Digital Signal Processing Reference
In-Depth Information
into fine one. Otherwise, it remains unchanged. Every time step, we compared these
two quantities in coarse grid and reset the fine grid to ensure detail enhancement
adaptive.
For the circumstance with obstacles, besides dividing some grids into fine ones by
above method, we would also divide the coarse grids around obstacles directly
without comparing the vorticity.
Figure 1 shows the grid division results using our method. (a) is the circumstance
with no obstacle, and (b) is with obstacle.
4
Vorticity Confinement Force with Helicity
4.1
Vorticity Confinement Force
Fedkiw et al. [2] points out that, fluid details can be enhanced by adding vorticity
confinement force. In incompressible fluid, we get the vorticity field through equation
(4). The normalized vorticity position vector
N
can be calculated through the gradient
of
||
as the following:
||
(5)
||||
At last, we get the vorticity confinement force form the the equation (6):
(6)
where
is the user defined parameter in order to control the vorticity, and
denotes
the grid size. The vorticity confinement force can be added as an external force to the
fluid simulation model.
4.2
The Helicity
Through equation (6) we can see that, if we want to get more details, we can set
larger. But it may lead to some non-realistic effects or even cause the “blow up”
phenomenon if
is too large. The main reason is that we have not considered the
helicity factor when calculating the vorticity confinement force:
(7)
It measures the amount of rotation of a fluid rotating around an axis that is parallel to
the main flow direction
[3]
, where
denotes volume. According to this physical
quantity, He et al. [3] improved equation (6) as follows:
||
(8)
where
| |
represents for helicity. Then the vorticity confinrment force is not only
determined by the value of grid size
, but also by the helicity.
