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My First Fluid Project

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

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My First Fluid Project

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  1. My First Fluid Project Ryan Schmidt

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

  3. 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!!!)

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

  5. 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

  6. 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

  7. Spatial Discretization • Staggered grid for u • Centered grid for p • (Cells)

  8. Equation Discretization • Central differences for spatial derivatives • Forward difference for time derivative • u component:

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

  10. 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

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

  12. 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

  13. Pressure Iteration • For each cell calculate change in pressure • Now update cell:

  14. Bad Formatting? • Does this: • Mean this?: • Inverse dependence on • But set to • If << , Di,j will be small? • If not, system explodes!

  15. How far did I get?

  16. Well…

  17. It’s not pretty…

  18. 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…

  19. What went wrong?

  20. 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…

  21. 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…

  22. 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

  23. Wall Boundary Problem • Assumption is made that there is only one adjacent fluid cell • What if there is morethan one? • Cannot do both…

  24. 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?

  25. 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

  26. 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

  27. Thanks! Questions?

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