C. C. Fang

Numerical investigation of vortical evolution in a backward-facing step expansion flow

Abstract A numerical investigation of laminar flow over a backward-facing step is presented for the Reynolds number in the range of 50⩽Re⩽2500. The objective of this numerical investigation is to add to the existing knowledge of the backward-facing step flow to deepen our understanding of the expansion flow structure. We proceed with the analysis by verifying the computer code through the Pearson vortex problem. We then perform a parametric study by varying the Reynolds number, with the aim of determining whether or not there exists a critical Reynolds number, above which reattachment length on the channel floor decreases. We also concentrate on subjects that have been little explored in the flow, examples of which are the onset of a single vortex in the primary eddy and how the recirculating bubble containing flow reversals is torn into smaller eddies. Eddy distortion, leading to mobile saddle points, and the merging of eddies are also discussed in this study.

High resolution finite-element analysis of shallow water equations in two dimensions

Abstract We present in this study the Taylor–Galerkin finite-element model to stimulate shallow water equations for bore wave propagation in a domain of two dimensions. To provide the necessary precision for the prediction of a sharply varying solution profile, the generalized Taylor–Galerkin finite-element model is constructed through introduction of four parameters. This paper also presents the fundamental theory behind the choice of free parameters. One set of parameters is theoretically determined to obtain the high-order accurate Taylor– Galerkin finite-element model. The other set of free parameters is determined using the underlying discrete maximum principle to obtain the low-order monotonic Taylor–Galerkin finite-element model. Theoretical study reveals that the higher-order scheme exhibits dispersive errors near the discontinuity while lower-order scheme dissipates the discontinuity. A scheme which has a high-resolution shock-capturing ability as a built-in feature is, thus, needed in the present study. Notice that lumping of the mass matrix equations is invoked in the low-order scheme to allow simulation of the hydraulic problem with discontinuities. We check the prediction accuracy against suitable test problems, preferably ones for which exact solutions are available. Based on numerical results, it is concluded that the Taylor–Galerkin-flux-corrected transport (TG-FCT) finite-element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles.

A NUMERICAL STUDY OF NONLINEAR PROPAGATION OF DISTURBANCES IN TWO-DIMENSIONS

In the spirit of the method of characteristics, we present in this paper a generalized Taylor-Galerkin finite element model to simulate the nonlinear propagation of finite-amplitude disturbances. In a nonlinear Euler system, the multi-dimensional formulation is constructed through the conservation variables. Noticeable is that the scheme is found to exhibit high-phase-accuracy, together with minimal numerical damping. This scheme, therefore, is best-suited to simulation of disturbances in an acoustic field. To begin with, we validate the characteristic model by simulating two transport problems amenable to analytic solutions. Motivated by the apparent success, we apply the proposed third-order accurate upwind model to investigate a truly nonlinear acoustic field. The present analysis is intended to elucidate to what extent the nondissipative, nondispersive and isotropic characteristics pertaining to three wave modes of the acoustic system are still valid.

Two Element-by-Element Iterative Solutions for Shallow Water Equations

In this paper we apply the generalized Taylor--Galerkin finite element model to simulate bore wave propagation in a domain of two dimensions. For stability and accuracy reasons, we generalize the model through the introduction of four free parameters. One set of parameters is rigorously determined to obtain the high-order finite element solution. The other set of free parameters is determined from the underlying discrete maximum principle to obtain the monotonic solutions. The resulting two models are used in combination through the flux correct transport technique of Zalesak, thereby constructing a finite element model which has the ability to capture hydraulic discontinuities. In addition, this paper highlights the implementation of two Krylov subspace iterative solvers, namely, the bi-conjugate gradient stabilized (Bi-CGSTAB) and the generalized minimum residual (GMRES) methods. For the sake of comparison, the multifrontal direct solver is also considered. The performance characteristics of the investigated solvers are assessed using results of a standard test widely used as a benchmark in hydraulic modeling. Based on numerical results, it is shown that the present finite element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles. Also, the GMRES solver is shown to have a much better convergence rate than the Bi-CGSTAB solver, thereby saving much computing time compared to the multifrontal solver.

On a monotonic convection-diffusion scheme in adaptive meshes

Abstract In this article we apply our recently proposed upwind model to solve the two-dimensional steady convection-diffusion equation in adaptive meshes. In an attempt to resolve high-gradient solutions in the flow, we construct finite-element spaces through use of Legendre polynomials. According to the fundamental analysis conducted in this article, we confirm that this finite-element model accommodates the monotonicity property. According to M-matrix theory, we know within what range of Peclet numbers the Petrov-Galerkin method can perform well in a sense that oscillatory solutions are not present in the flow, This monotonic region is fairly restricted, however, and limits the finite-element practioners choices of a fairly small grid size. This limitation forbids application to practical flow simulations because monotonic solutions are prohibitively expensive to compute. Circumvention of this shortcoming is accomplished by remeshing the domain in an adaptive way. To alleviate the excessive memory requ...

A HIGH RESOLUTION FINITE ELEMENT ANALYSIS FOR NONLINEAR ACOUSTIC WAVE PROPAGATION

We investigate the application of Taylor Galerkin finite element model to simulate the propagation of impulse disturbances governed by the nonlinear Euler equations. This formulation is based on the conservation variables rather than the primitive variables so that the slowly emerging sharp acoustic profiles due to the initial fluctuation can be sharply captured. We show that when the generalized Taylor Galerkin finite element model is combined with the flux corrected transport technique of Boris and Book, the acoustic field can be more accurately predicted. The proposed prediction method was validated first by simulating different classes of transport profiles before applying it to investigate the truly nonlinear acoustic field emanating from an initial square pulse.

Application of an element-by-element BiCGSTAB iterative solver to a monotonic finite element model

Abstract This paper deals with the advection-diffusion equation in adaptive meshes. The main feature of the present finite element model is the use of Legendre-polynomials to span finite element spaces. The success that this model gives good resolutions to solutions in regions of boundary and interior layers lies in the use of M-matrix theory. In the monotonic range of Peclet numbers, the Petrov-Galerkin method performs well in the sense that oscillatory solutions are not present in the flow. With proper stabilization, finite element matrix equations can be iteratively solved by the Lanczos method, used concurrently with local minimization provided by GMRES(1). The resulting BiCGSTAB iterative solver, supplemented with the Jacobi preconditioner, is implemented in an element-by-element fashion. This gives solutions which are computationally feasible for large-scale flow simulations. The results of two computations are presented in support of the ability of the present finite element model to resolve sharp gradients in the solution. As is apparent from this study is that considerable savings in computer storage and execution time are achieved in adaptive meshes through use of the preconditioned BiCGSTAB iterative solver.

On an adaptive monotonic convection—diffusion flux discretization scheme

Abstract The present paper concerns numerical investigation of a two-dimensional steady convection—diffusion scalar equation. Specific attention is given to resolving spurious oscillations in a confined region of high gradient. In smooth regions, a high-order accurate solution is desired, while in regions containing a sharp gradient, the strategy of applying a monotonic capturing scheme is preferred. In this paper, we model fluxes by means of a finite element model which has spaces spanned by Legendre polynomials. The propensity to yield an irreducible diagonal-dominant stiffness matrix is an attribute of this finite element flux discretization scheme. Fundamental analyses are extensively conducted in this paper in two areas. First, by conducting a modified equation analysis, we are led to confirm the consistency of the scheme. Both the stability and monotonicity of the solutions are also addressed. A guide for judging whether the stiffness matrix can yield a monotonic solution is rooted in the theory of the M -matrix. This monotonicity study provides us with greater numerical insight into the importance of the chosen Peclet number. Beyond its critical value, oscillatory solutions are present in the flow. This monotonic region is, however, fairly restricted. It is this shortcoming which limits the finite element practioners choices for fairly small grid size. The scope of application using the proposed upwind scheme is, thus, rather small because monotonic solutions are prohibitively expensive to compute. Circumvention of such deficiency is accomplished by making modifications to the structured grid. Several test cases have been employed to examine the grid adaption technique for the treatment of advection terms in flow regions having sharp gradients. Considerable savings in computer storage and execution time are achieved by the employed unstructured grid setting.

Finite Element Solution for Wave Propagation in Layered Fluids

A hyperbolic equation is considered for the propagation of pressure disturbance waves in layered fluids having different fluid properties. For acoustic problems of this sort, the characteristic finite element model alone does not suffice to ensure prediction of the monotonic wave profile across fluids having different properties. A flux corrected transport solution algorithm is intended for incorporation into the underlying Taylor–Galerkin finite element framework. The advantage of this finite element approach, in addition to permitting oscillation-free solutions, is that it avoids the necessity of dealing with medium discontinuity. As an analysis tool, the proposed monotonic finite element model has been intensively verified through problems which are amenable to analytic solutions. In modeling wave propagation in layered fluids, we have investigated the influence of the degree of medium change on the finite element solutions. Also, different finite element solutions are considered to show the superiority of using the flux corrected transport Taylor–Galerkin finite element model.