A POD-based reduced-order finite difference extrapolating model with fully second-order accuracy for non-stationary Stokes equations

A proper orthogonal decomposition (POD) reduced-order finite difference (FD) extrapolating model is established for the channel flow with local expansion denoted by non-stationary Stokes equations. The POD-based reduced-order numerical model to produce the solutions on the time span [T₀, T] (T₀ ? T) are obtained by extrapolation and iteration from the very short time span [0, T₀] information. The guides to choose the number of POD basis and renew POD basis are provided, and an implementation for solving the POD-based reduced-order FD extrapolating model is given. Some numerical experiments are used to show that the POD-based reduced-order FD extrapolating model is feasible and efficient for simulating the channel flow with local expansion.     

Solution of the incompressible Navier–Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.     

A high‐order compact finite‐difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows

A high‐order compact finite‐difference lattice Boltzmann method (CFDLBM) is proposed and applied to accurately compute steady and unsteady incompressible flows. Herein, the spatial derivatives in the lattice Boltzmann equation are discretized by using the fourth‐order compact FD scheme, and the temporal term is discretized with the fourth‐order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high‐order spectral‐type low‐pass compact filter is used to stabilize the numerical solution. An iterative initialization procedure is presented and applied to generate consistent initial conditions for the simulation of unsteady flows. A sensitivity study is also conducted to evaluate the effects of grid size, filtering, and procedure of boundary conditions implementation on accuracy and convergence rate of the solution. The accuracy and efficiency of the proposed solution procedure based on the CFDLBM method are also examined by comparison with the classical LBM for different flow conditions. Two test cases considered herein for validating the results of the incompressible steady flows are a two‐dimensional (2‐D) backward‐facing step and a 2‐D cavity at different Reynolds numbers. Results of these steady solutions computed by the CFDLBM are thoroughly compared with those of a compact FD Navier–Stokes flow solver. Three other test cases, namely, a 2‐D Couette flow, the Taylor's vortex problem, and the doubly periodic shear layers, are simulated to investigate the accuracy of the proposed scheme in solving unsteady incompressible flows. Results obtained for these test cases are in good agreement with the analytical solutions and also with the available numerical and experimental results. The study shows that the present solution methodology is robust, efficient, and accurate for solving steady and unsteady incompressible flow problems even at high Reynolds numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

An optimizing implicit difference scheme based on proper orthogonal decomposition for the two‐dimensional unsaturated soil water flow equation

An optimizing reduced implicit difference scheme (IDS) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) for the two‐dimensional unsaturated soil water flow equation is presented. An ensemble of snapshots is compiled from the transient solutions derived from the usual IDS for a two‐dimensional unsaturated flow equation. Then, optimal orthogonal bases are reconstructed by implementing SVD and POD techniques for the ensemble of snapshots. Combining POD with a Galerkin projection approach, a new lower dimensional and highly accurate IDS for the two‐dimensional unsaturated flow equation is obtained. Error estimates between the true solution, the usual IDS solution, and the reduced IDS solution based on POD basis are derived. Finally, it is shown by means of a numerical example using the technology of local refined grids that the computational load is greatly diminished by using the reduced IDS. Also, the error between the POD approximate solution and the usual IDS solution is proved to be consistent with the derived theoretical results. Thus, both feasibility and efficiency of the POD method are validated. Copyright © 2011 John Wiley & Sons, Ltd.     

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.     

Reduced order modelling method via proper orthogonal decomposition (POD) for flow around an oscillating cylinder

This paper presents reduced order modelling (ROM) in fluid–structure interaction (FSI). The ROM via the proper orthogonal decomposition (POD) method has been chosen, due to its efficiency in the domain of fluid mechanics. POD-ROM is based on a low-order dynamical system obtained by projecting the nonlinear Navier–Stokes equations on a smaller number of POD modes. These POD modes are spatial and temporally independent. In FSI, the fluid and structure domains are moving, owing to which the POD method cannot be applied directly to reduce the equations of each domain. This article proposes to compute the POD modes for a global velocity field (fluid and solid), and then to construct a low-order dynamical system. The structure domain can be decomposed as a rigid domain, with a finite number of degrees of freedom. This low-order dynamical system is obtained by using a multiphase method similar to the fictitious domain method. This multiphase method extends the Navier–Stokes equations to the solid domain by using a penalisation method and a Lagrangian multiplier. By projecting these equations on the POD modes obtained for the global velocity field, a nonlinear low-order dynamical system is obtained and tested on a case of high Reynolds number.     

A reduced‐order approach for optimal control of fluids using proper orthogonal decomposition

In this article, a reduced‐order modeling approach, suitable for active control of fluid dynamical systems, based on proper orthogonal decomposition (POD) is presented. The rationale behind the reduced‐order modeling is that numerical simulation of Navier–Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. The possibility of obtaining reduced‐order models that reduce the computational complexity associated with the Navier–Stokes equations is examined while capturing the essential dynamics by using the POD. The POD allows the extraction of a reduced set of basis functions, perhaps just a few, from a computational or experimental database through an eigenvalue analysis. The solution is then obtained as a linear combination of this reduced set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations (PDEs). It is used here in active control of fluid flows governed by the Navier–Stokes equations. In particular, flow over a backward‐facing step is considered. Reduced‐order models/low‐dimensional dynamical models for this system are obtained using POD basis functions (global) from the finite element discretizations of the Navier–Stokes equations. Their effectiveness in flow control applications is shown on a recirculation control problem using blowing on the channel boundary. Implementational issues are discussed and numerical experiments are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

On the stability and extension of reduced-order Galerkin models in incompressible flows

Proper orthogonal decomposition (POD) has been used to develop a reduced-order model of the hydrodynamic forces acting on a circular cylinder. Direct numerical simulations of the incompressible Navier–Stokes equations have been performed using a parallel computational fluid dynamics (CFD) code to simulate the flow past a circular cylinder. Snapshots of the velocity and pressure fields are used to calculate the divergence-free velocity and pressure modes, respectively. We use the dominant of these velocity POD modes (a small number of eigenfunctions or modes) in a Galerkin procedure to project the Navier–Stokes equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finite-dimensional nonlinear dynamical system in time. The solution of the reduced dynamical system is a limit cycle corresponding to vortex shedding. We investigate the stability of the limit cycle by using long-time integration and propose to use a shooting technique to home on the system limit cycle. We obtain the pressure-Poisson equation by taking the divergence of the Navier–Stokes equation and then projecting it onto the pressure POD modes. The pressure is then decomposed into lift and drag components and compared with the CFD results.     

Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method (FEM) based on a proper orthogonal decomposition (POD) theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save memory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element (FE) scheme is shown to be unconditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations (FDEs).     

Non‐intrusive reduced‐order modelling of the Navier–Stokes equations based on RBF interpolation

We present a new non‐intrusive model reduction method for the Navier–Stokes equations. The method replaces the traditional approach of projecting the equations onto the reduced space with a radial basis function (RBF) multi‐dimensional interpolation. The main point of this method is to construct a number of multi‐dimensional interpolation functions using the RBF scatter multi‐dimensional interpolation method. The interpolation functions are used to calculate POD coefficients at each time step from POD coefficients at earlier time steps. The advantage of this method is that it does not require modifications to the source code (which would otherwise be very cumbersome), as it is independent of the governing equations of the system. Another advantage of this method is that it avoids the stability problem of POD/Galerkin. The novelty of this work lies in the application of RBF interpolation and POD to construct the reduced‐order model for the Navier–Stokes equations. Another novelty is the verification and validation of numerical examples (a lock exchange problem and a flow past a cylinder problem) using unstructured adaptive finite element ocean model. The results obtained show that CPU times are reduced by several orders of magnitude whilst the accuracy is maintained in comparison with the corresponding high‐fidelity models. Copyright © 2015 John Wiley & Sons, Ltd.     

A Helmholtz pressure equation method for the calculation of unsteady incompressibl eviscous flows

A time-implicit numerical method for solving unsteady incompressible viscous flow problems is introduced. The method is based on introducing intermediate compressibility into a projection scheme to obtain a Helmholtz equation for a pressure-type variable. The intermediate compressibility increases the diagonal dominance of the discretized pressure equation so that the Helmholtz pressure equation is relatively easy to solve numerically. The Helmholtz pressure equation provides an iterative method for satisfying the continuity equation for time-implicit Navier–Stokes algorithms. An iterative scheme is used to simultaneously satisfy, within a given tolerance, the velocity divergence-free condition and momentum equations at each time step. Collocated primitive variables on a non-staggered finite difference mesh are used. The method is applied to an unsteady Taylor problem and unsteady laminar flow past a circular cylinder.     

Viscoelastic flows studied by smoothed particle dynamics

A viscoelastic numerical scheme based on smoothed particle dynamics is presented. The concept goes a step beyond smoothed particle hydrodynamics (SPH) which is a grid-free Lagrangian method describing the flow by fluid-pseudo-particles. The relevant properties are interpolated directly on the resulting movable grid. In this work, the effect of viscoelasticity is incorporated into the ordinary conservation laws by a differential constitutive equation supplied for the stress tensor. In order to give confidence in the methodology we explicitly consider the non-stationary simple corotational Maxwell model in a channel geometry. Without further developments the scheme is applicable to ‘realistic’ models relevant for three-dimensional (3D) viscoelastic flows in complex geometries.     

A Newton's method scheme for solving free-surface flow problems

A Newton's method scheme is described for solving the system of non-linear algebraic equations arising when finite difference approximations are applied to the Navier–Stokes equations and their associated boundary conditions. The problem studied here is the steady, buoyancy-driven motion of a deformable bubble, assumed to consist of an inviscid, incompressible gas. The linear Newton system is solved using both direct and iterative equation solvers. The numerical results are in excellent agreement with previous work, and the method achieves quadratic convergence.     

Reduced‐order modeling based on POD of a parabolized Navier–Stokes equation model I: forward model

A proper orthogonal decomposition (POD)‐based reduced‐order model of the parabolized Navier–Stokes (PNS) equations is derived in this article. A space‐marching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to construct the reduced‐order model. In order to improve the accuracy and the stability of the reduced‐order model in the presence of a high Reynolds number, we applied a Sobolev H₁ norm calibration to the POD construction process. Finally, some numerical tests with a high‐fidelity model as well as the POD reduced‐order model were carried out to demonstrate the efficiency and the accuracy of the reduced‐order model for solving the PNS equations compared with the full PNS model. Different inflow conditions and different selections of snapshots were experimented to test the POD reduction technique. The efficiency of the H₁ norm POD calibration is illustrated for the PNS model with increasingly higher Reynolds numbers, along with the optimal dissipation coefficient derivation, yielding the best root mean square error and correlation coefficient between the full and reduced‐order PNS models. Copyright © 2011 John Wiley & Sons, Ltd.     

On the application of pseudospectral FFT techniques to non-periodic problems

The reduction-to-periodicity method using the pseudospectral fast Fourier transform (FFT) technique is applied to the solution of non-periodic problems, including the two-dimensional incompressible Navier–Stokes equations. The accuracy of the method is explored by calculating the derivatives of given functions, one- and two-dimensional convective-diffusive problems, and by comparing the relative errors due to the FFT method with a second-order finite difference (FD) method. Finally, the two-dimensional Navier–Stokes equations are solved by a fractional step procedure using both the FFT and the FD methods for the driven cavity flow and the backward-facing step problems. Comparisons of these solutions provide a realistic assessment of the FFT method.     

数值摄动算法及其CFD格式

作者提出的数值摄动算法把流体动力学效应耦合进NS方程组和对流扩散（CD）方程离散的数学基本格式（MBS）,特别是耦合进最简单的MBS即一阶迎风和二阶中心格式之中,由此构建成一系列新格式,称呼方便和强调耦合流体动力学起见,称它们为流体力学基本格式（FMBS）。构建FMBS的主要步骤是把MBS中的通量摄动重构为步长的幂级数,利用空间分裂和导出的高阶流体动力学关系式,把结点变量展开成Taylor级数,通过消除重构格式修正微分方程的截断误差诸项求出幂级数的待定系数,由此获得非线性FMBS。FMBS的公式是MBS与 （及 ）之简单多项式的乘积, 和 分别是网格Reynolds数和网格CFL数。FMBS和MBS使用相同结点,简单性彼此相当,但FMBS精度高稳定范围大,例如FMBS包含了许多绝对稳定和绝对正型、高阶迎风和中心有限差分（FD）格式和有限体积（FV）格式,这些格式对网格Reynolds数的任意值均为不振荡格式。可见对不振荡CFD格式的构建,数值摄动算法提供了不同于调节数值耗散等常见的人为构建方法,而利用流体力学自身关系以及把迎风机制通过上、下游摄动重构引入中心MBS的解析构建方法,FMBS除了直接应用于流体计算外;对于通过调节数值耗散、色散和数值群速度特性重构高分辨率格式的研究,最简单FMBS提供了比最简单MBS更精确、但同样简单的基础和起步格式。FMBS用于计算不可压缩流,可压缩流,液滴萃取传质,微通道两相流等,均获得良好数值结果或与已有Benchmark解一致的数值结果。已有文献称数值摄动算法为新型高精度格式和高的算法和高的格式;本文FMBS比数值摄动格式的称呼可更好反映FMBS的物理内容。文中也讨论了值得进一步研究的一些课题,该法亦可用于其它一些数学物理方程（例如,简化Boltzmann方程、磁流体方程、KdV-Burgers方程等）MBS耦合物理动力学效应的重构。      

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

A combined analytical–numerical method based on a matching asymptotic algorithm is proposed for treating angular (sharp corner or wedge) singularities in the numerical solution of the Navier–Stokes equations. We adopt an asymptotic solution for the local flow around the angular points based on the Stokes flow approximation and a numerical solution for the global flow outside the singular regions using a finite‐volume method. The coefficients involved in the analytical solution are iteratively updated by matching both solutions in a small region where the Stokes flow approximation holds. Moreover, an error analysis is derived for this method, which serves as a guideline for the practical implementation. The present method is applied to treat the leading‐edge singularity of a semi‐infinite plate. The effect of various influencing factors related to the implementation are evaluated with the help of numerical experiments. The investigation showed that the accuracy of the numerical solution for the flow around the leading edge can be significantly improved with the present method. The results of the numerical experiments support the error analysis and show the desired properties of the new algorithm, i.e. accuracy, robustness and efficiency. Based on the numerical results for the leading‐edge singularity, the validity of various classical approximate models for the flow, such as the Stokes approximation, the inviscid flow model and the boundary layer theory of varying orders are examined. Although the methodology proposed was evaluated for the leading‐edge problem, it is generally applicable to all kinds of angular singularities and all kinds of finite‐discretization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

On wall pressure fluctuations and their coupling with vortex dynamics in a separated–reattached turbulent flow over a blunt flat plate

This study deals with the numerical predictions through Large-Eddy Simulation (LES) of the separated–reattached turbulent flow over a blunt flat plate for analyzing main coherent structure features and their relation to the unsteady pressure field. A compressible approach that inherently includes acoustic propagation is here followed to describe the relationship between pressure fluctuations and vortex dynamics around the separation bubble. The objective of the present work is then to contribute to a better understanding of the coupling between the vortex dynamics and the wall pressure fluctuations. The filtered compressible Navier–Stokes equations are then solved with a numerical method that follows a Lax–Wendroff approach to recover a high accuracy in both time and space. For validations, the present numerical results are compared to experimental measurements, coming from both the Pprime laboratory (Sicot el al., 2012) and the literature (Cherry et al., 1984; Kiya and Sasaki, 1985; Tafti and Vanka,1991; Sicot et al., 2012). Our numerical results very well predict mean and fluctuating pressure and velocity fields. Flapping, shedding as well as Kelvin–Helmholtz characteristic frequencies educed by present simulations are in very good agreement with the experimental values generally admitted. These characteristic modes are also visible on unsteady pressure signatures even far away from the separation. Spectral, POD and EPOD (extended POD) analyses are then applied to these numerical data to enhance the salient features of the pressure and velocity fields, especially the unsteady wall pressure in connection with either the vortex shedding or the low frequency shear-layer flapping. A contribution to the understanding of the coupling between wall pressure fluctuations and eddy vortices is finally proposed.     

Superconvergence of nonconforming finite element penalty scheme for Stokes problem using L 2 projection method

A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L ² projection method, the superconvergence results for the velocity and pressure are obtained with a penalty parameter larger than that of the classical penalty scheme. The numerical experiments are carried out to confirm the theoretical results.     

Wet/dry areas method for interfacial (free surface) flow

In this paper, a new numerical method is developed for two‐dimensional interfacial (free surface) flows, based on the control volume method and conservative integral form of the Navier–Stokes equations with a standard staggered grid. The new method deploys two continuity equations, the continuity equation of the mass conservation for better convergence of the implicit scheme and the continuity equation of the volume conservation for the equation of pressure correction. The convection terms (the total momentum flux) on the surfaces of control volume are accurately calculated from the wet area exposed to the water, and the dry area exposed to the air. The numerical results produced by the new numerical method agree very well with the analytical solution, experimental images and experimentally measured velocity. Copyright © 2012 John Wiley & Sons, Ltd.