Moving Interface Problems: Methods & Applications Tutorial Lecture II

Moving Interface Problems—Front Tracking

Moving Interface Problems: Methods & Applications Tutorial Lecture II Grétar Tryggvason Worcester Polytechnic Institute Moving Interface Problems and Applications in Fluid Dynamics Singapore National University, 2007

Moving Interface Problems—Front Tracking Outline Lecture 2: Motivation The One Fluid Formulation Solving the Navier-Stokes Equations Methods for the advection of a marker function Volume of Fluid (VOF) Level Sets Others methods

Moving Interface Problems—Front Tracking Numerical Method Front Tracking S.O. Unverdi, G. Tryggvason. A Front Tracking Method for Viscous Incompressible Flows. J. Comput. Phys, 100 (1992), 25-37. S.O. Unverdi and G. Tryggvason. Computations of Multi-Fluid Flows. Physica D, 60 (1992), 70-83. Review G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas, and Y.-J. Jan. A Front Tracking Method for the Computations of Multiphase Flow. J. Comput. Physics 169 (2001), 708–759

Moving Interface Problems—Front Tracking Numerical Method The conservation equations are solved on a regular fixed grid and the front is tracked by connected marker points

Moving Interface Problems—Front Tracking

The structure of the front

Moving Interface Problems—Front Tracking Numerical Method Data structure for the surface elements. The elements carry essentially all information about the structure of the front.

The points only “know” their locations

Moving Interface Problems—Front Tracking Numerical Method The right data structure makes it easier to work with the interface. In 2D it is a matter of convenience, in 3D it makes the difference between an algorithm that works and one that does not! add and delete front objects, change the topology, handle multiple interfaces

Moving Interface Problems—Front Tracking Numerical Method Working in barycentric coordinates simplifies the interpolations needed for the elements u + v + w =1

Quadratic interpolation !

1 p(u,v,w) = (1" u)("up5 + (1" v ) p3 + (1" w ) p2 ) 2 1 + (1" v )((1" u) p3 " vp6 + (1" w ) p1 ) 2 1 + (1" w )((1" u) p2 + (1" v ) p1 " wp4 ) 2

Moving Interface Problems—Front Tracking Numerical Method

In two-dimensions adding or deleting a point is a relatively simple operation. We generally split an element to add point and collapse an element to delete a point

Moving Interface Problems—Front Tracking Dynamic Regridding As the interface stretches and deforms, some parts are depleted of points while other parts become crowded by points. To maintain a nearly uniform resolution of the interface it is necessary to use dynamic regridding. Regridding can be achieved by 1. Adding elements 2. Deleting elements In 3D it is also often beneficial to 3. Reshape elements

Adding elements

e1 new1

Deleting elements

Reshaping elements

e2 new2

e5 e7

e6 e3 e4 e1 e2

Moving Interface Problems—Front Tracking Dynamic Regridding

Dynamic regridding of a buoyant bubble resolved on a 16 by 16 by 16 grid

Moving Interface Problems—Front Tracking

Transferring information between the fixed grid and the front

Moving Interface Problems—Front Tracking Numerical Method

Fluid 1

Tracked Front

Normal

Tangent

Fluid 2 Finite Volume Grid

!

Moving Interface Problems—Front Tracking Interpolating from grid The velocities are interpolated from the grid:

" l = # " ijk w ijk The front values are distributed onto the grid by

#sl !ijk = " !l wijk 3 h On the front: per length On the grid: per volume

the weights wijk can be selected in several different ways

Moving Interface Problems—Front Tracking Interpolating from grid

wijk (x p ) = d(x p ! ih) d(y p ! jh) d(z p ! kh) Area weighting

#% (r ! ih)/ h 0< r