1.8 Mathematical Classification of Flows
17
The Navier-Stokes equations are a system of non-linear second order equations in four independent variables. Consequently the classification scheme
does not apply directly to them. Nonetheless, the Navier-Stokes equations do
possess many of the properties outlined above and the many of the ideas used
in solving second order equations in two independent variables are applicable
to them but care must be exercised.
1.8.1 Hyperbolic Flows
To begin, consider the case of unsteady inviscid compressible flow. A compressible fluid can support sound and shock waves and it is not surprising
that these equations have essentially hyperbolic character. Most of the methods used to solve these equations are based on the idea that the equations
are hyperbolic and, given sufficient care, they work quite well; these are the
methods referred to briefly above.
For steady compressible flows, the character depends on the speed of the
flow. If the flow is supersonic, the equations are hyperbolic while the equations
for subsonic flow are essentially elliptic. This leads to a difficulty that we shall
discuss further below.
It should be noted however, that the equations for a viscous compressible
flow are still more complicated. Their character is a mixture of elements of
all of the types mentioned above; they do not fit well into the classification
scheme and numerical methods for them are difficult to construct.
1.8.2 Parabolic Flows
The boundary layer approximation described briefly above leads to a set of
equations that have essentially parabolic character. Information travels only
downstream in these equations and they may be solved using methods that
are appropriate for parabolic equations.
Note, however, that the boundary layer equations require specification of
a pressure that is usually obtained by solving a potential flow problem. Subsonic potential flows are governed by elliptic equations (in the incompressible
limit the Laplace equation suffices) so the overall problem actually has a
mixed parabolic-elliptic character.
1.8.3 Elliptic Flows
When a flow has a region of recirculation i.e. flow in a sense opposite to
the principal direction of flow, information may travel upstream as well as
downstream. As a result, one cannot apply conditions only at the upstream
end of the flow. The problem then acquires elliptic character. This situation
occurs in subsonic (including incompressible) flows and makes solution of the
equations a very difficult task.
17
The Navier-Stokes equations are a system of non-linear second order equations in four independent variables. Consequently the classification scheme
does not apply directly to them. Nonetheless, the Navier-Stokes equations do
possess many of the properties outlined above and the many of the ideas used
in solving second order equations in two independent variables are applicable
to them but care must be exercised.
1.8.1 Hyperbolic Flows
To begin, consider the case of unsteady inviscid compressible flow. A compressible fluid can support sound and shock waves and it is not surprising
that these equations have essentially hyperbolic character. Most of the methods used to solve these equations are based on the idea that the equations
are hyperbolic and, given sufficient care, they work quite well; these are the
methods referred to briefly above.
For steady compressible flows, the character depends on the speed of the
flow. If the flow is supersonic, the equations are hyperbolic while the equations
for subsonic flow are essentially elliptic. This leads to a difficulty that we shall
discuss further below.
It should be noted however, that the equations for a viscous compressible
flow are still more complicated. Their character is a mixture of elements of
all of the types mentioned above; they do not fit well into the classification
scheme and numerical methods for them are difficult to construct.
1.8.2 Parabolic Flows
The boundary layer approximation described briefly above leads to a set of
equations that have essentially parabolic character. Information travels only
downstream in these equations and they may be solved using methods that
are appropriate for parabolic equations.
Note, however, that the boundary layer equations require specification of
a pressure that is usually obtained by solving a potential flow problem. Subsonic potential flows are governed by elliptic equations (in the incompressible
limit the Laplace equation suffices) so the overall problem actually has a
mixed parabolic-elliptic character.
1.8.3 Elliptic Flows
When a flow has a region of recirculation i.e. flow in a sense opposite to
the principal direction of flow, information may travel upstream as well as
downstream. As a result, one cannot apply conditions only at the upstream
end of the flow. The problem then acquires elliptic character. This situation
occurs in subsonic (including incompressible) flows and makes solution of the
equations a very difficult task.