18
1. Basic Concepts of Fluid Flow
It should be noted that unsteady incompressible flows actually have a
combination of elliptic and parabolic character. The former comes from the
fact that information travels in both directions in space while the latter results
from the fact that information can only flow forward in time. Problems of
this kind are called incompletely parabolic.
1.8.4 Mixed Flow Types
As we have just seen, it is possible for a single flow to be described by equations that are not purely of one type. Another important example occurs in
steady transonic flows, that is, steady compressible flows that contain both
supersonic and subsonic regions. The supersonic regions are hyperbolic in
character while the subsonic regions are elliptic. Consequently, it may be
necessary to change the method of approximating the equations as a function of the nature of the local flow. To make matters even worse, the regions
can not be determined prior to solving the equations.
1.9 Plan of This Book
This book contains twelve chapters. We now give a brief summary of the
remaining eleven chapters.
In Chap. 2 an introduction to numerical solution methods is given. The
advantages and disadvantages of numerical methods are discussed and the
possibilities and limitations of the computational approach are outlined. This
is followed by a description of the components of a numerical solution method
and their properties. Finally, a brief description of basic computational methods (finite difference, finite volume and finite element) is given.
In Chap. 3 finite difference (FD) methods are described. Here we present
methods of approximating first, second, and mixed derivatives, using Taylor
series expansion and polynomial fitting. Derivation of higher-order methods,
and treatment of non-linear terms and boundaries is discussed. Attention is
also paid to the effects of grid non-uniformity on truncation error and to the
estimation of discretization errors. Spectral methods are also briefly described
here.
In Chap. 4 the finite volume (FV) method is described including the approximation of surface and volume integrals and the use of interpolation to
obtain variable values and derivatives at locations other than cell centers.
Development of higher-order schemes and simplification of the resulting algebraic equations using the deferred-correction approach is also described.
Finally, implementation of the various boundary conditions is discussed.
Applications of basic FD and FV methods are described and their use is
demonstrated in Chaps. 3 and 4 for structured Cartesian grids. This restriction allows us to separate the issues connected with geometric complexity
1. Basic Concepts of Fluid Flow
It should be noted that unsteady incompressible flows actually have a
combination of elliptic and parabolic character. The former comes from the
fact that information travels in both directions in space while the latter results
from the fact that information can only flow forward in time. Problems of
this kind are called incompletely parabolic.
1.8.4 Mixed Flow Types
As we have just seen, it is possible for a single flow to be described by equations that are not purely of one type. Another important example occurs in
steady transonic flows, that is, steady compressible flows that contain both
supersonic and subsonic regions. The supersonic regions are hyperbolic in
character while the subsonic regions are elliptic. Consequently, it may be
necessary to change the method of approximating the equations as a function of the nature of the local flow. To make matters even worse, the regions
can not be determined prior to solving the equations.
1.9 Plan of This Book
This book contains twelve chapters. We now give a brief summary of the
remaining eleven chapters.
In Chap. 2 an introduction to numerical solution methods is given. The
advantages and disadvantages of numerical methods are discussed and the
possibilities and limitations of the computational approach are outlined. This
is followed by a description of the components of a numerical solution method
and their properties. Finally, a brief description of basic computational methods (finite difference, finite volume and finite element) is given.
In Chap. 3 finite difference (FD) methods are described. Here we present
methods of approximating first, second, and mixed derivatives, using Taylor
series expansion and polynomial fitting. Derivation of higher-order methods,
and treatment of non-linear terms and boundaries is discussed. Attention is
also paid to the effects of grid non-uniformity on truncation error and to the
estimation of discretization errors. Spectral methods are also briefly described
here.
In Chap. 4 the finite volume (FV) method is described including the approximation of surface and volume integrals and the use of interpolation to
obtain variable values and derivatives at locations other than cell centers.
Development of higher-order schemes and simplification of the resulting algebraic equations using the deferred-correction approach is also described.
Finally, implementation of the various boundary conditions is discussed.
Applications of basic FD and FV methods are described and their use is
demonstrated in Chaps. 3 and 4 for structured Cartesian grids. This restriction allows us to separate the issues connected with geometric complexity