2.1. Problem set-up and estimation procedure

For stationary flow under a range of steady control commands, our objective is to estimate the instantaneous two-dimensional velocity field $\boldsymbol{u}_{\boldsymbol{b}}(\boldsymbol{x})$ from the known information, which includes the instantaneous sensor signal $\boldsymbol{s}$ and the control command $\boldsymbol{b}$. Here, the velocity vector is defined as $\boldsymbol{u}_{\boldsymbol{b}}=(u,v)$, and the spatial domain $\boldsymbol{x}=(x,y)$ is bounded in $\mathbb {R}^2$. The control command $\boldsymbol{b}=[b_1,\ldots,b_M]^\text {T}$ is composed of all parametrized actuation instructions from a total of $M$ actuators in the flow. Each actuation instruction is bounded in $\mathbb {R}$ such that the control command $\boldsymbol{b}$ is located in a finite high-dimensional space $\mathcal {D} \in \mathbb {R}^M$. For the fluidic pinball plant that will be examined in this study, $\boldsymbol{b}$ represents the rotation speeds of cylinders that are placed in the flow. For other relevant flow control applications, $\boldsymbol{b}$ may also represent duty cycles of the mini-jet/plasma actuators, shape parameters of the passive control devices, etc. A local sensor signal $\boldsymbol{s}=[s_1,\ldots,s_L]^\text {T}$ contains partial information about the flow field. The sensors are placed at sparse locations $\{\boldsymbol{x}_{\boldsymbol{s}}^{(i)}\}^{L}_{i=1}$, and the measurement is equivalent to a masking function $\mathcal {G}$ that preserves the flow state at sensor locations. The challenge of this study is to establish a mapping function $\mathcal {F}$ from local sensor signal $\boldsymbol{s}$ and the actuation command $\boldsymbol{b}$ to the estimation of the corresponding flow state $\hat {\boldsymbol{u}}_{\boldsymbol{b}}(\boldsymbol{x})$ under control.

To achieve this goal, we propose a data-driven procedure to estimate the flow state under different control laws. Figure 1 outlines the estimation procedure exemplified for the fluidic pinball, which is a benchmark problem that will be examined in this study. This plant contains multiple control commands ($b_1$ to $b_3$) and multiple sensor measurements ($s_1$ to $s_9$); details will be introduced in § 3. For most sensor-based flow state estimation methods, only sensor signals are utilized as the input. However, to include the influence of different control commands to the flow state, in this study we employ a combined input vector $\boldsymbol{q}$ that comprises the sensor signal $\boldsymbol{s}$ and the control command $\boldsymbol{b}$, such that

(2.1)\begin{equation} \boldsymbol{q}=[\boldsymbol{s},\boldsymbol{b}]^{\text{T}}\in\mathbb{R}^{L+M}. \end{equation}

The effectiveness of this algorithmically easy extension of the estimator input in the parameter direction has been proved in Hasegawa et al. (Reference Hasegawa, Fukami, Murata and Fukagata2020). In this pioneering work, a CNN-LSTM-based reduced-order model is established for flow at different Reynolds numbers. Here, we extend this approach and include high-dimensional actuation parameters to the input vector. For the purpose of establishing a data-driven mapping from the combined input $\boldsymbol{q}$ to the flow state ${\boldsymbol{u}}_{\boldsymbol{b}}$, we first construct a database that contains massive input–output pairs under a set of $N_b$ randomly generated control commands. These control commands are denoted as $\mathcal {B}=\{\boldsymbol{b}_j\}_{j=1}^{N_b}$. For numerical plants, the flow snapshots are calculated from the numerical solver, and the sensor signals $\boldsymbol{s}$ are obtained by masking the flow field according to the sensor locations. For experimental applications, the data can be acquired from synchronous measurements between particle image velocimetry and hotwire probes/pressure sensors (for example, see Tu et al. Reference Tu, Griffin, Hart, Rowley, Cattafesta and Ukeiley2013).

Figure 1.

A flow chart of control-oriented full-state estimation exemplified for the fluidic pinball plant and the DNN estimator.

Based on the well-established database described above, data-driven estimation can be performed to optimize and generalize the mapping function. Nevertheless, given the high-dimensional nature of the flow state space, it is computationally expensive to estimate the velocity vector at every point inside the flow domain. Hence a reduced-order representation of the flow field is required prior to the application of any estimation algorithm. Different types of dimensionality reduction techniques have been proposed and are proven to be successful, including but not limited to deep Autoencoder (Fukami, Nakamura & Fukagata Reference Fukami, Nakamura and Fukagata2020), the POD (Lumley Reference Lumley1967) and wavelet transform (Bruce, Koger & Li Reference Bruce, Koger and Li2002). In this work, a POD-based approach is employed to reduce the complexity of the flow snapshots. The application of POD does not involve any pre-training, which is typically computationally expensive, and the resulting POD eigenfunctions are human-interpretable. In this work, the database that includes all flow snapshots is pre-processed with a nearly lossless POD that compresses the high-dimensional data onto an energy-optimum, low-dimensional subspace (see § 2.2). Consequently, the estimation of instantaneous flow snapshots is equivalent to the estimation of the first $N$ POD modal coefficients $\boldsymbol{a}$, which take up the majority of fluctuating energy in the flow.

The generation of the database containing flow snapshots under various control laws, and the application of the dimensionality reduction technique, enable us to estimate the flow state via a spectrum of estimators that are capable of learning the mapping function $\mathcal {F}: \boldsymbol{q}\rightarrow \boldsymbol{a}$ from a total of $N_s$ input–output observation pairs $\mathcal {M} : \{(\boldsymbol{q}^{(j)},\boldsymbol{a}^{(j)})\}_{j=1}^{N_s}$ recorded under different control commands in the set $\mathcal {B}$. In this work, three representative estimators are proposed, and a graphical illustration of these data-driven methods can be found in figure 2. The $k$NN regression acts as a locally nonlinear approach; this method learns the mapping from its neighbourhood observations via a non-parametric, distance-based interpolating process. The DNN establishes a globally nonlinear model and optimizes the internal parameters of the model through an iterative optimization routine. For LSE, a linear relationship between inputs and outputs is first assumed, then a least-squares fit is performed to minimize the estimation error. These estimation methods, as well as the POD-based data compression technique, will be introduced in the following subsections.

Figure 2.

A graphical illustration of the three estimation methods. Scattered points represents true data. LSE (grey) fits the data in a linear manner. $k$NN (red) estimates from the neighbourhood observations (light blue). DNN (yellow) optimizes the input–output relationship by a globally nonlinear function.

2.2. Data pre-processing: the proper orthogonal decomposition

In this study, the snapshot POD (Sirovich Reference Sirovich1987) is employed as a data pre-processing technique, which reduces the computational cost of the estimation by compressing the high-dimensional flow snapshots onto a low-dimensional orthonormal subspace where energy is optimized. As we do not include the time coordinate for the control-oriented flow estimation in this study, Koopman-style decomposition techniques (Chen, Tu & Rowley Reference Chen, Tu and Rowley2012) are not available for the data pre-processing. Given a set of flow snapshots under various control commands in $\mathcal {B}$, a velocity decomposition is first performed that splits the velocity vectors $\boldsymbol{u}_{\boldsymbol{b}}$ into the mean and fluctuating components:

(2.2)\begin{equation} \boldsymbol{u}_{\boldsymbol{b}}(\boldsymbol{x},t) = {\boldsymbol{u}}_{\mathcal{B}} (\boldsymbol{x}) + \boldsymbol{u}'_{\boldsymbol{b}}(\boldsymbol{x},t). \end{equation}

Here, ${\boldsymbol{u}}_{\mathcal {B}}$ represents the mean velocity component, and $\boldsymbol{u}'_{\boldsymbol{b}}$ is the fluctuating counterpart. In this work, the mean velocity ${\boldsymbol{u}}_{\mathcal {B}}$ is defined as the ensemble average over time $t$ and all control laws in the set $\mathcal {B}$:

(2.3)\begin{equation} {\boldsymbol{u}}_{\mathcal{B}}(\boldsymbol{x}) = \langle \boldsymbol{u}_{\boldsymbol{b}}(\boldsymbol{x},t) \rangle_{\boldsymbol{b},t}, \end{equation}

where $\langle {\cdot }\rangle _{\boldsymbol{b},t}$ represents the ensemble average over both control commands and time instances. For the fluctuating velocity $\boldsymbol{u}'_{\boldsymbol{b}}$, we apply the snapshot POD and expand the velocity field based on a series of orthonormal POD eigenfunctions. As the first $N$ POD modes usually capture most of the energy and large-scaled coherent structures in the flow, an expansion of the fluctuating velocity can be formulated as

(2.4)\begin{equation} \boldsymbol{u}'_{\boldsymbol{b}}(\boldsymbol{x},t)=\sum_{k=1}^{N} a_k(\boldsymbol{b},t)\,\boldsymbol{u}_{k}(\boldsymbol{x}). \end{equation}

Here, $\boldsymbol{u}_{k}$ represents the orthonormal eigenfunction of the $k$th POD mode, and $a_k$ the corresponding modal coefficient which contains the energy trace. Since the eigenfunctions $\boldsymbol{u}_{k}$ are invariant with time and control command, the velocity expansion in (2.4) simplifies the full-state estimation problem to the estimation of the POD modal coefficients $\boldsymbol{a}=[a_1,\ldots,a_N]^{\text {T}}$. As a result, a significant reduction of the computational cost can be achieved. After the estimates of the POD coefficients are obtained, the flow field can be reconstructed according to (2.2) and (2.4).

2.3. $k$ nearest neighbours regression

To establish a mapping function $\mathcal {F}:\boldsymbol{q}\rightarrow \boldsymbol{a}$ for the purpose of flow estimation, we first introduce the $k$NN (Stone Reference Stone1977) regression algorithm. This method is a locally nonlinear estimation approach, and a schematic diagram of $k$NN is presented in figure 3. $k$NN adopts a non-parametric approach where the mapping function $\mathcal {F}$ is approximated locally by averaging a subset of $K$ observations in which the input data reside in the same neighbourhood of the observations. In this study, a weighted average over the first $K$ nearest neighbours is carried out based on the distance between the estimation input and the neighbourhood observations, such that the output can be approximated by the equation

(2.5)\begin{equation} {\hat{\boldsymbol{a}}}(\boldsymbol{q}) = \frac{\sum_{i=1}^K \boldsymbol{a}(\boldsymbol{q}^{(i)})\,1/d(\boldsymbol{q},\boldsymbol{q}^{(i)})}{\sum_{i=1}^K 1/d(\boldsymbol{q},\boldsymbol{q}^{(i)})}. \end{equation}

In this equation, $d(\boldsymbol{\alpha}_{1},\boldsymbol{\alpha}_{2})$ represents the distance between vectors $\boldsymbol{\alpha}_{1}$ and $\boldsymbol{\alpha}_{2}$, and the Euclidean distance $\lVert \boldsymbol{\alpha}_{2}-\boldsymbol{\alpha}_{1} \rVert _2$ is adopted throughout this study. Here, $\boldsymbol{a}(\boldsymbol{q}^{(i)})$ denotes the output in the observation set that corresponds to the combined input $\boldsymbol{q}^{(i)}$. The superscript $^{(i)}$ represents the $i$th nearest neighbour in the observation set based on the measurement of distance in the input space, and the parameter $K$ stands for the number of neighbourhood observations that are utilized during the averaging process. In general, the value of $K$ in (2.5) needs to be determined empirically. A small $K$ means that the result can be influenced greatly by the outlying observation and random noise, while a large value is usually computationally expensive. Therefore, a convergence study is carried out to determine the optimal value of $K$ used in this study. Figure 4 shows the normalized mean-squared estimation error (i.e. $E^2$ in (3.5)) of $k$NN under different $K$ values for the fluidic pinball plant, from which $K=10$ is selected in an effort to balance the accuracy and the computational efficiency.

Figure 3.

A schematic of $k$NN for the control-oriented flow estimation.

Figure 4.

Mean-squared error (MSE) of $k$NN under different neighbourhood numbers $K$ in (2.5).

2.4. Deep neural network

As the second approach, we employ DNN to approximate the mapping function $\mathcal {F}$ in a globally nonlinear manner. In general, DNN is composed of several layers of artificial neurons that are connected to one another. For each layer, a linear transformation of the input data $\boldsymbol{y}_{{in}}$ is first performed with tunable weights $\boldsymbol{\mathsf{W}}$ and biases $\boldsymbol{c}$, then a nonlinear activation function $\sigma$ is exerted, such that

(2.6)\begin{equation} \boldsymbol{y}_{{out}} = \sigma(\boldsymbol{\mathsf{W}}^{\rm T}\boldsymbol{y}_{{in}} + \boldsymbol{c}). \end{equation}

Here, $\boldsymbol{y}_{{in}}\in \mathbb {R}^{n_{{in}}}$, $\boldsymbol{y}_{{out}}\in \mathbb {R}^{n_{{out}}}$, $\boldsymbol{\mathsf{W}}\in \mathbb {R}^{n_{{in}} \times n_{{out}}}$ and $\boldsymbol{c}\in \mathbb {R}^{n_{{out}}}$. By stacking multiple layers of artificial neurons sequentially and assigning the output from one layer as the input to the next layer, DNN has the potential to accurately approximate complex nonlinear functions by learning weights and biases from data (Hornik, Stinchcombe & White Reference Hornik, Stinchcombe and White1989).

A schematic of the DNN architecture used for control-oriented flow estimation can be found in figure 5. Here, the sensor signal components $s_1$ to $s_L$, as well as the control command parameters $b_1$ to $b_M$, are concatenated and form the input of the neural network. The output of the neural network contains the truncated POD modal coefficients $a_1$ to $a_N$. Both the combined input vector $\boldsymbol {q}$ and output vector $\boldsymbol {a}$ are re-scaled into the range $[0,1]$ to facilitate convergence. Note that in $k$NN and LSE, the input vector is non-dimensionalized to an amplitude of ${O}(1)$ by the freestream velocity $U_{\infty }$, which is in accord with the recommendation in Burkov (Reference Burkov2019).

Figure 5.

A schematic of a DNN for control-oriented flow estimation.

A brief summary of the DNN architecture applied in this study can be found in table 1. The architecture of the neural network is determined empirically by balancing the estimation accuracy and the computational cost. Based on the dimension of the combined input vector and the POD modal truncation (see § 4.1), the sizes of the input and the output layers are 12 and 78, respectively. Three 256-node hidden layers are fully connected in between, for each neuron for which the rectified linear activation function (ReLU; Agarap Reference Agarap2018) is specified as the activation function.

Table 1.

The set-up of the DNN applied for the fluidic pinball plant.

To minimize the loss function that indicates the performance of the neural network, all tunable weights and biases are optimized with gradient-based back-propagation via an iterative training approach (LeCun, Bengio & Hinton Reference LeCun, Bengio and Hinton2015). In this study, the Adam optimization method (Kingma & Ba Reference Kingma and Ba2014) is adopted, and the corresponding loss function is defined as $L_{\varepsilon }=\langle \lVert \boldsymbol{a}-{\hat {\boldsymbol{a}}}\rVert _2^2 \rangle$, which represents the mean squared error between the desired output $\boldsymbol{a}$ and the actual output $\hat {\boldsymbol{a}}$ over the training samples. To accelerate the training phase and avoid the convergence to local minimum, the iterative back-propagation is achieved based on the evaluation of mini-batches that are partitioned randomly from the training data. In the present work, the batch size is set to 64, and a total number of 500 epochs were carried out to train the model. Other key settings related to the fluidic pinball dataset applied in this work, including the dataset partition and the iterative training process, will be detailed in § 3.4.

Before the iterative training, all weights are initialized randomly following the suggestion in Glorot & Bengio (Reference Glorot and Bengio2010). In this method, the randomly generated initial weights yield Gaussian distributions, and the initial biases are set to zero. To examine the influence of randomness from the network initialization to the estimation outcome, we trained the network 20 times, and for each time the initial weights are randomly generated. We calculate the averaged outputs and their variances based on the test set, and present these results in § 4.

2.5. Linear stochastic estimation

Besides the nonlinear, machine-learning-based methods, we apply LSE (Adrian Reference Adrian1994) as a reference point in this study. LSE represents the most general linear fit between fluctuating sensor signals and POD modal coefficients. The solution of LSE is equivalent to the least-squared estimation error for the estimation of POD modal coefficients. As presented in figure 6, this method analytically optimizes a linear input–output model, such that

(2.7)\begin{equation} {a}_i(\boldsymbol{b},t)=\sum_{j=1}^N \boldsymbol{\mathsf{T}}_{ij}\,q_j(t). \end{equation}

Multiplying by $q_k$ on both sides of (2.7) and taking the ensemble average over $\boldsymbol{b}$ and $t$, we get

(2.8)\begin{equation} \sum_{j=1}^N \boldsymbol{\mathsf{T}}_{ij} \langle q_jq_k\rangle_{\boldsymbol{b},t} = \langle a_i q_k \rangle_{\boldsymbol{b},t}. \end{equation}

Here, $\langle q_jq_k\rangle _{\boldsymbol{b},t}$ denotes the cross-correlation between inputs, and $\langle a_i q_k \rangle _{\boldsymbol{b},t}$ represents the autocorrelation between inputs and outputs. Both matrices can be computed directly from data, and a least-squares solution of $\boldsymbol{\mathsf{T}}_{ij}$ can be obtained by solving (2.8).

Figure 6.

A schematic of the LSE algorithm for the control-oriented flow estimation. The yellow column and light yellow column represent the sensor signal $\boldsymbol {s}$ and control command $\boldsymbol {b}$, respectively. The orange column represents the estimation of the POD modal coefficients $\boldsymbol {a}$. Dots in the transfer matrix correspond to the weights of $\boldsymbol{\mathsf{T}}$, which is visualized under a blue (negative)–white (0)–red (positive) colour scheme.

So far, we have described the control-oriented, sensor-based flow estimation problem, as well as a spectrum of machine learning flow estimation methods that are capable to tackle this problem. In the next section, we will present the numerical plant, i.e. the fluidic pinball, that will be utilized in this study to evaluate the performance of these estimation methods.

3.1. An overview of the fluidic pinball plant

The fluidic pinball plant is a classical control problem that contains rich dynamical behaviours thanks to different flow control inputs (Ishar et al. Reference Ishar, Kaiser, Morzyński, Fernex, Semaan, Albers, Meysonnat, Schröder and Noack2019). This plant is composed of a two-dimensional, uniform flow passing a set of three cylinders of the same diameter $D$. The centres of the three cylinders form an equilateral triangle pointing in the upstream direction, and the side length of the triangle is $1.5D$. The unforced flow exhibits six different Navier–Stokes solutions, including two stable statistically asymmetric periodic vortex sheddings (Deng et al. Reference Deng, Noack, Morzyński and Pastur2020, Reference Deng, Noack, Morzyński and Pastur2021a). The independent rotation of three cylinders allows the control of the wake formed downstream, and is capable of replicating most known control mechanisms, namely, phasor control, stagnation point control, boat tailing and base bleeding, as well as high- and low-frequency forcing (Ishar et al. Reference Ishar, Kaiser, Morzyński, Fernex, Semaan, Albers, Meysonnat, Schröder and Noack2019; Cornejo Maceda et al. Reference Cornejo Maceda, Li, Lusseyran, Morzyński and Noack2021). In this study, only steady, open-loop control commands are considered. Apart from the uncontrolled case with periodic vortex shedding, steady rotations of cylinders enables the realization of stagnation point control, boat tailing and base bleeding. The schematics of these control mechanisms can be found in figure 7. With the stagnation point control, a unified rotation is exerted on the three cylinders, from which the incoming flow is deflected following the cylinder rotation direction thus attenuated periodic vortex shedding (Seifert Reference Seifert2012). Both boat tailing and base bleeding mechanisms starts from the counter-rotation of the two rearward cylinders. With boat tailing, the top cylinder rotates clockwise and the bottom rotates counter-clockwise. This will guide the boundary layer flowing towards the centre region downstream of the cylinders, which delays the flow separation and leads to significant drag reduction (Geropp & Odenthal Reference Geropp and Odenthal2000; Evrard et al. Reference Evrard, Cadot, Herbert, Ricot, Vigneron and Délery2016). When the top cylinder rotates counter-clockwise and the bottom rotates clockwise, the base bleeding mechanism takes place, which generates a fluid jet on the centreline. The jet actuation will limit the interaction between upper and lower shear layers, hence impairing the development of large-scale vortices downstream. The ability of the base bleeding mechanism to reduce bluff-body aerodynamic drag has been reported in Wood (Reference Wood1964), Bearman (Reference Bearman1967) and Howell, Sheppard & Blakemore (Reference Howell, Sheppard and Blakemore2003), among many others. With the deployment of sparse sensors in the flow, the fluidic pinball forms an MIMO system. Here, we focus on the application of sensor signal, along with the control command, to estimate the instantaneous flow field under steady control.

Figure 7.

A graphical illustration of three flow control mechanisms in the fluidic pinball plant. Figures from Lin (Reference Lin2021). (a) Stagnation point control. (b) Boat tailing. (c) Base bleeding.

3.2. Flow configuration and numerical solver

The flow is depicted in a two-dimensional Cartesian coordinate system, where the origin is located on the midpoint of the two rear cylinders. The $x$ axis represents the streamwise direction, and the $y$ axis points perpendicular to the freestream direction. Both axes are scaled by the diameter of the cylinders. The velocity vector is scaled by the freestream velocity $U_{\infty }$, and is denoted by $\boldsymbol{u}=(u,v)$, where $u$ represents the streamwise velocity, and $v$ the transverse velocity. In this study, we consider an incompressible Newtonian fluid with constant density $\rho$ and kinematic viscosity $\nu$. The Reynolds number based on the cylinder diameter $D$ is defined as $Re_{D} = U_{\infty }D/\nu$. Throughout this study, the Reynolds number is set to 100, which corresponds to the asymmetric periodic vortex shedding according to Deng et al. (Reference Deng, Noack, Morzyński and Pastur2020). The computational domain $\varOmega$ is bounded by a rectangular box $[-6,20]\times [-6,6]$ externally and the three cylinders internally:

(3.1)\begin{align} \varOmega&=\{[x,y]^{\text{T}}\in \mathcal{R}^2:[x,y]^{\text{T}}\in[{-}6,20]\times[{-}6,6]\cap(x-x_i)^2\nonumber\\ &\quad +(y-y_i)^2\geqslant 1/4, i=1,2,3\}. \end{align}

Here, $x_i$ and $y_i$ represent the centres of the three cylinders positioned in the flow, where

(3.2)\begin{equation} \left. \begin{array}{@{}ll@{}} x_1={-}3/2\cos(30^\circ), & y_1=0,\\ x_2=0, & y_2={-}3/4,\\ x_3=0, & y_3=3/4.\\ \end{array} \right\} \end{equation}

The computational domain $\varOmega$ is discretized by the unstructured mesh shown in figure 8. This mesh contains 4225 triangles and 8633 vertices. The grid independence of the direct Navier–Stokes solutions has been established by Deng et al. (Reference Deng, Noack, Morzyński and Pastur2020).

Figure 8.

Computational grid for the fluidic pinball simulation.

The boundary conditions for the inflow, upper and lower boundaries are specified as the freestream velocity $U_{\infty }$ and a stress-free condition is assumed at the outflow boundary. To model the steady rotation of the cylinders, a no-slip condition is enforced on the cylinder boundaries. The flow adopts the circumferential velocities on the boundary of the front, bottom and top cylinders, which is specified by $b_1 = U_F$, $b_2 = U_B$ and $b_3 = U_T$. Here, $b_1$ to $b_3$ are scaled with $U_{\infty }$. For each cylinder, the rotation speed on the boundary is within the range $[-2, 2]$. A positive value corresponds to counter-clockwise rotation of the cylinders, and vice versa. The control command can then be written as $\boldsymbol{b}=[b_1,b_2,b_3]^{\textrm {T}}\in [-2,2]^3$. The initial condition for the numerical simulations is the symmetric steady solution calculated from the steady Navier–Stokes equations. We use an in-house implicit finite-element method solver UNS3 (Noack & Morzyński Reference Noack and Morzyński2017) that employs an implicit finite element method to perform numerical integration of Navier–Stokes equations with third-order accuracy in space and time. In this study, only the post-transient data are collected for the purpose of constructing the fluidic pinball database that will be used to model the input–output relationship between the sparse sensors and the flow state. We non-dimensionalize the time with $U_{\infty }$ and $D$. For every control command, 100 snapshots are recorded with a time interval $\Delta t = 1$. The numerical simulation under a single control command takes up around 1 core hour, and the data storage for 100 snapshots is approximately 78 MB. For the unforced flow, the vortex shedding period is approximately 8.5 times the non-dimensionalized time unit. We choose $\Delta t=1$ in an attempt to balance the computational cost and the overall sampling range in time. The generation of control commands, as well as the placement of virtual sensors, will be detailed in the next subsection.

3.3. Control command generation and sensor placement

To establish control-oriented flow estimators, a set of 1000 different control commands $\mathcal {B}=\{\boldsymbol{b}_j\}_{j=1}^{1000}$ are generated randomly inside the control parameter space $[-2,2]^3$. As mentioned in § 3.2, the control parameter $\boldsymbol{b}$ represents the circumferential rotation speed of three cylinders in the flow. Here, we employ the Latin hypercube sampling (McKay, Beckman & Conover Reference McKay, Beckman and Conover1979) to obtain a set of randomly distributed control commands $\mathcal {B}$. As presented in figure 9, this sampling method ensures a nearly uniform distribution of samples inside the multi-dimensional control parameter space. For each control command, we categorize the flow state into periodic and chaotic types after the flow snapshots are obtained from the numerical solver. The classification criteria are based on the relationship between two variables, $\gamma _1$ and $\gamma _2$, obtained from the sampled snapshots. Here, $\gamma _1 = C_L(t)$ and $\gamma _2 = (C_L(t+\Delta t)-C_L(t))/\Delta t$, and $C_L$ represents the lift coefficient. The lift coefficient is computed by integrating pressure over the surfaces of three cylinders. From these plots, different characteristics between periodic and chaotic flows can be distinguished clearly. Representative periodic and chaotic phase pattern approximations are displayed in figure 9. For periodic flows, the discrete points in the plots will form circular or periodic trajectories. In contrast, the patterns in chaotic flows are dispersed and disordered. The classification of the flow state will enable us to evaluate the performance of the flow estimators introduced in § 2 under different types of the flow states.

Figure 9.

Randomly sampled control commands $\mathcal {B}$ in the control space $[-2, 2]^3$ and their projections on the three orthogonal coordinate planes. Periodic and chaotic cases are rendered by red and blue dots, respectively. Representative patterns of $\gamma _1$ versus $\gamma _2$ in each category are also presented in this figure. See the text for the definitions of $\gamma _1,\gamma _2$.

To monitor the flow state, nine virtual probes are placed downstream to measure the discrete velocity signal, in either the streamwise or spanwise component. The velocity probes are deployed as a $3\times 3$ rectangular grid symmetric about the centreline, where the upper and lower sensors measure the streamwise velocity, and the middle ones on the centreline measure the spanwise velocity. The streamwise locations of the sensors are $x = 5, 6.5, 8$, and the spanwise locations are $y = -1.5,0,1.5$. This sensor placement has been validated by Cornejo Maceda et al. (Reference Cornejo Maceda, Li, Lusseyran, Morzyński and Noack2021), in a sense that such arrangement is capable of detecting large-scaled vortex shedding while encompassing phase information between sensors in its feedback flow control experiment. In other words, this arrangement can help to infer the flow state in the dynamically controlled flow with a wide range of control commands. Hence in this work, we continue to use this promising sensor placement to obtain flow state estimation. A summary of the sensor placement can be found in table 2, and a comprehensive visualization regarding the flow state, control parameters and downstream sensing is presented in figure 10.

Table 2.

Virtual sensor locations and the corresponding velocity components.

Figure 10.

Visualization of sensor placement in the flow domain. The background is rendered by an instantaneous vorticity field. The corresponding control command $\boldsymbol{b}$ is visualized by the colours on the three cylinders, under a blue (negative)–white (0)–red (positive) colour scheme.

3.4. Data partition and estimation performance evaluation

According to the steady control command and virtual probe placement described above, the fluidic pinball database is established aiming for sensor-based, control-oriented flow estimation. To preserve the generality of the data-driven estimation and prevent over-fitting, the collected input–output data pairs are partitioned into a training set, a validation set, and a test set based on a random split of the control commands $\mathcal {B}$. The partition of the control commands follows an 80 %–10 %–10 % ratio, and each set is composed of all time instances generated from the corresponding control commands. The training and validation sets compose the observation data $\mathcal {M}$, and will be utilized for the neighbourhood searching in $k$NN and the calculation of $T_{ij}$ in LSE. To prevent over-fitting the training data in DNN, the iterative training and gradient-based back-propagation are performed only on the mini-batched training set. After every epoch, we evaluate the model performance on the validation set. The network parameters corresponding to the minimum validation loss over the 500 epochs are outputted after training. The number of epochs is determined empirically by observing the fact that the training loss will hardly be minimized after 500 epochs. For all estimation methods, the test set is not accessible during the training stage. Therefore, with this separate set, we can evaluate the generalized performance of the proposed methods. All key parameters regarding the fluidic pinball database have been concluded and presented in table 3.

Table 3.

A conclusion of the fluidic pinball database.

To evaluate the performance of each estimation method from various perspectives, different types of errors are defined and calculated based on the data from the test set. In this work, different types of errors will be discussed; the definitions are presented in the following. To evaluate the estimation performance under each control command, the normalized estimation error of the POD modal coefficients is defined as

(3.3)\begin{equation} \varepsilon_{\boldsymbol{a}}(\boldsymbol{b}) = \sqrt{\frac{\langle \lVert \boldsymbol{a}-\hat{\boldsymbol{a}} \rVert_2^2 \rangle_t}{\langle \lVert \boldsymbol{a} \rVert_2^2 \rangle_t}}. \end{equation}

Similarly, the normalized velocity estimation error $\varepsilon _{\boldsymbol{u}}(\boldsymbol{x})$ evaluates the spatial distribution of the velocity estimation error that accounts for all time instances and control commands, which is normalized by the overall fluctuating energy, such that

(3.4)\begin{equation} \varepsilon_{\boldsymbol{u}}(\boldsymbol{x}) = \sqrt{\frac{\langle \lVert \boldsymbol{u}'_{\boldsymbol{b}}(\boldsymbol{x})-\hat{\boldsymbol{u}}'_b (\boldsymbol{x}) \rVert_{2}^2 \rangle_{\boldsymbol{b},t}}{\langle \lVert \boldsymbol{u}'_{\boldsymbol{b}}(\boldsymbol{x}) \rVert_{2}^2 \rangle_{\boldsymbol{b},t}}}. \end{equation}

Furthermore, we define the normalized overall estimation error $E$, which is a scalar representing the overall accuracy of each estimation method. The value of $E$ is determined as the cumulative estimation error over the spatial domain, all control commands, and all temporal instances in the test set normalized by energy, such that

(3.5)\begin{equation} E = \sqrt{\frac{\langle \int_{\varOmega} \lVert \boldsymbol{u}(\boldsymbol{x})-\hat{\boldsymbol{u}}(\boldsymbol{x}) \rVert_{2}^2\,\mathrm{d}\boldsymbol{x}\rangle_{\boldsymbol{b},t}}{\langle \int_{\varOmega} \lVert \boldsymbol{u}(\boldsymbol{x}) \rVert_2^2\,\mathrm{d}\boldsymbol{x}\rangle_{\boldsymbol{b},t}}}. \end{equation}