390
12. Special Topics
Fig. 12.6. Control volumes
under the free surface, whose
vertices lie on the free surface
are defined by the coordinates of the cell face centers
(open symbols); the volume
swept by the cell face during
the time step is shaded.
The volume swept by the free-surface cell face is then (see Eqs. (12.16) and
(12.21)):
where h is the distance the free surface-markers move during one time step;
h, = hi while hnw and h,, are obtained by linearly interpolating hi and
hi-1 or hi and hi+l. By expressing h,,, h, and h,, in terms of hi, hi- hi+l
and inserting the above expression into Eq. (12.38), we obtain a system of
equations for the locations of the cell-face centers, hi. In 2D, the system is
tridiagonal and can be solved directly by the TDMA method of Sect. 5.2.3. In
3D, the system is block-tridiagonal and is best solved by one of the iterative
solvers presented in Chap. 5. 'Boundary conditions' have to be specified a t
the CV vertices a t the edges of the free surface. If the boundary is not allowed
to move, h = 0. If the edge of the free surface is allowed to move, e.g. for
an open system, the boundary condition should be of the non-reflective or
'wave-transmissive' type that does not cause wave reflection; the condition
(9.4) is one appropriate possibility.
This approach was applied by Lilek (1995) to 2D and 3D problems on
structured grids. When the lateral bounding surface has an irregular shape
(e.g. a ship hull), the expressions for the volume 6f& become complicated
and require iterative solution on each outer iteration.
Muzaferija and PeriC (1997) suggested a simpler approach. They noted
that it is not necessary to calculate the swept volume from the geometry
of the cell vertices on the free surface; it can be obtained from Eq. (12.38).
The displacement of free-surface markers located above the cell-face center
is defined by the height h , obtained from the known volume and cell-face
area. The new vertex locations are then computed by interpolating h; the
resulting swept volume is not exact and iterative correction is necessary.
The method is suitable for implicit schemes, for which outer iterations are
required a t each time step anyway. The 'old' and 'new' locations shown in
Fig. 12.6 are now the values for the current and preceding outer iterations;
each outer iteration corrects the swept volume according to Eq. (12.38). At
12. Special Topics
Fig. 12.6. Control volumes
under the free surface, whose
vertices lie on the free surface
are defined by the coordinates of the cell face centers
(open symbols); the volume
swept by the cell face during
the time step is shaded.
The volume swept by the free-surface cell face is then (see Eqs. (12.16) and
(12.21)):
where h is the distance the free surface-markers move during one time step;
h, = hi while hnw and h,, are obtained by linearly interpolating hi and
hi-1 or hi and hi+l. By expressing h,,, h, and h,, in terms of hi, hi- hi+l
and inserting the above expression into Eq. (12.38), we obtain a system of
equations for the locations of the cell-face centers, hi. In 2D, the system is
tridiagonal and can be solved directly by the TDMA method of Sect. 5.2.3. In
3D, the system is block-tridiagonal and is best solved by one of the iterative
solvers presented in Chap. 5. 'Boundary conditions' have to be specified a t
the CV vertices a t the edges of the free surface. If the boundary is not allowed
to move, h = 0. If the edge of the free surface is allowed to move, e.g. for
an open system, the boundary condition should be of the non-reflective or
'wave-transmissive' type that does not cause wave reflection; the condition
(9.4) is one appropriate possibility.
This approach was applied by Lilek (1995) to 2D and 3D problems on
structured grids. When the lateral bounding surface has an irregular shape
(e.g. a ship hull), the expressions for the volume 6f& become complicated
and require iterative solution on each outer iteration.
Muzaferija and PeriC (1997) suggested a simpler approach. They noted
that it is not necessary to calculate the swept volume from the geometry
of the cell vertices on the free surface; it can be obtained from Eq. (12.38).
The displacement of free-surface markers located above the cell-face center
is defined by the height h , obtained from the known volume and cell-face
area. The new vertex locations are then computed by interpolating h; the
resulting swept volume is not exact and iterative correction is necessary.
The method is suitable for implicit schemes, for which outer iterations are
required a t each time step anyway. The 'old' and 'new' locations shown in
Fig. 12.6 are now the values for the current and preceding outer iterations;
each outer iteration corrects the swept volume according to Eq. (12.38). At