An Efficient Deep Learning Technique for the Navier-Stokes Equations: Application to Unsteady Wake Flow Dynamics

We present an efficient deep learning technique for the model reduction of the Navier-Stokes equations for unsteady flow problems. The proposed technique relies on the Convolutional Neural Network (CNN) and the stochastic gradient descent method. Of particular interest is to predict the unsteady fluid forces for different bluff body shapes at low Reynolds number. The discrete convolution process with a nonlinear rectification is employed to approximate the mapping between the bluff-body shape and the fluid forces. The deep neural network is fed by the Euclidean distance function as the input and the target data generated by the full-order Navier-Stokes computations for primitive bluff body shapes. The convolutional networks are iteratively trained using the stochastic gradient descent method with the momentum term to predict the fluid force coefficients of different geometries and the results are compared with the full-order computations. We attempt to provide a physical analogy of the stochastic gradient method with the momentum term with the simplified form of the incompressible Navier-Stokes momentum equation. We also construct a direct relationship between the CNN-based deep learning and the Mori-Zwanzig formalism for the model reduction of a fluid dynamical system. A systematic convergence and sensitivity study is performed to identify the effective dimensions of the deep-learned CNN process such as the convolution kernel size, the number of kernels and the convolution layers. Within the error threshold, the prediction based on our deep convolutional network has a speed-up nearly four orders of magnitude compared to the full-order results and consumes an insignificant fraction of computational resources. The proposed CNN-based approximation procedure has a profound impact on the parametric design of bluff bodies and the feedback control of separated flows.Comment: 49 pages, 12 figure

Decomposition of wake dynamics in fluid-structure interaction via low-dimensional models

We present a dynamic decomposition analysis of the wake flow in fluid-structure interaction (FSI) systems under both laminar and turbulent flow conditions. Of particular interest is to provide the significance of low-dimensional wake flow features and their interaction dynamics to sustain the free vibration of a square cylinder at a relatively low mass ratio. To obtain the high-dimensional data, we employ a body-conforming variational fluid-structure interaction solver based on the recently developed partitioned iterative scheme and the dynamic subgrid-scale turbulence model for a moderate Reynolds number. The snapshot data from high-dimensional FSI simulations are projected to a low-dimensional subspace using the proper orthogonal decomposition (POD). We utilize each corresponding POD mode for detecting features: the vortex street, the shear layer and the near-wake bubble. We find that the vortex shedding modes contribute solely to the lift force, while the near-wake and shear layer modes play a dominating role to the drag force. We further examine the fundamental mechanism of this dynamical behavior and propose a force decomposition technique via low-dimensional approximation. We ascertain quantitatively that the shear layer feeds the vorticity flux to the wake vortices and the near-wake bubble during the wake-body synchronization. Based on the decomposition of wake dynamics, we suggest an interaction cycle for the frequency lock-in during the wake-body interaction, which provides the inter-relationship between the high amplitude motion and the dominating wake features. Through our investigation of wake-body synchronization below critical Re range, we discover that the bluff body can undergo a synchronized high-amplitude vibration due to flexibility-induced unsteadiness. The interaction cycle for the wake-synchronization is found to be valid for the turbulent wake flow.Comment: 42 pages, 20 figure