Data-based, reduced-order, dynamic estimator for reconstruction of nonlinear flows exhibiting limit-cycle oscillations

We apply a data-based, linear dynamic estimator to reconstruct the velocity field from measurements at a single sensor point in the wake of an aerofoil. In particular, we consider a NACA0012 aerofoil at $\text{Re}=600$ and ${16}^{\circ }$ angle of attack. Under these conditions, the flow exhibits a vortex shedding limit cycle. A reduced-order model of the flow field is extracted using proper orthogonal decomposition (POD). Subsequently, a subspace system identification algorithm (N4SID) is applied to extract directly the estimator matrices from the reduced output of the system (the POD coefficients). We explore systematically the effect of the number of states of the estimator, the sensor location, the type of sensor measurements (one or both velocity components), and the number of POD modes to be recovered. When the signal of a single velocity component (in the stream wise or cross stream directions) is measured, the reconstruction of the first two dominant POD modes strongly depends on the sensor location. We explore this behavior and provide a physical explanation based on the nonlinear mode interaction and the spatial distribution of the modes. When, however, both components are measured, the performance is very robust and is almost independent of the sensor location when the optimal number of estimator states is used. Reconstruction of the less energetic modes is more difficult, but still possible. Finally, we assess the robustness of the estimator at off-design conditions, at $\text{Re}=550$ and 650.

Figure 1

Procedural steps of system identification techniques. Step 1: Numerical simulations/experiments are run and input $\left(u\right)$ and output data $\left(y\right)$ are acquired. Step 2: The model coefficients are computed by maximising the fit between the output of the system and the prediction of the model for part of the available data (training or learning data set). Step 3: The performance of the model is assessed on a different dataset (testing or validation dataset).

Figure 2

(Left) Computational domain and (right) control window for data extraction.

Figure 3

Snapshots of the (left) vorticity field and (right) velocity magnitude. The solid black line denotes the outline of the control window.

Figure 4

Streamwise component of the mean flow velocity. Black lines denote streamlines.

Figure 5

(Left) First 16 POD eigenvalues ${\lambda }_i$ of the correlation matrix and (right) spectrum of the first, third and fifth POD coefficients.

Figure 6

Contours of the (left column) streamwise and (right column) cross-stream velocity components of the (top) first and (bottom) second POD-modes.

Figure 7

Contours of the (left column) streamwise and (right column) cross-stream velocity components of the (top) third and (bottom) fourth POD modes.

Figure 8

Learning dataset: (top) the measurement signal $s\left(t\right)$ and (bottom) first four POD coefficients $y_i\left(t\right)$ obtained by projecting the flow field onto the POD modes (solid black) and predicted by the model (dashed red).

Figure 9

Validation dataset: Performance of the system-identified model. (Top) Input data $s$ and (bottom) comparison between the DNS (black solid) and model prediction for four POD coefficients $y_i$. The model is initialized by ${\mathbf{X}}_e={\mathsf{C}}^{†}\mathbf{Y}$ (red dashed) and ${\mathbf{X}}_{\mathbf{e}}=\mathbf{0}$ (blue dotted) at $t=40.$

Figure 10

Streamwise component of the true oscillatory velocity (left) obtained from the DNS and (right) model prediction. The gray squares represent the position of the estimation sensor. See Supplemental Material video [44].

Figure 11

Goodness of fit $\mathsf{FIT}[\%]$ between the actual oscillatory velocity field and the field estimated by the model at each point: (left) streamwise and (right) cross-stream components of the velocity.

Figure 12

Bode plots (gain and phase) of the transfer functions from the two inputs ($u$, $v$) to the two outputs (first and second POD coefficients).

Figure 13

Spectra of the streamwise $u$ and cross-stream $v$ components of the oscillatory velocity composing the input $s={[u,v]}^{\top }$.

Figure 14

Bode plots (gain and phase) of the transfer functions from the two inputs ($u$, $v$) to the two outputs (third and fourth POD coefficients).

Figure 15

$\mathsf{FIT}[\%]$ between the actual first pair of POD coefficients and the coefficients estimated by models composed of two POD modes and two states obtained from different learning datasets. The datasets lengths are (top) 1000 and (bottom) 2000 snapshots, respectively. Left and right columns correspond to different time periods. The input signal is $s=u$.

Figure 16

Influence of the number of states of the estimator on the quality of the identification. $\mathsf{FIT}[\%]$ between the actual first pair of POD coefficients and the coefficients estimated by models composed of two POD modes and (left) four and (right) six states. The input signal is $s=u$.

Figure 17

Influence of the input signal on the quality of the model. $\mathsf{FIT}[\%]$ between the actual first pair of POD coefficients and the coefficients estimated by models obtained using (left) the cross-stream $s=v,$ and (right) both the streamwise and cross-stream $s={\left[u,v\right]}^{\top }$ components of the oscillatory velocity. The models are composed of two POD modes and (top) two, (middle) four, and (bottom) six states.

Figure 18

(Left) Optimal number of states estimated by the N4SID algorithm for two POD modes and (right) $\mathsf{FIT}[\%]$ between the actual first pair of POD coefficients and the coefficient estimated by such models. The models are obtained using (top) the streamwise $s=u$, (middle) the cross-stream $s=v$, and (bottom) both the streamwise and cross-stream $s={\left[u,v\right]}^{\top }$ components of the oscillatory velocity.

Figure 19

Reynolds stress of the (top) streamwise and (bottom) cross-stream oscillatory velocity projected on (left) the first and second and (right) the third and fourth POD modes, respectively. Point $(x_{\mathrm{s}},y_{\mathrm{s}})=(1.82,-0.04)$ is denoted with an open square.

Figure 20

Square root of the ratio of Reynolds stresses contained in modes $3-m$ and $1-2$ superimposed with the locations of the sensor where the $\mathsf{FIT}[\%]$ is less than 60% (shown as red dots). (left) $\overline{u^2}$ Reynolds stress, (right) $\overline{v^2}$ Reynolds stress.

Figure 21

(Left) Optimal number of states estimated by the N4SID algorithm for four POD modes and (right) $\mathsf{FIT}[\%]$ for such models. The models are obtained using (top) the streamwise $s=u$, (middle) the cross-streamwise $s=v$ and (bottom) both the streamwise and cross-streamwise $s={[u,v]}^T$ components of the oscillatory velocity, respectively.

Figure 22

(Left) Optimal number of states estimated by the N4SID algorithm for the third and fourth POD modes and (right) $\mathsf{FIT}[\%]$ for such models. The models are obtained using (top) the streamwise $s=u$, (middle) the cross-streamwise $s=v$ and (bottom) both the streamwise and cross-streamwise $s={\left[u,v\right]}^{\top }$ components of the oscillatory velocity, respectively.

Figure 23

Performance of the estimator at off-design conditions: models extracted from data obtained at $\text{Re}=600$ are evaluated with datasets collected at (left) $\text{Re}=550$ and (right) $\text{Re}=650.\mathsf{FIT}[\%]$ between the actual (top) first and (bottom) second pair of POD coefficients and the predictions of the model. The models explicitly decouple the first and second pair of POD modes and use the optimal number of states given in the bottom row of Figs. 18 and 22 for the first and second pair of coefficients, respectively. We have used $s={\left[u,v\right]}^{\top }.$

Figure 24

Streamwise component of the (left) true oscillatory velocity obtained from a simulation at $\text{Re}=550$ and (right) the prediction of a model extracted from data measured at $\text{Re}=600$. The gray squares represent the position of the estimation sensor.