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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 - PowerPoint PPT Presentation
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My First Fluid ProjectRyan Schmidt
OutlineMAC MethodHow far did I get?What went wrong?Future Work
The MAC MethodMarker-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 pViscosity v, density d (constants)External force f
Navier-Stokes Equation:
Mass Conservation Condition:
Navier-Stokes EquationDerived from momentum conservation condition4 Components:
Advection/ConvectionDiffusion (damping)PressureExternal 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 subregionFlow in == flow outIncompressible fluid
Comes from continuum assumption
*gasses assumed to be locally incompressible
Spatial DiscretizationStaggered grid for uCentered grid for p
(Cells)
Equation DiscretizationCentral differences for spatial derivativesForward difference for time derivativeu component:
Mathematical TrickeryAdvection form different in literature:
These two are equivalent if the fluid is incompressible. Proof:
MarkersCell resolution very coarse (20-150)Want higher resolution surfaceAlso need to track which cells contain fluid
Solution: ‘Marker’ particlesMassless particles that flow freely in u fieldDo not contribute to computationVery fast to process
MAC AlgorithmInitialize u,p grids (easier said than done)
Forward-difference u to get new velocities
Enforce zero-divergence condition
Rinse and repeat
Enforcing Zero Divergence2 possibilities:
Iterative procedureProjection method of Stam99
Iterative Procedure – Pressure Iteration
Individually set each cell divergence to 0Calculate pressure change and modify velocities
Repeat over entire grid until maximum cell divergence < predefined tolerance
Pressure IterationFor each cell calculate change in pressure
Now update cell:
Bad Formatting?Does this:
Mean this?:
Inverse dependence onBut 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 BoundariesNormal velocity is 0
Prevents flow into boundary cellAlso have to set internal pressure
No-slipzero tangential velocity
Free-slip free tangential velocity
Wall Boundary ProblemAssumption is made that there is only one adjacent fluid cell
What if there is morethan one?
Cannot do both…
Free-Surface BoundariesHave to make sure that divergence in surface cells is 0
Lots of casesI think this is where my problem is28 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 WorkGo 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?