1.9 Plan of This Book
19
from the concepts behind discretization techniques. The treatment of complex geometries is introduced later, in Chap. 8.
In Chap. 5 we describe methods of solving the algebraic equation systems
resulting from discretization. Direct methods are briefly described, but the
major part of the chapter is devoted to iterative solution techniques. Incomplete lower-upper decomposition, conjugate gradients and multigrid methods
are given special attention. Approaches to solving coupled and non-linear
systems are also described, including the issues of under-relaxation and convergence criteria.
Chapter 6 is devoted to methods of time integration. First, the methods
of solving ordinary differential equations are described, including basic methods, predictor-corrector and multipoint methods, Runge-Kutta methods. The
application of these methods to the unsteady transport equations is described
next, including analysis of stability and accuracy.
The complexity of the Navier-Stokes equations and special features for
incompressible flows are considered in Chap. 7. The staggered and colocated variable arrangements, the pressure equation, pressure-velocity coupling and other approaches (streamfunction-vorticity, artificial compressibility, fractional step methods) are described. The solution methods for incompressible Navier-Stokes equations based on pressure-correction are described
in detail for staggered and colocated Cartesian grids. Finally, some examples
of two-dimensional and three-dimensional laminar flows are presented.
Chapter 8 is devoted to the treatment of complex geometries. The choices
of grid type, grid properties, velocity components and variable arrangements
are discussed. FD and FV methods are revisited, and the features special
to complex geometries (like non-orthogonal and unstructured grids, control
volumes of arbitrary shape etc.) are discussed. Special attention is paid to
pressure-correction equation and boundary conditions. One section is devoted
to F E methods, which are best known for their applicability to arbitrary
unstructured grids.
Chapter 9 deals with computation of turbulent flows. We discuss the nature of turbulence and three methods for its simulation: direct and large-eddy
simulation and methods based on Reynolds-averaged Navier-Stokes equations. Some models used in the latter two approaches are described. Examples
using these approaches are presented.
In Chap. 10 compressible flows are considered. Methods designed for compressible flows are briefly discussed. The extension of the pressure-correction
approach for incompressible flows to compressible flows is described. Methods
for dealing with shocks ( e g grid adaptation, total-variation-diminishing and
essentially-non-oscillating schemes) are also discussed. Boundary conditions
for various types of compressible flows (subsonic, transonic and supersonic)
are described. Finally, examples are presented and discussed.
Chapter 11 is devoted t o accuracy and efficiency improvement. The increased efficiency provided by multigrid algorithms is described first, followed
19
from the concepts behind discretization techniques. The treatment of complex geometries is introduced later, in Chap. 8.
In Chap. 5 we describe methods of solving the algebraic equation systems
resulting from discretization. Direct methods are briefly described, but the
major part of the chapter is devoted to iterative solution techniques. Incomplete lower-upper decomposition, conjugate gradients and multigrid methods
are given special attention. Approaches to solving coupled and non-linear
systems are also described, including the issues of under-relaxation and convergence criteria.
Chapter 6 is devoted to methods of time integration. First, the methods
of solving ordinary differential equations are described, including basic methods, predictor-corrector and multipoint methods, Runge-Kutta methods. The
application of these methods to the unsteady transport equations is described
next, including analysis of stability and accuracy.
The complexity of the Navier-Stokes equations and special features for
incompressible flows are considered in Chap. 7. The staggered and colocated variable arrangements, the pressure equation, pressure-velocity coupling and other approaches (streamfunction-vorticity, artificial compressibility, fractional step methods) are described. The solution methods for incompressible Navier-Stokes equations based on pressure-correction are described
in detail for staggered and colocated Cartesian grids. Finally, some examples
of two-dimensional and three-dimensional laminar flows are presented.
Chapter 8 is devoted to the treatment of complex geometries. The choices
of grid type, grid properties, velocity components and variable arrangements
are discussed. FD and FV methods are revisited, and the features special
to complex geometries (like non-orthogonal and unstructured grids, control
volumes of arbitrary shape etc.) are discussed. Special attention is paid to
pressure-correction equation and boundary conditions. One section is devoted
to F E methods, which are best known for their applicability to arbitrary
unstructured grids.
Chapter 9 deals with computation of turbulent flows. We discuss the nature of turbulence and three methods for its simulation: direct and large-eddy
simulation and methods based on Reynolds-averaged Navier-Stokes equations. Some models used in the latter two approaches are described. Examples
using these approaches are presented.
In Chap. 10 compressible flows are considered. Methods designed for compressible flows are briefly discussed. The extension of the pressure-correction
approach for incompressible flows to compressible flows is described. Methods
for dealing with shocks ( e g grid adaptation, total-variation-diminishing and
essentially-non-oscillating schemes) are also discussed. Boundary conditions
for various types of compressible flows (subsonic, transonic and supersonic)
are described. Finally, examples are presented and discussed.
Chapter 11 is devoted t o accuracy and efficiency improvement. The increased efficiency provided by multigrid algorithms is described first, followed