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Published byChase Blake Modified over 4 years ago

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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 – Harlow&Welch 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 < 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 <<, D i,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! d=1, v=0.01, g=981 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 more than 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?

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