My First Fluid Project

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					My First Fluid Project
     Ryan Schmidt
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
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)
   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

    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:
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:
          Bad Formatting?
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?
It’s not pretty…
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

  zero tangential velocity
  free tangential velocity
   Wall Boundary Problem
Assumption is made that there is only one
adjacent fluid cell

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…

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

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