Unstructured Grid Finite-Element Methods For Fluid Mechanics.pdf

Rep. Prog. Phys. 61 (1998) 569–638. Printed in the UK PII: S0034-4885(98)57471-4
Unstructured grid finite-element methods for fluid
mechanics
K Morgan† and J Peraire‡
† Department of Civil Engineering, University of Wales, Swansea SA2 8PP, UK
‡ Department of Aeronautics and Astronautics, MIT, Cambridge, MA 02139, USA
Received 9 October 1997
Abstract
The development of unstructured grid-based, finite-element methods for the simulation of
fluid flows is reviewed. The review concentrates on solution techniques for pressible
Euler and Navier–Stokes equations, employing methods which are based upon a Galerkin
discretization in space together with an appropriate finite-difference representation in time.
It is assumed that unstructured assemblies of triangles are used to achieve the spatial
discretization in two dimensions, with unstructured assemblies of tetrahedra employed
in the three-dimensional case. Adaptive grid procedures are discussed and methods for
accelerating the iterative solution convergence are considered. The areas of pressible
flow modelling and optimization are also included.
0034-4885/98/060569+70$
c 1998 IOP Publishing Ltd 569
570 K Morgan and J Peraire
Contents
Page
1. Introduction 571
2. Conservation equations in one space dimension 572
. Variational formulation 573
. Galerkin approximation 573
. Stabilization 575
. Discretization in time 578
. Treatment of discontinuities 580
. Extension to equation systems 585
3. Domain discretization for multi-dimensional problems 589
. Unstructured grids 590
. Unstructured grid generation 590
4. The multi-dimensional Euler equations 591
. Edge-based method with explicitly added diffusion 593
. Element-based Taylor–Galerkin method 596
. Edge-based LED methods 598
5. The Navier–Stokes equations 600
. Edge-based implementation 602
. Element-based implementation 602
. Turbulent flows 603
6. Adaptivity 603
. Error estim