International Journal for Numerical Methods in Fluids最新文献

This work presents a monolithic finite element strategy for the accurate solution of strongly-coupled fluid-structure-electrostatics interaction problems involving a compressible fluid. The complete set of equations for a compressible fluid is employed within the framework of the arbitrary Lagrangian–Eulerian (ALE) fluid formulation on the reference configuration. The proposed numerical approach incorporates geometric nonlinearities of both the structural and fluid domains, and can thus be used for investigating dynamic pull-in phenomena and squeeze film damping in high aspect-ratio micro-electro-mechanical systems (MEMS) structures immersed in a compressible fluid. Through various illustrative examples, we demonstrate the significant influence of fluid compressibility on the dynamics of MEMS devices subjected to constrained geometry and/or high-frequency electrostatic actuation. Moreover, we compare the proposed formulation with the nonlinear compressible Reynolds equation and highlight that, particularly at low pressures and high fluid viscosity, the Reynolds equation fails to provide a reliable approximation to the complete set of equations utilized in our proposed formulation.

A nonlinear correction technique for finite element methods of advection-diffusion problems on general triangular meshes is introduced. The classic linear finite element method is modified, and the resulting scheme satisfies discrete strong extremum principle unconditionally, which means that it is unnecessary to impose the well-known restrictions on diffusion coefficients and geometry of mesh-cell (e.g., “acute angle” condition), and we need not to perform upwind treatment on the advection term separately. Moreover, numerical example shows that when a discrete scheme does not satisfy the strong extremum principle, even if it maintains the global physical bound, non-physical numerical oscillations may still occur within local regions where no numerical result is beyond the physical bound. Thus, it is worth to point out that our new nonlinear finite element scheme can avoid non-physical oscillations around sharp layers in advection-dominate regions, due to maintaining discrete strong extremum principle. Convergence rates are verified by numerical tests for both diffusion-dominate and advection-dominate problems.

In this paper, a high-order compact finite difference method in general curvilinear coordinates is proposed for solving unsteady incompressible Navier-Stokes equations. By constructing the fourth-order spatial discretization schemes for all partial derivative terms of the pure streamfunction formulation in general curvilinear coordinates, especially for the fourth-order mixed derivative terms, and applying a Crank-Nicolson scheme for the second-order temporal discretization, we extend the unsteady high-order pure streamfunction algorithm to flow problems with more general non-conformal grids. Furthermore, the stability of the newly proposed method for the linear model is validated by von-Neumann linear stability analysis. Five numerical experiments are conducted to verify the accuracy and robustness of the proposed method. The results show that our method not only effectively solves problems with non-conformal grids, but also allows grid generation and local refinement using commercial software. The solutions are in good agreement with the established numerical and experimental results.

A local and hierarchical Koopman spectral analysis is proposed to extend Koopman spectral analysis typically used in a linear system or an ergodic process to its application in general nonlinear dynamics. The continuous and analytic Koopman eigenfunctions and eigenvalues, derived from operator perturbation theory, are capable of dealing with a nonlinear transition process with mathematical rigorousness. A proliferation rule is identified to derive high-order eigenvalues and eigenfunctions from lower-order ones, thus various spectral patterns may be generated through recursive proliferations. The locally linear map around each state constructs base local Koopman eigenvalues and eigenfunctions from an algebraic eigenvalue problem, and high-order ones are generated via the proliferation rule to express the systematic nonlinearity. The aforementioned hierarchy simplifies the Koopman spectral analysis and is verified by studying the development of Kármán vortex streets. The triangular chain and the lattice distribution of Koopman eigenvalues confirm the critical role of the proliferation rule and the hierarchy structure of Koopman eigenvalues. The local spectral analysis on the transition process shows that the periodic flow forms as the growth rates of the critical Koopman modes reduce to zero, and meanwhile, the Koopman modes at the same frequency superpose on each other to form the well-known Fourier or Floquet modes, where the latter are the enhanced nonlinear motions due to the alignment of Koopman eigenvalues with the critical ones.

This work presents a comprehensive framework for the sensitivity analysis of the Navier–Stokes equations, with an emphasis on the stability estimate of the discretized first-order sensitivity of the Navier–Stokes equations. The first-order sensitivity of the Navier–Stokes equations is defined using the polynomial chaos method, and a finite element-volume numerical scheme for the Navier–Stokes equations is suggested. This numerical method is integrated into the open-source industrial code TrioCFD developed by the CEA. The finite element-volume discretization is extended to the first-order sensitivity Navier–Stokes equations, and the most significant and original point is the discretization of the nonlinear term. A stability estimate for continuous and discrete Navier–Stokes equations is established. Finally, numerical tests are presented to evaluate the polynomial chaos method and to compare it to the Monte Carlo and Taylor expansion methods.