POD-based reduced-order hybrid simulation using the data-driven transfer function with time-resolved PTV feedback

Abstract

A data-driven system-identification technique is explored for proper orthogonal decomposition (POD)-based reduced-order unsteady simulation integrated with time-resolved particle-image-velocimetry/particle-tracking-velocimetry (PIV/PTV) feedback. Principal interaction pattern analysis is extended to calculate a nonlinear transfer function for the POD-mode evolution. Compared with the transfer function derived from the Galerkin projection of the Navier–Stokes equation, instability is suppressed in this approach by introducing a specific norm to be minimized. A feedback loop is implemented such that multiple POD modes obtained by the snapshot method can be stably tracked and assimilated into the PIV/PTV measurement over time. The proposed algorithm is demonstrated by solving a planar-jet problem at $Re \approx 2{,}000$. Suitable feedback gain is analyzed, and the capability for data assimilation is discussed.

Appendix

As an existing technique, we take a spectral viscosity method (Sirisup and Karniadakis 2004) to stabilize the reduced-order system and compare it with the other approaches. This technique is applicable to general nonlinear ODE systems and compatible with the system of our interest. On Eq. 11, we add an artificial viscous term as follows:

$$\begin{aligned} \frac{{\hbox{d}}a_k}{{\hbox{d}}t}&\approx \sum _{m = 1}^{N_{\mathrm{POD}}} L_{k m} a_{m} + \sum _{m = 1}^{N_{\mathrm{POD}}} \sum _{n = 1}^{N_{\mathrm{POD}}} Q_{k m n} a_{m} a_{n} \nonumber \\&\quad -\,\varepsilon \hat{q}_{k} \Biggl [ \left\langle \frac{\partial {\varvec{\phi }}_{k} }{\partial x}, \frac{\partial \mathbf{u}_0}{\partial x} \right\rangle + \left\langle \frac{\partial {\varvec{\phi }}_{k} }{\partial y}, \frac{\partial \mathbf{u}_0}{\partial y} \right\rangle \nonumber \\&\quad +\,\sum _{m = 1}^{N_{\mathrm{POD}}} a_{m} \left( \left\langle \frac{\partial {\varvec{\phi }}_{k} }{\partial x}, \frac{\partial {\varvec{\phi }}_{m} }{\partial x} \right\rangle + \left\langle \frac{\partial {\varvec{\phi }}_{k} }{\partial y}, \frac{\partial {\varvec{\phi }}_{m} }{\partial y} \right\rangle \right) \Biggr ], \end{aligned}$$

(26)

where the second derivative of the viscous operator is expanded with integration by parts, and the two-dimensional components are explicitly expressed above to clarify it (in addition, each ${\varvec{\phi }}_{k}$ and $\mathbf{u}_0$ has the $u$ and $v$ components). Here, $\hat{q}_{k}$ above is a POD coefficient of a viscosity kernel defined by

$$\begin{aligned} \hat{q}_{k} \equiv \left\{ \begin{array}{lll} 0, &{}\quad {\mathrm{if}}\quad \left| k \right| \le N_{\mathrm{C.O.}}, \\ \exp \left[ - \frac{ \left( k - N_{\mathrm{POD}} \right) ^2 }{ \left( k - N_{\mathrm{C.O.}} \right) ^2 } \right] ,&{}\quad {\mathrm{if}}\quad \left| k \right| > N_{\mathrm{C.O.}}, \end{array} \right.\end{aligned}$$

(27)

where the cut-off mode number is given by $N_{\mathrm{C.O.}} = 7$ referring to Fig. 4. Thus, the artificial viscous term is selectively activated at higher POD modes, mainly through the auto-correlation (i.e., $m = k$) terms due to high-wavenumber disturbances.

The form above is taken from the original (Sirisup and Karniadakis 2004); however, there is an ambiguity regarding the subscript for the viscosity kernel $\hat{q}$. Equation 26 applies $\hat{q}$ to the extracted mode $k$, but this creates a constant feedback term associated with the mean velocity (i.e., the second line), which overpowers the interaction term of the POD modes (i.e., the third line). Thus, this study applies the filter to the mode $m$ and deactivates the mean velocity term (corresponding to $m = 0$) according to Eq. 27, leading to the viscous term of

$$\begin{aligned} - \varepsilon \sum _{m = 1}^{N_{\mathrm{POD}}} \hat{q}_{m} a_{m} \left( \left\langle \frac{\partial {\varvec{\phi }}_{k} }{\partial x}, \frac{\partial {\varvec{\phi }}_{m} }{\partial x} \right\rangle + \left\langle \frac{\partial {\varvec{\phi }}_{k} }{\partial y}, \frac{\partial {\varvec{\phi }}_{m} }{\partial y} \right\rangle \right) . \end{aligned}$$

(28)

The differentiation is similarly computed as explained at the end of Sect. 3.2.

Fig. 17

Comparison of the pole distributions of the transfer matrix between the standard Galerkin projection and the spectral-viscosity method. Symbols: ○, Galerkin projection; *, spectral viscosity

The magnitude of $\varepsilon$ is determined such that the largest growth rate of the eigenvalues (i.e., $Re \left[ \lambda \right]$) is minimized, resulting in $\varepsilon \approx 1.01$. In Fig. 17, the pole distributions of the linear part in the transfer function are plotted before and after applying the viscous term for the optimal case. Although the system is still slightly unstable with the spectral viscosity, the magnitude of the growth rate now becomes comparable to that of the PIP estimation shown in Fig. 8.