A Spectral Stochastic Semi-Lagrangian Method for Convection-Diffusion Equations with Uncertainty

Real life convection-diffusion problems are characterized by their inherent or externally induced uncertainties in the design parameters. This paper presents a spectral stochastic finite element semi-Lagrangian method for numerical solution of convection-diffusion equations with uncertainty. Using the spectral decomposition, the stochastic variational problem is reformulated to a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic convection-diffusion equations, we implement a semi-Lagrangian method in the finite element framework. Once this representation is computed, statistics of the numerical solution can be easily evaluated. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the semi-Lagrangian method to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for a convection-diffusion problem driven with stochastic velocity and for an incompressible viscous flow problem with a random force. In both examples, the proposed method demonstrates its ability to better maintain the shape of the solution in the presence of uncertainties and steep gradients.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

最小二乘有限元法和有限体积法在CFD中的应用比较

为比较最小二乘有限元法（Least Square Finite Element Method,LSFEM）和有限体积法在CFD应用中的优劣,采用最小二乘法离散不可压N-S方程的有限元模型,得到正定对称线性系统,采用高效的预处理共轭梯度法求解方程组;利用LSFEM和基于有限体积法的FLUENT分别计算Kovasznay流动、定常二维和三维后台阶流动以及非定常圆柱绕流等4个实例并比较计算结果.结果表明,LSFEM比有限体积法的收敛性和精确性更好,在CFD领域的应用价值很高.     

Numerical simulation of Newtonian and non-Newtonian flows in bypass

This paper deals with the numerical solution of Newtonian and non-Newtonian flows with biomedical applications. The flows are supposed to be laminar, viscous, incompressible and steady or unsteady with prescribed pressure variation at the outlet. The model used for non-Newtonian fluids is a variant of power law. Governing equations in this model are incompressible Navier–Stokes equations. For numerical solution we use artificial compressibility method with three stage Runge–Kutta method and finite volume method in cell centered formulation for discretization of space derivatives. The following cases of flows are solved: steady Newtonian and non-Newtonian flow through a bypass connected to main channel in 2D, steady Newtonian flow in angular bypass in 3D and unsteady non-Newtonian flow through bypass in 2D. Some 2D and 3D results that could have application in the area of biomedicine are presented.     

A fourth-order accurate compact scheme for the solution of steady Navier-Stokes equations on non-uniform grids

This paper deals with the formulation of a higher-order compact (HOC) scheme on non-uniform grids in complex geometries to simulate two-dimensional (2D) steady incompressible viscous flows governed by the Navier-Stokes (N-S) equations. The proposed scheme which is spatially fourth-order accurate is then tested on three nonlinear problems, namely (i) a problem governed by N-S equations with a constructed analytical solution, (ii) lid-driven cavity flow problem, and (iii) constricted channel flow problem. In the process, we have also expanded the scope of fourth-order 9-point compact schemes to geometries beyond rectangular. It is seen to efficiently capture steady-state solutions of the N-S equations with Dirichlet as well as Neumann boundary conditions. In addition to this, it captures viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variations. Our results are in excellent agreement with analytical and numerical results whenever available and they clearly demonstrate the superior scale resolution of the proposed scheme.     

A stabilized finite element method for stochastic incompressible Navier–Stokes equations

The major emphasis of this work is the development of a stabilized finite element method for solving incompressible Navier–Stokes equations with stochastic input data. The polynomial chaos expansion is used to represent stochastic processes in the variational problem, resulting in a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic incompressible Navier–Stokes equations, we combine the modified method of characteristics with the finite element discretization. The obtained Stokes problem is solved using a robust conjugate-gradient algorithm. This algorithm avoids projection procedures and any special correction for the pressure. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the modified method of characteristics to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for the benchmark problems of driven cavity flow and backward-facing step flow. We also present numerical results for a problem of stochastic natural convection. It is found that the proposed stabilized finite element method offers a robust and accurate approach for solving the stochastic incompressible Navier–Stokes equations, even when high Reynolds and Rayleigh numbers are used in the simulations.     

Numerical solution of unsteady Navier–Stokes equations on curvilinear meshes

The objective of the present work is to extend our FDS-based third-order upwind compact schemes by Shah et al. (2009) [8] to numerical solutions of the unsteady incompressible Navier–Stokes equations in curvilinear coordinates, which will save much computing time and memory allocation by clustering grids in regions of high velocity gradients. The dual-time stepping approach is used for obtaining a divergence-free flow field at each physical time step. We have focused on addressing the crucial issue of implementing upwind compact schemes for the convective terms and a central compact scheme for the viscous terms on curvilinear structured grids. The method is evaluated in solving several two-dimensional unsteady benchmark flow problems.     

Incompressible flow computations over moving boundary using a novel upwind method

In this paper a novel method for simulating unsteady incompressible viscous flow over a moving boundary is described. The numerical model is based on a 2D Navier–Stokes incompressible flow in artificial compressibility formulation with Arbitrary Lagrangian Eulerian approach for moving grid and dual time stepping approach for time accurate discretization. A higher order unstructured finite volume scheme, based on a Harten Lax and van Leer with Contact (HLLC) type Riemann solver for convective fluxes, developed for steady incompressible flow in artificial compressibility formulation by Mandal and Iyer (AIAA paper 2009-3541), is extended to solve unsteady flows over moving boundary. Viscous fluxes are discretized in a central differencing manner based on Coirier’s diamond path. An algorithm based on interpolation with radial basis functions is used for grid movements. The present numerical scheme is validated for an unsteady channel flow with a moving indentation. The present numerical results are found to agree well with experimental results reported in literature.     

Mixed finite element method for analysis of viscoelastic fluid flow

In the present paper, numerical analysis of incompressible viscoelastic fluid flow is discussed using mixed finite element Galerkin method. Because Maxwellian viscoelasticity is assumed as the constitutive equation, stress components could not be eliminated from the governing equation system. Because of this, mixed finite element method is utilized to discretize the basic equations. For the solution procedures to solve discretized equation system, Newton-Raphson method for steady flow and perturbation method for unsteady flow is employed. As the numerical examples, comparison was made on the finite element computational results between by direct method and by mixed method. Effects of the viscoelasticity is analyzed for the flows at Reynold's numbers 30, 50 and 70.     

A comparison of various mixed-interpolation finite elements in the velocity-pressure formulation of the Navier-Stokes equations

It is generally recognized that mixed interpolation should be used in the velocity-pressure finite element formulation of incompressible viscous flow problems. In this paper, four types of mixed interpolation elements are considered and compared. These are namely: six-node triangular elements, eight-node serendipity elements, nine-node Lagrangian elements and four-node quadrilateral elements. The comparison is made via two numerical examples concerning steady flow through a sudden expansion and steady free thermal convection in a square cavity. Results indicate that for the same number of pressure unknowns, serendipity elements can give considerably less accurate pressure fields than most other types of elements. Lagrangian elements give the most accurate pressure and velocity distributions. The numerical performance of triangular elements is intermediate in accuracy and is dependent on the triangular pattern used. Finally, the four-node element may generate spurious pressure modes depending on the boundary condition specifications.     

Application of a posteriori error estimation to finite element simulation of incompressible Navier-Stokes flow

The main goal of this paper is to study adaptive mesh techniques, using a posteriori error estimates, for the finite element solution of the Navier-Stokes equations modeling steady and unsteady flows of an incompressible viscous fluid. Among existing operator splitting techniques, the θ-scheme is used for time integration of the Navier-Stokes equations. Then, a posteriori error estimates, based on the solution of a local system for each triangular element, are presented in the framework of the generalized incompressible Stokes problem, followed by its practical application to the case of incompressible Navier-Stokes problem. Hierarchical mesh adaptive techniques are developed in response to the a posteriori error estimation. Numerical simulations of viscous flows associated with selected geometries are performed and discussed to demonstrate the accuracy and efficiency of our methodology.     

On the Use of Compact Streamfunction-Velocity Formulation of Steady Navier-Stokes Equations on Geometries beyond Rectangular

In this paper, a new methodology has been proposed to solve two-dimensional (2D) Navier-Stokes (N-S) equations representing incompressible viscous fluid flows on irregular geometries. It is based on second order compact finite difference discretization of the fourth order streamfunction equation on computational plane. The important advantage of this formulation is not only to overcome the difficulties existing in the velocity-pressure and streamfunction-vorticity formulations, but also for being applicable to complex geometries beyond rectangular. We first apply the proposed scheme to a problem having analytical solution and then to the well-studied benchmark problem (problem of lid-driven cavity flow) in viscous fluid flow. Finally, we demonstrate the robustness of our proposed scheme on flow in a complex domain (e.g. constricted channel and dilated channel). It is seen to efficiently capture steady state solutions of the N-S equations with Dirichlet as well as Neumann boundary conditions. In addition to this, it captures viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variations. Estimates of the error incurred show that the results are very accurate on a coarser grid. The results obtained using this scheme are in excellent agreement with analytical and numerical results whenever available and they clearly demonstrate the superior scale resolution of the proposed scheme.     

MHD flow in a rectangular duct with a perturbed boundary

The unsteady magnetohydrodynamic (MHD) flow of a viscous, incompressible and electrically conducting fluid in a rectangular duct with a perturbed boundary, is investigated. A small boundary perturbation $\epsilon$ is applied on the upper wall of the duct which is encountered in the visualization of the blood flow in constricted arteries. The MHD equations which are coupled in the velocity and the induced magnetic field are solved with no-slip velocity conditions and by taking the side walls as insulated and the Hartmann walls as perfectly conducting. Both the domain boundary element method (DBEM) and the dual reciprocity boundary element method (DRBEM) are used in spatial discretization with a backward finite difference scheme for the time integration. These MHD equations are decoupled first into two transient convection–diffusion equations, and then into two modified Helmholtz equations by using suitable transformations. Then, the DBEM or DRBEM is used to transform these equations into equivalent integral equations by employing the fundamental solution of either steady-state convection–diffusion or modified Helmholtz equations. The DBEM and DRBEM results are presented and compared by equi-velocity and current lines at steady-state for several values of Hartmann number and the boundary perturbation parameter.     

On energy conservation for finite element approximation of flow-induced airfoil vibrations

In this paper the numerical approximation of a two-dimensional fluid–structure interaction problem is addressed. The fully coupled formulation of incompressible viscous fluid flow interacting with a flexibly supported airfoil is considered. The flow is described by the incompressible system of Navier–Stokes equations, where large values of the Reynolds number are considered. The Navier–Stokes equations are spatially discretized by the finite element method and stabilized with a modification of the Galerkin Least Squares (GLS) method; cf. [T. Gelhard, G. Lube, M.A. Olshanskii, J.-H. Starcke, Stabilized finite element schemes with LBB-stable elements for incompressible flows, Journal of Computational and Applied Mathematics 177 (2005) 243–267]. The motion of the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian (ALE) method and the stabilizing terms are modified in a consistent way with the ALE formulation.     

A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number

Numerical simulation of three-dimensional incompressible flows at high Reynolds number using the unsteady Navier–Stokes equations is challenging. In order to obtain accurate simulations, very fine meshes are necessary, and such simulations are increasingly important for modern engineering practices, such as understanding the flow behavior around high speed trains, which is the target application of this research. To avoid the time step size constraint imposed by the CFL number and the fine spacial mesh size, we investigate some fully implicit methods, and focus on how to solve the large nonlinear system of equations at each time step on large scale parallel computers. In most of the existing implicit Navier–Stokes solvers, segregated velocity and pressure treatment is employed. In this paper, we focus on the Newton–Krylov–Schwarz method for solving the monolithic nonlinear system arising from the fully coupled finite element discretization of the Navier–Stokes equations on unstructured meshes. In the subdomain, LU or point-block ILU is used as the local solver. We test the algorithm for some three-dimensional complex unsteady flows, including flows passing a high speed train, on a supercomputer with thousands of processors. Numerical experiments show that the algorithm has superlinear scalability with over three thousand processors for problems with tens of millions of unknowns.     

High order finite difference methods for unsteady incompressible flows in multi-connected domains

Using the vorticity and stream function variables is an effective way to compute 2-D incompressible flow due to the facts that the incompressibility constraint for the velocity is automatically satisfied, the pressure variable is eliminated, and high order schemes can be efficiently implemented. However, a difficulty arises in a multi-connected computational domain in determining the constants for the stream function on the boundary of the “holes”. This is an especially challenging task for the calculation of unsteady flows, since these constants vary with time to reflect the total fluxes of the flow in each sub-channel. In this paper, we propose an efficient method in a finite difference setting to solve this problem and present some numerical experiments, including an accuracy check of a Taylor vortex-type flow, flow past a non-symmetric square, and flow in a heat exchanger.     

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

In this paper we present a high-order Lagrangian-decoupling method for the unsteady convection diffusion and incompressible Navier-Stokes equations. The method is based upon Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem, implicit high-order backward-differentiation finite difference schemes for integration along characteristics, finite element or spectral element spatial discretizations and mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high-order accuracy and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.     

Least square finite element solution of stagnation point flow

A detailed analysis of the least square finite element solution of nonlinear boundary value problems is presented with reference to a particular example of nonlinear coupled differential equations governing the flow of an incompressible viscous fluid in the vicinity of a forward stagnation point at a blunt body. The numerical solutions are presented for different cases. The results obtained by the least square finite element method are in very good agreement with the results available in the literature confirming the versatility and usefulness of the application of the method to nonlinear boundary value problems governing the fluid flow problems.     

Finite element solutions of nonsteady incompressible viscous flow between two rotating concentric spheres

The problem of unsteady, incompressible viscous flow between two rotating concentric spheres has been investigated here. The full Navier-Stokes equations in terms of the velocity components u, v w and the pressure p, using spherical coordinates for axially symmetric flow, were solved by means of the finite element method in the spatial dimension and the alternating-direction method in the time dimension using Glowinski's algorithm. The element used is an annular-sector-type element with a bilinear approximation for the velocity components and with constant pressure within the element. Reynolds numbers in the range from 1 to 1000, gap size 0.5 and different combinations of the angular velocity of the inner and outer spheres were studied. In some of these cases a steady-state solution was possible, while in others only a transient solution was possible. This method proved to be successful and powerful in predicting the behavior of the flow for these nonlinear-type problems.     

The extension of incompressible flow solvers to the weakly compressible regime

Numerical simulation schemes for incompressible flows such as the Simple scheme are extended to weakly compressible fluid flow. A single time scale, multiple space scale asymptotic analysis is used to gain insight into the limit behavior of the compressible flow equations as the Mach number vanishes. Motivated by these results, multiple pressure variables (MPV) are introduced into the numerical framework. These account separately for thermodynamic effects, acoustic wave propagation and the balance of forces. Discretized analogues of the averaging and large scale differencing procedures known from multiple scales asymptotics allow accurate capturing of various physical phenomena that are operative on very different length scales. The MPV approach combines the explicit numerical computation of global compression from the boundary and the long wavelength acoustics on coarse grids with an implicit pressure or pressure correction equation that formally converges to the corresponding incompressible one when the Mach number tends to zero.