Network-theoretic modeling of fluid–structure interactions

Abstract

The coupling interactions between deformable structures and unsteady fluid flows occur across a wide range of spatial and temporal scales in many engineering applications. These fluid–structure interactions (FSI) pose significant challenges in accurately predicting flow physics. In the present work, two multi-layer network approaches are proposed that characterize the interactions between the fluid and structural layers for an incompressible laminar flow over a two-dimensional compliant flat plate at a 35$^{\circ }$ angle of attack. In the first approach, the network nodes are formed by wake vortices and bound vortexlets, and the edges of the network are defined by the induced velocity between these elements. In the second approach, coherent structures (fluid modes), contributing to the kinetic energy of the flow, and structural modes, contributing to the kinetic energy of the compliant structure, constitute the network nodes. The energy transfers between the modes are extracted using a perturbation approach. Furthermore, the network structure of the FSI system is simplified using the community detection algorithm in the vortical approach and by selecting dominant modes in the modal approach. Network measures are used to reveal the temporal behavior of the individual nodes within the simplified FSI system. Predictive models are then built using both data-driven and physics-based methods. Overall, this work sets the foundation for network-theoretic reduced-order modeling of fluid–structure interactions, generalizable to other multi-physics systems.

Graphical abstract

Appendices

Appendix A: Grid convergence study

The immersed boundary projection methodology used in our study is based on the work by Taira and Colonius [57] and Goza and Colonius [56]. It has been applied in numerous fluid–structure interaction investigations [21, 67, 68]. We conducted a grid convergence analysis focusing on a Reynolds number of $Re = 100$, an angle of attack $\alpha = 35^\circ$, a mass ratio $M_\rho = 3$, and a bending stiffness $K_\textrm{B} = 0.3125$. This was accomplished by implementing four different grid setups.

Our solver employs a multi-domain approach to expedite the computations [58]. The grid spacing we discuss here pertains to the innermost domain featuring the finest grid. We illustrate the mean drag coefficient ($C_D$), its standard deviation, and the duration of a single iteration for each grid in Fig. 9. Additionally, Fig. 9 shows the grid spacing for the various grids.

In our research, we use the grid denoted as G2 for simulations, with a corresponding $\Delta x/c=0.0077$, notably highlighted in red. The subsequent level of refinement, grid G3, results in a mere $0.6\%$ alteration in the mean $C_D$. The standard deviation for $C_D$ registers at 0.01845 for grid $G_2$ and slightly less at 0.01815 for grid $G_3$. Notably, the computational time for a single iteration on grid $G_3$ nearly doubles that of grid $G_2$. Consequently, grid $G_2$ delivers acceptable precision and shorter computational time, making it our choice for all simulations in this case, with a grid spacing of $\Delta x/c=0.0077$.

Fig. 9

Grid convergence study of direct numerical simulation of a flow past a flexible flat plate at Reynolds number $\hbox {Re} = 100$, angle of attack $\alpha = 35^\circ$, mass ratio $M_\rho = 3$, and stiffness $K_\textrm{B} = 0.3125$. The variation of the mean drag coefficient (left), the standard deviation of the drag coefficient (middle), and the time required for a single time step (right) with different grid spacing are shown. The grid resolution marked in red used in this work achieves a good compromise in simulation time and accuracy

Fig. 10

(top) The mean square error in prediction of the test data with different models of sparsity in SINDy. (bottom) The prediction of the vortexlet strength trajectory compared to DNS for $\lambda > 0.6$

Appendix B: Sensitivity of sparsity parameter

Sparse identification of nonlinear dynamics (SINDy) [64] employs an L$_1$ regularization which allows to construct a sparse model for the resulting dynamics. We present a sensitivity analysis of sparsity promoting factor $\lambda$ by first dividing the data set into training and test data. The mean square error in the prediction of test data set for different $\lambda$ is presented in Fig. 10. It is observed that $\lambda > 0.6$ produces minimum error for test data prediction. We employed a value $\lambda =0.6$ for the data-based prediction in the original manuscript.