My First Fluid Project

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My First Fluid Project

• Ryan Schmidt

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Outline

• MAC Method
• How far did I get?
• What went wrong?
• Future Work

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The MAC Method

• Marker-and-Cell HarlowWelch 1965
• Standard technique for simulating incompressible
  fluids w/Navier-Stokes fluid equations
• LANL Technical Report (access restricted!!!)

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Navier-Stokes Fluid Dynamics

• Velocity field u, Pressure field p
• Viscosity v, density d (constants)
• External force f
• Navier-Stokes Equation
• Mass Conservation Condition

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Navier-Stokes Equation

• Derived from momentum conservation condition
• 4 Components
• Advection/Convection
• Diffusion (damping)
• Pressure
• External force (gravity, etc)
• System of Nonlinear partial differential equations

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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
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Spatial Discretization

• Staggered grid for u
• Centered grid for p
• (Cells)

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Equation Discretization

• Central differences for spatial derivatives
• Forward difference for time derivative
• u component

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Mathematical Trickery

• Advection form different in literature
• These two are equivalent if the fluid is
  incompressible. Proof

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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

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MAC Algorithm

• Initialize u,p grids (easier said than done)
• Forward-difference u to get new velocities
• Enforce zero-divergence condition
• Rinse and repeat

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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 lt predefined tolerance

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Pressure Iteration

• For each cell calculate change in pressure
• Now update cell

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Bad Formatting?

• Does this
• Mean this?
• Inverse dependence on
• But set to
• If ltlt , Di,j will be small?
• If not, system explodes!

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How far did I get?
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Well
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Its not pretty
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Symmetry?

• Tried to reproduce experiments in literature
• Correct Physical Constants!
• d1, v0.01, g981 for breaking dam
• Inflow supposed to be symmetric

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What went wrong?
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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 dont know)
• Inflow is simple case, but it still doesnt work

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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

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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

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Wall Boundary Problem

• Assumption is made that there is only one
  adjacent fluid cell
• What if there is morethan one?
• Cannot do both

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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?

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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

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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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Thanks!

• Questions?