Spectral/Hp Element Methods for Cfd

by Karniadakis, George Em; Sherwin, Spencer J.

• ISBN13: 9780195102260
• ISBN10: 0195102266
• Format: Hardcover
• Copyright: 1999-04-15
• Publisher: Oxford University Press

Summary

Traditionally spectral methods in fluid dynamics were used in direct and large eddy simulations of turbulent flow in simply connected computational domains. The methods are now being applied to more complex geometries, and the spectral/ hp element method, which incorporates both multi-domain spectral methods and high-order finite element methods, has been particularly successful. This book provides a comprehensive introduction to these methods. Written by leaders in the field, the book begins with a full explanation of fundamental concepts and implementation issues. It then illustrates how these methods can be applied to advection-diffusion and to incompressible and compressible Navier-Stokes equations. Drawing on both published and unpublished material, the book is an important resource for experienced researchers and for those new to the field.

Table of Contents

Introduction	1	(15)
The Basic Equations of Fluid Dynamics	1	(5)
Numerical Discretizations	6	(9)
Fundamental Concepts in One Dimension	15	(47)
Method of Weighted Residuals	16	(3)
Galerkin Formulation	19	(12)
One-Dimensional Expansion Bases	31	(23)
Numerical Integration	54	(2)
Differentiation	56	(3)
Convergence Examples	59	(3)
Multidimensional Expansion Bases	62	(34)
Expansions in Structured Domains	63	(7)
Expansions in Unstructured Domains	70	(25)
Expansions in Homogeneous Domains	95	(1)
Multidimensional Formulation	96	(70)
Local Elemental Operations	96	(36)
Global Operations	132	(20)
Boundary Representation	152	(14)
Geometrically Non-Conforming Elements	166	(21)
The Need for Local Refinement	166	(2)
Interface Conditions and Implementation	168	(2)
Iterative Patching	170	(7)
Constrained Approximation	177	(2)
Mortar Patching	179	(8)
Advection Equation	187	(24)
Galerkin Discretization	189	(3)
Temporal Discretization	192	(3)
Eigen-Spectrum of the Galerkin Advection Operator	195	(9)
Discontinuous Galerkin Discretization	204	(3)
Convergence	207	(4)
Helmholtz Equation	211	(27)
Galerkin Discretization	211	(4)
Eigen-Spectrum of Laplacian Operator	215	(7)
Convergence	222	(3)
Non-Smooth Domains	225	(9)
Mixed and Discontinuous Galerkin Discretization	234	(4)
Incompressible Flows	238	(31)
Variational Formulation	238	(4)
Coupled Methods for Primitive Variables	242	(5)
Splitting Methods for Primitive Variables	247	(13)
Velocity-Vorticity Formulation	260	(8)
The Gauge Method	268	(1)
Flow Simulations	269	(35)
Exact Navier-Stokes Solutions	269	(7)
Direct Numerical Simulations-DNS	276	(14)
Large Eddy Simulations-LES	290	(8)
Dynamic (dDNS) versus Static DNS	298	(6)
Compressible Flows	304	(45)
Discontinuous Solutions and High Order	304	(2)
Conservative Formulation	306	(11)
Monotonicity	317	(9)
Euler Equations	326	(10)
Navier-Stokes Equations	336	(8)
Shock-Fitting Techniques	344	(5)
Appendices	349	(1)
A. Jacobi Polynomials	350	(3)
B. Gauss-Type Integration	353	(1)
B.1 Jacobi Formulae	354	(2)
B.2 Evaluation of the Zeros of Jacobi Polynomials	356	(2)
C. Collocation Differentiation	358	(1)
C.1 Jacobi Formulae	359	(3)
D. Continuous Expansion Basis	362	(1)
D.1 Modal Basis	362	(6)
D.2 Nodal Basis	368	(3)
E. Characteristic Flux Decomposition	371	(1)
E.1 One dimension	371	(1)
E.2 Two dimensions	372	(1)
E.3 Three dimensions	373	(2)
References	375	(11)
Index	386