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1. Basic Concepts of Fluid Flow
1.7.7 Modeling of Complex Flow Phenomena
Many flows of practical interest are difficult to describe exactly mathematically, let alone solve exactly. These flows include turbulence, combustion,
multiphase flow, and are very important. Since exact description is often impracticable, one usually uses semi-empirical models to represent these phenomena. Examples are turbulence models (which will be treated in some
detail in Chap. 9), combustion models, multiphase models, etc. These models, as well as the above mentioned simplifications affect the accuracy of the
solution. The errors introduced by the various approximations may either
augment or cancel each other; therefore, care is needed when drawing conclusions from calculations in which models are used. Due to the importance
of various kinds of errors in numerical solutions we shall devote a lot of attention to this topic. The error types will be defined and described as they
are encountered.
1.8 Mathematical Classification of Flows
Quasi-linear second order partial differential equations in two independent
variables can be divided into three types: hyperbolic, parabolic, and elliptic.
This distinction is based on the nature of the characteristics, curves along
which information about the solution is carried. Every equation of this type
has two sets of characteristics.
In the hyperbolic case, the characteristics are real and distinct. This means
that information propagates at finite speeds in two sets of directions. In
general, the information propagation is in a particular direction so that one
datum needs to be given at an initial point on each characteristic; the two sets
of characteristics therefore demand two initial conditions. If there are lateral
boundaries, usually only one condition is required at each point because one
characteristic is carrying information out of the domain and one is carrying
information in. There are, however, exceptions to this rule.
In parabolic equations the characteristics degenerate to a single real set.
Consequently, only one initial condition is normally required. At lateral
boundaries one condition is needed at each point.
Finally, in the elliptic case, the characteristics are imaginary or complex so
there are no special directions of information propagation. Indeed, information travels essentially equally well in all directions. Generally, one boundary
condition is required at each point on the boundary and the domain of solution is usually closed although part of the domain may extend to infinity.
Unsteady problems are never elliptic.
These differences in the nature of the equations are reflected in the methods used to solve them. It is an important general rule that numerical methods
should respect the properties of the equations they are solving.
1. Basic Concepts of Fluid Flow
1.7.7 Modeling of Complex Flow Phenomena
Many flows of practical interest are difficult to describe exactly mathematically, let alone solve exactly. These flows include turbulence, combustion,
multiphase flow, and are very important. Since exact description is often impracticable, one usually uses semi-empirical models to represent these phenomena. Examples are turbulence models (which will be treated in some
detail in Chap. 9), combustion models, multiphase models, etc. These models, as well as the above mentioned simplifications affect the accuracy of the
solution. The errors introduced by the various approximations may either
augment or cancel each other; therefore, care is needed when drawing conclusions from calculations in which models are used. Due to the importance
of various kinds of errors in numerical solutions we shall devote a lot of attention to this topic. The error types will be defined and described as they
are encountered.
1.8 Mathematical Classification of Flows
Quasi-linear second order partial differential equations in two independent
variables can be divided into three types: hyperbolic, parabolic, and elliptic.
This distinction is based on the nature of the characteristics, curves along
which information about the solution is carried. Every equation of this type
has two sets of characteristics.
In the hyperbolic case, the characteristics are real and distinct. This means
that information propagates at finite speeds in two sets of directions. In
general, the information propagation is in a particular direction so that one
datum needs to be given at an initial point on each characteristic; the two sets
of characteristics therefore demand two initial conditions. If there are lateral
boundaries, usually only one condition is required at each point because one
characteristic is carrying information out of the domain and one is carrying
information in. There are, however, exceptions to this rule.
In parabolic equations the characteristics degenerate to a single real set.
Consequently, only one initial condition is normally required. At lateral
boundaries one condition is needed at each point.
Finally, in the elliptic case, the characteristics are imaginary or complex so
there are no special directions of information propagation. Indeed, information travels essentially equally well in all directions. Generally, one boundary
condition is required at each point on the boundary and the domain of solution is usually closed although part of the domain may extend to infinity.
Unsteady problems are never elliptic.
These differences in the nature of the equations are reflected in the methods used to solve them. It is an important general rule that numerical methods
should respect the properties of the equations they are solving.