| POD and DMD methods are the two most commonly used methods for dimensionality reduction and mode decomposition in fluid mechanics to process flow field data.The resulting low-dimensional features and decomposed modes can be used for flow analysis,reduced-order modelling and flow control.However,these traditional linear methods cannot be always effective for the high Reynolds number flow which contains more complex flow features,and more decomposed modes are required for using linear methods to describe the main features of fluid flows.Therefore,this work proposed to extend the currently wide used linear mode decomposition and modelling methods based on the neural network frameworks which are the most popular and powerful nonlinear data processing tool.Firstly,the definition of linear mode decomposition method is extended to the nonlinear one.The decomposed modes describing the flow field(also known as basis function,linear transformation acting on the low-dimensional features)are extended to the nonlinear transformation acting on nonlinear low-dimensional features,which means that the flow field is decomposed into a series of nonlinear low-dimensional features and the corresponding nonlinear transformations.Then,a nonlinear mode decomposition framework for flow field is constructed based on convolutional autoencoder structure,corresponding to the definition of nonlinear mode decomposition.The nonlinear mode decomposition frameworks are designed for time-resolved flow fields and multi-variable flow fields respectively,and the prediction models have also been constructed based on the nonlinear low-dimensional features and the decomposed modes.For the time-resolved flow field,since there is a dynamical system describing the time evolution of the flow field snapshots,the basic idea of designing the nonlinear mode decomposition framework is assuming that there is also a simpler dynamical system describing the latent variables corresponding to the flow field snapshots.Referring to the DMD method used for extracting the frequency characteristics of the flow field snapshots,the projecting POD modes are extended to the nonlinear modes obtained by the nonlinear autoencoder,and the relationship between latent variables is chosen to be linear or nonlinear based on the flow patterns.The proposed method is applied to the periodic cylinder flow and transient cylinder flow respectively to verify the effectiveness,both the low-dimensional features with strong physical suggesting and the decomposed modes with more flow information are obtained.For the multi-variable flow field,the idea of designing nonlinear mode decomposition is to identify the nonlinear decomposed modes based on a series of independent lowdimensional features.It uses the variational autoencoder framework to identify the features from flow field data,and then learn the modes by the convolutional neural networks according to the definition of nonlinear mode decomposition.Several numerical cases are used to verify the performance of the independent parameter identification,the results show that VAE framework can successfully identify the minimal number of independent variables.Then,the periodic cylinder flows with different Reynolds numbers are tested for nonlinear mode decomposition.The resulting low-dimensional features show the strength of the flow stability,and the nonlinear decomposed fields show the specific contents of the low-dimensional features and modes.It can be concluded that the flow shows steady state when ri is close to 0,and unsteady vortex shedding when ri is far away from 0.Moreover,the prediction model of nonlinear dynamical system based on neural network method has also been studied in this work.In order to improve the performance of the neural network model for PDEs modelling,two improvements for conventional convolutional neural network have been proposed.Firstly,the hard boundary conditions are encoded into the convolutional neural networks based on the padding operation;Secondly,a linear and nonlinear separate convolutional neural network is suggested for learning the PDEs.A number of time-dependent PDEs equations are used to verify the accuracy and stability of the proposed method.The results show that the linear channel can significantly mitigate the overfitting problem,especially in high-dimensional or chaotic dynamic systems. |
