7.1 Special Features of the Navier-Stokes Equations
159
described in Chap. 3. However, as the pressure and velocity nodes on the
grid may not coincide, the approximations used for their derivatives may
differ.
In FV methods, the pressure term is usually treated as a surface force
(conservative approach), i.e. in the equation for ui the integral
is required. Methods described in Chap. 4 for the approximation of surface
integrals can then be used. As we shall show below, the treatment of this
term and the arrangement of variables on the grid play an important role in
assuring the computational efficiency and accuracy of the numerical solution
method.
Alternatively, the pressure can be treated non-conservatively, by retaining
the above integral in its volumetric form:
In this case, the derivative (or, for non-orthogonal grids, all three derivatives)
needs to be approximated at one or more locations within the CV. The nonconservative approach introduces a global non-conservative error; although
this error tends to zero as the grid size goes to zero, it may be significant for
finite grid size.
The difference between the two approaches is significant only in the FV
methods. In FD methods, there is no distinction between the two versions,
although one can produce both conservative and non-conservative approximations.
Other body forces, like the non-conservative ones arising when covariant
or contravariant velocities are used in non-Cartesian coordinate systems are
easy to treat in finite difference schemes: they are usually simple functions
of one or more variables and can be evaluated using techniques described in
Chap. 3. If these terms involve the unknowns, as for example, the component
of the viscous term in cylindrical coordinates:
they may be treated implicitly. This is usually done when the contribution of
this term t o the central coefficient Ap in the discretized equation is positive,
in order to avoid destabilization of the iterative solution scheme by reducing
the diagonal dominance of the matrix. Otherwise, the extra term is treated
explicitly.
In FV methods, these terms are integrated over the CV volume. Usually,
the mean value approach is used, so that the value a t CV center is multiplied
by cell volume. More elaborate schemes are possible but rarely used.
159
described in Chap. 3. However, as the pressure and velocity nodes on the
grid may not coincide, the approximations used for their derivatives may
differ.
In FV methods, the pressure term is usually treated as a surface force
(conservative approach), i.e. in the equation for ui the integral
is required. Methods described in Chap. 4 for the approximation of surface
integrals can then be used. As we shall show below, the treatment of this
term and the arrangement of variables on the grid play an important role in
assuring the computational efficiency and accuracy of the numerical solution
method.
Alternatively, the pressure can be treated non-conservatively, by retaining
the above integral in its volumetric form:
In this case, the derivative (or, for non-orthogonal grids, all three derivatives)
needs to be approximated at one or more locations within the CV. The nonconservative approach introduces a global non-conservative error; although
this error tends to zero as the grid size goes to zero, it may be significant for
finite grid size.
The difference between the two approaches is significant only in the FV
methods. In FD methods, there is no distinction between the two versions,
although one can produce both conservative and non-conservative approximations.
Other body forces, like the non-conservative ones arising when covariant
or contravariant velocities are used in non-Cartesian coordinate systems are
easy to treat in finite difference schemes: they are usually simple functions
of one or more variables and can be evaluated using techniques described in
Chap. 3. If these terms involve the unknowns, as for example, the component
of the viscous term in cylindrical coordinates:
they may be treated implicitly. This is usually done when the contribution of
this term t o the central coefficient Ap in the discretized equation is positive,
in order to avoid destabilization of the iterative solution scheme by reducing
the diagonal dominance of the matrix. Otherwise, the extra term is treated
explicitly.
In FV methods, these terms are integrated over the CV volume. Usually,
the mean value approach is used, so that the value a t CV center is multiplied
by cell volume. More elaborate schemes are possible but rarely used.