7.5 Solution Methods for the Navier-Stokes Equations
199
The source of the above problem may be traced to using 2Ax approximations to the first derivatives. Various cures have been proposed. In incompressible flows, the absolute pressure level is unimportant - only the differences
matter. Unless the absolute value of the pressure is specified somewhere, the
pressure equation is singular and has an infinite number of solutions, all differing by a constant. This makes a simple cure possible: filtering out the
oscillations, as was done by van der Wijngaart (1990).
We shall present one approach to dealing with the pressure-velocity coupling on colocated grids that has found widespread use in complicated geometries and is simple and effective.
On staggered grids, central difference approximations are based on Ax
differences. Can we do the same with the colocated arrangement? A Ax
approximation of the outer first derivative in the pressure equation (7.113)
has the form:
The problem is that the values of pressure derivatives and quantities H are
not available at cell face locations, so we have to use interpolation. Let us
choose linear interpolation, which has the same accuracy as the CDS approximation of the derivatives. Also let the inner derivatives of the pressure in
Eq. (7.113) be approximated by central differences. Linear interpolation of
cell center derivatives leads to:
With this interpolation the pressure equation (7.120) is recovered.
We could evaluate the pressure derivatives at cell faces using central differences and Ax spacing as follows:
If this approximation is applied a t all cell faces, we arrive a t the following
pressure equation (which is also valid on non-uniform grids):
PE - P; P; - P%
-Ax
Ax
Ax
which is the same as Eq.
obtained by interpolation:
(7.116), except that the right hand side is now
199
The source of the above problem may be traced to using 2Ax approximations to the first derivatives. Various cures have been proposed. In incompressible flows, the absolute pressure level is unimportant - only the differences
matter. Unless the absolute value of the pressure is specified somewhere, the
pressure equation is singular and has an infinite number of solutions, all differing by a constant. This makes a simple cure possible: filtering out the
oscillations, as was done by van der Wijngaart (1990).
We shall present one approach to dealing with the pressure-velocity coupling on colocated grids that has found widespread use in complicated geometries and is simple and effective.
On staggered grids, central difference approximations are based on Ax
differences. Can we do the same with the colocated arrangement? A Ax
approximation of the outer first derivative in the pressure equation (7.113)
has the form:
The problem is that the values of pressure derivatives and quantities H are
not available at cell face locations, so we have to use interpolation. Let us
choose linear interpolation, which has the same accuracy as the CDS approximation of the derivatives. Also let the inner derivatives of the pressure in
Eq. (7.113) be approximated by central differences. Linear interpolation of
cell center derivatives leads to:
With this interpolation the pressure equation (7.120) is recovered.
We could evaluate the pressure derivatives at cell faces using central differences and Ax spacing as follows:
If this approximation is applied a t all cell faces, we arrive a t the following
pressure equation (which is also valid on non-uniform grids):
PE - P; P; - P%
-Ax
Ax
Ax
which is the same as Eq.
obtained by interpolation:
(7.116), except that the right hand side is now