In this paper, we present a data-driven approach to construct a reduced-order model (ROM) for the unsteady flow field and fluid-structure interaction. This proposed approach relies on (i) a projection of the high-dimensional data from the Navier-Stokes equations to a low-dimensional subspace using the proper orthogonal decomposition (POD) and (ii) integration of the low-dimensional model with the recurrent neural networks. For the hybrid ROM formulation, we consider long short term memory networks with encoder-decoder architecture, which is a special variant of recurrent neural networks. The mathematical structure of recurrent neural networks embodies a non-linear state space form of the underlying dynamical behavior. This particular attribute of an RNN makes it suitable for non-linear unsteady flow problems. In the proposed hybrid RNN method, the spatial and temporal features of the unsteady flow system are captured separately. Time-invariant modes obtained by low-order projection embodies the spatial features of the flow field, while the temporal behavior of the corresponding modal coefficients is learned via recurrent neural networks. The effectiveness of the proposed method is first demonstrated on a canonical problem of flow past a cylinder at low Reynolds number. With regard to a practical marine/offshore engineering demonstration, we have applied and examined the reliability of the proposed data-driven framework for the predictions of vortex-induced vibrations of a flexible offshore riser at high Reynolds number.
In this paper, we discuss a fractional model arising in flow of two incompatible liquids through homogenous porous media with mean capillary pressure. The solution is derived by the application of the Sumudu transform and the Fourier sine transform. The results are received in compact and graceful forms in terms of the generalized Mittag-Leffler function, which are suitable for numerical computation. The mathematical formulation leads to generalized fractional derivative which has been solved by using a numerical technique by employing the iterative process with the help of appropriate boundary conditions. This problem has great importance in petroleum technology.
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