# My First Fluid Project

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```					My First Fluid Project
Ryan Schmidt
Outline
MAC Method
How far did I get?
What went wrong?
Future Work
The MAC Method
Marker-and-Cell – Harlow&Welch 1965

Standard technique for simulating
incompressible fluids w/Navier-Stokes
fluid equations

LANL Technical Report (access
restricted!!!)
Navier-Stokes Fluid Dynamics
Velocity field u, Pressure field p
Viscosity v, density d (constants)
External force f

Navier-Stokes Equation:

Mass Conservation Condition:
Navier-Stokes Equation
Derived from momentum conservation condition
4 Components:
Diffusion (damping)
Pressure
External force (gravity, etc)
System of Nonlinear partial differential equations
Incompressibility Condition
We want incompressible fluids*
Velocity field u has zero divergence
Mass conservation over any subregion
Flow in == flow out
Incompressible fluid

Comes from continuum assumption

*gasses assumed to be locally incompressible
Spatial Discretization
Staggered grid
for u
Centered grid
for p

(Cells)
Equation Discretization
Central differences for spatial derivatives
Forward difference for time derivative
u component:
Mathematical Trickery
Advection form different in literature:

These two are equivalent if the fluid is
incompressible. Proof:
Markers
Cell resolution very coarse (20-150)
Want higher resolution surface
Also need to track which cells contain fluid

Solution: ‘Marker’ particles
Massless particles that flow freely in u field
Do not contribute to computation
Very fast to process
MAC Algorithm
Initialize u,p grids   (easier said than done)

Forward-difference u to get new velocities

Enforce zero-divergence condition

Rinse and repeat
Enforcing Zero Divergence
2 possibilities:
Iterative procedure
Projection method of Stam99

Iterative Procedure – Pressure Iteration
Individually set each cell divergence to 0
Calculate pressure change and modify velocities
Repeat over entire grid until maximum cell
divergence < predefined tolerance
Pressure Iteration
For each cell calculate change in pressure

Now update cell:
Does this:

Mean this?:

Inverse dependence on
But   set to
If  <<       , Di,j will be small?
If not, system explodes!
How far did I get?
Well…
It’s not pretty…
Symmetry?
Tried to reproduce experiments in literature
Correct Physical Constants!
d=1, v=0.01, g=981 for breaking dam

Inflow supposed to
be symmetric…
What went wrong?
Initial Conditions ?!?
System becomes unstable as soon as there is
any large amount of divergence

How do we specify initial conditions that will
give us motion w/o immediately causing
unstable divergence?

(I don’t know…)
Inflow is simple case, but it still doesn’t work…
Boundary Conditions
Many, many cases
Too many to have special cases of finite difference equation

Solution: construct velocities & pressures in boundary cells
so that standard finite difference equation comes out right

I may have them wrong…
Not sure when to apply them
Unclear how order of application affects velocties…
Wall Boundaries
Normal velocity is 0
Prevents flow into boundary cell
Also have to set internal pressure

No-slip
zero tangential velocity
Free-slip
free tangential velocity
Wall Boundary Problem
Assumption is made that there is only one

What if there is more
than one?
Cannot do both…
Free-Surface Boundaries
Have to make sure that divergence in surface
cells is 0

Lots of cases
I think this is where my problem is
28 cases and counting…

Asymmetry?
Outer Tangential Velocities
Interpolation in surface cells reaches out into
empty cells
Finite difference equations may as well

Need to have same
velocity set there
Future Work
Go back and check boundary conditions

Harass Nick Foster

Finish report and put it on the web,
hope that someone reads it and has
some insight
Thanks!
Questions?

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