The document describes a level set method for simulating droplets using a Navier-Stokes equation. It defines the level set function φ such that φ<0 in the gas region and φ>0 in the liquid region. The interface is defined as the zero level set of φ. Density and viscosity are defined as constant in each region using the Heaviside function of φ. Dimensionless forms of the equations are presented. A projection method is used to handle high Reynolds number flows. The level set function is modified to have a smooth density and a thickness at the interface defined using a function of φ. An equation is added to correct volume during simulation.

1. Level set method
for Droplet Simulation
Jong su, Kim

2. They are
Different fluid...
������������ ������������

3. Equation of Motion
Navier-stokes equation for Incompressible flow
Boundary Condition

5. ������ < 0
������ = 0
������ > 0

6. Level Set Method
The interface is the zero level set of ������
Take ������ < 0 in the gas region and ������ > 0 in the liquid region.

7. Level set method
Define
Since the interface moves with the fluid particles,
then we can get
Surface tension force is interpreted as a body force localized on the interface

8. Level set method
Density and viscosity are constant in each field, define
Where ������(������) is
the Heaviside function
given by

9. Dimensionless form
+
Prime ′ means dimensionless

10. Dimensionless form
Gravity is −������ direction

11. Projection method
Use projection scheme coupled with high order upwind differencing of the
convective terms in order to handle high Reynolds number flow
Hodge decomposition
Therefore..
It eliminates pressure

12. Thickness of surface
To Solve
Modify
to (Smooth density)
with

13. Thickness of the interface
From
Thickness is
Assume
Define function

14. View level set function as distance function
Function d
Thickness is
Take

15. View level set function as distance function
To solve this, solve the following partial differential equation.

16. During simulation, volume cannot be retained.
To correct this, add this equation.

17. Overall Algorithm