\left[\mathit{U}(x,t,{\alpha }_1),\mathit{U}(x,t,{\alpha }_2),\cdots ,\mathit{U}(x,t,{\alpha }_m)\right].$$

(31)

The first POD is performed on the multi-parameter snapshot matrix to extract the spatial bases. By retaining the first I modes based on the value of E in Equation (7), the following is obtained as

$$\mathit{U}(x,t,\alpha )=\sum _{i=1}^I{\mathit{a}}_{\mathit{i}}(t,\alpha ){\mathit{\phi }}_{\mathit{i}}\left(x\right)=\mathit{A}\mathbf{\Phi },$$

(32)

where $\mathit{A}$ is the first POD mode coefficient matrix, which contains temporal and sample parameter information. It represents the response of the spatial modes to changes in time and parameters. The expansion of the i-th order coefficient is given by

$$\mathit{A}(i,:)=[{\mathit{a}}_{\mathit{i}}(t_1:t_n,{\alpha }_1)\mid {\mathit{a}}_{\mathit{i}}(t_1:t_n,{\alpha }_2)\mid \cdots \mid {\mathit{a}}_{\mathit{i}}(t_1:t_n,{\alpha }_m)],$$

(33)

the coefficient contains temporal information across all parameters. By reorganizing it along both the parameter and time dimensions, the result is obtained as

$${\mathit{M}}_{\mathit{i}}=[{\mathit{a}}_{\mathit{i}}{(t_1:t_n,{\alpha }_1)}^{\mathrm{T}}\mid {\mathit{a}}_{\mathit{i}}{(t_1:t_n,{\alpha }_2)}^{\mathrm{T}}\mid \cdots \mid {\mathit{a}}_{\mathit{i}}{(t_1:t_n,{\alpha }_m)}^{\mathrm{T}}]\subset {\mathbb{R}}^{n\times m}.$$

(34)

In the reorganized coefficient matrix, the column direction contains only the temporal information of a single sample, while the row direction represents the parameter information of the samples. Each column of the coefficient matrix $\mathit{A}$ obtained from the first POD is reorganized and concatenated along the column direction to form the coefficient matrix $\mathit{M}$. By performing the second POD on matrix $\mathit{M}$ and retaining the first J modes, the following is obtained:

$$\mathit{M}(\alpha ,t)=\sum _{i=1}^J{\mathit{c}}_{\mathit{i}}\left(\alpha \right){\mathit{\psi }}_{\mathit{i}}\left(t\right)=\mathit{C}\mathbf{\Psi },$$

(35)

where $\mathit{C}$ is the coefficient matrix related only to the sample parameters, and $\mathbf{\Psi }$ is the matrix composed of the temporal bases. At this point, the temporal, spatial, and parameter information has been successfully decoupled through the two POD operations.

During the dimensionality reduction of the spatial and temporal domains, mode truncation is performed. The error caused by mode truncation is still characterized in the form of energy proportion. The first POD retains the first I spatial modes, while the second POD retains the first J temporal modes. The relative error caused by the truncation of the first POD spatial modes is defined as

$$e_1=1-{\displaystyle \frac{{\sum }_{i=1}^I{\sigma }_i}{{\sum }_{i=1}^{m\times n}{\sigma }_i}}={\displaystyle \frac{{\sum }_{i=I+1}^{m\times n}{\sigma }_i}{{\sum }_{i=1}^{m\times n}{\sigma }_i}},$$

(36)

and the relative error caused by the truncation of the second POD temporal modes is defined as

$$e_2=1-{\displaystyle \frac{{\sum }_{j=1}^J{\sigma }_{ij}}{{\sum }_{j=1}^I{\sigma }_{ij}}}={\displaystyle \frac{{\sum }_{j=J+1}^I{\sigma }_{ij}}{{\sum }_{j=1}^J{\sigma }_{ij}}}.$$

(37)

The total relative error caused by the mode truncation is defined as

$$e=1-\sum _{i=1}^I\left({\displaystyle \frac{{\sigma }_i}{{\sum }_{k=1}^{m\times n}{\sigma }_k}}\times {\displaystyle \frac{{\sum }_{j=1}^J{\sigma }_{ij}}{{\sum }_{j=1}^I{\sigma }_{ij}}}\right).$$

(38)

2.2.2. RBF Interpolation

Radial basis function interpolation approximates a function based on the distances between data points. The target function is expressed as a weighted sum of several basis functions, where the center of each basis function is determined by the location of the data points. This multidimensional data interpolation method is particularly suitable for irregular data and datasets containing multiple parameters.

For a set of discrete training data points $\left\{\left({\theta }_i,{\beta }_i,P_i,c_i\right)\right\}$, where ${\theta }_i$, ${\beta }_i$, and $P_i$ are parameter inputs, and $c_i$ is the target value associated with these three parameters, that is, the coefficient matrix obtained after the second POD. In multi-parameter RBF interpolation, the form of the interpolation function can be written as

$$f({\theta }_i,{\beta }_i,P_i)=\sum _{i=1}^m{\lambda }_i\phi \left(\parallel (\theta ,\beta ,P)-({\theta }_i,{\beta }_i,P_i)\parallel \right),$$

(39)

where $\phi (\parallel (\theta ,\beta ,P)-({\theta }_i,{\beta }_i,P_i)\parallel )$ is the radial basis function, and $∥(\theta ,\beta ,P)-({\theta }_i,{\beta }_i,P_i)∥$ is the Euclidean distance between the target parameters and the sample parameters. Below, the specific forms of several common radial basis functions are provided.

Gaussian RBF:

$$\phi \left(∥(\theta ,\beta ,P)-({\theta }_i,{\beta }_i,P_i)∥\right)=e^{-\epsilon {∥(\theta ,\beta ,P)-({\theta }_i,{\beta }_i,P_i)∥}^2},$$

(40)

where $\epsilon$ is a constant that controls the smoothness of the interpolation. A smaller $\epsilon$ makes the interpolation result more localized, while a larger $\epsilon$ makes the interpolation function smoother. The Gaussian RBF exhibits excellent smoothness in response to changes in input data and is suitable for most nonlinear problems.

Polynomial RBF:

$$\phi \left(∥(\theta ,\beta ,P)-({\theta }_i,{\beta }_i,P_i)∥\right)={\left(∥(\theta ,\beta ,P)-({\theta }_i,{\beta }_i,P_i)∥\right)}^k,$$

(41)

where k is the order of the polynomial, which adjusts the smoothness of the interpolation function over the parameter domain. The polynomial RBF is often used to fit data with relatively smooth variations. By adjusting the polynomial order, it can also adapt to more complex nonlinear problems.

Inverse polynomial RBF:

$$\phi \left(∥(\theta ,\beta ,P)-({\theta }_i,{\beta }_i,P_i)∥\right)={\left({∥(\theta ,\beta ,P)-({\theta }_i,{\beta }_i,P_i)∥}^2+r^2\right)}^{-\beta },$$

(42)

where r is a constant that controls the smoothness of the function, and $\beta$ is the exponent of the polynomial, used to control the strength of the interpolation, typically ranging between 0 and 2. The inverse polynomial RBF generally assigns higher interpolation weights when the distances between data points are larger. This characteristic provides an advantage when dealing with systems involving long-range interactions or strong nonlinearities.

After selecting a specific form of the interpolation function and given m training data points, the solution process for RBF interpolation is based on a system of linear equations:

$$\mathit{D}\mathit{\lambda }=\mathit{C},$$

(43)

where $D_{ij}=\phi \left(∥({\theta }_i,{\beta }_i,P_i)-({\theta }_j,{\beta }_j,P_j)∥\right)$, $\mathit{\lambda }={\left[{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_m\right]}^T$, $\mathit{C}={\left[\begin{array}{c}{c}_1,c_2,\cdots ,c_m\end{array}\right]}^T$. The solution to this system of linear equations provides the weight coefficients ${\lambda }_i$ for each interpolation point. By solving this system, the specific form of the RBF interpolation function is obtained. For new parameters outside the sample points, the coefficients of the second POD can be directly obtained using this interpolation function.

2.2.3. Workflow of MPR-ROM

The workflow of the multi-parameter POD-based reduced-order model consists of two phases, offline and online.

Offline phase:

(1)

A certain number of samples from the target parameter space are generated using Latin Hypercube Sampling (LHS), and their high-fidelity unsteady flow fields are obtained through CFD methods.

(2)

The snapshot matrix for each sample is constructed and the multi-parameter sample snapshot matrix $\mathit{U}(x,t,\alpha )$ is assembled according to Equation (31).

(3)

The first POD is performed on $\mathit{U}(x,t,\alpha )$ to extract the spatial bases $\mathbf{\Phi }\left(x\right)$ and the first POD coefficients $\mathit{A}(t,\alpha )$.

(4)

The first I spatial bases are retained, the first I rows of $\mathit{A}(t,\alpha )$ are extracted, and they are all reorganized along the parameter and time dimensions according to Equation (34) to form the coefficient matrix $\mathit{M}(t,\alpha )$.

(5)

The second POD is performed on $\mathit{M}(t,\alpha )$ to extract the temporal bases $\mathbf{\Psi }\left(t\right)$ and the parameter coefficient matrix $\mathit{C}\left(\alpha \right)$.

(6)

The first J temporal bases are retained and RBF interpolation is used to construct the mapping relationship between the parameters and the parameter coefficients.

Online phase:

(1)

By taking the parameters of the new sample points as input, the parameter coefficients ${\mathit{C}}^{*}\left({\alpha }^{*}\right)$ are computed using the interpolation function constructed during the offline phase.

(2)

The parameter coefficients ${\mathit{C}}^{*}\left({\alpha }^{*}\right)$ are combined with the first J temporal bases of $\mathbf{\Psi }\left(t\right)$ to compute the matrix ${\mathit{M}}^{*}(t,{\alpha }^{*})$.

(3)

${\mathit{M}}^{*}(t,{\alpha }^{*})$ is reorganized and the coefficients ${\mathit{A}}^{*}(t,{\alpha }^{*})$ are obtained according to Equation (33).

(4)

The coefficients ${\mathit{A}}^{*}(t,{\alpha }^{*})$ are combined with the first I spatial bases of $\mathbf{\Phi }\left(x\right)$ to compute the predicted flow field ${\mathit{U}}^{*}(x,t,{\alpha }^{*})$.

A flowchart of the MPR-ROM workflow is presented in Figure 1.

3. Numerical Methods and Computational Setup

3.1. Governing Equations

The Unsteady Reynolds-Averaged Navier–Stokes (URANS) method is a widely used numerical approach for solving complex turbulent flows, particularly suitable for simulating unsteady flow fields with temporal variations. The URANS equations are derived based on the Reynolds-averaged assumption, in which the instantaneous velocity components in a turbulent flow are decomposed into a time-averaged component and a fluctuating component. By averaging the velocity and pressure fields over time, the URANS method reduces computational cost while still capturing the essential features of turbulent flows, making it appropriate for cases where fully resolving turbulent fluctuations is not required [38].

The URANS equations consist of the continuity equation and the momentum equations. In the momentum equations, the Reynolds stress terms arise due to the averaging process and must be modeled using an appropriate turbulence closure model. In this study, the simulation focuses on a two-dimensional rotor–stator interaction flow within a compressor cascade. The two-dimensional form of the continuity equation is given as follows [39]:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \overline{u}}{\partial x}}+{\displaystyle \frac{\partial \overline{v}}{\partial y}}=0\end{array}.$$

(44)

The two-dimensional momentum equations in the x- and y-directions are given as follows [39]:

$${\displaystyle \frac{\partial \overline{u}}{\partial t}}+\overline{u}{\displaystyle \frac{\partial \overline{u}}{\partial x}}+\overline{v}{\displaystyle \frac{\partial \overline{u}}{\partial y}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \overline{p}}{\partial x}}+\nu \left({\displaystyle \frac{{\partial }^2\overline{u}}{\partial x^2}}+{\displaystyle \frac{{\partial }^2\overline{u}}{\partial y^2}}\right)-{\displaystyle \frac{\partial \overline{u_i^{\prime }{u}_j^{\prime }}}{\partial x}},$$

(45)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \overline{v}}{\partial t}}+\overline{u}{\displaystyle \frac{\partial \overline{v}}{\partial x}}+\overline{v}{\displaystyle \frac{\partial \overline{v}}{\partial y}}=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \overline{p}}{\partial y}}+\nu \left({\displaystyle \frac{{\partial }^2\overline{v}}{\partial x^2}}+{\displaystyle \frac{{\partial }^2\overline{v}}{\partial y^2}}\right)-{\displaystyle \frac{\partial \overline{v_i^{\prime }{v}_j^{\prime }}}{\partial y}}\end{array},$$

(46)

where $\overline{u}$ and $\overline{v}$ are the time-averaged velocity components in the x- and y-directions, respectively; $\nu$ is the kinematic viscosity; $\overline{u_i^{\prime }{u}_j^{\prime }}$ represents the Reynolds stress term; $\overline{p}$ is the pressure; and $\rho$ is the fluid density.

3.2. Turbulence Model

When simulating complex flows, especially those involving boundary layer separation and laminar-to-turbulent transition as in the case of compressor cascades with RSI, the shear stress transport (SST) $k-\omega$ turbulence model is considered an ideal choice due to its favorable physical accuracy and computational efficiency. This turbulence model is designed to combine the advantages of the $k-\omega$ and $k-\epsilon$ models. It applies the $k-\omega$ formulation in the near-wall region and transitions to the $k-\epsilon$ formulation in the free-shear-flow region, allowing accurate representation of turbulent behavior near solid boundaries. It is well suited for simulating flow fields in turbomachinery. The governing equations of the model consist of the transport equations for turbulent kinetic energy k and specific dissipation rate $\omega$, which, respectively, characterize the two key properties of turbulence [40].

The turbulent kinetic energy k equation is used to describe the energy of turbulence, representing the amount of kinetic energy carried by the fluid due to turbulent motion. It is primarily used to evaluate the turbulence intensity. The equation is expressed as follows [39]:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+\nabla ·\left(\rho k\mathbf{U}\right)=P_k-{\beta }^{*}\rho k\omega +\nabla ·[(\mu +{\sigma }_k{\mu }_t)\nabla k].$$

(47)

The left-hand side of the equation represents the unsteady term and the convective transport term of turbulent kinetic energy, indicating how turbulence evolves over time and is transported spatially with the flow. On the right-hand side, $P_k$ is the production term, representing the transformation of mean shear flow into turbulence; $-{\beta }^{*}\rho k\omega$ is the dissipation term, describing how turbulent energy is lost due to viscous dissipation; and $\nabla ·[(\mu +{\sigma }_k{\mu }_t)\nabla k]$ is the diffusion term, illustrating the spatial diffusion of turbulent kinetic energy.

The specific dissipation rate $\omega$ equation is used to describe the characteristic scales of turbulence, including the size and decay rate of turbulent eddies. This equation governs how turbulence diffuses and evolves in the flow field. The equation is expressed as follows [39]:

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+\nabla ·\left(\rho \omega \mathbf{U}\right)={\displaystyle \frac{\gamma }{{\nu }_t}}{P}_k-\beta \rho {\omega }^2+\nabla ·[(\mu +{\sigma }_{\omega }{\mu }_t)\nabla \omega ].$$

(48)

Similar to the turbulent kinetic energy k equation, the left-hand side of the specific dissipation rate $\omega$ equation represents the unsteady and convective transport terms, while the right-hand side consists of the production, dissipation, and diffusion terms of $\omega$. In both equations, ${\sigma }_k$, ${\sigma }_{\omega }$, ${\beta }^{*}$, $\beta$, and $\gamma$ are empirical coefficients used to adjust the computation under various flow conditions. ${\nu }_t$, defined as the ratio of k to $\omega$, represents the turbulent viscosity, which characterizes the enhancement of fluid viscosity induced by turbulence. Together, the k and $\omega$ equations provide a complete description of turbulent behavior.

In the two-dimensional case, the fundamental formulation of the SST $k-\omega$ turbulence model remains unchanged. However, due to the absence of flow in the third direction, the convective and diffusive terms in the governing equations are reduced by eliminating the derivatives with respect to the third spatial dimension.

3.3. Mesh Generation and Numerical Setup

The compressor model selected in this study is the first 1.5 stages of a certain type of gas turbine compressor, including three blade rows: VIGV, R1, and VSV. The relevant parameters are shown in

Table 1. Parameters of 1.5-stage compressor.

Parameter	VIGV	R1	VSV
Number of blades	48	43	56
Mid-span chord length (m)	0.02993	0.04801	0.03713
Pitch (m)	0.037625	0.042	0.03225
Tip diameter (m)	0.65	0.65	0.65
Rotation speed (r/min)	0	9174	0
Inlet total pressure (kPa)	101.325	-	-
Inlet total temperature (K)	288.15	-	-

. To reduce the cost of sample acquisition, the blade height at 50% of the blade span is selected. The blade is unfolded along the flow direction to maintain its original three-dimensional characteristics as much as possible, ensuring that the chord length of the blade is consistent with the three-dimensional model to preserve aerodynamic similarity. The blade is then projected onto the two-dimensional plane. By adjusting the blade pitch, the number of moving blades is modified to 40, and the number of blades in the three rows becomes “48:40:56”. The blade is then reduced to form a two-dimensional blade row model with a blade count of “6:5:7”. It should be noted that when simulating the flow in a two-dimensional blade row, the blade row is transformed from rotation in three-dimensional space to translation in the two-dimensional plane. Therefore, the rotor’s motion is also converted from rotational motion in space to uniform translational motion in the plane. The translational speed of the blade row in the plane can be calculated based on the rotational speed and the radius at 50% of the blade height. The relevant parameters of the final planar blade row are shown in

Table 2. Parameters of 1.5-stage compressor planar blade row.

Parameter	VIGV	R1	VSV
Number of blades	6	5	7
Chord length (m)	0.02993	0.04801	0.03713
Pitch (m)	0.037625	0.04515	0.03225
Translational speed (m/s)	0	276.1374	0
Inlet total pressure (kPa)	101.325	-	-
Inlet total temperature (K)	288.15	-	-

.

The planar blade row is a structure composed of a series of repeating blades. Therefore, periodic boundary conditions are applied at the rotor–stator interface, with periodic boundary conditions specified at the circumferential boundaries to simulate the case of an infinite array of blades. The governing equations for the periodic boundary conditions are as follows:

$$U_{\mathrm{periodic},1}=U_{\mathrm{periodic},2}.$$

(49)

That is, the variables (such as velocity, pressure, turbulence parameters, etc.) on the periodic plane are equal at corresponding relative positions.

The geometric model of the planar blade row and the mesh near the rotor blade are shown in Figure 2. From left to right, the three rows of blades are the variable inlet guide vane, the first-stage rotor blade, and the variable stator vane.

The calculations were carried out using Fluent. Since the first 1.5 stages of the compressor operate in a low Mach number range where density variations have minimal impact on the flow, a pressure-based solver was employed. A steady-state case was first computed to verify mesh independence. In this case, both the VIGV and VSV angles were set to 0°, and the inlet boundary condition was specified by total pressure and total temperature values, as listed in Table 2. A static pressure boundary condition of 110 kPa was applied at the outlet, representing a commonly used operating condition for this setting. The SST $k-\omega$ turbulence model was adopted. Regarding convergence criteria, the residuals of all governing equations (including mass, momentum, and energy conservation) as well as the turbulence model equations (k and $\omega$) were set to a threshold of $1\times {10}^{-6}$, ensuring the stability and accuracy of the numerical solution. Three mesh densities were tested, labeled as Level 1 to Level 3 for coarse, medium, and fine grids, respectively. As shown in Figure 3, the analysis of the pressure variation along the characteristic interface of the rotor blade indicates that the Level 2 mesh meets the requirements, effectively capturing the pressure variation near the leading edge of the blade.

The distribution of the $y+$ values on the blade surface for the Level 2 mesh is shown in the contour plot in Figure 4 and the scatter plot in Figure 5. The maximum $y+$ value is 3.2, and the detailed mesh parameters are listed in

Table 3. Mesh parameters.

Parameter	Value
Number of elements $\times {10}^3$	201
Number of inflation layers	17
First layer thickness (m)	$1\times {10}^{-5}$
Inflation growth rate	1.2
Averaged $y+$	1.61
Min. skewness angle	54.43

. When using the SST $k-\omega$ turbulence model for numerical simulations of the compressor flow field, a $y+$ value less than 10 is generally considered acceptable. Therefore, the boundary layer mesh meets the requirements of the SST $k-\omega$ turbulence model.

3.4. Time Step Independence

The transient simulation was initialized using the steady-state flow field as the initial condition and solved using an implicit time integration scheme. In accordance with periodicity requirements, each rotor passage was divided into 100 time steps, corresponding to a time step size of $1.635056\times {10}^{-6}$ seconds, to ensure sufficient frequency resolution in the subsequently constructed snapshot matrix. The subsequent analysis focused primarily on the modal characteristics near the leading edge, the mid-span suction side, and the rotor–stator interface. The Courant–Friedrichs–Lewy (CFL) numbers in these local regions were calculated to determine whether the chosen time step size satisfies the requirements of the implicit integration scheme according to the CFL criterion. The formula for the CFL number is as follows [41]:

$$CFL={\displaystyle \frac{u·\Delta t}{\Delta x}}.$$

(50)

When calculating the local CFL number, u represents the local flow velocity, $\Delta t$ is the previously specified time step, i.e., $1.635056\times {10}^{-6}$ seconds, and $\Delta x$ refers to the local mesh size, which is taken as the average distance between the centers of two adjacent cells.

Table 4. CFL numbers corresponding to three different mesh levels.

Mesh Level	Leading Edge	Suction Side Mid-Chord	Rotor–Stator Interface
Level 1	1.27	1.8	1.48
Level 2	3.29	4.93	3.82
Level 3	15	22.06	17.65

shows the local CFL numbers in the characteristic regions of interest discussed later. When using an implicit time integration scheme, the CFL number is generally required to be below 10. It can be seen that the CFL numbers of the Level 2 mesh meet this requirement.

4. Results and Discussion

This section presents the mode-based spatiotemporal characterization and ROM of the RSI flow field.

4.1. Spatial–Temporal Characteristics Analysis

Set the angle of VIGV and S1 to 0°, with an outlet pressure of 110 kPa (typical operating condition). After the flow field stabilizes, extract pressure data from the rotor region over 5 rotor passage periods (500 time steps) and assemble them into an instantaneous snapshot matrix $\mathit{u}$ according to Equation (1). It should be noted that the pressure fluctuations in the flow field are significantly smaller than the mean pressure values, making it difficult to directly observe unsteady pressure variations from the original flow field pressure, as shown in Figure 6a. Therefore, the time-averaged pressure over one period is subtracted from the original flow field pressure to obtain the fluctuating flow field data, as illustrated in Figure 6b.

The fluctuating pressure snapshot matrix is decomposed into spatial modes and their corresponding temporal coefficients. Using Equation (7), it is found that the first ten modes capture 99.2% of the total energy, effectively representing the flow field characteristics. Thus, only these ten modes are considered. Figure 7 illustrates their cumulative energy. The RSI flow field in the compressor exhibits approximate periodicity in space and time, as discussed in the theoretical section. However, due to unsteady factors such as shock waves and flow separation, the periodicity is not strict. Consequently, each pair of POD modes has similar modal energy and exhibits orthogonal spatial distributions.

The POD modes represent the spatial distribution characteristics of the excitation in the assembled data matrix. The spatial modes of the flow field in the rotor region under this operating condition primarily consist of two types of features: the wake characteristics of the upstream VIGV and the potential flow characteristics of the downstream VSV. Figure 8 illustrates the distribution of the first four spatial modes. Specifically, Figure 8a shows the first and second modes, which represent the wake characteristics of the upstream VIGV. It can be observed that the upstream wake is concentrated near the leading edge of the rotor blade and extends toward the trailing edge of the suction side. Figure 8b depicts the third and fourth modes, which represent the potential flow characteristics of the downstream VSV. The unsteady disturbances induced by the downstream potential flow manifest as periodic flow structures distributed along the rotor–stator interface.

Figure 9 presents the spatial distribution of the modal amplitudes of the third and fourth modes sampled along the blade rotation direction at the interface, where half of the interface length is considered. It can be observed that the amplitudes of these two modes are similar, and their spatial distributions exhibit a phase difference of 90°, confirming the orthogonality of adjacent modes in space.

The adjusted blade count is “48:40:56”, and the phase difference $\Delta \varphi$ of the upstream wake components in the unsteady flow field of adjacent passages in the rotor region is defined as

$$\Delta \varphi ={\displaystyle \frac{(48-40)\times 2\pi }{40}}={\displaystyle \frac{2}{5}}\pi .$$

(51)

The sampling spatial length must be an integer multiple of the number of passages. The minimum number of passages required to capture one full period of the upstream wake ($2\pi$) is five. Similarly, the minimum number of passages required to capture one full period of the downstream potential flow ($2\pi$) is also five. Consequently, the blade count can be reduced to “6:5:7”. Spatially, sampling is conducted over five passages, while temporally, it is performed over five passage periods, ensuring that the snapshot matrix satisfies the minimum periodicity in both time and space. However, this sampling method only satisfies the minimum spatial period length of the upstream wake and downstream potential flow. As a result, the circumferential spatial periodicity cannot be observed in the modes shown in Figure 8.

To visually observe the spatial periodicity of the POD modes, a full-annulus “48:43:56” simulation was conducted. The computational setup and mesh treatment are consistent with those of the “6:5:7” model, and the data were also extracted at 50% span. The blade pitch values are listed in Table 1. Figure 10 shows the spatial distribution of POD modes in the full-annulus simulation, where the phase difference of the upstream wake in adjacent passages is given by

$$\Delta \varphi ={\displaystyle \frac{(48-43)\times 2\pi }{43}}={\displaystyle \frac{10}{43}}\pi .$$

(52)

The minimum number of passages that approximately satisfies an integer multiple of one spatial period $2\pi$ is nine. As shown in Figure 10a for passages A and B, the spatial periodicity of the mode distribution in the full-annulus model corresponds to a passage length of nine passages. However, the phase difference over nine passages is approximately $2.09\pi$, which is not an exact integer multiple. Therefore, the mode does not exhibit perfect spatial periodicity. Similarly, the phase difference of the downstream potential flow in adjacent passages is given by

$$\Delta \varphi ={\displaystyle \frac{(56-43)\times 2\pi }{43}}={\displaystyle \frac{26}{43}}\pi .$$

(53)

The minimum number of passages that approximately satisfies an integer multiple of one spatial period $2\pi$ is 10. The phase difference over 10 passages is approximately $6.04\pi$, which is closer to an integer multiple of $2\pi$, resulting in better spatial periodicity compared to the upstream wake modes, as shown for passages A and B in Figure 10b.

On the other hand, during the POD process, the modal coefficients associated with each mode are also obtained. These modal coefficients represent the temporal evolution of the spatial modes. By performing a Fourier transform on the modal coefficients, the frequency components corresponding to the modes can be determined. The POD modes encompass multiple frequency components, with low-amplitude frequencies excluded to retain focus on the dominant ones. Figure 11 shows the dominant frequencies and their amplitudes obtained from the Fourier transform of the first ten modal coefficients in the “6:5:7” model, which include two types of frequency components: 7339.2 Hz, 8562.4 Hz, and their harmonics. The frequencies of the first and second modes correspond to the VIGV passing frequency:

$$f_{1,2}={\displaystyle \frac{9174\times 48}{60}}=7339.2\mathrm{Hz}.$$

(54)

The frequencies of the third and fourth modes correspond to the VSV passing frequency:

$$f_{3,4}={\displaystyle \frac{9174\times 56}{60}}=8562.4\mathrm{Hz}.$$

(55)

The spatial distribution of modes characterizes their excitation in space, while the modal frequency explains their temporal excitation source. From the distribution of subsequent modal frequencies, it can be observed that the fifth and sixth modes correspond to the second harmonic of the VIGV passing frequency, the seventh and eighth modes to the third harmonic of the VIGV passing frequency, and the ninth and tenth modes to the second harmonic of the VSV passing frequency. This phenomenon stems from the nonlinearity of the rotor–stator interference flow field, which induces spatiotemporal characteristics in the POD flow modes analogous to frequency multiplication observed in nonlinear vibration systems. Figure 12 illustrates the spatial distribution of the second harmonic modes associated with the VIGV and VSV passing frequencies.

In Figure 12a, the fifth and sixth modes are shown, where the second harmonic of the VIGV frequency is concentrated at the leading edge of the blade tip, amplifying the excitation intensity of the VIGV wake in this region. Unlike the first harmonic mode of the VIGV frequency, the second harmonic mode primarily excites the leading edge of the blade. Due to the relatively fast dissipation rate of low-intensity disturbance clusters, the excitation intensity of the second harmonic mode is insufficient to support its propagation toward the trailing edge of the suction side. Figure 12b presents the spatial distribution of the ninth and tenth modes. The second harmonic of the VSV frequency exhibits a spatial distribution pattern similar to its fundamental harmonic (first harmonic), with both concentrating at the rotor–stator interface. However, the modal structure appears more fragmented, corresponding to a larger Fourier wavenumber in Equation (30), reflecting the distribution characteristics of higher-frequency components within the same flow structure. Additionally, amplitude comparisons across mode frequencies indicate that the IGV wake exerts a stronger influence on the rotor region compared to the VSV potential flow, highlighting the dominance of wake-driven instabilities in rotor–stator systems.

With the VIGV and VSV angles set to 0°, the outlet pressure was reduced to 105.5 kPa to approach the compressor choke boundary. The frequency distribution of the first ten modes obtained from the POD of the RSI flow field is shown in Figure 13. It can be observed that, in addition to the VSV passing frequency, the VIGV passing frequency, and their harmonics, a modal frequency component of 611.6 Hz appears in the third and fourth modes.

Based on the modal spatial distribution shown in Figure 14, it is evident that the modal characteristics are primarily concentrated at the blade leading edge and the mid-section of the suction side. This distribution pattern is closely related to the dominant aerodynamic phenomena that occur near the choke boundary [42]. Shock waves tend to form near the leading edge of the blade because, under high mass flow conditions, this region experiences strong flow deflection and steep pressure gradients. The local Mach number may reach or even exceed the sonic speed, triggering intense compressive effects and making the leading edge a typical location for shock formation. Meanwhile, the mid-section of the suction surface is prone to flow separation. After accelerating near the leading edge, the flow encounters a significant adverse pressure gradient as it moves downstream. If the boundary layer lacks sufficient kinetic energy to overcome the increasing static pressure, separation occurs and the flow detaches from the wall. As the operating point approaches the choke boundary, the inlet Mach number and mass flow rate further increase, enhancing compressibility effects. This not only intensifies the strength of the leading-edge shock but also strengthens the shock–boundary layer interaction, thereby aggravating flow separation on the suction side. The combined effects significantly increase flow instability, leading to stronger fluctuations and oscillations within the passage. As a result, these features become more pronounced closer to the choke boundary, and higher-order modes begin to emerge. Moreover, such phenomena disrupt the periodicity of the flow field. Therefore, at 105.5 kPa, the modal symmetry is significantly weaker than at 110 kPa, especially for the third and fourth modes associated with this behavior, where the relative difference in modal frequencies increases markedly at 105.5 kPa.

Table 5. The mode order at different outlet pressures.

Outlet Pressure (kPa)	Mode Order
110	13, 14
107.5	11, 12
106.5	9, 10
106	7, 8
105.8	5, 6
105.5	3, 4
105	1, 2

presents the variation of this modal characteristic order with the outlet pressure. It can be observed that the closer the operating condition is to the choke boundary, the higher the ranking of this modal characteristic. Near the typical operating condition, the energy proportion of this modal characteristic is extremely low, resulting in a lower ranking. However, near the choke boundary, the ranking of this modal characteristic increases sharply. When the outlet back pressure decreases from 106 kPa to 105 kPa, a change of 1 kPa, the mode order rises from the seventh and eighth modes to the first and second modes. This is because the intensity of shock waves and flow separation does not vary linearly with changes in the outlet pressure but instead rises abruptly when approaching the choke boundary.

Figure 15 shows the fluctuating pressure distribution of the flow field at an outlet pressure of 105 kPa, a condition close to the choke boundary. Compared to the mean-removed pressure distribution under the commonly used operating condition shown in Figure 6b, the pressure fluctuations near the leading edge and the mid-section of the suction side are more intense. By extracting various flow characteristic modes through the POD method, this change can be observed more intuitively, providing a convenient method for vibration analysis and optimization under fluid–structure interaction in compressors.

4.2. MPR-ROM of 1.5-Stage Compressor Cascade

4.2.1. Offline Phase

Using the 1.5-stage compressor model introduced in the previous section, the VIGV and VSV angles are constrained within the range of 0° to 10°. Thirty training samples are generated for these two parameters using the LHS method. For each sample, the outlet pressure at the choke boundary and surge boundary is determined, which is then used as the boundary condition for sampling the outlet pressure parameter of each training sample. Additionally, two test samples are selected from the parameter space. Test sample A has a VIGV angle of 8°, a VSV angle of 8°, and an outlet pressure of 108 kPa, while test sample B has a VIGV angle of 8°, a VSV angle of 8°, and an outlet pressure of 104 kPa. Sample A represents a typical operating condition, whereas sample B is a condition near the compressor choke boundary. The distribution of the samples in the parameter space is shown in Figure 16.

The high-fidelity unsteady flow fields of 30 training samples are obtained using CFD numerical methods. After stabilization, 500 time steps of pressure data from the rotor region are extracted and assembled into the multi-parameter snapshot matrix $\mathit{U}$ according to Equation (31). Using the multi-parameter POD method, two-step POD is performed: the first POD extracts spatial bases from $\mathit{U}$; and the second POD extracts temporal bases from the coefficient matrix $\mathit{M}$. During both POD steps, mode truncation is applied. While retaining more bases reduces truncation error, it increases the computational cost of ROM training. Thus, the objective is to minimize the number of retained bases while ensuring acceptable truncation error.

Figure 17 shows the energy proportion of the first five spatial and temporal bases. The first few bases capture most of the energy, especially the first-order basis, with 84% and 87.5% energy proportions, respectively. This indicates that a small number of bases can effectively reconstruct $\mathit{U}$ and $\mathit{M}$.

To intuitively determine the number of retained bases, the relative errors caused by truncation are calculated using Equations (36)–(38) defined earlier.

Table 6. The relative error value of mode truncation.

Truncation	${\mathit{e}}_1$	${\mathit{e}}_2$	e
$I=3,J=3$	0.036	0.022	0.057
$I=5,J=5$	0.020	0.005	0.025
$I=10,J=10$	0.005	0.000	0.005
$I=20,J=10$	0.001	0.000	0.001
$I=30,J=10$	0.000	0.000	0.000

shows the relative errors under different truncation orders, where $e_1$ is the relative error caused by spatial mode truncation. When $I=30$, that is, retaining 30 spatial bases, the truncation error is below 1‰. And $e_2$ is the relative error caused by temporal mode truncation. When $J=10$, that is, retaining 10 temporal bases, the truncation error is below 1‰. Therefore, 30 spatial bases and 10 temporal bases are retained, ensuring that the total error e caused by mode truncation is kept at an extremely low level, effectively eliminating the impact of truncation on the accuracy of the MPR-ROM method. Finally, RBF interpolation is employed, where a third-order polynomial is used to establish the relationship between the parameter coefficients and the parameters, completing the offline phase.

4.2.2. Online Phase

Taking test sample A as an example, the three parameters of sample A are used as inputs, and the constructed RBF interpolation is employed to calculate the parameter coefficients ${\mathit{C}}^{*}$. After reorganizing ${\mathit{C}}^{*}$, the predicted flow field ${\mathit{U}}^{*}$ is obtained by combining it with the retained 10 temporal bases and 30 spatial bases. Figure 18a shows the unsteady flow field fluctuating pressure obtained through CFD methods, while Figure 18b shows the pressure obtained through the MPR-ROM method. The comparison results demonstrate that the MPR-ROM method can accurately predict the fluctuating characteristics of the RSI flow field in the compressor cascade. Points A and B are selected, with their locations shown in Figure 18a. Point A is located at the rotor–stator interface between R1 and VSV, where the fluctuating pressure amplitude is relatively large, while Point B is situated at the leading edge of the blade, where the fluctuating pressure amplitude is smaller. Figure 19 compares the time trajectories of fluctuating pressure over one period obtained by CFD and MPR-ROM methods at these two points. The results show that, regardless of the intensity of the pressure fluctuations, the pressure solutions obtained from MPR-ROM agree well with the CFD results.

To visually observe the average error across the entire field, the root-mean-squared relative error is defined as

$$e_t^{rms}=\sqrt{{\displaystyle \frac{1}{N_s}}\sum _{i=1}^{N_s}{\left({\displaystyle \frac{P_{t,i}-P_{t,i}^{\prime }}{P_{t,i}}}\right)}^2},$$

(56)

where $N_s$ represents the number of mesh nodes in the rotor region, $P_{t,i}$ is the pressure at the i-th node at time t in the CFD results, $P_{t,i}^{\prime }$ is the pressure at the i-th node at time t in the MPR-ROM results, and $e_t^{rms}$ represents the average relative error across all nodes in the rotor region at time t. Figure 20 shows a variation of the root-mean-squared relative error over one period under different numbers of spatial bases, and the results are consistent with those in Table 6. The error by retaining 30 spatial bases is nearly identical to that obtained by retaining 35 spatial bases, and further increasing the number of spatial bases no longer improves the prediction accuracy.

The MPR-ROM method is used to solve the flow field of test sample B, and the relative errors between MPR-ROM and CFD results are calculated for both samples A and B. Figure 21a shows the relative error distribution for sample A, while Figure 21b shows that for sample B. The maximum relative error near the typical operating point in sample A is 1.4%, whereas near the choke boundary in sample B, it reaches 2.9%, indicating lower prediction accuracy near the choke boundary. Errors are primarily concentrated near the leading edge and the mid-section of the suction side, consistent with pressure fluctuation modes caused by shock waves and flow separation. The leading-edge pressure gradient is high due to shock waves, making predictions more challenging. The lower accuracy in the mid-section of the suction side is due to the significance of the pressure fluctuation gradient, and the low-pressure region amplifies relative errors. The modal analysis under different outlet pressures in Section 4.1 showed that shock waves and flow separation are highly sensitive to outlet pressure near the choke boundary, explaining the lower prediction accuracy near the choke boundary.

4.2.3. Comparison of Computational Efficiency

All computations in this paper were performed on a computer with 8 cores (Intel Core i7-10700, 2.90GHz) and 16GB of memory.

Table 7. Comparison of computational costs between CFD and MPR-ROM methods.

Method	Extraction of Bases	RBF Construction	Prediction/Solution
CFD	0	0	1197.5
MPR-ROM	40.08	13.02	0.12

lists the computational time requirements (CPU time) for the main steps of the MPR-ROM and CFD methods. The computational time for the MPR-ROM method is mainly consumed during the offline phase, including the extraction of spatial and temporal bases and the construction of the radial basis interpolation function. The base extraction is related to the number of training samples and the truncation order. In this study, 30 training samples were selected, and the time required for extracting 30 spatial bases and 10 temporal bases was 40.08 s, while the construction of the radial basis interpolation function took 13.02 s. After completing the offline phase, the time required for prediction was only 0.12 s, while a single CFD calculation took approximately 1197.5 s. For computations under a new parameter, the MPR-ROM method improves efficiency by four orders of magnitude compared to CFD.

4.2.4. Application of ROM in Excitation Pattern Analysis

In compressor stator regulation, the adjustment mode typically involves the coordinated control of multiple stages of stator vanes. Here, the established ROM is utilized to rapidly obtain the unsteady flow field of RSI in the compressor cascade, analyzing the influence of angle adjustment on the excitation characteristics. Figure 22 illustrates the variation in excitation amplitude of upstream wake and downstream potential flow with angle when the regulation coefficient is 1 (where the angle ratio between VSV and VIGV is 1). It can be observed that the variation amplitude of the upstream wake is significantly higher than that of the downstream potential flow. The excitation amplitude of the upstream wake gradually decreases as the angle increases, showing a more pronounced change at smaller angles (0° to 4°), a relatively smoother variation between 4° and 6°, and a slight increase in the amplitude of change beyond 6°. In contrast, the excitation amplitude of the downstream potential flow initially decreases and then increases with the angle, exhibiting a turning point in its trend at 7°.

Figure 23 illustrates the variation trends of upstream wake excitation amplitude with respect to the angle under four different regulation coefficients (1, 0.95, 0.9, 0.85). It can be observed that, across different regulation coefficients, the variation of excitation amplitude generally follows the same trend. However, a turning point emerges near 6°. Before this turning point, the excitation amplitude at the same angle increases as the regulation coefficient increases, whereas after the turning point, the excitation amplitude decreases as the regulation coefficient increases.

Figure 24 illustrates the variation of downstream potential flow excitation amplitude with respect to the angle under these four regulation coefficients. It can be observed that, at the same angle, the excitation amplitude increases as the regulation coefficient increases. Moreover, the excitation amplitude generally exhibits a trend of first decreasing and then increasing as the angle increases. However, the positions of the turning points differ: for regulation coefficients of 1 and 0.95, the turning point occurs at 7°, while for regulation coefficients of 0.9 and 0.85, the turning point appears at 8°.

5. Conclusions

In this study, the POD method is used to decompose the RSI flow field to identify and extract the primary flow characteristics, and the spatial symmetry and phase difference of the POD modes in a periodic flow field are demonstrated in terms of mathematical derivation and physical phenomena. To address the challenges of excessive computational time and high costs in high-fidelity unsteady flow simulations for large-scale multi-parameter problems, a ROM for a 1.5-stage compressor cascade is constructed by combining multi-parameter POD with RBF interpolation. The accuracy and effectiveness of the MPR-ROM are validated, and the influence of angle adjustment on flow field excitation is analyzed based on the established ROM. The following conclusions are drawn:

(1)

The POD method can accurately identify the flow characteristics in the RSI flow field. Under periodic flow conditions, the snapshot matrix should satisfy the minimum periodicity of the flow field in both time and space. The resulting POD modes always appear in pairs, exhibit a 90° phase difference in space, and inherit the spatial periodicity of the flow field.

(2)

At the typical operating point, the primary excitation sources of the 1.5-stage compressor cascade are the upstream wake influence and the downstream potential flow influence, with the upstream wake playing a dominant role. The first harmonic mode of the upstream wake acts on the leading edge of the blade and extends toward the trailing edge along the suction side, while the second harmonic mode rapidly decays as it propagates downstream. As the outlet pressure gradually approaches the choke boundary, pressure fluctuations near the leading edge and the mid-section of the suction side become more intense due to the effects of shock waves and flow separation. The modal characteristics in these regions gradually become dominant as the outlet pressure changes.

(3)

The MPR-ROM method can accurately predict the RSI flow field. Under commonly used operating conditions, the relative prediction error is less than 1.4%, while near the choke boundary, it is less than 2.9%. While ensuring accuracy, this method reduces the time cost of obtaining the flow field by four orders of magnitude. By integrating the POD method, it enables fast and accurate identification of excitation components and their variation patterns in the flow field, providing an effective approach for fluid–structure interaction vibration optimization.