An Analysis of the Discontinuous Galerkin Method for Wave Propagation Problems

A numerical investigation of laminar flow over a three-dimensional backward-facing step is presented with comparisons with detailed experimental data, available in the literature, serving to validate the numerical results. The continuity constraint method, implemented via a finite element weak statement, was employed to solve the unsteady three-dimensional Navier–Stokes equations for incompressible laminar isothermal flow. Two-dimensional numerical simulations of this step geometry underestimate the experimentally determined extent of the primary separation region for Reynolds numbers Re greater than 400. It has been postulated that this disagreement between physical and computational experiments is due to the onset of three-dimensional flow near Re ≈ 400. This paper presents a full three-dimensional simulation of the step geometry for 100⩽ Re⩽ 800 and correctly predicts the primary reattachment lengths, thus confirming the influence of three-dimensionality. Previous numerical studies have discussed possible instability modes which could induce a sudden onset of three-dimensional flow at certain critical Reynolds numbers. The current study explores the influence of the sidewall on the development of three-dimensional flow for Re greater than 400. Of particular interest is the characterization of three-dimensional vortices in the primary separation region immediately downstream of the step. The complex interaction of a wall jet, located at the step plane near the sidewall, with the mainstream flow reveals a mechanism for the increasing penetration (with increasing Reynolds number) of three-dimensional flow structures into a region of essentially two-dimensional flow near the midplane of the channel. The character and extent of the sidewall-induced flow are investigated for 100⩽Re⩽ 800. © 1997 John Wiley & Sons, Ltd.

Numerical simulations of laminar flow over a 3D backward‐facing step

An explicit Galerkin/least-squares formulation is introduced for a quasilinear transport equation in field dislocation mechanics (FDM) and applied to the study of the kinematics of dislocation density evolution in the following physical contexts: annihilation of dislocations, expansion of a polygonal dislocation loop and simulation of a Frank–Read source. Stability analysis is carried out for the corresponding linear one-dimensional (1D) case. The formulation reduces to the Lax–Wendroff finite difference scheme for the 1D equation when equal weighting is used for the Galerkin and least-squares terms and the shape functions are linear. This conditionally stable method leads to a symmetric well-conditioned system of equations with constant coefficients, making it attractive for large-scale problems.It is shown that the transport equation, in the contexts mentioned above simplifies to the Hamilton–Jacobi equations governing geometrical optics and level-set methods. The weak solutions to these equations are not unique, and the numerical method is able to capture solutions corresponding to shock as well as rarefraction waves by appropriate algorithmic modifications.

https://iopscience.iop.org/article/10.1088/0965-0393/14/7/011/pdf

Dislocation transport using an explicit Galerkin/least-squares formulation

Development of a robust finite element CFD procedure for predicting indoor room air motion

A collaborative research project, established between the University of Tennessee (UT) computational fluid dynamics (CFD) Lab and the Federal Aviation Administration (FAA) Civil Aerospace Medical Institute (CAMI), sought to validate CFD prediction of the environmental control system (ECS)-generated ventilation velocity vector flow field of a wide-body aircraft passenger cabin. Measurements generating the experimental database of the time-dependent, three-dimensional velocity vector field were conducted in the CAMI Aircraft Environmental Research Facility (AERF) and in an identical wide-body aircraft in cruise flight at altitude. The CFD simulations were conducted on the CFD Lab resident PC cluster using both commercial and proprietary CFD computer codes. The results of this velocity field validation study reported herein quantitatively assess CFD algorithm/code strengths and weaknesses, and fully confirm the dominance of unsteadiness in the aircraft ventilation velocity field. To be reported in a separate...

Aircraft Passenger Cabin ECS-Generated Ventilation Velocity and Mass Transport CFD Simulation: Velocity Field Validation

SUMMARY ATaylor series augmentation of a weak statement (a ‘Taylor weak statement’ or ‘Taylor-Galerkin’ method) is used to systematically reduce the dispersion error in a finite element approximation of the one-dimensional transient advection equation. A frequency analysis is applied to determine the phase velocity of semi-implicit linear, quadratic and cubic basis one-dimensional finite element methods and of several comparative finite difference/ finite volume algorithms. The finite element methods analysed include both GaIerkin and Taylor weak statements. The frequency analysis is used to obtain an improved linear basis Taylor weak statement finite element algorithm. Solutions are reported for verification problems in one and two dimensions and are compared with finite volume solutions. The improved finite element algorithms have sufficient phase accuracy to achieve highly accurate linear transient solutions with little or no artificial diffusion. Application of the Galerkm finite element method (FEM) to parabolic and hyperbolic differential equations has presented difficulties with the control of dispersion error. Dispersion error results fiom shorter-wavelength solution components travelling at the wrong speed, usually too slowly. Waves travelling at the wrong speed eventually appear in the wrong place as extraneous short waves and can lead to instability in a non-linear problem statement. The ‘upwind’ finite volume method has been extensively applied either to reduce dispersion error or to artificially &&se the resulting short waves. The interpolation is biased for greater contribution fiom the direction of the velocity. It originated with the donor cell method of Courant et af. in 1952 and was applied to the FEM in the 1970s as the Petrov-Galerkin methods of Christie et ~1.~9 and Heinrich et ~1.~3~ Early Petrov-Galerkin methods suffered from excess diffusion of the solution and later work has been oriented towards reducing the excess diffusion, including the SUPG method of Brooks and Hughes6 in 1980 and the methods of Dick7 in 1983, Westerink and Sheas in 1989, Bouloutas and Celia9 in 199 1 and Konda et al. lo in 1992. Similar work in the finite volume method is typified by the QUICK methods of Leonard” in 1979 and Leonard and Mokhtari12 in 1992. Many upwind methods still cause excess diffusion and some also cause an undesirable increase in the width of the matrix stencil, thus increasing the computational effort. An alternative approach which avoids these problems originated for the finite volume method with Lax and WendroP3 in 1960. The

On Taylor weak statement finite element methods for computational fluid dynamics

SUMMARY The quest continues for accurate and efficient computational fluid dynamics (CFD) algorithms for convection-dominated flows. The boundary value ‘optimal’ Galerkin weak statement invariably requires manipulation to handle the disruptive character introduced by the discretized first-order convection term. An incredible variety of methodologies have been derived and examined to address this issue, in particular, seeking achievement of monotone discrete solutions in an efficient implementation. The UT CFD research group has participated in this search, leading to the development of consistent, encompassing theoretical statements exhibiting quality performance, including generalized Taylor series (Lax‐ Wendroff) methods, characteristic flux resolutions, subgrid embedded high-degree Lagrange bases and assembled stencil optimization for finite element weak statement implementations. For appropriate model problems, recent advances have led to accurate monotone methods with linear basis efficiency. This contribution highlights the theoretical developments and presents quantitative documentation of achieved high-quality solutions. Copyright © 1999 John Wiley & Sons, Ltd.

D. J. Chaffin

Papers

On Taylor weak statement finite element methods for computational fluid dynamics

Finite elements for cfd—how does the theory compare?