388
12. Special Topics
Many level-set methods have been proposed; they differ in the choices
for the various steps. Zhang et al. (1998) describe one such method, which
they applied to bubble-merging and mold-filling, including melt solidification.
They used a FV-method on structured, non-orthogonal grids to solve the
conservation equations, and a FD-method for the level-set equation. An ENOscheme was used to discretize the convective term in the latter.
Another version of this method is used to study flame propagation. In
this case, the flame propagates relative to the fluid and this introduces the
possibility that the surface will develop cusps, locations at which the surface
normal is discontinuous. This will be discussed further below.
More details on level-set methods can be found in a book by Sethian
(1996); see also Smiljanovski et al. (1997) and Reinecke et al. (1999) for
examples of similar approaches used for flame tracking.
12.5.1 Interface-Tracking Methods
In the calculation of flows around submerged bodies, many authors linearize
about the unperturbed free surface. This requires introduction of a height
function, which is the free surface elevation relative to its unperturbed state:
z = H ( x , y, t) .
(12.35)
The kinematic boundary condition (12.25) then becomes the following equation describing the local change of the height H :
This equation can be integrated in time using the methods described in Chap.
6. The fluid velocity at the free surface is obtained either by extrapolation
from the interior or by using the dynamic boundary condition (12.27).
This approach is usually used in conjunction with structured grids and
explicit Euler time integration. Many authors use a FV method for the flow
calculation and a FD method for the height equation and enforce both boundary conditions at the free surface only at the converged steady state (see
e.g. Farmer et al., 1994).
Hino (1992) used a FV method with the enforcement of the SCL, thus
satisfying all conditions at each time step and ensuring volume conservation.
Similar methods were developed by Raithby et al. (1995), Th6 et al. (1994)
and Lilek (1995). One fully-conservative FV method of this type consists of
the following steps:
Solve the momentum equations using the specified pressure at the current
free surface to obtain velocities u,f .
Enforce local mass conservation in each CV by solving a pressure-correction
equation, with a zero pressure correction boundary condition at the current
free surface (see Sect. 10.2.2). Mass is conserved both globally and in each
12. Special Topics
Many level-set methods have been proposed; they differ in the choices
for the various steps. Zhang et al. (1998) describe one such method, which
they applied to bubble-merging and mold-filling, including melt solidification.
They used a FV-method on structured, non-orthogonal grids to solve the
conservation equations, and a FD-method for the level-set equation. An ENOscheme was used to discretize the convective term in the latter.
Another version of this method is used to study flame propagation. In
this case, the flame propagates relative to the fluid and this introduces the
possibility that the surface will develop cusps, locations at which the surface
normal is discontinuous. This will be discussed further below.
More details on level-set methods can be found in a book by Sethian
(1996); see also Smiljanovski et al. (1997) and Reinecke et al. (1999) for
examples of similar approaches used for flame tracking.
12.5.1 Interface-Tracking Methods
In the calculation of flows around submerged bodies, many authors linearize
about the unperturbed free surface. This requires introduction of a height
function, which is the free surface elevation relative to its unperturbed state:
z = H ( x , y, t) .
(12.35)
The kinematic boundary condition (12.25) then becomes the following equation describing the local change of the height H :
This equation can be integrated in time using the methods described in Chap.
6. The fluid velocity at the free surface is obtained either by extrapolation
from the interior or by using the dynamic boundary condition (12.27).
This approach is usually used in conjunction with structured grids and
explicit Euler time integration. Many authors use a FV method for the flow
calculation and a FD method for the height equation and enforce both boundary conditions at the free surface only at the converged steady state (see
e.g. Farmer et al., 1994).
Hino (1992) used a FV method with the enforcement of the SCL, thus
satisfying all conditions at each time step and ensuring volume conservation.
Similar methods were developed by Raithby et al. (1995), Th6 et al. (1994)
and Lilek (1995). One fully-conservative FV method of this type consists of
the following steps:
Solve the momentum equations using the specified pressure at the current
free surface to obtain velocities u,f .
Enforce local mass conservation in each CV by solving a pressure-correction
equation, with a zero pressure correction boundary condition at the current
free surface (see Sect. 10.2.2). Mass is conserved both globally and in each