2.1. Parameters of numerical simulation

The detector of the turbulent region in the flow past a circular cylinder is trained and validated using the data obtained from direct numerical simulation (DNS) and large-eddy simulation (LES). Figure 1 illustrates the computational domain and coordinates of the simulations. As shown, $x$, $y$ and $z$ denote the streamwise, cross-flow and spanwise directions, respectively. In the $x$–$y$ plane, the origin of the coordinates is located at the centre of the cylinder. The computational domain size is $L_x\times L_y \times L_z = 50D \times 30D \times 3.2 D$, where $D$ is the diameter of the cylinder. The inlet of the computational domain is $13.5D$ away from the centre of the cylinder. The number of grid points is $N_x \times N_y \times N_z = 768 \times 512 \times 256$. Around the cylinder, 80 grid points are used within each diameter of the cylinder in the $x$- and $y$-directions, giving a resolution of $\varDelta _x = \varDelta _y = 1.25\times 10^{-2}D$. The region with the finest grid resolution is shown using the dashed box in figure 1(b). To show the refined region clearly, only part of the computational domain is plotted in figure 1(b). The grid is stretched gradually to the ends of the computational domain in the $x$–$y$ plane. In the $z$-direction, the grid is evenly spaced, and the resolution is $\varDelta _z = 1.25\times 10^{-2}D$.

Figure 1.

Computational domain and coordinate system of DNS and LES of flow past a circular cylinder. (a) Three-dimensional view and (b) two-dimensional view in an $x$–$y$ plane near the cylinder. The boxes in panel (b) show the regions where training samples are chosen. The samples for the non-turbulent state are chosen from the dashed box for $Re=100$ and the dash-dotted box for $Re=3900$. The turbulent samples are chosen from the solid box for $Re=3900$.

The simulations were conducted by solving the Navier–Stokes equations for incompressible flows using DNS and LES. In the $x$-direction, the inflow velocity is uniform, and a convective condition is applied at the outlet. The boundary conditions in the $y$- and $z$-directions are free-slip and periodic, respectively. A second-order central difference scheme is utilized for spatial discretization, and a second-order Runge–Kutta method is employed for time integration. A sharp-interface immersed-boundary method is used to capture the geometry of the cylinder. The details of the numerical methods are given in Cui et al. (Reference Cui, Yang, Jiang, Huang and Shen2018).

We have run simulations of two cases for $Re = UD/\nu = 100$ and 3900, respectively, where $U$ is the inflow velocity and $\nu$ is the kinematic viscosity. The flow for $Re=100$ is non-turbulent, while the wake region of the flow for $Re=3900$ is turbulent. Owing to the coexistence of turbulent and non-turbulent regions at $Re=3900$, this case is suitable for examining the proposed method for turbulence detection. The dynamic Smagorinsky model (Germano et al. Reference Germano, Piomelli, Moin and Cabot1991; Lilly Reference Lilly1992) is used to calculate the subgrid-scale stresses in the case for $Re = 3900$, while no subgrid-scale model is needed in the case for $Re = 100$. We note here that while DNS is ideal for generating the data, it is unfeasible to conduct DNS for the $Re=3900$ case due to the limitation in the computational power. However, the data for training the detectors are all chosen inside the dashed box in figure 1(b), where the grid resolution is high. Specifically, the value of $\varDelta / \eta$ is smaller than 2.0 inside the dashed box, where $\varDelta = (\varDelta _x \varDelta _y \varDelta _z)^{1/3}$ is the grid size and $\eta$ is the Kolmogorov length scale. We have also examined the value of the subgrid-scale eddy viscosity inside the box, and found it to be two orders of magnitude smaller than the kinematic viscosity, indicating that the simulation inside the box is essentially DNS. Previous numerical studies of flow past a circular cylinder also indicate that LES can make reliable predictions on the dominant large-scale physical processes, including vortex shedding and transition (Kravchenko & Moin Reference Kravchenko and Moin2000). Therefore, we believe that the data resolution is sufficient to capture the main flow physics in the wake of a circular cylinder. Actually, in many flows in nature, such as oceanic and atmospheric flows, the data obtained from either numerical simulation or field measurement cannot resolve the Kolmogorov length scale. Therefore, detecting turbulence with under-resolved data is an important research topic, which is rather challenging due to the lack of information of small-scale turbulent motions and the uncertainty in the flow data. Haller (Reference Haller2002) proposed methods for detecting Lagrangian structures using the Okubo–Weiss criterion and finite-time Lyapunov exponents, and found that even though a large error may exist in the flow data, the predictions on Lagrangian coherent structures are reliable. In the present study, we focus on turbulent detection with well-resolved data as in most previous studies in the context of Eulerian approaches.

2.2. Validation of LES data

Before proceeding to apply the machine learning method for turbulence detection, it is necessary to examine the quality of the LES data. The turbulent statistics are validated against the experimental results of Lourenco & Shih (see Ma, Karamanos & Karniadakis Reference Ma, Karamanos and Karniadakis2000) and numerical results of Kravchenko & Moin (Reference Kravchenko and Moin2000). Figure 2 shows the profiles of the mean square of streamwise velocity fluctuations $\overline {u^\prime u^\prime }$ at three locations, $x/D=1.06$, 1.54 and 2.02, respectively, in the wake of a circular cylinder at $Re=3900$. The prime denotes fluctuations, defined as $\phi ^\prime = \phi - \bar {\phi }$, where $\phi$ is an arbitrary physical variable, and the overline represents time averaging. It is seen from the figure that the present results are in agreement with previous numerical and experimental results. We have also examined other quantities including the mean streamwise velocity $\bar {u}$ and mean cross-flow velocity $\bar {v}$. The results are also in good agreement with previous results. We note here that the novelty of the numerical scheme for conducting DNS and LES is not the focus of the present study. We choose the second-order spatial discretization scheme with the incorporation of the immersed-boundary method because of its high computational speed. As we use a fine grid resolution that resolves the Kolmogorov length scale near the cylinder, we generate reliable data of flow field for the present study of turbulence detection.

Figure 2.

Transverse profiles of the mean square of streamwise velocity fluctuations $\overline {u^\prime u^\prime }$ at three downstream locations in the wake of a circular cylinder at $Re=3900$.

3.1. Input features

In the present study, we propose to use flow invariants as the input features for turbulence detection. The input vector $\boldsymbol {X}$ consists of eight features,

(3.1)\begin{align} \boldsymbol{X} & = [X_0,X_1,\ldots\ldots,X_7] \nonumber\\ &=[k,I_2(\boldsymbol{S}^\prime),I_3(\boldsymbol{S}^\prime), I_2(\boldsymbol{\varOmega}^\prime), \nonumber\\ &\qquad\ I_2(\boldsymbol{S}^\prime \boldsymbol{\cdot} \boldsymbol{S}^\prime), I_3(\boldsymbol{S}^\prime \boldsymbol{\cdot} \boldsymbol{S}^\prime), I_2(\boldsymbol{\varOmega}^\prime \boldsymbol{\cdot} \boldsymbol{\varOmega}^\prime), I_2(\boldsymbol{S}^\prime \boldsymbol{\cdot} \boldsymbol{\varOmega}^\prime+ \boldsymbol{\varOmega}^\prime \boldsymbol{\cdot} \boldsymbol{S}^\prime)] , \end{align}

where $k= u_i^\prime u_i^\prime / 2$ is the kinetic energy of instantaneous velocity fluctuations; $I_i(\,)$ denotes the $i$-th invariant of a tensor; $\boldsymbol {S}^\prime = (\boldsymbol {A}^\prime + \boldsymbol {A}^{\prime \textrm {T}})/2$ and $\boldsymbol {\varOmega }^\prime = (\boldsymbol {A}^\prime - \boldsymbol {A}^{\prime \textrm {T}})/2$ are the fluctuations of the strain-rate tensor and rotation-rate tensor, respectively, with $\boldsymbol {A}^\prime = \boldsymbol {\nabla } \boldsymbol {u}^\prime$ being the gradient tensor of velocity fluctuations. The superscript ‘T’ represents the transpose of a matrix. Note that we have considered all invariants of the first- and second-order algebraic polynomials of $\boldsymbol {S}^\prime$ and $\boldsymbol {\varOmega }^\prime$. However, each input sample includes only eight features, because some invariants are trivial. Specifically, the values of some invariants are zero (for example, $I_1(\boldsymbol {S}^\prime ) \equiv 0$) and some invariants are not independent (for example, $I_1(\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime ) \equiv -2I_2(\boldsymbol {S}^\prime )$). Each feature is normalized by its standard deviation over all samples. Besides the eight input features, a supervising label $L$ is needed as an input in the supervised classification methods. If the flow state is known as ‘non-turbulent’ or ‘turbulent’, the value of $L$ is given as 0 or 1, respectively.

3.2. Mathematical background of the choice of input features

The input features are chosen based on flow physics. In this section, we introduce the physical background of the choice of input features. To capture characteristic physical processes of turbulent flows, we determine the input features according to the following governing equation of the velocity fluctuation:

(3.2)\begin{equation} \frac{\partial \boldsymbol{u}^\prime}{\partial t} = - \boldsymbol{u}^\prime \boldsymbol{\cdot} \boldsymbol{\nabla} \bar{\boldsymbol{u}} - \bar{ \boldsymbol{u}} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}^\prime - \boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{u}^\prime \boldsymbol{u}^\prime) + \boldsymbol{\nabla} \boldsymbol{\cdot} \overline{\boldsymbol{u}^\prime \boldsymbol{u}^\prime} - \frac{\boldsymbol{\nabla} p}{\rho} + \nu {\rm \Delta} \boldsymbol{u}^\prime , \end{equation}

where $p$ represents the pressure, $\rho$ and $\nu$ are the density and kinetic viscosity of the fluid, respectively, and $\varDelta = \boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {\nabla }$ is the Laplacian operator. In (3.2), there are two important tensors $\boldsymbol {u}^\prime \boldsymbol {u}^\prime$ and $\overline {\boldsymbol {u}^\prime \boldsymbol {u}^\prime }$. The former one, $\boldsymbol {u}^\prime \boldsymbol {u}^\prime$, has one non-trivial invariant, i.e. the first invariant $I_1(\boldsymbol {u}^\prime \boldsymbol {u}^\prime ) = 2k$. As a result, the kinetic energy of the velocity fluctuation $k$ is chosen as an input feature. The latter one, $\overline {\boldsymbol {u}^\prime \boldsymbol {u}^\prime }$, is not considered for turbulence detection, because it is a time-averaged quantity, but the flow state should be time dependent. For the same reason, the gradient of the mean flow $\boldsymbol {\nabla } \bar {\boldsymbol {u}}$ is not considered. Further from (3.2), it is understood that the fluctuation of the velocity gradient tensor $\boldsymbol {\nabla } \boldsymbol {u}^\prime$ also participates in the transport of the velocity fluctuation. To choose the invariants corresponding to $\boldsymbol {\nabla } \boldsymbol {u}^\prime$, we decomposed it into the symmetric part $\boldsymbol {S}^\prime$ and anti-symmetric part $\boldsymbol {\varOmega }^\prime$. Therefore, the invariants of $\boldsymbol {S}^\prime$ and $\boldsymbol {\varOmega }^\prime$ are chosen as the input features. To choose more input features, we consider the transport equations of $\boldsymbol {S}^\prime$ and $\boldsymbol {\varOmega }^\prime$, which can be written, respectively, as

(3.3)\begin{align} \frac{\mathrm{D} \boldsymbol{S}^\prime}{\mathrm{D} t} &= - \boldsymbol{u}^\prime \boldsymbol{\cdot} \boldsymbol{\nabla} \bar{\boldsymbol{S}} - (\bar{\boldsymbol{S}} \boldsymbol{\cdot} \boldsymbol{S}^\prime + \boldsymbol{S}^\prime \boldsymbol{\cdot} \bar{\boldsymbol{S}}) - (\bar{\boldsymbol{\varOmega}} \boldsymbol{\cdot} \boldsymbol{\varOmega}^\prime + \boldsymbol{\varOmega}^\prime \boldsymbol{\cdot} \bar{\boldsymbol{\varOmega}}) - \boldsymbol{S}^\prime \boldsymbol{\cdot} \boldsymbol{S}^\prime - \boldsymbol{\varOmega}^\prime \boldsymbol{\cdot} \boldsymbol{\varOmega}^\prime\nonumber\\ &\quad + \boldsymbol{\nabla}\left(\boldsymbol{\nabla} \boldsymbol{\cdot} \overline{\boldsymbol{u}^\prime \boldsymbol{u}^\prime}\right) - \frac{\boldsymbol{\nabla}(\boldsymbol{\nabla} p)}{\rho} + \nu {\rm \Delta} \boldsymbol{S}^\prime \end{align}

and

(3.4)\begin{align} \frac{\mathrm{D} \boldsymbol{\varOmega}^\prime}{\mathrm{D} t} &= - \boldsymbol{u}^\prime \boldsymbol{\cdot} \boldsymbol{\nabla} \bar {\boldsymbol{\varOmega}} - (\bar{ \boldsymbol{S}} \boldsymbol{\cdot} \boldsymbol{\varOmega}^\prime + \boldsymbol{\varOmega}^\prime \boldsymbol{\cdot} \bar{\boldsymbol{S}}) - (\bar{ \boldsymbol{\varOmega}} \boldsymbol{\cdot} \boldsymbol{S}^\prime + \boldsymbol{S}^\prime \boldsymbol{\cdot} \bar{\boldsymbol{\varOmega}}) - (\boldsymbol{S}^\prime \boldsymbol{\cdot} \boldsymbol{\varOmega}^\prime + \boldsymbol{\varOmega}^\prime \boldsymbol{\cdot} \boldsymbol{S}^\prime) \nonumber\\ &\quad + \nu {\rm \Delta} \boldsymbol{\varOmega}^\prime . \end{align}

It is seen that the symmetric tensors $\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime$ and $\boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime$ occur in the transport equation of $\boldsymbol {S}^\prime$, while the anti-symmetric tensor $\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime + \boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime$ occurs in the transport equation of $\boldsymbol {\varOmega }^\prime$. Therefore, the non-trivial invariants of $\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime$, $\boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime$ and $\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime + \boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime$ are chosen as the input features. The tensors corresponding to fluid flow are actually unlimited due to the turbulence closure problem. For example, beyond the above second-order algebraic polynomials, the third-order algebraic polynomials $\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime$, $\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime + \boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime$, $\boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime$, $\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime + \boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime$, $\boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime$ and $\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime$ occur on the right-hand side of the transport equations of $\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime$, $\boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime$ and $\boldsymbol {S}^\prime \boldsymbol {\cdot } \boldsymbol {\varOmega }^\prime + \boldsymbol {\varOmega }^\prime \boldsymbol {\cdot } \boldsymbol {S}^\prime$. From the test results shown in § 4.5, it is understood that the inclusion of the invariants of the third-order algebraic polynomials of $\boldsymbol {S}^\prime$ and $\boldsymbol {\varOmega }^\prime$ does not alter the detected turbulent/non-turbulent interface. Therefore, we confine our choice of the tensors up to the second-order algebraic polynomials of $\boldsymbol {S}^\prime$ and $\boldsymbol {\varOmega }^\prime$.

3.3. Training samples

The turbulence detector is trained using XGBoost, a supervised classification method. The samples used for training and validating the detector must be chosen from the flow region where the flow is known as ‘turbulent’ or ‘non-turbulent’. The samples of non-turbulent flow are chosen from both the $Re=100$ case and the $Re=3900$ case. It is known that at $Re=100$, the entire flow field is non-turbulent, while at $Re=3900$, the flow field in the upstream of the cylinder is also non-turbulent. Therefore, we choose the flow field in the boxes for $(x,y) \in ([-1.0D, 3.0D], [-1.5D, 1.5D])$ (the dashed box in figure 1b) at $Re=100$ and $(x,y) \in ([-1.0D, 0.5D], [-1.5D, 1.5D])$ (the dash-dotted box in figure 1b) at $Re=3900$ as the samples of non-turbulent flow state. The samples of turbulent flow state are all chosen from the case for $Re=3900$. According to previous studies of the flow past a circular cylinder at $Re = 3900$ (see e.g. Kravchenko & Moin Reference Kravchenko and Moin2000), the transition takes place after flow separation, extending for approximately one cylinder diameter. To choose the samples representing the turbulent flow state, we focus on the wake flow around the centreline of the cylinder for $(x,y) \in ([2.0D, 3.0D] , [-0.5D, 0.5D])$ (the solid box in figure 1b), where the flow is known as turbulent. We note here that the turbulent region is actually much larger than the solid box. However, there are two restrictions in the choice of turbulent samples. The first is that the box should be away from the turbulent/non-turbulent interface. If the box is too close to the interface, the non-turbulent samples would be mixed into the turbulent samples due to the wake meandering. The second restriction is the grid resolution. As described in § 2, the grid is refined near the cylinder, and the simulation is essentially DNS in the dashed box in figure 1(b). Therefore, the turbulent samples are confined in the dashed box to minimize the influences of subgrid-scale model. Considering these two restrictions, the solid box is chosen as the representation of the turbulent flow region. We have examined the impact of the choice of turbulent samples on the results of the detected turbulent/non-turbulent interface. It is found that if the flow region for turbulent samples is doubled or halved, the detected turbulent/non-turbulent interface remains almost unchanged. This indicates that the choice of turbulent samples imposes little influence on the detector obtained from the machine learning methods. The details of the test results are shown in § 4.4. We have also tested other machine learning methods, including the full-connected neural network (FCN) and SOM. It is found that XGBoost provides the most reasonable turbulence detection among various machine learning methods. Therefore, in the main content of this paper, we focus on analysing the results of XGBoost, while those of FCN and SOM are given in appendix A.

Once the training process is finished, the detector can be applied at each grid point to identify the flow state at that point. Such a point-by-point method of turbulence detection is similar to Wu et al. (Reference Wu, Lee, Meneveau and Zaki2019b). In some situations, the point-based method may produce small ‘patches’ (unphysical small turbulent regions in non-turbulent flow, see appendix A). Another type of method that is potentially useful for flow identification is a region-based one, which is used by Ströfer et al. (Reference Ströfer, Wu, Xiao and Paterson2019) to identify the vortices in the wake of an airfoil. The region-based method treats flow structures as objects, and as such it does not generate the ‘patches’. However, a region-based method does not provide a sharp edge to the object as the point-based method does; instead, it outputs an approximate region that contains the flow structures of interest.

3.4. Cost function

In the training process, the following cost function is minimized:

(3.5)\begin{equation} E = H(L,\hat{L}) + \tfrac{1}{2} ||{\boldsymbol{w}}||^2 , \end{equation}

where $H(L,\hat {L})$ is the cross-entropy between the values of the supervising label $L$ (which is either 0 or 1) and the identification label $\hat {L}(s)$ (which is the output of the model, ranging from 0 to 1), defined as

(3.6)\begin{equation} H(L,\hat{L}) = - \sum_s \left[ L(s) \log(\hat{L}(s)) + (1- L(s)) \log(1-\hat{L}(s)) \right], \end{equation}

where $s$ represents the sample index, and the summation is performed over all training samples. Because the value of $L$ is either 0 or 1, the cross-entropy can also be expressed as

(3.7)\begin{equation} H(L,\hat{L}) = \begin{cases} - \log(1-\hat{L}(s)), & \mathrm{if}\ L=0, \\ - \log(\hat{L}(s)), & \mathrm{if}\ L=1. \end{cases} \end{equation}

From the expression for $H(L,\hat {L})$, it is known that its value decreases as the value of $\hat {L}(s)$ approaches that of $L(s)$. The second part of the cost function $E$ is used to penalize the complexity of the model. Specifically, $\boldsymbol {w} = [w_1, w_2, \ldots ]$ is a vector consisting of trainable variables $w_i$ in the model. The value of $w_i$ is randomly initialized, and is trained to minimize the cost function $E$. The details of the model are described in Chen & Guestrin (Reference Chen and Guestrin2016).

The supervised machine learning method, to some extent, mimics the human learning process. In conventional methods of flow state identification, the flow statistics are first studied. The main differences between the turbulent and non-turbulent regions are then summarized. Based on these investigations, some variables that look the most different between the turbulent and non-turbulent regions, such as kinetic energy or vorticity of velocity fluctuations, together with specified threshold values, are proposed as the criteria for identifying the turbulent/non-turbulent interface. In a supervised machine learning method, the first step is to choose training samples. To minimize the human interference with the machine learning process, the supervising label value of the training samples ($L=0$ and 1 for non-turbulent and turbulent samples, respectively) should be given without ambiguity. Therefore, as introduced in § 3.3, the training samples in this study are chosen in the flow region away from the turbulent/non-turbulent interface, where the flow state is exactly known according to the knowledge gained from previous studies. After choosing the training samples, the machine learning method is used to train a detector, which outputs an identification label $\hat {L}$ ranging from 0 to 1. To apply the detector, the input features at an arbitrary location in the flow field are given to the detector. If the value of the identification label $\hat {L}$ is greater than 0.5, it means that the machine learning method ‘thinks’ that the flow at that location is ‘closer’ to the turbulent state; otherwise, the flow state is identified as non-turbulent.

From the above descriptions of the detector, it is understood that the machine learning algorithm essentially identifies the flow state by examining if the testing sample is closer to the turbulent or non-turbulent training samples in the feature space. Although there must exist a threshold between the turbulent and non-turbulent samples, this threshold does not need to be specified explicitly as in the conventional criteria, while instead, it depends implicitly on the samples used for training the detector, of which the flow state is determined according to the knowledge gained from previous studies. Furthermore, the machine learning method can treat multiple input features (eight in the present study) as a combination, while the conventional method is usually proposed based on one or two flow quantities. In these senses, the criteria obtained from machine learning methods are more objective.

3.5. Implementation and hyperparameters of XGBoost

The XGBoost algorithm is compiled into an open source package, of which the Python version is used in the present study. The hyperparameters of XGBoost are summarized in table 1. Note that we only list some of the hyperparameters. If a hyperparameter does not appear in the table, its default value preset in the package is used. The definitions of the hyperparameters of XGBoost are introduced in Chen & Guestrin (Reference Chen and Guestrin2016). In the present study, the hyperparameters are specified to result in the best training accuracy. To be specific, we found that the performance of the detector is mainly influenced by the learning rate, number of trees and the maximum depth of an individual tree. After testing various values of these three hyperparameters, we found an ideal combination of them as listed in table 1, which results in a high training accuracy.

Table 1.

Hyperparameters of XGBoost applied in the present study.

The training accuracy can be examined using the confusion matrix as shown in figure 3. The row and column of the matrix represent the labelled and predicted flow states, respectively. The diagonal and off-diagonal entries show the percentages of consistent and inconsistent classifications, respectively. For example, the entry value of the first row and second column gives the percentage of the samples labelled as the non-turbulent state but identified as the turbulent state. It is seen from the figure that the percentage of consistent identifications is greater than $99\,\%$, indicating a high training accuracy.

Figure 3.

Accuracy of the detector trained by XGBoost. Diagonal entries show percentages of consistent classification and off-diagonal entries show percentages of inconsistent classification.