# Canonical correlation decomposition of numerical and experimental data for observable diagnosis

B. Lyu $^{1}$ † 

$^{1}$ State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China 

(Received xx; revised xx; accepted xx) 

A flow decomposition method based on canonical correlation analysis is proposed in this paper to optimally dissect complex flows into mutually orthogonal modes that are ranked by their cross-correlation with an observable. It is particularly suitable for identifying the observable-correlated flow structures while effectively excluding those uncorrelated, even though they may be highly energetic. Therefore, this method is capable of extracting coherent flow features under low signal-to-noise ratios. A numerical validation is conducted and shows that the method can robustly identify the observable-correlated flow events even though the underlying signal is corrupted by random noise that is four orders of magnitude stronger. The temporal sampling frequency and duration of the observable determine the maximum and minimum frequencies to be resolved in the cross-correlation respectively, while those of the flow are to ensure convergence. These criteria are validated using synthetic examples. The decomposition method is subsequently used to analyse a turbulent channel flow, a subsonic turbulent jet and an unsteady vortex shedding from a cylinder, showing the effectiveness of observable-correlated structure identification and order reduction. This decomposition represents a data-driven method of effective order reduction for highly noisy numerical and experimental data and is suitable for identifying the source and descendant events of a given observable. It is hoped that this method will join the existing flow diagnosis tools, in particular for observable-related diagnosis and control. 

# 1. Introduction

Many natural flows exhibit complex behaviour, such as the boundary layer formed over a sand dune or compressed air inside an aeroengine. This is particularly true at high Reynolds numbers, where most realistic engineering flows occur, because turbulence comes into play exhibiting a wide range of temporal and spatial scales. To understand, model, and possibly exert control on these flows, it is crucial to extract dominant structures and reduce the systems' degrees of freedom. 

Extensive research has been conducted to extract coherent features and decompose complex flows into a collection of simple modes. Well-established methods include the Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD), resolvent/input-output analysis and global stability analysis (Taira et al. 2017). Among these, POD and DMD fall into the category of data-driven approaches, while the resolvent/input-output analysis and global stability analysis are model-based. 

POD (Lumley 1967; Berkooz et al. 1993) is a particularly well-known data-driven method and represents a powerful tool for feature extraction and order reduction. Originating from Principal Component Analysis (PCA) in classical statistics, POD decomposes a complex flow into mutually orthogonal modes ranked by their fluctuation energy. If a flow is comprised of a few energetic coherent structures, POD effectively identifies them as leading-order modes. A linear combination of these leading-order modes then forms an optimal reduced-order representation of the total flow. POD may be used to extract the spatial or temporal structures (Lumley 1970; Sirovich 1987). These two structures are coupled, with the temporal structures representing the temporal variation of their corresponding spatial modes, and the spatial modes representing the spatial distribution of their corresponding temporal modes (Aubry 1991). This leads to the so-called Bi-orthogonal Decomposition (BOD). Recent years have also seen the increasingly widely-used Spectral Proper Orthogonal Decomposition (SPOD) in studying turbulent flows (Towne et al. 2018). In addition, to better capture the structures in transient and intermittent flows, conditional space-time POD (Schmidt and Schmid 2019) and multidimensional empirical mode decompositions (Souza et al. 2024) are proposed. These techniques are used to examine the acoustic bursts, the onset and evolution of the dynamic stall and intermittent vortex pairs, showing advantageous capability in resolving transient and intermittent events. It is worth noting that since POD relies on the underlying coherence within the flow to work, it is capable of identifying the flow structures that are dynamically nonlinear compared to linear model-based approaches. 

While POD aims to identify the coherent structures within a complex flow, DMD aims to extract temporal evolutionary information of the underlying dynamics captured in the data (Schmid 2010). The resulting representation is a dynamical system of fewer degrees of freedom. DMD starts by assuming a linear mapping between a sequence of the flow data, and the dynamics is extracted by examining the eigenvalues of a similarity matrix. For a linear system, this amounts to identifying the eigenmodes of the system. For nonlinear systems, DMD is connected with the modes of the so-called Koopman operator (Koopman 1931; Mezić 2013; Schmid 2022). Unlike POD, DMD modes capture the main “contributions” to the overall dynamics embedded in the data sequence. Recent years have seen numerous variants of DMD such as the extended DMD (Williams et al. 2015) and Residual DMD (Colbrook et al. 2023). More details on the recent development of DMD can be found in the recent review by Schmid (2022). 

As mentioned above, both POD and DMD are data-driven, while the resolvent analysis is based on the modal analysis of a linear operator. The resolvent analysis has an early origin in control theory and is based on the pseudospectrum of an operator (Trefethen et al. 1993; Taira et al. 2017), rather than the spectrum. For example, when the flow is decomposed into a base part and a fluctuation part, the Navier-Stokes equations can be rewritten and interpreted as a forced linear system, by which the evolution of the fluctuation part is governed. The nonlinear terms are collected on the right-hand side and interpreted as the forcing of the system. The resolvent modes are ranked by the energy gain between the response and forcing. Therefore, the resolvent analysis examines the gain properties of the linearized operator and has been successfully used to study turbulence from a linearized Navier-Stokes equation point of view (Farrell and Ioannou 1993; Mckeon 2010). Recent studies also show that the leading-order resolvent modes match the leading-order SPOD modes extracted from a numerically simulated high-speed jet (Schmidt et al. 2018). The input-output analysis (Jovanović 2021) is similar to the resolvent analysis in that a modal analysis is performed on a linearized operator. Input-output analysis differs from the conventional resolvent analysis in that a weight may be added to the operator to bias both the forcing and response towards interested domains or observables (Jeun et al. 2016). Therefore, input-output analysis may be regarded as a weighted resolvent analysis. 

In contrast to the resolvent analysis, model-driven global stability analysis (Theofilis 2011) examines the eigenvalue properties of an operator linearized around a base flow with multiple inhomogeneous spatial directions. In particular, it pays special attention to unstable modes, which would dominate the linear response of the system at large times. Note that through global stability analysis, the stable modes can also be obtained, which may play an important role in determining the transient dynamics of underlying flows. This is particularly true in fluid mechanics, where the linearized operators are often non-normal (Trefethen et al. 1993) and the transient growth can become crucial in determining the flow stability. In addition, an adjoint analysis of the operator may be performed to examine the receptivity problem, yielding modes that are similar to the optimal forcing modes in the resolvent analysis. 

POD and DMD, together with their variants, are common data-driven flow decomposition methods used in fluid mechanics. These provide important tools for probing the structures and dynamics of an underlying dynamical system. The ultimate goal of identifying the dominant structures or dynamics is, however, often to understand and possibly control some observables of the flow, such as to reduce the drag of a cylinder, minimise the unsteady force of a wing, or abate the noise emission from a jet. However, because POD modes are ranked by their fluctuation energy, the leading-order modes are not necessarily the most important structures as far as the observable is concerned, although they do carry the largest energy. For example, a large coherent structure effectively extracted from a turbulent subsonic jet using POD may be very inefficient at generating noise. In other words, the leading-order POD mode may not be the leading-order noise-generating flow structure. For example, it has been shown that a substantial number of near-field POD modes are required to reconstruct the acoustic field (Freund and Colonius 2009). Similarly, DMD extracts the dominant dynamics embedded within the flow without taking their connection with any observable into account. Consequently, the leading-order dynamic mode does not necessarily represent the flow events connected with the leading-order dynamics of the observables. 

That the energy rank may not be an appropriate measure, in particular for an observable-related diagnosis, is a well-recognised limitation of POD (Rowley 2005; Schmid 2010). One widely-used approach to overcome this difficulty is to use different norms to bias the decomposition towards interested observables or to use the extended POD (Maurel et al. 2001; Borée 2003). For example, Freund and Colonius (2009) performed the POD decomposition of a turbulent jet using various norms, including the near-field turbulent kinetic energy, near-field pressure, and far-field pressure. When the far-field pressure norm is used, the near-field flow quantities drop out in the correlation matrix and the resulting modes are effectively ranked only by the far-field pressure. Although the near-field flow can still be projected onto the far-field basis, the resulting near-field mode does not necessarily form a direct continuation of the far-field physics, particularly when the near- and far-field exhibit completely different dynamics or the far-field and near-field variables are characterised by pronounced phase delays. Note that the balanced POD proposed by Rowley (2005) is another similar technique to overcome the energy norm limitation of POD, which may be viewed as a special form of POD when the observability Gramian is used as the norm. 

On the other hand, the resolvent and input-output analyses decompose the flow to maximise the energy gain between the output and forcing based on the spectral theory of linear operators. Hence, the observable may be directly included in the choice of output. The resolvent and input-output analysis represent powerful tools to diagnose the flow structure and are capable of providing insightful understanding into a variety of turbulent flows (Mckeon 2010; Sharma and Mckeon 2013). In order to do so, a linearized operator describing the underlying system is often needed. In some cases, however, such an operator may not be readily known, while in others the linearized operator may not be an appropriate representation of the dynamical system, particularly in highly nonlinear systems. For example, an input-output analysis was performed on compressible subsonic and supersonic jets and found that a considerable number of modes were required to reconstruct the acoustic energy of subsonic jets (Jeun et al. 2016), which may be partly due to the limitation imposed by linearity. Such a limitation is also applicable to global stability analysis, where a linearized operator must be known in advance. 

In this paper, we aim to develop a data-driven flow decomposition method that is suitable for observable diagnosis based on flow and observable snapshots instead of linear operators. Instead of redefining the POD energy norm to bias towards the observable, the decomposition aims to introduce a rank based on a cross-correlation norm between the resulting modes and the observable, hence including both the flow and observable data in the correlation matrix. The decomposition method falls under the framework of canonical correlation analysis (CCA) (Hotelling 1936) in classical statistics. This paper is structured as follows: section 2 shows a mathematical formulation of the decomposition method. The physical significance of the resulting modes, the frequency and wavenumber resolutions, the effect of including multiple observables and the connection of the present decomposition to POD and the extended POD are discussed in detail sequentially. Section 3 validates the method by performing the decomposition on multiple synthetic flow fields. The effects of varying sampling frequency, duration and including multiple observables are also thoroughly validated. Section 4 applies this technique to both numerical and experimental data, demonstrating the potential use of such a method. The following section concludes the paper and lists some future work. 

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/75159b30fee58ee8c37eab1182a7d5ababa12d31b362597a80ac0b808f8a8b31.jpg)



Figure 1: Schematic illustration of the two-dimensional flow snapshots sampled at time $t_{i}, i = 1, 2, 3, \ldots, N$ . Each snapshot contains flow data in both x and y directions, where x and y denotes the Cartesian coordinates of the flow domain.


# 2. The canonical correlation decomposition

# 2.1. The decomposition procedure

Assume that we have a sequence of snapshots $u_{i}$ obtained by sampling a flow field $u(\boldsymbol{x}, t)$ at time $t = t_{i}$ , where x represents the coordinates of the flow domain and i is an integer that takes the value of $1, 2, 3, \ldots, N$ . If the snapshots $u_{i}$ are sampled in time, $t_{i}$ increases sequentially as i increases, as shown in figure 1. If $u_{i}$ are, however, sampled in the ensemble space, each $t_{i}$ refers to the sampling time in its corresponding independent realisation and can therefore be independent of each other. In the most general case, $u_{i}$ can be sampled both in the time and ensemble space. Each snapshot of this sequence is obtained by discretizing the spatial domain on a mesh and represented by a column vector of length M. We write this snapshot sequence compactly in a matrix notation as 

$$
\boldsymbol {U} = \left[ \boldsymbol {u} _ {1}, \boldsymbol {u} _ {2}, \boldsymbol {u} _ {3}, \dots , \boldsymbol {u} _ {N} \right]. \tag {2.1}
$$

For each snapshot $u_{i}$ , which is obtained by sampling a flow field at $t_{i}$ , assume that we can simultaneously sample an interested observable $p(t)$ of this flow at time $t_{i} + \tau_{j}$ , where j is an integer and takes the values of 1, 2, 3, ...Q with Q being a positive integer. We therefore obtain a sequence of the sampled observable $p_{i,j}$ , $j = 1, 2, 3...Q$ . Note that the sequence $p_{i,j}$ can be sampled at an earlier or later time of $t_{i}$ , depending on whether $\tau_{1}$ is a negative or positive value, respectively. This is important, and we will discuss its significance in the rest of this paper. 

For each integer i, we can define a column vector $p_{i}$ such that 

$$
\boldsymbol {p} _ {i} = [ p _ {i, 1}, p _ {i, 2}, p _ {i, 3}, \dots p _ {i, Q} ] ^ {T}, \tag {2.2}
$$

where T denotes transpose. We then form a matrix P such that 

$$
\boldsymbol {P} = [ \boldsymbol {p} _ {1}, \boldsymbol {p} _ {2}, \dots , \boldsymbol {p} _ {N - 1}, \boldsymbol {p} _ {N} ]. \tag {2.3}
$$

The key step is to construct a matrix A, representing the cross-correlation matrix between the flow and the observable, such that 

$$
\boldsymbol {A} = \frac {1}{N \sqrt {Q}} \boldsymbol {P} \boldsymbol {U} ^ {\dagger}, \tag {2.4}
$$

where $\dagger$ denotes the Hermitian adjoint. The Hermitian adjoint here allows both P and U to be complex matrices. This is useful because both the observable and flow field can be just a Fourier component of the total fields (see the end of section 2.1 for more details). In the case where only real matrices are involved, the Hermitian adjoint $\dagger$ reduces to the simple transpose T. 

We then perform the standard Singular Value Decomposition (SVD) of matrix A, such that 

$$
\boldsymbol {A} = \boldsymbol {R} \boldsymbol {\Sigma} \boldsymbol {V} ^ {\dagger}, \tag {2.5}
$$

where R and V are $Q \times Q$ and $M \times M$ unitary matrices respectively, while $\Sigma$ is a diagonal matrix of $Q \times M$ with the singular values $\sigma_{j}$ ( $j = 1, 2, 3 \ldots, \min(M, Q)$ ) as its diagonal elements. The column vectors of V represent the desired modes of the flow field $u_{i}$ , while those of R represent the normalised cross-correlation functions between the resulting modes and the observable. From SVD, it can be readily shown that these modes are mutually orthonormal and form a complete basis of $R^{M}$ . Therefore, the flow field $u_{i}$ can be conveniently decomposed as 

$$
\boldsymbol {u} _ {i} = \sum_ {k = 1} ^ {N} a _ {k} (t _ {i}) \boldsymbol {v} _ {k}, \tag {2.6}
$$

where $v_{k}$ denotes the k-th column of V while $a_{k}(t_{i})$ denotes its corresponding expansion coefficient at time $t_{i}$ , or equivalently, 

$$
u (\boldsymbol {x}, t) = \sum_ {k = 1} ^ {N} a _ {k} (t) \phi_ {k} (\boldsymbol {x}), \tag {2.7}
$$

where $\phi_{k}(\boldsymbol{x})$ denotes the basis function corresponding to $v_{k}$ , while $a_{k}(t)$ is the expansion coefficient of $u(\boldsymbol{x},t)$ using the basis $\phi_{k}(\boldsymbol{x})$ . As shown in section 2.2, these modes are ranked by their cross-correlation with the observable, and a significant order reduction may be expected if only a small number of modes are pronouncedly correlated with the observable. As will be shown in section 2.2, the decomposition method falls under the framework of CCA, therefore it will be referred to as the canonical correlation decomposition (CCD) in the rest of this paper. 

Note that, as mentioned above, both the observable and the flow can be just a Fourier component of the total fields. For example, the observable may be $\tilde{p}_{\omega}$ while the flow may be $\tilde{u}_{\omega}$ , where $\tilde{p}_{\omega}$ and $\tilde{u}_{\omega}$ represent the temporal Fourier components of the observable and flow at angular the frequency $\omega$ , respectively. In practice, a long flow snapshot sequence $u(\boldsymbol{x}, t_{k})$ ( $k = 1, 2, 3 \ldots$ ) obtained in experiments or simulations may be first partitioned into N segments; each segment may be regarded as a realisation in the ensemble space and then Fourier transformed in time and/or space to form the $u_{i}$ ( $i = 1, 2 \ldots N$ ) shown in (2.1). Similarly, a long observable sequence $p(t_{k} + \tau_{1})$ obtained in experiments or simulations may be partitioned into N segments; the ith segment is then Fourier transformed with respect to $t_{k}$ ( $\tau_{1}$ is a constant) to obtain $\tilde{p}_{\omega i}(\tau_{1})$ . In a similar manner, $\tilde{p}_{\omega i}(\tau_{2})$ , $\tilde{p}_{\omega i}(\tau_{3}), \ldots$ $\tilde{p}_{\omega i}(\tau_{Q})$ can be obtained, which are just $p_{i,2}$ , $p_{i,3}, \ldots p_{i,Q}$ shown in (2.2) ( $\tilde{p}_{\omega i}(\tau_{1})$ constitutes $p_{i,1}$ ). Note that when the observable and the flow are sampled at different frequencies, proper temporal alignment of them for each realisation must be ensured according to those described in section 2.1. Care must also be taken regarding the frequency resolutions of the flow and the observable when the Fourier transform is performed. CCD can then be performed according to (2.3) to (2.5), which may be regarded as a form of CCD decomposition in the spectral space. 

# 2.2. Physical significance of CCD modes

The CCD represents an optimal decomposition that maximises the cross-correlation between the flow field u and the observable p. This can be shown mathematically as follows. Assuming the flow field is described by the function $u(\boldsymbol{x}, t)$ while the observable by $p(t + \tau)$ , where $\tau$ represents the time delay between flow and the observable. We form the cross-correlation $R(\tau, \boldsymbol{x})$ using 

$$
R (\tau , \boldsymbol {x}) = \left\langle p ^ {*} (t + \tau) u (\boldsymbol {x}, t) \right\rangle , \tag {2.8}
$$

where $*$ represent the complex conjugate, while $\langle\cdot\rangle$ represents the temporal or ensemble average. In the latter case, the statistical processes represented by u and p are assumed to be stationary. For non-stationary processes, (2.8) explicitly depends on t, but the following derivation can still proceed. 

First, let us define an inner product in the Hilbert space defined on a domain $\Omega$ such that 

$$
(f, g) = \int_ {\Omega} f (\boldsymbol {x}) g ^ {*} (\boldsymbol {x}) \mathrm{d} x ^ {n}, \tag {2.9}
$$

where $f(\boldsymbol{x})$ and $g(\boldsymbol{x})$ denote two functions within this space and n represents the dimension of $\Omega$ . A norm is therefore defined as $||f|| = (f, f)^{1/2}$ . Suppose we wish to find a function $\phi(\boldsymbol{x})$ of unit norm, such that the inner product between $R(\tau, \boldsymbol{x})$ and $\phi(\boldsymbol{x})$ , i.e. $(R, \phi)$ , obtains its maximum value in the $L_{2}$ norm. Mathematically, this is equivalent to 

$$
\max _ {| | \phi | | = 1} \frac {1}{T} \int_ {\tau_ {0}} ^ {\tau_ {0} + T} | (R, \phi) | ^ {2} \mathrm{d} \tau , \tag {2.10}
$$

where $|\cdot|$ represents the complex modulus, and $\tau_{0}$ and T are two constants chosen such that the integration includes the entire interval where the integrand obtains non-negligible values. 

Physically, this amounts to finding the optimal function $\phi$ that most correlates with the observable. This is because the ensemble average in (2.8) commutes with the inner product in (2.10), i.e. 

$$
(R, \phi) = \langle p ^ {*} (t + \tau) a _ {\phi} (t) \rangle , \tag {2.11}
$$

where $a_{\phi}(t)$ represents the expansion coefficient of the flow field u using the basis $\phi$ , i.e. 

$$
a _ {\phi} (t) = (u, \phi). \tag {2.12}
$$

Clearly, we see from (2.11) and (2.12) that $(R,\phi)$ represents the cross-correlation function between the mode $\phi$ and the observable p. The $L_{2}$ norm of $(R,\phi)$ defined over an interval of length T is a natural measure of the correlation level between $\phi$ and p. We therefore define the correlation strength $C_{e}$ as the average of $|(R,\phi)|^{2}$ over the interval $[\tau_{0},\tau_{0}+T]$ , i.e. 

$$
C _ {e} = \frac {1}{T} \int_ {\tau_ {0}} ^ {\tau_ {0} + T} | (R, \phi) | ^ {2} \mathrm{d} \tau . \tag {2.13}
$$

Evidently, if $\phi(\boldsymbol{x})$ maximises $C_{e}$ , it represents a flow structure that most correlates with the observable p. 

The function $\phi(\boldsymbol{x})$ that we seek can be obtained from an eigenvalue problem as follows. We know that $\phi(\boldsymbol{x})$ is a function of unit norm that yields a maximum $C_{e}$ , i.e. $\phi(x)$ satisfies 

$$
\max _ {| | \phi | | = 1} \frac {1}{T} \int_ {\tau_ {0}} ^ {\tau_ {0} + T} | (R, \phi) | ^ {2} \mathrm{d} \tau . \tag {2.14}
$$

Classic calculus of variation shows that a necessary condition for (2.14) to hold is that $\phi$ is an eigenfunction of the correlation tensor, i.e. 

$$
\int_ {\Omega} B (\boldsymbol {x}, \boldsymbol {x} ^ {\prime}) \phi (\boldsymbol {x} ^ {\prime})   \mathrm{d} x ^ {\prime n} = \lambda \phi (\boldsymbol {x}), \tag {2.15}
$$

where the correlation tensor is defined by 

$$
B (\boldsymbol {x}, \boldsymbol {x} ^ {\prime}) = \frac {1}{T} \int_ {\tau_ {0}} ^ {\tau_ {0} + T} R (\tau , \boldsymbol {x}) R ^ {*} (\tau , \boldsymbol {x} ^ {\prime}) \mathrm{d} \tau , \tag {2.16}
$$

and the eigenvalue $\lambda$ corresponds to $C_e$ defined in (2.13) (Riesz and Nagy 1955). Clearly, the maximum $C_e$ is given by the largest eigenvalue. 

When the flow field and the observable are discretized, we can show that after multiplied by $\sqrt{Q}$ the matrix A defined in section 2.1 is identical to a discretized form of $R^{*}(\tau, \boldsymbol{x}')$ . The correlation tensor $B(\boldsymbol{x}, \boldsymbol{x}')$ then reduces to $A^{\dagger}A$ because 

$$
B (\pmb {x}, \pmb {x} ^ {\prime}) = \frac {1}{T} \int_ {\tau_ {0}} ^ {\tau_ {0} + T} R (\tau , \pmb {x}) R ^ {*} (\tau , \pmb {x} ^ {\prime}) \mathrm{d} \tau \approx \frac {1}{Q} \sum_ {i = 1} ^ {Q} R (\tau_ {i}, \pmb {x}) R ^ {*} (\tau_ {i}, \pmb {x} ^ {\prime}) = \pmb {A} ^ {\dagger} \pmb {A}, (2. 1 7)
$$

where $\tau_{i}$ is the discretized values of $\tau$ . Equation 2.15 therefore reduces to a discretized eigenvalue problem of the matrix $A^{\dagger}A$ , i.e. 

$$
\boldsymbol {A} ^ {\dagger} \boldsymbol {A} \boldsymbol {v} _ {k} = \lambda_ {k} \boldsymbol {v} _ {k}, \tag {2.18}
$$

where $v_{k}$ as defined in section 2.1 is the discretized form of the k-th eigenfunction $\phi(\boldsymbol{x})$ , while $\lambda_{k}$ is the k-th $\lambda$ in (2.15) subject to a discretization constant, whose exact value often carries no significance in practice. The eigenvalue problem of (2.18) is equivalent to the singular value decomposition shown in (2.5). Therefore, the column vectors of V are these optimal modes, while the corresponding column vectors of R are the normalised cross-correlation functions. In addition, the squares of the singular values $\sigma_{k}^{2}$ are precisely $\lambda_{k}$ , representing the correlation strength $C_{e}$ between the CCD modes and observable (subject to a discretization constant). In particular, when the components of p that correlate with their corresponding CCD modes of u are of equal energy, $\sigma_{k}^{2}$ also represent the observable-correlated energy of their corresponding CCD modes (subject to a constant), and the correlation ranking is identical to the ranking of the observable-correlated flow energy. In summary, instead of decomposing the flow field u based on its energy ranking using the classical POD, (2.5) yields a decomposition that is based on a cross-correlation ranking with an observable, or the observable-correlated energy ranking in the special case where the correlated components of p are of equal energy. 

Mathematically, the flow decomposition method can be shown to fall under the framework of CCA (Hotelling 1936) as follows. Given two column vectors $\boldsymbol{X} = (x_{1}, x_{2}, \ldots, x_{n})^{T}$ and $\boldsymbol{Y} = (y_{1}, y_{2}, \ldots, y_{m})^{T}$ of random variables with finite second moments, CCA seeks two vectors $\boldsymbol{a} (\boldsymbol{a} \in \mathbb{R}^{n})$ and $\boldsymbol{b} (\boldsymbol{b} \in \mathbb{R}^{m})$ such that the random variables $a^{T} X$ and $b^{T} Y$ yield the maximum correlation. The process may be continued in a subspace to yield a sequence of vector pairs. In the context of CCD, the flow field u may be regarded as the Y vector. However, the key part of the decomposition is to find a proper X vector. There are many ways X can be specified, such as the flow within a specific subdomain of interest. However, the essence and novelty of the present decomposition is to construct an X that consists of the observable sampled in a synchronised manner with the flow but at different time delays. Compared to POD or the extended POD, this time shift is an additional dimension used in CCD. As will be shown, the additional information embedded in this “hidden” shifted-time dimension is the key to yielding a more observable-targeting decomposition. More importantly, this permits independent sampling rates between the observable and the flow, which can be of great advantage. 

CCD possesses a number of key features that would be particularly useful for targeted flow diagnosis. First, the decomposition modes are not ranked by their energy, but by the correlation strength with the observable. Flow features that are not correlated with the observable can be effectively suppressed, while those correlated are promoted and ranked according to their correlation strength with the observable. This targets exclusively the observable and is, therefore, very useful in finding the sources or descendant structures of the observable. Second, as will be shown in section 3, the decomposition is robust even when the signal-to-noise ratio (SNR) is low. This is useful when only a small portion of the flow energy correlates with the observable, for example in the classical problem of aeroacoustic emission due to turbulence. Moreover, this robustness can be continuously improved when a longer time duration is used. This is therefore suitable for experimental diagnosis, where an arbitrarily long measurement may be readily performed. 

Third, as will be shown in section 3, the decomposition appears more capable of order or dimensionality reduction compared to POD. This is because CCD aims to decompose the flow only in the observable-correlated subspace, rather than the entire $\mathbb{R}^M$ . In fact, this fact may be used to estimate the convergence of the decomposition by examining how well observable can be reconstructed only using modes corresponding to non-zero singular values. Last but not least, the flexibility to use different sampling frequencies for the flow and the observable enables one to fully exploit the instrument's capabilities in experiments and numerical simulations. For example, it is well known that acoustic signals can often be sampled much faster using a microphone than the entire flow field using PIV. Similarly, in numerical simulations, the observable can also be sampled much faster than the flow due to limited storage requirements imposed by the observable at only a number of probe positions. Note that in general the sampling frequency of the observable is independent of that of the flow, provided $p_i$ properly aligns with $u_i$ as prescribed in section 2.1. In practice, if the sampling frequency of the observable is an integer multiple of that of the flow, it would be trivial to achieve such alignment. In other cases, clock-triggered synchronisation may be used to meet such a requirement in experiments. 

# 2.3. Frequency and wavenumber resolution and sampling delay

In section 2.1 we mention that the matrix P is assumed to have Q rows and each adjacent row is shifted by time $\Delta\tau = \tau_{j+1} - \tau_j$ (assuming a constant sampling frequency). Moreover, p is sampled temporally behind u by a time $\tau_1$ (or ahead of u if $\tau_1$ is negative). In practice, the choice of Q, $\Delta\tau$ , and $\tau_1$ has significant physical implications. 

First, we show that $\Delta\tau$ and Q determine the maximum and minimum frequencies that can be resolved in the cross-correlation between the flow and the observable, respectively. To see this, we start by defining the correlation tensor $C(\tau', \tau)$ as 

$$
C (\tau , \tau^ {\prime}) = \int_ {\Omega} R ^ {*} (\tau , \boldsymbol {x}) R (\tau^ {\prime}, \boldsymbol {x})   \mathrm{d} x ^ {n}. \tag {2.19}
$$

Similar to that shown in section 2.2, we can show that after discretization $C(\tau, \tau')$ reduces to $AA^{\dagger}$ subject to a scaled constant. The eigenvalue $\lambda$ defined in (2.15) can also be found by 

$$
\frac {1}{T} \int_ {\tau_ {0}} ^ {\tau_ {0} + T} C (\tau , \tau^ {\prime}) \psi (\tau^ {\prime}) \mathrm{d} \tau^ {\prime} = \lambda \psi (\tau), \tag {2.20}
$$

where $\psi (\tau)$ corresponds to the column vectors of matrix $\pmb{R}$ defined in (2.5) in a discretized form. When the function $C(\tau ,\tau^{\prime})$ is of a homogeneous (stationary) form, i.e. 

$$
C (\tau , \tau^ {\prime}) = C _ {0} (\tau - \tau^ {\prime}), \tag {2.21}
$$

equation (2.20) reduces to a Fourier expansion (Berkooz et al. 1993), i.e. 

$$
\int_ {\tau_ {0}} ^ {\tau_ {0} + T} C _ {0} (\tau - \tau^ {\prime}) \mathrm{e} ^ {\mathrm{i} 2 \pi f \tau^ {\prime}} \mathrm{d} \tau^ {\prime} = \lambda T \mathrm{e} ^ {\mathrm{i} 2 \pi f \tau}, \tag {2.22}
$$

or equivalently, 

$$
C (\tau , \tau^ {\prime}) = \sum_ {n} \lambda_ {n} T \mathrm{e} ^ {\mathrm{i} 2 \pi f _ {n} (\tau - \tau^ {\prime})}. \tag {2.23}
$$

Equation (2.23) indicates that the function $C(\tau, \tau')$ can be expanded into a Fourier series. When $C(\tau, \tau')$ is discretized, the well-known Nyquist's theorem demands that the sampling frequency $f_s^p \equiv 1 / \Delta \tau$ of the observable must be at least twice as large as the highest frequency to be resolved. Similarly, the total sampling duration $Q\Delta \tau$ determines the frequency resolution to be $1 / Q\Delta \tau$ . When the function $C(\tau, \tau')$ is not a homogeneous function, there are no general theorems, but we expect that the frequency requirement remains similar to the homogeneous case. In summary, $\Delta \tau$ determines the maximum frequency while $Q$ determines the frequency resolution similar to those in the Discrete Fourier Transform (DFT). 

Second, we note that the choice of $\tau_{1}$ depends on the physical time delay between p and u. In general, the observable may be temporally ahead of or behind the flow events depending on the causal relations between the two. For example, if p represents the upstream forcing imposed near the nozzle lip of a turbulent jet, then there must exist a finite time delay between the evolved downstream structure and p due to the finite propagation speed of jet instability waves. In this case, p is preferably sampled ahead of u in order to capture the physical correlation within a reasonably short sampling duration of p. A good estimation of $\tau_{1}$ would be around $-d/U_{c}$ , where d and $U_{c}$ represent the maximum distance between the flow and the observable and the convection velocity of the instability waves, respectively. Conversely, if p is temporally behind u then it must be sampled after u. For example, if the observable p represents the acoustic pressure at a distance r from the jet flow, a good estimate of $\tau_{1}$ would be around r/c, where c represents the speed of sound. In other more general flows, a good estimate of $\tau_{1}$ may be obtained by examining the cross-correlation function between the flow and the observable. $\tau_{1}$ should be chosen such that the correlation matrix A captures the entire correlation peaks. 

In addition to the sampling rate, sampling duration, and sampling delay of p, the temporal and spatial sampling of u also have physical implications. First, the spatial sampling rate of u has the conventional implication that it determines the maximum spatial wavenumber, whereas the length of the spatial sample determines the minimum wavenumber that can be resolved. This can be shown in a similar manner to those shown from $(2.19)$ to $(2.23)$ by considering the expansion of $B(\boldsymbol{x}, \boldsymbol{x}')$ . We omit a repetitive presentation here for brevity. 

Second, the temporal sampling rate of the flow $f_{s}^{u}$ (when $u_{i}$ is obtained via temporal sampling), however, has a different implication. The sampling rate here is not to determine the frequency limit, but mainly to ensure the convergence of the correlation between p and u. In particular, there is no need for the flow field and the observable to be sampled at the same frequency. This is an important advantage, because, as discussed in section 2.2, in experiments PIV can only be sampled at a much slower rate than that using a hot-wire or a microphone, whereas in numerical simulations sampling the flow field fast is impractical because of storage limit. However, such limitation does not exist for a number of interested observables. Therefore, the much higher sampling rate of the observable can be fully exploited by CCD in both experiments and numerical simulations. The fact that the sampling rates of the flow and observable are independent of each other is evident in the case that $u_{i}$ is obtained in the ensemble space. 

# 2.4. Inclusion of multiple observables

In many applications, the appropriate observable is not necessarily limited by one. For example, to examine the dominant flow structures in a subsonic round jet that generates sound at $90^{\circ}$ to the jet centreline, the acoustic pressure at any azimuthal position is an appropriate choice due to the azimuthal statistical homogeneity. In such cases, upon defining a local coordinate system, each observable and the flow field in the local coordinates may be treated as an independent realisation. In such cases, using multiple observables is trivial by following section 2.1, i.e. allowing $u_{i}$ to be sampled both in the temporal and ensemble space. By doing so, the number of flow snapshots is increased by $N_{rl}$ fold, where $N_{rl}$ denotes the number of independent realisations. This would be very useful in improving the convergence of the resulting CCD modes. 

In cases where there is no apparent statistical homogeneity in the flow, multiple observables may still be included. For example, when a turbulent jet is forced in an upstream position (Crow and Champagne 1971), the introduced disturbance evolves downstream. One may wish to extract the coherent structures induced by the forcing using observable measurements downstream of the jet. In such cases, velocity fluctuations at any location within a reasonable distance from the forcing location may be used. However, each observable is likely to be heavily contaminated by turbulence. Using multiple observables are expected to improve the converge of the resulting modes. In such case, suppose that the matrix $P_{i}$ ( $i = 1, 2, 3 \ldots L$ ) can be formed using the i-th observable according to (2.3), then a straightforward way to include multiple observables 

is to form the total matrix P such that 

$$
\boldsymbol {P} = \left[ \begin{array}{l} \boldsymbol {P} _ {1} \\ \boldsymbol {P} _ {2} \\ \dots \\ \boldsymbol {P} _ {L} \end{array} \right]. \tag {2.24}
$$

The normalisation constant Q in $(2.4)$ should be replaced by LQ. However, it is important to note that although P has L times as many rows as $P_{i}$ , this does not improve the temporal frequency resolution of the decomposition, which is still determined by $P_{i}$ . This is because, as illustrated in section 2.3, the temporal frequency resolution is determined by the duration of the time shift $Q\Delta\tau$ when $(2.8)$ is truncated and discretized; including more observables does not increase the length of this duration. Nevertheless, convergence of the resulting CCD modes may improve due to the effective inclusion of more data, particularly when highly noisy observables are used. For highly complicated flow data with a limited sampling duration, such as those obtained in numerical turbulent simulations, including multiple observables is expected to improve the convergence, i.e. reduce the uncertainty or noise of the resulting CCD modes. 

The choice of multiple observables, non matter in statistical homogeneous or inhomogeneous flows, must be made with care. As mentioned, the observables must be expected to resolve the same structures either due to statistical homogeneity or well-defined sources of the underlying problem. In the case where the multiple observables chosen are correlated with different events, including more observables would effectively seek an average between these flow structures, which may not be one's intention. For example, if one is interested in identifying the flow structures that are most correlated with the skin friction under a turbulent boundary layer, observables sampled at various streamwise stations are expected to resolve different structures. In such cases, using multiple observables may not be a worthwhile technique. 

# 2.5. Connection to POD and extended POD

As shown in section 2.2, CCD is different from POD in that the decomposition is based on a cross-correlation rather than an energy norm. This difference is similar to that between CCA and its sister method PCA in classical statistics. Physically, CCA aims to find the “common parts” between two sets of variables, while PCA aims to find the main energetic structures. Mathematically, instead of decomposing the matrix $U^{\dagger}$ , a projection onto P is performed first in CCD. This shows that the decomposition takes into account the space spanned by P. Note this projection may result in a rank that is lower than that of the original flow; however, this is intended as one seeks to decompose $U^{\dagger}$ in the subspace correlated with the observable only. One could argue that this projection leads to a “lower-rank” behaviour by construction, as this would yield fewer singular values. However, the low-rank behaviour we discuss in the following sections is not characterised by fewer singular values, but rather characterised by a quick decay of singular values as the mode number increases and, perhaps more importantly, by a rapid reconstruction of the observable using fewer flow modes. 

As mentioned in section 1, the extended POD (Maurel et al. 2001; Borée 2003) is developed with a similar aim as the present decomposition, i.e. to better target the observable. One can show that the extended POD using a subdomain s is closely related to the degenerate case of CCD when no time shift is allowed between the observable and flow (using multiple observables in s). Mathematically, this implies Q = 1, $\tau_{1} = 0$ and the observable matrix P shown in (2.3) is a degenerate row vector of rank 1. In the special case where the subdomain of the extended POD only includes one observable point and only one mode results, the extended POD and degenerate CCD are identical subject to a normalisation constant. This can be shown as follows. 

Suppose that there exist L observables in the subdomain s. Since no time shift is allowed between the flow and observable, the matrix $P_{i}$ for each observable is a row vector. Hence, the assembled matrix P is a matrix of dimension $L \times N$ . Written in the matrix convention used in the present paper, the essential steps of the spatial extended POD start by decomposing P using POD or, equivalently, by SVD, i.e. 

$$
\boldsymbol {P} ^ {\dagger} = \boldsymbol {R} _ {s} \boldsymbol {\Sigma} _ {s} \boldsymbol {V} _ {s} ^ {\dagger}, \tag {2.25}
$$

where both $R_{s}$ and $V_{s}$ are unitary matrices, the subscript s represents that this is a POD performed in the subdomain s. Note that it is the $P^{\dagger}$ that is decomposed. Right-multiplying (2.25) by $V_{s}$ , one obtains 

$$
\boldsymbol {P} ^ {\dagger} \boldsymbol {V} _ {s} = \boldsymbol {R} _ {s} \boldsymbol {\Sigma} _ {s}. \tag {2.26}
$$

Taking the $k$ th column of both sides of (2.26) yields 

$$
\boldsymbol {P} ^ {\dagger} \boldsymbol {V} _ {s, k} = \boldsymbol {R} _ {s, k} \sigma_ {s, k}, \tag {2.27}
$$

where $\sigma_{s,k}$ represents the kth diagonal element of $\Sigma_{s}$ . The right-hand side of (2.27) represents the temporal coefficient of the kth subdomain POD mode $V_{s,k}$ . The kth extended POD mode $V_{e,k}$ is obtained by projecting the flow U in the extended domain defined in (2.1) onto the kth temporal coefficient, followed by a normalisation, i.e. 

$$
\boldsymbol {V} _ {e, k} = \frac {1}{\sigma_ {s , k} ^ {2}} \boldsymbol {U} \boldsymbol {P} ^ {\dagger} \boldsymbol {V} _ {s, k}. \tag {2.28}
$$

Following the procedure introduced in sections 2.1 and 2.4, the multiple-observable CCD yields 

$$
\frac {1}{N \sqrt {L}} \boldsymbol {P} \boldsymbol {U} ^ {\dagger} = \boldsymbol {R} \boldsymbol {\Sigma} \boldsymbol {V} ^ {\dagger}, \tag {2.29}
$$

where R, $\Sigma$ and V are defined earlier in section 2.1. Left-multiplying (2.29) by $R^{\dagger}$ and then taking the Hermitian adjoint of both sides of the resulting equation yields, 

$$
\frac {1}{N \sqrt {L}} \boldsymbol {U} \boldsymbol {P} ^ {\dagger} \boldsymbol {R} = \boldsymbol {V} \boldsymbol {\Sigma}. \tag {2.30}
$$

Taking the kth column of both sides of $(2.30)$ yields the kth multi-observable degenerate CCD mode 

$$
\boldsymbol {V} _ {k} = \frac {1}{N \sigma_ {k} \sqrt {L}} \boldsymbol {U} \boldsymbol {P} ^ {\dagger} \boldsymbol {R} _ {k}. \tag {2.31}
$$

Comparing (2.28) and (2.31), one sees that the kth extended POD and degenerate CCD modes share much similarity. In particular, since both $V_{s}$ and R are unitary matrices of size $L \times L$ , $V_{s,k}$ and $R_{k}$ are of similar forms. This shows that both modes can be written as a projection of $UP^{\dagger}$ onto a unitary matrix of the same size. However, since $V_{s}$ is obtained by decomposing $P^{\dagger}$ while R by decomposing $PU^{\dagger}/N\sqrt{L}$ , in general, they are not the same. This represents the key difference between the two methods, i.e. one uses an energy-like rank in the subdomain only, while the other uses a correlation rank involving both the subdomain and full domain. It is also this difference that ensures the resulting CCD modes are orthogonal, while it is not necessarily so for the extended POD. 

However, in the special case where only one observable exists in the subdomains s and only one extended POD mode results, both $V_{e}$ and R reduce to 1. Clearly, in this case, the kth extended POD and degenerate CCD modes are identical, subject to a normalisation constant. This also suggests that a key difference between the two is that an extra dimension of time shift is allowed in CCD. It is in fact this difference that results in a more effective order-reduction, which will be discussed in the following sections. 

In summary, one can see that CCD is different from the extended POD in the following ways. First, CCD uses a norm involving both the subdomain and full domain, while the extended POD uses a norm defined in a subdomain space. Second, it is not the energy of the flow within a subdomain that is maximised, but the cross-correlation between the flow and the observable, which is the key difference from the extended POD. Last, the matrix P is formed by consecutively shifting the temporal delay between the flow and the observable. This is why although $A^{\dagger}A$ can be written as $U(P^{\dagger}P)U^{\dagger}$ , CCD is not weighted POD as $P^{\dagger}P$ is a non-diagonal matrix formed by time shifting the observable, instead of a diagonal weight independent of the flow variables. Note that P does not have to be within the flow field; instead, it can represent a variable outside the flow field, a Fourier component of the flow, a particular event in a complex flow, or an observable obtained by integrating the entire flow field. 

Apart from these differences, the connections between POD, extended POD and CCD can also be shown. For example, mathematically POD can be regarded as a special case of CCD when the observable is just an impulse exhibiting no spectral preferences. Specifically, if $p_{ij} = \delta_{i(N+1-i)}$ where $i = 1, 2, \ldots, N$ and $\delta_{ij}$ is the Kronecker delta function, we see that matrix A is a reversed $U^{\dagger}$ and CCD reduces to POD. Physically, this implies that p contains identical frequency components, and therefore exhibits no preferences in the spectral space. Therefore, U is decomposed into modes ranked purely by their energy. Similarly, mathematically CCD may reduce to the extended POD if P is a mathematically constructed simple diagonal weight matrix independent of any flow variables. The exact diagonal elements of course depend on the specific subdomain to be interrogated in the extended POD. 

# 3. Validation

# 3.1. One-dimensional deterministic flow fields

To validate that CCD can effectively extract flow events that correlate with an observable, even under very low SNR, we create an artificial one-dimensional unsteady flow field 

$$
\begin{array}{l} u (x, t) = 2 \cos (t - x) + 1. 5 \cos (2 t) \cos (2 x) + \cos (3 t) \cos (3 x) \\ + 0. 5 \cos (4 t) \cos (4 x) + \cos (6 t) \cos (6 x) \exp (- 0. 1 (x - \pi) ^ {2}) + 1 0 0 r (t, x), \tag {3.1} \\ \end{array}
$$

where $r(t,x)$ represents a random noise field with a uniform probability distribution over $[-0.5,0.5]$ , while other terms represent given flow structures with different amplitudes. Note that the energy of the random noise field is deliberately chosen to be around $10^{4}$ times stronger than the defined flow structures. 

Suppose that p represents an observable of interest at a specific point of the flow field, for example, it may represent the skin friction fluctuations at one point on the bottom wall within a turbulent channel flow. It is known that some flow structures are the primary cause of the skin friction fluctuations while others have minimal effects on them. Therefore, as an illustration we suppose that p is generated by the flow events represented by the first, second, fourth and fifth terms in $(3.1)$ , but not by the third and last terms. For instance, p may be given by 

$$
p (t) = \cos (t - \frac {\pi}{4}) + \sin (2 t - \frac {\pi}{3}) + \cos (4 t) + \cos (6 t - \frac {\pi}{1 2}). \tag {3.2}
$$

Note that the amplitudes of the terms shown in $(3.2)$ are chosen to be identical, although this is not at all necessary. In fact, they may be changed arbitrarily without affecting the validity of the decomposition, for instance, the amplitudes shown in $(3.1)$ can be used should one be interested. 

Suppose that the flow field u is sampled over $t \in [0, 2N\pi]$ at a sample frequency $f_{s}^{u} = 128/2\pi$ , where N is an integer representing the number of periodic cycles. Given the strong random noise in (3.1), N is chosen to be a large number (only necessary when strong noise is present). p, on the other hand, is sampled at the same sample frequency $f_{s}^{p} = f_{s}^{u}$ but for a slightly longer duration of $2(N + 1)\pi$ . According to section 2.1, by choosing $\tau_{1} = 0$ and Q = 128, we can construct a matrix P with 128 rows straightforwardly. Within each snapshot, the flow field is discretized on a mesh of 128 points uniformly distributed between $[0, 2\pi]$ . In this example, $\Delta\tau = 2\pi/128$ , therefore the maximum frequency that can be resolved is limited by around $64/2\pi$ . Similarly, Q = 128 implying that the frequency resolution is around $1/2\pi$ . 

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/d9d99ac379847f175f6232b864e3bc539ae9171bf99fd22936981ae37e7d3c88.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/f35c492029a591bd7abab3e1a8320a1acadd4577449f4895b0ab5b9ed38817b5.jpg)



Figure 2: Comparison of the spectra of CCD (a) and POD (b) when $N = 10^{4}$ , Q = 128 and $f_{s}^{p} = f_{s}^{u} = 128/2\pi$ . CCD is capable of effectively extracting the observable-correlated events leading to a low-rank spectrum, while POD results in a flat spectrum.


Following the procedures introduced in section 2.1, both matrix U and P can be easily constructed, where U is of a size of $128 \times 128N$ while P is of a size $128 \times 128N$ . Upon constructing the matrix A, the CCD can be carried out in a straightforward manner. The resulting CCD spectrum, i.e. the magnitude of the singular values against the mode number, is shown in figure 2(a). To facilitate a direct comparison, the POD spectrum is also shown in figure 2(b). 

Figure 2 shows the CCD spectrum with a desired low-rank behaviour. From figure 2(a) we see that the five modes that correlate with the observable can be robustly identified, even though the energy of the random noise is up to $10^{4}$ times stronger. Specifically, the first two identical singular values form a pair, revealing a flow event of travelling-wave nature, i.e. $\cos(t - x)$ . The first mode of the pair corresponds to $\sin(t + \phi)\sin(x + \phi)$ while the other to $\cos(t + \phi)\cos(x + \phi)$ ( $\phi$ is an arbitrary phase delay), as demonstrated in figure 3(a). The third, fourth, and fifth singular values correspond to the flow events described by the second, fifth, and fourth terms in (3.1), respectively. These can be confirmed by examining the corresponding mode vectors shown in figure 3(b-d). Most importantly, the $\cos(3x)$ mode, which does not correlate with the observable, is robustly removed in the CCD spectrum. This shows that CCD can effectively remove those uncorrelated flow events while only keeping those correlated, and therefore works well for an observable-targeted feature extraction and order reduction. 

The sixth to the ninth singular values $(\sigma_{j}^{2})$ shown in figure 2(a), which are two orders of magnitude weaker than the first few modes, are artefacts introduced by the strong random noise. Note, however, that these unphysical modes can be further suppressed robustly if the flow field is sampled for a longer duration (larger N). All other values of $\sigma_{j}^{2}$ are below $10^{-24}$ and therefore not shown within the given range. As discussed in section 2.2, the singular values represent the correlation strengths between corresponding CCD modes and the observable. In this illustrative case, the observable is comprised of four modes of equal amplitude, as shown in (3.2), therefore the singular values in figure 2(a) are precisely the observable-correlated fluctuation energy (subject to a fixed constant), as evidenced in figure 3 (for example $\sigma_{1}^{2}:\sigma_{2}^{2}:\sigma_{3}^{2}=2^{2}:2^{2}:1.5^{2}$ ). 

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/f9d4d3688ccfda05ec3b750e581d1b55461d83c70a23f13a277578fc45232eb3.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/09200b6a65af1b7c860c05cc93d12cf46f5519325a13aac274dff76240343715.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/b1274b88dc1871df86a5616ecc6333da8d2b868230695d475db0bb4fcf975e05.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/258c99e1ad45bc9fcf905fef10804234fc32d070203d12f4e0b75d193e1bff73.jpg)



Figure 3: Extracted CCD modes when $N = 10^{4}$ , Q = 128 and $f_{s}^{p} = f_{s}^{u} = 128/2\pi$ . They are most correlated with the observable p (corresponding to the first, second, fifth and fourth terms in (3.2), respectively).


On the other hand, figure 2(b) shows that due to the strong random noise the POD spectrum is completely corrupted and shown as a flat line. The low-rank behaviour embedded within the data therefore cannot be identified. This is expected, because POD modes are ranked by their corresponding fluctuation energy. The random noise present in the flow field is up to $10^{4}$ times stronger than the observable-correlated events, and therefore completely dominates the POD spectrum. More importantly, even though POD may be able to extract the coherent structures when weaker noise is present, it cannot separate the observable-correlated flow structures from those uncorrelated in the same way as CCD does, since no information of $p$ is used. For example, the second term of (3.1) would stay in the POD spectrum and also exhibit as a dominant mode. 

Having validated the decomposition, one can straightforwardly demonstrate the effects of varying the sampling frequency, duration, time shift and including multiple observables. The results agree well with the arguments discussed in sections 2.3 and 2.4. For conciseness, however, we do not include them in this section, but rather have it shown in Appendix A. 

# 3.2. One-dimensional statistical flow fields

The example shown in figures 2 and 3 illustrates the capability of CCD in extracting flow events from highly noisy data. The temporal signals given in (3.1) are deterministic; we can show in a similar manner that CCD can also effectively extract the observable-correlated flow events when the temporal variation is statistical, such as those exhibited in many turbulent flows. To show this, we construct an artificial one-dimensional flow field 

$$
u (x, t) = 3 s _ {1} (t) \cos x + 2 s _ {2} (t) \cos 3 x + s _ {3} (t) \cos 6 x \exp (- 0. 1 (x - \pi) ^ {2}) + 1 0 r (t, x), \tag {3.3}
$$

where $s_{i}(t)$ (i = 1, 2, 3) represent three statistical processes. The $s_{i}(t)$ series are generated by a random number generator with different seeds in MATLAB and then filtered using three different 6th-order Butterworth filters. More specifically, $s_{1}(t)$ is filtered using a bandpass filter with lower and upper cut-off frequencies of $0.2f_{s}$ and $0.4f_{s}$ , respectively. The $s_{2}(t)$ and $s_{3}(t)$ series are filtered using low-pass filters with cut-off frequencies of $0.2f_{s}$ and $0.15f_{s}$ , respectively. For illustrative purposes, we also add a random noise field that is two orders of magnitude more energetic than $s_{3}(t)$ . Suppose that the observable p is generated by the flow events represented by the second and third terms in (3.3), but not by the first, i.e. 

$$
p (t) = s _ {2} (t - \frac {\pi}{3}) + s _ {3} (t) + 2 \left[ s _ {3} (t) ^ {2} - \overline {{s _ {3} (t) ^ {2}}} \right] + 3 \left[ s _ {3} (t) ^ {3} - \overline {{s _ {3} (t) ^ {3}}} \right] + r (t). \qquad (3. 4)
$$

Note that because the observable may be non-linearly related to the flow dynamics, we also add in (3.4) two nonlinear terms of $s_3(t)$ , as shown by the two bracket terms. Similarly, the observable may be also subject to noise contamination. A statistical random noise $r(t)$ , with a uniform distribution over $[-0.5, 0.5]$ , is therefore also added. The flow field is again sampled at $f_s^u = 128 / 2\pi$ on a uniform spatial mesh of 128 points over the time interval $[0, 2N\pi]$ , while $p$ is sampled over $[0, 2(N + 1)\pi]$ using the same frequency $f_s^p = 128 / 2\pi$ . 

Routine use of the decomposition yields the CCD spectrum and the first two modes, as shown in figures 4(a) and 4(b), respectively. Clearly, the leading-order mode corresponds to the second term in (3.3), while the second-order mode the third. This can be clearly seen from figure 4(b). It is worth noting that the observable also contains the square and cube of $s_3(t)$ , but this does not appear to affect the identification of the second mode. Indeed, CCD works by maximizing the correlation between the flow field and the observables, but in general it does not limit the observable being a linear function of the flow field. Additionally, the first term of (3.3), due to it being uncorrelated with $p$ , is effectively removed in the CCD spectrum. Other higher-order modes are more than two orders of magnitude lower than the first two. Again, as $N$ increases, these unphysical modes can be further suppressed, while the physical modes resolved more accurately. Note that in this illustrative example, the observable p is also corrupted by the random noise, but CCD continues to work robustly. 

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/03108d0c82d261a4cb2cbcd8380144ae5c5fe14e0bfead9f68c1bce1bbb30360.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/cc91f66662bd8489047a732a67289db47468d885f224eaa3351740b57b16ae86.jpg)



Figure 4: The CCD spectrum (a) and the first and second modes (b). N takes the value of $10^{4}$ but $10^{3}$ may also be used at the expense of convergence.


# 4. Applications to numerical and experimental data

Having validated the method, in this section CCD is used to decompose numerical and experimental data in order to demonstrate its potential use in fluid mechanics. Three flows are used, including a turbulent channel flow, a subsonic jet and a wake flow past a cylinder. Where possible, POD results are also included for comparison. In all cases, the simple $L_{2}$ norm of the flow $u_{i}$ is used in POD. 

# 4.1. Turbulent channel flow

As an illustrative example, we first apply CCD to a Direct Numerical Simulation (DNS) database of turbulent channel flows. The database was obtained from a turbulent channel flow using the code developed by Lee and Moser (2015). The computational domain is of $4\pi H \times 2H \times 2\pi H$ in the streamwise $(x)$ , wall-normal $(y)$ and spanwise $(z)$ directions, respectively, where H denotes the half-height of the channel. The domain is discretized using 192, 128 and 192 points in x, y and z directions, respectively. The friction Reynolds number $Re_{\tau}$ defined as $\rho u_{\tau}H/\mu$ , where $\rho$ , $\mu$ and $u_{\tau}$ denote the fluid density, dynamic viscosity and friction velocity at the wall respectively, is around 180. The time step is fixed at $0.01H/U_{b}$ , where $U_{b}$ is the bulk flow velocity. The flow is sampled every 100 time steps, resulting in a sampling frequency of $f_{s}^{u} = U_{b}/H$ . In total, 1687 snapshots of the flow field are recorded. 

In turbulent channel flows, skin friction represents a significant operational cost in applications such as long-range oil transport (Kim 2011). The control of turbulent skin friction is therefore of particular interest and has been studied extensively in the literature (Gad-el Hak 2007). To understand the physical mechanism concerning its generation and suppression, it is crucial to extract the turbulent flow structures that determine the skin friction. CCD is therefore suitable for such a diagnosis. As mentioned in section 2, without the data storage limit, the observable is allowed to be sampled at a much higher frequency than the flow field. In this example, the sampling frequency $f_{s}^{p} = 10f_{s}^{u}$ , resulting in an interval of $\Delta\tau = 0.1H/U_{b}$ and 16870 samples for the skin friction. 

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/55f0684256957b3b9f104ebd3d7e303e68f8cab3d6ecb15bacc7ca3ae231abed.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/bc88a20258ee3a01741790eccc7dac177e4799bea7e6b3cba953fa9df6fa6e21.jpg)



Figure 5: The spectra of (a) CCD and POD; The CCD spectrum exhibits a much steeper decay as mode number n increases, indicating a more effective order reduction.


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/b9b11fc53d3a151dd7cdb4d517d89357530d251cc44def4400b002a38cefbaba.jpg)



Figure 6: Reconstruction of the dimensionless wall friction coefficient using the first 6 CCD (a) and POD (b) modes, respectively.


Considering the statistical homogeneity of the flow, we use the skin friction sampled at $x = 2\pi H$ and $z = \pi H$ on the lower wall (y = -H) as the observable and choose the streamwise velocity as the flow variable in order to extract the coherent structures. Considering the short temporal correlation scale, we choose Q = 100 and $\tau_{1} = -50\Delta\tau$ . Using the procedures described in section 2, we perform CCD and obtain the resulting singular values and CCD modes. The singular values are shown in figure 5(a). Also shown is the spectrum from POD in figure 5(b), where the streamwise velocity is decomposed. Comparing the two we see that the CCD spectrum is markedly different from that of POD. In particular, the CCD spectrum exhibits a much quicker decay. For example, higher-order modes ( $\geqslant$ 5) are one order of magnitude lower, whereas the POD spectrum is rather flat. This signals a quicker reconstruction of the skin friction using CCD modes. Indeed, using the first 6 modes recovers more than 80% of the total skin friction at the observer point, as shown in figure 6(a). The high-frequency deviation may be further reduced if the observable is allowed to be sampled faster. In contrast, the first 6 POD modes only recover less than 5% energy, as shown in figure 6(b). 


B. Lyu


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Figure 7: Front views of the first four CCD (a,c,e,g) and POD (b, d, f, h) modes. The z coordinate is fixed at $\pi H$ for (a, c, e, g). The spanwise widths of the streaks in (a,c,e,g) are around 0.2H.


The resulting CCD modes are shown in figure 7. We see that the CCD modes take the form of streamwise streaks slightly above the bottom wall, in accordance with current understanding. More importantly, figure 7 also shows that they are spatially localized around the observer point. This is particularly true in the spanwise direction with a streak width of less than 0.2H. Moreover, higher-order modes have increasingly short spatial and temporal scales. To the best knowledge of the authors, such a quantitative and unambiguous characterization of these structures specifically targeting the skin fluctuation in the middle of the wall has not been reported in the literature. In contrast, although the POD modes take the form of streaks, they are not localized around the observer point, but stretched in the streamwise direction and scattered in the spanwise direction instead. Moreover, the first few modes do not exhibit a clear decrease of either spatial or temporal scales, signalling a slower reconstruction of the skin friction. 

Although CCD focuses on examining the flow structures that contribute to the skin friction at one individual point, it in fact does not lose generality. This is because the flow is homogeneous in the streamwise direction; the structures that generate the skin friction at other locations on the wall remain identical (subject to a shift in space). However, by focusing on the observable at a specific point, one expects to obtain a more effective order reduction since our interest is more focused. The fact that the flow is homogeneous can also be exploited to improve the convergence of the resulting flow. Instead of using the observable at one point, one can use multiple points along different spanwise or streamwise locations. They can be treated as independent realisations, with which the resulting mode indeed converges better. However, since the structures remain similar to those in figure 7, we omit showing their contours repetitively. 

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Figure 8: The spectrum of extended POD using (a) a subdomain of size $4\pi H \times 2\pi H$ and (b) a subdomain consisting of the observable point only; Reconstructed skin friction fluctuations using (c) a subdomain of size $4\pi H \times 2\pi H$ (using the first 6 extended POD modes) and (d) a subdomain of only one point (only 1 mode results).


Note that part of the reason why the skin friction reconstruction using POD is slow is due to its use of energy within the entire domain as the norm. Since the extended POD can be used to target more towards the observable, it is interesting to compare it with CCD in detail. To show this, we first perform the extended POD using skin friction on the wall. The resulting singular values and reconstruction of the skin friction using the first 6 extended POD modes are shown in figures 8(a) and 8(c), respectively. The resulting spectrum of singular values exhibits a similar slow decay to that shown in POD. This is consistent with a similar reconstruction of the skin friction, as shown in figure 8(c), where a limited time range from 650 to 1150 is shown for clarity. However, comparing to figure 6(b), the skin friction reconstruction appears slightly improved when the wall shear stress is used as the subdomain in the extended POD. 

One is, therefore, interested in seeing how much the reconstruction can improve by using increasingly small subdomains centring around the observable. In the ultimate case, the subdomain can be chosen to consist of the observable point only. We choose to perform extended POD using such a special subdomain. Note that this is identical to the degenerate CCD where no time shift is included between the observable and the flow. We expect the resulting mode to better target the observable, which is indeed the case, as shown in figure 8(d). The extended POD modes can capture an overall trend in the skin friction variation. However, it is important to note that since there is only one mode available, as can be seen in its spectrum shown in figure 8(b), this is the best reconstruction one can achieve using the extended POD. 

On the other hand, since the decomposition is also a degenerate case of CCD, this represents the worst reconstruction one would obtain using CCD. Indeed, by including the dimension of time shifts, P would have a rank of more than 1, and the reconstruction using CCD improves considerably, as shown in figure 6(a). Note that the reconstruction further improves as one includes more CCD modes, the family of which forms a complete orthonormal set. In addition, figure 8(d) shows that only an overall trend of the skin friction is captured in the reconstruction, and the deviation occurs mainly in the high-frequency regime. This is expected, since this degenerate CCD corresponds to a sampling interval $\Delta\tau = \infty$ for the observable, hence a failure to resolve high-frequency components. 

In summary, using only one point where the observable is located in the extended POD better targets the observable, but the resulting one mode limits the capability of separating multiple flow structures that possibly coexist within the flow. To do that, a sufficiently large region is preferred, compromising the observable specificity. This appears a trade-off between targeting a local observable and separating multiple flow structures. CCD does not have this limitation, and this relaxation is enabled by exploiting the “hidden” time-shift dimension. This reflects a key difference between CCD and the extended POD. More importantly, this also adds the flexibility of fully exploiting a different (possibly much higher) sampling frequency. 

Figures 5 and 7 show that CCD works well in extracting the coherent structures that are most correlated with the given observable. This is further evidenced by a quick reconstruction of the skin friction using the first few CCD modes. Note again that in this example the observable is sampled at a much higher frequency than the flow. This flexibility, as mentioned in section 2.5, plays an important role in the successful feature extraction and order reduction. 

# 4.2. Turbulent subsonic round jets

In this section, we apply CCD to a numerical dataset of a turbulent subsonic round jet. Using the pressure fluctuations as the observable we can examine the flow structures that are most correlated with them. Directly resolving far-field pressure fluctuation is rarely possible in numerical turbulence simulations, hence in this example we examine the near-field pressure fluctuation instead. This is expected to suffice for the purposes of demonstrating the potential use of CCD. The near-field dynamics of turbulent jets is expected to connect with their mixing and acoustic characteristics, and is therefore studied extensively in the literature. Order reduction techniques are widely used. This includes POD or the extended POD with a variety of norms (Freund and Colonius 2009; Sinha et al. 2014; Schmidt and Schmid 2019), the resolvent/input-output analysis (Jeun et al. 2016; Pickering et al. 2021; Bugeat et al. 2024) and other source identification methods that we do not aim to show exhaustively. Moreover, the near-field pressure fluctuations are crucial in determining installed jet noise (Lyu et al. 2017; Lyu and Dowling 2019), therefore its modelling and control have practical uses. 

The numerical data is extracted from an earlier work (Lyu et al. 2017), where an LES simulation of a subsonic round jet was performed. Only a slice of data on one azimuthal plane is used, but it should be sufficient for illustration purposes. The jet Mach number is $M_{j} = 0.5$ while the nozzle diameter D is 2 inches. The computational domain is axisymmetric, with the streamwise coordinate x extending from 0 to 20D and lateral coordinate r extending to 4D. The computational domain is discretized using 512 and 97 points in the x and r directions, respectively. 

We choose the near-field pressure fluctuation at x/D = 10 and r/D = 4 as the observable. At this close distance, the observable is likely to include both acoustic and hydrodynamic pressure fluctuations. In addition, we choose the pressure field as the flow variable to be decomposed. The same is used in a reference POD decomposition. The flow is sampled at a frequency of $f_{s}^{u} = 4U_{j}/D$ for a duration of $200D/U_{j}$ , where $U_{j}$ is the jet exit velocity. The near-field pressure p is sampled at the same frequency but for a longer duration of $280D/U_{j}$ . This results in a Q value of 320. Due to the short distance between the flow field and the near-field pressure fluctuations, we choose the time delay $\tau_{1}$ to be 0. With the procedure described in section 2, the CCD spectrum is shown in figure 9. Also shown is the POD spectrum to facilitate a direct comparison. Only the first 50 singular values are shown. Compared to POD, the CCD spectrum exhibits a more rapid decay as the mode number n increases. In particular, at small mode numbers the CCD spectrum shows a clear low-rank behaviour. The first two modes are almost one order of magnitude stronger than higher-order modes. This is in direct contrast to the POD spectrum, where the low-rank behaviour is not pronounced. 


B. Lyu


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Figure 9: The CCD (a) and POD (b) spectra of the unsteady pressure field on a x - r plane. The CCD spectrum shows a clear low-rank behaviour compared to POD.


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Figure 10: The first 4 CCD modes (a,c,e,g) and PSD spectra (b,d,f,h) of their corresponding temporal coefficients, where the blue dashed line in (b) represents the spectrum of the observable with its magnitude scaled for an easier comparison; mode 1 (a-b), mode 2 (c-d), mode 3 (e-f), mode 4 (g-h). As the mode number increases the CCD modes are characterised by increasingly short spatial scale and high frequency components.


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Figure 11: The first 4 POD modes (a,c,e,g) and PSD spectra (b,d,f,h) of their corresponding temporal coefficients, where the blue dashed line in (b) represents the spectrum of the observable with its magnitude scaled for an easier comparison; mode 1 (a-b), mode 2 (c-d), mode 3 (e-f), mode 4 (g-h). As the mode number increases the POD modes have larger spatial scales with more low-frequency components.


This can be understood from figures 10 and 11, where the first 4 CCD and POD modes $\phi_{k}(\boldsymbol{x})$ and the corresponding Power Spectral Densities (PSDs) of their temporal expansion coefficients $a_{k}(t)$ are shown, respectively. Clearly, the first two CCD modes are large flow structures exhibiting relatively low-frequency behaviour, whereas the leading-order POD modes have much shorter scales with a well-known dominant frequency at around St = 0.3, where St is the Strouhal number defined using $U_{j}$ and D. The PSD spectrum of the observable is also included with its magnitude scaled for an easier comparison. Since the observer is located at x/D = 10 and r/D = 4, the pressure fluctuations inevitably include the signatures of the downstream large coherent structures. The similar first two singular values shown in figure 9 and similar mode shapes shown in figure 10 indicate a convection behaviour of this large structure. CCD decomposition can take this into consideration and yield an observable-relevant low-frequency fluctuation mode. The leading-order POD modes, on the other hand, are ranked only by the fluctuation energy and, therefore, are not as relevant as the CCD modes. 

Note that the singular value represents a measure of the cross-correlation in the $L_{2}$ norm and, therefore, is in general not equal to the energy of the CCD modes contained in the flow, nor is it equal to the energy of the corresponding correlated component in the observable. Nevertheless, since the decomposition targets more at the observable, we expect that it can reconstruct the pressure fluctuations at the observable position using much fewer modes. This is indeed the case, as shown in figure 12, where only the first two CCD and POD modes are included to calculate the reconstructed pressure fluctuations at the observable position, respectively. As can be seen from figure 12(a), using two CCD modes can yield a good reconstruction, which is in contrast to POD shown in figure 12(b). Note that in this application we use the pressure field as the flow u, whereas in general a combined velocity and pressure field may be used. We can show that a similar result may also be obtained when a combination of velocity and pressure fluctuations is used in CCD. 

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Figure 12: The reconstructed pressure fluctuations at the observable location using the first 2 CCD (a) and POD modes (b), respectively.


At large mode numbers, the CCD spectrum shows a steeper decay, and higher-order modes tend to have increasingly short scales together with higher frequencies, as shown in figures 9 and 10, respectively. Note that the singular values represent the correlation strength between the CCD modes and the observable, therefore the decay of the singular values is determined by both the energy of the flow and the observable and the coherence decay between them. Therefore, the steeper CCD spectrum suggests that although the pressure fluctuations consist of energetic structures of various scales, they may not be equivalently important in contributing to the observable, therefore the coherence between the two may decrease rapidly. On the other hand, the POD spectrum decays much more slowly, and as the mode number increases the POD mode starts to capture more downstream large structures with more low-frequency content, as shown in figure 11. That the POD spectrum decays more slowly is attributed to the fact that the POD spectrum is determined solely by the energy of flow and, therefore, does not depend on its coherence with the observable. 

Figure 10 shows that each CCD mode corresponds to a unique temporal variation. 

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Figure 13: The mean (a) and instantaneous (b) streamwise velocity distributions of the immediately downstream wake over a cylinder flow (Renn et al. 2023). The black diamond, red circle and white square are located at x/D = 1.5 but y/D = 6.1, y/D = 5.7, and y/D = 4.9, respectively.


Unlike the Fourier analysis, each of these temporal variations is spectrally broadband. In essence, CCD decomposition works as a special spectral transform of the flow based on its correlation with the observable. Note, however, that the sampling frequency and duration are limited in this simulation, and further analysis using longer samples is needed for better statistical convergence. In addition, due to current data availability, we only decompose the near-field pressure, and it would be interesting to apply this technique to extract acoustically dominant flow features in future studies. Nevertheless, it suffices for the purpose of demonstrating the potential application of CCD. 

# 4.3. Unsteady wake flows over cylinders

In this example, we apply CCD to the experimental data of an unsteady wake flow behind a cylinder. The experiment was performed in a water tunnel using the two-dimensional time-resolved Particle Image Velocimetry (PIV) technique. The cylinder had a diameter of $D = 9.53 \mathrm{~mm}$ while the Reynolds number was fixed at 650. The interrogation window was a rectangle immediately behind a cylinder in the wake and measured $13D \times 9D$ in the streamwise ( $x$ ) and cross-stream ( $y$ ) directions, respectively. Details of the experimental setup can be found in Renn et al. (2023). The velocity field was sampled at a frequency of around $50 \mathrm{~Hz}$ on a mesh of $N_x = 133$ and $N_y = 89$ , and in total $N = 8250$ snapshots were obtained. The mean and instantaneous streamwise velocity fields are shown in figure 13 for reference. As shown in figure 13, the mean flow exhibits the expected symmetry across the wake, while the instantaneous velocity field shows a clear vortex shedding behaviour behind the cylinder. The vortex shedding occurring when the Reynolds number exceeds a critical number is one iconic feature of the flow over cylinders. Given its wide applications such as wind blowing over chimneys and high-rise buildings, its control has attracted significant attention in the fluid mechanics community (Choi et al. 2008). Many techniques exist, including both passive and active controls. Earlier studies show that using the feedback signal recorded in the wake, vortex shedding can be successfully suppressed or even eliminated at low Reynolds numbers (Williams and Zhao 1989; Roussopoulos 1993; Park et al. 1994). In designing a closed-loop active control system such as the one in Park et al. (1994), one primary interest is to identify the optimal location to place the feedback sensor. Ideally, the observable, such as the cross-stream velocity, at the feedback sensor location should maintain a strong correlation with the vortex structures shed from the cylinder. CCD may be used to give an initial assessment of the correlation between the sensed signal and the vortex structures. 

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Figure 14: The CCD (a) and POD (b) spectra of the streamwise velocity fluctuations on an x-y plane with the cross-stream velocity fluctuation at y/D = 5.7 as the observable.


As an illustration, we choose the observable to be the cross-wake velocity (Park et al. 1994) in the initial shear layer behind the cylinder, for example at x/D = 1.5 and y/D = 5.7 as shown by the red circular dot in figure 13(a). Using this observable, we may decompose the streamwise velocity field using CCD. Again, the streamwise velocity is correspondingly used in POD. The time shift $\tau_{1}$ is chosen to be $-4Q\Delta\tau/5$ while Q is chosen to be N/3. Figures 14(a) and (b) show the CCD and POD spectra, respectively. Clearly, both CCD and POD capture the dominant vortex shedding behaviour, and the two nearly identical singular values reflect a convecting behaviour of the shed vortices. Figures 15 and 16 show the corresponding first three CCD and POD modes and their corresponding PSDs, respectively. Clearly the first two vortex shedding modes from both CCD and POD are virtually identical, which can be seen from both the mode shape and their corresponding PSD spectra. The CCD spectrum shows a slightly smaller singular value for the third mode, which is somewhat more symmetric, whereas the similar mode resulting from POD obtains a similar singular value compared to the leading-order mode. This suggests that although this mode carries one of the largest energy, it is slightly less correlated with the cross-stream velocity fluctuation at x/D = 1.5 and y/D = 5.7. 

If we keep x/D = 1.5 but move the observable position further away from the shear layer, for example, to y/D = 4.9 and y/D = 6.1 as shown by the white square and black diamond symbols respectively in figure 13, these three modes can still be identified using CCD, but their relative singular values changed significantly, as shown in figure 17. This implies that these modes correlate differently to different observables. Evidently, the first and second modes in figure 17(a) represent the vortex shedding modes. Their mode shapes are similar to those shown in figure 15(a) and (b), so we omit a repetitive presentation. 

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Figure 15: The first 3 CCD modes (a,c,e) and PSD spectra (b,d,f) of their corresponding temporal coefficients; mode 1 (a-b), mode 2 (c-d) and mode 3 (e-f).


However, it is important to note that these singular values are much larger compared to those shown in figure 14(a), suggesting they are more strongly correlated with the observable. More importantly, the third singular value drops rapidly, almost one order of magnitude weaker than the leading-order mode. From the feedback control point of view, this would be a good candidate for placing the feedback sensor owing to its higher correlation with our interested flow events and simultaneously a higher SNR. Figure 17(b) shows that the singular values corresponding to the vortex shedding modes are slightly lower than those shown in figure 14(a) with an even stronger leading-order non-shedding mode. Consequently, this would be a position to be avoided for placing the feedback sensor. This may be why the wake centreline was used to place the feedback sensors in Park et al. (1994). 

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/4e131b3f4172e290869bc6016058fde48507d4a60d8b65427485e11baaf1f530.jpg)


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Figure 16: The first 3 POD modes (a,c,e) and PSD spectra (b,d,f) of their corresponding temporal variation coefficients; mode 1 (a-b), mode 2 (c-d) and mode 3 (e-f).


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/6f2fecafe70c5574de69b22829965a75b748cd4d373bd72d2e4e6abf3dbd5473.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/c47f038310a2071c4a49a9f662ea1ced7e79a0997f56d497029ab36e3cd47f80.jpg)



Figure 17: The CCD spectra of the streamwise velocity fluctuations when the observable is located at $y / D = 4.9$ (a) and $y / D = 6.1$ (b), respectively.


# 5. Conclusion

A data-driven method referred to as CCD is proposed in this paper to decompose complex flows into modes ranked by their correlation strength with an observable. The method is based on the canonical correlation analysis in classical statistics. The method is validated for both deterministic and statistical flow events. First, the results show that CCD can effectively extract the observable-correlated flow features while suppressing those uncorrelated in both cases. CCD, therefore, results in more low-rank spectra compared to POD. Second, CCD can effectively extract those observable-correlated flow structures even under low SNRs. Third, numerical validation shows that the sampling frequency and duration of the observable determine the frequency limit and resolution while that of the flow are to ensure the convergence of the cross-correlation. Longer sampling of the flow and including multiple observables can improve the convergence of the resulting CCD modes. Therefore, CCD is particularly suitable for experimental data because long samples can be more conveniently obtained. Lastly, as no linearity is assumed, CCD is capable of extracting nonlinear flow events similar to POD, provided a non-negligible correlation exists between the flow and the observable. 

As an illustrative example, the method is first used to analyse a turbulent channel flow obtained using DNS. The flow structures that are most correlated with the skin friction at the point in the middle of the bottom wall are extracted. It is shown that CCD yields a spectrum of singular values that decays rapidly as the mode number increases compared to POD. The first 6 CCD modes effectively recover more than 80% of the skin friction fluctuations. The extended POD using only one observable point can better target the observable, and is found to be equivalent to the degenerate case of CCD when no time shift between the flow and observable is used. The CCD modes take the form of streamwise streaks slightly above the wall. More importantly, the streamwise and spanwise extent of these streaks are unambiguously determined. As the mode number increases, CCD modes have increasingly short spatial and temporal scales. 

In a subsequent example, CCD is used to decompose the unsteady pressure field of a turbulent subsonic jet using a near-field pressure fluctuation as the observable. Results show that CCD results in a steeper spectrum compared to POD. In particular, the CCD spectrum exhibits a clear low-rank behaviour and the corresponding modes correspond to the large coherent flow structures that convect downstream. The first two CCD modes recover 80% of the energy of the near-field pressure fluctuations. The method is subsequently applied to analyse the unsteady vortex shedding behind a cylinder. It shows that similar modes to POD can be robustly identified, but their strengths depend crucially on the observable and its locations, suggesting that these modes correlate differently with observables at different locations. This diagnosis would be useful for determining the optimal location for placing the feedback sensor in a closed-loop control of the vortex shedding behind a cylinder. 

Note that the examples shown in the paper are only for illustrations. They suffice for the purpose of demonstrating the potential uses of CCD, but further improvements are needed for a more in-depth analysis. For example, we can see that both the sampling frequency and sampling duration in the jet example are rather limited; therefore, a faster sampling of the observable, together with a longer sampling duration, is needed for more accurate diagnosis. In addition, a possible far-field noise diagnosis using pressure or the Lighthill stress tensors as the flow variables may be conducted. These, together with the application of CCD in the spectral space such as that shown in section 2.1, form some of our future work. 

# Acknowledgements

The author wishes to thank Prof. Ann Dowling and Prof. Tim Colonius for the stimulating discussions on including multiple observables and the low-rank properties of CCD modes. The author would like to gratefully thank Prof. Jie Yao for sharing the DNS data of channel flows. The author is very grateful to Dr. Cong Wang and his collaborators for agreeing to use their PIV data on cylinder wakes. The author wishes to thank Dr. I. Naqavi who performed the LES simulations in our last collaborative publication (Lyu et al. 2017), from which part of the data is extracted and reused. The author also wishes to gratefully acknowledge the National Natural Science Foundation of China (NSFC) under the grant number 12472263. 

Declaration of interests. The authors report no conflict of interest. 

# Appendix A. The effects of sampling frequency, duration and multiple observables

In this appendix, we aim to demonstrate the effects of sampling frequency, duration and multiple observables using the one-dimensional synthetic example introduced in section 3.1. We first demonstrate the effects of varying the sampling frequency of the observable $f_{s}^{p}$ . In figure 3, a large sampling frequency of $128 / 2\pi$ is used. This is a sufficiently large number considering one only needs to resolve the approximately maximum frequency of $6/2\pi$ required by the fifth term in (3.1). According to Nyquist's theorem, a minimum sampling frequency of $12/2\pi$ is needed. To demonstrate the validity of this requirement, we first perform CCD using $f_{s}^{p} = 12/2\pi$ , resulting in extracted modes virtually the same as those in figure 3. Subsequently, we use an under-resolved sampling frequency of $10/2\pi$ while are other parameters remain unchanged. The resulting modes are shown in figure 18. Clearly, although the low-frequency modes shown in figure 18(a) and (b) can still be correctly captured, the high-frequency modes shown in figure 18(c) and (d) start to differ from their correct forms. This is because the under-sampling causes aliasing effects in the decomposition, leading to incorrect modes 4 and 5. 

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/8caa0e061600aba4bfc66fef0b215ecb9cd5e035b8b68936d44066475f7a17d6.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/29af093c51a7cb9ae90c3d7d1a632a023f0277ea93bc2089593633b6ad228e5f.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/8d956b65ab4cd42412075bf0cca3e20aed43639cb59e41fb4a9c5cea2dfe8e92.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/5222f11258621410683f089893b4205cd6273a29d957276156075a37beb08546.jpg)



Figure 18: Extracted CCD modes when the observable is sampled using an under-resolved frequency $f_{s}^{p}=10/2\pi$ while $N=10^{4}$ , Q=128 and $f_{s}^{u}=128/2\pi$ . With this low sampling rate, modes 4 and 5 cannot be captured accurately.


We then demonstrate the effects of varying the sampling duration of the observable. In figure 3, Q is taken as 128 as this is the minimum value to use in order to resolve the flow structures given in (3.1), resulting in a frequency resolution of $f_{s}/Q = 1/2\pi$ . If a smaller Q such as 64 is used, one would expect the failure of resolving the low-frequency structures. This is precisely the case, as shown in figure 19, where Q = 64 while all other parameters remain the same. As shown in figure 19(a), the mode at the lowest frequency of $1/2\pi$ is extracted incorrectly. In addition, the next mode at the frequency of $1/\pi$ seems resolved incorrectly as well (see figure 19(b)). In fact, the modes shown in figure 19(a) and (b) appear to have somehow mixed. This is expected, since the frequency resolution is only $1/\pi$ , but the two modes differ from each other only by $1/2\pi$ . On the other hand, modes shown in figure 19(c) and (d) are characterised by frequencies of $3/\pi$ and $2/\pi$ respectively, and therefore have been resolved correctly. Figure 19 clearly shows that the number of shifted rows Q determines the frequency resolution in a similar manner to that in DFT. 

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/b542af658dcb7e3fbfb053b05b57cc726ddd96a330af4f418020a4f450ec1a7a.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/231fd69522c6ea1fa65d7ae079cccc6e77630c3de88daae4f89209cbc84811da.jpg)


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Figure 19: Extracted CCD modes when the observable is sampled for a short duration, i.e. Q = 64, while $N = 10^{4}$ and $f_{s}^{p} = f_{s}^{u} = 128/2\pi$ . With this short sampling duration, modes 1 and 2 cannot be resolved correctly.


We are now in a position to illustrate the effects of sampling frequency and duration of the flow u. Figure 3 shows that the extracted modes are subject to small random noise. This noise decreases rapidly as N increases. As mentioned in section 2.3, this is because the sampling rate and duration of the flow are to ensure the convergence of the correlation tensor (they do not affect the frequency limit and resolution). In figures 2 and 3, N is taken to be 10000. This is a large number because we deliberately chose an SNR that is as low as $10^{-4}$ . A small N can also be used at the expense of augmented noise in the resolved modes. For example, figure 20 shows the extracted CCD modes when N = 1000 (a-b) and N = 100 (c-d), respectively. Only modes 1, 2 and 4 are shown for brevity. Clearly, we see that using a duration of N = 1000 results in CCD modes that converge less well but are still unambiguously identified. When N reduces to 100, the resolved CCD modes are further corrupted by the random noise, nevertheless, the structures of the modes can still be recognized. Note that N here denotes the number of cycles of the sampled flow. At the lowest frequency of 100 Hz widely used in the fluid mechanics literature, N = 100 yields a sampling duration of 1 s. In the experiments, a record of 100 s can be easily managed. 

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/2b63b306e6f06875a6f646c5b9f2df292aba101e31f07d7b414f8a0f96454439.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/3a1b364cfb2f8137241ba3187610264e9f3a3c7d90403567919081740506b26c.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/b7bec8203f82af7afc047aaaf00c0621b96d2a666cc518a3b18cf753a55a0b11.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/87919f87582ba19f15ba4cb43d057b185d0673b5d0f3e1960d31179c9e0d65ba.jpg)



Figure 20: The extracted CCD modes when N = 1000 (a-b) and N = 100 (c-d) while other parameters remain the same as those in figure 3. Convergence increasingly deteriorates as N decreases.


Figure 20 shows that the CCD modes are corrupted significantly by random noise at an SNR $< 10^{-4}$ when $N = 100$ . However, when the random noise is only two orders of magnitude more energetic, $N = 100$ yields sufficiently well-resolved CCD modes. In general, we find that to obtain the same level of convergence, $N$ scales roughly as 1/SNR. Conversely, if longer samples are readily available, CCD can robustly extract the observable-correlated flow events at the same level of convergence at an even lower SNR. Therefore, CCD is especially suitable for analysing data acquired in experiments, where the data may be recorded for as long as desired. 

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/ff840f5d3e2bb2d3c8effb1047afa59ad6eb1f74606f1d6d935828cf74750804.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/08b39d3e2e764d3e4e8c3fd9caa05e394c22a5b863b70c947269fb8ac42e31ed.jpg)


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Figure 21: The extracted CCD modes using a low flow sampling frequency of $f_{s}^{u} = 16/2\pi$ while $N = 8 \times 10^{4}$ , Q = 128, $f_{s}^{p} = 128/2\pi$ . Negligible change occurs compared to figure 3 because the same number of flow snapshots are used for the temporal average.


It is worth noting that although the extracted modes in figure 20 are subject to stronger noise contamination due to the insufficient convergence level, they do not suffer from aliasing effects due to under sampling. The fact that the sampling frequency and duration of the flow do not affect the frequency limit and resolution can be even more clearly demonstrated in figure 21. Figure 21 shows the extracted modes when the flow is only sampled at $16 / 2\pi$ . However, to exclude the effects of insufficient convergence, the total number of flow snapshots is kept the same. Comparing figures 3 and 21 reveals that the extracted modes are virtually identical, demonstrating the independence of the frequency resolution on the temporal duration and frequency of sampling. 

We are now in a position to demonstrate the effects of including multiple observables. The flow field u takes the same form of $(3.1)$ . However, more observables need to be defined. To ensure that the observables are not only similar to $(3.2)$ but also exhibit variations, we construct up to ten observables $p_{i}(t)$ ( $i = 1, 2, 3 \ldots 10$ ) such that 

$$
\begin{array}{l} p _ {i} (t) = \left(1 + 0. 2 r _ {1 i}\right) \cos \left(t - \frac {\pi}{4}\right) + \left(1 + 0. 2 r _ {2 i}\right) \sin \left(2 t - \frac {\pi}{3}\right) + \left(1 + 0. 2 r _ {3 i})\right) \cos (4 t) \tag {A1} \\ + (1 + 0. 2 r _ {4 i}) \cos (6 t - \frac {\pi}{1 2}) + 1 0 0 r _ {i} (t), \\ \end{array}
$$

![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/40244f4a1338849948b2a9e818a3f6883f8b412db8f10921fc1ef33c754ae8f1.jpg)


![image](https://cdn-mineru.openxlab.org.cn/result/2026-06-07/b70089ee-fa1a-4e57-9e24-8f0805a1690f/424e568b81c5f4645fa4445be64905ffd44aa417a66d7d119493976b7e75698d.jpg)


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Figure 22: The extracted CCD modes 1 and 4 when 1 (a-b) and 10 (c-d) observables are included. Including multiple observables improves the convergences of the resulting CCD modes.


where $r_{ji}$ (j = 1, 2, 3, 4) are random numbers between [-0.5, 0.5] and $r_{i}(t)$ a random function with a uniform distribution over [-0.5, 0.5]. Note that a very strong random noise (SNR $\sim 10^{-4}$ ) is also added in each observable in order to demonstrate the validity of the decomposition with strongly contaminated observables. 

Following the procedure listed in section 2.4, multiple observables can be included straightforwardly to form the matrix P. When identical parameters to those in figure 3 are used, the extracted modes using either 1 or 10 observables are shown in figures 22(a,b) and 22(c,d), respectively. Only modes 1, 2 and 4 are shown for brevity. Clearly, when the observables are also strongly contaminated, the extracted modes converge less satisfactorily, as shown in figure 22(a-b). However, by including 10 observables, the quality of the extracted modes improves significantly. Figure 22 shows that including more observables can indeed improve the convergence of the resulting CCD modes, particularly when the observables are corrupted by noise. 



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