12.5 Free-Surface Flows
381
pletely analogous to that encountered in block-structured grids with nonmatching interfaces and can be handled by the method described in Sect.
8.6.5; the only difference is that the cell connectivity changes with time and
has to be re-established after each time step. This approach is more flexible
than the one described above; the grids can be of different kinds and/or fineness, and the interface can be an arbitrary surface. This approach can also be
applied to flows around bodies passing each other, entering a tunnel, or moving in an enclosure with a known trajectory. Examples of such applications
were presented by Lilek et al. (1997~) and DemirdiiC et al. (1997).
The third approach is t o use overlapping (Chimera) grids. Again, one
grid is attached to the fixed part of the domain and the other t o the moving
body. This approach can be used even if the trajectory of the moving body is
not known in advance, when it is very complicated, or when the surrounding
domain is of a complex shape (e.g. when a sliding interface can not be constructed). The fixed grid may cover the whole "environment" in which the
body is moving. The overlap region changes with time and the relationship
between the grids needs to be re-established after each time step. Except for
difficulties in ensuring exact conservation, there are almost no limitations on
the applicability of this approach.
As noted above, the same equations and discretization methods apply to
both the fixed and moving grids, the only difference being that on the fixed
grid, the grid velocity vb is obviously zero. Sometimes it may be advantageous
to use different coordinate systems on the two domains; for example, one may
use Cartesian velocity components in one part and polar components on the
other grid. This is possible provided: (i) one adds the body forces due t o
frame acceleration and (ii) one transforms the vector components from one
system to another a t the interface or in the overlap region. Both of these are
easy to do in principle but the programming may be tedious.
12.5 Free-Surface Flows
Flows with free surfaces are an especially difficult class of flows with moving
boundaries. The position of the boundary is known only at the initial time;
its location a t later times has to be determined as part of the solution. To
accomplish this, the SCL and the boundary conditions a t the free surface
must be used.
In the most common case, the free surface is an air-water boundary but
other liquid-gas surfaces occur, as do liquid-liquid interfaces. If phase change
a t the free surface can be neglected, the following boundary conditions apply:
0 The kinematic condition requires that the free surface be a sharp boundary
separating the two fluids that allows no flow through it, i.e.:
[(v - v b ) . nIfs = 0 or mf, = 0 ,
(12.25)
381
pletely analogous to that encountered in block-structured grids with nonmatching interfaces and can be handled by the method described in Sect.
8.6.5; the only difference is that the cell connectivity changes with time and
has to be re-established after each time step. This approach is more flexible
than the one described above; the grids can be of different kinds and/or fineness, and the interface can be an arbitrary surface. This approach can also be
applied to flows around bodies passing each other, entering a tunnel, or moving in an enclosure with a known trajectory. Examples of such applications
were presented by Lilek et al. (1997~) and DemirdiiC et al. (1997).
The third approach is t o use overlapping (Chimera) grids. Again, one
grid is attached to the fixed part of the domain and the other t o the moving
body. This approach can be used even if the trajectory of the moving body is
not known in advance, when it is very complicated, or when the surrounding
domain is of a complex shape (e.g. when a sliding interface can not be constructed). The fixed grid may cover the whole "environment" in which the
body is moving. The overlap region changes with time and the relationship
between the grids needs to be re-established after each time step. Except for
difficulties in ensuring exact conservation, there are almost no limitations on
the applicability of this approach.
As noted above, the same equations and discretization methods apply to
both the fixed and moving grids, the only difference being that on the fixed
grid, the grid velocity vb is obviously zero. Sometimes it may be advantageous
to use different coordinate systems on the two domains; for example, one may
use Cartesian velocity components in one part and polar components on the
other grid. This is possible provided: (i) one adds the body forces due t o
frame acceleration and (ii) one transforms the vector components from one
system to another a t the interface or in the overlap region. Both of these are
easy to do in principle but the programming may be tedious.
12.5 Free-Surface Flows
Flows with free surfaces are an especially difficult class of flows with moving
boundaries. The position of the boundary is known only at the initial time;
its location a t later times has to be determined as part of the solution. To
accomplish this, the SCL and the boundary conditions a t the free surface
must be used.
In the most common case, the free surface is an air-water boundary but
other liquid-gas surfaces occur, as do liquid-liquid interfaces. If phase change
a t the free surface can be neglected, the following boundary conditions apply:
0 The kinematic condition requires that the free surface be a sharp boundary
separating the two fluids that allows no flow through it, i.e.:
[(v - v b ) . nIfs = 0 or mf, = 0 ,
(12.25)