12.5 Free-Surface Flows
387
Another class of interface-capturing methods is based on the level-set
formulation, introduced by Osher and Sethian (1988). The surface is defined
as the one on which a level-set function 4 = 0. Other values of this function
have no significance and t o make it a smooth function, 4 is typically initialized
as the signed distance from the interface i.e. its value at any point is the
distance from the nearest point on the surface and its sign is positive on one
side and negative on the other. This function is then allowed to evolve as a
solution of the transport equation:
84
- + div (4v) = 0 ,
at
where v is the local fluid velocity and, at any time, the surface on which
4 = 0 is the interface. If the function 4 becomes too complicated, it can be
re-initialized in the manner described above. The advantage of this approach
relative to the VOF-method is that 4 varies smoothly across the interface
while the volume fraction c is discontinuous there.
As in VOF-like methods, fluid properties are determined by the local value
of 4 but here, only the sign of 4 is important.
When solving for the volume fraction c, its step-wise variation across the
interface is usually not maintained--the step is smeared by the numerical
approximation. As a result, the fluid properties experience a smooth change
across the interface. In level-set methods, the step-wise variation of the properties is maintained, since we define
However, this usually causes problems when computing viscous flows so one
needs to introduce a region of some finite thickness 6 over which a smooth
but rapid change of the properties occurs across the interface.
As noted above, the computed 4 needs to be re-initialized every now
and then. Sussman et al. (1994) proposed that this be done by solving the
following equation:
until steady state is reached. This guarantees that 4 has the same sign and
zero level as 4o and fulfills the condition that Jgrad 41 = 1, making it similar
to a signed distance function.
Since 4 does not explicitly occur in any of the conservation equations, the
original level-set method did not exactly conserve mass. Mass conservation
can be enforced by making the right-hand side of Eq. (12.34) a function
of the local mass imbalance A m as was done by Zhang et al. (1998). The
more frequently one solves this equation, the fewer iterations are needed to
reach steady state; of course, frequent solution of this equation increases the
computational cost so there is a trade-off.
387
Another class of interface-capturing methods is based on the level-set
formulation, introduced by Osher and Sethian (1988). The surface is defined
as the one on which a level-set function 4 = 0. Other values of this function
have no significance and t o make it a smooth function, 4 is typically initialized
as the signed distance from the interface i.e. its value at any point is the
distance from the nearest point on the surface and its sign is positive on one
side and negative on the other. This function is then allowed to evolve as a
solution of the transport equation:
84
- + div (4v) = 0 ,
at
where v is the local fluid velocity and, at any time, the surface on which
4 = 0 is the interface. If the function 4 becomes too complicated, it can be
re-initialized in the manner described above. The advantage of this approach
relative to the VOF-method is that 4 varies smoothly across the interface
while the volume fraction c is discontinuous there.
As in VOF-like methods, fluid properties are determined by the local value
of 4 but here, only the sign of 4 is important.
When solving for the volume fraction c, its step-wise variation across the
interface is usually not maintained--the step is smeared by the numerical
approximation. As a result, the fluid properties experience a smooth change
across the interface. In level-set methods, the step-wise variation of the properties is maintained, since we define
However, this usually causes problems when computing viscous flows so one
needs to introduce a region of some finite thickness 6 over which a smooth
but rapid change of the properties occurs across the interface.
As noted above, the computed 4 needs to be re-initialized every now
and then. Sussman et al. (1994) proposed that this be done by solving the
following equation:
until steady state is reached. This guarantees that 4 has the same sign and
zero level as 4o and fulfills the condition that Jgrad 41 = 1, making it similar
to a signed distance function.
Since 4 does not explicitly occur in any of the conservation equations, the
original level-set method did not exactly conserve mass. Mass conservation
can be enforced by making the right-hand side of Eq. (12.34) a function
of the local mass imbalance A m as was done by Zhang et al. (1998). The
more frequently one solves this equation, the fewer iterations are needed to
reach steady state; of course, frequent solution of this equation increases the
computational cost so there is a trade-off.